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Bio-inspired tendon-driven systems: computational analysis, optimization, and hardware implementation
(USC Thesis Other)
Bio-inspired tendon-driven systems: computational analysis, optimization, and hardware implementation
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Content
BIO-INSPIRED TENDON-DRIVEN SYSTEMS: COMPUTATIONAL
ANALYSIS, OPTIMIZATION, AND HARDWARE IMPLEMENTATION
by
Joshua M. Inouye
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulllment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(BIOMEDICAL ENGINEERING)
May 2012
Copyright 2012 Joshua M. Inouye
Epigraph
But grow in the grace and knowledge of our Lord and Savior Jesus Christ. To him be
the glory both now and to the day of eternity. Amen.
2 Peter 3:18
ii
Dedication
To my father, my mother, and my Heavenly Father.
iii
Acknowledgements
There have been many people that have provided invaluable contributions to my research
and personal encouragement in dicult times. My father and mother originally instilled
in me a very strong work ethic from a very young age, and this set the foundation for
all of the perseverance that was necessary to get to this point. They always were very
encouraging throughout my life and I owe them much love and thanks. My brother and
sister have likewise been very encouraging in my journey the past several years.
I would like to thank Laila Guessous and Brian Sangeorzan for accepting me into
the summer NSF Research Experience for Undergraduates at Oakland University during
my undergraduate studies. I rst became interested in academic research after my great
experience there, and I am indebted to them for their generosity in their time and eort
mentoring me.
I would like to thank Francisco Valero-Cuevas so much for his incredible guidance
in the past few years and for believing in me and funding me even when I had a very
slow start in nding a lab during my rst few years at USC. If I had not connected
with him, I would have dropped out of the PhD program. I have learned so much from
him about doing solid research, obtaining funding, networking, managing a versatile and
interdisciplinary lab, and appreciating many dierent cultures.
iv
I would also like to thank Jason Kutch for all of his guidance and help. He provided
some very crucial advice at many important turning points in my research, and his
encouragement and experience were huge factors that led to quick progress once I settled
on a research project.
Manish Kurse was also extremely helpful to me nearly every day. We worked on some
similar research areas, being that we are both mechanical engineers, and he was able
to teach me many things about linear algebra, ecient programming, and experimental
testing.
Thanks also to Alex Reyes, who has become one of my best friends in the past few
years, for always listening to my ideas and giving helpful feedback and encouragement.
In addition, he has taught me a lot about circuit design and electrical engineering, which
hopefully will come in handy someday.
To the other lab members to which I am extremely indebted for their advice, opin-
ions, and encouragement: Sudarshan Dayanidhi, Evangelos Theodorou, Cornelius Raths,
Brendan Holt, Emily Lawrence, and Srideep Musuvathy.
I would also like to thank Edward Ebramzadeh for his advice, encouragement, and
friendship as well. I worked in his lab for about a year and learned a great deal about
orthopaedic research as well as perseverance in my doctoral studies.
Thanks to Jill McNitt-Gray, Chris Powers, and Phil Requejo for allowing me to do
laboratory rotations in their labs, and the students Joe Munaretto and Sean Farrokhi, for
helping me work on their projects, and Noom Somboon also for helping me with research
projects.
v
I would like to thank the members of my guidance committee for their time and eorts
in providing suggestions to ensure a respectable thesis submission: Terry Sanger, Tansu
Celikel, Stefan Schaal, and Gaurav Sukhatme. Additional thanks to Tansu and Stefan
for taking positions on my dissertation committee.
Thanks to Mischal Diasanta for all of her help and guidance in class selection as well
as encouragement and advice in the lab selection process. She has been an absolutely
fantastic graduate advisor.
Lastly and most importantly, all of my work would not have been possible without
the hand of the Lord in my life.
vi
Table of Contents
Epigraph ii
Dedication iii
Acknowledgements iv
List Of Tables xi
List Of Figures xii
Abstract xviii
Chapter 1: Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Signicance of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Dissertation outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Chapter 2: Fundamentals of Tendon Actuation 7
2.1 General serial linkage mechanisms . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Tendon routing and moment arms . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Denition of necessary actuation: versatility . . . . . . . . . . . . . . . . . 24
2.3.1 Moment arm matrices . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 Necessary conditions for admissible moment arm matrices . . . . . 27
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Chapter 3: Fundamentals of Feasible Sets 33
3.1 Forward analysis I: from feasible activation sets to feasible force sets using
vertex enumeration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Forward analysis II: from basis vectors to feasible sets using Minkowski
sums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Why SVD is not the correct approach for analysis of tendon-driven systems 39
3.4 Other cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.1 Feasible wrench sets . . . . . . . . . . . . . . . . . . . . . . . . . . 44
vii
3.4.2 Feasible velocity and twist sets . . . . . . . . . . . . . . . . . . . . 47
3.4.3 Feasible acceleration sets . . . . . . . . . . . . . . . . . . . . . . . 49
Chapter 4: A Novel Synthesis of Computational Approaches Enables Opti-
mization of Task-Independent Grasp Quality of Tendon-Driven Hands 51
4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.1 Finding the set of feasible grasp wrenches and computing grasp
quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.1.1 Select initial grasp parameters . . . . . . . . . . . . . . . 56
4.3.1.2 Build ngertip feasible force set . . . . . . . . . . . . . . 56
4.3.1.3 Find feasible object force set . . . . . . . . . . . . . . . . 57
4.3.1.4 Simplify feasible object force set . . . . . . . . . . . . . . 58
4.3.1.5 Translate contact forces to object wrenches . . . . . . . . 60
4.3.1.6 Find feasible grasp wrench set . . . . . . . . . . . . . . . 61
4.3.1.7 Compute grasp quality . . . . . . . . . . . . . . . . . . . 62
4.3.2 Computing grasp quality metrics for specic manipulator designs . 63
4.3.2.1 Finger topology . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.2.2 Grasp conguration . . . . . . . . . . . . . . . . . . . . . 65
4.3.2.3 Calculating grasp quality . . . . . . . . . . . . . . . . . . 66
4.3.2.4 Monte Carlo simulations . . . . . . . . . . . . . . . . . . 67
4.3.2.5 Regression analysis . . . . . . . . . . . . . . . . . . . . . 67
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4.1 Baseline results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.4.2 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4.3 Regression analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.6 Appendix: Calculation of the feasible force sets of tendon-driven manipu-
lators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Chapter 5: Quantitative Prediction of Grasp Impairment in Peripheral
Neuropathies of the Hand 81
5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Chapter 6: Bettering the Human Hand: Anthropomorphic Tendon-Driven
Robotic Hands can Exceed Human Grasping Capabilities 86
6.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3.1 Calculating human hand grasp quality . . . . . . . . . . . . . . . . 89
6.3.2 Calculating anthropomorphic hand grasp quality . . . . . . . . . . 90
6.3.3 Optimizing anthropomorphic hand grasp quality . . . . . . . . . . 94
viii
6.3.3.1 Monte Carlo on structure matrices . . . . . . . . . . . . . 94
6.3.3.2 Optimization of joint centers of rotation . . . . . . . . . . 95
6.3.3.3 Optimization of maximal tendon tensions . . . . . . . . . 97
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4.1 Random Monte Carlo designs . . . . . . . . . . . . . . . . . . . . . 99
6.4.2 Optimization of joint centers of rotation and distribution of maxi-
mal tendon tensions . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Chapter 7: Asymmetric Routings With Fewer Tendons Can Oer Both
Flexible Endpoint Stiness Control and High Force-Production Capa-
bilities in Robotic Fingers 109
7.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.3.1 Analysis and Synthesis of Stiness . . . . . . . . . . . . . . . . . . 116
7.3.1.1 Joint Stiness Adjustability . . . . . . . . . . . . . . . . . 116
7.3.1.2 Endpoint Stiness Eccentricity . . . . . . . . . . . . . . . 119
7.3.2 Analysis and Optimization of Force Polytopes . . . . . . . . . . . . 121
7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.4.1 Joint Stiness Adjustability . . . . . . . . . . . . . . . . . . . . . . 124
7.4.2 Optimized Endpoint Stiness Eccentricity vs. Maximal Isotropic
Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.4.3 Optimizing MIV for 3 Specic Routings . . . . . . . . . . . . . . . 126
7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Chapter 8: A Novel Computational Approach Helps Explain and Recon-
cile Con
icting Experimental Findings on the Neural Control of Arm
Endpoint Stiness 132
8.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.3.1 Arm Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.3.2 Theoretical Formulation . . . . . . . . . . . . . . . . . . . . . . . . 136
8.3.3 Checking for Realizable Endpoint Stiness Matrices . . . . . . . . 141
8.3.4 Varying Moment Arms and Synergies . . . . . . . . . . . . . . . . 144
8.3.5 Exploring Energy Expenditure Within the Stiness Redundancy
Solution Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.4.1 Realizable Fraction of Endpoint Stiness Ellipses . . . . . . . . . . 146
8.4.2 Realizable Orientations in the Presence of Synergies . . . . . . . . 148
8.4.3 Energy Expenditure Ranges . . . . . . . . . . . . . . . . . . . . . . 149
8.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
ix
Chapter 9: Optimization of Tendon Topology for Robotic Fingers: Pre-
diction and Implementation 154
9.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
9.3 Finger Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
9.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
9.4.1 Force polytope analysis . . . . . . . . . . . . . . . . . . . . . . . . 159
9.4.2 Evaluating tendon routings . . . . . . . . . . . . . . . . . . . . . . 164
9.4.3 Experimental testing of tendon routings . . . . . . . . . . . . . . . 167
9.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9.5.1 Calculating maximum isotropic values . . . . . . . . . . . . . . . . 170
9.5.2 Theoretical predictions vs. experimental results . . . . . . . . . . . 171
9.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Chapter 10:Computational Optimization and Experimental Evaluation of
Grasp Quality for Tendon-Driven Hands Under Constraints 183
10.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
10.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
10.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.3.1 Hand Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.3.2 Grasp Quality Analysis . . . . . . . . . . . . . . . . . . . . . . . . 188
10.3.3 Optimization of Grasp Quality . . . . . . . . . . . . . . . . . . . . 191
10.3.3.1 Optimizing Pulley Sizes . . . . . . . . . . . . . . . . . . . 192
10.3.3.2 Optimizing Tendon Tension Distribution . . . . . . . . . 192
10.3.4 Experimental testing of tendon routings . . . . . . . . . . . . . . . 195
10.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
10.4.1 Computational grasp quality predictions . . . . . . . . . . . . . . . 197
10.4.2 Theoretical predictions vs. experimental results . . . . . . . . . . . 199
10.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Chapter 11:Conclusions and Future Work 206
Bibliography 208
x
List Of Tables
4.1 Baseline grasp quality results. Coecient of static friction
s
= 0:5. Units
of grasp quality are in Newtons. . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 Average evaluation times (standard deviations) during Monte Carlo simu-
lations, in seconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 Signicant normalized regression coecients for grasp quality with 95%
condence intervals on N+1 topology, 2-nger grasp. `{' denotes not sig-
nicant at the cuto p-value of 0.05. Moment arms expressed as (Joint
number,tendon number). R
2
= 0.930. . . . . . . . . . . . . . . . . . . . . 72
4.4 Expected (from linear regression on Monte Carlo iterations) and actual
(from computational method implementation) eects of moment arm ad-
justments by 10% on grasp quality of N+1 design, 2-nger grasp. Moment
arms expressed as (Joint number,tendon number). . . . . . . . . . . . . . 73
5.1 Muscles in each nerve pathology group. M: median, R: radial, U: ulnar,
CTS: Carpal Tunnel Syndrome. . . . . . . . . . . . . . . . . . . . . . . . . 83
6.1 Maximal joint diameters (i.e., dierences between largest moment arms for
each joint) according to data obtained from the literature. . . . . . . . . . 93
xi
List Of Figures
2.1 (a) Human arm modeled as planar 2-joint serial manipulator. (b) Human
thumb modeled as 3-D 5-joint serial manipulator. . . . . . . . . . . . . . 9
2.2 Coordinate frames assigned to planar 2-link arm with the D-H convention
and corresponding D-H parameters. . . . . . . . . . . . . . . . . . . . . . 11
2.3 Homogeneous transformation matrix in terms of i
th
set of D-H parameters. 13
2.4 Coordinates of hand obtained from homogeneous transformation matrix. . 13
2.5 Illustration of Jacobian for arm system. . . . . . . . . . . . . . . . . . . . 16
2.6 Calculation of joint torques using perpendicular distances of line of action
from joint axes. Force units are in N. . . . . . . . . . . . . . . . . . . . . . 19
2.7 A one-DOF manipulator congurations and associated endpoint force pro-
duction capabilities. (a) Torque motor. (b) Tendons with equal max ten-
sions, equal moment arms. (c) Tendons with unequal max tensions, equal
moment arms. (d) Tendons with equal max tensions, unequal moment arms. 24
2.8 Fundamental relationship between tendon routing and moment arm ma-
trices. Tendon layout of 2 4-DOF robotic ngers and their associated
moment arm matrices. (a) All equal moment arms. (b) Varying moment
arms. Not to scale. Note that this gure underscores how a given tendon
routing drawing has a corresponding moment arm matrix, and vice versa. 26
2.9 (a) A one-DOF manipulator. (b) Determination of null space ofR with sin-
gular value decomposition. (c) Calculation of solution for tendon tensions,
given desired joint torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Forward mapping of feasible activation set to feasible force set. Note: the
feasible activation set and the feasible tendon tension set for this example
are both inR
6
, but only 3 dimensions are shown. . . . . . . . . . . . . . . 34
xii
3.2 3-D Feasible force set of 5-DOF nger using parameters shown. . . . . . . 38
3.3 Demonstration of the duality between basis vectors and feasible sets. . . . 40
3.4 Illustration of why manipulating force ellipsoids do not accurately capture
the force-production capabilities of tendon-driven manipulators. . . . . . 43
3.5 Diering endpoint conditions aect feasible wrench set analysis for 3-DOF
planar manipulator. (a) Fixed boundary condition, 3 DOFs constrained.
(b) Pin joint boundary condition, 2 DOFs constrained. . . . . . . . . . . . 45
3.6 One-DOF manipulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Illustration of integration of techniques that were previously isolated. . . . 54
4.2 Flowchart of steps for nding feasible grasp wrench set and computing
grasp quality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 An example of a ngertip feasible force set and its intersection with a
friction cone to produce a feasible object force set. . . . . . . . . . . . . . 57
4.4 (a) An example of an edge collapse operation. The vertices v
1
and v
2
are collapsed into a new vertex v
new
. (b) Example of using edge collapse
operations to simplify the feasible object force set from 19 vertices down
to 10 vertices. Note: this view is of the underside of the feasible object
force set shown in Figure 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5 Grasp congurations analyzed. (a) 4-DOF robotic nger, 2N tendon ar-
rangement, with endpoint wrench description. (b) 4-DOF robotic nger,
N+1 tendon arrangement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6 Grasp congurations analyzed. (a) Isometric view of 2-nger grasp. (b)
Front view of 2-nger grasp. (c) Side view of 3-nger grasp. . . . . . . . . 66
4.7 Uniform sampling distribution used for each independent parameter value
perturbation in Monte Carlo simulations. . . . . . . . . . . . . . . . . . . 68
4.8 Histogram of grasp quality values from Monte Carlo simulations for two-
nger grasp, 2N and N+1 designs, with Gaussian curves overlaid. The
N+1 topologies exceeding the baseline 2N topology are shaded gray. . . . 70
5.1 Grasp quality calculation steps. (a) Basis vectors. (b) Feasible force sets.
(c) Feasible object force sets. (d) Feasible object wrench illustrated in 3-D.
(e) Grasp quality metric of radius of largest ball (illustrated in 3-D; actual
calculation is in 6-D). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
xiii
5.2 Grasp quality deterioration as a function of % advancement of neuropathy. 85
6.1 Computation of human hand grasp quality. Index nger and thumb basis
vectors and feasible force sets not equal scales. (a) Fingertip basis vectors.
(b) Feasible force sets built from basis vectors. (c) Feasible object force
sets: intersection of feasible force sets with friction cones. (d) Feasible
object wrench (only 3-D feasible forces shown). (e) Examples of maintained
and failed grasps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Anthropomorphic hand grasp. . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.3 Procedure for nding randomized designs via Monte Carlo sampling. . . . 96
6.4 Crossover of the 10 structure matrices with the highest MIVs for each nger. 96
6.5 Illustration of perturbations of the joint center of rotation. . . . . . . . . . 97
6.6 Illustration of Markov-Chain Monte Carlo algorithm for distribution of
maximal tendon tensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.7 Comparison of human grasp quality with boxplots of anthropomorphic
hand designs. Optimization step #1 is for the joint centers of rotation.
Optimization step #2 is for the distribution of maximal tendon tensions. . 100
6.8 Various index nger tendon routings. Not to scale. (a) Best crossover
index nger design selected for optimization. (b) Example of an index
nger tendon routing that produced a very low grasp quality. (c) Typical
2N design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.9 Markov-Chain Monte Carlo optimization. (a) Visualization of optimization
paths of 10 random seeds and one center seed. (b) Optimization progress
over 150 iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.10 Index nger tendon layout, maximal tendon tension distribution, and fea-
sible force sets for top 2 optimized designs and the human hand. Tendon
layouts shown roughly to scale. Feasible force sets shown to scale. . . . . 103
6.11 Power grasp of a heavy bar. The hand must resist a force that is opposite
the distal direction of the index nger. . . . . . . . . . . . . . . . . . . . . 106
7.1 Procedure for nding admissible structure matrices. N+1 structure matrix
shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Postures analyzed for each tendon routing. Link lengths and joint diame-
ters shown to scale (i.e., with kinematic parameters of the DLR hand). . . 115
xiv
7.3 Reformulation of variables in Equation 7.4 for use in Equation 7.6. ()
denotes element-by-element multiplication. R
i
is the i
th
row of R. Joint
stiness adjustability (JSA) is equal to rank(
~
R). . . . . . . . . . . . . . . 119
7.4 Joint stiness adjustability versus number of tendons, plotted for all ad-
missible routings. Mathematically, JSA is he the rank of
~
R, which is the
number of free parameters of the joint stiness matrix that can be inde-
pendently chosen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.5 Optimized endpoint stiness eccentricity (ESE*) vs. unoptimized maximal
isotropic value (MIV), averaged over the 3 postures. Note: only 524 out
of 1,200 data points shown (all other designs had higher ESE* than 8 and
lower MIV than 16). Large circles mark the averages of posture 1 (small
squares), posture 2 (small triangles), and posture 3 (small diamonds) for
the routings shown in Figure 7.6. . . . . . . . . . . . . . . . . . . . . . . . 125
7.6 Illustration of 3 routings along with stiness ellipsoids and feasible force
sets. JSA: Joint stiness adjustability. MIV: Maximal isotropic value
before optimization, in N. MIV*: Maximal isotropic value after optimiza-
tion, in N. ESE: Endpoint stiness eccentricity before optimization. ESE*:
Endpoint stiness eccentricity after optimization.
~
K
t
: tendon stinesses
producing ESE*, normalized so that maximal stiness is 1.
~
T
max
: maxi-
mal tendon tensions producing MIV*, in N. Note: all results shown are for
posture 2 only, and values correspond with the small triangles in Figure 7.5.127
8.1 (a) Arm model. (b) Workspace of arm model. SVD used to transform
endpoint stiness matrix to a stiness ellipse. . . . . . . . . . . . . . . . . 137
8.2 Endpoint stiness variables reformulated. () denotes element-by-element
multiplication, and R
i
is the i
th
row of R. . . . . . . . . . . . . . . . . . . 140
8.3 The endpoint stiness ellipse as dened by the desired endpoint stiness
vector, corresponding to an orientation angle and a set condition number. 142
8.4 Example of realizable orientations in various postures for a condition num-
ber of 2. Red lines indicate that the orientation of the major axis in that
position is realizable, and black lines indicate that the orientation is not
realizable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.5 Fraction of realizable orientations given various modeled conditions. . . . 147
8.6 Range of orientations in the presence of synergies. . . . . . . . . . . . . . 148
8.7 Maximal possible energy reduction for any orientation given the condition
number and posture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
xv
9.1 2-D and 3-D views of nger model in SolidWorks, and the actual nger. . 160
9.2 Pulleys used in nger design. (a) Turcite rotating small pulley. (b) Alu-
minum terminating small pulley. (c) Turcite rotating large pulley. (d)
Aluminum terminating large pulley. . . . . . . . . . . . . . . . . . . . . . 161
9.3 Illustration of calculation of MIV (maximum isotropic value) from feasible
force set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
9.4 Base moment arm matrices used when nding admissible and unique ten-
don routings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
9.5 Finger posture used in computations and experimental testing. . . . . . . 166
9.6 Experimental system for feasible force set testing. . . . . . . . . . . . . . . 169
9.7 Maximum isotropic values for various routings. (a) Boxplot of MIV for all
designs before and after pulley-size optimization. (b) Boxplot of MIV vs.
design (includes optimized and unoptimized pulley sizes). . . . . . . . . . 172
9.8 Results from experimental testing of various routings. (a) The 6 dierent
routings tested. Shown to scale. R matrix values are in mm. (b) Experi-
mental vs. theoretical MIV. Parity line is where experimental MIV would
be exactly equal to theoretical MIV (intercept of 0, slope of 1). Regression
has an R
2
value of 0.987. (c) Table of averages from 3 tests for each de-
sign in main posture. (d) 3-D visualization of experimental and theoretical
feasible force sets for designs 1 and 6. . . . . . . . . . . . . . . . . . . . . 173
9.9 Illustration of a simple (but intelligent) tendon re-routing that drastically
increases MIV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
10.1 Top and side views of nger design and kinematics. . . . . . . . . . . . . . 187
10.2 Finger placements for each grasp. . . . . . . . . . . . . . . . . . . . . . . . 190
10.3 Base moment arm matrices used when nding realizable, unique tendon
routings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10.4 Illustration of Markov-Chain Monte Carlo algorithm for distribution of
maximal tendon tensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
10.5 Experimental system for grasp testing. . . . . . . . . . . . . . . . . . . . 196
10.6 (a) Computational results of grasp quality for hand designs. Optimization
paths shown. (b) Pulley-size and max tension optimized designs. . . . . . 198
xvi
10.7 Computational predictions of tness for unoptimized and optimized naive
2N design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
10.8 Results from experimental testing of various routings. Experimental vs.
theoretical grasp quality for both grasps. Parity line is where experimental
grasp quality would be exactly equal to theoretical grasp quality (intercept
of 0, slope of 1). Regression line constant term forced to zero. 3-D force
portions of grasp wrench set for two dierent tests shown on right (torque
constrained to zero). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
10.9 Experimental vs. predicted volumes of grasp wrench sets. . . . . . . . . . 204
xvii
Abstract
This thesis work focuses on the design and optimization of tendon-driven systems. One
of the chief reasons these systems can be termed \bio-inspired" is due to the fact that
they produce forces and movements via tendons which are connected to actuators and
have uni-directional action (they can only pull, not push). In particular, the human
hand sets itself apart from the rest of the body's neuromuscular systems in that there
are no muscles in the ngers themselves: all the muscles are proximal to the ngers. The
study and analysis of tendon-driven hands and ngers suggest that i) the human hand
is mechanically optimized for grasping capabilities and ii) use of bio-inspired principles
in robotic systems can drastically improve force-production capabilities, grasp strength,
and stiness control performance. Some of the bio-inspired principles which are proven
throughout this work to be very benecial in robotic systems are asymmetry and skewed
combinations of design parameters. This work not only has strong implications for the
design of commercial robotic and prosthetic hands, but also broadly seeks to inspire
creativity in design and optimization routines for complex problems in many dierent
engineering disciplines.
xviii
Chapter 1
Introduction
1.1 Background
Biomimetic tendon-driven systems have been designed over the past few decades for the
purposes of grasping and manipulation (Ambrose, Aldridge, Askew, Burridge, Blueth-
mann, Diftler, Lovchik, Magruder & Rehnmark 2000, Jacobsen, Iversen, Knutti, Johnson
& Biggers 1986, Jau 1995, Massa, Roccella, Carrozza & Dario 2002, Salisbury 1982).
These systems (including dexterous prosthetic and robotic hands) take their inspiration
from the tendons present in the entirety of the human body. The most fantastic example
of a tendon-driven system either in nature or in robotics is the human hand. The com-
plexity, intricacy, and functionality of its elaborate architecture and 46 muscles have never
been duplicated, even without consideration of its spectacular sensory and regeneration
capabilities. It is unrivaled in it ability to grasp and manipulate objects, particularly in
unstructured environments, while having very compact size and excellent durability when
compared with robotic and prosthetic hands available today.
1
While bio-inspired articial hands have been criticized for issues such as friction and
tendon compliance (Chang, Lee & Yen 2005), they have distinct advantages over torque-
driven hands including light weight, low backlash, small size, high speed, and remote
actuation (Tsai 1995). Another important advantage is that they oer signicant design
exibility in setting moment arms and maximal tendon tensions (Pons, Ceres & Pfeier
1999), which allows optimization of capabilities for particular families of tasks (such as
grasping).
1.2 Previous Work
Several studies have addressed the problem of designing the topology, tendon routing,
or link design of tendon-driven manipulators (or ngers). According to Flavio Firmani,
for example, \The knowledge of maximum twist (velocity-producing) and wrench (force-
producing) capabilities is an important tool for achieving the optimum design of ma-
nipulators" (Firmani, Zibil, Nokleby & Podhorodeski 2008). Special attention has been
given to the design of manipulators with isotropic transmission characteristics (i.e., abil-
ity to transmit forces equally in all directions at the end eector) (Lee 1991, Chen, Su &
Yao 1999, Ou & Tsai 1996, Sheu, Huang & Lee 2009). A few advantages of this isotropic
property include more uniform tendon force distribution and minimization of the disper-
sion of noise through the system (Salisbury & Craig 1982, Ou & Tsai 1996). However, it
has been noted that it may be advantageous to design a manipulator with non-isotropic
characteristics (Tsai 1995), as is the case in the human hand (Valero-Cuevas 2005). In
2
addition, prior work on isotropic transmission does not consider tension limits of the
tendons, which is of particular importance when designing small, dexterous hands.
There is a vast body of literature on dening and computing grasp quality metrics.
Some studies emphasize the full set of grasp wrenches, but do not take into account
the force capabilities of each nger (Miller & Allen 1999). Many other grasp quality
metrics can be computed based on other criteria, but do not take into account any
tendon-driven architectures, which limits their applications to the design of tendon-driven
mechanisms (Surez Feijo, Cornell & Roa Garzn 2006). Previous research on grasp quality
for tendon-driven hands has enabled computation of a grasp quality metric based on a
very specic, pre-dened task wrench space (Fu & Pollard 2006). However, they note that
their methodology, which utilizes a linear programming approach, does not generalize to
the full set of feasible grasp wrenches.
Several studies have analyzed the kinetostatic performance of tendon-driven and
torque-driven manipulators (Bouchard, Gosselin & Moore n.d., Chiacchio, Bouard-
Vercelli & Pierrot 1997, Firmani et al. 2008, Finotello, Grasso, Rossi & Terribile 1998,
Gouttefarde & Krut 2010, Zibil, Firmani, Nokleby & Podhorodeski 2007, Tsai 1995, Lee
& Tsai 1991, Lee 1991) (determining the kinetostatic capabilities given design parame-
ters), and several others have addressed their optimization or synthesis (specifying the
design parameters given desired capabilities) (Fu & Pollard 2006, Chen et al. 1999,
Ou & Tsai 1993, Ou & Tsai 1996, Sheu et al. 2009, Angeles 2004, Aref, Taghirad
& Barissi 2009, Chablat & Angeles 2002, Khan & Angeles 2006). These studies are
based on mathematical theory. The fabrication of robotic ngers has been widely ac-
complished for robotic hands (Jacobsen et al. 1986, Salisbury & Craig 1982, Shadow
3
Robot Company n.d., Grebenstein, Albu-Scher, Bahls, Chalon, Eiberger, Friedl, Gru-
ber, Hagn, Haslinger & Hppner n.d., Ambrose et al. 2000, Jau 1995, Massa et al. 2002,
Lin & Huang 1996, Kawasaki, Komatsu & Uchiyama 2002, Namiki, Imai, Ishikawa &
Kaneko 2003, Yamano & Maeno 2005, Gaiser, Schulz, Kargov, Klosek, Bierbaum, Py-
latiuk, Oberle, Werner, Asfour & Bretthauer 2008). Experimental testing of kinetostatic
performance can be found in the biomechanics literature (Kuxhaus, Roach & Valero-
Cuevas 2005, Valero-Cuevas, Zajac & Burgar 1998), but these do not implement a system
whose parameters can be altered. We combine these three areas of theory, fabrication,
and testing to optimize and validate hardware implementations of alternative robotic
nger and hand designs.
1.3 Signicance of Research
Borrowing principles from robotics and mathematics (Yoshikawa 2002) and utilizing com-
putational geometry tools, this dissertation research is devoted to developing and imple-
menting computational analyses of tendon-driven systems. This thesis presents novel
analysis methods and the results from associated experimental studies.
The signicance of using this methodology for analyzing human hands is that it allows
the quantication of the deterioration of grasping abilities of hands that are subjected
to nerve trauma (e.g., an injury to the innervation pathways of the hand due to a trac
accident) or disease (e.g., carpal tunnel syndrome, or low nerve palsies resulting from
incomplete spinal cord injury). In addition, due to impairment or loss of grasping abilities
resulting from disease or trauma, tendon-transfer surgeries (where a tendon from one
4
muscle is cut and then attached to another area to restore function) are commonly used
to reinstate these abilities. Part of this work now enables comparison of various tendon-
transfer surgeries to calculate their predicted success. Comparison of patient outcomes
with these quantitative predictions of functional impairment and surgical success will
allow more rapid development of new treatment modalities.
The signicance of this research for designing robotic and prosthetic hands is that
it will allow optimization of the strength and minimization of the size of these hands.
More specically, the design methodology developed in this reserach can be used to
design dexterous, tendon-driven hands with higher grasp capabilities than are currently
available, and simpler hands with specic capabilities.
In addition to robotic and prosthetic hand design, the application of this methodology
is very broad and is applicable to the design of any industrial and commercial tendon-
driven system. One example of such a commercial system is the da Vinci Surgical System,
which is the most successful commercially-available surgical robotic system and whose
instruments are driven by tendons. Moreover, the small size aorded by tendon-driven
devices make them ideal for minimally invasive surgery, rescue robot and service robot
applications.
1.4 Dissertation outline
There are 11 chapters in this thesis including the introductory chapter. Chapters 2
and 3 are dedicated to introductory material on tendon-driven systems which is the
mathematical and computational foundation for the remaining chapters. These were my
5
parts of a book chapter submitted in 2011 with Francisco Valero-Cuevas, Jason Kutch,
and Evangelos Theodorou. Chapter 4 is based on a journal paper which develops a novel
method for grasp quality analysis of tendon-driven hands, and it is the core analysis
methodology for chapters 5, 6, and 10. Chapter 5 is based on a conference paper on
human hand grasp quality following nerve disorders. Chapter 6 is based on a journal
paper which compares the human hand with optimized robotic hands that have the
same kinematics and constraints. Chapter 7 is based on a full-length conference paper
which compares hundreds of robotic nger designs in their force-production and passive
stiness control characteristics. Chapter 8 is based on a journal paper which analyzes the
capabilities of humans to control the stiness characteristics of their arms. Chapters 9
and 10 are based on journal papers which experimentally test bio-inspired, recongurable
nger designs in hardware. Francisco Valero-Cuevas is the last author on the papers
based on Chapters 2-10, and Jason Kutch is the second author on the papers based on
Chapters 4, 5, and 9. Chapter 11 discusses conclusions and future research.
6
Chapter 2
Fundamentals of Tendon Actuation
A thorough understanding of the kinematics and kinetics of tendon-driven actuation is
necessary to fully analyze the capabilities and limitations inherent in these biological
and robotic systems. This section gives a brief overview of i) the theory behind the basic
kinematic and kinetic analysis of serial robotic manipulators and ii) the aspects of tendon-
driven systems that set them apart from traditional torque-driven systems. It should be
noted that this section is intended to give the reader a sense of the mathematical principles
of these topics and some associated issues, but is by no means comprehensive and further
study is likely required to use this knowledge in practice. Our prior publications (Valero-
Cuevas, Homann, Kurse, Kutch & Theodorou 2009, Valero-Cuevas 2005) also contain a
wealth of references to background material.
2.1 General serial linkage mechanisms
The term \serial linkage mechanism" refers to a mechanism (or manipulator as is called
in robotics) that has two basic types of joints that connect dierent links in series to
grant them DOFs: prismatic and revolute. Prismatic joints (i.e., translational) are those
7
that can extend or retract a link along a xed axis with no rotation (e.g., a pneumatic or
hydraulic cylinder), and revolute (i.e., rotational) joints are those that revolve a certain
axis with no other motion (e.g., a hinge joint). In this section, we will be concentrating
on revolute joints, as these are the types of joints that are used to represent the DOFs of
vertebrate limbs. Most vertebrate limbs are approximated by revolute joints articulating
rigid links
1
. Universal joints such as those used to represent 2-DOF joints like in the
metacarpophalangeal (MCP) joint of the index nger (which is the base knuckle of the
nger) can be represented by two revolute joints with intersecting rotational axes. Other
joints with 3 DOFs like the shoulder or hip can be represented by spherical joints with
three revolute joints with rotational axes perpendicular to each other. Other joints like
the knee are more complex and involve both rotation and sliding.
The primary goals of robotic (and biological) manipulators are to i) produce a certain
desired kinematics at the end-eector (e.g., the hand is the end-eector if the arm is
considered as the manipulator, or the ngertip is the end-eector if the nger is considered
as the manipulator) or ii) to produce a desired force at the end-eector. Combinations of
motion and force are also desired, but they are an important special case that is treated
in detail elsewhere (Keenan, Santos, Venkadesan & Valero-Cuevas 2009).
1
In biological systems, joint kinematics arise from the interaction of the contact of bony articulating
surfaces held by ligamentous structures. In such cases, the kinematics can be load dependent (Valero-
Cuevas & Small 1997)
8
2.1.1 Kinematics
Kinematics refers to the motions and positions of a manipulator without regard to the
forces that produce them. Very straightforward methods can be used to calculate ve-
locities and positions of the end-eector. The basic kinematic problem can be stated as
follows: given that we have a mechanical model and known the joint variables (i.e., joint
angles and angular velocities) of a robotic or biological manipulator, what is the endpoint
position and velocity? We will rst describe the method for calculating endpoint position.
x y
z
CMC Adduction-
abduction
CMC Flexion-
extension
MCP Flexion-
extension
MCP Adduction-
abduction
IP Flexion-
extension
x
y
(a)
(b)
Figure 2.1: (a) Human arm modeled as planar 2-joint serial manipulator. (b) Human
thumb modeled as 3-D 5-joint serial manipulator.
A human arm modeled as a planar 2-joint serial manipulator is shown in Figure 2.1a.
The variables used to calculate the hand position are the lengths of the forearm and upper
arm and the joint angles of the shoulder and elbow joint. It requires very little knowledge
of geometry to be able to calculate that the endpoint (in cm) is at
9
(x;y) = (25:4 cos(135
) + 30:5 cos(15
); 25:4 sin(135
) + 30:5 sin(15
)) = (11:5; 25:9)
Simple enough. However, consider the 3-D model of the thumb shown in Figure 2.1b,
in which there is one universal joint at the CMC (carpometacarpal) joint, one universal
joint at the MCP (metacarpophalangeal), and one hinge joint at the IP (interphalangeal)
joint. Say the metacarpal bone (closest to the wrist) has length 5.08cm, the proximal
phalanx (next bone toward the thumbtip) has length 3.18cm, and the distal phalanx (the
bone on the thumbtip) has length 2.54cm, and the joint angles are (as dened by the
Denavit-Hartenberg convention, explained below):
Joint Angle
CMC Adduction-abduction 45
CMC Flexion-extension 20
MCP Adduction-abduction 10
MCP Flexion-extension 30
IP Flexion-extension 20
Where is the endpoint? The analytical expression for calculating the endpoint co-
ordinates is quite complicated, and the position is very dicult to calculate by cursory
geometric examination. However, we are able to calculate the endpoint position in relation
to a global reference frame in a straightforward, step-by-step manner if we use homoge-
neous transformations of coordinate systems, and attach a unique coordinate system to
each link of the serial manipulator. The placement of the coordinate system is most con-
veniently and concisely done for serial manipulators using the Denavit-Hartenberg (D-H)
convention (Denavit 1955), and is shown for the arm example in Figure 2.2.
10
y
x
y
x
x
y
Figure 2.2: Coordinate frames assigned to planar 2-link arm with the D-H convention
and corresponding D-H parameters.
Each coordinate system (after the global coordinate system x
0
;y
0
;z
0
) is placed in a
systematic manner to obey the convention and accurately dene each of the 4 parameters.
Thei
th
z-axis of each joint is always aligned with the i
th
joint axis. In the arm example,
the axes of rotation are all out of the page in the positive z-axis. The x-axis is dened
as the directed line perpendicular to the z-axes from frame i 1 to framei. For the arm
example, this means that the i
th
x-axis is aligned with the i
th
link. The y-axis is then
determined from the already-dened x- and z-axes using the right-hand rule.
Brie
y, the Denavit-Hartenberg convention involves 4 parameters for each coordinate
system transformation dened by a joint-link combination: a, , d, and . For serial
manipulators with all revolute joints, these parameters are easily dened. The link length
for thei
th
link isa
i
, the rotation of the z-axis from framei1 toi around thei
th
x-axis is
i
(typically in 90
increments for serial robotic manipulators), the perpendicular distance
11
between the x-axes of link i 1 and i is d
i
(typically 0 for serial robotic manipulators),
and the joint angle for thei
th
joint is
i
. The parameters for the arm example are shown
in Figure 2.2 and the parameters for the thumb example are (note that the 5th frame,
not shown in Figure 2.1b, is at the thumbtip):
Thumb D-H Paramters
Frame a
i
i
d
i
i
1 0 90
0 45
2 2" 0 0 20
3 0 90
0 10
4 1.25" 0 0 30
5 1" 0 0 20
Once the D-H parameters are determined, the procedure for nding the endpoint is
standardized (in fact, it was developed to be implemented computationally for arbitrary
systems) by means of homogeneous transformations expressed in terms of D-H param-
eters. A homogeneous transformation matrix is a 4 4 matrix composed of a 3 3
rotation matrix, a 3 1 translation vector, and a 1 4 row vector whose elements are
always
0 0 0 1
, as shown in Figure 2.3. The i
th
parameter set can be used to
transform points from frame i back to frame i 1. For our purposes, we can use one
homogeneous transformation matrix for each set of D-H parameters, multiply them to-
gether, and then read o the 31 translation vector which has the x, y, and z coordinates
of the endpoint.
Let us take the arm as a short example. The transformation matrix T
2
1
uses the D-H
parameters from frame 2 and the transformation matrixT
1
0
uses the D-H parameters from
12
3x3 Rotation Matrix 3x1 Translation Vector
1x4 Constant Vector
Figure 2.3: Homogeneous transformation matrix in terms of i
th
set of D-H parameters.
frame 1 to produce the following transformation T
2
0
matrix, from which we can directly
get the coordinates of the endpoint, shown in Figure 2.4.
(x,y,z) position of hand
Figure 2.4: Coordinates of hand obtained from homogeneous transformation matrix.
Using this simple method, we can determine that the coordinates (in cm) of the
thumb endpoint are (8.92, -3.94, -2.49). While this is not an exhaustive treatment of
the kinematics of serial manipulators, homogeneous transformations, or the D-H param-
eter convention, it suces for the purposes of this chapter to know that the endpoint
coordinates of any manipulator can be obtained in analytical form, into which the D-H
parameters can be substituted to get the endpoint position. For a more detailed explana-
tion of these topics, the reader is referred to (Valero-Cuevas 2005, Denavit 1955, Murray,
13
Li & Sastry 1994, Yoshikawa 1990). In addition, an interesting analysis of anatomical D-H
parameters found in the human thumb can be reviewed in (Santos & Valero-Cuevas 2006).
Let us now turn to the issue of calculating endpoint velocity, given joint angular
velocities of the joints. The endpoint velocity vector can be calculated using the following
equation:
_
~ x =J(~ q)
_
~ q (2.1)
where
_
~ x is the endpoint velocity vector (generally
_ x _ y _ z
T
, but can be modied to
_ x _ y
T
if manipulator is planar), J(~ q) is the manipulator Jacobian, ~ q is the vector of
joint angles, and
_
~ q is the vector of joint angular velocities.
The manipulator Jacobian is fundamental to the calculation of the feasible motions
and forces that a manipulator can generate. It is a matrix of partial derivatives of the
endpoint position coordinates with respect to each of the joint angles, as shown in the
following equation (x(~ q) denotes the analytical expression of x-position of the endpoint
as a function of ~ q, as determined by the homogeneous transformations described above
or other methods):
J(~ q) =
2
6
6
6
6
6
6
4
@x(~ q)
@q
1
@x(~ q)
@q
2
@x(~ q)
@qn
@y(~ q)
@q
1
@y(~ q)
@q
2
@y(~ q)
@qn
@z(~ q)
@q
1
@z(~ q)
@q
2
@z(~ q)
@qn
3
7
7
7
7
7
7
5
(2.2)
where n is the number of joints of the manipulator. The last row can be eliminated if
analyzing a planar manipulator, since all the entries will be zero anyways. In addition,
14
it should be noted that this is a translational Jacobian (i.e., involving only the endpoint
position). Jacobians with up to 6 rows which describe the endpoint translational and
angular velocity can be similarly calculated (Yoshikawa 2002), but we will work primarily
with translational velocities and forces in this chapter, in which case we only require the
calculation of the translational Jacobian.
Each column of the Jacobian is the endpoint velocity vector produced by one unit
of the corresponding joint angular velocity (i.e., the rst column of the Jacobian is the
endpoint velocity vector produced by an angular velocity of 1 radian per second at the rst
joint if other joint angular velocities are zero, the second column is the velocity vector
produced by a 1 radian per second angular velocity at the second joint if other joint
velocities are zero, etc.). Let us look at the planar arm example again. If we eliminate
the last row since the arm is planar, then the Jacobian is illustrated in Figure 2.5.
It can be noted that the translational Jacobian for the thumb example has 3 rows (for
x, y, and z coordinates) and 5 columns (one for each joint, or rotational DOF). Further
treatment of the Jacobian can be found in (Yoshikawa 2002, Murray et al. 1994). For now,
it suces for the reader to know that the Jacobian relates joint velocities to endpoint
velocities, and that it can be derived in a straightforward manner for any arbitrary serial
manipulator by taking partial derivatives of the analytical expressions derived from the
use of D-H parameterization (or other methods) and homogeneous transformations.
2.1.2 Kinetics
Kinetics refers to the forces and/or energy transmitted through a system without regard
to its motion or positioning capabilities. We will mainly address the force-production
15
x
y
Endpoint velocity direction
produced by a positive
angular velocity in joint 1
x
y
Endpoint velocity direction
produced by a positive
angular velocity in joint 2
Figure 2.5: Illustration of Jacobian for arm system.
capabilities of manipulators. The main equation involved in determining the forces that
can be produced by a serial manipulator is given in the following static equilibrium
equation:
~ =J(~ q)
T
~
F (2.3)
where~ is the vector of joint torques,J(~ q)
T
is the transpose of the Jacobian at a specic
posture (i.e., set of joint angles,~ q), and
~
F is the 3-D vector of forces (or 2-D if analyzing a
planar system) produced at the endpoint. This equation is derived using the principle of
virtual work. For simplicity we will write J instead ofJ(~ q). We begin with the following
basic physics equations:
16
W =
~
F~ x (2.4)
W =~ ~ q (2.5)
(2.6)
Which, for conservation of energy means that the internal (joint torques) and external
(endpoint forces) must equal each other in the absence of deformation of the rigid links:
~
F~ x =~ ~ q (2.7)
where W is the work done,
~
F is a vector of forces, ~ x is a vector of translational
displacements, ~ is a vector of joint torques, and ~ q is a vector of angular displacements.
Taking dierentials on both sides, we get
@W =
~
F@~ x (2.8)
@W =~ @~ q (2.9)
~
F@~ x =~ @~ q (2.10)
Substituting the dot product operator by its equivalent vector inner product, we get
~
F
T
@~ x =~
T
@~ q (2.11)
17
If we then divide both sides by @t then we get
~
F
T
_
~ x =~
T
_
~ q (2.12)
Now, since we have already seen in Equation 2.1 that
_
~ x =J
_
~ q, then we can produce
~
F
T
J
_
~ q =~
T
_
~ q (2.13)
~
F
T
J =~
T
(2.14)
And following the rules of transposition for matrix multiplication we get
~ =J
T
~
F (2.15)
Recall that although it is not explicitly shown for simplicity, J is really J(~ q), a function
of ~ q (i.e., joint conguration ).
Let us look at the arm example. If we would like to exert a 1N force in the positive
y-direction (or rather, that we want to resist a 1N force in the negative y-direction),
we can uniquely specify the joint torques which will make this happen. It is simple to
do this by using the basic equation = Fd, where F is the force applied and d is the
perpendicular distance of the line of action from the joint. Using this equation, we see
that
1
=d
1
F =0:115 Nm and
2
=d
2
F =0:295 Nm, as shown in Figure 2.6.
However, using Equation 2.2, we can do this by simple matrix multiplication:
18
x
y
Figure 2.6: Calculation of joint torques using perpendicular distances of line of action
from joint axes. Force units are in N.
~ =J
T
~
F =
2
6
6
4
0:259 0:115
0:0787 0:295
3
7
7
5
2
6
6
4
0
1
3
7
7
5
=
2
6
6
4
0:115
0:295
3
7
7
5
(2.16)
What if we specify an arbitrary joint torque vector,~ ? Can we nd the force generated
at the endpoint,
~
F ? The answer is yes, in certain situations, although we will brie
y take
a look at an example in which this does not work. The simplest case is when we have a
square (i.e., invertible, full rank, or non-singular) Jacobian matrix, in which case we can
invert the Jacobian transpose matrix in Equation 2.15 and obtain the following equation:
~
F =J
T
~ (2.17)
In this way, we can specify the joint torques found in the arm example and get the
force output at the endpoint. An important thing to note, however, is that for this
case, nding the force output is dependent on a pin joint connection at the endpoint, in
which no torque can be applied. Consider the following situation: suppose that there is
19
a pure moment exerted on the endpoint of 1 Nm. Free-body static analysis will simply
reveal that the required joint torques (in Nm) to resist this moment are~ =
1 1
T
.
However, if we plug this joint torque vector into Equation 2.17, we get that the endpoint
force (in N) is
2:67 2:67
T
! Remember that a pure moment was applied with zero
force, but using the equation indicates that there was force. However, if the joint is a pin
joint, then no torque can be exerted at the endpoint, and Equation 2.17 holds because
static equilibrium holds. Note that equations 2.15 and 2.17 are simply a compact way to
write the static equilibrium equations for a serial kinematic chain in matrix form (try it!);
which implies that to use them properly you must rst make sure that you are considering
the correct boundary conditions and number of equations to satisfy
~
F = 0 and ~ = 0.
Much confusion has arisen in the use of equations 2.15 and 2.17 because some people
tend to assume that using those equation suces to solve the problem (i.e., Jacobians are
easy to calculate, rows are easy to delete to make the Jacobian invertible as if by magic),
when in reality they represent a very specic version of the equilibrium equationsand the
user needs to make sure that they are correctly implemented.
The situation becomes even more complicated when the Jacobian is inherently non-
square. Take the thumb as an example, with a ball joint connection at the endpoint (i.e.,
the endpoint is constrained from translation but not rotation). The Jacobian is a 3 5
matrix, and is not uniquely invertible. Only the pseudoinverse may be taken. Without
getting into too much detail, the torque vector is 5-dimensional, but the 3-dimensional
linear subspace (or hyperplane) spanned by the columns of J
T
is the only portion of
this 5-dimensional torque space in which pure static x, y, and z-forces will be produced
without any self-motion of the thumb (recall that the thumb has 5 DOFs and the spherical
20
connection only constrains 3 DOFs). Any applied torque combination outside of this 3-
D hyperplane will cause the thumb to move along the 2 remaining degrees of freedom
that are not constrained. Therefore, if the specied torque vector is located on this 3-D
hyperplane, then the pseudoinverse may be used to calculate the endpoint force that is
generated by the torque vector
2
. A more detailed explanation may be found in (Fu &
Pollard 2006), but it benets the reader to at least be aware of this issue with redundant
manipulators (or those whose Jacobian is non-square). It can be noted that redundant
systems are found throughout the human body, but commercial robotic manipulators
tend to be non-redundant to simplify these (and other) issues. In addition, certain cases
arise in which the endpoint Jacobian can include one or more angular components (i.e., up
to 3 rows, one for each rotational DOF at the endpoint), and then the endpoint boundary
conditions become crucial in the analysis. For an example of a special case (the index
nger), the reader is referred to (Valero-Cuevas et al. 1998).
2
At this point it is useful to underscore properties of non-square matrices that are often not well
explored in introductory linear algebra classes. One should build intuition about non-square matrices so
that one can begin to relate the dierent names used for them: A non-square matrix is also non-invertible,
which is also called a singular matrix, which is also rank decient, which also has a determinant equal
to zero, etc. If the matrix has more columns than rows, then it also has a nullspace, which means it has
a large condition number, or singular values equal to zero, etc. All of these dierent properties point
to the fact that such matrices map from spaces of dierent dimensions, such that there is no longer a
one-to-one mapping between theses spaces, or that some points in one space map into the zero point in
the other, etc. One should not think of Jacobian matrices as simply an array of numbers, but rather as
lters that transform energy from one kind to another (Valero-Cuevas 2005). In this case, we have the
mechanical transformation of energy provided by the joint torques into work done by the endpoint forces
and vice versa (as per the law of conservation of energy used to derive the Jacobian: Equation 2.7). Thus
matrices have gains, preferred input and output directions, etc. Several of these issues will be explored
in the present chapter, as when we discuss singular value decomposition, but it is highly recommended
that the reader develop an intuition for the properties of rank decient matrices.
21
2.2 Tendon routing and moment arms
The purpose of this section is to explore the nuances of tendon-driven systems. Many
robotic manipulators are driven by torque motors at each of the joints either by linkages,
cables, or gears. These motors are able to exert forces at the various joints in both
clockwise and counterclockwise directions, with equal torques and velocities in either
direction. While robotic tendon-driven systems can also be actuated via torque motors,
they are only able to pull on the tendons and not push. This makes tendon-driven systems
distinctly dierent from their idealized torque-driven counterparts. While this one-way
actuation might seem like a disadvantage, a tendon-driven manipulator can be designed to
have advantages such as light weight, low backlash, remote actuation, and perhaps most
importantly, design
exibility. As the reader will discover, varying the tendon routing
and moment arms of a tendon-driven manipulator enables optimization (which is much
more limited with torque-driven systems) for various task requirements, especially when
size and power constraints are critical.
As a simple example, Figure 2.7 shows a one DOF manipulator that can rotate a
beam around a single joint. We will see how we can oset the bias caused by the mass of
a link by using the
exibility of tendon-driven systems by varying moment arms and/or
motor strengths. The beam has massm and length`. If a single torque motor is attached
to the beam with maximal torque in either direction
max
, then the motor has to exert
m`
2
units of torque to simply support the weight of the beam, and then remaining torque
capacity (in the counterclockwise direction) is
max
m`
2
units of torque, generating up to
max
m`
2
`
units of force in the upward direction at the endpoint. Conversely, the maximal
22
force that is able to be generated in the downward direction at the endpoint is
max+
m`
2
`
,
since the manipulator can use the weight of the beam to exert downward force. This is
all well and good. However, if we want the manipulator to be able exert equal forces
up and down given the maximal torque constraint, this cannot be accomplished with a
single torque motor.
Now say that we use 2 tendons, driven by one motor each, and the torque capabilities
are identical (i.e., the motors are able to deliver
max
units of torque to the joint in either
direction). The pulley size (also, moment arm) of radius r for each tendon is used to
actuate the manipulator. This conguration will allow the manipulator to be able to
exert the same range of forces as the torque-driven conguration. However, we are easily
able to congure the system to be able to exert equal forces in both directions at the
endpoint. One solution is divide up the torque capacity so that the motor on the top
has
m`
2
maximal units of torque more than the bottom one. This would allow the joint
to be able to produce
max
+
m`
4
units of torque counterclockwise and
max
m`
4
units
of torque clockwise at the joint, translating to the endpoint to be able to exert
max
`
units of force in the upward and downward direction after the mass of the beam is taken
into account. Alternatively, if size constraints are in place on the total pulley size, we
could also let the motors have equal maximal tensions and shift the center of rotation
downward so that the larger moment arm is on the top, allowing the torque produced in
the counterclockwise direction to be greater than in the clockwise direction in the same
manner that we could divide the motor strengths to be able to produce equal force in
both upward and downward directions. These designs are illustrated in Figure 2.7.
23
Torque motor
mounted at joint
Max torque produced at
joint by motor 1 =
Max torque produced at
joint by motor 2 =
Moment arms equal
Max torque produced at
joint by motor 1 =
Max torque produced at
joint by motor 2 =
Moment arms equal
*Motors have
unequal max
tensions
*Motors have
equal max
tensions
Max torque produced at
joint by motor 1 =
Max torque produced at
joint by motor 2 =
Moment arms unequal
*Motors have
equal max
tensions
(a) (b)
(c)
(d)
Figure 2.7: A one-DOF manipulator congurations and associated endpoint force produc-
tion capabilities. (a) Torque motor. (b) Tendons with equal max tensions, equal moment
arms. (c) Tendons with unequal max tensions, equal moment arms. (d) Tendons with
equal max tensions, unequal moment arms.
While this is not an involved or necessarily realistic scenario, it partially demonstrates
the design
exibility oered by a tendon-driven system. In this section, we will brie
y
discuss i) the denition of versatility, ii) moment arm matrices, which arise from the spe-
cic tendon routing and pulley size and type, and iii) necessary conditions for admissible
moment arm matrices.
2.3 Denition of necessary actuation: versatility
We will dene versatility as the ability to produce a joint torque vector in any direction in
torque space (i.e., cover every quadrant of the space). Versatility is always desirable and in
most cases necessary for adequate function of a manipulator. Versatility of the endpoint is
easy to check once the feasible torque set of the manipulator is determined. If the feasible
torque set (described in Section 3) encloses the origin, then the endpoint can produce at
24
least some force in any direction (if the Jacobian is full rank). In certain situations in
which joints are coupled (e.g., coupling of the distal and proximal interphalangeal joints
of the index nger), the manipulator can produce force in any direction without being
necessarily versatile, as dened here.
For a tendon-driven manipulator to be versatile, it must have a tendon routing de-
scribed by an admissible moment arm matrix (further details in a section below). Admis-
sible moment arm matrices are those that enable independent control of all of the joints
in both directions. If the Jacobian of the manipulator is full rank, then an admissible
moment arm matrix implies versatility, and vice-versa.
3
2.3.1 Moment arm matrices
A moment arm matrix, R, is an mn matrix, where m is the number of joints and n is
the number of tendons of the manipulator. The entries arer
i;j
, which is a signed moment
arm value (positive values indicate positive torque generated at a joint when tension is
applied to the tendon, and v-v), i is the joint number and ranges from 1 to m, and j is
the tendon number, which ranges from 1 to n. The moment arm matrix can be used to
transform tendon tensions to joint torques using the following equation:
~ =R
~
T (2.18)
3
The nuance here is that one can have a non-convex feasible torque set that is made of thin lines that
enter each quadrant, but cannot go everywhere (i.e., cover an area) in the neighborhood of the origin.
25
Therefore, if tendon tensions are specied, the joint torques are uniquely specied.
This is not true in the reverse case, as we will soon see. Two example robotic ngers and
their associated moment arm matrices are shown in Figure 2.8.
Joint 1
Ad-abduct
Joint 2
Flex-extend
Joint 3
Flex-extend
Joint 4
Flex-extend
d=0
Tendon 2
Tendon 1
Tendon 3
Tendon 4
Tendon 5
Tendon
1 2 3 4 5
Joint
1
2
3
4
Tendon 7
Tendon 5
Tendon 1
Tendon 2
Tendon 3
Tendon 4
Tendon 6
Tendon 8
(a)
(b)
Figure 2.8: Fundamental relationship between tendon routing and moment arm matrices.
Tendon layout of 2 4-DOF robotic ngers and their associated moment arm matrices. (a)
All equal moment arms. (b) Varying moment arms. Not to scale. Note that this gure
underscores how a given tendon routing drawing has a corresponding moment arm matrix,
and vice versa.
We can observe the systematic way in which the moment arm matrix is constructed.
In the simple nger in Figure 2.8a, let us look at Tendon 1. It crosses joints 1 and 2, and
its moment arm values are contained in the rst column of the R matrix. It will produce
a negative motion or force around the rst joint (using the right-hand rule), so the entry
in the rst row is -1. It will produce a positive motion or force around the second joint,
so the entry in the second row is +1. It does not cross the other joints, so the moment
arm value at those joints is zero. The rest of the R matrix is similarly constructed.
We can see that the design space for the moment arm matrix of a multi-joint, tendon-
driven system is very high-dimensional. The nger in Figure 2.8a has a 20-dimensional
26
design space (i.e., there are 20 entries that dene the tendon routing and moment arms)
and the nger in Figure 2.8b has a 32-dimensional design space. Even if we only use
3 possible values for the moment arm matrix entries{ 1, 0, and -1, the 20-dimensional
design space will have 3
20
3:5 billion possibilities. If we use the 32-dimensional design
space, there will be 3
32
1:9 10
15
possibilities! And keep in mind that this is not
considering that moment arm values are not discreet and can be varied to any desired
size. How is a designer ever to choose a moment arm matrix for a system given such a large
design space? Thankfully, the mathematical theory behind moment arm matrices limits
the design space of moment arm matrices to a subset of matrices which are admissible
(i.e., to make to system versatile). In addition, the computational analysis which we
will discuss later in the chapter can be used to optimize manipulator performance based
on certain task requirements. In this next section, we will brie
y go over the necessary
conditions for an admissible moment arm matrix.
2.3.2 Necessary conditions for admissible moment arm matrices
The three necessary conditions for an admissible moment arm matrix R (i.e., versatile
in the sense that provides for independent controllability of all of the degrees of freedom
of the manipulator) for a manipulator with n degrees of freedom are as follows (Sheu
et al. 2009, Tsai 1999):
1. The rank of the moment arm matrix R must be n (i.e., full row rank, even if
non-square
4
).
4
Recall that for a matrix to be full row rank it needs all its rows to be linearly independent (i.e., no
one row can be found by linear combinations of the other rows). Thus even if the matrix is `wide and has
many more columns than rows (i.e., more tendons than joints, and is therefore non-square), it can still
have full row rank
27
2. The null space of R must contain at least one vector in which all the elements are
of the same sign.
3. There are at least 2 nonzero elements in each row and there must be at least one
sign change between elements in each row.
The rst condition is necessary for manipulator versatility. In this context, we dene
manipulator versatility as the ability to control each joints angular position, velocity,
and/or torque independently. Technically, if the tendons were able to both push and pull
(i.e., if they were rods like for the swash plate of helicopter rotors, or Stewart platforms),
this condition alone would be necessary and sucient for manipulator versatility. The
second and third conditions are necessary due to the tendon-driven nature of the system.
The second condition ensures that there is at least one combination of tendon tensions
that will produce no net torque on the joints (and hence no net force at the endpoint).
We will soon see that this is necessary for the calculation of feasible solutions for tendon
tensions given a desired output, since none of the tendons may exert negative tension.
The third condition ensures that there is a minimum of one tendon routed on each side
of the joint so that it can exert torque (and produce angular velocity) in both directions
for each joint, due to the tension-only nature of tendons.
Let us now examine an equation that relates joint torques to tendon tensions. As-
suming that the moment arm matrix satises the above 3 conditions, we can nd a
least-squares solution for tendon tensions given a desired set of joint torques using the
pseudoinverse of R:
28
~
T =R
+
~ (2.19)
However, given that the pseudoinverse provides the least squares solution to the in-
vertibility problem, there is bound to be at least one negative element in the predicted
solution for the vector of tendon tensions. But again, because tendons can only pull, any
implementable solution requires that all of the elements of the tension vector be non-
negative. Therefore one should also use the basis vectors for the null space of R to nd
solutions that satisfy this non-negativity requirement
5
. The following equation contains
both theparticular solution, also given in Equation 2.19, and thehomogeneous solution
(the second term on the right side of the following equation):
~
T =R
+
~ +H
~
(2.20)
whereH is a matrix whose columns are the null vectors ofR, and
~
is an arbitrary vector
of constants. Any
~
vector may be chosen without aecting the joint torques. Therefore,
it can be adjusted freely to make all tensions positive, given that the second necessary
condition is satised.
As an extremely simple example, let us turn again to a one-DOF manipulator, shown
in Figure 2.9a. We will assume the beam is massless for this example. The 2-D Jacobian
for this manipulator is
0 `
T
. If we want to exert 1 unit of force in the upward direction,
then we can us Equation 2.15 to get the necessary torque =J
T
~
F =
0 `
0 1
T
=
` at the joint. The moment arm matrix R for this system is a 1 2 matrix equal
5
Recall that the null space is the set of all ~ x vectors that satisfy the equation A~ x =
~
0
29
to
1 1
, assuming a moment arm radius of 1 unit for each tendon. Although the
necessary conditions may seem trivial for this example, they are all satised. The matrix
is clearly full row rank (i.e., it only has one row. It is non-square because it has more
tendons than joints). To determine a basis for the null space of R (which is only one-
dimensional in this case. See (Strang 2003) for a review of this topic), we can use singular
value decomposition (See Section 3.3 and (Strang 2003)) which yields a null vector of
0:707 0:707
T
, shown in Figure 2.9b. This satises the second necessary condition.
The third necessary condition is also satised since there is a sign change between elements
in the only row of R. Since we want a torque of `, we can use the particular solution
of Equation 2.20 and the pseudoinverse R
+
=
0:5 0:5
T
to determine that the least-
squares solution is
~
T = R
+
~ =
0:5 0:5
T
` =
0:5` 0:5`
T
. Now, since we must
make all tensions non-negative, we can use the homogeneous solution with the arbitrary
constant . We have the null space of R, which we substitute in for H, shown in Figure
2.9c.
Now if we choose =0:5=0:707, then we get a tension vector
~
T =
` 0
. Now
the tendon tensions are non-negative and the desired force output is generated! We
could also have chosen a lambda that was higher than the one chosen, which would have
produced equally higher tensions in both tendons, which would cancel each other out and
the output force would be the same. In biomechanical terms, this would be known as
co-contraction, where muscles on either side of a joint are active with no net endpoint
output force generated.
30
Moment arms equal
This vector spans
the null space of R
(a)
(b)
(c)
Figure 2.9: (a) A one-DOF manipulator. (b) Determination of null space of R with sin-
gular value decomposition. (c) Calculation of solution for tendon tensions, given desired
joint torque.
2.4 Discussion
In summary, while we see that these 3 necessary conditions denitely eliminate some de-
signs from the large, high-dimensional design space that was discussed earlier, the space
is still very large and suers from the curse of dimensionality when trying to optimize
tendon driven systems that have more than a few tendons. This necessitates computa-
tional tools for quickly evaluating the tness of a particular moment arm matrix given a
specic or general set of tasks. In addition, it begins to suggest that analytical, closed
form solutions are not available for tendon-driven systems because tools such as singular
31
value decomposition assume symmetry of actuation. Thus, in Section 3 we go on to in-
troduce an alternative approach based on computational geometry that enables a more
complete and accurate treatment of these systems.
32
Chapter 3
Fundamentals of Feasible Sets
3.1 Forward analysis I: from feasible activation sets to feasible
force sets using vertex enumeration.
Having introduced the fundamentals of tendon driven systems, we now begin to describe
in detail several analysis approaches and tools to understand the capabilities of a given
manipulator design. Knowing the workspace of a manipulator (i.e., the set of postures it
can achieve and the endpoint locations it can reach) is critical to the design and control of
its kinematics, see (Yoshikawa 1990) for details. Similarly, knowing the set of all possible
endpoint forces and torques a manipulator design can produce is critical to the design
and control of its kinetics. We can analyze the kinetostatic (i.e., static force production)
capabilities of manipulators using forward analysis. The term `forward' is used because it
recapitulates the order in which a robotic manipulator is usually controlled: you deliver
select torques to the joints to produce desired endpoint forces and torques. This approach
uses a computational geometry analysis in which we map the set of all feasible commands
to its actuators (the feasible activation set, which we will describe shortly) through the
33
manipulator to nd the set of endpoint forces that the mechanism can generate. To
perform this forward analysis, we need the manipulator Jacobian J, the moment arm
matrix R, and the diagonal matrix of maximal tendon tensions F
0
(See (Valero-Cuevas
2005)). This is the opposite of inverse analysis, which maps forces at the endpoint back to
activation vectors, and which will be described in a later section. Figure 3.1 illustrates the
forward mapping of feasible activation sets to feasible force sets, using the arm example
shown in previous sections. We will give it 6 tendons (actuated by muscles).
1
2
3
4
5
6
1
1
1
Feasible
activation set
Feasible tendon
tension set
Feasible joint
torque set
Feasible
force set
Figure 3.1: Forward mapping of feasible activation set to feasible force set. Note: the
feasible activation set and the feasible tendon tension set for this example are both inR
6
,
but only 3 dimensions are shown.
Please note that we use the term `activation' to mean (in the physiological sense) the
intensity with which a muscle is driven by the neural system), or (in the robotic sense)
to mean the tension level in a tendon.
34
Thus, a feasible activation set is the set of all feasible activations of muscles, or tensions
in tendons where activation levels are normalized to range between 0 and 1, with 0 being
no activation of a tendon (and therefore no tension) and 1 being full activation of a tendon
(or maximal tension). The mapping from an activation vector ~ a to a tendon tension
vector
~
T is given by the following equation (for an overview of this topic see(Valero-
Cuevas 2005)):
~
T =F
0
~ a (3.1)
As was noted earlier, F
0
is a diagonal matrix whose elements correspond to the max-
imal tensions (or muscle forces, in the physiological sense) of each tendon. This equation
transforms a vector of normalized activations to a vector of tendon tension scaled to the
capabilities of the muscles or motors driving the system. To nd the entire feasible ten-
don tension set, all we need to do is map each vertex of the feasible activation set (i.e.,
each vertex of the unit hypercube in the positive orthant) into the feasible tendon tension
set via the F
0
matrix (See Figure 3.1, (Valero-Cuevas et al. 1998, Valero-Cuevas 2005)).
Essentially, we scale the n-cube
1
that denes and contains the convex set of all possi-
ble combinations of activation vectors
2
in to an n-dimensional parallelepiped of tendon
tensions. This is because F
0
matrix is a diagonal matrix, so each activation level gets
mapped into a tension value in units of force.
1
A unit cube in the n-dimensional space of number of tendons
2
Where a given activation pattern is a point belonging to this n-dimensional cube
35
After we have the feasible tendon tension set, we can map it to the feasible joint
torque set through the moment arm matrix R using the following equation (See Figure
3.1, (Valero-Cuevas et al. 1998, Valero-Cuevas 2005)):
~ =R
~
T (3.2)
Once again, we can map each vertex of the feasible tendon tension set to a vertex
of the feasible joint torque set. After mapping all the vertices in this way, the feasible
joint torque set is the convex hull of these vertices. Note that at this point we are able
to dene the set of all possible net joint torques the tendons are able to produce. The
size and shape of the feasible torque set will depend on the properties of the moment arm
matrix R
3
. A geometric analogy is that of casting the shadow of a parallelepiped (like
a white- or black-board eraser) onto a plane. This would be a mapping from R
3
to R
2
(three muscles and two joints). Try then to change the orientation of the parallelepiped
to see how the number of vertices, size and shape of the shadow can change. In all cases,
however, the vertices of the 2D projection are a function of the vertices of the 3D object.
Next, we can map the feasible joint torques to endpoint forces (if the proper endpoint
boundary conditions are in place and the Jacobian is square and invertible, as discussed
in Section 2; in this example, we assume a pin joint which is the proper condition for
this analysis, and the Jacobian is square{2 2). To do this, and assuming the proper
conditions, we can use the following equation to map feasible joint torque set vertices to
feasible force set vertices:
3
for the sake of simplicity we restrict ourselves to constant moment arms (i.e., which would arise from
circular pulleys, but in reality these can change with posture so one would dene R(~ q).
36
~
F =J
T
~ (3.3)
Then the feasible force set will be the convex hull of these vertices in endpoint force
space. Figure 3.1 shows all of these steps, given the parameters shown. It can be observed
that the moment arm matrix satises all of the necessary conditions for full controlla-
bility, discussed in the previous section. First, it is full rank. Second, SVD gives 4 null
vectors:
~ v
1
=
0:289 0:5 0:789 0:211 0 0
T
~ v
2
=
0:289 0:5 0:211 0:789 0 0
T
~ v
3
=
0:289 0:5 0:289 0:289 0:5 0:5
T
~ v
4
=
0:289 0:5 0:289 0:289 0:5 0:5
T
.
Adding~ v
1
,~ v
2
, and~ v
3
gives a vector in the null space~ v
null
=
0:289 0:5 1:289 0:711 0:5 0:5
T
that satises the second necessary condition of all elements being of the same sign. Lastly,
there is at least one sign change between elements in each row of R, which satises the
third necessary condition.
The entire analysis, from feasible activation sets to feasible force sets (for square,
invertible Jacobians{not the general case), can be summarized by the following equation
(Valero-Cuevas et al. 1998):
~
F =J
T
RF
0
~ a (3.4)
37
What happens in the general case when the Jacobian is non-square and therefore
the last transformation using J
T
between feasible torque sets and feasible force sets is
not possible? Well, we must modify this last transformation from feasible joint torques
to feasible endpoint forces by rst intersecting the feasible torque set with the linear
subspace spanned by the columns of J
T
(Fu & Pollard 2006). The vertices of this
reduced-dimensionality set can then be transformed to endpoint force space using the
pseudoinverse so that
~
F = J
+T
~ . The 5-DOF thumb in Figure 3.2 is a case of a non-
square Jacobian, and determination of the feasible force set using this method produces
the feasible force set shown in Figure 3.2 using the parameters shown.
x y
z
Feasible
force set
Figure 3.2: 3-D Feasible force set of 5-DOF nger using parameters shown.
38
3.2 Forward analysis II: from basis vectors to feasible sets
using Minkowski sums.
As an alternative to the vertex enumeration approach described above, we can also think
of the capability of each tendon as being represented by a \basis vector" in the following
spaces:
1. Activation space
2. Tendon tension space
3. Torque space
4. Force space
Taking the convex hull of the Minkowski sum (all positive additions) of these basis
vectors in their respective spaces can then be used to build the feasible set in that space
4
. Basis vectors and their corresponding feasible sets are shown in Figure 3.3.
3.3 Why SVD is not the correct approach for analysis of
tendon-driven systems
Singular value decomposition (SVD) has been used typically in the past for analysis of the
force and velocity-production capabilities of manipulators at their endpoint (Yoshikawa
2002, Valero-Cuevas 2005). This approach produces what are called the manipulability
4
It is important to note that the notion of basis vectors in force space in only valid if the Jacobian
is square and invertible, since the basis vectors in torque space can only be transformed to basis vectors
in force space through J
T
. If the Jacobian is non-square, then the vertex enumeration approach is
necessary to build the feasible force set, as described in the previous section.
39
Tendon 1
2
3
4
5
6
Tendon 1
2
3
4
5
6
1
2
3
4
5
6
1
1
1
3
1
1
1
1
1
1
1
3
Tendons 1,4,5
Basis
Vectors
Feasible
Sets
Minkowski Sums
Activation
Space
Tendon Tension
Space
Torque
Space
Force
Space
Figure 3.3: Demonstration of the duality between basis vectors and feasible sets.
and manipulating force ellipsoid that, respectively, describe the directions in which the
endpoint velocities and forces can be produced with greatest and least ease. While very
valuable, this analysis assumes that the feasible joint angular velocity set (i.e., the set
of possible combinations of joint angular velocities) is a sphere in joint angular velocity
space (i.e.,k _ qk = ( _ q
2
1
+ _ q
2
2
+ + _ q
2
n
)
1=2
1, where n is the number of joints in the
manipulator), and that the feasible joint torque set is a sphere in joint torque space (i.e.,
k~ k = (
2
1
+
2
2
++
2
n
)
1=2
1). These assumptions are somewhat reasonable with torque
motors at the joints, and enables a quick and easy method for determining the feasible
endpoint velocity or force ellipsoids (manipulability and manipulating force ellipsoids,
respectively). The manipulability ellipsoid comes from the SVD of the Jacobian J, and
40
the manipulating force ellipsoid from the SVD of the pseudoinverse (or inverse, if the
Jacobian is square) of the Jacobian transpose J
T
. The SVD of a matrix A is:
A =UV
T
(3.5)
Where is a diagonal matrix containing the singular values, in descending order, that
indicate the highest and lowest gains the matrix can produce. The rows ofV
T
contain the
right singular vectors. These row vectors are the directions in the input space, ordered
from top to bottom, that are magnied by the corresponding singular values. The columns
ofU contain the left singular vectors. These row vectors are the directions in the output
space, ordered from left to right, that are magnied by the corresponding singular values.
Thus, if we assume the input space is a unit sphere, SVD describes the direction in output
space where the output will be greatest (i.e., the leftmost column ofU) and smallest (i.e.,
the rightmost column of U). Thus, the left singular vectors of J or J
+T
(or J
T
, if
the Jacobian is square) dene the major axes of the feasible velocity or force ellipsoid,
respectively, of the endpoint, and the corresponding singular values dene the length of
these axes. For an example of this analysis, the reader is referred to (Valero-Cuevas 2005).
It has been noted that analysis based on ellipsoids is not as accurate a representation
as using polytopes describing the feasible output set, in which the feasible joint torque
or velocity set is a hypercube or parellelpiped rather than a sphere (i.e.,j _ q
i
j _ q
i;max
or
j
i
j
i;max
) (Finotello et al. 1998). Furthermore, even if the vertex enumeration analysis
is carried out to nd such polytopes of feasible output, it is only correct for torque-driven
manipulators in which the motors can produce equal amounts of force and/or velocity
41
in both directions at the joints. For the case of tendon-driven manipulators where the
feasible inputs are hypercubes or parallelpipeds in the rst, or all positive, octant the
assumptions of input unit spheres is violated and thus SVD does not oer an accurate
representation of their output capabilities.
We illustrate the ways in which analysis with manipulating force ellipsoids can lead to
false impressions of the endpoint force capabilities of tendon-driven manipulator using the
simple arm example, shown in Figure 3.4. We see that the manipulating force ellipsoid
generated from singular value decomposition of the J
T
matrix for the arm generates a
manipulating force ellipsoid with a certain alignment (given by the left singular vectors of
J
T
) and length (given by the singular values of J
T
) of the major axes of the ellipsoid.
This is the feasible force set if the Euclidean norm of the joint torque vector is bounded
by a constant value (i.e., the feasible torque set lies inside the unit circle, or generally,
sphere).
However, the feasible torque set for the tendon-driven arm is not a circle, but is a
polygon (or generally polytope, if the dimensionality is higher than 2) in torque space,
shown in Figure 3.4. We see that the manipulating force ellipsoid diers greatly in size,
shape and orientation from its feasible force set, Figure 3.4. We see that changing the
moment arm matrix, the maximal tendon tensions, or any combination of the two results
in diering shapes and sizes of the feasible torque sets and feasible force sets. The arm
in Figure 3.4a is better at producing force in the positive-y direction, while the arm in
Figure 3.4b is better at producing force in the negative-y direction. This distinction is
not possible with manipulating force ellipsoid analysis using SVD.
42
Equal moment arm matrices,
unequal maximal tendon tensions
Unequal moment arm matrices,
unequal maximal tendon tensions
Unequal moment arm matrices,
equal maximal tendon tensions
Feasible
force set
Manipulating force
ellipsoid
(a) (b)
(c)
Feasible
torque set
Torque sphere
(circle in 2-D)
Figure 3.4: Illustration of why manipulating force ellipsoids do not accurately capture
the force-production capabilities of tendon-driven manipulators.
3.4 Other cases
While we have focused on the feasible force sets of serial chain manipulators up until this
point, there are other analyses that involve other feasible sets, including feasible wrench
sets, feasible velocity sets, feasible twist sets, and feasible acceleration sets. We will brie
y
discuss each of these feasible sets.
43
3.4.1 Feasible wrench sets
A wrench refers to a force-torque combination, and it can be up to 6-dimensional at a
manipulator endpoint (3-D forces and 3-D torques). The analysis we have presented thus
far involves 2-D or 3-D forces only, with a pin joint or ball joint connection boundary
condition at the endpoint. However, if we have a redundant manipulator (i.e., more than
2 DOFs for a planar manipulator or more than 3 DOFs for a 3-D manipulator), we can
normally exert one or more torques about the endpoint if it is constrained from rotation.
As a simple example, let us examine a planar 3-DOF manipulator, shown in Figure
3.5. The endpoint is constrained from any rotation or translation by a xed boundary
condition in Figure 3.5a. Translations in the x- and y-directions are constrained and
rotation at the endpoint around the z-axis is constrained. Therefore, all 3 DOFs have
been constrained. This is distinctly dierent from a pin joint, in which rotation is allowed.
If a pin joint is present, shown in Figure 3.5b, then only the x- and y-translations are
constrained and therefore there is an extra degree of freedom in the rotation of the
endpoint, so the manipulator may take on multiple congurations, each having a dierent
endpoint orientation angle but the same translational coordinates, shown also in Figure
3.5b. The translational Jacobian J for this manipulator is a 2 3 matrix, which has a
null space of one dimension. If a joint angular velocity vector
_
~ q is a scalar multiple of the
null vector (or in the general case, if the joint angular velocity vector lies in the linear
subspace spanned by the null vectors ofJ), then the endpoint velocity (translational) will
be zero and there will be what is termed self-motion of the manipulator (i.e., motion of the
44
nger without translational movement of the endpoint (Valero-Cuevas et al. 1998, Valero-
Cuevas 2005)).
x
y
x
y
Rotation and translation
constrained (xed
boundary condition)
Only translation
constrained (pin joint
boundary condition)
(a) (b)
Figure 3.5: Diering endpoint conditions aect feasible wrench set analysis for 3-DOF
planar manipulator. (a) Fixed boundary condition, 3 DOFs constrained. (b) Pin joint
boundary condition, 2 DOFs constrained.
Due to this possibility of self-motion, there is also the possibility of controlling the
endpoint torque by using specic combinations of joint torques if the rotation of the
endpoint is constrained. A Jacobian that has 2 translational components and one ro-
tational component can be dened if the endpoint position/rotation vector is dened
as
x y
T
(where is the angle of the endpoint in relation to the global reference
frame) rather than only a position vector
x y
T
. Also, since the manipulator has 3
DOFs, it can independently control the x-force, y-force, and endpoint torque
endpoint
.
If the Jacobian is constructed to have 2 translational components and one rotational
component, it is then a 3 3 matrix and can be inverted (unless the manipulator is in a
singular conguration in which case the matrix is also singular and not invertible). The
static endpoint output is no longer just a force vector with x- and y-components, but also
has the endpoint torque component. This force/torque combination is called a wrench
45
vector ~ w, and for this manipulator it is the vector
F
x
F
y
endpoint
T
. Equation 2.15
can then be written as
~ =J
T
~ w (3.6)
The last column of J
T
is the joint torque vector that, when applied (or any scalar
multiple), will result in only endpoint torque and zero endpoint force.
For this manipulator, since the Jacobian is square and invertible, then Equation 3.4
can be rewritten to calculate the endpoint wrench as
~ w =J
T
RF
0
~ a (3.7)
and therefore the transformation from feasible activation sets to feasible wrench sets is
simplied. An example of a feasible wrench set analysis using a square 4 4 Jacobian
can be found in (Valero-Cuevas et al. 1998). It should be noted that that example
as well as the example presented above are special cases, and further analysis of self-
motion manifolds (Burdick 2002) are necessary in general to create a square Jacobian in
the general case of translational redundancy. There will be 2 \instantaneous" (i.e., for a
specic set of joint angles) self-motion manifolds of the 5-DOF thumb for a given posture,
corresponding to 2 dimensions along which torque may be exerted. The 2 dimensions that
torque may be exerted, however, are not of constant global orientation and they change
with posture. Therefore, it is not possible to simply use a 5 5 square Jacobian (3 global
translational components and 2 global rotational components) for use in Equation 3.7
46
that holds for all sets of joint angles. To the best of our knowledge, no one has ever
published results on using square Jacobians in the general case for obtaining the feasible
wrench set for (translationally) redundant manipulators.
3.4.2 Feasible velocity and twist sets
As mentioned earlier, singular value decomposition (SVD) has been used to analyze the
manipulability ellipsoids of the endpoint of a manipulator. Now we are referring to the
feasible velocities that the endpoint can generate and not the forces (which are termed
manipulating force ellipsoids). SVD of the manipulator JacobianJ can be used to obtain
the orientation (given by the left singular vectors of J) and length (given by the non-
vanishing singular values ofJ) of the major axes of the ellipsoid which encloses the feasible
velocities of the endpoint, given that the joint angular velocity vector lies within the unit
ball (i.e.,k _ qk = ( _ q
2
1
+ _ q
2
2
+ + _ q
2
n
)
1=2
1). However, as noted before, the assumption
that the Euclidean norm of the joint velocity vector is limited is not realistic, and for
torque-driven systems, a more realistic assumption is that each joint velocity is limited
(i.e.,j _ q
i
j _ q
i;max
).
However, for tendon-driven systems, the analysis of feasible velocity sets becomes
much more complex than the analysis of feasible force sets. This is due to the mathematics
of the analysis: in the force domain, superposition may be used and therefore vector
addition types of operations are appropriate, whereas in the velocity domain, min()
types of operations are necessary for analysis rather than addition. To illustrate, we will
turn to our trusty one-DOF manipulator, shown in Figure 3.6. We shall assume that the
the maximal tendon forces for each of the three tendons are equal and also that their
47
maximal velocities are also equal. If we are doing a force analysis, it is trivial to see
that twice as much force can be exerted downward as upward, since we can activate both
tendons 2 and 3 at the same time, producing twice the force downward that would be
exerted by tendon 1 upwards if it was fully activated (note that we are neglecting the
weight of the beam). So the force production capabilities of the endpoint are clearly
asymmetrical. If we activate the maximal velocity of tendon 1, we get a certain maximal
upward velocityv
max
. However, we cannot get 2v
max
velocity in the downward direction
since the velocities will not add like forces do. If one of either tendons 2 or 3 has less
maximal velocity, then the maximal endpoint velocity will be limited by the tendon with
less maximal velocity. This is why amin() operator is more appropriate than an addition
operator We are assuming that we do not want any of the tendons to go slack at any time,
and that dynamical eects are not involved (the latter is a very questionable assumption,
see below).
A specic joint angle velocity vector
_
~ q requires a specic tendon excursion velocity
vector
_
~ s. If the desired joint angle velocity vector is known, then the maximal magnitude
of this vector is limited by one or more tendons when they reach their maximal velocities.
To the best of our knowledge, the problem of computing the complete set of feasible
velocities of the endpoint for a tendon-driven system has never been solved.
Twists are velocity vectors that have translational and rotational velocity compo-
nents. Feasible force sets and feasible velocity sets are only concerned with translational
components, whereas feasible wrench and twist sets are concerned with both translational
and rotational components. The calculation of feasible twist sets is more involved than
48
Moment arms equal
Endpoint force
or velocity
Tendon 1
Tendon 2
Tendon 3
Figure 3.6: One-DOF manipulator.
the calculation of feasible velocity sets. Diering formulations of the Jacobian to include
rotational components can be used to employ ellipsoidal measures of feasible twist sets.
3.4.3 Feasible acceleration sets
Feasible acceleration sets can be generated for manipulators if the kinetic characteristics
of the systems are known (i.e., mass of each link and gravity constant). A general for-
mulation for the dynamics of a tendon-driven system is given by the following equation
(Kuo & Zajac 1993):
M(~ q)
~ q =R(~ q)
~
T +V (~ q;
_
~ q) +G(~ q) (3.8)
whereM is the mass matrix,~ q is the vector of joint angles,R is the moment arm matrix,
~
T is the column vector of tendon tensions, V (~ q;
_
~ q) is the vector of velocity-dependent
terms (i.e., Coriolis forces), and G is the vector of gravity-dependent terms. If the initial
conguration is static (i.e.,
_
~ q = 0, then the velocity term V (~ q;
_
~ q) is equal to zero and the
joint angular accelerations can be written as a linear function of tendon tensions:
~ q =M(~ q)
1
(R(~ q)
~
T +G(~ q)) (3.9)
49
The feasible acceleration set can then be built up from the feasible tendon tension
set (or alternatively, we may substitute F
0
~ a for
~
T in Equation 3.9 and then build up the
feasible acceleration set from the feasible activation set). In Equation 3.9, the angular
acceleration vector
~ q is linear with respect to the tension vector
~
T (also, the activation
vector~ a) if the manipulator is in a quasi-static conguration initially (so the Coriolis force
terms are 0), and therefore it is only necessary to transform the vertices of the feasible
activation set or the feasible tendon tension set to vertices in the angular acceleration
space to nd the feasible acceleration set. Once the set of feasible joint accelerations
has been found, then an endpoint feasible acceleration set can also be constructed (if the
manipulator is originally in a quasi-static state) since the endpoint accelerations provided
by each joint angular accelerations will add linearly if the Coriolis acceleration term is 0.
50
Chapter 4
A Novel Synthesis of Computational Approaches Enables
Optimization of Task-Independent Grasp Quality of
Tendon-Driven Hands
4.1 Abstract
We propose a complete methodology to nd the full set of feasible grasp wrenches and
the corresponding wrench-direction-independent grasp quality for a tendon-driven hand
with arbitrary design parameters. Monte Carlo simulations on two representative designs
combined with multiple linear regression identied the parameters with the greatest po-
tential to increase this grasp metric. This synthesis of computational approaches now
enables the systematic design, evaluation and optimization of tendon-driven hands.
4.2 Introduction
Tendon-driven hands have been designed for the purposes of grasping and manipulation
(Jacobsen et al. 1986, Salisbury & Craig 1982, Shadow Robot Company n.d., Grebenstein
51
et al. n.d., Ambrose et al. 2000, Jau 1995). While their shortcomings can include friction
and tendon compliance (Chang et al. 2005), in certain applications (such as dexterous
hands) they have distinct advantages over torque-driven systems including light weight,
low backlash, small size, high speed, and remote actuation(Pons et al. 1999, Tsai 1995).
They can also oer signicant design
exibility in setting moment arms and maximal
tendon tensions (Pons et al. 1999), which allows optimization of system output capabilities
for a particular task while minimizing size and weight.
Several studies have addressed the problem of designing the topology, tendon rout-
ing, or link design of tendon-driven manipulators (or ngers) (Lee & Tsai 1991, Chen
et al. 1999, Ou & Tsai 1996, Ou & Tsai 1993, Sheu et al. 2009, Salisbury & Craig 1982, Fir-
mani et al. 2008, Tsai 1995, Tsai & Lee 1988). According to (Firmani et al. 2008),
for example, \The knowledge of maximum twist and wrench capabilities is an impor-
tant tool for achieving the optimum design of manipulators". Optimization of kinematic
hand parameters, such as nger placements, link lengths, and joint limits is addressed in
(Salisbury & Craig 1982), but we still lack comprehensive methodologies to do large-scale
optimization in these high-dimensional parameter spaces. In addition, special attention
has been given to the design of manipulators with isotropic transmission characteris-
tics (i.e., ability to transmit forces equally in all directions at the end eector) (Lee &
Tsai 1991, Chen et al. 1999, Ou & Tsai 1996, Ou & Tsai 1993, Sheu et al. 2009, Sal-
isbury & Craig 1982). Advantages of this isotropy include more uniform tendon force
distribution and minimization of the dispersion of noise through the system (Salisbury &
Craig 1982, Ou & Tsai 1996). However, it may be advantageous to design a nger with
non-isotropic characteristics (Tsai 1995), as in the human hand (Valero-Cuevas 2005). In
52
addition, prior work on isotropic transmission does not consider limits on tendon tensions,
which is critical when designing small, dexterous hands.
While there has been progress in designing and controlling tendon-driven robotic
hands, a complete methodology for the evaluation and renement of alternative topolo-
gies based on general-purpose grasp quality (i.e., wrench-direction-independent) has not
yet been synthesized or implemented. Our novel synthesis of computational approaches
now allows us to integrate and expand prior work to eliminate the following shortcomings
of using previous techniques in isolation for optimization of wrench-direction-independent
grasp quality of tendon-driven hands. The previously-isolated computational approaches
and the integration we have accomplished are illustrated graphically in Figure 4.1.
1. Optimization intractability
2. Not considering tendon-driven architecture
3. Inability to calculate wrench-direction-independent grasp quality
The rst shortcoming has been previously circumvented by using an approximation
of the full grasp wrench set itself using mathematically convenient operations (Miller &
Allen 1999, Miller & Allen 2004). If desired, our method can make computations more
ecient by a dierent method: mesh simplication of the full grasp wrench set. This
allows more accurate grasp quality calculations than prior approximations. The second
shortcoming has not been addressed in several studies that only consider independent
and identical contact points for grasp planning or analysis (Markcnsco & Yapadimitriou
53
Feasible Force Sets
Chiacchio et. al 1997
Finotello et. al 1998
Tendon-driven
Analysis
Lee and Tsai 1991,
Murray et. al 1994
Quality Metrics
Li and Sastry 1988,
Zhu and Wang 2003
Vertex Enumeration:
Qhull Algorithm
Barber et. al 1996
Global Grasp
Quality Metrics
Li and Sastry, 1988,
Ferrari and Canny 1992
Qslim Algorithm
Garland and Heckbert 1997,
Garland 2004
**Never before used in grasp analysis
Deformable Finger
Tangential Torque
Howe et. al 1988
Valero-Cuevas et. al 1998
Fu and Pollard 2006
Miller and Allen 1999
Ciocarlie et. al 2005
Inouye et. al 2012
Figure 4.1: Illustration of integration of techniques that were previously isolated.
1987, Kirkpatrick, Mishra & Yap 1992, Ferrari & Canny 1992, Miller & Allen 1999,
Miller & Allen 2004, Lin, Burdick & Rimon 2000, Mishra 1995). We have incorporated
complete characterization of the force production capabilities of arbitrary tendon-driven
hands. The third shortcoming was encountered in (Fu & Pollard 2006). They used an
ecient linear programming approach to calculate a grasp quality metric for tendon-
driven hands based on a very specic, pre-dened task wrench space, in which a nite
number of required wrench magnitudes and directions was specied. They note that their
methodology does not generalize to the full set of feasible grasp wrenches. Our integrated
method does generalize to the full set of feasible grasp wrenches and allows ecient
calculation of wrench-direction-independent grasp quality for tendon-driven hands.
Many other studies have addressed multi-ngered grasp (Murray et al. 1994, Surez Feijo
et al. 2006, Dai & Kerr 1996, Ghafoor, Dai & Duy 2004, Dai & Kerr 2000, Dai &
Kerr 2002). Several other grasp quality metrics can be computed based on other crite-
ria, but their application to the design of tendon-driven mechanisms is extremely lim-
ited (Surez Feijo et al. 2006). Compliances are included in grasp analysis for statically
54
4. Simplify feasible object force set (optional)
5. Translate contact forces to object wrenches
6. Find feasible grasp wrench set
7. Compute grasp quality
2. Build ngertip feasible force set
1. Select initial grasp parameters
3. Find feasible object force set
Procedural Steps
Figure 4.2: Flowchart of steps for nding feasible grasp wrench set and computing grasp
quality.
indeterminate grasps in (Dai & Kerr 1996) and for grasp stiness analysis in (Ghafoor
et al. 2004, Dai & Kerr 2000). We calculate the boundaries of the grasp wrench set, where
the forces are deterministic. A software environment for grasp synthesis is presented in
(Dai & Kerr 2002), but it does not consider tendon-driven architecture.
We demonstrate this novel synthesis of techniques and compare grasp quality among
two tendon-driven nger topologies, two grasp congurations, and thousands of parameter
combinations. We then use Monte Carlo simulations to demonstrate how this computa-
tionally ecient method can be used to optimize grasp quality metrics by tuning specic
design parameters.
55
4.3 Procedure
4.3.1 Finding the set of feasible grasp wrenches and computing grasp
quality
Assessing the quality of a specic grasp with a specic hand/manipulator topology re-
quires computing the feasible grasp wrench set and its associated grasp quality. A
owchart is in Figure 4.2.
4.3.1.1 Select initial grasp parameters
The calculation of grasp quality involves a few preliminary parameters to be specied,
based on the nger geometry, number of ngers, and placement of grasping points. Grasp
qualities will dier when these parameters are altered (although not substantially if they
are not greatly altered, in general). So the nger geometry (i.e., D-H parameters of the
nger), nger placements, nger postures, and object size and shape must all be specied
before the rest of the steps of the procedure are carried out. Finger geometry is used
to nd the analytical manipulator Jacobian (see the appendix in Section 4.6 for further
details) and the nger postures are determined from the nger geometry and choice of
nger placements (which is based on object size and shape) on the object.
4.3.1.2 Build ngertip feasible force set
The next step is to build the set of 3-D forces that each nger can produce while
maintaining a static posture. This set has been called the feasible force set (Valero-
Cuevas 2005, Valero-Cuevas et al. 1998), or force manipulability set in the strong sense
56
(i.e., zero endpoint torque) using the language of (Finotello et al. 1998, Yoshikawa 2002)
1
. The user must specify the nger input parameters of topology (i.e., tendon routing),
maximal tendon tensions, moment arm values, nger posture, and link lengths. Then the
feasible force set can be calculated using the method described in detail in the appendix
in Section 4.6. A visual example of a feasible force set is in Figure 4.3.
Feasible
force set
Friction cone
Intersection
Feasible object
force set
x
y
z
Figure 4.3: An example of a ngertip feasible force set and its intersection with a friction
cone to produce a feasible object force set.
4.3.1.3 Find feasible object force set
The ngertip feasible force set does not represent the actual forces that can be applied to
the surface of an object by the nger because ngertips can generally only push against
surfaces. To nd these feasible object forces, we must nd the portion of the feasible
1
The force manipulability set in the weak sense is the set of all Cartesian forces that can be exerted
by a manipulator with no constraints on endpoint torque. The strong sense force manipulability set is a
subset of weak sense set with the added constraint of zero endpoint torque.
57
force set that also lies inside a Coulomb friction cone. We approximate this cone by using
the convex hull of 8 vectors around the perimeter of the base of the cone, plus the origin,
as in (Miller & Allen 2004, Murray et al. 1994). We intersect this cone with the feasible
force set to nd the convex hull of feasible forces that may be applied to the object. We
call this set the feasible object force set, and an example is in Figure 4.3.
The inputs required for this step are the static coecient of friction and the angle of
nger contact (which is determined by object shape and nger placement). We use the
Qhull vertex enumeration algorithm to complete the intersection of these convex sets.
4.3.1.4 Simplify feasible object force set
Due to the complexity and high number of vertices that may dene the feasible object
force set for each contact point, we may wish to simplify the set to make the analysis more
computationally ecient
2
. The analysis presented in this paper can still be completed
without this step, but for thousands or millions of calculations, this step can be very
benecial with minimal loss in accuracy. To this end, we use edge collapse operations
to perform 3-D mesh simplication, see Figure 4.4a. Due to the nature of tendon-driven
feasible force sets, there may be many vertices that are very near each other. The edge
collapse operations, in eect, combine these very close points into a few points or one
point, as can be seen in Figure 4.4.This procedure was developed in computer graphics
to reduce the processing and display time for 3-D objects (Garland & Heckbert 1997).
2
The number of vertices of the grasp wrench set is on the order ofm
n
, wheren is the number of feasible
object force set vertices and m is the number of ngers (Miller & Allen 2004). So the computation time
can become intractable for high numbers of vertices.
58
Mesh
Simpli cation
(a)
(b)
Edge Collapse
y
x
z
Close vertices,
short edge
Figure 4.4: (a) An example of an edge collapse operation. The vertices v
1
and v
2
are
collapsed into a new vertex v
new
. (b) Example of using edge collapse operations to
simplify the feasible object force set from 19 vertices down to 10 vertices. Note: this view
is of the underside of the feasible object force set shown in Figure 4.3.
While some of the ner details of the feasible object force set are eliminated after this
process, this algorithm accomplishes the simplication in a theoretically optimal manner
(when considering the minimization of quadric error). Because of this, the algorithm
automatically selects close vertices for edge collapse operations. Figure 4.4b shows a
feasible object force set before and after simplication. We nd it reduces computation
time considerably with minimal eect on the results (see Results).
When the routing of the tendons is complex, such as in the human hand or in robotic
hands with complex interconnections among tendons such as in the ACT Hand(Deshpande,
Balasubramanian, Lin, Dellon & Matsuoka 2008), the mesh simplication will improve
performance even more drastically than with simple routings. For example, simplication
of the human nger feasible object force sets in (Inouye, Kutch & Valero-Cuevas 2011b)
from approximately 60 down to 12 vertices reduced computation time from 50 s to 1.37
s, a 97% reduction!
59
The single parameter input for this step is the number of desired vertices for the
simplied feasible object force set. Qslim is the program used to implement the edge
collapse operations for mesh simplication (Garland 2004).
4.3.1.5 Translate contact forces to object wrenches
The combined forces of the ngertips produce a resultant wrench on the object. An
object wrench vector, w
i;j
, produced statically by a point-contact force with friction, f
i;j
,
at ngertip location i, is given by the following equation (Miller & Allen 1999):
w
i;j
=
2
6
6
4
f
i;j
(d
i
f
i;j
)
3
7
7
5
(4.1)
where is the scaling factor that converts units of torque to comparable units of force,
d
i
is the vector from the torque origin to theith contact point,i = 1;:::;n, wheren is the
number of ngertip contact locations, andj = 1;:::;m
i
, wherem
i
refers to the number of
points dening the convex hull of the feasible object force set at ngertip locationi. Each
m
i
may be unique, in contrast to analyses that treat all contact points equally and for
which allm
i
are equal. A reasonable choice for is 1=r, wherer is the distance from the
torque origin to the furthest point on the object from that origin. As noted in (Miller &
Allen 2004), this choice of guarantees that the feasible object wrench, and hence grasp
quality metrics, are independent of object scale.
We use a soft nger model for two-nger grasp so that the grasp can produce force
closure by withstanding tangential torque (Howe, Kao & Cutkosky 1988). The nger
model assumes a certain contact area for the calculation of a rotational coecient of
60
friction, but the contact is still considered to be a point contact that can withstand
tangential torque, as described in (Murray et al. 1994). Past work has shown that an
approximately elliptical friction limit suces to encloses all combinations of tangential
torque and shear force that the ngertip can withstand without slipping or rotating.
However, a linear approximation of the friction limit surface is a valid conservative way
to model a soft nger (Howe et al. 1988), which we use to make calculations more ecient:
all we need to do is add and subtract the tangential torque limit to the appropriate object
wrench torque component for each vertex of the feasible object force set. This process
is similar to that used in (Ciocarlie, Miller & Allen 2005), but they do not consider any
feasible force set (only a simple friction cone). We assume that the ngertip can resist
any combination of tangential torque and tangential force for a constant normal force
underneath the boundary of the linear approximation.
The inputs to this step are the nger placements (for an arbitrary grasp), and coef-
cient of rotational friction (which can be specied directly or calculated from the soft-
nger contact radius) if the grasp is with two ngers, and linearization of the tangential
torque capabilities is utilized for two-nger grasp.
4.3.1.6 Find feasible grasp wrench set
After computing all the feasible object wrenches that can be applied by each nger, these
wrench vectors in 6-D are combined to form the set of all wrenches in 6-D space that can
be applied to the object which the grasp can resist. This set is a convex polytope found by
taking the convex hull of the Minkowski sum of the sets of feasible object wrench vectors,
61
where each set corresponds to a ngertip contact location. This operation is given by the
following equation (Ferrari & Canny 1992):
FGWS =ConvexHull(
n
M
i=1
fw
i;1
;:::; w
i;m
i
g) (4.2)
where FGWS is the feasible grasp wrench set,
L
is the Minkowski sum operator,n is the
number of contact points, andfw
i;1
;:::; w
i;m
i
g denotes them
i
wrench vectors dening the
feasible forces at the ith contact point. It should be noted that often the union and not
the Minkowski sum is used in grasp quality calculations to greatly reduce computation
time (Miller & Allen 1999, Miller & Allen 2004)
3
.
4.3.1.7 Compute grasp quality
Once we have calculated the feasible object wrench set, we can compute a grasp quality
based on that set. The user can specify their own grasp quality metric of choice. We
chose as an example the wrench-direction-independent grasp quality metric known as
the radius of the largest ball. It was originally proposed in (Ferrari & Canny 1992).
Determination of this grasp quality metric involves calculating the minimum oset (from
the origin) of the halfspaces that dene the convex hull of feasible grasp wrenches. The
minimum of these osets is equal to the radius of the largest ball, centered at the origin,
that the hull can contain. The metric, in eect, is equal to the maximal magnitude of a
wrench that can be applied to the object in all directions in wrench space without it losing
3
The union limits the sum of nger forces (i.e., if one nger is exerts more force at a given time,
then the other cannot produce as much force), while the Minkowski sum limits each nger force (i.e., the
feasible object force sets are independent). While the union is computationally easier and still can provide
important information about a grasp, for this study we concentrated on the more realistic Minkowski sum.
For more discussion see (Miller & Allen 1999).
62
force closure (i.e., causing the grasp to fail). A wrench vector whose magnitude is less
than the grasp quality can be applied to the object in any direction in 6-D wrench space
without losing force closure. These calculations have been completed for independent and
identical contact points in (Miller & Allen 1999).
We use the Qhull vertex enumeration algorithm for the calculation of grasp quality
and it can also easily be implemented for 2-D or 3-D visualizations of the feasible object
wrench set (Miller & Allen 2004).
4.3.2 Computing grasp quality metrics for specic manipulator designs
Here we describe the specications of the designs we analyzed and the parameters that we
used in the computations and Monte Carlo simulations presented in the results section.
4.3.2.1 Finger topology
We performed this analysis on the two dierent nger topologies in Figure 4.5a and
4.5b. Both of them had 4 kinematic DOFs: one universal joint at the base of the nger
and 2 parallel hinge joints distally. For the purposes of kinematic clarity, the nger
ad-abduction (i.e., side-to-side) axis was considered to be immediately proximal to the
perpendicular axis of the rst
exor-extensor joint, as demonstrated in Figure4.5a. The
rst nger topology had a \2N" tendon arrangement, in which there are 2 opposing (or
antagonistic) tendons for each degree of freedom, Figure 4.5a. This topology is similar to
that in the Utah/MIT, DLR, and Shadow Hands (Jacobsen et al. 1986, Shadow Robot
Company n.d., Grebenstein et al. n.d.)
4
. The second nger topology had an \N+1"
4
These hands are not all fully actuated, and some have coupled joints. However, they are 2N designs
in the sense that they use 2 antagonistic, symmetrically routed tendons to actuate each independent joint.
63
Tendon 1
Tendon 2
Tendon 4
Tendon 6
Tendon 8
Tendon 3
Tendon 5
Tendon 7
Joint 1
Ad-abduct
Joint 2
Flex-extend
Joint 3
Flex-extend
Joint 4
Flex-extend
d=0
Fingertip
(a)
(b)
Tendon 1
Tendon 2
Tendon 3
Tendon 4
Tendon 5
2N Design
N+1 Design
Figure 4.5: Grasp congurations analyzed. (a) 4-DOF robotic nger, 2N tendon ar-
rangement, with endpoint wrench description. (b) 4-DOF robotic nger, N+1 tendon
arrangement.
tendon arrangement, which has one more tendon than degree of freedom, and it is the
minimum number of tendons that can be used to fully control the nger (Lee & Tsai 1991)
5
. N+1 topologies are analyzed for isotropic transmission in (Ou & Tsai 1993, Chen
et al. 1999) and analyzed for implementation in the Stanford-JPL hand (Salisbury &
Craig 1982). The particular N+1 topology we analyzed (there are many possible N+1
topologies) is in Figure 4.5b.
For the baseline results, each of the three links of each nger had length of 2 cm. The
posture of the nger was 0
ad-abduction, 45
extension on joint 2, and 45
exion on
both joints 3 and 4. This is the posture in Figure 4.5. The link lengths and the posture
It should also be noted that there are many possible 2N, symmetric, antagonistic designs, and that we
simply chose this particular one for demonstration purposes.
5
Any more than N+1 tendons is considered tendon redundancy, and typically not more than 2N tendons
are used in dexterous robotic ngers. Manipulators or ngers with more than 2N tendons can have very
interesting redundancy properties, as in (Kobayashi, Hyodo & Ogane 1998), and can be analyzed using
our method as well.
64
were used to calculate the Jacobian matrix for these ngers. All of the moment arms for
both topologies and all joints were given a value of 5 mm, which, along with the tendon
conguration, dened the R matrix. This matrix was either 4 8 (2N design) or 4 5
(N+1 design). The sum of the maximal tendon tensions was 1000 N and divided up
evenly among the tendons. This dened the F
0
matrix, which was a diagonal 8 8 (2N
topology) or 5 5 (N+1 topology) matrix. TheJ,R, andF
0
matrices were then used to
calculate the feasible force sets of the ngers.
The sum of maximal tendon tensions being equal is an important constraint due to
the size, weight, and motor torque (and therefore tendon tension) limitations inherent in
dextrous hands. For example, the torque capacity of motors is roughly proportional to
motor weight, and minimization of weight was an important consideration in the design of
the DLR Hand II(Butterfa, Grebenstein, Liu & Hirzinger 2001). In addition, the maximal
force production capabilities of McKibben-style muscles are roughly proportional to cross-
sectional area (Pollard & Gilbert 2002). Since the actuators typically will be located
in the forearm, then the total cross-sectional area will be limited to the forearm cross-
sectional area. In this rst presentation of the methodology, we do not consider alternative
constraints on the actuation system (e.g., electrical current capacity, tendon velocities,
etc).
4.3.2.2 Grasp conguration
Both two- and three-nger grasps were analyzed for each of the two topologies, and the
nger placements are in Figure 4.6. The two-nger grasp simply had both ngertips on
opposite sides of a sphere of radius 6 cm. The two-nger conguration is in Figure 4.6a
65
(c)
(b)
(a)
x
y
z
y
z
Figure 4.6: Grasp congurations analyzed. (a) Isometric view of 2-nger grasp. (b) Front
view of 2-nger grasp. (c) Side view of 3-nger grasp.
and 4.6b. The three-nger grasp had one ngertip at the bottom and the other two
ngers were placed so that they were 30
from a vertical line going through the bottom
nger, Figure 4.6c.
4.3.2.3 Calculating grasp quality
For the two-nger grasps, the linear coecient of friction was set to 0.5. The rotational
coecient of friction was set to 2.5 times the linear coecient of friction (in mm). This
corresponds to a very-soft nger contact radius of 5 mm (Howe et al. 1988).
The grasp analysis was performed in MATLAB c
(R2010a, The MathWorks) on an
Apple desktop computer (2 x 2.66 GHz Dual-Core Intel
R
Xeon
TM
) running OS X Ver-
sion 10.6.4. The programs Qhull (
oating-point arithmetic vertex enumeration), LRS
(exact arithmetic vertex enumeration), and Qslim (edge collapse operations) were used
66
as compiled binaries for Mac OS X and were called through the MATLAB `system' com-
mand (Avis 2000, Barber, Dobkin & Huhdanpaa 1996, Garland 2004). The rest of the
computations were completed using custom MATLAB code.
4.3.2.4 Monte Carlo simulations
To demonstrate the computational utility of our method, the baseline parameters of mo-
ment arms, maximal tendon tensions, and link lengths were perturbed simultaneously
and independently (Valero-Cuevas et al. 2009). To do so, we drew from uniform distri-
butions with the lower bound being 20% below each particular baseline parameter value
and the upper bound being 20% above the baseline parameter value, for a total range of
40% variation, Figure 4.7. For each nger there are 14 non-zero moment arm values, 3
link lengths and 8 (for 2N topology) or 5 (for N+1 topology) maximal tendon tensions,
for a total of 25 (2N topology) or 22 (N+1 topology) total independent parameters that
were perturbed for each iteration. We performed 1000 iterations (each having their own
set of parameters) for each of the 2 topologies and each of the 2 grasp congurations.
This number of iterations was found to be sucient for convergence, as in (Santos &
Valero-Cuevas 2006) (discussed further in Results).
4.3.2.5 Regression analysis
To demonstrate the utility of these Monte Carlo simulations for design and analysis
purposes, the grasp quality was regressed on the independent parameters that were varied
during the simulations. Stepwise regression on only the linear terms was performed
(i.e., no interaction or higher-order terms were used) using an initial model with no
67
Nominal
parameter value
120 % nominal
parameter value
80 % nominal
parameter value
Probability
Uniform probability
distribution
Figure 4.7: Uniform sampling distribution used for each independent parameter value
perturbation in Monte Carlo simulations.
predictors, and predictors were added to the model with a cuto p-value of 0.05. This
was performed in MATLAB. Prior to the regression analysis, the independent parameters
were normalized so that the baseline value was equal to 1. In addition, the dependent
parameters were normalized so that their average was also 1. Therefore, the regression
coecients represent the expected percentage increase in the grasp quality with a 1%
increase in the independent parameter.
4.4 Results
4.4.1 Baseline results
Table 4.1 shows the grasp quality results for the two \baseline" topologies (i.e., those
with the nominal values for each design parameter) for the two- and three-nger grasps.
Despite the fact that both topologies have the same sum of maximal tendon tensions
(i.e., system input), the 2N topology is clearly superior to the N+1 topology (using the
nominal parameters) in grasp quality, and hence can resist wrenches of higher magnitude
in all directions.
68
2N Topology (N) N+1 Design (N)
2-nger 2.59 1.71
3-nger 8.66 5.60
Table 4.1: Baseline grasp quality results. Coecient of static friction
s
= 0:5. Units of
grasp quality are in Newtons.
In addition, as expected, the grasp quality is higher for the three-nger grasp than
the two-nger grasp for both topologies. These baseline results were veried using the
exact arithmetic vertex enumeration code LRS (Avis 2000), where the evaluation time
was around 100 times greater than the Quickhull algorithm (Barber et al. 1996).
4.4.2 Monte Carlo simulations
Given that computation times for the baseline cases were fairly long (about 30 s), espe-
cially for the 3-nger grasp, we simplied the feasible object force sets to make Monte
Carlo simulations feasible. We found that simplifying the feasible object force set down
to 12 vertices reduced computation time by a minimum of 46% (reduction from 23.7 sec-
onds to 12.7 seconds for N+1, 3-nger case) and a maximum of 77% (reduction from 39.9
seconds to 9.10 seconds for 2N, 3-nger case) out of the 4 baseline cases, and resulted in
less than 2% error in grasp quality.
The 1000 Monte Carlo simulations reached \convergence" in the sense that the run-
ning mean and coecient of variation varied less than 2% in the last 20% of iterations,
similar to the criteria used in (Santos & Valero-Cuevas 2006). Average evaluation time
for each of the 4 congurations is shown in Table 4.2. Figure 4.8 shows histograms of the
Monte Carlo grasp quality results for 2-nger grasp. The dierent nger topologies for
this grasp certainly have dierent mean characteristic lengths (p < 0.00001) when the
69
2N Topology (s) N+1 Topology (s)
2-nger 1.46 (0.30) 1.29 (0.42)
3-nger 9.79 (1.98) 9.77 (2.66)
Table 4.2: Average evaluation times (standard deviations) during Monte Carlo simula-
tions, in seconds.
0 1 2 3 4 5
0
100
200
Iteration Count
2N Design
0 1 2 3 4 5
0
100
200
Grasp Quality (N)
Iteration Count
N+1 Design
19 N+1
topologies
exceed
baseline 2N
topology
2N topology with baseline
parameter values
Figure 4.8: Histogram of grasp quality values from Monte Carlo simulations for two-
nger grasp, 2N and N+1 designs, with Gaussian curves overlaid. The N+1 topologies
exceeding the baseline 2N topology are shaded gray.
parameter values are perturbed by 20%. However, for the N+1 topology we nd that
19 parameter combinations exceed the grasp quality of the 2N topology with baseline
parameter values.
4.4.3 Regression analysis
The signicant regression coecients at a cuto p-value of 0.05 for grasp quality for the
N+1, 2-nger case are shown in Table 4.3, grouped by parameter type. The coecient
of determination R
2
is 0.930, signifying a good t for the linear model. Link lengths,
maximal tendon tensions, and moment arms should be adjusted according to Table 4.3
to produce the N+1 topologies that exceed the baseline 2N topology. We nd that
70
decreasing the link lengths understandably increases grasp quality (because it improves
the moment-arm:lever-arm ratio of the tendons), and decreasing the length of link 2 has
the greatest predicted eect on grasp quality. As would be expected, the one signicant
regression coecient for maximal tendon tension is positive (i.e., grasp quality can never
be worsened by increasing one of the maximal tensions). However, not all maximal tendon
tensions improve this grasp quality metric if increased. While all maximal tendon tensions
change the size and shape of the feasible grasp wrench set, some have insignicant eects
on grasp quality because increasing them increases the size of the feasible grasp wrench
set in directions that do not increase the weakest wrench capability of the grasp. That
is, they do not push out the boundary of the feasible grasp wrench set that is closest
to the origin. However, the maximal tension of tendon 1 does aect that boundary and
increasing it enhances this metric of grasp quality. Therefore, one of the \weak links"
in this topology is the maximal tension of tendon 1, which if increased leads to better
performance. Moment arms exhibit both positive and negative regression coecients in
their eect on grasp quality, as they aect the direction and magnitude of the wrench basis
vectors (Valero-Cuevas 2005). The best N+1 topology from the Monte Carlo simulations
(grasp quality of 3.31{94% greater than the N+1 baseline) has maximal tension of tendon
1 15% higher than the baseline and the moment arm of tendon 1 across the ad-abduction
axis is 16% above baseline. These parameters have the greatest eect on grasp quality,
as can be seen in Table 4.3.
Table 4.4 shows the eects of adjusting the signicant moment arm parameters indi-
vidually by 10% in the direction that increases grasp quality while keeping all the other
71
Expected Percentage
Increase in Quality
for a 1% Increase 95% Condence
Parameter in Parameter Value Interval
Link length
Link 2 -0.436 (-0.466, -0.406)
Link 1 -0.333 (-0.363, -0.302)
Link 3 -0.290 (-0.320, -0.260)
Max tension
Tendon 1 0.995 (0.265, 1.03)
Tendon 2 { {
Tendon 3 { {
Tendon 4 { {
Tendon 5 { {
Moment arm
1,1 1.01 (0.975, 1.04)
1,5 -0.593 (-0.623, -0.564)
2,5 0.553 (0.522, 0.583)
2,4 0.272 (0.243, 0.302)
1,4 -0.259 (-0.289, -0.228)
2,3 0.159 (0.128, 0.190)
1,3 -0.143 (-0.174, -0.112)
1,2 { {
2,2 { {
3,3 { {
3,4 { {
3,5 { {
4,4 { {
4,5 { {
Table 4.3: Signicant normalized regression coecients for grasp quality with 95% con-
dence intervals on N+1 topology, 2-nger grasp. `{' denotes not signicant at the cuto
p-value of 0.05. Moment arms expressed as (Joint number,tendon number). R
2
= 0.930.
72
Grasp Normalized Expected Actual
Quality Coecient Increase Increase
Baseline 1.709 { { {
Moment arm 1,1 (+10%) 1.880 1.01 10.1% 10.0%
Moment arm 1,5 (-10%) 1.812 -0.593 5.93% 6.03%
Moment arm 2,5 (+10%) 1.809 0.553 5.53% 5.88%
Moment arm 2,4 (+10%) 1.756 0.272 2.72% 2.79%
Moment arm 1,4 (-10%) 1.758 -0.259 2.59% 2.86%
Moment arm 2,3 (+10%) 1.733 0.159 1.59% 1.40%
Moment arm 1,3 (-10%) 1.734 -0.143 1.43% 1.47%
Table 4.4: Expected (from linear regression on Monte Carlo iterations) and actual (from
computational method implementation) eects of moment arm adjustments by 10% on
grasp quality of N+1 design, 2-nger grasp. Moment arms expressed as (Joint num-
ber,tendon number).
parameters at baseline levels. We see that the predictions from even a simple linear
regression are validated.
4.5 Discussion
In this work, we have demonstrated a novel synthesis of computational approaches for
evaluating the grasp quality of arbitrary tendon-driven hand designs. Our formulation is
ecient enough to consider all nger design parameters (number and routing of tendons,
tension limits, and posture) and grasp (number and conguration of ngers, friction
characteristics, and object shape and size) and computes the full feasible grasp wrench set,
from which a variety of grasp quality metrics can be obtained. In this rst demonstration
of our methodology, we compared the wrench-direction-independent grasp quality for
two topologies, two grasp congurations, and thousands of parameter combinations when
grasping a sphere, and we present the steps for extending this methodology to completely
arbitrary hand designs, objects, and nger placements.
73
Our Monte Carlo exploration of the design space demonstrates the computational
eciency and utility of our method and shows that, as expected, the 2N topology is
generally superior to the N+1 topology in grasp quality and hence can resist wrenches
of higher magnitude in all directions. This is because this 2N topology can exert a
wider range of forces on the object than the N+1 topology, resulting in higher grasp
quality. Importantly, however, our parameter exploration found certain designs (within
the allowed20% variability) for which the N+1 topology can outperform the nominal 2N
topology. If a designer favors the N+1 topology due to actuator/space/weight constraints,
there are N+1 topologies that can meet or exceed the performance of a nominal 2N
topology (which may have less design
exibility because of more tendons). These results
would apply to most objects of similar size since the main dierence would be a small
change in nger contact angle.
In addition, the extensive exploration of the high dimensional parameter spaces (i.e.,
22 or 25 dimensions) allows us to identify some critical design parameters for grasp quality
(i.e., with a highR
2
value of 0.930, noted in Table 4.3). Regressions for our N+1, 2-nger
case (Table 4.3), for example, it is clear that one tendon and one moment arm are, from
among 22 parameters, the most critical individual parameters in the design; altering
them in isolation has the greatest eect on characteristic length. Exploring second and
third order parameter sensitivities is likely intractable with this or most other techniques
because of the geometric growth of iterations needed. Second order terms in a regression
would bring the number of regressed independent variables to over 400, and third order
terms would raise that number to over 8000.
74
Nevertheless, our approach demonstrates sucient computational eciency to enable,
for the rst time, exploring large-dimensional design spaces. Optional adjustments in
mesh simplication procedures or friction cone approximations can and do bring improve-
ments to speed with minimal loss in accuracybut they are not central to our methodology.
Additionally, other techniques, such as hull approximation or Voronoi ltering, could be
used to simplify the grasp wrench set. Importantly, we tested and found that our com-
putationally streamlined
oating-point computations produced results equivalent to the
100 times slower exact arithmetic calculations.
This approach is innovative because it now enables optimizing the design of dexterous
tendon-driven hands by testing hundreds or thousands of alternative hand topologies
quickly. For anthropomorphic hands or prosthetic hands, link geometry is relatively
xed, but all tendon routing and moment arm values can be varied. For general-purpose
manipulators, everything from number and arrangement of ngers, to DOFs and link
lengths of each nger, to number, routing and strength of tendons may be varied and
evaluated. Any number of optimization algorithms, including gradient-descent, genetic,
or random search algorithms, could be employed with this methodology to explore the
design space and optimize the topology of dexterous hands. The ecacy and eciency
of random search algorithms are being explored in current research.
This method can also be used to determine the optimal grasping points of a particular
object for a particular set of tendon-driven hand design parameters. If this is desired,
then many nger placements can be tested to determine the one with the optimal grasp
quality.
75
We calculate the grasp quality for precision grasp (i.e., grasp by the ngertips) in this
study. This is the grasp that is necessary to manipulate an object. Power grasp capa-
bilities (where the ngers are wrapped completely around an object) could be calculated
with a modied version of this algorithm. However, in general, power grasp quality and
precision grasp quality will tend to be highly correlated due to the fact that a high
exion
force in the ngers is desirable for both grasps.
The shaping of the feasible output of a robotic system via variation of mechani-
cal design parameters has been of interest for several decades (Lee & Tsai 1991, Chen
et al. 1999, Ou & Tsai 1996, Ou & Tsai 1993, Sheu et al. 2009, Salisbury & Craig 1982).
Our novel synthesis of computational approaches now enables its pursuit for large dimen-
sional, tendon-driven systems. Grasp quality, manipulability metrics, and hand complex-
ity metrics such as number of ngers, number of joints per nger, and number of tendons
could also be integrated into a multi-objective optimization algorithm.
Many other grasp quality metrics are easily computed using the basic procedure
we have described. One example is the volume of the feasible wrench set (Miller &
Allen 1999). Qhull can be easily queried to calculate this volume at the same time it
is calculating the weakest wrench metric we analyzed in this study. Another example
is task-specic grasp quality metrics such as those used in (Fu & Pollard 2006, Zhu &
Wang 2003, Li & Sastry 1988). Once the grasp wrench set is calculated, the straightfor-
ward linear programming technique used in (Fu & Pollard 2006) can be used to calculate
this metric for polytopes or using singular value decomposition (Li & Sastry 1988) for
ellipsoids.
76
Future work will use this methodology to design dexterous, tendon-driven hands with
higher grasp capabilities than are currently available, and simpler hands with specic
capabilities. Furthermore, this work on static grasp can be extended to manipulability
sets or feasible acceleration sets, which quantify the velocities or accelerations with which
an object can be manipulated. This methodology could also be used in grasp planning,
where an optimal or near-optimal grasp found for a specic tendon-driven hand may
actually be a bad grasp for another tendon-driven hand. This methodology also enables
the quantitative analysis of biological hands and grasps (including human (Inouye et al.
2011b)), and also can help to answer questions about its anatomical structure, so we can
perhaps draw inspiration from it for novel robotic designs. Lastly, this analysis can also be
applied to design and optimize arbitrary tendon-driven and recongurable robots, such
as tensegrity structures, to perform complex manipulation and locomotion tasks(Paul,
Valero-Cuevas & Lipson 2006, Rieel, Valero-Cuevas & Lipson 2009).
4.6 Appendix: Calculation of the feasible force sets of tendon-
driven manipulators
Fundamental to feasible force set analysis is the calculation of the posture-dependent
manipulator Jacobian, J(q). q is the vector of joint angles (i.e., nger posture). The
Jacobian represents a linear mapping from angular velocities of the joints to endpoint
velocity as shown in the following equation:
_ x =J(q) _ q (4.3)
77
where _ x is the endpoint velocity vector (it can include both translational and rotational
components, and so can be up to 6-dimensional, see (Yoshikawa 2002) for more details),
J(q) is the manipulator Jacobian, q is the vector of joint angles (i.e., nger posture), and
_ q is the vector of joint angle velocities.
If an underactuated nger is being analyzed, then the Jacobian is only constructed
with columns that correspond with joint angles that can be independently actuated, and
the analytical expressions for each entry of the Jacobian matrix, which would normally
include all joint angles, will only include the actuated joint angles. If the last 2 joints are
coupled such as in the human hand or Shadow Hand (Leijnse, Quesada & Spoor 2010,
Shadow Robot Company n.d.), then the last joint angleq
4
would be a (presumably) linear
function ofq
3
(e.g.,q
4
=q
3
=2). The Jacobian could be reduced from 4 to 3 columns, and
the analytical expressions for each entry of the Jacobian matrix could be constructed as a
function of 3 joint angles by substituting in for the last joint angle (e.g., substitutingq
3
=2
in forq
4
). The Jacobian would then be 33 (instead of 34) even though there are 4 joint
angles. Advanced kinetostatic analysis of underactuated ngers is performed in (Birglen
& Gosselin 2004), although the simple procedure just described should be sucient for the
calculation of feasible force sets for most robotic applications. Furthermore, minimally
underactuated hands with, for example, one tendon for
exion and springs for extension
could be analyzed in the torque domain and appropriate dimensionality reduction of the
Jacobian matrix.
Once the Jacobian is calculated, using the principle of virtual work, we can nd the
linear mapping between endpoint wrench (i.e., generalized forces which can include force
and torque components and therefore can be up to 6-dimensional, depending on the
78
formulation of the Jacobian used), w, and joint torques, , as shown in the following
equation:
=J
T
w (4.4)
Since we are analyzing tendon-driven systems, we also need the moment arm matrix,
R, which contains the values of the moment arms for each of the tendons across each of
the joints. It is an n` matrix, where n is the number of joints and ` is the number
of tendons of the manipulator. The entries are r
i;j
, which is a signed moment arm value
(positive values indicate positive torque generated at a joint when tension is applied to
the tendon, and v-v), i is the joint number and ranges from 1 to n, and j is the tendon
number, which ranges from 1 to `. The moment arm matrix can be used to transform
tendon tensions, T, to joint torques using the following equation:
=RT (4.5)
We can use an activation vector, a, to represent the degree to which a tendon is
activated. Each element of a ranges between 0 (no activation) and 1 (full activation).
Further discussion may be found in (Valero-Cuevas 2005). If we dene F
0
as a diagonal
matrix of maximal tendon tensions, then we get the following relation between activations
and tendon tensions:
T =F
0
a (4.6)
79
The rst step to calculating the feasible force set is to nd the feasible torque set
by taking the convex hull of points generated by mapping each vertex of the feasible
activation set to joint torque space by combining Equations 4.5 and 4.6:
=RF
0
a (4.7)
The feasible 3-D force set can be found from this feasible torque set by intersecting the
feasible torque set with the linear subspace spanned by the columns of J
T
(Fu & Pollard
2006, Chiacchio et al. 1997). This can be accomplished with any vertex enumeration
algorithm. The vertices of this reduced-dimensionality set can then be transformed to
endpoint force space using the Moore-Penrose pseudoinverse so that
w =J
+T
(4.8)
where J
+T
denotes the Moore-Penrose pseudoinverse of J
T 6
.
If the 3-D feasible force set is being calculated (as in this study), then the wrench
vector in Equation 4.8 will be of length 3 and will have components of F
x
, F
y
, and F
z
.
Acknowledgements
The authors gratefully acknowledge the useful comments by M. Kurse, B. Holt, and C.
Raths.
6
If the Jacobian is square and invertible, the inverse J
T
can be taken and the feasible torque set
subspace intersections are unnecessary.
80
Chapter 5
Quantitative Prediction of Grasp Impairment in Peripheral
Neuropathies of the Hand
5.1 Abstract
5.2 Introduction
Grasping is a fundamental hand function that is impaired or eliminated following periph-
eral neuropathies of the hand (Riordan 1969). Using a novel computational framework for
calculating grasp quality of tendon-driven hands (Inouye, Kutch & Valero-Cuevas 2012),
we predicted grasp quality for various degrees of simulated peripheral neuropathies: (i)
carpal tunnel syndrome, (ii) low median nerve palsy, (iii) low ulnar nerve palsy, and (iv)
low radial nerve palsy.
5.3 Methods
Calculation of grasp quality for tendon-driven hands involves several steps (Inouye et al.
2012). The rst is determination of the ngertip forces that each tendon produces when
81
FPL
AbPB
AbPL
EPB
EPL
ADDo
ADDt
OPP
DIO
FS
FP
EC
EI
PI
DI
LUM
y
x
z
(a) (b) (c)
Fingertip basis vectors Feasible force sets Feasible object force sets Feasible object wrench
(d)
Largest Ball (Grasp quality)
(e)
Figure 5.1: Grasp quality calculation steps. (a) Basis vectors. (b) Feasible force sets. (c)
Feasible object force sets. (d) Feasible object wrench illustrated in 3-D. (e) Grasp quality
metric of radius of largest ball (illustrated in 3-D; actual calculation is in 6-D).
tension is applied. We use previously-published cadaveric data from the thumb and index
nger (Pearlman, Roach & Valero-Cuevas 2004, Valero-Cuevas, Towles & Hentz 2000) to
determine these basis vectors (Figure 5.1a), from which the feasible force set (the set of
3-dimensional forces that the ngertip can produce) can be calculated, shown in Figure
5.1b. These sets are then intersected with friction cones, producing feasible object force
sets: the sets of forces that the ngertip is able to apply to the object, shown in Figure
5.1c. From these sets, the set of all forces and torques on the object (i.e., the feasible
object wrench) that the grasp can resist may be calculated, shown in Figure 5.1d.
The rst grasp quality metric we used was the weakest wrench (combination of force
and torque) magnitude (in Ntorque is scaled to N with the radius of the object) that could
be resisted by the grasp. This is equivalent to the radius of the largest ball, centered at
the origin, that the feasible object wrench set can contain, and is illustrated in Figure 5.1e
in 3-D, although the actual measure is in 6-D (3 dimensions for force and 3 for torque).
For example, if the grasp quality is 5, then the grasp can resist at least 5N of force or
scaled torque in any direction. We call this the radius of largest ball.
82
Finger Muscle Innervation group
Index
Flexor digitorum profundus (FDP) M
Flexor digitorum supercialis (FDS) M
Extensor indicis proprius (EIP) R
Extensor digitorum communis (EDC) R
First lumbrical (LUM) M, CTS
First dorsal interosseous (FDI) U
First palmar interosseous (FPI) U
Thumb
Abductor pollicis brevis (AbPB) M, CTS
Abductor pollicis longus (AbPL) R
Adductor pollicis oblique (ADDo) U
Adductor pollicis transverse (ADDt) U
First dorsal interosseous (DIO) U
Extensor pollicis brevis (EPB) R
Extensor pollicis longus (EPL) R
Flexor pollicis brevis (FPB) M, CTS
Flexor pollicis longus (FPL) M
Opponens pollicis (OPP) M, CTS
Table 5.1: Muscles in each nerve pathology group. M: median, R: radial, U: ulnar, CTS:
Carpal Tunnel Syndrome.
The second grasp quality metric we used was the radius of the 6-D ball with the same
volume as the feasible object wrench, which we call the characteristic length.
Using this framework, we simulated various degrees of nerve palsies and carpal tunnel
syndrome by progressively weakening muscles controlled by these innervation groups from
their maximal force (Pearlman et al. 2004, Valero-Cuevas et al. 2000). We modeled carpal
tunnel syndrome as low median nerve palsy that does not aect extrinsic muscles, since
they are innervated proximal to the wrist (Katz & Simmons 2002). The muscles for each
innervation group are shown in Table 5.1.
83
5.4 Results and Discussion
The impairment of grasp quality with simulated advancement of peripheral neuropathy
is shown in Figure 5.2. We see that low median nerve palsy quantitatively aects both
measures of grasp quality most severely. In addition, we observe that complete loss of
any innervation group causes the radius of largest ball to be zero. This means that there
are some directions of perturbation in 6-D wrench space that the grasp cannot resist, and
therefore the grasp does not have force closure, which is considered to be a maximally
decient grasp. As expected, carpal tunnel syndrome decreases grasp quality less than
full low median nerve palsy. Although low radial nerve palsy aects the extensors of the
ngers, it has a comparable eect to carpal tunnel syndrome because they, counterintu-
itively, also contribute to grasp, as described earlier (Valero-Cuevas et al. 1998).
Acknowledgements
NSF EFRI 0836042 and NIH AR050520 and AR052345 to FVC. Thanks to Sudarshan
Dayanidhi for helpful discussions
84
0 50 100
0
2
4
6
8
Radius of Largest Ball (N)
Median Nerve Palsy
Radial Nerve Palsy
Ulnar Nerve Palsy
Carpal Tunnel Syndrome
0 50 100
0
5
10
15
20
% Advancement of Neuropathy
Characteristic Length (N)
Figure 5.2: Grasp quality deterioration as a function of % advancement of neuropathy.
85
Chapter 6
Bettering the Human Hand: Anthropomorphic
Tendon-Driven Robotic Hands can Exceed Human
Grasping Capabilities
6.1 Abstract
There is great debate about how eectively the human hand is able to grasp and manip-
ulate objects and whether it is optimized in any sense. Here we compare the grasping
capabilities of the physiological human hand and thousands of tendon-driven anthropo-
morphic hand designs. The anthropomorphic hands are given constraints that allow for
fair comparisons to the human hand (friction coecients, maximal moment arms, and
maximal tendon tensions). The layout of the anthropomorphic hand and the D-H pa-
rameters of each nger (a 4-DOF index nger and a 5-DOF thumb) are set to those
of the commercially available Shadow Hand. We use a previously-developed computa-
tional technique to assess the grasp quality of each anthropomorphic hand design. We
initially tested designs that had randomly-generated, admissible structure matrices along
with equal moment arms and maximal tendon tensions. Next, we used an optimization
86
scheme that employed crossover operations (similar to those used in genetic algorithms)
and greedy Markov-Chain Monte Carlo optimization. We nd that none of the randomly-
generated designs are able to exceed the grasp quality of a human hand and the best one
has a grasp quality 45% below that of the human hand, with a mean grasp quality 78%
below the human hand. However, optimization of the joint centers of rotation and the
distribution of maximal tendon tensions produces hands with grasp qualities that exceed
the human hand by 13-45%. In addition, one optimized design was able to outperform
a na ve 2N design by 501%. This huge dierence implies that grasping performance of
dexterous prosthetic or anthropomorphic hands can be vastly improved by altering some
of the numerous parameters. In addition, we conclude that the human hand is opti-
mized for grasping, at least to an extent, when considering that it vastly outperforms
randomly-generated designs.
6.2 Introduction
Anthropomorphic tendon-driven hands have been designed over the past few decades for
the purposes of grasping and manipulation (Ambrose et al. 2000, Jacobsen et al. 1986, Jau
1995, Massa et al. 2002, Salisbury & Craig 1982). These dexterous prosthetic and robotic
hands take their inspiration from the tendons present in the entirety of the human body.
The most fantastic example of a tendon-driven system either in nature or in robotics is
the human hand. The complexity, intricacy, and functionality of its elaborate architecture
and 46 muscles have never been duplicated, even without consideration of its spectacular
sensory and regeneration capabilities. It is unrivaled in it ability to grasp and manipulate
87
objects, while having very compact size and excellent durability when compared with
robotic and prosthetic hands available today. One part of the elegance of the human
hand is due to the control imparted to the muscles via the nervous system. The other
part is the mechanics of the tendon and muscle arrangements. This paper focuses on the
latter, in relation to grasp quality.
While bio-inspired articial hands have been criticized for issues such as friction and
tendon compliance (Chang et al. 2005), they have distinct advantages over torque-driven
hands including light weight, low backlash, small size, high speed, and remote actuation
(Tsai 1995). Another important advantage is that they oer signicant design
exibility
in setting moment arms and maximal tendon tensions (Pons et al. 1999), which allows
optimization of capabilities for particular families of tasks (such as grasping).
Previous research on grasp quality for tendon-driven hands has enabled computation
of a grasp quality metric based on a very specic, pre-dened task wrench space (Fu &
Pollard 2006). However, they note that their methodology, which utilizes a linear pro-
gramming approach, does not generalize to the full set of feasible grasp wrenches. A
recent study (Inouye et al. 2012) has developed a comprehensive framework for evaluat-
ing global grasp quality metrics of tendon-driven hands. We use this methodology for
computing and optimizing grasp quality in this paper.
A previous study has optimized a robotic nger to mimic the force production capa-
bilities of the human index nger (Pollard & Gilbert 2002). This optimization was based
on minimizing total maximal tendon tensions while meeting or exceeding the static forces
that the human nger was able to produce. Size constraints were in place to allow for
fair comparison. They also performed some optimization techniques on moment arms.
88
However, to the best of our knowledge, there has been no study to compare the grasp
quality of the human hand with anthropomorphic hands which have the same mechanical
constraints (e.g., size, muscle strength).
Here we compare the grasp quality of the human hand with that of random and
optimized anthropomorphic hands with human hand constraints. We show that the
human hand has much higher grasp quality than random designs, but that careful use of
optimization techniques can produce designs that meet or exceed the grasp qualities of
the human hand.
6.3 Procedure
6.3.1 Calculating human hand grasp quality
In order to nd the grasp quality of the human hand, we combined previously-published
cadaveric data (Valero-Cuevas et al. 2000, Pearlman et al. 2004) with a computational
method for calculating grasp quality that was recently developed (Inouye et al. 2012).
The cadaveric data are given by basis vectors in ngertip endpoint wrench space that
each of the tendons produce when tension is applied. Maximal tendon tensions are also
given. For the index nger, the data are given in 4 dimensions: 3 in force space and one in
torque space. For the thumb, the data are given in 3 dimensions, all in force space. The
force space components of the basis vectors for each nger are shown in Figure 6.1a. The
feasible force set of each ngertip is built from these basis vectors and maximal tendon
tensions using the approach outlined in (Inouye et al. 2012, Valero-Cuevas 2005) and is
shown in Figure 6.1b. The remaining procedure of intersecting the feasible force sets
89
with friction cones (Figure 6.1c) and calculating the feasible grasp wrench set (Figure
6.1d) and associated global grasp metrics is described in (Inouye et al. 2012). Since the
grasp quality measures of characteristic length and radius of largest ball are both linear
measures and have identical units, we assign a single grasp quality metric to the human
hand that simply sums the two measures. Figure 6.1e shows how computation of the
feasible object wrench predicts whether a grasp can be maintained against an external
force.
The linear coecient of friction is set to 0.5, and the ratio of the linear coecient
of friction to the rotational coecient of friction is set to 7. Both of these values are in
the range of physiologically-reported friction coecients for human ngertips (Kinoshita,
Backstrom, Flanagan & Johansson 1997). The hand is assumed to grasp a ball of diameter
6cm with the thumb ngertip on the bottom and the index nger on the top, as pictured
in Figure 6.1c.
6.3.2 Calculating anthropomorphic hand grasp quality
We used the kinematic layout of the Shadow Hand (a tendon-driven, commercially-
available, anthropomorphic hand) (Shadow Robot Company n.d.) in order to determine
nger link lengths, placements, and postures for grasp. The grasping posture of the an-
thropomorphic hand was selected to closely resemble that of human posture while still
grasping the 6cm ball, shown in Figure 6.2.
The index nger has 4 degrees of freedom: 3
exion-extension and one ad-abduction.
The thumb has 5 degrees of freedom: 3
exion-extension and 2 ad-abduction. The
process for nding the feasible force set of the 4-DOF index nger has been studied
90
FPL
AbPB
AbPL
EPB
EPL
ADDo
ADDt
OPP
DIO
FS
FP
EC
EI
PI
DI
LUM
y
x
z
(a)
(b) (c)
Fingertip basis vectors Feasible force sets Feasible object force sets
Feasible object wrench
(d) (e)
F
External force outside feasible
object wrench: grasp fails!!
F
External force inside feasible object
wrench: grasp is maintained!!
Figure 6.1: Computation of human hand grasp quality. Index nger and thumb basis
vectors and feasible force sets not equal scales. (a) Fingertip basis vectors. (b) Feasible
force sets built from basis vectors. (c) Feasible object force sets: intersection of feasible
force sets with friction cones. (d) Feasible object wrench (only 3-D feasible forces shown).
(e) Examples of maintained and failed grasps.
91
Figure 6.2: Anthropomorphic hand grasp.
and implemented previously (Inouye et al. 2012, Valero-Cuevas et al. 1998). However,
while the procedure for determining the feasible force set of a tendon-driven, redundant,
serial-chain manipulator has been outlined theoretically in (Fu & Pollard 2006), to our
knowledge no study has ever actually implemented an algorithm for this purpose. We
have developed and implemented a practical algorithm that uses vertex enumeration
(Barber et al. 1996) to perform the necessary computations and nd the feasible force
set of the 5-DOF thumb (or any tendon-driven, redundant, serial-chain manipulator for
that matter).
To allow a fair comparison of the grasp qualities of the anthropomorphic hand designs
with the human hand, we tried to adhere as much as possible to the constraints that the
human hand is subjected to. We set the coecients of friction for the ngertips equal to
those of human ngertips. Moreover, the sum of the maximal tendon tensions for each
nger was set to that of the human ngers: 764N for the index nger and 478N for the
thumb (Valero-Cuevas et al. 2000, Pearlman et al. 2004). Also, the moment arm values
were constrained so that the size of each joint would not exceed that of the corresponding
92
Joint Joint Diameter (mm)
Index nger
MCP Adduction-Abduction 10.9
MCP Flexion-extension 21.0
PIP Flexion-extension 9.25
DIP Flexion-extension 5.14
Thumb
CMC Adduction-Abduction 35.6
CMC Flexion-extension 40.4
MP Adduction-Abduction 15.5
MP Flexion-extension 20.2
IP Flexion-extension 11.0
Table 6.1: Maximal joint diameters (i.e., dierences between largest moment arms for
each joint) according to data obtained from the literature.
human joints. The moment arms for the human index nger and thumb were taken from
the literature (Valero-Cuevas et al. 1998, Valero-Cuevas, Johanson & Towles 2003), and
the implied \joint diameters" (i.e., the maximal dierences between moment arm values
for a joint) were used to constrain the joint diameters for the anthropomorphic hand
designs. For example, the maximal positive moment arm value for the index nger MCP
joint is 13.2mm and the maximal negative moment arm value for the index nger MCP
joint is -7.8mm (Valero-Cuevas et al. 1998). Therefore, the \joint diameter" is 21.0mm.
The anthropomorphic hand designs were constrained so that their joint diameters were
equal to those of their human hand counterparts. The joint diameters are shown in Table
6.1.
Moreover, after some initial testing and optimization, it was found that using only
the above constraints resulted in optimized designs that were extremely good at grasping
(more than 100% greater grasp quality than the human hand), but were extremely poor
in exerting force in all directions: the maximum isotropic value, or MIV (the maximum
force that can be exerted at the end eector in all directions, described in (Finotello
93
et al. 1998)), for optimized designs was less than 0.05 N. This was due to the optimization
process placing all the emphasis on
exion force and none on extension force. Since this
is such a low force, we felt it was necessary to constrain the MIV for anthropomorphic
hand designs to be at least that of the human hand. The MIV for the human index nger
is 2.89N and the MIV for the thumb is 5.37N. This ensures ability to release a grasp and
for each nger to exert at least as much force in all directions as human ngers.
6.3.3 Optimizing anthropomorphic hand grasp quality
We used 3 separate steps for initializing and optimizing anthropomorphic hand designs:
Monte Carlo on the structure matrices, Markov-Chain Monte Carlo on the joint centers of
rotation, and Markov-Chain Monte Carlo on the distribution of maximal tendon tensions.
Monte Carlo over the structure matrices was used due to the fact that the feasible force
sets of the ngers, and therefore the grasp quality of the hand, are complex functions
of the high-dimensional structure matrices. The Markov-Chain Monte Carlo methods
were deemed appropriate due to the high dimensionality of the system and the large
computational cost of computing approximate gradients (i.e., for using steepest-ascent,
Newton's method, etc.).
6.3.3.1 Monte Carlo on structure matrices
For nding tendon layouts on which to perform optimization, we rst performed a Monte
Carlo exploration of the space of structure matrices for the ngers. We explored the
space of 4 dierent tendon designs: N+1, N+2, N+3, and 2N, where N is the number of
degrees of freedom for each nger. In order to do so, we randomly generated 1000 pairs of
94
admissible structure matrices (one for the index nger, one for the thumb) for each of the
4 designs. These structure matrices were in a pseudotriangular form, as described in (Lee
& Tsai 1991). The process of generating a random matrix for each nger is illustrated
in Figure 7.1. We rst began with a xed matrix consisting of some zeros and some
\#" entries where non-zero moment arm values would be inserted. Next, we randomly
replaced each \#" with either a 1 or -1. We then would check feasibility conditions for
controllability of a tendon-driven nger, also described in (Lee & Tsai 1991). Basically,
we checked if the matrix was full rank, there was at least one sign change in each row,
and we found a null vector whose elements all had the same sign using SVD. If the
structure matrix was found to satisfy these conditions, we would calculate the MIV for
that structure matrix. If the structure matrix was not admissible, we would randomize
the structure matrix again. After a pair of admissible structure matrices were found for
the index nger and thumb, we calculated the grasp quality for the pair. Therefore, we
found 4000 randomized hand designs and calculated the grasp quality of each.
6.3.3.2 Optimization of joint centers of rotation
After we computed the MIV for each structure matrix, we took the 10 index nger struc-
ture matrices and the 10 thumb structure matrices with the highest MIVs and crossed
them over with each other, creating 100 combinations of index nger and thumb struc-
ture matrices. This crossover operation resembles that performed in a standard genetic
algorithm. The crossover we performed is illustrated in Figure 6.4. The grasp quality was
then calculated for these 100 hand designs produced from crossover operations. It was
reasoned that the structure matrices with the highest MIVs would be the most
exible
95
No
Begin with xed matrix
(N+1, N+2, N+3, or 2N)
Randomly replace each
“#” with either 1 or -1
Admissible?
Perform analyses
Yes
Figure 6.3: Procedure for nding randomized designs via Monte Carlo sampling.
Index
Thumb
1 2 3 10
1 2 3 10
Figure 6.4: Crossover of the 10 structure matrices with the highest MIVs for each nger.
during the later stages of the optimization. In addition, the designs with the very highest
grasp quality either did not meet the minimum MIV requirements or were very close to
violating them.
The top 10 designs by grasp quality were then selected from the crossover results for
optimization of the joint centers of rotation. This is in the sense of moment arms on
one side of the joint becoming larger than those on the other side of the joint, with the
total range of the moment arms being equal to the joint diameter. A greedy Markov-
Chain Monte Carlo algorithm was employed on each nger separately. The initial centers
96
Center of rotation
Extensor
moment arm
Flexor
moment arm
Joint
diameter
Lower bound
Upper bound
Center of rotation perturbations
Figure 6.5: Illustration of perturbations of the joint center of rotation.
of rotation were in the middle of the joints and the grasp quality was calculated. The
index nger joint centers of rotation were then perturbed simultaneously (all 4 at once)
by independent, normally distributed random numbers u
i
(for the ith joint) with zero
mean and a standard deviation of 2% of the joint diameter. The centers of rotation were
constrained so that they did not go outside the joint diameters, and a re
ection technique
was used similar to that in (Santos, Bustamante & Valero-Cuevas 2009). Perturbation of
the center of rotation is demonstrated in Figure 6.5.
6.3.3.3 Optimization of maximal tendon tensions
The last step in our optimization process involved performing a similar Markov-Chain
Monte Carlo algorithm on the maximal tendon tension distribution. Starting with all
the tendons having equal maximal tendon tensions, we perturbed the distribution of
the maximal tendon tensions using a multivariate normal distribution with standard
deviation of 2% of the maximal tendon tension sum (764N for the index nger, 478N for
97
Constraint:
Max Tendon
Tension Sum
Starting point
Rejected step
Accepted step
Random perturbation direction
Random perturbation endpoint
Projection of perturbation
endpoint onto constraint
1
Max Tendon
Tension Sum
2
3
Figure 6.6: Illustration of Markov-Chain Monte Carlo algorithm for distribution of max-
imal tendon tensions.
the thumb). This perturbation was eectively like one inside ann-dimensional hypercube
in the positive orthant with side length equal to maximal tendon tension sum, where n
is the number of tendons for each nger. After perturbation inside the hypercube, we
projected the point onto the hyperplane given by the following equation:
n
X
i=1
F
i;max
=MaxTendonTensionSum (6.1)
where F
i;max
is the maximal tension of tendon i. This would give us a new distribution
of maximal tendon tensions, and we would then evaluate the grasp quality. If it was
higher, we would take that point as the starting point for the next perturbation. The
same re
ection technique as that of the previous section was used. The overall process is
shown graphically for a simplied 2-tendon example in Figure 6.6.
98
A detailed explanation of the eects of dierent structure matrices and distributions
of maximal tendon tensions on the kinetostatic (i.e., force-production) capabilities of
manipulators and biological hands can be found in (Valero-Cuevas 2005, Inouye et al.
2012, Lee & Tsai 1991, Finotello et al. 1998, Ou & Tsai 1993, Ou & Tsai 1996, Tsai 1995).
6.4 Results
The results of the initial Monte Carlo and the two Markov-Chain Monte Carlo optimiza-
tion steps are shown in Figure 6.7. We see that random Monte Carlo designs do very
poorly in relation to the human hand. However, optimization of the joint centers of ro-
tation and the distributions of maximal tendon tensions produces designs whose grasp
quality exceeds that of the human hand, while still adhering to human hand constraints.
In addition, we see that intelligently choosing parameters for hand design can result in
grasp quality that is 501% higher than that of a na ve 2N design!
6.4.1 Random Monte Carlo designs
The structure matrix Monte Carlo eectively randomized the routing of the tendons prior
to the other optimization procedures. We see from Figure 6.7 that there is a large range of
grasp qualities for fully controllable hands. Figure 6.8 shows 3 dierent tendon routings
for the index nger. Figure 6.8a shows the best index nger crossover design (i.e., after
crossing over the ngers with the highest MIVs, as described in the methods section)
that was selected for optimization. We do not show the design that had the highest grasp
quality from the Monte Carlo designs because the MIVs of the ngers were too low to
meet human hand constraints. Figure 6.8b shows an index nger tendon routing that
99
0
10
20
30
40
Original
Monte Carlo
Grasp Quality (N)
Optimization
Step #1
Optimization
Step #2
Human
13% below human
45% below
human
165%
increase after
optimization
45% above
human
Naive 2N Design
501%
greater than
naive 2N
design
Figure 6.7: Comparison of human grasp quality with boxplots of anthropomorphic hand
designs. Optimization step #1 is for the joint centers of rotation. Optimization step #2
is for the distribution of maximal tendon tensions.
produced a very poor grasp quality. Figure 6.8c shows a typical 2N design, which is
similar to that used in the current Shadow Hand design and other tendon-driven designs.
Its grasp quality (combined with a similar 2N routing for the thumb) is only about half
of that of the design from Figure 6.8a. It should be noted that this gure does not draw
the moment arms to scale. It only shows which sides of the joints the tendons cross. In
addition, for the purposes of kinematic clarity, the nger ad-abduction (i.e., side-to-side)
axis was considered to be immediately proximal to the perpendicular axis of the rst
exor-extensor joint, as demonstrated in Figure 6.8a.
100
d=0
Very good Monte Carlo design.
Grasp Quality: 11.9 (before optimization).
Very bad Monte Carlo design.
Grasp Quality: 2.33
Typical 2N design.
Grasp Quality: 5.96
(a)
(b)
(c)
Figure 6.8: Various index nger tendon routings. Not to scale. (a) Best crossover index
nger design selected for optimization. (b) Example of an index nger tendon routing
that produced a very low grasp quality. (c) Typical 2N design.
101
6.4.2 Optimization of joint centers of rotation and distribution of maximal
tendon tensions
We optimized the top 10 hand designs produced from the crossover operations. We used
150 iterations for the joint centers of rotation optimization for each nger separately.
Results from using 10 random starting locations for the index nger within the space of
allowable centers of rotation as well as a starting location in the middle of all the joints
showed that the random seeds all converged to roughly the same point, with a small
range in nal grasp quality, as shown in Figure 6.9a. The range of grasp qualities for the
random seeds was 16.08-16.19N, or 0.7% of the mean.
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
5
10
15
20
DIP center of rotation
(normalized units)
PIP center of rotation
(normalized units)
Grasp Quality (N)
Initial seeds
Ending solutions
Paths
(a) (b)
0 50 100 150
11
12
13
14
15
16
17
Iteration
Highest grasp quality found (N)
Figure 6.9: Markov-Chain Monte Carlo optimization. (a) Visualization of optimization
paths of 10 random seeds and one center seed. (b) Optimization progress over 150 itera-
tions.
Therefore, we deemed that the 150 iterations allowed sucient convergence for our
study, and also that running multiple Markov Chains for each design was unnecessary.
It should be noted that although the 10 random seeds were randomly selected from a
102
EIP
EDC
DI
LUM
FDS
PI
FDP
Extensor mechanism
tendon connections
EIP EDC DI PI LUM FDS FDP
0
50
100
150
200
250
Tendon
Max tension (N)
Physiological
Nominal
Human. Grasp Quality: 24.0
Tendon 5
Tendon 1
Tendon 2
Tendon 3
Tendon 4
Tendon 6
Tendon 7
Tendon 5
Tendon 1
Tendon 2
Tendon 3
Tendon 4
Tendon 6
Tendon 7
1 2 3 4 5 6 7
0
50
100
150
200
250
300
Tendon #
Max tension (N)
Optimized
Nominal
1 2 3 4 5 6 7
0
50
100
150
200
250
Tendon #
Max tension (N)
Optimized
Nominal
Result #2. Grasp Quality: 33.1
Result #1. Grasp Quality: 34.8
Figure 6.10: Index nger tendon layout, maximal tendon tension distribution, and feasible
force sets for top 2 optimized designs and the human hand. Tendon layouts shown roughly
to scale. Feasible force sets shown to scale.
uniform distribution in the theoretically allowable range (i.e., within the unit hypercube;
2 selected dimensions shown in Figure 6.9a), some seeds did not meet minimum MIV
requirements, and they were discarded and re-randomized. The nal solutions shown in
Figure 6.9a are slightly dispersed in the PIP center of rotation, but the range of grasp
qualities is very small, suggesting a very
at tness landscape in that vicinity along that
direction.
In addition, we see the progress of the best found result versus iteration number in
Figure 6.9b. Graphically, it can be seen that over the course of 150 iterations, most of
the improvement (almost 85%) comes in the rst 50 iterations. While we allowed the
103
optimization to run for 3 times that long, we felt that any more iterations would result
in diminishing returns and marginal improvements in results. It took about 3.7 seconds,
on average, to compute the grasp quality for each iteration. The optimization of the
joint centers of rotation produced designs with grasp qualities between 19.3N and 20.8N,
falling short of the human grasp quality by 13-20%, and 17% on average, as shown in
Figure 6.7.
We used 150 iterations as well for the Markov-Chain Monte Carlo optimization of
the distribution of maximal tendon tensions for each nger separately. We tried random
seeding, but the result of using 10 random seeds (same technique as that used in the joint
center of rotation random seeds) produced grasp qualities with a very small range (5.5% of
the mean). Again, this indicated that multiple chains were unnecessary for our purposes.
A plot of typical optimized grasp quality versus iteration number was very similar to that
shown in Figure 6.9b (in fact, with 96% of the improvement coming within the rst 50
iterations), and therefore the 150 iterations were deemed sucient. The optimization of
maximal tendon tension distributions produced designs with grasp qualities that were on
average 55% higher than those with evenly distributed maximal tendon tensions.
The full optimization of the best crossover designs produced hands whose grasp quality
exceeded that of the human hand by 13-45%, shown in Figure 6.7. The index nger
routing, the distribution of maximal tendon tensions, and the feasible force sets for the
best 2 designs and the human hand design are shown in Figure 6.10. Interestingly, the
feasible force set of the human index nger is much larger than that of the thumb, while
the opposite is true with the optimized designs. It can be seen that the optimization
104
caused the feasible force sets to be weighted more heavily toward the palmar direction
(i.e., the direction of force production during grasping).
6.5 Discussion
In this study, we have successfully compared thousands of random and optimized hand
designs to the human hand in terms of grasp quality. We demonstrate the use and power
of the methodology developed in (Inouye et al. 2012) to compare and optimize hand
designs in a high-dimensional parameter space, while still adhering to specic design
constraints.
As mentioned earlier, the optimization caused the feasible force sets to be weighted
more heavily toward the palmar direction. In the human index nger however, the feasible
force set is weighted both in the palmar direction and also in the distal direction, as can
be seen in Figure 6.10. The feasible force set being weighted in this direction would
indicate optimization for other tasks that would utilize this direction. Consider a hand
picking up a heavy bar, shown in Figure 6.11. In this case, we have a power grasp, and
the ngertip is not supporting much of the weight of the bar, but the nger itself must
exert a force in that direction. This could suggest that power grasp of very heavy objects
in this manner is also something the human hand, to a certain extent, is optimized for.
As can also be seen in Figure 6.10, the optimized thumb feasible force sets were much
larger than that of the human. One explanation for this is that all the tendons in the
optimized designs were at the boundaries of their joint diameters. In the human, the
extreme moment arms dened the joint diameters, but the rest of the tendons typically
105
Distal direction
F
Palmar direction
Figure 6.11: Power grasp of a heavy bar. The hand must resist a force that is opposite
the distal direction of the index nger.
were not nearly as large as these extremes. For example, the maximal human MCP
exion moment arm was 13.2mm and the maximal extension moment arm was 7.77mm,
giving a joint diameter of 21.0mm. If this was an optimized robotic hand, all the
exion
moment arms would be 13.2mm and all the extension moment arms would be 7.77mm.
However, in the human, only one
exor has a moment arm this large and only one other
extensor has a moment arm this large. The other 4 tendons have smaller moment arms
at this axis. Due to the fact that the joint diameters for the thumb CMC joint were
quite large, the optimized designs were able to take full advantage of having many large
moment arms, while the human thumb only used 2 moment arms to the maximum extent
possible.
Our study concentrated on the grasp of one object of a reasonable size in a specic
posture. We did not perform extensive sensitivity analysis on certain factors such as dif-
fering postures, nger placements, friction coecients, surface curvatures, object shapes,
106
etc. The methodology that we employed would allow simulations to take most of these
factors into account, but we felt that this was not necessary to arrive at our basic con-
clusions. In addition, we assumed no force-length interaction in the human muscles to
aect the maximal tendon tensions (Zajac 1989).
Furthermore, we have only scratched the surface of the optimization techniques that
could be employed in order to even further improve the results. Varying the optimization
techniques, parameters, or orders of steps could aect the upper limit of grasp qualities
found. We did not feel that it was necessary or feasible to nd a global optimum, and we
felt that our results would not be signicantly aected by spending much more time and
eort trying to get, say, 10-20% higher grasp quality than we found for the optimized
designs. Also, the optimization only occurred on specic initial structure matrices for
the various designs. However, thousands of other initial structure matrices could also be
used and optimized (Sheu et al. 2009).
This paper concentrates on the upper limit of the mechanical capabilities of hands
in relation to grasp quality. Nearly all (grasping) tasks performed are far from the me-
chanical boundaries dened by theoretical or practical means. However, it is generally
desirable to perform tasks that are not too close to the mechanical boundary limit and
to have some sort of \safety factor" in place. If the task is well-dened, then the lin-
ear programming approach used in (Fu & Pollard 2006) is well-suited to calculating this
safety factor. In the general case where many tasks must be performed or the hand is
performing tasks in unstructured environments, global grasp metrics (such as those used
in this study and in (Inouye et al. 2012) must be used to assess the tness of various
designs.
107
We note that in any useful biological or robotic hand, the mechanics and control
must both be considered and developed. The best control algorithms cannot perform any
task that the mechanics are not designed for. Moreover, a globally-optimal hand from
a mechanics standpoint is useless without eective control to utilize these capabilities.
The consequences to control of various designs have not been considered in this paper,
and this is the subject of future work: is a tendon-driven hand with an optimized grasp
quality dicult to control?
One exciting conclusion of this study is that utilizing even the fairly simple Monte
Carlo exploration and optimization techniques we employed can improve grasp quality of
robotic hands by nearly an order of magnitude (501% increase) when compared with the
na ve 2N design. While the scientic insight (involving the human hand) of this study is
interesting from an intellectual and academic perspective, we feel that the implications
for the industrial design of robotic hands are perhaps the most useful ndings from this
study.
Acknowledgements
The authors gratefully acknowledge the useful discussions with J. Kutch.
108
Chapter 7
Asymmetric Routings With Fewer Tendons Can Oer Both
Flexible Endpoint Stiness Control and High
Force-Production Capabilities in Robotic Fingers
7.1 Abstract
The force-production and passive stiness capabilities of ngers are two critical design
specications for dexterous robotic hands. We used the link and joint kinematic param-
eters of the 4-DOF DLR index nger to explore the tradeo between these two design
specications as a function of the number, routing, stiness, and strength of each tendon.
Our innovative computational approach allowed building the Pareto front of optimized
passive endpoint stiness (measured by the eccentricity of the endpoint stiness ellip-
soids) vs. maximal force-production capabilities (measured by the size and shape of the
force polytope) for 1,200 randomly generated valid routings with 5, 6, 7, or 8 tendons.
Our results show that this parametric optimization can increase realizable isotropic forces
by up to 80% compared to the default tendon tension distribution. In addition, designs
109
with 5 or 6 tendons can have endpoint stiness ellipsoids with optimized low eccentric-
ities and with force production capabilities comparable to designs with 7 or 8 tendons.
Interestingly, we did not nd a systematic tradeo between force-production and passive
stiness capabilities, given a specic routing. However, the choice of number, routing and
strength of each tendon greatly aects force and passive stiness capabilities of robotic
nger, which reveals the many design opportunities aorded by tendon-driven manipula-
tors and oers insight into the anatomical features of the human musculoskeletal system.
7.2 Introduction
Robotic ngers and hands have been designed for the past few decades for the purposes
of grasping and manipulation (Jacobsen et al. 1986, Salisbury & Craig 1982, Shadow
Robot Company n.d., Grebenstein et al. n.d., Ambrose et al. 2000). There are many
factors involved in the design decisions for these hands, but two important ones are force-
production capabilities and passive stiness. The ngers clearly must be able to generate
suciently high forces to perform a specic or general task. In addition, the integration
of passive stiness control into the design of robotic hands is important for preventing
damage to itself and its surroundings, enabling the ability to perform highly dynamic
tasks, and increasing the safety of interacting humans (Wolf & Hirzinger 2008, Pratt &
Williamson 1995, Grebenstein & van der Smagt 2008).
Several studies have addressed the problem of identifying the force-production (or
more formally, wrench-production) capabilities of both parallel and serial manipulators
(Bouchard et al. n.d., Chiacchio et al. 1997, Firmani et al. 2008, Finotello et al. 1998,
110
Gouttefarde & Krut 2010, Zibil et al. 2007, Kuxhaus et al. 2005, Valero-Cuevas et al.
1998). According to (Firmani et al. 2008), "The knowledge of maximum twist and wrench
capabilities is an important tool for achieving the optimum design of manipulators". Two
common approaches to the problem of quantifying these capabilities are manipulating
force ellipsoids (which apply accurately only to torque-driven systems) and force polytopes
(also known as feasible force sets, which apply exactly to tendon driven systems (Valero-
Cuevas et al. 2009). From these force-production capabilities, a performance metric can
then be assigned based on the size and/or shape of the ellipsoid or polytope.
For tendon-driven robotic ngers, one key design element is the tendon routing, which
denes the structure matrix, of the nger. This structure matrix denes the torque and
force produced by the nger based on tensions of the tendons. Certain studies have
addressed the problem of designing a structure matrix for isotropic force transmission
characteristics (i.e., ability to transmit forces equally in all directions at the end eector)
(Tsai 1995, Sheu et al. 2009, Lee & Tsai 1991, Chen et al. 1999, Ou & Tsai 1996, Ou
& Tsai 1993). However, these studies have not considered the distribution of maximal
tensions across tendons, which is certainly important in small, dexterous hands where
weight and size minimization are signicant priorities. Altering the maximal tendon
tension distribution in tendon-driven hands is known to have a signicant eect on force-
production capabilities (Pollard & Gilbert 2002, Inouye et al. 2011b). Additionally, the
number of tendons used in the design is fundamental to the design of the structure matrix,
and using fewer tendons \has the advantage of reducing the number of tendons and
actuators and therefore reduces the weight, size, and complexity of the manipulator..."
(Tsai 1995).
111
The importance of stiness control of manipulators has been widely recognized in the
literature(Wolf & Hirzinger 2008, Pratt & Williamson 1995, Grebenstein & van der Smagt
2008, Grebenstein et al. n.d., Hashimoto & Imamura 1994, Hogan 1985a, Ott 2008, Starr
1988). Manipulators can have active or passive stiness control, or a combination of both.
Active stiness control can be programmed using a feedback control law (Hogan 1985a),
but is limited by the control loop frequency, and a sudden impact to the manipulator
can cause damage to the robot or its surroundings before the control loop is activated
to absorb the energy (Grebenstein et al. n.d.). Thus, passive stiness is also important,
especially in unstructured environments where unexpected obstacles, objects, or humans
may make contact with the manipulator. Passive stiness control is typically implemented
by variable-stiness actuators (Laurin-Kovitz, Colgate & Carnes 1991, Wimbock, Ott,
Albu-Schaer, Kugi & Hirzinger 2008, Wolf & Hirzinger 2008, Pratt & Williamson 1995,
Sugano, Tsuto & Kato 1992, Grebenstein, Chalon, Hirzinger & Siegwart 2010, Kobayashi
et al. 1998). Synthesis of endpoint stiness for serial manipulators with adjustable joint
stinesses is studied in (Huang & Schimmels 2000). An extensive analysis of the joint
stiness matrices for tendon-driven manipulators is conducted in (Kobayashi et al. 1998).
Therefore, it is clearly desirable to design a robotic hand with both adequate passive
stiness and high force-production capabilities. In addition, it can be benecial to design
a nger with as few tendons as possible. Utilizing computational methods and applying
theoretical analyses, we quantify the ability of 1,200 tendon routings to produce maximal
isotropic forces and endpoint stiness ellipsoids with low eccentricity. This novel approach
enables the systematic exploration of the design space. For example, we show that fewer
tendons does not imply worse passive stiness, but designs with fewer tendons typically
112
cannot produce as much isotropic force as designs with more tendons. Tuning tendon
stinesses can lead to endpoint stiness ellipsoids with low eccentricity, and adjusting
the distribution of maximal tension across tendons can lead to large increases in isotropic
force-production capabilities. Our study demonstrates, to the best of our knowledge, the
rst practical computational exploration of the eect of tendon routing simultaneously
on these two characteristics.
7.3 Methods
he minimal number of tendons required to fully control all of the degrees of freedom
(DOFs) of an n-joint robotic nger is n + 1 (Ou & Tsai 1993). Because tendons have
unidirectional actions (i.e., they can only pull), this minimal number of tendons must
also be routed judiciously (Valero-Cuevas et al. 2009). A nger with this many tendons
employs what is called an \N+1" design. The DLR nger and most anthropomorphic
ngers have 4 DOFs (that do not use coupled joints), which means that the minimal
number of tendons for full controllability of the nger is 5. However, many hands have
been designed using a \2N" design, which uses a number of tendons equal to 2 times
the DOFs{with a pair of agonist-antagonist tendons dedicated to each joint (Grebenstein
et al. n.d., Jacobsen et al. 1986, Shadow Robot Company n.d.). In general, increasing the
number of tendons beyond 2N is impractical or undesirable for robotic ngers because
of size constraints
1
. In addition, any number of tendons between N+1 and 2N may be
1
However, most vertebrate limbs have more than 2N muscles{which is a subject of continual debate.
See (Valero-Cuevas et al. 2009, Kutch & Valero-Cuevas 2011)
113
used. We carried out analyses on 4 categories of designs: having 5 (N+1), 6 (N+2), 7
(N+3), or 8 (2N) tendons.
The routing and moment arms of tendons in a nger are critical, and can be mathe-
matically described by an nm structure matrix (also called a moment arm matrix) R,
wheren is the number of DOFs of the nger andm is the number of tendons. The entries
r
i;j
are signed moment arm values for thei
th
joint andj
th
tendon (Tsai & Lee 1988). For
simulation purposes, we randomly selected 300 admissible structure matrices from each of
the tendon categories by randomizing the signs of the non-zero entries and then checking
for controllability conditions
2
as described in (Lee & Tsai 1991). This process is shown
in Figure 7.1, with the non-zero entries represented by `#'. The number of admissible
structure matrices for all categories combined is 222,208 (using a combinatoric search of
the `#' entries in Figure 7.1 for each category). Therefore, evaluating all of these designs
is relatively intractable for our purposes and we deemed that randomly selecting 300 from
each category for a total of 1,200 evaluated routings was sucient to prove the point of
this study.
To compare and contrast the force-production and stiness capabilities of various
nger designs, we identied a tness metric describing each aspect. We calculated these
metrics at 3 dierent postures for each of the 1,200 routings, and averaged the metrics
over the 3 postures, which are shown in Figure 7.2.
The tness metric used for stiness control was the ESE (endpoint stiness eccen-
tricity) and the metric used for force production was the MIV (maximal isotropic value).
The calculation of these metrics is described in the next sections.
2
The basic idea behind the controllability conditions is that each joint can be actuated independently
in torque and motion, given that tendons can only pull and not push.
114
No
Begin with xed matrix
(N+1, N+2, N+3, or 2N)
Randomly replace each
“#” with either 1 or -1
Admissible?
Perform analyses
Yes
Figure 7.1: Procedure for nding admissible structure matrices. N+1 structure matrix
shown.
Joint 1
Ad-abduct
Joint 2
Flex-extend
Joint 3
Flex-extend
Joint 4
Flex-extend
Posture 1
2
3
y
x
Figure 7.2: Postures analyzed for each tendon routing. Link lengths and joint diameters
shown to scale (i.e., with kinematic parameters of the DLR hand).
115
7.3.1 Analysis and Synthesis of Stiness
7.3.1.1 Joint Stiness Adjustability
The endpoint Cartesian stiness matrix, K
end
, relates the endpoint displacements (from
an equilibrium position),
~
@x, to endpoint forces,
~
F , as shown in the following equation:
~
F
end
=K
end
~
@x (7.1)
The joint stiness matrix,K
joint
, relates the joint displacements (from an equilibrium
position),
~
@, to joint torques,~ , as shown in the following equation:
~ =K
joint
~
@ (7.2)
The endpoint stiness matrix can be found from the joint stiness matrix using the
following well-known equation (Pashkevich, Klimchik & Chablat 2011, Hogan 1990, Chen
& Kao 2000, Alici & Shirinzadeh 2005, McIntyre, Mussa-Ivaldi & Bizzi 1989):
K
end
=J
+T
(K
joint
@J
T
~
@
~
F
tip
)J
+
(7.3)
where J is the posture-dependent Jacobian relating joint angle velocities to endpoint
velocities,J
+
is the Moore-Penrose pseudoinverse ofJ (if the manipulator is redundant, as
is the case in this study), and
~
F
tip
is the external force vector on the tip of the nger. The
joint stiness matrix for a tendon-driven nger may be found from the structure matrix
R and the diagonal tendon stiness matrix K
t
using the following equation (McIntyre,
Mussa-Ivaldi & Bizzi 1996):
116
K
joint
=RK
t
R
T
(7.4)
For the purposes of this computational study, we assume that the external force on
the ngertip is zero. However, similar computational studies could be conducted with
an external ngertip force. Combining Eqs. 7.3 and 7.4 with the external force being
zero, we get the endpoint stiness matrix as a function of tendon stinesses and tendon
routing:
K
end
=J
+T
(RK
t
R
T
)J
+
(7.5)
For the DLR index nger in an unloaded conguration (Equation 7.3 changes if there is
a constant load applied to the endpoint (McIntyre et al. 1996)),K
end
is a 33 symmetric,
positive semi-denite matrix, K
joint
is a 4 4 symmetric, positive semi-denite matrix,
J is a 3 4 matrix, R is a 4m matrix (m is the number of tendons ranging from 5-8),
and K
t
is an mm diagonal matrix.
We can see clearly from Equation 7.3 that the endpoint stiness is a function of joint
stiness, and that realizing a completely arbitrary endpoint stiness can be dicult in
general due to the multiplication by the Jacobian and the inversions involved. Depending
on the conguration, it may or may not be possible to realize an arbitrary endpoint sti-
ness matrix (Huang & Schimmels 2000) because of the constraint that the joint stiness
matrix must be positive denite. It can be noted that an arbitrary 3-D endpoint stiness
matrix involves 6 free parameters. Therefore, if there are not at least 6 free parameters
in the joint stiness matrix, it is not possible to realize an arbitrary endpoint stiness.
117
For the 4-DOF DLR nger, the joint stiness matrix is 4 4, and so there are 10
free parameters. Reformulating Equation 7.4 enables quantication of the joint stiness
adjustability (JSA), which can be interpreted as the
exibility of realizing a joint stiness
matrix for a specic routing when tendon stiness selection is arbitrary. The reformula-
tion (Kobayashi et al. 1998) involves rearranging the independent parameters of the joint
stiness matrix into a vector, which is then a linear function of the tendon stiness, also
rearranged into a vector:
~
K
joint
=
~
R
~
K
t
(7.6)
where
~
K
joint
,
~
R, and
~
K
t
are reformulated as shown in Figure 7.3. () denotes element-by-
element multiplication, andR
i
is thei
th
row ofR. Note that
~
K
joint
has lengthn(n+1)=2
(where n is the number of DOFs of the nger),
~
R is an n(n + 1)=2m matrix (where
m is the number of tendons of the nger), and
~
K
t
has length m, and all of its elements
must be positive. Mathematically, the rank of
~
R is the number of free parameters of the
joint stiness matrix that can be independently chosen. Of course, the tendon stinesses
must be positive, and the range of realizable joint stinesses is constrained by a particular
routing, but this is nevertheless an estimation of the freedom in choosing an arbitrary
joint stiness matrix (Kobayashi et al. 1998), which in turn aects the freedom to choose
an arbitrary endpoint stiness matrix. The rank of
~
R is the joint stiness adjustability
(JSA):
JSA = rank(
~
R) (7.7)
118
Figure 7.3: Reformulation of variables in Equation 7.4 for use in Equation 7.6. () denotes
element-by-element multiplication. R
i
is the i
th
row of R. Joint stiness adjustability
(JSA) is equal to rank(
~
R).
Note again that this measure assumes that each tendon stiness can be independently
chosen, regardless of tendon tension.
7.3.1.2 Endpoint Stiness Eccentricity
As suggested above, higher JSA will, in general, translate to a larger set of realizable
endpoint stinesses. We quantify the ability of a specic routing to realize an endpoint
119
stiness ellipsoid
3
with low eccentricity by formulating the following optimization prob-
lem:
minimize
Kt
(K
end
)
subject to K
t
0
where () denotes the condition number: the ratio of the largest to the smallest sin-
gular values of the matrix. It is a measure of the eccentricity of the endpoint stiness
ellipsoid(Stroeve 1999). We will call this the endpoint stiness eccentricity, or ESE.
ESE =(K
end
) (7.8)
We implemented the above optimization in Matlab using the `fmincon' command.
Condition number minimization is a dicult problem in general (Elsner, He & Mehrmann
1995, Lu & Pong 2010, Chen, Womersley & Ye 2011). It is quasi-convex over the entries of
the matrix, but the entries of the matrix are non-convex functions of the elements of
~
K
t
in
our problem due to the matrix inversions. However, taking the best result from 5 random
starting points in the positive unit hypercube (i.e., positive orthant) seemed to give
good, repeatable results. The optimized endpoint stiness eccentricity, ESE*, quanties
the eccentricity of the best-conditioned stiness ellipsoid that the optimization was able
to nd. It can be noted that minimizing the eccentricity of the ellipsoid is equivalent to
3
The stiness ellipsoid is formed by projecting a unit sphere from dierential displacements to dier-
ential endpoint forces using the linear transformation K
end
, as in Equation 7.1. It assumes innitesimal
displacements that have negligible eects on the Jacobian matrix. Large displacements will not be as
accurately represented by ellipsoids due to larger changes in the Jacobian matrix.
120
maximizing its isotropy. A perfectly spherical stiness ellipsoid has a condition number
of 1.
7.3.2 Analysis and Optimization of Force Polytopes
The feasible force set is the convex polytope of all forces that can be exerted by the
endpoint of a tendon-driven nger, given a posture, tendon routing, and maximal tendon
tensions (Valero-Cuevas et al. 1998, Pollard & Gilbert 2002, Chiacchio et al. 1997). Any
force vector outside of this 3-D set (or 2-D set, for planar analyses) cannot be achieved by
the endpoint. A quality metric that can be assigned to this set is known as the maximal
isotropic value (MIV) (Finotello et al. 1998). It is the radius of the largest ball, centered
at the origin, that the feasible force set can contain. A nger can exert at least that many
units of force in any direction.
To nd the feasible force set, we rst specify the posture (which allows computation of
the JacobianJ) and tendon routingR of the nger. These matrices involve the following
relations:
_ x =J
_
(7.9)
~ =R
~
T (7.10)
where _ x is the endpoint translational velocity vector,
_
is the joint velocity vector, and
~
T
is the vector of tendon tensions.
We can use an activation vector, ~ a, to represent the degree to which a tendon is
activated. Each element of ~ a ranges between 0 (no activation) and 1 (full activation).
121
Further discussion may be found in (Valero-Cuevas 2005). If we deneT
max
as a diagonal
matrix of maximal tendon tensions, then we get the following relation between activations
and tendon tensions:
~
T =T
max
~ a (7.11)
If we combine Eqs. 7.10 and 7.11, then we get:
~ =RT
max
~ a (7.12)
The feasible 3-D force set can be found from this feasible torque set by intersecting
the feasible torque set with the linear subspace spanned by the columns of J
T
(Fu &
Pollard 2006). The vertices of this reduced-dimensionality set can then be transformed
to vertices in endpoint force space:
~
F =J
+T
~ (7.13)
where J
+T
denotes the Moore-Penrose pseudoinverse of J
T
. The convex hull of all of
these vertices in force space is a polytope and denes the feasible force set. We use the
Quickhull algorithm (Barber et al. 1996) implemented in the Qhull software package to
nd the MIV.
Dierent routings and maximal tendon tensions both aect the size and shape of the
feasible torque set shown in Equation 7.12, which in turn aects the size and the shape
of the feasible force set. If we have a xed routing and posture, then we can change the
122
feasible force set and MIV by varying the maximal tendon tensions (given by diagonal
matrixT
max
). If we constrain the sum of the maximal tendon tensions to be constant (a
reasonable constraint due to the size and weight constraints inherent in dexterous hands
(Pollard & Gilbert 2002)), then we can optimize the MIV using the following formulation:
maximize
F
0
MIV =f(J;R;T
max
)
subject to T
max
0;
trace(T
max
) = T
max sum
where T
max sum
is a constant.
Evaluating the MIV given J, R, and T
max
is fairly expensive computationally when
compared with function evaluations for the stiness problem. Therefore, we utilized
a custom, greedy Markov-Chain Monte Carlo optimization algorithm which was fairly
eective at nding a local maximum within 300 iterations. We denote this maximum by
MIV*.
It is worth noting that MIV is a function of J, R, and T
max
, while the ESE is a
function ofJ,R, andK
t
. Therefore, changes inT
max
will not aect the ESE and changes
inK
t
will not aect the MIV. However, the Jacobian matrix and the routing have eects
on both of these characteristics. This study is focused mostly on the eects of routing
on these two characteristics. It can also be noted that with a torque-driven manipulator,
the analysis of force production and stiness synthesis becomes much less interesting,
as there are very few design parameters that can be altered compared with the tendon-
driven manipulator. Furthermore, torque-driven manipulators are not able to utilize the
advantages of tendon-driven manipulators as stated in the introduction.
123
5 6 7 8
3
4
5
6
7
Number of Tendons
Joint Stiffness Adjustability (JSA)
N+1
N+2
N+3
2N
Figure 7.4: Joint stiness adjustability versus number of tendons, plotted for all admis-
sible routings. Mathematically, JSA is he the rank of
~
R, which is the number of free
parameters of the joint stiness matrix that can be independently chosen.
7.4 Results
7.4.1 Joint Stiness Adjustability
We were able to determine the JSA of all 222,208 admissible tendon routings. The
results are shown in Figure 7.4. We see that designs with 5 or 6 tendons have a JSA of
5, while designs with 7 or 8 tendons can have a JSA of up to 7 (but never 8!). However,
some designs with 8 tendons can only have a JSA of 4, which corresponds to symmetric
routings (a 2N design where the moment arms of one tendon are the opposite sign and
equal magnitude of those from another tendon), as noted in (Kobayashi et al. 1998). A
symmetric routing which controls all of the degrees of freedom of an N-DOF manipulator
requires at least 2N tendons, so routings of a 4-DOF nger with 5, 6, or 7 tendons cannot
be symmetric.
124
0 5 10 15 20 25
1
2
3
4
5
6
7
8
Average MIV (Force, in N)
Average ESE* (Stiffness)
N+1
N+2
N+3
2N
Routing 3
Routing 2
Routing 1
Figure 7.5: Optimized endpoint stiness eccentricity (ESE*) vs. unoptimized maximal
isotropic value (MIV), averaged over the 3 postures. Note: only 524 out of 1,200 data
points shown (all other designs had higher ESE* than 8 and lower MIV than 16). Large
circles mark the averages of posture 1 (small squares), posture 2 (small triangles), and
posture 3 (small diamonds) for the routings shown in Figure 7.6.
7.4.2 Optimized Endpoint Stiness Eccentricity vs. Maximal Isotropic
Value
For the 300 randomly-selected routings from each category, we found the ESE* (optimized
ESE) and the unoptimized MIV, shown in Figure 7.5. The MIV was calculated with all
maximal tendon tensions being equal and the sum being constant at 1000N. We did not
optimize the MIV for every design due to computational tractability considerations and
because it is not crucial for the purposes of this study. (See below for some examples of
optimized MIV).
We see that, in general, the best routings with 7 or 8 tendons have a substantially
higher (unoptimized) MIV than the best routings with 5 or 6 tendons. However, routings
with fewer tendons are not less able to produce low ESE values.
125
7.4.3 Optimizing MIV for 3 Specic Routings
To demonstrate that optimization of MIV is possible, we did optimize the MIV, in posture
2, for the three routings marked with large circles in Figure 7.5. Routing 1 is the one
with the highest unoptimized MIV, and it has 8 tendons. Routing 2 was chosen as
an N+1 design that had both low average ESE* and high MIV compared with other
N+1 designs. Routing 3 was chosen as a reference point, having a mathematically even,
symmetric moment arm matrix of a 2N design (the matrix values are only indicative of
the sign of the moment arm and not the magnitude):
R
ROUTING 3
=
2
6
6
6
6
6
6
6
6
6
6
4
1 1 1 1 1 1 1 1
0 0 1 1 1 1 1 1
0 0 0 0 1 1 1 1
0 0 0 0 0 0 1 1
3
7
7
7
7
7
7
7
7
7
7
5
The optimization was able to improve the MIV in posture 2 from 21.2N to 26.0N
for Routing 1 (a 23% increase), from 8.68N to 15.6N for Routing 2 (an 80% increase),
and from 10.0N to 18.0N for Routing 3 (an 80% increase). Figure 7.6 shows the rout-
ings, unoptimized feasible force sets, optimized feasible force sets, unoptimized endpoint
stiness ellipsoids (with all tendons having equal stinesses), and optimized endpoint
stiness ellipsoids. Two 3-D views are shown of the feasible force sets. The optimized
tendon stiness values shown are normalized so that the highest stiness has a value of
1 (multiplying all the stinesses by a scalar does not aect the condition number).
126
JSA = 5
MIV = 8.68
MIV* = 15.6
ESE = 1.8
ESE* = 1.15
Routing 2 (N+1)
x
y
z y
x x
y
z Unoptimized
ellipsoid (red)
Optimized
(purple)
Unoptimized
feasible force
set (green)
Optimized
feasible
force set
(wireframe)
JSA = 7
MIV = 21.2
MIV* = 26.0
ESE = 3.19
ESE* = 1.14
Routing 1 (2N)
JSA = 4
MIV = 10.0
MIV* = 18.0
ESE = 13.3
ESE* = 6.37
Routing 3 (2N)
Figure 7.6: Illustration of 3 routings along with stiness ellipsoids and feasible force sets.
JSA: Joint stiness adjustability. MIV: Maximal isotropic value before optimization,
in N. MIV*: Maximal isotropic value after optimization, in N. ESE: Endpoint stiness
eccentricity before optimization. ESE*: Endpoint stiness eccentricity after optimization.
~
K
t
: tendon stinesses producing ESE*, normalized so that maximal stiness is 1.
~
T
max
:
maximal tendon tensions producing MIV*, in N. Note: all results shown are for posture
2 only, and values correspond with the small triangles in Figure 7.5.
127
We see that routings 1 and 2 are able to produce very low ESE* values, while Routing
3 is only able to produce an ESE* of 6.37 (in posture 2). It can be noted that some of the
optimized tendon stinesses are 0, which may only be realizable in practice with a direct
drive DC motor actuator (in which case also the motor inertia would not necessarily allow
for instantaneous extension of the tendon with zero resistance). However, we feel that
this is not extremely important in our simulation results and general conclusions, since
this analysis and optimization could easily be implemented with additional constraints
(such as minimal and maximal values for tendon stinesses).
We can also observe in Routing 1 that the two tendons that only cross the rst joint
could be easily combined into one tendon in strength and stiness, resulting in a routing
with 7 tendons that has the exact same characteristics as the routing shown with 8
tendons.
7.5 Discussion
The main purpose of this study was to demonstrate the large eect of tendon routing,
number, and properties on force-production and stiness realization capabilities. We show
that tendon routings with fewer than 2N tendons (which are necessarily asymmetric) can
have high force-production capabilities as well as low eccentricity of endpoint stinesses.
Our optimization of the endpoint stiness assumed that it is desirable to produce
nearly-isotropic endpoint stiness, as we assumed no knowledge about the task or poten-
tial obstacles to the ngertip. In some practical cases, it may be desirable to adjust the
endpoint stiness characteristics asymmetrically according to the task or situation (Wolf
128
& Hirzinger 2008) to be compliant in one direction and sti in another. For example,
if the task is to push a button, guide a rod, etc, then it may be benecial to have high
stiness in the direction of force application but low stiness in the directions perpendic-
ular to the direction of force application. Any specic task requirements could easily be
incorporated into an optimization routine.
For any practical application, the analyses used in this study would need to take
into account the actuation system, and whether it incorporates non-linear or adjustable
stinesses. The calculation of JSA and the optimization of ESE and MIV in this study
assume that the tendon stinesses can be controlled independently of tendon tension and
that the stinesses are linear.
Even if the actuation scheme used in a physical system does not allow for tendon
stiness control apart from tendon tension, the analyses used here could be used to guide
the designation of spring constants for linear springs in series with actuators (i.e., some
tendons could use sti springs and others more compliant springs for a desired generic
endpoint stiness). Non-linear springs (Laurin-Kovitz et al. 1991) could be designed also
with varying properties among tendons (e.g., with dierent elasticity constants and biases
(Kobayashi et al. 1998)).
While the MIV was used as the tness metric for the force-production capabilities,
some hands or ngers may only need strong
exing force for use in grasping and the
maximal extension force requirements may be low. In this case, the distribution of max-
imal tendon tensions could be adjusted or optimized according to grasping or other task
requirements (Pollard & Gilbert 2002), possibly signicantly reducing the total weight
or volume of the actuation system when compared with only using identical actuators
129
for all tendons. Also, if non-linear stinesses are used in series with actuators, then the
calculation of feasible force sets may need to be adjusted to account for the fact that
pre-tension (possibly very high) will need to be applied to obtain a desired stiness.
We only analyzed routings where the tendons routed around every joint that they
passed (i.e., that the structure matrix is pseudo-triangular, as in (Lee & Tsai 1991)) and
where all moment arms were equal in magnitude for a particular joint. We acknowledge
that many of the routings that we analyzed may not be realizable in practice. Routings
can be designed where tendons pass through the center of joints (Grebenstein et al.
2010), or where moment arms for dierent tendons on the same joint can have dierent
magnitudes. Varying moment arms can add more potential
exibility to force-production
and stiness characteristics, while on the other hand, practical design considerations may
preclude realization of some routings. However, the analyses could be run on a set of
practical routings, spring stinesses, and maximal tendon tensions to guide in the design
process.
While we have analyzed the passive control of stiness and the bounds of force pro-
duction in a nger, we have not considered directly the consequences of nger design to
active control, which is of huge importance when constructing a useful system. In ad-
dition, a physical system subject to friction, estimation errors, actuator inconsistencies,
and other factors may mandate certain design constraints that we have not analyzed.
Lastly, it is natural to compare our results to the number, routing and strength of
the musculotendons of the human index nger. That index nger has 4 DOFs, and 7
tendons (6 tendons for the middle and ring ngers) (Valero-Cuevas et al. 1998, Valero-
Cuevas, Anand, Saxena & Lipson 2007). Interestingly, that anatomy has fewer than 2N
130
actuators and exhibits cross-over tendons such as those seen in Routings 1 and 2. In
addition, muscles have dierent strengths and stinesses (muscles with longer tendons
are naturally more compliant). Future work will apply this analysis to the anatomy of
biological ngers.
In this study we have shown that there is a very wide range of force-production and
stiness capabilities of dierent tendon routings with varying numbers of tendons. We
feel that the methods presented here could be used to guide in the design process for
tendon-driven ngers, hands, or other manipulators, to maximize force production for
various tasks, minimize the size and weight of the actuation system, and design tendon
stiness characteristics to realize various joint and endpoint stinesses. Furthermore,
analysis of the human musculoskeletal system from the perspective of stiness control
and force-production simultaneously could elucidate the advantages and disadvantages of
its anatomical features.
Acknowledgements
The authors gratefully acknowledge the help of M. Grebenstein in providing the kinematic
parameters for the DLR index nger and the helpful discussions with J. Kutch, S. Schaal,
M. Kurse, and P. Pastor.
131
Chapter 8
A Novel Computational Approach Helps Explain and
Reconcile Con
icting Experimental Findings on the Neural
Control of Arm Endpoint Stiness
8.1 Abstract
Much debate has arisen from the experimental ndings of limb impedance control during
reaching movements, and particularly around the regulation of stiness characteristics,
and its relation to minimization of energy expenditure for a particular task. The two chief
divergent experimental ndings are i) that the CNS has very limited control over endpoint
stiness orientation and ii) that the CNS has almost complete control over endpoint
stiness ellipsoid orientation and eccentricity. In this study, we provide the results from
novel theoretical analyses and computational experiments that oer explanations for both
of these divergent ndings, using only the passive stiness characteristics of muscles, the
132
arm posture, and a standard 6-muscle planar arm model. There are three chief conclusions
from this study. The rst is that the mechanical ability to orient stiness ellipsoids is
heavily dependent on even small changes in posture, as well as moment arm ratios of
the bi-articular muscles. The second is that neuromuscular synergies drastically reduce
endpoint stiness
exibility. The third is that in the complete absence of synergies,
for any desired and realizable endpoint stiness matrix, there exists a one-dimensional
manifold in muscle activation space that can produce that stiness (i.e., there exists
stiness redundancy). This provides a solution space which the CNS can then search
to minimize energy consumption. In summary, this computational study helps to shed
light on the diering conclusions of limb stiness experiments, and its insights also can
be used to design new experiments that can further elucidate the mechanisms of learning
and plasticity present in the human motor system.
8.2 Introduction
Limb stiness control by the central nervous system has been a subject of much study and
debate over the past 3 decades. Numerous experiments and theoretical analyses have been
conducted on the biomechanical and neuromuscular capabilities of the CNS to regulate
endpoint stiness of a limb (Burdet, Osu, Franklin, Milner & Kawato 2001, Burdet,
Osu, Franklin, Yoshioka, Milner & Kawato 2000, Flash & Mussa-Ivaldi 1990, Franklin,
133
So, Kawato & Milner 2004, Franklin, Liaw, Milner, Osu, Burdet & Kawato 2007, Hogan
1984, Hogan 1985a, Hogan 1985b, Hu, Murray & Perreault 2011, Kadiallah, Liaw, Kawato,
Franklin & Burdet n.d., McIntyre et al. 1996, Milner 2002, Mussa-Ivaldi, Hogan & Bizzi
1985, Perreault, Kirsch & Crago 2001, Perreault, Kirsch & Crago 2002, Stroeve 1999, Tee,
Franklin, Kawato, Milner & Burdet n.d., Shin, Kim & Koike 2009, Osu & Gomi 1999,
Gomi & Osu 1998, Darainy, Malfait, Gribble, Towhidkhah & Ostry 2004). These studies
chie
y analyze the stiness of the hand in reaching-like postures, as this is a simple case
to analyze and experiments with this limb are standardized and fairly easy to perform.
In addition, there is extensive literature on the analysis and synthesis of stiness in
robotic manipulators (Ciblak & Lipkin 1994, Ciblak & Lipkin 1999, Huang & Schimmels
2000, Kobayashi et al. 1998, Hogan 1985a). The theoretical contributions and conclusions
of these studies are unencumbered by extensive discussion of the mechanisms and limita-
tions of motor control by the CNS, and hence can form a good foundation for analyzing
the bounds of the biomechanical capabilities of the human musculoskeletal system. It
can therefore aid in interpretation of human subjects experiments as well as stimulating
creativity in novel experiments.
One set of experimental literature nds that the CNS can arbitrarily regulate the ori-
entation and eccentricity of arm stiness ellipsoids following training in order to perform
a task more reliably and with less energetic expenditure than before training (Burdet
et al. 2001, Franklin et al. 2004, Franklin et al. 2007, Kadiallah et al. n.d.). Another
134
set of experiments concludes that the CNS cannot arbitrarily regulate endpoint stiness,
and that it is only able to rotate the orientation of the stiness ellipsoid around 30
(Gomi & Osu 1998, Darainy et al. 2004, Perreault et al. 2002). While these are very
con
icting results, one large dierence in the experimental conditions is that the former
literature trains the arm and measures stiness during reaching movements, while the
latter literature measures stiness without training in reaching movements.
Using the analyses developed in the robotics literature, we are able to resolve these
divergent ndings by computational experiments which oer explanations for both sets
of results. Our novel formulation of the stiness synthesis problem allows us to easily
and eciently analyze stiness synthesis
exibility in the presence as well as absence
of muscle synergies. We nd that stiness
exibility is very sensitive to small changes
in arm posture and moment arm ratios, especially in certain portions of the workspace.
Our experiments also emphasize the importance of bi-articular muscles in the human
arm for stiness control which has been highlighted extensively in the literature(Franklin
et al. 2007, Flash & Mussa-Ivaldi 1990, Hogan 1985a, Hogan 1985b, Kobayashi et al. 1998).
Moreover, our formulation reveals a one-dimensional manifold (which is a convex set)
where a realizable stiness can be attained in the absence of synergies. This can be called
stiness redundancy, and once the CNS solves the problem of tuning endpoint stiness
to its desired conguration, then it further faces the problem of energy minimization
135
within the constraints of the desired stiness. This implies that conclusions about energy
minimization due to changes in the stiness ellipsoid should be used with caution.
8.3 Methods
8.3.1 Arm Model
We use a simplied planar arm model with 6 muscles similar to those that have been used
in other theoretical and computational studies (Fagg, Sitko, Barto & Houk 1997, Flash
& Mussa-Ivaldi 1990, Hogan 1985b, Milner 2002, Mussa-Ivaldi et al. 1985). It is shown
in Figure 8.1a. We use workspace constraints identical to those used in (Hogan 1985b)
to produce the workspace shown in Figure 8.1b. In the same gure, we illustrate that
we use singular value decomposition (SVD) to transform the endpoint stiness matrix
(K
end
) or the manipulator Jacobian (J) to an ellipse that represents the characteristics
of either of these matrices (e.g., condition number, orientation, size, etc.).
8.3.2 Theoretical Formulation
We begin our formulation with the endpoint stiness matrix, K
end
, which relates the
vector of dierential endpoint displacements to dierential endpoint forces:
@
~
F =K
end
@~ x (8.1)
136
x
y
=
Mono-articular
shoulder muscles
Mono-articular
elbow muscles
Bi-articular
muscles
WORKSPACE
SVD
(a)
(b)
or
Figure 8.1: (a) Arm model. (b) Workspace of arm model. SVD used to transform
endpoint stiness matrix to a stiness ellipse.
137
where@
~
F is the endpoint force vector resulting from a displacement vector@~ x. The joint
stiness matrix relates the vector of dierential joint angle displacements to dierential
joint torques:
@~ =K
joint
@
~
(8.2)
where @~ is the joint torque vector resulting from a joint angle displacement vector @
~
.
The endpoint stiness matrix is dependent on the joint stiness matrix as well as the
manipulator Jacobian J (which is posture dependent: a vector of joint angles
~
uniquely
denes the posture):
_
~ x =J(
~
)
_
~
(8.3)
where
_
~ x denotes the endpoint velocity vector and
_
~
denotes the joint angle velocity vector.
The endpoint stiness matrix, in the absence of an external tip force, is given by
(Hogan 1985b):
K
end
=J
T
K
joint
J
1
(8.4)
Furthermore, the joint stiness matrix is given by (Hogan 1985b):
138
K
joint
=RK
tendon
R
T
(8.5)
whereR is the moment arm matrix relating joint angle changes to tendon displacements,
@~ s:
@~ s =R@
~
(8.6)
andK
tendon
is the diagonal matrix of tendon stinesses, which is assumed to be a linearly
related to muscle force (Cui, Perreault, Maas & Sandercock 2008):
K
tendon
=diag(
~
F
muscles
) (8.7)
Combining Equations 8.4, 8.5, and 8.7, we get:
K
end
=J
T
R(diag(
~
F
muscles
))R
T
J
1
(8.8)
From the above equation, we can reformulate the endpoint stiness matrix, the moment
arm matrix, and the Jacobian in order to make the endpoint stiness vector,
~
K
end
a linear
function of the muscle force vector
~
F
muscles
. We show these reformulations in Figure 8.2.
() denotes element-by-element multiplication, and R
i
is the i
th
row of R. The Jacobian
139
Figure 8.2: Endpoint stiness variables reformulated. () denotes element-by-element
multiplication, and R
i
is the i
th
row of R.
reformulation is specic to the 2-link planar arm model, but similar expressions can be
formulated for Jacobians of higher dimensions.
The endpoint stiness and the moment arm matrix have been previously reformulated
in this way (Kobayashi et al. 1998), but to the best of our knowledge, no study has ever
reformulated the Jacobian in this way to allow for the following simple set of linear
equations, which is equivalent to Equation 8.8:
~
K
end
=
~
J
T
~
R
~
F
muscles
(8.9)
140
8.3.3 Checking for Realizable Endpoint Stiness Matrices
Given a desired endpoint stiness condition number (the ratio of the length of the major
axis of the ellipse to the length of the minor axis) and orientation (angle from the x-axis
to the major axis of the ellipse), we can determine if the arm is able to realize this stiness
matrix given the constraints that the muscle forces are non-negative, the net joint torque
vector is zero, and the endpoint stiness vector corresponds to that which is desired. We
can express this as follows:
if9
~
F
muscles
s.t.
~
K
end;desired
=
~
J
T
~
R
~
F
muscles
(8.10)
R
~
F
muscles
= 0 (8.11)
~
F
muscles
0 (8.12)
Then if9
~
F
muscles
s.t. Equations 8.10, 8.11, and 8.12,
~
K
end;desired
is realizable.
We would like to estimate the angular range of achievable endpoint stiness ellipse
orientations given the arm posture and a desired condition number. To this end, we
x the condition number and the posture, and then determine a set of desired endpoint
141
Orientation angle
Major axis
Minor axis
Checked for realizable
orientation every 5
degrees
x
y
Figure 8.3: The endpoint stiness ellipse as dened by the desired endpoint stiness
vector, corresponding to an orientation angle and a set condition number.
stiness vectors
~
K
end
, where each corresponds to a dierent ellipse orientation. We do
this every 5
around the full range of orientations and then check the fraction of these
orientations that are realizable. This is shown in Figure 8.3.
In our computational experiments, we formulated the realizability problem above
as a constrained quadratic programming problem, with the optimization criteria being
minimizing the sum of squares of the muscle forces. If the optimization found a solution,
then the orientation (for that specic condition number and posture) is realizable.
An example of the fraction of realizable orientations for all postures in the workspace
is shown in Figure 8.4.
142
Fraction of
achievable
orientations
Desired condition
number for each
orientation = 2
Fraction = 1
Fraction = 0.5
Fraction = 0
Realizable (red)
Not realizable (black)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 8.4: Example of realizable orientations in various postures for a condition number
of 2. Red lines indicate that the orientation of the major axis in that position is realizable,
and black lines indicate that the orientation is not realizable.
143
8.3.4 Varying Moment Arms and Synergies
We did this realizability experiment with all muscles having equal moment arms and also
with varying ratios of elbow to shoulder moment arms. In addition, we also performed
the experiment with synergies as reported from an EMG study (Osu & Gomi 1999).
This synergy basically couples the bi-articular muscles with the mono-articular elbow
muscles, and quantitatively it was found that the elbow stiness from co-contraction of
the bi-articular muscles was approximately one half of the elbow stiness from the mono-
articular elbow muscles. The mono-articular shoulder muscles were not found to have
synergies with the bi-articular muscles.
In the presence of these synergies, then only one parameter can be varied to change
the orientation and condition number of the endpoint stiness ellipse: the ratio of elbow
muscle co-contraction (which includes the synergistic bi-articular muscles' co-contraction)
to shoulder muscle co-contraction. Increasing both co-contractions simultaneously only
increases the size of the ellipse but not the condition number or orientation. As we will
see in the results, none of the desired endpoint stiness vectors can be achieved because
there are two parameters that must by satised: the condition number and orientation.
In the presence of synergies, as just described, there is only one free parameter that
can be varied and therefore changing orientation independently of condition number is
impossible.
144
Due to this in
exibility, we use another method to analyze the eect synergies have
on realizable endpoint orientations, without considering the specic condition number: we
vary the ratio of shoulder stiness to elbow stiness over a range of 2 orders of magnitude
(1/10 to 10) and see how much the orientation of the ellipse is able to change.
8.3.5 Exploring Energy Expenditure Within the Stiness Redundancy
Solution Space
The constraints in the realizability tests have 5 equality constraints (Equations 8.10
and 8.11). Since there are 6 muscles, if there is any solution which satises the non-
negative constraint (Equation 8.12), in general there will be a one-dimensional manifold
(or nullspace) of solutions for the desired endpoint stiness vector in the 6-dimensional
muscle force space. Vertex enumeration algorithms can be used to determine the ver-
tices of this one-dimensional manifold (which is a convex set). However, we are most
interested in the maximal and minimal energy expenditures within this space. Therefore,
we can use opposite quadratic programming optimization criteria to determine both of
these energy expenditures. For the minimal energy expenditure, as already described,
our optimization criteria is to minimize the sum of squares of the muscle forces. For
maximal energy expenditure, our optimization criteria is to maximize the sum of squares
of the muscle forces. From these numbers we can then determine the maximal amount of
energy reduction that is possible. For example, if the maximal energy expenditure is 10
145
(normalized) and the minimal energy expenditure is 7, then there is a maximum of 30%
reduction in energy possible.
Our rationale for quantifying these ratios is that for an observed stiness ellipsoid in
human subjects experiments, we want to know if we can infer that the central nervous
system is minimizing energy expenditure. If there is a large possible range of energies ex-
pended for the exact same endpoint stiness ellipse, then additional experimental means
such as EMG must be used to make precise and accurate conclusions about energy min-
imization.
8.4 Results
8.4.1 Realizable Fraction of Endpoint Stiness Ellipses
Figure 8.5 shows the results of various condition numbers, synergies, and moment arm
ratios on the fraction of realizable orientations.
Our computational results for condition number 1, in the equal moment arms, no
synergies case, is identical to the theoretical result determined by (Hogan 1985b). We
can make several observations from Figure 8.5. First, posture has a huge eect on the
range of realizable orientations. Second, the ratio of shoulder to elbow moment arms also
has a large eect on realizable orientations. Third, as discussed before, synergies prevent
any orientation from being obtained given a set condition number because that requires
146
Condition
Number = 1
Equal moment
arms, no synergies
1/2 shoulder
moment arms,
no synergies
1/2 elbow
moment arms,
no synergies
Equal moment
arms, synergies
present
Condition
Number = 2
Condition
Number = 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of
achievable
orientations
Figure 8.5: Fraction of realizable orientations given various modeled conditions.
147
Figure 8.6: Range of orientations in the presence of synergies.
2 free parameters, while in the synergies case, there is only one free parameter. Fourth,
in general, a higher condition number results in less orientation range realizability.
8.4.2 Realizable Orientations in the Presence of Synergies
Figure 8.6 shows the orientations achieved by varying the ratio of shoulder to elbow
stiness from 1/10 to 10, with the arm endpoint being in normalized x-y position (0,1):
each link of the arm has length of 1. The volumes of the ellipses are normalized to be
equal to each other, allowing observation of the orientation and condition number of the
ellipse.
148
The range of orientations is approximately 70
, which represents a realizable fraction
of orientations of about 0.39. This is a much more limited range than the rst column
of plots shown in Figure 8.5. In this posture, for all 3 condition numbers, the fraction
of realizable orientations is 1 (all orientations are achievable) in the absence of synergies.
In addition, we see that as the orientation of the ellipse in Figure 8.6 changes, the con-
dition number of the ellipse changes. The range of physically-realizable ratios of elbow
to shoulder co-contraction are likely much less extreme than two orders of magnitude,
which would result in an even smaller range of possible ellipse orientations.
8.4.3 Energy Expenditure Ranges
The maximal ratios between maximal and minimal energy expenditure given a condition
number and posture for any orientation produced the results shown in Figure 8.7. The
maximum possible energy reduction at a given posture is shown. That is, for a fraction
of 0.5, the maximal energy expended for a certain orientation was 2 times that of the
minimal energy expended. It can be seen that the maximum possible energy reduction
for any posture for these condition numbers is around 0.5 (in some very specic postures).
In much of the workspace, very little energy reduction is possible.
149
Condition
Number = 1
Condition
Number = 2
Condition
Number = 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Maximum
possible
energy
reduction
Black region: condition
number not achievable
Figure 8.7: Maximal possible energy reduction for any orientation given the condition
number and posture.
150
8.5 Discussion
We have explored the eects of posture, synergies, and moment arm ratios on the achiev-
able endpoint stiness ellipses for the human arm as well as energy minimization pos-
sibilities. Our novel formulation allows us to do this with ease and eciency. We see
that the range of achievable orientations is very sensitive to posture, and that synergies
drastically limit the ability to orient the ellipse as well as control the condition number
(since it cannot be controlled independently of orientation). In addition, signicant en-
ergy minimization is possible once certain achievable endpoint ellipses have been realized
by a co-contraction pattern in the solution space.
We would like to point out an important dierence in our formulation compared with
other modeling studies for arm stiness (Franklin et al. 2007, Hogan 1985b, Hogan 1985a,
McIntyre et al. 1996, Flash & Mussa-Ivaldi 1990, Perreault et al. 2002). The general form
of the joint stiness matrix in these studies is (for all equal moment arms):
K
joint
=
2
6
6
4
K
s
+K
b
K
b
K
b
K
e
+K
b
3
7
7
5
(8.13)
whereK
s
is the shoulder stiness provided by co-contraction of the mono-articular shoul-
der muscles,K
b
is the bi-articular joint stiness provided by co-contraction of bi-articular
muscles, andK
e
is the elbow stiness provided by co-contraction of mono-articular elbow
151
muscles. This implies 3 equality constraints, one for each pair of muscles. This is one
more constraint than the condition of zero endpoint force (i.e., R
~
F
muscles
= 0, which is
two equalities). If these 3 equality constraints are in place, then in addition to the 3
provided by
~
K
end;desired
=
~
J
T
~
R
~
F
muscles
, we have a total of 6 equality constraints. If a
solution exists given these constraints, it is unique (in general), since there are 6 muscles.
Therefore, no energy minimization is possible if these 3 \synergies" are present.
This study only looks at the mechanical capabilities of the arm model given the pas-
sive stiness properties of muscles, which were assumed to be linear and proportional to
muscle force. This model does not take into account active re
ex pathways (or feedback
pathways), which clearly can be used to minimize energy further depending on the fre-
quency content of a perturbation or motor noise during a task. It has been suggested
(Hu et al. 2011) that some studies involving endpoint stiness analysis may incorporate
active re
ex contributions (Burdet et al. 2001, Franklin et al. 2007, Darainy et al. 2004).
If only passive stiness properties are used and there is no net force at the endpoint, then
the endpoint stiness matrix is symmetric. It has been noted that any non-symmetric
component of endpoint stiness \can only be due to heteronymous inter-muscular feed-
back" (Hogan 1985b). However, our study is still able to reconcile con
icting ndings
even if only passive stiness is considered.
Our results suggest ways in which future arm stiness experiments may be conducted
in order to analyze stiness synthesis strategies used by the CNS such as synergies,
152
energy minimization, posture adjustment, and active re
ex pathways. Reaching exper-
iments could test stiness ellipsoids in various postures during the reaching movement,
since stiness ellipsoid orientation
exibility is very sensitive to small changes in posture.
Findings that con
ict with the results of this study could be studied in more detail as
this would suggest signicant feedback pathways that were developed as a result of motor
learning and neural plasticity.
Acknowledgements
The authors gratefully acknowledge the useful comments from Jason Kutch.
153
Chapter 9
Optimization of Tendon Topology for Robotic Fingers:
Prediction and Implementation
9.1 Abstract
Tendon-driven mechanisms in general, and tendon-driven ngers in particular, are an
important class of bio-inspired mechatronic systems. However, their mechanical com-
plexity and high-dimensional design space has not been fully harnessed to optimize their
performance, control, and construction. In this study we describe a novel, systematic
approach to analyze and optimize the routing of tendons for force-production capabili-
ties of a recongurable 3D tendon-driven nger. Our results show that these capabilities
could be increased by up to 277% by rerouting tendons, and up to 82% by changing
specic pulley sizes for specic routings. Experimental results for 6 implemented tendon
routings correlated very highly with theoretical predictions with an R
2
value of 0.987,
154
and the average eect of unmodeled friction decreased performance an average of 12%.
We not only show that, as expected, functional performance can be highly sensitive to
tendon routing and pulley size, but also that informed design of ngers with fewer ten-
dons can exceed the performance of some ngers with more tendons. This now enables
the systematic simplication and/or optimization of the design and construction of novel
robotic ngers. Lastly, this design and analysis approach can be used to model complex
anatomical systems such as the human hand to understand it in detail.
9.2 Introduction
Bio-inspired robotic hands employ multiple robotic ngers for dexterous grasping and ma-
nipulation tasks (Jacobsen et al. 1986, Salisbury & Craig 1982, Shadow Robot Company
n.d., Grebenstein et al. n.d., Ambrose et al. 2000, Jau 1995, Massa et al. 2002, Lin &
Huang 1996, Kawasaki et al. 2002, Namiki et al. 2003, Yamano & Maeno 2005, Gaiser
et al. 2008). Bio-inspiration can refer to their tendon-driven nature, but also the asymme-
try of the routings, the variation in moment arm sizes, and the non-uniform distribution
of maximal tendon tensions. Robotic nger kinematics may be anthropomorphic or they
may be less complex to simplify the construction and control of the ngers. Moreover, two
fundamental classes of actuation are typically used: i) those that use remote actuation
(e.g., motors outside the ngers which actuate tendons, cables, or gears) and ii) those
155
that use internal actuation (e.g., motors inside the ngers). Tendon-driven limbs and
ngers are ubiquitous in vertebrates, and such bio-inspired tendon-driven actuation has
proven engineering design advantages such as light weight, low inertia, small size, back-
drivability, and design
exibility (Pons et al. 1999). However, the mechanical complexity
of tendon-driven system (e.g., the large number of design parameters) has precluded
the development of modeling, design, and analysis tools to optimize their performance,
control, and construction. In this paper, we analyze, optimize, and test alternative im-
plementations of a 3D tendon-driven robotic nger. We validate this approach with
physical hardware implementations from the functional perspective of maximizing the
set of feasible endpoint static forces.
Many considerations go into the design of robotic ngers and hands, such as force and
velocity production, control, ease of construction, design simplicity, and cost. Adequate
force-production capabilities are a necessary element of the multidimensional design puz-
zle: according to Firmani,\The knowledge of maximum twist and wrench capabilities is an
important tool for achieving the optimum design of manipulators" (Firmani et al. 2008).
In fact, if a nger cannot produce sucient endpoint force while meeting other critical
design requirements such as size and number of motors (for example in space, hazardous
or surgical applications), then the mechatronic system is useless regardless of the at-
tributes of the controller or ease of manufacturing. Therefore, as a demonstration of our
156
novel modeling, analysis and optimization approach we concentrates on the kinetostatic
(endpoint force-production) capabilities for robotic ngers.
Several studies have analyzed the kinetostatic performance of tendon-driven and
torque-driven manipulators (Bouchard et al. n.d., Chiacchio et al. 1997, Firmani et al.
2008, Finotello et al. 1998, Gouttefarde & Krut 2010, Zibil et al. 2007, Tsai 1995, Lee
& Tsai 1991, Lee 1991) (determining the kinetostatic capabilities given design parame-
ters), and several others have addressed their optimization or synthesis (specifying the
design parameters given desired capabilities) (Fu & Pollard 2006, Chen et al. 1999, Ou
& Tsai 1993, Ou & Tsai 1996, Sheu et al. 2009, Angeles 2004, Aref et al. 2009, Chablat
& Angeles 2002, Khan & Angeles 2006). These studies are based on mathematical the-
ory. The fabrication of robotic ngers has been widely accomplished for robotic hands
(Jacobsen et al. 1986, Salisbury & Craig 1982, Shadow Robot Company n.d., Grebenstein
et al. n.d., Ambrose et al. 2000, Jau 1995, Massa et al. 2002, Lin & Huang 1996, Kawasaki
et al. 2002, Namiki et al. 2003, Yamano & Maeno 2005, Gaiser et al. 2008). Experimental
testing of kinetostatic performance can be found in the biomechanics literature (Kuxhaus
et al. 2005, Valero-Cuevas et al. 1998), but these do not implement a system whose pa-
rameters can be altered. We combine these three areas of theory, fabrication, and testing
to optimize and validate hardware implementations of alternative robotic nger designs.
157
9.3 Finger Construction
We had several design requirements when designing a recongurable robotic nger as a
test bed for analysis, optimization, and testing. They were
1. Ability to arbitrarily change tendon routing (i.e., the joints each tendon crosses,
and whether they produce positive or negative torque at each joint).
2. Ability to vary pulley sizes (i.e., moment arms of the tendons)
3. Low friction
4. Sucient and well arranged degrees-of-freedom (DOFs) to allow three-dimensional
endpoint motion and force production
5. Robust, durable, rigid
We designed the nger in SolidWorks 2010 (Dassault Syst emes), as shown in Figure
9.1. The actual recongurable nger is also shown. It was constructed with one ad-
abduction DOF and two
exion-extension DOFs. The primary materials were aluminum
(for the links and terminating pulleys), turcite (for the spacers and rotating pulleys), and
ball bearings with extended inner rings (mounted on all pulleys and link axes). All of
the pulleys were custom-machined and two sizes were constructed for recongurability:
a radius of 8.0mm for the large pulleys and 4.4mm for the small pulley, as shown in
158
Figure 9.2. There were multiple pulleys that had to be added between the axes to ensure
recongurability of the tendon routing (i.e., that each tendon could rout on either side of
every joint and no bowstringing of the tendons would occur).
The selection of link lengths and pulley sizes was otherwise fairly arbitrary, and since
our study did not involve optimization of incremental changes in these parameters (ex-
cept the 2 pulley sizes), we simply constructed it to have reasonable size that could be
fabricated and tested.
9.4 Methods
After construction of the nger with the desired capabilities, we were then able to an-
alyze and optimize tendon routing and pulley sizes based on the actual kinematics and
reconguration options of the nger.
9.4.1 Force polytope analysis
Quantication of the force-production capabilities of a robotic nger (or manipulator)
can be accomplished by determination of the feasible force set (or force polytope) of
the nger. This convex set encloses all feasible forces that the ngertip can exert given
kinematic parameters, tendon routing and pulley sizes, and maximal tendon tensions. A
quality metric that can be assigned to this set is known as the maximal isotropic value
159
Side Views
Top View
SolidWorks Student Edition.
For Academic Use Only.
3-D View
Actual Finger
83mm 64mm 57mm
Joint 1
Ad-abduct
Joint 2
Flex-extend
Joint 3
Flex-extend
Figure 9.1: 2-D and 3-D views of nger model in SolidWorks, and the actual nger.
160
(a) (b)
(d)
(c)
Figure 9.2: Pulleys used in nger design. (a) Turcite rotating small pulley. (b) Aluminum
terminating small pulley. (c) Turcite rotating large pulley. (d) Aluminum terminating
large pulley.
(MIV) (Finotello et al. 1998). Since we are not assuming any specic task that this nger
must perform, then we chose to use this metric. We could have used any other metric
instead of the MIV. Further comments can be found in the discussion section. The MIV
is the radius of the largest ball, centered at the origin, that the feasible force set can
contain, as illustrated in Figure 9.3 for a 2-D feasible force set. A nger can exert at least
that many units of force in any direction.
We can use an activation vector, ~ a, to represent the degree to which a tendon is
activated. Each element of ~ a ranges between 0 (no activation, zero force) and 1 (full
activation, maximal force). Further discussion may be found in (Valero-Cuevas 2005).
If we dene F
0
as a diagonal matrix of maximal tendon tensions, R as the moment
arm matrix (or structure matrix) relating tendon tensions to joint torques, and J as the
posture-dependent Jacobian relating joint velocities to ngertip velocities, then we can
161
Force in x
Force in y
-50 -25 0 25
0
100
200
Radius = MIV
(maximum
isotropic value)
Feasible force set
Figure 9.3: Illustration of calculation of MIV (maximum isotropic value) from feasible
force set.
get the ngertip force vector
~
f from tendon activations (Valero-Cuevas et al. 1998) if the
Jacobian is square and invertible:
~
f =J
T
RF
0
~ a =A~ a (9.1)
For a given xed nger posture, the J
T
, R, and F
0
matrices can be grouped into
a linear mapping from activations into ngertip force, which we call an action matrix
A (Valero-Cuevas et al. 1998, Valero-Cuevas 2005). Each column of A represents the
force vector each tendon produces at the ngertip in that posture if fully activated. The
collection of all such forces (i.e., all columns of matrix A) forms a set of output force
162
basis vectors. Linearity of this mapping holds true for static forces because the Jacobian
and moment arms remain constant. The Minkowski sum of these basis vectors forms the
feasible force set of the ngertip, and can be computed by taking the convex hull of the
points generated by mapping each vertex of the activation hypercube (i.e., each vertex
of the unit hypercube in the positive orthant) to ngertip wrench space via the action
matrix A (Valero-Cuevas et al. 1998).
There are two ways to describe a convex hull: i) a set of vertices and ii) a set of linear
inequalities. Vertex enumeration methodologies can calculate one description given the
other. The Qhull software package uses the Quickhull algorithm (Barber et al. 1996) and is
used to perform the MIV calculations in this study. Other vertex enumeration algorithms
that can perform these calculations easily include CDD (Fukuda & Prodon 1996) and LRS
(Avis 2000).
The description involving a set of linear inequalities (similar to a linear programming
inequality constraint formulation) takes the form
Axb (9.2)
where A is a matrix of constants dening the inequalities, x is a vector of variables of
length d, where d is the dimensionality of the convex hull, and b is a vector of constants.
If we denoteA
i
as thei
th
row ofA, then the linear inequalityA
i
xb
i
denes a halfspace,
163
which also denes a facet of the convex hull. The perpendicular (i.e., shortest) Euclidean
distance (or oset) of this facet from the origin, in general, will be given by
b
i
kA
i
k
2
(9.3)
The Qhull output, however, automatically sets eachkA
i
k
2
equal to 1, so thei
th
oset
from the origin is simply the signed constant b
i
. Calculation of the MIV in this study
involves simply nding the minimum of b corresponding to the feasible force set.
For our nger, the Jacobian is a 3 3 matrix which is square and invertible in our
experimental postures, R is a 3` matrix (` is the number of tendons, which is 4, 5, or
6), and F
0
is an `` diagonal matrix of maximal tendon tensions.
9.4.2 Evaluating tendon routings
The construction of the nger allowed for various moment arm matrices to be implemented
which had 4, 5, or 6 tendons. These designs are known as N+1, N+2, and 2N designs,
where N is the degrees of freedom of the nger. We enumerated all possible moment
arm matrices beginning with the \base" matrices shown in Figure 9.4. The N+2a and
N+2b designs dier only in that the second tendon terminates at the rst joint in the
N+2a designs and at the second joint in the N+2b designs. We replaced each `#' with
either a 1 or -1 (in accordance with the sign of the moment exerted on a joint when the
164
corresponding tendon is under tension; see Figure 9.1 for denition of joint axes) in a
full combinatoric search and then checked the controllability (i.e., that all of the joints
could be actuated independently in torque and motion) conditions as described in (Lee
& Tsai 1991). We then calculated the MIV for these routings using the large pulleys in
the main posture: 0
at joint 1,45
at joint 2, and45
at joint 3, as shown in Figure
9.5. To make comparisons feasible across nger designs with dierent number of tendons,
we used a uniform maximal tendon tension distribution, with the sum being constrained
1
to 60N (i.e., for designs with 4, 5, and 6 tendons, the maximal tensions were 15N, 12N,
and 10N, respectively). We found that many of the admissible routings produced the
exact same MIVs and feasible force set volumes, likely corresponding with structurally
isomorphic routings (Lee & Tsai 1991). The number of routings that produced unique
MIVs was a very small subset of the admissible routings, as can be seen from the numbers
in Figure 9.4. In cases of optimization of a more complex nger or manipulator where the
number of unique MIVs may be orders of magnitude higher, methods for selection of a
subset for further optimization such as in (Taguchi, Konishi & American Supplier 1987)
may be used.
1
The sum of maximal tendon tensions being equal is an important constraint due to the size, weight,
and motor torque (and therefore tendon tension) limitations inherent in dexterous hands. For example,
the torque capacity of motors is roughly proportional to motor weight, and minimization of weight was an
important consideration in the design of the DLR Hand II (Butterfa et al. 2001). In addition, the maximal
force production capabilities of McKibben-style muscles are roughly proportional to cross- sectional area
(Pollard & Gilbert 2002). Since the actuators typically will be located in the forearm, then the total
cross-sectional area will be limited to the forearm cross-sectional area. In this study, for simplicity and
without aecting the generalizability of our approach or results, we do not consider alternative constraints
on the actuation system (e.g., electrical current capacity, tendon velocities, etc).
165
N+1 design:
24 admissible,
3 unique
N+2a design:
88 admissible,
6 unique
N+2b design:
296 admissible,
11 unique
2N design:
872 admissible,
20 unique
Figure 9.4: Base moment arm matrices used when nding admissible and unique tendon
routings.
Joint 3
Joint 1
Joint 2
Main posture
Auxiliary
posture
Figure 9.5: Finger posture used in computations and experimental testing.
166
The total number of routings producing distinct feasible force sets
2
was 40. For each of
those routings, we calculated the MIV for all combinations of large and small pulleys. For
example, the N+1 design has 9 moment arm values. Therefore, there are 2
9
combinations
of large and small pulleys for that case. Taking the combination with the highest MIV for
each routing gave 40 moment-arm-optimized routings. Therefore, we had 40 unoptimized
routings and 40 optimized routings. Out of these 80, we chose 6 dierent routings to test
experimentally in a fashion that permitted testing of a large range of MIVs, and included
the design with the highest predicted MIV. Otherwise the selection was arbitrary.
9.4.3 Experimental testing of tendon routings
For each of the tendon routings tested, we rst arranged the pulleys and strings (0.4mm
braided polyester twine) to match the desired conguration. We then mounted the nger
onto a base that was part of a motor array system as shown in Figure 9.6. The DC motors
were coupled to capstans on which the string wound. Each string was then routed around
pulleys that were attached to load cells (Interface SML 25, Scottsdale, AZ) which provided
force measurements for the closed-loop controller implemented in Realtime LabView. The
endpoint of the nger was xed to a custom made gimbal which constrained translational
2
Due to the nature of our full combinatoric search, moment arm matrices that produced mirrored
feasible force sets about a plane passing through the origin (which would have the same MIV) were
discarded and also those moment arm matrices that were produced by a rearrangement of the columns.
For example, in Figure 9.4, interchanging columns 5 and 6 does not change the feasible force set, it only
reverses the \numbering" of the tendons. But in the full combinatoric search, both of these numberings
would be dierent matrices producing identical feasible force sets.
167
motion but not rotational motion (we did not want the ngertip to be over-constrained).
The gimbal was attached to a 6-axis load cell (JR3, Woodland, CA). The sampling rate
and control loop frequency were both 100Hz.
A small pretension of 1N was applied to each string to remove slack and prevent it from
falling o of the pulleys. Then each vertex of the activation hypercube (as described in the
previous section) was applied to the strings (in addition to the pretension) in ramp-up,
hold, and ramp-down phases to nd the feasible force set (Kutch & Valero-Cuevas 2011).
As in prior work, (Kutch & Valero-Cuevas 2011), vertices of this experimental feasible
force set were determined from the hold phases and then used to nd the MIV using Qhull
as described earlier. The experimental MIV could then be compared with the theoretical
MIV (from computational results).
To compare the shapes of the experimental and theoretical feasible force sets, we
rst normalized the volume of the experimental feasible force set to make it equal to
the volume of the theoretical feasible force set. We then calculated the mean Euclidean
distance of each vertex from the theoretical feasible force set from the corresponding
vertex in the experimental feasible force set. We did this because there was always
friction loss in the experiment
3
. In addition, we calculated the average angle between the
3
Take an extreme case in which friction loss was 50% exactly for every tendon. The theoretical feasible
force set is a unit cube. While the shape of the experimental feasible force set would be also an exact
cube, it would be 50% contracted in every direction and therefore the corresponding vertices would be far
from each other. If we normalized the volume, the corresponding vertices would be in the same location,
and the mean distance (in shape similarity) would be zero.
168
DC Motors
Gimbal
JR3 Load Cell
Pulleys
attached to
load cells
Figure 9.6: Experimental system for feasible force set testing.
169
two vectors (starting and the origin and ending at the corresponding vertex) formed from
corresponding vertices.
We tested the nger in the main posture (for which we optimized MIV) and an
auxiliary posture (to validate the predictions more fully). These postures are shown in
Figure 9.5. For each design and posture, we did three repetitions of tests. Since there
were 6 designs, 2 postures, and 3 repetitions, we conducted a total of 36 tests.
9.5 Results
9.5.1 Calculating maximum isotropic values
The 40 unique unoptimized and 40 unique optimized routings produced the MIVs shown
in Figures 9.7a and b. Optimization of the pulley sizes increased the average MIV from
0.60N to 0.78N, a 30% increase as shown in Figure 9.7a, and the maximum increase for a
routing by this optimization was 82%. It is interesting to note that this force-production
capability increase is achieved by simply decreasing specic pulley sizes in an informed
manner. We can see from Figure 9.7b that designs with 4 tendons could not produce
MIVs higher than the best designs with 5 or 6 tendons. However, the best design with 4
tendons did have a higher MIV than many alternative routings that had more tendons. In
addition, the maximal increase from only rerouting tendons (no pulley size optimization)
170
was 277% (i.e., the increase from the worst admissible routing to the best admissible
routing for a given number of tendons).
9.5.2 Theoretical predictions vs. experimental results
The experimental results and the routings tested are shown in Figure 9.8. The data
points shown in Figure 9.8b are averages of the three test repetitions in both the main
posture and the auxiliary posture for each of the designs. The average standard devia-
tion from the three test repetitions was very low at 0.0090N, showing that the results for
each design/posture combination were extremely repeatable. We see a consistent linear
relationship between theoretical and experimental MIVs with an R
2
value of 0.987. This
result shows that the theoretical calculations are very good at predicting actual perfor-
mance. The slope of the line is 0.879, which we interpret to represent an average loss
of performance of about 12% from theoretical predictions, likely due mainly to friction
in the system. We show experimental and theoretical feasible force sets for one test of
designs 1 and 6 in the main posture in Figure 9.8d, and we can see visually that the shape
of the theoretical feasible force sets was extremely similar to those of the experimental
feasible force sets, although the experimental ones were contracted to an extent.
We can see several interesting features in Figure 9.8. First of all, in Figure 9.8a, we
see that routings 1 and 5 are identical except that 2 of the signs in the moment arm
matrix are reversed (i.e., 2 of the tendons are switched from one side of the ad-abduction
171
0
0.5
1
1.5
All Large Pulleys
(all designs)
Maximum Isotropic Value (N)
Large and Small Pulleys
(all designs)
(b)
(a)
0
0.5
1
1.5
Design
Maximum Isotropic Value (N)
N+1 N+2a N+2b 2N
Figure 9.7: Maximum isotropic values for various routings. (a) Boxplot of MIV for all
designs before and after pulley-size optimization. (b) Boxplot of MIV vs. design (includes
optimized and unoptimized pulley sizes).
172
6 5
4
2
3
0 0.5 1 1.5
0
0.5
1
1.5
Theoretical MIV (N)
Experimental MIV (N)
Main posture
Auxiliary posture
Parity (Slope = 1).
Regression (Slope = 0.88)
6
5
4
2
3
1
y
z
1N
x
y
1N
x
6
Main posture
Test1
1
(a)
(b)
(c)
(d)
y
z
1N
x
1
y
1N
x
Main posture
Test1
Experimental
feasible force
set (green)
Theoretical
feasible
force set
(blue)
Figure 9.8: Results from experimental testing of various routings. (a) The 6 dierent
routings tested. Shown to scale. R matrix values are in mm. (b) Experimental vs. theo-
retical MIV. Parity line is where experimental MIV would be exactly equal to theoretical
MIV (intercept of 0, slope of 1). Regression has an R
2
value of 0.987. (c) Table of aver-
ages from 3 tests for each design in main posture. (d) 3-D visualization of experimental
and theoretical feasible force sets for designs 1 and 6.
173
joint to the other). However, we can see in Figure 9.8b and the table in Figure 9.8c
that the MIV of routing 5 is more than twice that of routing 1 both theoretically and
experimentally. Figure 9.9 emphasizes this large change in MIV for a small, intelligent
(but perhaps counterintuitive before performing the analyses) change in tendon routing.
Secondly, the MIVs of routings 5 and 6 are very similar, but routing 6 has one less tendon.
Thirdly, routings 2 and 3 have two fewer tendons than routing 1 but still outperform it
in terms of MIV. Figure 9.8d demonstrates visually that the experimental feasible force
sets corresponded very closely with the theoretical feasible force sets in shape, and that
the size was similar but contracted by a small amount due to friction. While the side
views of both of these feasible force sets look similar, the isometric views show clearly
that the feasible force set of routing 1 is quite thin along one direction (which results in a
low MIV) and the feasible force set of routing 6 is much more expanded in all directions
(so the MIV is much higher).
In Figure 9.8b, we see that the data points lie underneath but fairly close to the parity
line (if the theoretical and experimental MIVs were identical, the data points would lie
exactly on the parity line). As would be expected, none of them are above the parity line.
In the table in Figure 9.8c, we see that the error in the prediction of MIV ranged roughly
between 4% and 16% which is the percentage MIV below the parity line where the data
points are located. As far as shape goes, we see that the average dierence in angles
between corresponding vertices of the feasible force sets was between 2:56
and 6:5
. A
174
small amount of angular error would be expected due to error in the positioning and
alignment of the JR3 axes relative to the nger axes, so this contributes to the average
dierence in angles. The mean distance of corresponding vertices is less than 0.35N. The
normalization factors were less than 1.15, and it is the factor by which the experimental
feasible force set had to be expanded in every direction to have the same volume as the
theoretical feasible force set. It roughly corresponds with the MIV error.
9.6 Discussion
In this work, we have investigated the eect of various tendon routings on the set of
feasible forces that can be exerted by a robotic nger both computationally and in a
physical system. We see that routing has a very dramatic eect on the shape and size
of the feasible force set. We also see that computational predictions are quite accurate
and that they can be useful when making informed design decisions. Therefore the main
conclusions of this study are twofold:
1. Dierent routings in robotic ngers can result in extremely dierent force-production
capabilities.
2. Theoretical feasible force set analyses predict experimental force-production perfor-
mance quite well and therefore they are a useful design tool.
175
Original
(Design 1)
New
(Design 5)
157% increase in
MIV from
switching 2
moment arms
Re-route
New (Design 5) Original (Design 1)
0
0.5
1
1.5
Experimental MIV (N)
Figure 9.9: Illustration of a simple (but intelligent) tendon re-routing that drastically
increases MIV.
176
One application of this work is to design tendon-driven ngers and manipulators for
a given task that are optimized in terms of minimal size, weight, complexity, and cost.
Since tendon-driven systems are linear for xed postures, if we double all the moment
arms (or the maximal tensions of all tendons), we double the size of feasible force set
in every direction. Our results showed that the MIV for routing 6 was more than 100%
greater than that of routing 1. Therefore, one could either reduce all of the maximal
tendon tensions or all of the moment arms in routing 6 by 50% and still have a greater
MIV than routing 1. In a robotic hand system, if the maximal tendon tensions were
cut in half (by implementing smaller motors), then the weight of the actuators could be
roughly halved and cost would be reduced. This would be very desirable, in general.
If tendons are driving a minimally-invasive surgical instrument, then the moment arms
could be halved and therefore the diameter of the instrument would be halved (which
actually would reduce the cross-sectional area by 75%!). The instrument would then be
much smaller and could be much better suited for certain surgical procedures.
We have used MIV as the tness metric for our analyses since no prior assumption of
task specication was made. We acknowledge that in general, the MIV would typically
be more practically used for a tendon-driven, multi-purpose manipulator rather than
a robotic nger. These analyses and optimizations that we described apply equally to
tendon-driven manipulators and ngers regardless of size. Since we were only testing one
nger in this paper, we decided to use the MIV. Current work has accomplished design
177
optimization and validation for grasp quality of multiple ngers of identical design to the
one in this paper.
If a necessary task or set of tasks is known (e.g., to have high
exion force for strong
grasps) then the analyses could assign a tness metric to a routing based on that task
specication. The optimization could then be based on that metric. For example, if it
is desired to have a very strong
exion force with low extension force requirements, then
linear programming can easily be used to determine the maximal force possible in the
exing direction(s) after the feasible force set has been calculated (e.g., using the generic
procedure outlined in (Fu & Pollard 2006)). We have previously described that the feasible
force sets of the human ngers are asymmetrically biased towards endpoint forces in the
exion direction than in the extension which is anatomically reasonable for grasping tasks
(Inouye et al. 2011b, Valero-Cuevas et al. 1998, Valero-Cuevas et al. 2000, Valero-Cuevas
& Hentz 2002). If strong grasp and minimal size/weight/cost is desired for a set of ngers,
then analyses like those used in (Inouye, Kutch & Valero-Cuevas 2011a) can be used to
design an optimized tendon-driven robotic hand.
We have investigated force-production capabilities in this paper, but there are many
other considerations that go into the design of a robotic hand. Other signicant consid-
erations include the robustness and eectiveness of control algorithms, passive stiness
178
characteristics, sensitivity to friction and positioning errors, and maximal endpoint veloc-
ities. We acknowledge that force-production capabilities are only one piece of the design
puzzle for optimized robotic ngers.
For reasons of practicality, we only analyzed and constructed routings where the
tendons routed around every joint that they passed (i.e., that the structure matrix is
pseudo-triangular, as in (Lee & Tsai 1991)) and where there were only two sizes of
pulleys that could be chosen. Routings can, however, be designed where tendons pass
through the center of joints (Grebenstein et al. 2010), or where moment arms can have
many feasible magnitudes. This opens up the design space even more, and exhaustive
searches like the ones we performed in this study may be more laborious, or even not
be feasible given the exponential growth of design options. In addition, tendon-driven
ngers or manipulators with more than 3 degrees of freedom will tend to suer from
the curse of dimensionality in the design space, and a designer may have to use various
optimization algorithms (Inouye et al. 2012) in a search for a \good enough design which
could then be selected for physical construction. Alternatively, a designer could come up
with a handful of feasible, physically-realizable routings and then search in the vicinity
of that region of the parameter space to determine feasible improvements with aordable
computational cost (Valero-Cuevas et al. 2009).
The optimization process we used in this study only addressed this realization of a
robotic nger. If a general robotic nger or manipulator has more joints or tendons, the
179
dimensionality of the design space increases dramatically and nding a globally optimum
solution for a specic tness metric (of which any task-specic metric may be used, not
only the general MIV metric) may be computationally infeasible due to the curse of
dimensionality. Other custom or typical optimization algorithms could be used to nd
solutions with a high tness. Furthermore, in the case when the optimization may involve
link lengths and D-H parameters in addition to tendon routing, number of tendons, and
pulley sizes, then nding a locally optimal solution or just a good-enough solution could
still be very useful. The main purpose of this study was to investigate the correlation of
predictions with experiments, as opposed to identifying a general optimization method
for tendon-driven robotic ngers and manipulators.
Tendon friction was a signicant factor in our experiment, (as it is for any tendon
driven system), especially for the tendons at the last joint that had to wrap around as
many as 12 pulleys. The main source of friction seemed to be the pulleys, as general
observation of the data indicated that tendons attached to the ad-abduction joint (which
wrapped around 4 pulleys) suered from very little friction loss (less than a few %) while
the tendons that attached to the last joint (which wrapped around 12 pulleys and routed
through the fairly complicated tendon redirection between the rst and second joints)
suered from as much as 20% friction loss.
Future work will extend this experimental validation approach to routings of multiple
ngers for optimized grasp quality. In addition, this work is easily applicable to rene the
180
design of generic tendon-driven manipulators. Furthermore, investigation of the control
and structure of biological tendon-driven systems is now made possible using a similar
framework.
9.7 Conclusions
We conclude from this validation that these computational methods are eective at pre-
dicting the performance of drastically dierent tendon-driven robotic nger (or manipu-
lator) designs, and are therefore a useful design tool. Various benets of fully utilizing
this design tool include
1. Minimization of weight: if a superior design has a force-production performance
twice that of an inferior design, the superior design's actuators only need to be
half the strength of the inferior design's to match the inferior design's performance,
which in general corresponds to a large reduction in weight of the actuators.
2. Minimization of size: if a superior design has a force-production performance twice
that of an inferior design, the superior design's moment arms only need to be
half the size of the inferior design's to match the inferior design's performance,
which could be used to half the overall thickness of the nger (or manipulator, or
minimally-invasive surgical device).
181
3. Minimization of number of tendons (and therefore actuators): If a design with less
tendons (such as an N+1 design) can be synthesized with the same force-production
performance as that of one with more tendons (such as a 2N design), then the
actuator system can be simplied and less space to rout the tendons inside the
nger (or manipulator, or minimally-invasive surgical device) is needed.
Acknowledgements
The authors gratefully acknowledge the help of Manish Kurse in providing the data acqui-
sition routine for the experimental procedure, and Dr. Veronica Santos for construction
of the gimbal used in the experiments.
182
Chapter 10
Computational Optimization and Experimental Evaluation
of Grasp Quality for Tendon-Driven Hands Under
Constraints
10.1 Abstract
The chief tasks of robotic and prosthetic hands are grasping and manipulating objects,
and size and weight constraints are very in
uential in their design. In this study we
use computational modeling to both predict and optimize the grasp quality of a recong-
urable, tendon-driven hand. Our computational results show that grasp quality, measured
by the radius of the largest ball in wrench space, could be improved up to 259% by simply
making some pulleys smaller and redistributing the maximal tensions of the tendons. We
experimentally evaluated several optimized and unoptimized designs, which had either
4, 5, or 6 tendons, and found that the theoretical calculations are eective at predicting
183
grasp quality, with an average friction loss in this system of around 30%. We conclude
that this optimization can be a very useful design tool, and that using biologically-inspired
asymmetry and parameter variability can be used to maximize performance.
10.2 Introduction
Robotic and prosthetic hands have been designed for many years, and their essential tasks
are grasping and manipulating objects (Jacobsen et al. 1986, Salisbury & Craig 1982,
Shadow Robot Company n.d., Grebenstein et al. n.d., Ambrose et al. 2000, Jau 1995,
Massa et al. 2002, Lin & Huang 1996, Kawasaki et al. 2002, Namiki et al. 2003, Yamano
& Maeno 2005, Gaiser et al. 2008). Many robotic and prosthetic hands also are designed
with approximately the same shape and/or size as the human hand in order to be able to
perform tasks in place of a human. Weight and size constraints are two of the paramount
design constraints for these manipulators. Actuators for the ngers, typically located
proximal to the hand, are generally either larger or heavier if they are able to produce
more tension. This is also the case in the human hand: larger muscles are both heavier and
stronger. In addition, the pulley sizes in the ngers cannot be made too big, otherwise the
nger itself will become too large. In this paper, we utilize two recongurable ngers to
test computational predictions of grasp quality for a given tendon routing whose pulley
184
sizes are constrained and the sum of maximal tendon tensions is constrained (due to
weight and size constraints on the actuation system).
In addition to maximizing performance for a set of given constraints, the optimization
techniques presented here can also be used to minimize size or weight given performance
requirements. This is a useful tool when designing certain tendon-driven systems, such
as minimally-invasive surgical devices (minimization of size and number of actuators
desired) and prosthetic hands (minimization of weight desired). The consequences of
over-designing the capabilities of these systems are increased weight, cost, size, and power
consumption. Other capabilities important in the design of these systems are position
control, force control, velocity production, and design simplicity.
A large body of literature exists which addresses grasp quality of objects by robotic
hands and manipulators. Some studies have looked at grasp quality in order to determine
the optimal nger placement on an object by a robotic hand (Miller & Allen 1999, Miller
& Allen 2004, Surez Feijo et al. 2006, Borst, Fischer & Hirzinger 2003). However, they
do not take into account the mechanical capabilities of the ngers and therefore are only
useful in optimizing grasp placement and hand kinematics, not in optimizing the design
parameters. Mechanical design parameters were taken into account for tendon-driven
hands in (Fu & Pollard 2006), but they did not optimize over the parameters. Opti-
mization of parameters given requirements on force production for a single nger was
examined in (Pollard & Gilbert 2002), but they did not implement any of the optimized
185
designs in hardware, and they did not optimize for grasp quality. The hardware im-
plementation of robotic hands has been widely accomplished (Jacobsen et al. 1986, Sal-
isbury & Craig 1982, Shadow Robot Company n.d., Grebenstein et al. n.d., Ambrose
et al. 2000, Jau 1995, Massa et al. 2002, Lin & Huang 1996, Kawasaki et al. 2002, Namiki
et al. 2003, Yamano & Maeno 2005, Gaiser et al. 2008), but their choice of design param-
eters has not been guided by a systematic optimization or analyses of grasp quality.
Our study uses a previously-developed computational framework (Inouye et al. 2012)
to evaluate and optimize the grasp quality of a recongurable tendon-driven hand (tak-
ing into account all design parameters and constraints). The mechanical design of the
recongurable ngers is identical to that of a nger used in another study for single nger
force-production analysis and optimization (Inouye, Kutch & Valero-Cuevas n.d.). We
show that under constraints, optimization of design parameters can improve grasp per-
formance by more than 200%, and our predictions of grasp quality are corroborated by
experimental results.
10.3 Methods
10.3.1 Hand Construction
The robotic hand we optimized and tested consisted of two recongurable ngers, which
were designed for a previous study (Inouye et al. n.d.). 2-D views of the nger design
186
Side Views
Top View
83mm 64mm 57mm
Joint 1
Ad-abduct
Joint 2
Flex-extend
Joint 3
Flex-extend
Figure 10.1: Top and side views of nger design and kinematics.
are shown in Figure 10.1. The ngers were able to accept arbitrary tendon routings that
were analyzed and optimized computationally. In addition, the pulley size was variable,
consisting of two options: a large pulley, with radius 8mm, and a small pulley, with radius
4.4mm. Each of the custom pulleys was tted with ball bearings to minimize friction.
187
10.3.2 Grasp Quality Analysis
Our calculation of grasp quality was based on the wrench-direction-independent metric
known as the radius of largest ball, originally proposed by (Ferrari & Canny 1992). The
metric, in eect, is equal to the maximal magnitude of a wrench that can be applied to
the object in all directions in wrench space without it losing force closure (i.e., causing
the grasp to fail). A wrench vector whose magnitude is less than the grasp quality can
be applied to the object in any direction in wrench space without losing force closure.
The process for eciently calculating the grasp quality and optimizing the parameters of
a tendon-driven hand was originally developed in (Inouye et al. 2012).
Brie
y, the grasp quality analysis rst involves selecting the initial grasp parameters:
the nger geometry, object size and shape, grasping points, and number of ngers. These
are shown for this experiment in Figure 10.2. We analyzed grasp quality for two dierent
nger placements, also in Figure 10.2. Next, the ngertip force-production capabilities
for each nger are determined by calculating the feasible force set, which is a function of
nger posture and geometry, tendon routing and pulley sizes, and maximal tensions of the
tendons (Valero-Cuevas 2005). The nger Jacobian, J, relating joint angle velocities to
endpoint velocities, is determined from the geometry (i.e., D-H parameters) of the nger
and the nger posture. The tendon routing and pulley sizes determine the moment arm
matrixR. The maximal tensions of the tendons dene the diagonal F
0
matrix. After the
188
feasible force sets are calculated, they are intersected with friction cones whose orientation
is dependent on ngertip contact angle in order to produce a feasible object force set.
We used a coecient of friction of 0.5 for this study. This set represents the forces that
can be applied to the object by the ngertip. The feasible object force set is calculated
for each nger, using the nger placements determined initially. These sets are used to
determine all the forces and torques that can be resisted in wrench space (i.e., the grasp
wrench set), from which the grasp quality metric is then calculated using the Quickhull
algorithm (Barber et al. 1996) implemented in the software program Qhull.
The construction of the nger allowed for various moment arm matrices (which dene
the tendon routing and pulley sizes of the nger) to be implemented which had 4, 5, or 6
tendons. These designs are known as N+1, N+2, and 2N designs, where N is the degrees
of freedom of the nger. We enumerated all possible moment arm matrices beginning
with the \base" matrices shown in Figure 10.3. We replaced each `#' with either a 1 or
-1 (in accordance with the sign of the moment exerted on a joint when the corresponding
tendon is under tension; see Figure 10.1 for denition of joint axes) in a full combinatoric
search and then checked the controllability conditions as described in (Lee & Tsai 1991).
This resulted in a total of 252 realizable, unique routings (with all large moment arms).
The construction of the nger only allowed for routings where the tendons routed around
every joint that they passed (i.e., that the moment arm matrix is pseudo-triangular, as
in (Lee & Tsai 1991)).
189
50 mm 57.2 mm
Side View
(Grasp 1 only)
y
x
Front View
y
z
Grasp 1 (black)
Grasp 2 (red)
Isometric View
y
z x
Figure 10.2: Finger placements for each grasp.
190
N+1 design N+2 design
2N design
Figure 10.3: Base moment arm matrices used when nding realizable, unique tendon
routings.
We then calculated the grasp quality for these routings using the large pulleys and an
even distribution of maximal tendon tension. The sum of maximal tendon tensions was
limited to 60N for each nger
1
(i.e., for designs with 4, 5, and 6 tendons, the maximal
tensions were 15N, 12N, and 10N, respectively).
10.3.3 Optimization of Grasp Quality
The tness metric we used for optimization was the sum of the grasp qualities for grasp
1 and grasp 2 (they were both weighted equally).
1
The sum of maximal tendon tensions being equal is an important constraint due to the size, weight,
and motor torque (and therefore tendon tension) limitations inherent in dextrous hands. For example,
the torque capacity of motors is roughly proportional to motor weight, and minimization of weight was an
important consideration in the design of the DLR Hand II (Butterfa et al. 2001). In addition, the maximal
force production capabilities of McKibben-style muscles are roughly proportional to cross-sectional area
(Pollard & Gilbert 2002). Since the actuators typically will be located in the forearm, then the total
cross-sectional area will be limited to the forearm cross-sectional area. In this study, we do not consider
alternative constraints on the actuation system (e.g., electrical current capacity, tendon velocities, etc).
191
10.3.3.1 Optimizing Pulley Sizes
We calculated the tness for all combinations of large and small pulleys for the best
routing from each of the base moment arm matrices shown in Figure 10.3. We also
optimized the moment arms for the following naive 2N design (where 1 and -1 denote the
large moment arm sizes), for a total of 4 pulley-size-optimized routings:
R
NAIVE 2N
=
2
6
6
6
6
6
6
4
1 1 1 1 1 1
0 0 1 1 1 1
0 0 0 0 1 1
3
7
7
7
7
7
7
5
A combinatoric search for the 2N designs involved 12 moment arms, so 2
12
= 4096
tness evaluations were performed. Similarly, the N+2 design had 2048 evaluations and
the N+1 design had 512 evaluations.
10.3.3.2 Optimizing Tendon Tension Distribution
The last step in our optimization process involved performing a greedy Markov-Chain
Monte Carlo algorithm on the maximal tendon tension distribution. Starting with all
the tendons having equal maximal tendon tensions, we perturbed the distribution of
the maximal tendon tensions using a multivariate normal distribution with standard
deviation of 1% of the maximal tendon tension sum. This perturbation was eectively
like one inside ann-dimensional hypercube in the positive orthant with side length equal
192
to maximal tendon tension sum, where n is the number of tendons for each nger. After
perturbation inside the hypercube, we projected the point onto the hyperplane given by
the following equation:
n
X
i=1
F
i;max
=MaxTendonTensionSum (10.1)
where F
i;max
is the maximal tension of tendon i and one of the diagonal entries in the
F
0
matrix. This would give us a new distribution of maximal tendon tensions, and we
would then evaluate the tness. If it was higher, we would take that point as the starting
point for the next perturbation. The maximal tendon tensions were constrained so that
they did not go negative or above the total maximum, and a re
ection technique at those
boundaries was used similar to that in (Santos et al. 2009). The overall process is shown
graphically for a simplied 2-tendon example in Figure 10.4.
A detailed explanation of the eects of dierent structure matrices and distributions
of maximal tendon tensions on the kinetostatic (i.e., force-production) capabilities of
manipulators and biological hands can be found in (Valero-Cuevas 2005, Inouye et al.
2012, Lee & Tsai 1991, Finotello et al. 1998, Ou & Tsai 1993, Ou & Tsai 1996, Tsai 1995).
193
Constraint:
Max Tendon
Tension Sum
Starting point
Rejected step
Accepted step
Random perturbation direction
Random perturbation endpoint
Projection of perturbation
endpoint onto constraint
1
Max Tendon
Tension Sum
2
3
Figure 10.4: Illustration of Markov-Chain Monte Carlo algorithm for distribution of max-
imal tendon tensions.
194
10.3.4 Experimental testing of tendon routings
We tested each of the pulley-size-optimized layouts for each of the 4 routings (the best
from each of the base matrices, and then the naive 2N). We tested them with optimized
tendon tension distributions and unoptimized tendon tension distributions. This gave 8
nger design congurations, and we tested each of these 8 designs in both grasp congura-
tions. This gave 16 tests. In addition, we tested the naive 2N design in both postures with
unoptimized pulley sizes and unoptimized tendon tension distribution. So we obtained a
total of 18 data points.
For each of the designs tested, we rst arranged the pulleys and strings (0.4mm braided
polyester twine) to match the desired conguration. We then mounted the ngers onto a
base that was part of a motor array system as shown in Figure 10.5. The DC motors were
coupled to shafts which string wound around. The string was then routed around pulleys
that were attached to Interface SML 25 load cells which provided force measurements for
the closed-loop controller implemented in LabView. The ends of the ngers had a ball
which t into a socket on the object that we designed to attach to the 6-axis JR3 load
cell and also provide the correct nger placements. This ball-and-socket joint constrained
translational motion but not rotational motion (we did not want the ngertip to be over-
constrained). One object was printed for each grasp conguration. Both are shown in
Figure 10.5. The sampling rate and control loop frequency were both 100Hz.
195
Pulleys attached
to load cells
DC motors
Ball and
socket joint
JR3 load cell
Grasp 2 Grasp 1
Figure 10.5: Experimental system for grasp testing.
196
A small pretension of 1N was applied to each string to remove slack and prevent the
string from falling o of the pulleys. Then the Minkowski sum
2
of each vertex of both
feasible object force sets (described in the previous section) was applied to the strings (in
addition to the pretension) in ramp-up, hold, and ramp-down phases to nd the grasp
wrench set. The vertices of this experimental grasp wrench set were determined from
the hold phases and then used to nd the grasp quality. The experimental grasp quality
could then be compared with the theoretical grasp quality (from computational results).
10.4 Results
10.4.1 Computational grasp quality predictions
The 252 unoptimized routings produced the grasp qualities shown in Figure 10.6. The
optimization paths for the best N+1, N+2, and 2N designs, plus the naive 2N design,
are also shown. Each optimization step produced a higher tness, and iso-tness dashed
lines are shown.
We see that the naive 2N design improved the most from the optimization steps. This
would be expected since the other designs that were optimized already had the highest
tness from their base matrix. In fact, the naive 2N design improved its tness by 259%.
2
The Minkowski sum in this context refers to the combination of each vertex of one feasible object
force set with every vertex of the other feasible object force set
197
Max Tension (N)
10 0 20
Best N+1
Fitness = 1.54
Max Tension (N)
10 0 20
Best N+2
Fitness = 1.78
Max Tension (N)
10 0 20
Best 2N
Fitness =1.90
Max Tension (N)
10 0 20
Naive 2N
Fitness = 1.65
Optimized Designs
(b) (a)
0 0.5 1 1.5
0
0.5
1
1.5
Radius of Largest Ball, Posture 1 (N)
Radius of Largest Ball, Posture 2 (N)
Best N+1
Best N+2
Best 2N
Naive 2N
Unoptimized
Pulley−size optimized
Max tension optimized
Fitness = 0.5
Fitness = 1
Fitness = 1.5
Figure 10.6: (a) Computational results of grasp quality for hand designs. Optimization
paths shown. (b) Pulley-size and max tension optimized designs.
198
Unptimized
Naive 2N.
Fitness = 0.46
Max Tension (N)
10 0 20
259% increase in
grasp quality!
Max Tension (N)
10 0 20
Optimized
Naive 2N.
Fitness = 1.65
Figure 10.7: Computational predictions of tness for unoptimized and optimized naive
2N design.
We see the optimization for the naive 2N design made 6 out of the 12 moment arms
smaller and redistributed the maximal tensions severely, as seen in Figure 10.7.
Furthermore, we can see that the optimized N+1 design had a tness that was higher
than any of the unoptimized N+2 and 2N designs, even though it had fewer tendons.
Fewer tendons, in general, would be desirable due to simplication of the actuation system
and less complexity in design and manufacturing.
10.4.2 Theoretical predictions vs. experimental results
The experimental results are shown in Figure 10.8. We see that the data points lie under-
neath the parity line (if the theoretical and experimental grasp qualities were identical,
199
the data points would lie exactly on the parity line). As would be expected, none of
them are above the parity line. We ran a regression on the data points and we felt it
was reasonable to constrain the intercept to zero. The slope of the line was 0.70, which
indicated around a 30% loss in quality, on average, due to friction and other experimental
error. The coecient of determination is not well dened for regressions whose constants
are constrained to zero, but visually the t is fairly good, and the main point is that
in general, a prediction of higher (or at least much higher) grasp quality resulted in an
experimental result of higher grasp quality. We show the 3-D force portions of the grasp
wrench set in Figure 10.8 for two of the designs and we can see that the theoretical and
experimental 3-D force portions of the grasp wrench sets are very similar in size and
shape, with the experimental sets being contracted, mainly due to friction. We see that
the optimized best 2N design has a 223% greater experimental grasp quality than the
unoptimized naive 2N design in grasp 1.
10.5 Discussion
In this work, we have investigated the eect of tendon routings, pulley sizes, and distri-
bution of maximal tendon tensions on the grasp quality of a tendon-driven robotic hand
under design constraints. We see that altering all of these design parameters has a very
200
0 0.5 1 1.5
0
0.5
1
1.5
Theoretical Grasp Quality (N)
Experimental Grasp Quality (N)
Grasp 1
Grasp 2
Regression (Slope = 0.70)
Parity (Slope = 1)
Unoptimized Naive 2N Design
Grasp Wrench Set
Experimental Grasp Quality = 0.22 N
Optimized Best 2N Design
Grasp Wrench Set
Experimental Grasp Quality = 0.71 N
Theoretical
(blue)
Experimental
(green)
Figure 10.8: Results from experimental testing of various routings. Experimental vs.
theoretical grasp quality for both grasps. Parity line is where experimental grasp quality
would be exactly equal to theoretical grasp quality (intercept of 0, slope of 1). Regression
line constant term forced to zero. 3-D force portions of grasp wrench set for two dierent
tests shown on right (torque constrained to zero).
201
dramatic eect on grasp quality while still satisfying the constraints. We also see that
computational predictions can be useful when making design decisions.
We have used radius of largest ball as the grasp quality metric for our analyses since
no prior assumption of wrench direction specication was made. If a necessary wrench
or set of wrenches is known (Borst, Fischer & Hirzinger n.d.) (e.g., to pick up a heavy
object, pull on a cord, or turn a knob) then the analyses could assign a tness metric to
a routing based on that task specication, using a procedure similar to that used in (Fu
& Pollard 2006). The optimization could then be based on that metric. In addition, any
other grasp quality metric based on kinetostatic performance (such as the volume of the
grasp wrench set (Miller & Allen 1999)) could be used to optimize a tendon-driven hand
under constraints.
As in a previous study which utilized a single nger of the same design (Inouye
et al. n.d.), we only analyzed and constructed routings where the tendons routed around
every joint that they passed (i.e., that the structure matrix is pseudo-triangular, as in
(Lee & Tsai 1991)) and where there were only two sizes of pulleys that could be chosen.
Routings can be designed where tendons pass through the center of joints (Grebenstein
et al. 2010), or where moment arms can have many feasible magnitudes. This opens up
the design space even more, and exhaustive searches like the ones we performed in this
study may not be feasible. In addition, tendon-driven ngers or manipulators with more
than 3 degrees of freedom will tend to suer from the curse of dimensionality in the design
202
space, and a designer may have to use various optimization algorithms (Inouye et al. 2012)
in a search for a very good design which could then be designed for physical construction.
Alternatively, a designer could come up with a handful of feasible, physically-realizable
routings and then run these analyses to determine the best one.
Friction was a signicant factor in our experiment, especially for the tendons at the
last joint that had to wrap around as many as 12 pulleys. This high number of pulleys
was necessary to ensure total recongurability. Fingers for a commercial robotic hand
under constraints (such as a dexterous prosthetic hand) could use these computational
methods to design routings where very few pulleys are necessary, and hence the friction
could be reduced.
Moreover, the 30% reduction in grasp quality compared with predictions due to fric-
tional loss and other experimental error can partly be explained by the fact that the
grasp quality metric used was the \worst case scenario" (i.e., the magnitude of weakest
wrench). This would result in the grasp quality being lowered by one or more of many
sources of friction loss, small controller errors, or positioning errors of the load cell or
ngers, and be similar to a \max error" operator. The experimental volume of the grasp
wrench set, which can be viewed as an approximation of the average wrench that can be
produced by the ngers, is much more consistent with predictions, as shown in Figure
10.9. This could be viewed as an \average error" operator. We have plotted the volume
to the one-fth power to linearly scale the volume for inspection.
203
0 1 2 3
0
0.5
1
1.5
2
2.5
3
Theoretical Volume
(1/5)
Experimental Volume
(1/5)
Grasp 1
Grasp 2
Regression (Slope = 0.98)
Parity (Slope = 1)
Figure 10.9: Experimental vs. predicted volumes of grasp wrench sets.
We have investigated grasp quality in this paper, but there are many other consider-
ations that go into the design of a robotic hand. Other signicant considerations are the
eectiveness of control algorithms, passive stiness characteristics, sensitivity to friction
and positioning errors, and maximal nger velocities. We acknowledge that grasp quality
is only one piece of the design puzzle for optimized robotic hands.
This work is also easily extended to the case of underactuated hands. These hands
are simpler to design and control, and current users of prosthetic hands can only transfer
a few reliable signals to the hand. This limited control bandwidth currently restricts the
applicability of these results to prosthetic hands at the current stage of neural interface
204
development. However, as the number of reliable neural or cortical signals from amputees
increases, the market for dexterous prosthetic hands will increase dramatically.
This study also gives us insight into the anatomy of the human hand. The human
hand's tendon routing is very asymmetric, the distribution of maximal tendon tensions
is extremely skewed, and the magnitudes of the moment arms vary widely. In the index
nger, the smallest muscle is only 10% of the strength of the largest muscle, the magnitude
of the smallest moment arm is approximately 10% of that of the largest moment arm,
and it utilizes one less tendon than a 2N design (Valero-Cuevas et al. 2000). In fact, our
optimized results show that none of the optimized designs had much symmetry. We feel
that the biological inspiration from these parameter variances can be used in the design
of tendon-driven robotic systems to maximize performance.
Acknowledgements
The authors gratefully acknowledge the help of Dr. Jason Kutch and Manish Kurse in
providing the data acquisition routine for the experimental procedure, and also for their
helpful comments.
205
Chapter 11
Conclusions and Future Work
The central contribution of the work in this dissertation is providing and testing ad-
vanced computational methods for the systematic design and optimization of bio-inspired
tendon-driven systems. Some of the systems that can be analyzed and optimized with
the computational methods developed here include minimally-invasive surgical devices,
robotic and prosthetic hands, lightweight robotic arms, miniature robots, rescue robots,
and service robots. As the market for advanced, dexterous robotic and prosthetic de-
vices grow, optimization of cost, weight, and size will be important considerations in a
competitive commercial market.
In addition, in-depth analysis of the neuromuscular systems of the human body can
be performed using these methods to oer insight and inspiration for robotic mechanisms,
medical devices, and physical therapy. A thorough understanding of biomechanics and the
206
eects of various neuromuscular and cognitive disorders on normal function is necessary
to develop and implement new and eective treatment modalities.
207
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Abstract (if available)
Abstract
This thesis work focuses on the design and optimization of tendon-driven systems. One of the chief reasons these systems can be termed “bio-inspired” is due to the fact that they produce forces and movements via tendons which are connected to actuators and have uni-directional action (they can only pull, not push). In particular, the human hand sets itself apart from the rest of the body’s neuromuscular systems in that there are no muscles in the fingers themselves: all the muscles are proximal to the fingers. The study and analysis of tendon-driven hands and fingers suggest that i) the human hand is mechanically optimized for grasping capabilities and ii) use of bio-inspired principles in robotic systems can drastically improve force-production capabilities, grasp strength, and stiffness control performance. Some of the bio-inspired principles which are proven throughout this work to be very beneficial in robotic systems are asymmetry and skewed combinations of design parameters. This work not only has strong implications for the design of commercial robotic and prosthetic hands, but also broadly seeks to inspire creativity in design and optimization routines for complex problems in many different engineering disciplines.
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Asset Metadata
Creator
Inouye, Joshua M.
(author)
Core Title
Bio-inspired tendon-driven systems: computational analysis, optimization, and hardware implementation
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Biomedical Engineering
Publication Date
04/13/2012
Defense Date
02/14/2012
Publisher
Los Angeles, California
(original),
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
biomechanics,Biomedical Engineering,design optimization,grasping,mechanical design,Mechanical Engineering,multi-fingered hands,OAI-PMH Harvest,prosthetic hands,robotic finger design,robotic hands,robotics,stiffness control,tendon-driven systems
Language
English
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Electronically uploaded by the author
(provenance)
Advisor
Valero-Cuevas, Francisco J. (
committee chair
), Celikel, Tansu (
committee member
), Schaal, Stefan (
committee member
)
Creator Email
josh700@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-5708
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UC1109031
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usctheses-c3-5708 (legacy record id)
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etd-InouyeJosh-603.pdf
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5708
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Dissertation
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Inouye, Joshua M.
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texts
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(contributing entity),
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The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
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Tags
biomechanics
design optimization
grasping
mechanical design
multi-fingered hands
prosthetic hands
robotic finger design
robotic hands
robotics
stiffness control
tendon-driven systems