University of Southern California Dissertations and Theses

Development and control of biologically-inspired robots driven by artificial muscles

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Development and control of biologically-inspired robots driven by artificial muscles

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Content Development and Control of Biologically-Inspired Robots Driven by Articial Muscles by Joey Zaoyuan Ge 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 (Mechanical Engineering) May 2020 Copyright 2020 Joey Zaoyuan Ge To my parents ii Acknowledgements First and foremost, I would like to express my deep gratitude to my advisor Professor N estor O. P erez-Arancibia for being a great teacher and mentor to me. His thirst for knowledge, vision and strive for excellence have inspired me to work hard. In my research, he gave me a lot of freedom and provided invaluable advice, guidance, and critical remarks. I am also very grateful to Professor Henryk Flashner and Professor Heather Culbertson for serving on my dissertation as well as qualifying exam committees. My gratitude also goes to other members of my qualifying exam committee, Professor Satyandra K. Gupta and Professor Paul D. Ronney. I have beneted immensely from their advices on my research. Many thanks go to the Aerospace and Mechanical Engineering Department and USC for pro- viding not only the resources supporting my research but also the sense of a great community. The research presented in this dissertation has been funded by the Viterbi School of Engineer- ing, National Science Foundation (NSF) and the Defense Advanced Research Projects Agency (DARPA) through awards and contract awarded to my advisor. Their nancial supports are greatly acknowledged here. I could not have asked for better lab mates. Special thanks to Dr. Longlong Chang and Xiufeng Yang. The collaborations with them have brought forth invaluable solutions and ideas on topics ranging from dynamics modeling to micro-fabrication, which have contributed to the works presented in this manuscript. I would also like to thank Luiz F. Toledo, Juan M. Oxoby and Ariel A. Calder on for initiating me into the study of shape memory alloys and soft robotics. Besides, it has been a genuine pleasure sharing lab space, learning and experiencing the ups and downs iii of research with Dr. Ying Chen, Emma K. Singer, Xuan-Truc Nguyen and Coco Ke Xu. I will always cherish our bonding over the PhD marathon. I thank my friends James Croughan, William Li, Kwendy Lau, Rishabh Parekh, Samuel Wong and Gerry Linsheng Fan, who contributed most of my social life during my graduate school. I am forever grateful for your friendships. There are no words that can express my gratitude to the Capparellis and their extended families. I truly lucked out when you became my host family during my high school exchange program. I could not have asked for more than a family thousands miles away from home. I would like to thank my girlfriend Ivy, for the immeasurable joy and love you have brought me during the last stretch of graduate school. Finally and most importantly, I would like to thank my parents for their unconditional love and support. None of these works would have been possible without your encouragement and support. iv Table of Contents Dedication ii Acknowledgements iii List Of Tables viii List Of Figures ix Abstract xx Chapter 1: Introduction 1 1.1 Earthworm-Inspired Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 SMA-Based Articial Muscles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Articial Muscles High-Frequency Actuation . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 2: Earthworm-Inspired Soft Crawling Robot 13 2.1 Earthworm-Inspired Locomotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Design and Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Dynamic Modeling and Controllability Analysis . . . . . . . . . . . . . . . . . . . . 18 2.4 Locomotion Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5 Real-Time Locomotion Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 v 2.6 Experimental Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Chapter 3: Shape-Memory-Alloy (SMA) Based Articial Muscle 39 3.1 Characterization of SMA Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.1 Overview of SMA Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.2 SMA Wire Hysteresis Modeling Overview . . . . . . . . . . . . . . . . . . . 41 3.2 Preisach Model of SMA Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.1 Classic Preisach Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.2 Modied Preisach Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.3 Geometrical Interpretation of the Preisach Model . . . . . . . . . . . . . . . 46 3.2.4 The Wiping-Out and Congruency Property . . . . . . . . . . . . . . . . . . 47 3.2.5 Numerical Implementation of the Preisach Model . . . . . . . . . . . . . . . 48 3.3 Experimental Setup and Controller Design . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.2 Temepature Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.3 Apparatus and Temperature Controller Update . . . . . . . . . . . . . . . . 57 3.4 Hysteresis Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.5 Preisach Model Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.5.1 Method Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.5.2 Inversion Method Realization . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.6 Position Control Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.6.1 Position Control Simulations and Results . . . . . . . . . . . . . . . . . . . 70 3.6.2 Position Control Experiments and Results . . . . . . . . . . . . . . . . . . . 72 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 vi Chapter 4: Fast Actuation of SMA-Based Articial Muscles 82 4.1 Major and Minor Hysteresis Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Multiplier Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2.1 Displacement and Frequency Multiplication . . . . . . . . . . . . . . . . . . 84 4.2.2 Preliminary Multiplier Design . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2.3 Multiplier Experimental Results and Discussion . . . . . . . . . . . . . . . . 88 4.3 SMA Wire Based Bending Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.3.1 Actuator Design and Fabrication . . . . . . . . . . . . . . . . . . . . . . . . 92 4.3.2 Actuator Testing and Application . . . . . . . . . . . . . . . . . . . . . . . 94 Chapter 5: Conclusion and Direction of Future Work 98 5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.2 Direction of Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Reference List 101 Appendix A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 A.1 Actuation Method of the Earthworm-Inspired Crawling Robot . . . . . . . . . . . 114 A.2 Additional SMA Wire Position Control Results . . . . . . . . . . . . . . . . . . . . 116 A.3 SMA Wire Long-Duration Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 vii List Of Tables 2.1 Reference Pressures During Locomotion (kPa). . . . . . . . . . . . . . . . . . . . . 33 2.2 Experimental Results on Horizontal Surfaces. . . . . . . . . . . . . . . . . . . . . 36 3.1 Performance of Gain-Scheduled Controllers. . . . . . . . . . . . . . . . . . . . . . 57 3.2 Summary of position control results under constant stresses. . . . . . . . . . . . . 73 3.3 Position control results under time-varying stress. . . . . . . . . . . . . . . . . . . 77 3.4 Sample position control results neglecting time-varying stress (SP IV). . . . . . . 78 viii List Of Figures 1.1 The earthworm-inspired soft robot capable of bidirectional locomotion presented in this thesis. The system is composed of two hard casings, one central actuator and two extremal actuators constrained by o-rings, two machined steel plates and o-the-shelf pneumatic components. . . . . . . . . . . . . . . . . . 3 1.2 Conceptual crawling robot design. A catalyst-coated SMA wire is preloaded with a spring and heated through controlled catalytic combustion. As the SMA con- tracts, an array of slanted spikes is pulled forward while the robot remains station- ary. This is possible because the spikes conguration exploits friction anisotropy, resulting a signicantly smaller frictional force when it slides forward than back- ward. During cooling, the loaded spring facilitates the SMA's phase transition from austenite to martensite, causing the wire to extend. Meanwhile, the anisotropic fric- tion in this case anchors the array of spikes to the ground and thus pushing the robot forward. The sharp arrows indicate the direction of motion of the compo- nents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Major and minor SMA hysteresis loops, shown in red and blue respec- tively. This set of temperature-strain curves were obtained from a 127 m thick SMA wire with a nominal phase transition temperature of 90 C under a constant stress of 155 MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 (a) Illustration of a nightcrawler's metamere unit: A metamere expands ra- dially when its longitudinal muscles contract (circular muscles relax) and expands longitudinally when its circular muscles contract (longitudinal muscles relax). The four pairs of setae on its ventral and lateral surface will protrude and retract accord- ingly with longitudinal muscles' contraction and relaxation, respectively, to provide variable tractions that enable locomotion. (b) Peristaltic crawling kinemat- ics: Following the convention in [1], we dene a stride as a complete cycle of peri- stalsis and describe the crawling kinematics as a function of four variables: stride length, protrusion time, stance time and stride period. . . . . . . . . . . . . . . . . 14 ix 2.2 (a) Fabrication of the extremal actuators. In Step 1, liquid silicone is poured into a 3D-printed mold; then, the lower half of a symmetrical double-cylindrical core is submerged in the silicone. The silicone within the mold is then cured at 65 C for 15 min. In Step 2, the cured silicone is released before liquid silicone is added to cast the other half of the shell (Step 3 and Step 4). Step 5 shows the complete shell structure after being peeled o from the 3D-printed core. In Step 6, o-rings are tted onto the shell's imprinted grooves, and a layer composed of silicone and a berglass net is employed to seal one end of the shell. Step 7 shows a extremal actuator once the fabrication process is completed. (b) & (c) Fabrication of the central actuator and connecting modules. These procedures are identical to those employed in (a) with the exception that the connecting modules do not require an additional reinforcing layer and o-rings. Additionally, an orice is perforated from the bottom of both connecting modules to allow the ow of air into both of extremal actuators. (d) Final assembly. In Step 1, the two extremal actuators, the central actuator, two connecting modules and three air-supplying lines are integrated together. In Step 2, a pair of 3D-printed casings are xed over the connecting modules of both extremal actuators. Step 3 depicts the cross-sectional view of the robot in two states: all the actuators in ated and all the actuators de ated. Finally, Step 4 shows the robot after the fabrication process of the entire robot is completed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Reduced-complexity double-mass-spring-damper model of the robot in Fig. 1.1. The associated mathematical description of this model is employed to study the controllability of the robot in the absence and presence of forces f 1 and f 2 . To model friction, the values of f 1 and f 2 are considered to be positive when the associated force vectors act in the same direction as that of i; consistently, the values of f 1 and f 2 are considered to be negative when the force vectors act in the opposite direction as that of i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 (a) Block diagram of the discrete-time model employed in the numerical simulations. The plant ^ G d (z) =C d (zIA d ) 1 B 1d +D d is the discretized version of ^ G(s) =C(sIA) 1 B 1 +D. The friction function block computes the friction forcesf 1 [n] andf 2 [n] using the algorithm in (2.5) and the inputsu 1 [n],u 2 [n],f a [n] andx[n]. Zero initial conditions are set to start the simulations. (b) Example in- put signals. In this case, the central actuation force,f a , is modeled as a sinusoidal wave with its magnitude and bias consistent with the empirically-estimated inter- nal pressures of the central actuator during periodic operation (see Appendix A.1). Specically,f a is set to oscillate between 0:66 N and 19:9 N. The indicator function u i [n], for i = 1; 2, switches the friction of the corresponding actuator i from high (u i [n] = 1) to low (u i [n] = 0), and vice versa. The frequencies of all the inputs are set to 1 Hz (2 rad s 1 ) and the phase dierence between the extremal actuators is set to = 1:64 rad (equivalent to 0:82 s) . . . . . . . . . . . . . . . . . . . . . . 26 x 2.5 (a) & (b) Time-series of the robot's state and associated close-ups. Po- sitions and velocities of the two extremal actuators when the input frequencies and are set to 1 Hz and 1:64 rad, respectively. The displacement of the cen- ter of mass x CM is also shown in magenta, calculated post-simulation according to (2.4). (c) Simulated displacement of block 1. This 3D-plot shows the position reached by the mass m 1 after 60 s of locomotion when the constant frequencies of the actuation force (f a freq.) and friction input (f i freq., fori = 1; 2) are taken from the interval [0:02 : 2] rad s 1 while is maintained at 1:64 rad. (d) Exper- imental relationship between displacement and phase dierence of the extremal actuators. The simulated positionsx 1 andx 2 after 60 s are plotted over the interval [0 : 2] rad. All the input frequencies are set to 2 rad s 1 while the phase dierence is taken from the interval [0 : 2] rad andm 1 =m 2 = 0:2 kg. 29 2.6 Actuation sequence employed to generate forward locomotion on a at surface. Here, red indicates in ation and gray de ation; the 3D-printed casings are shown in sectional views so that the state of each actuator can be clearly seen in each phase of the locomotion sequence. When both extremal actuators are de ated, the smooth 3D-printed casings are in direct contact with the supporting surface and support the entire weight of the robot. To crawl forward, in Phase 1, the back actuator is in ated and anchors to the ground; in Phase 2, the central actuator is in ated to expand forward while the back actuator remains anchored to the ground; in Phase 3, the front actuator is in ated and anchors to the ground; nally, in Phase 4, both the back and central actuators are de ated and contract in order to complete a locomotion cycle. . . . . . . . . . . . . . . . . . . . . . . . 32 2.7 Pressure signals of the robot's actuators during a forward locomotion test. From the upper left (in red) to the bottom left (in blue) to the upper right (in black) the plots show the references (F-RP, C-RP and B-RP) and controlled outputs (F-P, C-P and B-P) of the front, central and back actuators. In this case, the protrusion time is 1:2 s, the stance time is 1:8 s and corresponding stride time is 3 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.8 Photographic sequence composed of movie stills showing the soft robot crawling bidirectionally on a at HDPE surface. During this test, the trav- eling direction is reversed every 32:5 s and the forward and backward locomo- tion is implemented employing the actuator pressure references listed in Table 2.1. The complete experiment can be found in the supplementary movie softRobot.mp4 (http://www.uscamsl.com/resources/gethesis2020/softRobot.mp4). . . . . . . . . 35 2.9 (a) Trajectories of the extremal actuators during the bidirectional loco- motion test. The time-series corresponding to the back block is shown in red; the time-series corresponding to the front block is shown in blue. The direction of locomotion is reversed every 32:5 s; a zoomed-in view of one of the controlled reversals (from 56 s to 72 s) is shown in the superposed window. (b) Trajecto- ries of the extremal actuators of the robot while climbing on the HDPE surface held at a 7:5 angle with respect to the horizontal. The time-series corresponding to the back block is shown in red; the time-series corresponding to the front block is shown in blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 xi 2.10 Photographic sequence composed of movie stills showing the soft robot climbing on an inclined HDPE surface positioned at an angle of 7:5 with respect to the horizontal. A cup containing dyed water indicates the direction of the gravitational force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.1 Graphic illustration of the constitution of SMA's hysteresis. Once SMA is consisted of pure Austenite, hysteresis in its strain is known as pseudoelasticity, which is caused by variation in loading condition. When no loading condition is applied, hysteresis behavior resulted from heating and cooling is known to be the shape memory eect (SME). However, as pointed out in the diagram, to transfer from a purely Austenitic state to a purely detwinned Martensitic state, loading is required in addition to cooling. Otherwise, the SMA wire can only reach the state in which its crystal structure is known as the twinned Martensite. Evidently, SMA's hysteretic behavior is usually a combined outcome of pseudoelasticity and SME. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Temperature-strain hysteresis loops under dierent stresses. A 127 m thick SMA wire with a nominal phase transition temperature of 90 C is heated to follow an identical temperature prole bounded between 30 C and 130 C under 5 dierent constant stresses: 105; 115; 125; 135 and 145 MPa. The corresponding temperature-strain hysteresis loops, shown in red, blue, green, magenta and cyan, respectively, exhibit considerable disparities. . . . . . . . . . . . . . . . . . . . . . 41 3.3 Hysteron operator. ^ outputs 1 once the increasing input U surpasses and starts to output 0 when the decreasing input reaches . . . . . . . . . . . . . . . . 44 3.4 The graphic description of the modied Preisach model. In case of SMA wires, the total input u consists both temperature (T ) and loading ((t)) informa- tion since both contribute to the deformation of an SMA wire. The stress value is re ected in the weighting function (;;) while the hysteron is only a function of the temperature input T .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.5 Geometrical interpretation of the Preisach memory mechanism (a), (b), (c) The parameters 0 and 0 are the upper and lower saturation temperatures determined empirically, respectively. In (a), the temperatureT is rst raised from 0 to 1 , before being reduced to 1 in (b). Successive heating and cooling will create a staircase-shaped L(t), as illustrated using bold line in (b) and (c). In all three gures, S + , the region within which the hysterons switched `on' is shown in shade; correspondingly, the region where the hysterons are turned `o' is labeled using S 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.6 (a) Wiping-out property. At t = t 1 and t = t 2 , all past local maxima and minima are erased. For a current local maximum or minimum , all previous local maxima k or minimum k , where k < or k >, respectively, are wiped out. (b) Congruency property. All minor loops dened by the same local minimum and maximum bounds, [;], are congruent. . . . . . . . . . . . . . . . 48 xii 3.7 (a) & (b) Graphical interpretation of the rst order reversal curve (FORC) and its associated strain increment. (a) demonstrates the con- struction of the FORC on the Preisach plane while (b) is a temperature-strain diagram that illustrates the corresponding strain output of the same FORC. The shaded ( 0 ; 0 ) is the region within which the hysterons are switched `o' after the temperature reversal process. Integrating weighted hysterons over this region will lead to the strain increment value F ( 0 ; 0 ;). Consistently, S + is the area within which the hysterons remained `on' and L(t) the bold line is the interface resulted from the temperature reversal. . . . . . . . . . . . . . . . . . . . . . . . . 49 3.8 Numerical implementation of the Preisach model. In the case shown here, T is the monotonically increasing at time t and the last link of the staircase- shaped interface L(t), shown as the bold line is horizontal. Specically, T (t) = n . Accordingly, S + is divided into n 1 trapezoids and 1 triangle, ( n ; n1 ). Each trapezoid is equivalent to the dierence between its two adjacent triangles, as exemplied by the shaded Q k , which equals to the dierence between ( k ; k1 ) and ( k ; k ). If T is monotonically decreasing at time t, the S + region can then be divided to n trapezoids. In both cases, n is the number of local temperature extrema recorded by T (t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.9 (a) CAD model and (b) picture of the experimental setup. The SMA wire is electrically heated and the temperature is measured by thermocouples, as high- lighted by the red boxes in (a) and (b), and controlled using the adaptive scheme presented in Section 3.3.2. The stress is applied, measured, and controlled using an electromagnet, a load cell, and a PID controller, respectively, as highlighted by the blue boxes in (a) and (b). The contraction due to phase transformations is measured by a laser sensor, shown on the right in (a) and (b). . . . . . . . . . . . 53 3.10 Temperature controller eect. Temperature tracking performance comparison between the initial setup, composed of a PID controller with one type K thermocou- ple, and the nal setup, consisting of the adaptive controller presented in Section 3.3.2, three type T thermocouples, and electrically-insulating epoxy. . . . . . . . . 54 3.11 (a) Adaptive control scheme for temperature tracking. P is the open-loop thermodynamics of the SMA wire,y r andy are the desired and true temperatures, respectively, and w is a disturbance modeling the combined eect from the phase transformation nonlinearities, input, output, and sensor disturbances. (b) PID controller tuning scheme. For the online tuning ofK i , the future adaptive con- trol command u is set to zero. The closed-loop transfer functions M i , G i , and S i are dened so that y =M i y i;r +G i u +S i w and the closed-loop disturbance w i is dened as w i = S i w. (c) Filter tuning Control scheme showing (i) the imple- mentation of a FIR lter F i and (ii) the adaptation of F i using a RLS algorithm, where the RMSE of y is minimized . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.12 (a), (b) and (c) Results of the rate-independency, wiping-out, and con- gruency experiments, respectively. All of these tests were conducted when a constant stress of 150 MPa was maintained at wire using the electromagnet mecha- nism and corresponding stress controller. The input reference (RT) and true (TT) temperatures are shown on the left while the resulting strain is shown on the right. 58 xiii 3.13 Adaptive gain-scheduling scheme for the control of the SMA wire's tem- perature. Here, T r (t d ) is the discrete-time temperature reference; T m (t d ) is the sampled measured temperature; each lter K i ; i = 1 21, composing the adap- tive controller has a PID structure; and the blocks D/A and A/D denote digital- to-analog and analog-to-digital conversion, respectively. In this conguration, the discrete time t d coincides exactly with the continuous time t at each sampling in- stant; therefore, for purposes of analyzing the main inverse scheme, we assume t d =t. During operation, the lters K i ; i = 1 21, are gain-scheduled according to the values of T r , from 30 C to 130 C at 5 C intervals. . . . . . . . . . . . . . . 59 3.14 Sample measurement of an output increment along a FORC from a set of system identication experiments: F(75 C,50 C,150 MPa). Consistent with Fig. 3.12, the input reference (RT) and true (TT) temperatures are shown on the left; the resulting strain is shown on the right. Stress in this case is held constant at 150 MPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.15 (a) Example experimentally obtained mesh surface of FORC strain in- crement values. (b) Set of identied output increment surfaces. Specif- ically, these are F (;; m ), for m 2f100; 125; 150; 175; 200 MPag. The results presented here were obtained on the 150 m SMA wire by employing the heating proles described in Fig. 3.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.16 Results of the system validation experiments for (a) static and (b) dy- namic loading conditions. The input reference temperature (RT) and stress (RS) and the true temperature (TT) and stress (TS) are shown on the left. The true strains are calculated using the measured displacements and plotted together with the Preisach model calculated strain. . . . . . . . . . . . . . . . . . . . . . . 61 3.17 Example FORC increment surface generation procedure. (a) Sample heat- ing prole used in the identication process. Here, the SMA wire is heated between 30 C and 130 C following a sinusoidal curve under a constant stress of 145 MPa. The almost perfect matching between reference temperature (RT) and measured temperature (MT) validates the eectiveness of both the hardware setup and tem- perature control method. The strain measured is shown in blue. (b) The corre- sponding FORC strain increments extracted/interpolated from the data collected in (a). (c) Experimentally-identied discrete surface of FORC strain increments over a square-gridded mesh on the Preisach plane. . . . . . . . . . . . . . . . . . . 62 3.18 Set of experimentally identied FORC strain increment surfaces. F ( 0 ; 0 ; m ), for m = 105; 115; 125; 135; 145 MPa. Note that the results shown here are collected from the 127 C thick SMA wire with a nominal phase transition temperature of 90 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.19 Inversion bases. (a) -inverse basis and (b) -inverse basis for m = 145 MPa. (c) The set of discrete-inverse bases and (d)-inverse bases formulated at m = 105; 115; 125; 135; 145 MPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 xiv 3.20 Block diagram illustrating the feedforward position control structure used in real-time simulation. is the Preiscah hysteresis model identied in Section 3.4. The reference temperature T r obtained through the inverse process together with the time-varying reference stress r are the input signals evaluated by to generate the simulated strain output s . . . . . . . . . . . . . . . . . . . . 70 3.21 Real-time simulation results of purposed SMA position control scheme. (a) Simulation results for tracking a sinusoidal-shaped reference strain under a sinusoidal-shaped stress prole. (b) Simulation results for tracking a strain se- quence that oscillates between randomly selected peaks and troughs while subject to a random stress prole. In both cases, reference strain (RS) is shown in blue and simulated strain (SS) is shown in red in Part I, while the reference loading (RL) and simulated temperature (ST) are presented in black and magenta in Part II and III, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.22 Real-time simulation results of purposed SMA position control scheme under constant stress. (a) Simulation results for a sinusoidal-shaped reference strain signal. (b) Simulation results for a randomly generated reference strain signal. In both (a) and (b), the reference and simulated strain signals are shown in blue, and the simulated temperature signal is shown in red. Note that the identied hysteresis model of a dierent wire (same dimension and nominal phase transition temperature) was used in these simulations. . . . . . . . . . . . . . . . . . . . . . 72 3.23 Sample position control under constant stress results. Here, SP III (am- plifying sinusoidal signal) is tracked as the reference strain when a constant stress of 135 MPa is maintained on the wire. The reference and measured strains, RS and MS, shown in blue and red, are presented in Part I; the reference and mea- sured loadings, RL and ML, shown in green and black, are demonstrated in Part II; and the simulated and measured temperatures, ST and MT, shown in cyan and magenta, are displayed in Part III. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.24 Sample position control under time-varying stresses results. Here, SP IV is selected as the reference strain to be tracked under 6 dierent LPs. Experimental results corresponding to each of the 6 LPs are displayed in (a) through (f). Con- sistent with Fig. 3.23, the strain, stress and temperature tracking performances are presented in Part I, II and III, respectively in each subgure. Similarly, RS and RL, marked using black circles, are the reference strain and loading signals; MS1, ML1, ST1 and MT1, shown in red, are the measured strain and loading, simulated and measured temperature recorded in the experiments. To further demonstrate the necessity and ecacy of the proposed position control method in adjusting for the dynamic stress, we performed comparative tests in which we attempted to track the same SP while overlooking the loading condition; specically in these cases, temperature output was computed solely based on RS. Corresponding re- sults are shown in blue and labeled as MS2, ML2, ST2 and MT2 in each subgure. Note that in these example experiments, the FORC strain increment surface and inverse bases extracted at the constant stress 145 MPa were utilized to compute temperature output, ST2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 xv 4.1 From top to bottom, the temperature and displacement histories of an electrically-heated SMA wire led to cool naturally, respectively. A bang- bang controller was implemented to oscillate the temperature of the wire between 77 C and 80 C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2 Idealized illustration of the proposed the multiplier mechanism. (a) A SMA wire measures l 0 in length in the fully autenitic state when heated. As it is cooled to room temperature, it recovers to the fully martensitic state, producing a maximum instantaneous displacement of 1 (t). (b) Compound SMA mechanism with an actuation wire of lengthnl 0 , which is capable of producing a displacement of n 1 (t) or nl 0 1 (t) when heated within its major hysteresis loop, multiplying the amplitude of displacement attainable in (a) by a factor of n. . . . . . . . . . . . . 85 4.3 (a) The CAD drawing of over experimental setup. The multiplier design is highlighted in a red dashed line box and can be better observed in closed-up views in (b) and (c). (b) is a CAD rendering while (c) shows the actual apparatus. (d) oers an overview of the complete setup together with associated power electronics. 88 4.4 (a), (b) & (c) Displacement magnication verication of the multiplier design. (a) The temperature tracking (Part I), displacement (Part II) as well as the temperature-strain hysteresis loop (Part III) results of the multiplier (shown in red) and the short wire (shown in blue) undergoing the exact same major loop heating sequence (between 30 and 130 C). (b) & (c) The displacement, tem- perature tracking, as well as the temperature-strain hysteresis loop results of the multiplier and the short wire undergoing the same major and minor loop heating sequences. The minor loop temperature bandwidths are chosen to be [50 : 100] C and [60 : 90] C for cases shown in (b) and (c), respectively. . . . . . . . . . . . . 90 4.5 (a) & (b) Temperature and deformation measurements of the multiplier and the short wire when heated between 75 and 90 C adopting the bang-bang control scheme. The switching temperatures are marked in black and blue in the temperature prole plot and the actual measurement is shown in red. Additionally, zoomed-in views of segments enclosed within dashed lines are displayed accordingly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.6 SMA wire displacement, temperature measurements and their respec- tive zoomed-in views when the heating pulse period is chosen to be 0:05 s (20 Hz). In this case, the heating pulse has an amplitude of 13:125 V and duty cycle of 15%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.7 (a) & (b) CAD rendering of the bending actuators. A still actuator is shown in (a) and a bent actuator is shown in (b). Note that the magnitude of beam de ection shown in (b) is for illustration purpose only. (c) & (d) Image of a fabricated actuator prototype. The caps and central beam are made of FR4 and carbon ber, respectively. In this iteration, the distance between the two caps is set to be 10 mm and the two wires are spaced 1:7 mm apart from the central beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 xvi 4.8 A quadruped crawling robot employing the described bending actuator. (a) & (b) CAD rendering of the proposed robot shown in stationary and mo- tion, respectively. (c) CAD rendering of the foot design stemming from the top of the bending actuator. A slanted hook is introduced at the tip of foot to induce anisotropic friction. (d) The leg in bending conguration. Consistently, the mag- nitude of beam de ection shown is for illustration purpose only. (e) The image of a manufactured robot prototype before wiring. . . . . . . . . . . . . . . . . . . . . 94 4.9 Testing setup for the bending-actuator-based leg. The laser mea- sures the de ection of a spot 17 mm from the base of actuator, which is held stationary onto a 3D-printed frame. Video recording of this test series can be found in the supplementary movie bending actuator.mp4 (http://www.uscamsl.com/resources/gethesis2020/bending actuator.mp4). . . . . 95 4.10 (a) & (b) Tip de ection recorded when only either one of of two SMA wires was excited. (c) Tip de ection recorded when both SMA wire were heated alternately. (d) Temperature measurement of one SMA wire during the test when both SMA wires are excited alternately. In all cases presented, the excitation pulse signal employed had a frequency, amplitude and duty cycle of 4 Hz, 5 V and 5%, respectively. When alternating heating was enabled, the excitation signal was set apart with a 0:125 s delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.11 Movie stills of a testing gait realized on the crawler pro- totype. Cartoons demonstrating the desired gait are also presented. Footage of the test can be found in the supplementary movie crawler.mp4 (http://www.uscamsl.com/resources/gethesis2020/crawler.mp4). . . . . . . . . . . 97 A.1 LTI scheme used to independently control each actuator (i = 1; 2; 3) of the soft robot. Here, the discrete-time plant P i (z) represents the entire SISO dynamics of the actuator i, including those associated with the corresponding in- atable silicone-rubber structure, pressure sensor, solenoid valve and DAQ board. The input to P i (z) is the duty cycle of the PWM signal that opens and closes the corresponding solenoid valve; p m;i is the measured internal air pressure; p r;i is the reference pressure; and ^ K i (z) is the PID controller that controls the internal air pressure of the corresponding actuator i. . . . . . . . . . . . . . . . . . . . . . . . 115 A.2 Results of SMA wire position control experiments under a constant stress of 105 MPa. In (a), (b), (c), and (d), the reference strain (RS) to be tracked are SPI, II, III and IV, respectively. . . . . . . . . . . . . . . . . . . . . 117 A.3 Results of SMA wire position control experiments under a constant stress of 115 MPa. In (a), (b), (c), and (d), the RS to be tracked are SPI, II, III and IV, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 A.4 Results of SMA wire position control experiments under a constant stress of 125 MPa. In (a), (b), (c), and (d), the RS to be tracked are SPI, II, III and IV, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 A.5 Results of SMA wire position control experiments under a constant stress of 135 MPa. In (a), (b), (c), and (d), the RS to be tracked are SPI, II, III and IV, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 xvii A.6 Results of SMA wire position control experiments under a constant stress of 145 MPa. In (a), (b), (c), and (d), the RS to be tracked are SPI, II, III and IV, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 A.7 Results of SMA wire position control experiments under time-varying stress and SP I is the RS. In (a), (b), (c), (d), (e), and (f), the reference loadings (RLs) introduced are LP I, II, III, IV, V and VI, respectively. . . . . . . 120 A.8 Results of SMA wire position control experiments under time-varying stress and SP II is the RS. In (a), (b), (c), (d), (e), and (f), the RLs introduced are LP I, II, III, IV, V and VI, respectively. . . . . . . . . . . . . . . . 121 A.9 Results of SMA wire position control experiments under time-varying stress and SP III is the RS. In (a), (b), (c), (d), (e), and (f), the RLs introduced are LP I, II, III, IV, V and VI, respectively. . . . . . . . . . . . . . . . 122 A.10 SMA wire position control results when time-varying stress is neglected I. SP IV is the sole input evaluated to calculate the simulated temperature 2 (ST2), utilizing the FORC strain increment and inverse bases procured at 105 MPa. Consistently, In (a), (b), (c), (d), (e), and (f), the RLs introduced are LP I, II, III, IV, V and VI, respectively. measured strain 2 (MS2), measured loading 2 (ML2) and measured temperature 2 (MT2), shown in blue, are the strain, loading and temperature measured in each case. Moreover, following the same convention of Fig. 3.24, as a comparison, the results from experiments where dynamic loading is adjusted for are also exhibited here, labeled as MS1, ML1, ST1 and MT1, shown in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 A.11 SMA wire position control results when time-varying stress is ne- glected II. SP IV is the sole input evaluated to calculate ST2, utilizing the FORC strain increment and inverse bases procured at 115 MPa. Consistently, In (a), (b), (c), (d), (e), and (f), the RLs introduced are LP I, II, III, IV, V and VI, respectively. MS2, ML2 and MT2, shown in blue, are the strain, loading and temperature measured in each case. Moreover, following the same convention of Fig. 3.24, as a comparison, the results from experiments where dynamic loading is adjusted for are also exhibited here, labeled as MS1, ML1, ST1 and MT1, shown in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 A.12 SMA wire position control results when time-varying stress is neglected III. SP IV is the sole input evaluated to calculate ST2, utilizing the FORC strain increment and inverse bases procured at 125 MPa. Consistently, In (a), (b), (c), (d), (e), and (f), the RLs introduced are LP I, II, III, IV, V and VI, respectively. MS2, ML2 and MT2, shown in blue, are the strain, loading and temperature measured in each case. Moreover, following the same convention of Fig. 3.24, as a comparison, the results from experiments where dynamic loading is adjusted for are also exhibited here, labeled as MS1, ML1, ST1 and MT1, shown in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 xviii A.13 SMA wire position control results when time-varying stress is neglected IV. SP IV is the sole input evaluated to calculate ST2, utilizing the FORC strain increment and inverse bases procured at 135 MPa. Consistently, In (a), (b), (c), (d), (e), and (f), the RLs introduced are LP I, II, III, IV, V and VI, respectively. MS2, ML2 and MT2, shown in blue, are the strain, loading and temperature measured in each case. Moreover, following the same convention of Fig. 3.24, as a comparison, the results from experiments where dynamic loading is adjusted for are also exhibited here, labeled as MS1, ML1, ST1 and MT1, shown in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 A.14 Preliminary SMA wire long-duration test results. The temperature-strain major hysteresis loops obtained at constant stresses 105; 115; 125; 135; 145 MPa, are presented in (a), (b), (c), (d) and (e), respectively. In all subgures, the results of the rst 13 sets of tests, T1 through T13 are shown on the left side while the results of the second 13 sets of tests (including T13), T13 through T25, are presented on the right side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 xix Abstract For decades, researchers have been creating robots that can mimic the locomotion patterns or functional movements of animals. However, most of these biologically-inspired robots are driven by conventional actuation technologies including combustion engines and electric motors, which usually require additional complex transmission systems. Such deciencies are further exposed in the development of small size (mm to cm scale) robots of which the spacing and loading capacities are strictly limited. Additionally, these mechanisms can potentially be hazardous in applications where human-machine interactions are involved. Intuitively, `muscle-like' actuation technologies are highly desirable in these situations and have garnered immense research attentions. In this work, we present our research progresses made on the study on two types of articial muscle designs. We rst created a pneumatically-driven earthworm-inspired soft robot capable of bidirectional locomotion on both horizontal and inclined platforms. In this approach, the locomo- tion patterns are controlled by actively varying the coecients of friction between the contacting surfaces of the robot and the supporting platform, thus emulating the limbless locomotion of earthworms at a conceptual level. Using the notion of controllable subspace, we show that fric- tion plays an indispensable role in the generation and control of locomotion in robots of this type. Based on this analysis, we introduce a simulation-based method for synthesizing and implement- ing feedback control schemes that enable the robot to generate forward and backward locomotion. From the set of feasible control strategies studied in simulation, we adopt a friction-modulation- based feedback control algorithm which is implementable in real time and compatible with the hardware limitations of the robotic system. xx Another type of articial muscle of interest is shape-memory-alloy (SMA)-based actuators. SMA wires are known for their unmatched power-to-weight ratio. Under tension, they can generate repeatable extension and contraction characterized by nonlinear hysteresis when subject to cyclic heatings according to the shape-memory eect (SME). Empirical evidence has also shown that such hysteretic deformation is also a function of the loading condition. Achieving accurate position control of an SMA wire thus requires the hysteresis phenomenon in the presence of time-varying stress to be modeled and compensated for. To this end, we create a Preisach-model-based a priori forward mapping from time varying temperature and stress inputs to the strain output through a series of system identication processes. The identied parameters are then used in a Preisach-model-inversion algorithm which is integrated into a feedforward controller designed to mitigate the hysteresis eect. The proposed deformation tracking approach employs light-weight sensing apparatuses and has proven to be computationally ecient; both features are well-suited for small-size autonomous robots. Furthermore, the temperature-stress-strain mapping procured in this study can be easily utilized to modulate the deformations of SMA actuators powered by means other than joule heating, for example, catalytic reaction. One of the deciencies of SMA-based actuators is known to be their relatively slow actuation speed. To address this issue, we propose to excite these articial muscles to operate within selected hysteresis minor loops based on hysteresis characterization results attained earlier. This leads to signicant reduction in heating/cooling time at the cost of disproportionally minor decrease in the actuator displacement output. A displacement and frequency multiplier mechanism is also devised to be employed jointly with the minor loop excitation method. In addition, a fast bending actuator driven by SMA wires is developed. In this case, at high frequency, limited displacements of SMA wire are exploited to induce substantially greater deformation at the coupling structure. A quadruped crawling robot prototype employing actuators of this type is designed and manufac- tured accordingly. Preliminary test results have proven the eectiveness of these actuation speed improvement methods. xxi Chapter 1 Introduction Conventional actuation mechanisms, including the electric motor family, hydraulic cylinders and solenoids, have already evolved into reliable and indispensable driving forces behind people's daily lives as well as the society's everyday production. Consistently, these actuation technologies have been successfully integrated into almost all of the most advanced robots [2, 3, 4]. Nevertheless, the search for novel and more specialized actuation technologies has never ceased. Amongst the new technologies, more and more emphases have been placed onto downsizing the mechanisms so they can be integrated into small (centimeter to millimeter scales) robots and devices. Furthermore, areas including medical practices and search/rescue missions where human-machine interactions are becoming more common and tasks have to be carried out in complex and hazardous environ- ment, the employed robots/devices are envisioned to be able to adapt to their surroundings, while precisely executing their assignments without causing undesirable damages. Silicone-based soft robots [5] (usually driven pneumatically) as well as Shape-memory-alloy-(SMA)-based actuators have emerged to becoming promising options [6]. Naturally, nding eective and ecient meth- ods to control and optimize the performances of these technologies have garnered considerable research interests. Here, in this manuscript, we present our ndings and results from exploring these topics. 1 1.1 Earthworm-Inspired Robot Limbless locomotion has long been a source of inspiration for robotic research as it appears to be the most eective method of traveling on unstructured terrains [7, 8, 9]. One of the most stud- ied limbless gaits is peristalsis-based, typically observed in earthworms (Lumbricidae). Within the earthworm family, Lumbricus terrestris, also known as the nightcrawlers, gained such name since they typically remain underground during day time but crawl out onto the ground to feed at night [1, 10]. As a result of such behavior, nightcrawlers are capable of switching between burrowing and crawling while maneuvering underground and covering complex terrains, respec- tively. In both cases, the eective gaits are peristalsis-based, produced by the coordinated and successive contraction and relaxation of the longitudinal and circular muscles embedded within nightcrawlers' metameres (segmented body parts) [11]. This repeating pattern can be perceived as retrograde waves propagating along the nightcrawler's body opposite to its traveling direction, providing the thrust that propels the worm either forward or backward. This thrust is then trans- lated to locomotion through traction with the ground, modulated by nightcrawlers' microscopic bristle-like skin structures known as setae [1, 12]. Numerous research projects have focused on creating robots that can replicate earthworms' peristalsis-based locomotion, adopting a variety of dierent actuation technologies, including elec- tric motors [13, 14, 15], magnetic uids [16] and SMAs [17, 18, 19]. In addition, structures that incorporate origami-inspired folding patterns have proven to be a feasible basis for mechanical actuators as specic folding patterns inherently allow for large bidirectional deformations. Cou- pled with actuation mechanisms, structures of this type are shown to be capable of mimicking the motions of earthworms' circular and longitudinal muscles [20, 21]. Furthermore, other recent innovations in fabrication and sensing technologies have enabled the development of biologically- inspired soft actuators, soft sensors and exible electronics [22, 23, 24]. 2 Fig. 1.1: The earthworm-inspired soft robot capable of bidirectional locomotion pre- sented in this thesis. The system is composed of two hard casings, one central actuator and two extremal actuators constrained by o-rings, two machined steel plates and o-the-shelf pneumatic components. Here, employing these technologies, we introduced a new modular soft robot, of which the conceptual design was inspired by the functionalities of earthworms' muscle and setae structures. Specically, the robot consists of two 3D-printed casings, three air lines, and one central and two extremal pneumatic actuators as shown in Fig. 1.1. In this design, in order to produce the peristalsis-based crawling motion, the axial longitudinal actuator in ates and de ates to generate the deformation and force comparable to that of earthworms' longitudinal muscles. Meanwhile, the two pneumatic extremal actuators, serving a function similar to that of earthworms' circular muscles, are coupled with the casings. Together, such mechanisms modulate the frictional forces needed to alternately anchor the robot's extremes to the ground, which is deterministic to the crawling motion of robots of this type. Almost all forms of animal and robot terrestrial locomotion rely on friction or friction-induced traction [25, 26]. For example, frictional anisotropy is the underlying principle for many animal gaits, including serpentine slithering [27]. The geometric pattern in snakeskin leads to frictional forces with dierent magnitudes when measured in dierent directions, subsequently allowing snakes to travel in the orientation leading to low frictional forces. Drawing inspiration from nature, researchers have developed a class of robots that exploits frictional anisotropy to travel [20, 28, 29, 30]. However, for most of these robots, frictional force is a passive mechanism as it is not 3 modulated actively to realize locomotion. Moreover, anisotropic-friction-induced locomotions are inherently unidirectional, limiting the animals and their inspired robots to travel only in xed orientations. In the context of soft robots, it is easy to see that bidirectional locomotion has the obvious advantage of agility over unidirectional locomotion as it no longer requires the robots to re-orient themselves in order to travel backwards. Nighcrawlers are capable of bidirectionally crawling because their protruded setae can anchor one segment against the pulling forces generated by the deformations of adjacent segments on either side [12]. In attempt to replicate this feature on the proposed robot, we introduced a simple but reliable friction control mechanism. Specically, the robot can actively modulate its frictional force by switching its contacting surfaces, made of dierent materials, with the supporting surfaces through controlled in ation and de ation of its extremal actuators [31, 32]. This design has been experimentally proven to be capable of generating robust bidirectional locomotion on both at and inclined surfaces. Another contribution of this work is a simplied dynamic modeling and associated controllability analysis we conducted on the developed robot. Dry friction is an inherently nonlinear phenomenon [33, 34, 35]. Here, by treating the actuation force generated by the central actuators as well as the nonlinear and time-varying frictional forces produced by the extremal actuators as system inputs, we described the robot's dynamics using a linear time- invariant (LTI) state-space model. Using this reduced-complexity model, we were able to analyze the robot's controllability characteristics and arrive at the conclusion that friction is indispensable for robots of this type to generate locomotion. We explicitly showed that in the absence of friction, the controllable subspace contains only states that characterize a static center of mass with respect to an inertial frame dened for the system. We further showed through numerical simulation, that if all central actuator and frictional force inputs can be chosen without constraints, the system would become fully controllable [31, 32]. This nding, despite being based on physically unattainable assumptions, indicates that there exists an innite number of theoretically feasible 4 traveling modes. Thus, it can be deduced that biologically-inspired locomotion modes represent only a small set of what can possibly be achieved with this framework. 1.2 SMA-Based Articial Muscles Another aspect of the proposed thesis involves in the characterization and control of articial muscles, particularly ones made of SMAs. Shape-memory-alloys (SMAs) are a type of alloys that can, after undergoing seemingly plastic deformation, memorize and retain their original shapes under proper stimulations. This behavior re ects the fact that SMAs can transition between two dierent crystal phases: the austenite and martensite states [36]. The austenite state, also known as the parent phase, is characterized by its stability at relatively high temperatures with respect to the value of the nominal transition point; the martensite state, also referred to as the product phase, is characterized by its stability at relatively low temperatures with respect to the value of the nominal transition point. During a phase transformation from austenite to martensite, the crystal structure of the SMA material changes from a cubic (austenitic) conguration to a combined cubic-monoclinic (twinned martensitic) conguration, and then into a purely mono- clinic (detwinned martensitic) arrangement [6]. A complete phase transformation can be induced mechanically according to the pseudoelasticity eect (or superelasticity), or by a combination of thermal and mechanical stimuli according to the shape-memory eect (SME). Similarly, SMA ma- terials can be induced to transition from the martensite state to the austenite state according to the SE and SME dynamics. Thus, complete austenite{martensite{austenite cycles, characterized by marked hysteretic features, can be completed. The SMA material most widely used in mechatronics and robotics research is a nickel- titanium (NiTi) compound known as nitinol. This material has been demonstrated capable of deforming periodically with strains in the range [2 : 4]% for over 10 5 thermal cycles under loading stress [37]. Thus, due to this proven functionality, NiTi-based structures and mechanisms have been profusely 5 Fig. 1.2: Conceptual crawling robot design. A catalyst-coated SMA wire is preloaded with a spring and heated through controlled catalytic combustion. As the SMA contracts, an array of slanted spikes is pulled forward while the robot remains stationary. This is possible because the spikes conguration exploits friction anisotropy, resulting a signicantly smaller frictional force when it slides forward than backward. During cooling, the loaded spring facilitates the SMA's phase transition from austenite to martensite, causing the wire to extend. Meanwhile, the anisotropic friction in this case anchors the array of spikes to the ground and thus pushing the robot forward. The sharp arrows indicate the direction of motion of the components. employed as actuators [6]. Furthermore, what makes SMA-based actuators very attractive is their unmatched power-to-weight ratio compared to those of other actuation technologies [38, 39]; an extremely desirable feature for actuators that drive small robots. Currently, batteries are the primary source of energy employed to power autonomous robots. However, at the cm and mm scales, the applicability of batteries is highly limited due to their relatively low specic energies; approximately 0:9 MJ kg 1 for state-of-the-art lithium-ion (Li- ion) batteries and approximately 1:8 MJ kg 1 for experimental lithium-metal (Li-Po, Li-Hv) batteries. As a result, most of existing small-scale robots are tethered to external power sup- plies [30, 40, 41]. In contrast, the specic energy of animal fat is approximately 38 MJ kg 1 and those of hydrocarbon fuels approximately range from 20 MJ kg 1 to 50 MJ kg 1 . Therefore, there are high incentives for nding novel techniques that employ hydrocarbon fuels instead of electricity to thermally drive SMA-based actuators. At the mm-scale, this idea can be realized by developing NiTi-catalyst composite wires on whose surfaces ameless catalytic combustion [42] can be initiated, sustained and controlled. According to this notion, chemical energy can be di- rectly converted into mechanical work; for example, Fig. 1.2 depicts a conceptual design in which this idea is used to drive a crawling robot. In this case, the periodic strain variation of a catalytic- combustion-driven NiTi-catalyst composite wire is transformed into a locomotion pattern through 6 a surface-array of spikes that exploits friction anisotropy to propel the microrobot forward. In this context, it is crucial to devise control methods that directly utilize temperature for feedback, which motivates and justies our research. Periodic phase transformations of SMA-based structures and mechanisms are inherently non- linear processes characterized by irreversible hysteretic paths [6]. Therefore, to achieve precise SMA-based actuation by employing non-electrical thermal stimuli, such as catalytic combustion, it is required to accurately model the hysteretic behavior of the system dynamics in order to synthesize compensating controllers. Since the precise proportion of austenitic and martensitic crystalline structure of an SMA wire during actuation is inaccessible, phenomenological models that employ mathematical expressions to directly characterize observed phenomena are often se- lected to model SMA wire hysteresis. One of the most well-studied phenomenological models in the Preisach model [43]. Existing versions of Preisach-model-based characterizations of SMA wires' hysteretic behaviors describe mappings from an electrical current (or voltage) input to the resulting strain output [44, 45, 46]; therefore, they are not applicable to catalytic-combustion-driven SMA- based actuation. To this end, we created an empirically identied Preisach-model-based mapping that describes the relationship between time-varying temperature and time-varying strain in [47]. Extending upon the nding of [47], we introduced a Preisach-model-based inversion algorithm in [48] that enable us to synthesize a displacement controller for an SMA wire under regulated constant loading. Consistently, one major dierence between the proposed method and exist- ing inverse schemes for SMA-actuator control [44, 46] is that we select temperature instead of electrical current, as the control signal inputted to the nonlinear hysteretic mapping. Typically, Preisach inversion problems have been solved using two dierent approaches; meth- ods of the rst type directly search for numerical approximations of the inverse hysteretic mappings using analytical descriptions of the forward dynamics of the actuators [49, 50], while methods of the second type aim to nd almost-exact inverses of the empirically-estimated hysteretic map- pings of the actuators through computationally intensive recursions and iterations [44, 51, 52]. 7 Here, we introduce a new procedure, with reduced mathematical complexity, which is based on the inversion of the surface of strain increments associated with a set of rst-order reversal curves (FORCs). The main advantage of the proposed approach is that the inversion algorithm only re- quires the computation of simple interpolations and can be readily implemented into a feedforward open-loop control scheme. One fact of SMA-wire-based actuators that cannot be overlooked is that in order for the ac- tuators to complete the austenite-martensite-austenite full cycle, they need to be preloaded with external stress. More importantly, existing literatures [53, 54, 55] and our own experimental results have shown that change in loading conditions would lead to change in SMA actuators' hysteresis characteristics. Evidently, the contribution of time-varying stress to SMA's displacement needs to be accounted for by the displacement controller. To this end, [53] utilizes a neural network model using training data obtained under cyclic loading; an additional sliding-mode based feedback con- troller is adopted to reduce the tracking error. Stress variations are also treated as disturbances in [56] and [57], to `reject' which, a sliding-mode controller with time-delay estimation and an adaptive scheme are introduced, respectively. All these methods are made possible with feedback position controllers, which require additional deformation measurement apparatuses that are likely impractical for small-size autonomous robots. In [47], we demonstrated promising results in mod- eling SMA wire's strain output under dynamic loading by treating stress as a correction factor and interpolating between the FORC strain increment surfaces identied at a set of discrete constant stresses. By integrating this method into the Preisach-model-inversion scheme introduced in [48], we synthesized a position controller capable of precisely regulating SMA actuators' deformations under dynamic loading. The only sensing apparatuses required by this controller are lightweight and compact thermocouples and load cells, which can be tted onto small-size autonomous robots with simple modications. 8 Fig. 1.3: Major and minor SMA hysteresis loops, shown in red and blue respectively. This set of temperature-strain curves were obtained from a 127m thick SMA wire with a nominal phase transition temperature of 90 C under a constant stress of 155 MPa 1.3 Articial Muscles High-Frequency Actuation Our research ndings showed that Nitinol-based SMA wires in general can output strain up to 4% of its martensitic-state length [47, 48]. To generate cyclic deformation of this level of magnitude, the SMA wire needs to be heated between its lower and upper saturation. The corresponding strain-temperature relationship is commonly known as the hysteresis major loop. In Fig. 1.3, the major hysteresis loop obtained from a sample experiment is shown in red. Intuitively, as the SMA is heated between narrower temperature bandwidths, its deformation magnitude is expected to reduce, leading to smaller temperature-strain hysteresis loops, referred to as minor loops. Two dierent minor loops are shown in blue in Fig. 1.3. Evidently, the overall shape of the hysteresis loops are characterized with steeper slope near transition temperature and atter slope near saturation temperatures. Such property determines that within certain range, major temperature bandwidth reduction would only lead to much less substantial decrease in the strain output. Subsequently, narrower temperature bandwidth would make possible faster heating cycles and therefore higher frequency actuation. 9 We propose two methods to exploit such property in creating fast moving SMA-based mech- anisms. The rst one is referred to as an SMA multiplier. Intuitively, in an ideal setting, SMA wires with identical diameter and chemical composition will display the same temperature-strain relationship if excited with the exact temperature prole under equal loading. Consequently, the longer wire is expected to generate a longer displacement than the shorter wire when heated to follow the same temperature signals holding the stress condition identical. Hence, the displace- ment can be magnied, or `multiplied' by adopting a longer SMA wire. More importantly, this suggests that in order to produce the major loop displacement achievable on the shorter wire, the longer wire only needs to be excited to follow a minor loop temperature prole. Given that the time required to complete a minor loop is always shorter than that needed for a major loop, realizing equal displacement outcome theoretically will always be faster on a longer wire excited to follow a carefully selected minor loop, hence allowing for higher actuation frequency. However, employing a long SMA wire to drive a compact mm-cm scale robot requires special wire installa- tion conguration. To that end, we designed and tested a capstan-based multiplier mechanism, detailed in Chapter 4. The second method focuses on utilizing limited high-frequency displacements of SMA wires excited within minor hysteresis loops to induce signicantly larger deformation of a coupling structure. Following this concept, we developed an SMA-driven bending actuator based on a phenomenon known as buckling due to tilting of forces, or tilt buckling [58, 59]. In the proposed design, a pair of pretensioned SMA wires are installed eccentrically from the elastic center beam structure at equal distance. The alternating excitation of each SMA wire will cause them to deform, which is then translated to the cyclic bidirectional de ection of center beams. Note that the moment induced by the SMA wires at the beam could potentially lead to its buckling, hence `tilt buckling', but there is an extremely large range of beam de ections can exploited for actuation purpose before buckling occurs. Moreover, the contraction of the SMA wire on one side could further accelerate the relaxation (extension) of the SMA wire on the other side under 10 this conguration. In addition, a quadruped crawling robot employing these bending actuators as legs were developed to validate the actuator's application in small-scale robot's locomotion. All related details and preliminary test results are also discussed in Chapter 4. 1.4 Summary of Contributions The main contributions of this dissertation can be summarized as follows: • A soft robot capable of crawling bidirectionally on at and inclined surfaces was developed, drawing inspiration from earthworm and its locomotion scheme. A novel silicone-casting fabrication method integrating multiple materials were introduced to create the prototype. In addition, using the concept of controllable subspace, we showed that friction can be introduced as input to control the locomotion for robots of the described type. • A modied classical Preisach model was designed and implemented to model the hysteresis behavior of a Nitinol SMA actuator. Unlike most of the existing SMA hysteresis models that chose electric current/voltage as input, our approach creates a mapping between time- varying temperature and SMA strain output. Such mapping could be adopted to model hysteresis observed from SMA wires powered by means other than electric heating, for instance catalytic reaction, in which case temperature will be the control signal. We also developed the proper experimental apparatuses and novel control algorithms to regulate heating and loading conditions on the SMA wire. • Based on the procured Preisach hysteresis model, we proposed a inversed-Preisach-model- based SMA position control method. Upon receiving a reference strain sequence, the in- versed Preisach model computes the corresponding temperature prole that once realized on the SMA wire, would generate the displacement that tracks the reference strain. The proposed scheme was integrated into a feedforward position controller. 11 • SMA's hysteresis is a function of both the temperature and stress conditions. To account for dynamic loading's eect in SMA hysteresis modeling, we treated stress as a correcting factor and weighted the parameters of the identied Preisach model by stress value at each instance. This approach was also incorporated into the inversed Preisach model to realize position control of SMA wire subject to time-varying stress. The proposed method was validated through both real-time simulation and experiments. • Based on the characteristics of SMA's hysteresis, we devised methods to exploit the concept of hysteresis minor loop to enable high-frequency actuation on SMA-based articial muscles. We proposed two approaches, one relies on a mechanism described as the SMA multiplier and the other one utilizes the phenomenon known as the tilt buckling. Both approaches delivered promising test results. To further prove the application of the latter concept, we designed and fabricated a quadruped crawling robot incorporating the technology of interest. 12 Chapter 2 Earthworm-Inspired Soft Crawling Robot 2.1 Earthworm-Inspired Locomotion Earthworms belong to the phylum Annelida, characterized by their segmented body composed of metameres [1]. During locomotion, each metamere is actively recongured by contraction and relaxation of the longitudinal and circular muscle layers. Specically, as illustrated in Fig. 2.1(a), a metamere will expand radially (shorten longitudinally) as the longitudinal muscle contracts and elongate (shrink radially) when its circular muscle contracts [11]. To locomote, each metamere alternately cycles through longitudinal elongation and radial expansion, generating a wavelike motion along the body, as depicted in Fig. 2.1(b). To model and realize such peristalsis-based locomotion for worm-inspired robots, researchers have explored a range of options, including nite state machines [60], genetic algorithms [61], actuation phase coordination [62] and adaptive controllers that track prescribed reference lengths between segments [63]. Earthworms also bear setae in each metamere, a feature found in Oligochaetas, a subclass of Annelida [11]. Setae perform the critical function of providing adequate traction so the peri- staltic body motion can be translated to either forward or backward locomotion, especially when earthworms are crawling above ground. For earthworms that can both crawl and burrow (nigth- crawlers), they have evolved setae only on the ventral and lateral surfaces of each metamere, as 13 Fig. 2.1: (a) Illustration of a nightcrawler's metamere unit: A metamere expands radially when its longitudinal muscles contract (circular muscles relax) and expands longitudinally when its circular muscles contract (longitudinal muscles relax). The four pairs of setae on its ventral and lateral surface will protrude and retract accordingly with longitudinal muscles' contraction and relaxation, respectively, to provide variable tractions that enable locomotion. (b) Peristaltic crawling kinematics: Following the convention in [1], we dene a stride as a complete cycle of peristalsis and describe the crawling kinematics as a function of four variables: stride length, protrusion time, stance time and stride period. demonstrated in Fig. 2.1(a). When a metamere undergoes radial expansion (longitudinal muscle contraction), its setae protrude and anchor that segment to the substratum to prevent slipping while adjacent metameres go through reconguration. Once the metamere's circular muscle starts to contract, its longitudinal muscle will subsequently relax, pulling the setae o from the ground to allow the metamere to slide. The proposed gait of the presented robot is based on the nightcrawlers' crawling sequence illustrated in Fig. 2.1(b). Here, following [1], we dene a stride as one cycle of peristalsis and stride length as the total distance covered during one stride. In addition, protrusion time is associated with the time span during which the nightcrawler's head locomotes forward and covers the stride length. Subsequently, stance time describes the period during which the nightcrawler's head remains anchored to the ground while the rest of its body moves forward and returns to its initial state. Therefore, the stride period, dened as the time it takes to complete one full stride, is equal to the sum of the protrusion time and the stance time. Despite the fact that 14 nightcrawlers have numerous segments with staggered stride periods, we choose these kinematic variables associated with a nightcrawler's rst segment. During backward locomotion, protrusion time and stance time are dened in reference to the tail metamere (the last segment). 2.2 Design and Fabrication In this section, we present the design and fabrication tools developed for creating the proposed soft crawling robot shown in Fig. 1.1, which is capable of locomoting bidirectionally on horizontal and inclined at surfaces, employing a peristalsis-based friction-controlled crawling mode inspired by that performed by earthworms. In this case, the problems of robotic design, fabrication, loco- motion planning and control are strongly coupled, as the proposed bioinspired friction-based loco- motion scheme relies not only on a digital controller but also on control functions preprogrammed in the physical components of the robot. A key innovation in this approach is the apparatus that enables the active variation of the friction coecients associated with the sliding of the extremal actuators, which can switch their surfaces of contact with the ground in order to modulate friction forces. This technique is used to replicate the combined active{passive mechanisms employed by earthworms for locomotion and control; consistently, as seen in Fig. 1.1, the robot is powered by one pneumatically-driven central longitudinal actuator while the pneumatically-driven extremal actuators are required for control. The central and extremal actuators emulate the functionalities of an earthworm's longitudinal and circular muscles, respectively. These three actuators are fed through independent air lines, and were designed and built to expand and contract axially according to empirically-estimated functions of their internal pressures. The radial expansions are constrained with black butadi- ene rubber elastomeric o-rings as seen in Fig. 1.1. As discussed in Section 2.1, during locomo- tion, nightcrawlers adjust traction with the ground by combining the eects produced by the protraction{retraction of setae and deformations of their hydrostatic skeletons. Similarly, the 15 Fig. 2.2: (a) Fabrication of the extremal actuators. In Step 1, liquid silicone is poured into a 3D-printed mold; then, the lower half of a symmetrical double-cylindrical core is submerged in the silicone. The silicone within the mold is then cured at 65 C for 15 min. In Step 2, the cured silicone is released before liquid silicone is added to cast the other half of the shell (Step 3 and Step 4). Step 5 shows the complete shell structure after being peeled o from the 3D-printed core. In Step 6, o-rings are tted onto the shell's imprinted grooves, and a layer composed of silicone and a berglass net is employed to seal one end of the shell. Step 7 shows a extremal actuator once the fabrication process is completed. (b) & (c) Fabrication of the central actuator and connecting modules. These procedures are identical to those employed in (a) with the exception that the connecting modules do not require an additional reinforcing layer and o-rings. Additionally, an orice is perforated from the bottom of both connecting modules to allow the ow of air into both of extremal actuators. (d) Final assembly. In Step 1, the two extremal actuators, the central actuator, two connecting modules and three air-supplying lines are integrated together. In Step 2, a pair of 3D-printed casings are xed over the connecting modules of both extremal actuators. Step 3 depicts the cross-sectional view of the robot in two states: all the actuators in ated and all the actuators de ated. Finally, Step 4 shows the robot after the fabrication process of the entire robot is completed. proposed robot adjusts the frictional forces acting on the extremal actuators by changing the surfaces of contact with the ground; therefore, also changing the associated coecients of friction when the materials of these surfaces are dierent. According to the literature on friction [64], the values that kinetic coecients of friction can take vary widely depending on the type of surfaces 16 in contact; for example, the coecient for Te on on steel is approximately 0:04 while the coe- cient for rubber on concrete is approximately 0:8, which indicates that there is a wide range of possibilities for the design and implementation of locomotion strategies. The design of the extremal actuators is such that their in atable soft structures are made of silicone rubber whose contact with any other surface is characterized by a friction coecient with a high value while the contacts of the enclosing 3D-printed casings with any surface are characterized by friction coecients with low values. Thus, as shown in the bottom illustration of Fig. 2.2, since these in atable soft actuators are xed to the upper ceilings of their corresponding casings, their rubber surfaces do not touch the ground when de ated and the robot's body is supported by the hard casings. On the other hand, the rubber surfaces of the actuators make contact with the ground when these in atable structures are in ated, thus inducing high values of friction. In this scheme, the switching between high and low friction states, and vice versa, is accomplished by simply in ating and de ating the friction actuators, and vice versa. Unlike existing robots which employ frictional anisotropy [20, 28, 29, 30], for which the direc- tion of friction is prescribed by the morphology and skin patterns of the structures in contact with the ground, the proposed robotic design was conceived to generate controllable isotropic frictional forces. This method for active friction control has the potential to enable earthworm-like crawling in any conceivable direction in a plane by either increasing the number of modules or/and the number of friction actuators. In this case, given the linear two-metamere axially-powered con- guration of the robot, it can crawl and be controlled bidirectionally (forward and backward). Note that even though the functionalities of the extremal friction actuators are inspired by those of earthworms muscles and setae, the underlying working principles are fundamentally dierent. For example, earthworms protract setae to increase traction because their skins are smooth and low-friction while the robot employs the direct contact of its rubber skin with the ground to increase friction. In addition, natural muscles contract actively and elongate passively while the pneumatic actuators of the robot elongate actively and contract passively. 17 The methods and construction sequences employed to fabricate the soft robot are depicted in Fig. 2.2. As shown in Fig. 2.2(a), (b) and (c), the extremal actuators, the central actuator and connecting modules are fabricated by casting liquid silicone (Eco ex ® 00-50, Smooth-On) in 3D-printed acrylonitrile butadiene styrene (ABS) molds. The yellow casings are also 3D-printed using the same ABS material employed to create the casting molds. The nal assembly of the robot, in which all the individual parts are put together, is illustrated in Fig. 2.2(d). At this fabrication stage, all the interfaces are airtight sealed using liquid silicone and extendable coiled air supply lines are inserted into the central and front actuators. Note that the coil shape of the internal air lines is the key design feature that allows the pneumatic actuators to freely expand and contract. The in atable structures of the central actuator and both extremal actuators are 35 mm in diameter and have homogeneous walls with a thickness of either 2:5 mm or 3 mm depending on the location in the soft structure. The central actuator has a length of 83 mm and weighs 51 g including the o-rings and air supply lines; each extremal actuator combined with a connecting module has a total length of 26 mm along its axial direction and weighs 19 g; and each 3D-printed casing weighs 18 g. These dimensions were initially selected to replicate those of the robotic design in [5] and then modied to t the required o-the-shelf pneumatic components. Finally, as can be observed in Fig. 1.1, after the nal assembling in Fig. 2.2(d) is completed, 130 g steel plates are xed to the tops of the yellow casings in order to increase the maximum friction forces attainable with the proposed actuation scheme. 2.3 Dynamic Modeling and Controllability Analysis In this section, employing a reduced-complexity dynamic model and linear systems theory, we develop the analytical tools necessary to study the motion and controllability properties of the robot described in Section 2.2. From an abstract perspective, the dynamics of the proposed robot 18 Fig. 2.3: Reduced-complexity double-mass-spring-damper model of the robot in Fig. 1.1. The associated mathematical description of this model is employed to study the con- trollability of the robot in the absence and presence of forces f 1 and f 2 . To model friction, the values off 1 andf 2 are considered to be positive when the associated force vectors act in the same direction as that of i; consistently, the values off 1 andf 2 are considered to be negative when the force vectors act in the opposite direction as that of i. can described by a double-mass-spring-damper model [65, 40, 63] excited by active actuation forces and time-varying friction forces, as illustrated in Fig. 2.3. Accordingly, the extremal actuators are modeled as two blocks, with masses m 1 and m 2 , capable of varying their coecients of friction with the supporting ground surface in real time in order to modulate the values (signed magnitudes) of the frictional forces, f 1 and f 2 in Fig. 2.3. Similarly, given its function and the elastic nature of its soft structure, the central axial actuator is modeled as a system composed of a massless elastic spring with stiness constant k, a dissipative element with constant c and two pneumatically-generated actuation forces with opposite directions and identical magnitudes f a (see Fig. 2.3). In this model, the radial expansion of the actuator is considered to be completely negligible; therefore, f a can be estimated as f a (t) =s a p a (t); (2.1) where p a and s a are the instantaneous internal air pressure and constant cross-sectional area of the actuator, respectively. Note that in agreement with the ndings in [5], we assume a constant value ofk, which is suciently accurate for the purposes of controllability analysis and controller synthesis; the true stiness, however, is nonlinear, time-varying and most likely depends on the internal air pressure of the actuator. 19 To study the controllability of the system, we rst consider the frictionless case in which the sole input to the system is the force magnitude f a . Consistent with Fig. 2.3, we dene x 1 and x 2 as the position variations of m 1 and m 2 with respect to the inertial frame of reference; and correspondingly, the associated speeds as v 1 = _ x 1 andv 2 = _ x 2 . Thus, for the purpose of analysis, the system is described with the single-input{multiple-output (SIMO) state-space realization _ x(t) =Ax(t) +B 0 u 0 (t); y(t) =Cx(t) +Du 0 (t); (2.2) where A = 2 6 6 6 6 6 6 6 6 4 0 1 0 0 k=m 1 c=m 1 k=m 1 c=m 1 0 0 0 1 k=m 2 c=m 2 k=m 2 c=m 2 3 7 7 7 7 7 7 7 7 5 ; B 0 = 2 6 6 6 6 6 6 6 6 4 0 1=m 1 0 1=m 2 3 7 7 7 7 7 7 7 7 5 ; C = 2 6 6 6 6 6 6 6 6 4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 3 7 7 7 7 7 7 7 7 5 ; D = 2 6 6 6 6 6 6 6 6 4 0 0 0 0 3 7 7 7 7 7 7 7 7 5 ; x =y = 2 6 6 6 6 6 6 6 6 4 x 1 _ x 1 x 2 _ x 2 3 7 7 7 7 7 7 7 7 5 ; and u 0 =f a . In this case, for any conceivable choice of parameters m 1 > 0, m 2 > 0 and k > 0, the controllability matrixC 0 = B 0 AB 0 A 2 B 0 A 3 B 0 has rank 2; therefore, the system is not controllable. This implies that there exists a set of states that cannot be reached from any given initial state by employing an input signal [66]. To determine if the uncontrollability of the system prevents locomotion, we further investigate the controllable subspace associated with the pair fA;B 0 g, dened asC AB0 = ImagefC 0 g. As explained in [67], it can be shown thatC AB0 is also 20 the set of states that can be reached from the initial condition x(0) = 0 by employing an input signal,R t . In particular, it follows thatR t =C AB0 = Spanf 1 ; 2 g, in which 1 = 1 0 m1 m2 0 T ; 2 = 0 1 0 m1 m2 T : (2.3) Therefore, every state inC AB0 can be written as 1 1 + 2 2 for some 1 , 2 2 R, which im- plies that all the positions of m 1 and m 2 that can be reached from x(0) = 0 have the form n x 1 = 1 ;x 2 = 1 m1 m2 o . Thus, we conclude that for an initial statex(0) = 0, the position varia- tion of the system's center of mass x CM with respect to the inertial frame will remain unchanged regardless of what input signal f a is employed to excite the system, because x CM = m 1 x 1 +m 2 x 2 m 1 +m 2 = 0; (2.4) for all x2R t . Hence, in the absence of frictional forces, locomotion is impossible for robots of the type in Fig. 2.3. This nding is consistent with basic physical intuition (the sum of forces acting on the system is zero) and biological observations [68, 69]. When friction is included in the model of the robot, the sliding blocks are alternately subjected to either static or kinetic friction, depending on their relative motions with respect to the surface; a rapid variation between these two types of friction is a phenomenon commonly referred to as the stick-slip eect. Here, we model the friction force associated with the sliding blocki,f i , according to the method in [34], as f i (t) = 8 > > > > > > > > < > > > > > > > > : f k;i (t); if j _ x i j>v s;i ; f n;i (t); if j _ x i j<v s;i and jf n;i (t)j6f s;i ; f k;i (t); if j _ x i j<v s;i and jf n;i (t)j>f s;i ; (2.5) 21 where, f k;i is the kinetic friction aecting the mass; f n;i is the net force jointly exerted by the spring, damper and actuator on the block; (v s;i :v s;i ), with 0<v s;i 2R, denes a narrow band of speed near zero, outside which the mass slips and is aected by the kinetic friction f k;i ; and 0<f s;i 2R is the maximum value that the magnitude of the static friction force can take. The kinetic friction associated with each block is modeled as f k;i (t) = sign[ _ x i (t)] k;i m i g cos> 0; (2.6) fori = 1; 2, where 0< k;i 2R is the corresponding kinetic friction coecient;g is the acceleration of gravity constant; and is the angle of the surface that supports the block with respect to the horizontal. Similarly, the critical value f s;i is estimated as f s;i = s;i m i g cos; (2.7) for i = 1; 2, where 0< s;i 2R is the corresponding static friction coecient and m i g cos is the magnitude of the static normal force exerted by the mass m i on the supporting surface. Also, from Newton's laws, it immediately follows that the signed (positive or negative) magnitude of the net force is simply f n;i (t) = (1) i [k (x 1 x 2 ) +c( _ x 1 _ x 2 ) +f a ]; (2.8) for i = 1; 2. Thus, according to this idealization, when the signed speed of the block i lies within the range (v s;i :v s;i ) and the frictional force equals the sum of all the other external forces acting on the mass m i , f n;i , the block is considered to be sticking. When the magnitude of f n;i exceeds that of f s;i (i.e, the static frictional force is too small to counteract all the other external forces), slipping occurs and the friction acting on the block i is f k;i according to (2.6). 22 Following the model in (2.5), if the masses of the blocks remain constant, f 1 and f 2 can only be modulated by varying the associated coecients of friction. Using linear-system-theory-based analyses only, it is not possible to determine whether the system would become fully controllable when the inputs are chosen to beff a ; s;1 ; k;1 ; s;2 ; k;2 g. However, to study the importance of friction in the control of locomotion, we analyze the case in which the inputsff a ;f 1 ;f 2 g are assumed to be unconstrained and arbitrarily choosable. Under this assumption, the system can be described using the multi-input{multi-output (MIMO) state-space representationfA;B 1 ;C;Dg, where the new input state matrix and input signal are given by B 1 = 2 6 6 6 6 6 6 6 6 4 0 0 0 1=m 1 1=m 1 0 0 0 0 1=m 2 0 1=m 2 3 7 7 7 7 7 7 7 7 5 ; u 1 = 2 6 6 6 6 4 f a f 1 f 2 3 7 7 7 7 5 : (2.9) The controllability matrix associated with the augmented state-space realizationfA;B 1 ;C;Dg, C 1 = B 1 AB 1 A 2 B 1 A 3 B 1 , has rank 4; therefore, the controllable subspace C AB1 = ImagefC 1 g spans R 4 , i.e., the system dened by (2.9) is fully controllable. This analysis shows that if the inputu 1 could be selected without restriction, any desired nal state (and any position of the system's center of mass) could be reached in a nite amount of time. In the case of the robotic system in Fig. 2.3, however, u 1 is strongly constrained by the limitations of the actuators and the time-varying nonlinear nature of the frictional forces acting on the sliding blocks. Despite these restrictions, controlled locomotion is achievable by varying the coecients of friction in (2.5), which is demonstrated through the numerical simulations and experiments discussed in the next sections. 23 2.4 Locomotion Simulation In this section, through numerical simulations, we demonstrate that the proposed robotic system can generate locomotion using feedforward-controlled time-varying friction. In this case, a set of feasible control inputs is selected via an exhaustive search followed by an iterative numerical process. According to (2.6) and (2.7), the values of the frictional forces f i are functions of the normal forces exerted by the blocks on the supporting surface and the corresponding coecients of friction (static or kinetic). As discussed in Section 2.2, the robot in Fig. 1.1 regulates the friction forcesf 1 andf 2 by varying the associated friction coecients in real time while the normal forces remain constant. Specically, the in atable structures of the extremal actuators, made of silicone rubber, induce a high friction coecient, k;i or s;i , when in contact with the supporting surface while the 3D-printed casings induce a low friction coecient, k;i or s;i , when in contact with the same surface. To estimate the achievable range of values for the friction coecients induced by the silicone rubber and ABS-printed casings when in contact with a supporting surface, we place the robot on an inclined plane covered with a test material; then, we slowly increase the angle of inclination until the robot starts to slip and eventually slides down. To explain the details of the method, as an example, here we describe the tests and estimation process corresponding to a supporting surface made of high-density polyethylene (HDPE). This procedure was repeated twice for each pair of surfaces in contact, i.e., silicone rubber in contact with the ground when the in atable structures of the extremal actuators are in ated and ABS material in contact with the ground when the in atable structures of the extremal actuators are de ated. The specics of the experimental method to estimate the friction coecients are as follows. In static equilibrium, the mechanics of the system can be described by f d;i =m i g sin =f i ; (2.10) 24 for i = 1; 2, where f d;i is the tangential component of the weight m i g, which equals the static frictional force acting on the blocki,f i . At the critical equilibrium angle ? (right before slipping occurs), (2.10) becomes m i g sin ? = s;i m i g cos ? : (2.11) Thus, the static coecient of friction can be estimated as s;i = sin ? cos ? = tan ? : (2.12) Experimental data obtained employing the complete robotic prototype in Fig. 1.1 indicate that the critical angle ? for the contact of silicone rubber with the HDPE surface is approximately 30 and for the contact of ABS material with the HDPE surface is approximately 7:5 . These angle values translate to static coecients of friction of approximately s;i = 0:13 and s;i = 0:58, respectively. In these experiments, however, the tangential weight components of the robot's steel plates (shown in Fig. 1.1) induce bending and twisting on the soft structures, thus disrupting the contacts between the extremal actuators and the inclined supporting surface. If the steel plates are removed, the shear forces exerted on the actuators can be drastically reduced, enabling the robot to maintain a rm contact with the supporting HDPE board at much steeper slopes. In experiments, the robot remained stationary on a HDPE surface with a slope as large as 45 with respect to the horizontal before starting to slip. Correspondingly, the static friction coecient was estimated be as large as s;i = 1, which is likely an estimation closer to the true value. Further details about the experiment-based estimation process can be found in the supplementary movie softRobot.mp4 (http://www.uscamsl.com/resources/gethesis2020/softRobot.mp4). The purpose of performing an experiment-based estimation of the friction coecients is to obtain a guideline in the selection of the parameters employed in the numerical simulations of 25 Fig. 2.4: (a) Block diagram of the discrete-time model employed in the numerical simulations. The plant ^ G d (z) = C d (zIA d ) 1 B 1d +D d is the discretized version of ^ G(s) = C(sIA) 1 B 1 +D. The friction function block computes the friction forcesf 1 [n] andf 2 [n] using the algorithm in (2.5) and the inputsu 1 [n],u 2 [n], f a [n] and x[n]. Zero initial conditions are set to start the simulations. (b) Example input signals. In this case, the central actuation force, f a , is modeled as a sinusoidal wave with its magnitude and bias consistent with the empirically- estimated internal pressures of the central actuator during periodic operation (see Appendix A.1). Specically, f a is set to oscillate between 0:66 N and 19:9 N. The indicator functionu i [n], for i = 1; 2, switches the friction of the corresponding actuator i from high (u i [n] = 1) to low (u i [n] = 0), and vice versa. The frequencies of all the inputs are set to 1 Hz (2 rad s 1 ) and the phase dierence between the extremal actuators is set to = 1:64 rad (equivalent to 0:82 s) the robot's locomotion, for which we have to dene static and kinetic coecients for the two types of contacts employed by the extremal actuators. Since here we are not trying to exactly replicate experimental data through simulations but to study the proposed method of locomotion, consistent estimates with values in the same range as those in the literature are sucient for numerical implementation. Following this principle, despite expected experimental inaccuracies, we also estimate the kinematic coecients of friction informed by the fact that their values should be smaller than the corresponding parameters for the static case [70]. Specically, we measure the forces required to initiate and maintain the sliding motion of the robot using a spring scale, for the two types of contacts employed by the extremal actuators (i.e., silicone rubber sliding on HDPE and ABS material sliding on HDPE). The obtained empirical results indicate that regardless of the contact surfaces involved in the experiments, the estimated value for the kinetic friction coecient in every case is approximately 75% the estimated value for the corresponding static friction coecient. Consistently, in the implementation of the simulations, we employ k;1 = k;2 = 0:75 s;1 = 0:75 s;2 = 0:75 1 = 0:75 and k;1 = k;2 = 0:75 s;1 = 0:75 s;2 = 0:75 0:13 = 0:0975. Additional experiments performed to identify the dynamic characteristics of the in ation-de ation-based friction switching mechanism employed by the extremal actuators 26 indicate that the transitions between k;i (or s;i ) and k;i (or s;i ), fori = 1; 2, can be completed within 0:4 s. This nding allows us to model the cyclic switching between high and low friction coecients assuming a quasi square-wave signal. The numerical simulations of the system's dynamics during locomotion are implemented ac- cording to the block diagram in Fig. 2.4(a). Here, ^ G d (z) =C d (zIA d ) 1 B 1d +D d is the discrete- time input-output dynamic description of the robot with input u 1 [n] = [ f a [n] f 1 [n] f 2 [n] ] T and output x[n] = [ x 1 [n] x 2 [n] v 1 [n] v 2 [n] ] T , obtained from the continuous-time transfer ma- trix ^ G(s) = C(sIA) 1 B 1 +D by using the zero-order hold (ZOH) method and a sampling rate of 1 kHz (T s = 10 3 s). From a theoretical viewpoint, the input u 1 [n] can be thought of as the sequence obtained from sampling the continuous-time signal u 1 (t) at 1 kHz. In this case, however, f a [n] is directly computed according to the discrete-time version of (2.1), and f 1 [n] and f 2 [n] are computed according to the discretized version of (2.5) by the friction function block (FFB). Besidesx[n] andf a [n], the inputs to the FFB are the indicator functionsu 1 [n] andu 2 [n], which take either the value 1 or 0 in the denition of the time-varying friction coecients as ?;i [n] = 8 > > > < > > > : ?;i if u i [n] = 1 ?;i if u i [n] = 0 ; (2.13) for i = 1; 2 and ? is either k or s. During the execution of the simulations, at each computational step, the algorithm running inside the FFB evaluates the displacement and velocity outputs from ^ G d (z) to determine the type and values of the instant frictional forces acting on the two extremal actuators. Specically, v s;i , fori = 1; 2, is set to 0:001 m s 1 , which implies that when the simulated speedv i [n], fori = 1; 2, lies in the range (0:001 : 0:001) m s 1 , the associated mass i is considered to be static and jf n;i j is compared to f s;i to determine the value of f i [n] according to (2.5). On the other hand, whenjv i [n]j>v s;i , the friction force acting on the mass i is immediately considered to be of the 27 kinetic type. Also, in agreement with the empirical relationship between k;i and s;i discussed above, for the execution of the simulation we dene s;i [n] = 4 3 k;i [n] and s;i [n] = 4 3 k;i [n]; (2.14) for i = 1; 2. Considering the experimentally-identied mechanical characteristics of the central axial actuator presented in [5], we model the magnitude of the force generated by a cyclic in ation{ de ation process, f a , using a biased sinusoidal signal with its amplitude and bias estimated from the minimum and maximum internal measured pneumatic pressures of actuation. Specically, for simulation purposes we set the internal pressure of the central actuator to oscillate between 0:7 kPa and 20:7 kPa, which in agreement with (2.1) translates to a force value that oscillates between 0:66 N and 19:9 N; a sample f a signal is shown in Fig. 2.4(b). In the simulations, we limit the frequencies of all the inputs to be less or equal than 1 Hz. This parameter selection re ects the observed behaviors of the robot's actuators during real-time experiments. While the frequencies of the indicator signals,u 1 [n] andu 2 [n], that determine the instantaneous friction forces are kept constant, they are set apart by a phase dierence which is actively varied between 0 rad and 2 rad. The stiness constant k is set to 200 N m 1 , which is the value estimated from a series of tensile tests conducted on a detached central actuator using an Instron Universal Testing Machine (Instron 5567). Also, a damping coecient c = 1:3 N s m 1 was implemented to account for energy dissipation; with this value, in agreement with empirical observations, the simulation results show that high-frequency displacement oscillations are mostly eliminated. All the simulations discussed in this paper were performed employing the xed-step 4 th -order Runge-Kutta method. A set of simulation results, obtained with the parameters k, c, k;i , s;i , k;i and s;i (for i = 1; 2) selected above, are presented in Fig. 2.5. Here, Fig. 2.5(a) and (b) show the displacements and velocities of the two blocks for m 1 = m 2 = 0:2 kg, the frequencies of f a [n], u 1 [n] and 28 Fig. 2.5: (a) & (b) Time-series of the robot's state and associated close-ups. Positions and velocities of the two extremal actuators when the input frequencies and are set to 1 Hz and 1:64 rad, respectively. The displacement of the center of mass x CM is also shown in magenta, calculated post-simulation according to (2.4). (c) Simulated displacement of block 1. This 3D-plot shows the position reached by the mass m 1 after 60 s of locomotion when the constant frequencies of the actuation force (f a freq.) and friction input (f i freq., for i = 1; 2) are taken from the interval [0:02 : 2] rad s 1 while is maintained at 1:64 rad. (d) Experimental relationship between displacement and phase dierence of the extremal actuators. The simulated positionsx 1 andx 2 after 60 s are plotted over the interval [0 : 2] rad. All the input frequencies are set to 2 rads 1 while the phase dierence is taken from the interval [0 : 2] rad and m 1 =m 2 = 0:2 kg. u 2 [n] set to 1 Hz and a phase dierence = 1:64 rad. The instantaneous x CM is calculated according to (2.4), compared tox 1 andx 2 , and used to quantify the amount of locomotion. In the example of Fig. 2.5(a), the system covers a total distance of approximately 2:4 m in 60 s, which corresponds to an average velocity of 0:04 ms 1 . Overall, the displacements of both masses follow a trajectory composed of a linear component and a periodic oscillation, while the position of x CM varies according to a staircase-like pattern. The corresponding instant velocities of both masses exhibit periodic oscillation patterns. Interestingly, it can be observed that in this simulation to generate forward locomotion, both blocks slide simultaneously in opposite directions. Note that this resulting locomotion pattern is dierent from that commonly observed in the gaits of earthworms in which a segment rmly anchors before its adjacent segment moves forward. 29 Figure 2.5(c) shows the observed relationship between the input frequency and amount of motion for the proposed pattern of actuation. Here, form 1 =m 2 = 0:2 kg and = 1:64 rad, the numerical simulation was repeatedly performed across a wide range of frequency combinations of the actuation force input f a (f a freq:) and the friction inputs f 1 and f 2 (f i freq:). Note that in all the cases, the frequencies ofu 1 [n] andu 2 [n] are identical. The simulated displacement of the block 1 after 60 s of operation are plotted along the z-axis. This result strongly suggests that the selection of the input frequencies plays an essential role in locomotion generation and eciency. Specically, for inputs of this form, it is clear that substantial locomotion is achieved when the frequencies of all the inputs are closely matched. Also, in general, actuation inputs with higher frequencies generate faster locomotion. To demonstrate the relationship between the phase 2 [0 : 2] rad and velocity of locomotion, Fig. 2.5(d) shows the displacements x 1 and x 2 after 60 s of operation, for all the input signals synchronized at 1 Hz (2 rad s 1 ) and m 1 =m 2 = 0:2 kg. This result indicates that is crucial for locomotion generation and that its modulation can be directly employed to induce direction reversal. Even though these ndings are limited to the specic actuation pattern and set of inputs employed in the discussed cases, they clearly exemplify the challenges and potentials of friction-controlled locomotion. 2.5 Real-Time Locomotion Planning In the previous section, we showed that high-performance locomotion can be achieved by em- ploying perfectly-shaped periodic driving and frictional forces with closely matched frequencies. These conditions are not yet realizable with the pneumatically driven soft actuators designed and fabricated as shown in Fig. 2.2. Thus, the replication of the high-speed simulated locomotion behaviors using the physical robot is, at this moment, not an attainable objective. Consistently, to enable the robot to crawl, we implement a real-time strategy compatible with low actuation frequencies. This locomotion mode closely emulates earthworms' peristalsis-based crawling and 30 can be readily applied to similar modular systems. From Fig. 2.3 it immediately follows that the block m 1 remains stationary while the block m 2 slides, as the central actuator in ates, if f s;1 >jf 1 j =jf n;1 j =jf n;2 j>jf 2 j =jf k;2 j: (2.15) In this case, the signal f 1 corresponds to static friction and f 2 corresponds to kinetic friction. Similarly, the block m 2 stays anchored to the ground and the block m 1 slides forward, as the central actuator de ates, if f s;2 >jf 2 j =jf n;2 j =jf n;1 j>jf 1 j =jf k;1 j: (2.16) In this case, the signal f 2 corresponds to static friction and f 1 corresponds to kinetic friction. Thus, forward locomotion can be generated by alternately activating the conditions described by (2.15) and (2.16). Similarly, backward locomotion can be realized by simply reversing the conditions dened in (2.15) and (2.16). In this way, a complete stride (forward or backward) can be generated by implementing a four-phase actuation sequence; Fig. 2.6 illustrates the forward motion case. Here, following the biological terminology on earthworm kinematics reviewed in Section 2.1, we dene the protrusion time as the span during which the central actuator expands (Phase 2). Similarly, the stance time is dened as the span within a locomotion cycle during which the central actuator is not expanding. Note that during the stance time, the front actuator remains horizontally static while completing an in ation{de ation cycle (Phase 3 + Phase 4 + Phase 1). 2.6 Experimental Results and Discussion A set of real-time experiments was conducted to validate the proposed locomotion method. The real-time locomotion control strategy is based on the implementation of a pressure tracking con- troller for each actuator. In this scheme, the reference pressure signal for each controller is 31 Fig. 2.6: Actuation sequence employed to generate forward locomotion on a at surface. Here, red indicates in ation and gray de ation; the 3D-printed casings are shown in sectional views so that the state of each actuator can be clearly seen in each phase of the locomotion sequence. When both extremal actuators are de ated, the smooth 3D-printed casings are in direct contact with the supporting surface and support the entire weight of the robot. To crawl forward, in Phase 1, the back actuator is in ated and anchors to the ground; in Phase 2, the central actuator is in ated to expand forward while the back actuator remains anchored to the ground; in Phase 3, the front actuator is in ated and anchors to the ground; nally, in Phase 4, both the back and central actuators are de ated and contract in order to complete a locomotion cycle. determined o-line through an actuator characterization process. The details of the procedure and the corresponding experimental setup are presented in Appendix A.1. Sets of experimentally- determined actuator pressure references corresponding to feasible forward and backward crawling motions are shown in Table 2.1; the resulting controlled actuator pressure outputs for desired pro- trusion and stance times of 1:2 s and 1:8 s are shown Fig. 2.7. Here, it can be seen that the three actuators can track their prescribed reference pressures reasonably well. Unlike the forward case, during backward crawling, the stance time is given by the span during which the back actuator remains stationary with respect to the at supporting surface after the central actuator has been fully in ated. The protrusion and stance states corresponding to backward locomotion are de- ned exactly as in the forward case. As an example, the photographic sequence in Fig. 2.8 shows stills from a movie in which the robot performs bidirectional crawling on a horizontal at HDPE surface. The complete experiment can be seen also in the supporting movie softRobot.mp4. For purposes of analysis, the experimental locomotion data is extracted and processed using Open Source Computer Vision (OpenCV). Figure 2.9(a) presents the displacement history of the robot during the bidirectional crawling experiment in Fig. 2.8, which were obtained by tracking 32 Table 2.1: Reference Pressures During Locomotion (kPa). Fig. 2.7: Pressure signals of the robot's actuators during a forward locomotion test. From the upper left (in red) to the bottom left (in blue) to the upper right (in black) the plots show the references (F-RP, C-RP and B-RP) and controlled outputs (F-P, C-P and B-P) of the front, central and back actuators. In this case, the protrusion time is 1:2 s, the stance time is 1:8 s and corresponding stride time is 3 s. and processing the left edges of both yellow casings. A close-up of the estimated motion signals clearly demonstrates the working principle of the robot, as it can be observed that the front and back actuators are controlled to alternately anchor (stick) and slide (slip), according to the conditions dened in (2.15) and (2.16). In this experimental test, the robot is controlled to reverse its direction every 32:5 s. During forward locomotion, the stride length is approximately 0:028 m with an average speed of 0:0089 m s 1 (HDPE-F in Table 2.2). During backward locomotion, the stride length is approximately 0:022 m with an average speed of 0:007 m s 1 (HDPE-B in Table 2.2). 33 The discrepancy between the forward and backward speeds is caused by a noticeable dier- ence between the amount of slippage experienced by the front actuator when sliding in dierent directions, most likely due to fabrication errors. Also note that, as seen in Table 2.1, the front actuator requires a higher internal pressure threshold than the back actuator to create a rm contact with the supporting surface. The observed inverse correlation between the amount of displacement generated by the robot and unwanted slippage is in agreement with the ndings in [71]. Employing the same actuation sequence in Fig. 2.6 and parameters in Table 2.1, the robot is thoroughly demonstrated to crawl eectively on several surfaces with dierent associated friction coecients, including a laboratory benchtop, wood, a foam pad and aluminum (see supplemen- tary movie). In addition, we show that the robot is capable of transitioning between surfaces with dierent coecients of friction; for example, from a foam pad to an HDPE plate, as can be seen in the supplementary movie. The relevant OpenCV-estimated locomotion parameters corresponding to these experimental tests are summarized in the last ve rows of Table 2.2. Note that dierent stride periods are required for the robot to adapt and achieve robust crawling on dierent types of surfaces. For consistency of analysis, the recorded speed is that of the system's center of mass and for the case in which the robot crawls from a foam pad to an HDPE plate (Foam+HDPE), only the time spans during which both extremal actuators are on a given surface are employed in the estimation of the locomotion parameters. Among all the tested surfaces, the slowest speed is obtained when the robot crawls on wood, probably because microscopic timber particles accumulate on the silicone surfaces of the actuators, thus altering the corresponding friction coecients and progressively causing more undesired slippage. To assess the ability of the robot to travel on inclined surfaces, we performed a set of tests using the same parameters in Table 2.1 but with the robot crawling upward on an HDPE plane positioned at a 7:5 angle with respect to the ground. In these tests, the protrusion and stance times, as in the horizontal cases, were set to be 1:2 s and 1:8 s, respectively. The plot in Fig. 2.9(b) 34 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Fig. 2.8: Photographic sequence composed of movie stills showing the soft robot crawling bidirectionally on a at HDPE surface. During this test, the traveling direction is reversed every 32:5 s and the forward and backward loco- motion is implemented employing the actuator pressure references listed in Table 2.1. The complete experiment can be found in the supplementary movie softRobot.mp4 (http://www.uscamsl.com/resources/gethesis2020/softRobot.mp4). shows the OpenCV-estimated positions of the left bottom corners of both yellow casings and a photographic sequence of the experiment is displayed in Fig. 2.10. In this case, the robot achieves a stride length of approximately 0:018 m and an average speed of 0:0061 m s 1 . It can be observed that the downward components of the gravitational forces acting on the casings and the steel plates induce shear forces that cause the anchored extremal actuators to bend and curl. Thus, the contact areas between the actuators and the inclined plane are reduced, which noticeably increases the slippage at both ends of the robot. As in the horizontal cases, slippage appears 35 Fig. 2.9: (a) Trajectories of the extremal actuators during the bidirectional loco- motion test. The time-series corresponding to the back block is shown in red; the time-series corresponding to the front block is shown in blue. The direction of locomotion is reversed ev- ery 32:5 s; a zoomed-in view of one of the controlled reversals (from 56 s to 72 s) is shown in the superposed window. (b) Trajectories of the extremal actuators of the robot while climbing on the HDPE surface held at a 7:5 angle with respect to the horizontal. The time-series corresponding to the back block is shown in red; the time-series corresponding to the front block is shown in blue. Table 2.2: Experimental Results on Horizontal Surfaces. to aect more severely the front actuator than the back actuator. The complete set of inclined experimental tests can also be found in the supporting movie softRobot.mp4. Through the experiments presented in this section, we clearly demonstrated the suitability and eectiveness of the proposed friction-controlled scheme for generating bidirectional peristalsis- based crawling using the robot introduced in Section 2.1. Note that the locomotion speeds ob- served in the experiments are not comparable to those obtained in the numerical simulations. This discrepancy is expected because the actuation methods adopted in both types of implementations are substantially dierent. In the experimental cases, locomotion control is achieved indirectly by modulating the deformation of the actuators via feedback control of their internal air pressures; in the simulations, the locomotion modes are directly controlled. Direct displacement control of the robot will be implemented in the future by employing faster actuators and a real-time 36 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 Fig. 2.10: Photographic sequence composed of movie stills showing the soft robot climbing on an inclined HDPE surface positioned at an angle of 7:5 with respect to the horizontal. A cup containing dyed water indicates the direction of the gravitational force. motion-capture system as the main sensor in a feedback loop. Finally, it is important to em- phasize that the main characteristic of the locomotion approach introduced in this work is that, unlike existing methods based on anisotropic friction, the actuator switching mechanisms enable 37 active control of the isotropic frictions between the actuators and contacting surface; thus, the basic design of the robotic prototype in Fig. 1.1 can be readily modied to create active steering capabilities by simply enabling the central actuator to bend and twist and not only extend and contract. Furthermore, note that the structural conguration of the proposed robot enables the adoption of other mechanisms for the generation of traction and anchoring forces, such as suction and magnetism, to enable locomotion on at perpendicular or uneven surfaces. 2.7 Summary In Chapter 2, we presented an earthworm-inspired soft crawling robot capable of bidirectional locomotion on both at and inclined surfaces. The robot consists of pneumatic actuators made of silicone rubber and 3D-printed casings that were designed to emulate the functionalities of an earthworm's longitudinal and circular muscles as well as its bristle-like setae structures. The locomotion was made possible through active switching of the robot's coecients of friction with the contacting surface driven by the controlled in ation and de ation sequences of its extremal actuators. Furthermore, we modeled the robot as a double-mass-spring-damper system and de- scribed its crawling dynamics using an LTI state-space representation. Based on this model, we were able to prove mathematically that frictional forces can be employed as inputs that lead to system controllability. This nding was validated through numerical simulations. The robot's real-time locomotion capability was tested experimentally. 38 Chapter 3 Shape-Memory-Alloy (SMA) Based Articial Muscle 3.1 Characterization of SMA Hysteresis 3.1.1 Overview of SMA Dynamics The repeatable deformation displayed by Shape-Memory-Alloys (SMA) is due to the fact that SMAs can transit between two phases characterized by dierent crystal structure compactnesses: austenite and martensite. The austenitic phase, also known as the parent phase, exhibits a compact high-symmetry cubic structure while the product martensitic phase has a low-symmetry monoclinic structure [6, 72]. During the transformation from austenite to martensite, the crystal structure of SMA transforms from cubic (austenite), to a combination of cubic and monoclinic (twinned martensite), before nally developing into a purely monoclinic arrangement (detwinned martensite) [6]. Such conversion can also be reversed, so that martensite can transform back to austenite and thus completing the austenite-martensite-austenite cycles. Consequent to the micro- structural change occurred during phase transformation, recoverable macro-structural deformation with marked hysteretic features is generated. Note that the phase transformation of SMA wires is inherently a complex nonlinear process characterized by hysteresis [6]. Moreover, since the full phase transformation can be induced thermomechanically according to pseudoelasticity eect (PE) and shape-memory eet (SME), the 39 Fig. 3.1: Graphic illustration of the constitution of SMA's hysteresis. Once SMA is consisted of pure Austenite, hysteresis in its strain is known as pseudoelasticity, which is caused by variation in loading condition. When no loading condition is applied, hysteresis behavior resulted from heating and cooling is known to be the shape memory eect (SME). However, as pointed out in the diagram, to transfer from a purely Austenitic state to a purely detwinned Martensitic state, loading is required in addition to cooling. Otherwise, the SMA wire can only reach the state in which its crystal structure is known as the twinned Martensite. Evidently, SMA's hysteretic behavior is usually a combined outcome of pseudoelasticity and SME. corresponding hysteresis is a function of both the temperature and stress conditions, as demon- strated in Fig. 3.1. Specically, as indicated in Fig. 3.1, under the temperature which SMAs are in the fully austenitic state void of stresses, SMAs can complete the full austenite-martensite- austenite transition by undergoing an isothermal loading and unloading cycle. This feature of SMA materials are widely exploited in structures as damping mechanisms [73, 74]. In contrast, in the absence of external loading, change in temperature could only lead to partial phase trans- formation in SMAs; namely the crystalline structure can only vary between the austenite and twinned martensite states, as illustrated in Fig. 3.1. Consequently, when exploited for its SME property to function as an actuator, an SMA wire needs to be preloaded so that the detwinned martensitic state can be restored each time during cooling after the wire's crystalline structure has morphed into the austenitic state due to heating [73, 74, 75]. Specically, the preloading has been realized through either employing a deadweight [76, 77] or a tensioned bias spring [56, 78, 53]. The latter is arguably better suited to be implemented 40 Fig. 3.2: Temperature-strain hysteresis loops under dierent stresses. A 127 m thick SMA wire with a nominal phase transition temperature of 90 C is heated to follow an identi- cal temperature prole bounded between 30 C and 130 C under 5 dierent constant stresses: 105; 115; 125; 135 and 145 MPa. The corresponding temperature-strain hysteresis loops, shown in red, blue, green, magenta and cyan, respectively, exhibit considerable disparities. on small-size autonomous robots for its light weight and simple installation, as demonstrated in the conceptual robotic design in Fig. 1.2. However in this case, the cyclic deformation occurred at the SMA wire will simultaneously trigger contraction and elongation at the bias spring, which will ultimately cause the loading condition on the SMA wire to uctuate according to Hooke's law. Existing research literatures have shown that change in stress could aect an SMA wire's hysteresis properties [53, 54, 55], a phenomenon we have also observed in our experiments. Fig- ure 3.2 presents 5 temperature-strain hysteresis loops obtained under 5 dierent constant stresses: 105; 115; 125; 135; 145 MPa, shown in red, blue, green, magenta and cyan, respectively. Since in these tests, a 127 m thick SMA wire with a nominal phase transition temperature of 90 C is heated to follow an identical temperature prole between 30 C and 130 C, the considerable discrepancies between each hysteresis loop are results of the dierent stresses applied. Evidently, the time-varying loading condition needs to be accounted for in the hysteresis modeling process rst so that adequate control strategy can be devised accordingly. 3.1.2 SMA Wire Hysteresis Modeling Overview Ultimately we envision achieving controlled deformation on catalytic-combustion powered SMA- based articial muscles. Here, for the purposes of characterizing the dynamics of SMA wires 41 subject to thermomechanical loads and devising suitable controllers to regulate their displace- ments, we rst conducted experiments in which SMA wires are electrically heated (joule heating). Therefore, following rst principles, the thermodynamics of an electrically-heated SMA wire can be described as the † mC p _ T =R()I 2 hA(TT 1 ); (3.1) whereC p is the specic heat of the NiTi material, h is the heat transfer coecient for convection in air,T 1 is the ambient temperature for convective losses, andm,R,I,A,T , and are the mass, resistivity, current, surface area, temperature, and mole fraction of Martensite (or Austenite) of the SMA wire, respectively. Consistently, the constitutive model of SMA's hysteretic behavior is generally formulated using _ = 1 E() _ +() _ T + _ (T;); (3.2) where and are the strain and stress of the SMA wire, respectively, E is the Young's modulus, is the coecient of linear thermal expansion, and is the contribution factor due to phase transformations [79]. Clearly, calculating instantaneous SMA deformation using (3.2) relies on an explicit knowledge of real-time value. However, as pointed out in Section 3.1.1, is a nonlinear time-varying variable that re ects SMA wire's hysteretic property when the wire undergoes loading and heating cycles, which is extremely challenging to quantify exactly in reality. In [79, 80, 81] for instances, was approximated using trigonometric, exponential relations and the Fermi-Dirac statistics. In general, two schools of methods have been exploited in hysteresis modeling: operator-based and dierential-based [82]. These models are phenomenological in nature as they are formulated to describe the empirical relationships between variables involved using mathematical expressions † Denotations of variables employed in Chapter 2 do not carry over to Chapter 3. 42 (dierent types of operators in these cases) instead of attempting to explain the physics behind the observed phenomena. Operator-based hysteresis models include the Preisach model [47, 48] and its derivatives, the Prandtl-Ishlinskii model [83, 84] and the Krasnosel'skii-Pokrovskii (KP) model [85, 57]. Dierential-based hysteresis models namely utilize dierential equations to characterizes hysteresis. Some of the widely-used dierential-based models include the Bouc-Wen model [86] and the Duhem model [87, 88]. The dierential hysteresis models are usually used within the constitutive relationship framework to estimate the composite of [88, 89], while the operator- based models can also be employed to formulate direct mappings from stimuli to strain output [47, 48, 44, 45]. In the past, we have selected to use a modied Preisach model to characterize SMA wire's hysteresis for its reliability and performance. More importantly, we developed a computationally ecient method to procure the inverse of the Preisach model in realizing precise deformation control of SMA wire-based actuators. The description and implementation of the proposed Preisach model is detailed in the following sections. 3.2 Preisach Model of SMA Hysteresis 3.2.1 Classic Preisach Model The classical Preisach model was developed in 1935 to describe hystereses with nonlocal memory found in the study of magnestism. Since then, many modications and generalizations have been published to describe hysteresis observed from a wide range of materials, including piezoceramics [51] and electromagnets [52], other than SMAs. As mentioned in Section 3.1.2, the operator-based Preisach model is formulated upon the concept of a rectangular relay operator ^ , known as a hysteron, depicted in Fig. 3.3. The hysteron operator maps the input U to a binary output, 0 or 1, at two switching values of input, and . In particular, as an increasing U exceeds the `on' value, ^ switches `on' and outputs 1; similarly whenU is decreasing, ^ will switch `o' and outputs 0 once U drops below . 43 Fig. 3.3: Hysteron operator. ^ outputs 1 once the increasing input U surpasses and starts to output 0 when the decreasing input reaches . The hysteron operator ^ is also assigned with an weighing function (;) that describes the relative contribution of each relay operator to the overall hysteresis. Therefore, provided a set of time-varying input signalU(t), the Preisach model calculates the outputf(t) of the system of interest following f(t) = ^ u(t) = ZZ > (;)^ u(t)dd: (3.3) Evidently, the classical Preisach model is suited to represent single-input{single-output (SISO) hysteresis, of which the future output only depends on past history of a single input. However, as we have pointed out in Section 3.1.1, when subject to time-varying stresses during actuation poten- tially due spring loading, SMA wire's hysteresis is a multiple-input{single-output (MISO) system with two dynamic inputs: temperature and stress. A theoretical method to model hysteresis with two inputs is briefed in [43] which relies on a substantially more demanding characterization pro- cess, making it impractical for real-time implementation. To this end, we introduced modication to the classical Preisach model. 3.2.2 Modied Preisach Model In the case of SMA wires, the phase transformation induced hysteresis is MISO, with the output strain being a function of two inputs, temperatureT and stress. When functioning as actuators, 44 Fig. 3.4: The graphic description of the modied Preisach model. In case of SMA wires, the total input u consists both temperature (T ) and loading ((t)) information since both contribute to the deformation of an SMA wire. The stress value is re ected in the weighting function (;;) while the hysteron is only a function of the temperature input T .. SMA wires are usually exploited for SME, in which case the temperature variation is the principal cause of the phase transformation. This feature justies modeling the hysteresis yet as an SISO system, with input T and output . Stress in our proposed modeling method, is treated as a correction factor. Specically, the Preisach hysteresis model is identied at a set of discrete constant stresses. To compute the instantaneous strain output under dynamic stress, the hysteresis model parameters are weighted by the stress applied and interpolated. The graphical presentation of the modied Preisach model is shown in Fig. 3.4 Under this scheme, the hysteron operator ^ , for any given stress value , maps the temper- ature input T (t) to a binary output of 0 or 1. Specically, during heating, ^ will switch `on' to output 1 uponT reaching; reciprocally during cooling, ^ will switch `o' to output 0, whenT drops below . Here, the set of and is dened as S =f(;)j 0 >>> 0 g, where 0 and 0 are the empirically determined upper and lower boundaries of T (t), referred to as upper and lower saturation temperatures, respectively. Therefore, provided with a set of time-varying 45 Fig. 3.5: Geometrical interpretation of the Preisach memory mechanism (a), (b), (c) The parameters 0 and 0 are the upper and lower saturation temperatures determined empirically, respectively. In (a), the temperature T is rst raised from 0 to 1 , before being reduced to 1 in (b). Successive heating and cooling will create a staircase-shaped L(t), as illustrated using bold line in (b) and (c). In all three gures, S + , the region within which the hysterons switched `on' is shown in shade; correspondingly, the region where the hysterons are turned `o' is labeled using S 0 . temperature and stress signals T and , the Preisach model calculates the instantaneous strain output following (t) = (T;)(t) = ZZ S (;;)^ [T (t)]dd; (3.4) where (;;) weighs ^ [T (t)] forf;g2 S at each instant t; is incorporated in the weight function to adjust for the stress condition. 3.2.3 Geometrical Interpretation of the Preisach Model A geometrical interpretation of (3.4) constructed upon the concept of a Preisach plane is usually employed to compute(t). A Preisach plane is dened as a triangle bounded by the line = as well as 0 and 0 , as shown in Fig. 3.5. Each pointf;g within the Preisach plane corresponds to a unique ^ , and ^ = 0;8f;g = 2S . 46 Under these premises, the Preisach model's memory mechanism can be described as follows. Under a constant stress, starting fromT = 0 , which corresponds to the state ^ = 0;8f;g2 S , T (t) is raised to 1 , during which, all the hysterons of 6 1 are then switched `on', reaching the state ^ = 1;8 6 1 , as illustrated in Fig. 3.5(a). Note that the Preisach plane is subsequently divided by the horizontal line T (t) = 1 into two regions: S + , within which all the hysterons are `on', and S 0 , within which hysterons remain `o'. Then, as T (t) is lowered from 1 to 1 , all the hysterons enclosed withinf;g2 ( 1 ; 1 ) are turned `o', i.e., = 0;8f;g2 ( 1 ; 1 ). Consequently, the updated S + region illustrated as the shaded area in Fig. 3.5(b) is now dened by the horizontal line 1 and a vertical line 1 . Following this convention, as T (t) alternates according to a sequence of successive local maxima and minima, k and k with k2 N, satisfying k < k1 and k > k1 , a staircase-shaped interface L(t) separating S + and S 0 will be formed, as demonstrated in Fig. 3.5(c). Meanwhile, (3.4) can be re-written as (t) = ZZ S + (t) (;;)dd: (3.5) Evidently, the shape of the interface L(t) stores the temperature input histories required to cal- culate the instantaneous strain output. 3.2.4 The Wiping-Out and Congruency Property Note that according to this memory mechanism, during heating, if T (t) exceeds any previously recorded local maxima at the discrete counter k (i.e., k > i , for i = 1; ;k 1), then the segment ofL(t) containing all pairsf i ; i g, fori = 1; ;k1 is erased. Similarly, during cooling, ifT (t) drops below any previously recorded local minima so that k < i , fori = 1; ;k1, the segment of L(t) associated withf i ; i g, for i = 1; ;k 1, is also erased. This phenomenon 47 Fig. 3.6: (a) Wiping-out property. At t =t 1 and t =t 2 , all past local maxima and minima are erased. For a current local maximum or minimum , all previous local maxima k or minimum k , where k < or k >, respectively, are wiped out. (b) Congruency property. All minor loops dened by the same local minimum and maximum bounds, [;], are congruent. re ects the wiping-output property of the Preisach model [43] and is graphically described in Fig. 3.6(a). Another inherited characteristics of the Preisach hysteresis model is known as the congruency property. It states that all minor hysteresis loops dened by the same minimum and maximum bounds [;] are congruent, as demonstrated in Fig. 3.6(b). Both the wiping-out and congru- ency properties are the necessary and sucient conditions to represent a rate-independent SISO hysteresis using the Preisach model [43]. In addition, the classical Preisach model requires the hysteresis to be rate-independent, which we would present the verication in later sections. 3.2.5 Numerical Implementation of the Preisach Model According to (3.5), solving (t) relies on the construction of a proper weight function (;;). This involves examining the increment of a rst order reversal curve (FORC), which is the strain dierence recorded from a heating curve originated fromT = 0 , reversed once to cool back down to T = 0 after reaching T = 0 , mathematically expressed as F (;;) =f 0()f 0 0(); (3.6) 48 Fig. 3.7: (a) & (b) Graphical interpretation of the rst order reversal curve (FORC) and its associated strain increment. (a) demonstrates the construction of the FORC on the Preisach plane while (b) is a temperature-strain diagram that illustrates the corresponding strain output of the same FORC. The shaded ( 0 ; 0 ) is the region within which the hysterons are switched `o' after the temperature reversal process. Integrating weighted hysterons over this region will lead to the strain increment valueF ( 0 ; 0 ;). Consistently,S + is the area within which the hysterons remained `on' and L(t) the bold line is the interface resulted from the temperature reversal. where F (;;) is the increment value, f 0() and f 0 0() are the strain recorded at T = 0 and T = 0 respectively, all weighted by stress. More importantly, the increment value is also equivalent to the surface integral of the weight function over the triangle ( 0 ; 0 ) F (;;) = ZZ ( 0 ; 0 ) (;;)dd: (3.7) This relationship is also graphically illustrated in Fig. 3.7. Accordingly, the weight function can be deduced by taking derivatives of a nite set of experimentally obtained FORC increments with respect to and . However, this process could further amplify the error and noise embedded in the experimental data [43]. To avoid this issue, we employ a numerical method that exploits the geometric conguration of S + region in the Preisach plane. Specically, when T (t) with n recorded local maxima and minima, is monotonically increasing, the last link ofL(t) is horizontal. 49 In this case, S + can be divided into n 1 trapezoidsQ k and 1 triangle, as illustrated in Fig. 3.8. Mathematically, ZZ S + (t) (;;)dd = n(t)1 X k=1 ZZ Q k (t) (;;)dd + ZZ (n;n) (;;)dd; (3.8) Furthermore, each trapezoidQ k is equivalent to the dierence between the two adjacent triangles, ( k ; k1 ) and ( k ; k ), as illustrated by the shaded exemplaryQ k in Fig. 3.8. Thus, according to (3.7), ZZ Q k (t) (;;)dd =F ( k ; k1 ;)F ( k ; k ;): (3.9) Subsequently, by substituting (3.7) and (3.8) into (3.9), the strain output associated with a monotonically increasing T (t) can be expressed as (t) = n(t)1 X k=1 [F ( k ; k1 ;)F ( k ; k ;)] +F ( n ; n1 ;); (3.10) where n =T (t). Similarly, for a monotonically decreasing T (t) withn memorized local maxima and minima, the last link ofL(t) is vertical. Accordingly,S + can then be divided inton trapezoids, yielding (t) = n(t) X k=1 [F ( k ; k1 ;)F ( k ; k ;)]; (3.11) where n =T (t). Evidently, this numerical method avoids dierentiating experimentally measured data. The actual implementation of (3.10) and (3.11) is carried out in the following steps. Firstly, under a constant stress m ; m2N, a nite set of FORC strain increments F ( 0 ; 0 ; m ), evenly spaced 50 Fig. 3.8: Numerical implementation of the Preisach model. In the case shown here, T is the monotonically increasing at time t and the last link of the staircase-shaped interface L(t), shown as the bold line is horizontal. Specically,T (t) = n . Accordingly,S + is divided inton1 trapezoids and 1 triangle, ( n ; n1 ). Each trapezoid is equivalent to the dierence between its two adjacent triangles, as exemplied by the shaded Q k , which equals to the dierence between ( k ; k1 ) and ( k ; k ). If T is monotonically decreasing at time t, the S + region can then be divided ton trapezoids. In both cases,n is the number of local temperature extrema recorded by T (t). across a set off 0 ; 0 g, is collected experimentally. Secondly, these values are then arranged to procure the square-gridded discretized surface over the Preisach plane to approximate the continuous surface of FORC increments, which contains the hysteresis parameters at a particular m . Strain increment values not included in the discrete setf 0 ; 0 g (i.e. within each square grid) can be calculated through linear interpolation. Step 1 and 2 are then repeated at a selected number of m s to create a set of discretized FORC strain increment surfaces. The complete set of the hysteresis parameters with stress bounded within the extrema of m is formulated by linearly interpolating between each individual FORC increment surface along the direction and takes the form of a hypersurface. Therefore when calculating the strain output using (3.10) and (3.11), impact to the hysteresis from time-varying loading is estimated by weighing each F (;;) term by the stress applied at the instance when and values are recorded. 51 3.3 Experimental Setup and Controller Design 3.3.1 Experimental Setup Ultimately, our goal is to realize deformation control of SMA wire actuators powered by hydro- carbon fuels through catalytic combustion, in which case temperature would be treated as the controlling signal. Evolved around this objective, the Preisach model we employed in our research maps temperature and stress inputs instead of current (or voltage) to strain outputs. Conse- quently, precise temperature and loading regulations of SMA wire are critical to collecting the FORC strain increment values, and therefore crucial to achieving accurate real-time deformation control. To this end, we developed the experimental setup shown in Fig. 3.9. In the earlier stage of the study, we electrically heated a 70 C Dynalloy (150 m NiTi) SMA wire by applying a potential (0 2:5 V) between the end supports, as shown in the red boxes of Fig. 3.9(a) and (b). To measure and control the temperature of the SMA wire, we initially installed a 127m diameter bare wire Omega type K thermocouple, at the midpoint of the SMA wire and employed a PID controller. For this arrangement, the root mean square error (RMSE) of T was between 6 C and 8 C during regulation. Such poor performance was primarily due to hardware limitations. The imsy nature of the bare wire thermocouple made it dicult to place and maintain its bead perfectly in contact with the SMA wire during phase transformations. Furthermore, since the SMA wire is electrically heated, there can only be one point of contact between the bare wire thermocouple and the SMA wire without creating a common ground and a short circuit with the thermoelectric eect. The limited contact surface area and installation uncertainty greatly reduced the accuracy of the temperature measurement. To improve the hardware setup, we installed three 127 m diameter Omega type T bare wire thermocouples evenly spaced out along the SMA wire, as shown in the red boxes of Fig. 3.9(a) and 52 Fig. 3.9: (a) CAD model and (b) picture of the experimental setup. The SMA wire is electrically heated and the temperature is measured by thermocouples, as highlighted by the red boxes in (a) and (b), and controlled using the adaptive scheme presented in Section 3.3.2. The stress is applied, measured, and controlled using an electromagnet, a load cell, and a PID controller, respectively, as highlighted by the blue boxes in (a) and (b). The contraction due to phase transformations is measured by a laser sensor, shown on the right in (a) and (b). (b). We averaged the temperature measurements from the three thermocouples, ltering out high- frequency disturbances. Each thermocouple was calibrated by measuring water at dierent steady- state temperatures between 20 C and 80 C using an Omega HH81A thermometer. Additionally, the SMA wire was coated with an M.G. Chemicals high-temperature epoxy that is electrically- insulating and thermally-conductive. To coat the SMA wire, the epoxy was brushed on under a microscope, inspected for smoothness, and cured at 65 C for 60 minutes. This type of coating allows the thermocouples to be wrapped around the SMA wire, increasing the contact surface area and therefore eectively improving the measurement accuracy without creating any short circuits. 53 30 90 150 210 270 330 Time (s) 20 40 60 80 Temperature (°C) RT initial final Fig. 3.10: Temperature controller eect. Temperature tracking performance comparison between the initial setup, composed of a PID controller with one type K thermocouple, and the nal setup, consisting of the adaptive controller presented in Section 3.3.2, three type T thermocouples, and electrically-insulating epoxy. Finally, the adaptive scheme presented in Section 3.3.2 was implemented. The nal RMSE ofT (t) was reduced to the range 0:09 0:25 C during regulation and to the range 0:14 0:36 C during tracking. A comparison between the performance of the initial setup and that of the nal set up is shown in Fig. 3.10. To load the SMA wire, we initially attached calibration weights to the wire. However, this arrangement could achieve neither dynamic loading nor precise loading control. To vary and measure the load, we designed and installed an electromagnetic loading system employing a Futek Miniature S Beam LSB200 load cell, as shown in the blue boxes of Fig. 3.9(a) and (b). The electromagnet loading system consists of an Apex rare earth magnet placed within a current carrying coil made of copper wire. According to Ampere's law, the electromagnet generates a magnetic eld whose magnitude and direction depends on the current. Its magnetic eld interacts with that of the permanent magnet, varying the stress in the SMA wire. The current in the electromagnetic coil is regulated by an Arduino Uno controller using pulse-width modulation. In addition, a 200 g calibration weight is attached to the SMA wire to increase the tension limit from 135 MPa to 270 MPa. The stress tracking and regulation in the SMA wire is realized with a PID controller. Here, we are neglecting the change in the wire's radius caused by strain variation during 54 SMA's isochoric phase transition process and therefore assuming a constant cross-sectional area of the wire for the stress calculation. The contraction of the wire is measured with a Keyence LK-503 laser sensor. All sensors sample at a frequency of 1 kHz and are connected to a Measurement Computing PCIM-DAS1602/16 data acquisition board, which is mounted on a target PC that communicates with a host PC via xPC Target 5.5 (R2013b). 3.3.2 Temepature Controller Design As discussed in Section 3.3.1, controlling the temperature of the SMA wire presents many chal- lenges. To reject the disturbance w and improve the quality of the reference tracking, we im- plemented an adaptive scheme based on [90], shown in Fig. 3.11(a). In Fig. 3.11(a), each H i is composed of a set of LTI controllers, including a PID structure K i , a disturbance model ^ G i , and an adaptively-tuned nite impulse response (FIR) lter F i . To form the adaptive scheme H, all H i are gain-scheduled over the temperature range 25 75 C. The parameters for K i , ^ G i , andF i are identied at 5 C increments along the operating temperature range. The PID controllerK i is tuned online to track the desired temperaturey r;i for the conguration presented in Fig. 3.11(b). There, the presence of w introduces variation in the temperature y, so K i is tuned to have zero-mean error. Although y generally tracks y r;i , the resulting RMSE of y is not small enough to allow for the precise temperature control required to identify output increments along FORCs. To further improve the performance of the controller, we identied a LTI ^ G i and ltered the modeled disturbance ^ w i with F i so that the RMSE of y i is minimized, thus creating the adaptive control commandu, as shown in Fig. 3.11(c). We accomplished this by rst performing least squares tting for over-parametrized SISO discrete-time systems. We then used balanced realization and truncation theory [67, 91, 92] to obtain a reduced-order model of 55 Fig. 3.11: (a) Adaptive control scheme for temperature tracking. P is the open-loop thermodynamics of the SMA wire, y r and y are the desired and true temperatures, respectively, andw is a disturbance modeling the combined eect from the phase transformation nonlinearities, input, output, and sensor disturbances. (b) PID controller tuning scheme. For the online tuning of K i , the future adaptive control command u is set to zero. The closed-loop transfer functionsM i ,G i , andS i are dened so thaty =M i y i;r +G i u+S i w and the closed-loop disturbance w i is dened as w i = S i w. (c) Filter tuning Control scheme showing (i) the implementation of a FIR lter F i and (ii) the adaptation of F i using a RLS algorithm, where the RMSE of y is minimized ^ G i . To adaptively tune F i , we used a recursive least squares (RLS) algorithm to minimize the cost function J e = p Ef^ e 2 ()g; (3.12) whereEfg represents the expected value of the random variable ^ e(t) at timet, associated with a random process assumed to be stationary and ergodic. For each temperature y r;i , F i of dierent lengths and initial variances are tested. Finally, once the RMSE ofy i is minimized for a particular y r;i , the parameters for K i , ^ G i , and F i are implemented in the gain-scheduling scheme shown in Fig. 3.11(a). At each operating temperature for which a gain-scheduled controller is designed, the improvement in performance from implementing the controller H i is substantial, as shown 56 Table 3.1: Performance of Gain-Scheduled Controllers. Reference Controller ConstantT K i (z) ConstantT H i (z) LinearT±2.5 ◦ C H i (z) T ( ◦ C) RMSE ( ◦ C) RMSE ( ◦ C) RMSE ( ◦ C) 25 0.12 0.09 0.29 30 0.27 0.12 0.14 35 0.19 0.11 0.28 40 0.20 0.10 0.36 45 0.19 0.16 0.30 50 0.31 0.21 0.25 55 0.35 0.25 0.31 60 0.44 0.25 0.27 65 0.25 0.14 0.27 70 0.32 0.13 0.27 75 0.33 0.24 0.29 in Table 3.1. There, the regulation performance of K i and H i are compared and the tracking performance of H i is presented. Once we have developed a reliable temperature and stress regulation setup and control algo- rithm, we rst performed tests to verify that the SMA's SME-induced hysteresis satises the 3 properties needed to be modeled using the Preisach model, namely rate-dependency, wiping-out and congruency. Evidently, Fig. 3.12(a) proves that at the heating rates we conducted our ex- periments, the SMA hysteresis is indeed rate-independent. Similarly, Fig. 3.12(b) and (c) show that SMA wire's hysteresis also satises the wiping-out and congruency properties. As it can be observed from all the temperature-time plots, the temperature tracking results are highly satisfying. 3.3.3 Apparatus and Temperature Controller Update For later stage in our study of SMA wires, we switched to electrically heating a a 58:42 mm long Dynalloy NiTi SMA wire with a diameter of 127m and a nominal phase transition temperature of 90 C since SMA wires with higher transition temperature were found out to be better suited for catalytic reaction. The lower and upper saturation temperatures 0 and 0 of this new wire 57 Fig. 3.12: (a), (b) and (c) Results of the rate-independency, wiping-out, and congru- ency experiments, respectively. All of these tests were conducted when a constant stress of 150 MPa was maintained at wire using the electromagnet mechanism and corresponding stress controller. The input reference (RT) and true (TT) temperatures are shown on the left while the resulting strain is shown on the right. were experimentally identied to be 30 C and 130 C, respectively. Accordingly, we reduced the calibration weight in the electromagnet setup from 200 g to 100 g so that comparable range of stress can be realized at the thinner wire. 58 Fig. 3.13: Adaptive gain-scheduling scheme for the control of the SMA wire's temper- ature. Here, T r (t d ) is the discrete-time temperature reference; T m (t d ) is the sampled measured temperature; each lter K i ; i = 1 21, composing the adaptive controller has a PID structure; and the blocks D/A and A/D denote digital-to-analog and analog-to-digital conversion, respec- tively. In this conguration, the discrete time t d coincides exactly with the continuous time t at each sampling instant; therefore, for purposes of analyzing the main inverse scheme, we assume t d = t. During operation, the lters K i ; i = 1 21, are gain-scheduled according to the values of T r , from 30 C to 130 C at 5 C intervals. In addition, we replaced the 127 m thick type T thermocouples with the thinner 50 m thick type K thermocouples to further reduce the thermocouple response time and improve mea- surement accuracy. As a result, a reduced-complexity adaptive-gain-scheduled control scheme shown in Fig. 3.13 was veried to be capable of producing temperature regulation results with comparable level of accuracy. This controlled is composed of 21 PID lters which are uniformly gain-scheduled over the temperature range [30 : 130] C (between identied 0 and 0 ) at 5 C increments. Each controller K i for i = 1; 2 21 is tuned experimentally to track the reference temperature when it falls within the corresponding set range. Intuitively, it makes sense that im- provement in the hardware setup and prociency in the wire coating as well as the thermocouple installation processes would demand less controller correction. 3.4 Hysteresis Characterization As explained in Section 3.2.5, the Preisach modeling of an SMA wire's hysteresis requires the formulation of surfaces comprised of FORC strain increment values identied at a set of discrete constant stresses. For the 150 m wire with a phase transition temperature of 70 C, we rst discretized the Preisach plane at 5 C increments along and . For each point ( i ; j ) in the discretized Preisach plane, an output increment associated with the FORCf i;j ( m ) is measured 59 Fig. 3.14: Sample measurement of an output increment along a FORC from a set of system identication experiments: F(75 C,50 C,150 MPa). Consistent with Fig. 3.12, the input reference (RT) and true (TT) temperatures are shown on the left; the resulting strain is shown on the right. Stress in this case is held constant at 150 MPa. at a particular stress m . Figure 3.14 illustrates an experiment measuring the output increment associated with the point (75 C; 50 C). There, the steady state-state strain is measured at the low saturation limit of the wire, 0 = 25 C, the FORC reversal point temperature, 0 = 75 C, as well as the end point temperature, 0 = 50 C. These quantities are used to calculatef 75 C (150 MPa), f 75 C;50 C (150 MPa) and F (75 C; 50 C; 150 MPa). The output increment surfaces f i ( k ), f ij ( m ), and F ( i ; j ; m ) are measured at all ( i ; j ) in the discretized Preisach plane for each m inf100; 125; 150; 175; 200 MPag. Then, smooth and continuous surfacesf ( m ),f ( m ), andF (;; m ) are interpolated from the out- put increment measurements using thin-plate splines. Figure 3.15(a) presents the experimentally obtainedF (;; 150 MPa). The complete set of parameters is identied by interpolating the con- stant stress surfaces F (;; m ), presented in Fig. 3.15(b) along the direction to formulate the corresponding hypersurfaces F (;;). These hypersurfaces are the parameters of the proposed modied Preisach model described by (3.4). The results from the model validation experiments conducted on the 150 m thick SMA wire with a 70 C nominal phase transition temperature are presented in Fig. 3.16(a) and (b) for the static and dynamic loading conditions, respectively. In the case of constant stress, the proposed model is shown to have predicted the strain output almost perfectly. For the case with dynamic 60 Fig. 3.15: (a) Example experimentally obtained mesh surface of FORC strain in- crement values. (b) Set of identied output increment surfaces. Specically, these are F (;; m ), for m 2f100; 125; 150; 175; 200 MPag. The results presented here were obtained on the 150 m SMA wire by employing the heating proles described in Fig. 3.14. 50 150 250 350 450 550 Time (s) 25.0 37.5 50.0 62.5 75.0 75 100 125 150 175 Stress (MPa) RT TT RS TS 50 150 250 350 450 550 Time (s) 0 0.01 0.02 0.03 0.04 Strain (mm/mm) true model 50 150 250 350 450 550 Time (s) 25.0 37.5 50.0 62.5 75.0 165 170 175 180 185 Stress (MPa) RT TT RS TS 50 150 250 350 450 550 Time (s) 0 0.01 0.02 0.03 0.04 Strain (mm/mm) true model Fig. 3.16: Results of the system validation experiments for (a) static and (b) dy- namic loading conditions. The input reference temperature (RT) and stress (RS) and the true temperature (TT) and stress (TS) are shown on the left. The true strains are calculated using the measured displacements and plotted together with the Preisach model calculated strain. loading, the general trend is tracked closely despite noticeable discrepancies observed at several peaks and troughs. Overall, the eectiveness of proposed SMA hysteresis modeling method is validated. 61 Fig. 3.17: Example FORC increment surface generation procedure. (a) Sample heating prole used in the identication process. Here, the SMA wire is heated between 30 C and 130 C following a sinusoidal curve under a constant stress of 145 MPa. The almost perfect matching between reference temperature (RT) and measured temperature (MT) validates the eectiveness of both the hardware setup and temperature control method. The strain measured is shown in blue. (b) The corresponding FORC strain increments extracted/interpolated from the data collected in (a). (c) Experimentally-identied discrete surface of FORC strain increments over a square-gridded mesh on the Preisach plane. Note that experiment procedure adopted to procure the FORC strain increments mesh surface is highly time-consuming as each grid value was identied following a heating path similar to that displayed in Fig. 3.14(a). This poses a drawback when we switched to experiment with the thinner SMA wire (127 m) with an identied working temperature range of [30 : 130] C. To that end, we introduce a more ecient FORC increment value identication method. Similarly, the Preisach plane is discretized along the and axes at 5 C increments to form a square mesh grid, setting up a total of 231 FORC increment values to be identied experimentally under each m . To obtain these values, we performed 21 experiments at each m , during which we heated up the SMA wire from 0 to each of the 21 grid values along the axis, before cooling the wire back down to 0 , following a sinusoidal temperature prole. A sample heating signal, together with the temperature tracking results and corresponding strain measurements are presented in Fig. 3.17(a). In this case, a constant stress of 145 MPa is maintained on the wire. Note that here, we choose to calculate the strain based on the absolute deformation of the SMA wire with respect to its length measured under 150 g of preload at room temperature. Consistently, contraction of the SMA is dened to be positive strain throughout this work. 62 Consequent to the heating prole selected, each experimentally obtained temperature-strain curve dened by its peak temperature 0 contains all the possible FORCs associated with 0 under m . Therefore, the descending segment of each time-strain curve contains the complete set of f 0 0( m ) values for 0 ranging between 0 and 0 . As a result, for each FORC dened by 0 and m , all f 0 0( m ) values of a discretized set of 0 can be extracted or estimated through interpolation, from the associated time-temperature and time-strain curves. The increment val- ues can then be computed by invoking (3.6). As an example, the 21 FORC strain increments, ranging fromF (130 C; 30 C; 145 MPa) toF (130 C; 130 C; 145 MPa) are obtained from the tem- perature signal and strain measurement presented in Fig. 3.17(a) and plotted against the axis in Fig. 3.17(b). Following the same procedure, the strain increment values from the other 20 FORCs under m = 145 MPa, with 0 ranging between 30 C and 125 C are procured and assembled to formulate the discretized FORC increment surface shown in Fig. 3.17(c). For pairsf 0 ; 0 g within any mesh grid, their increment values can be simply interpolated, using for example, the MATLAB function interp2. Consistent with proposed method to address dynamic stress's impact on hysteresis, the same identication process is then repeated for m = 105; 115; 125; 135 MPa to generate a total of 5 discretized FORC increments surfaces, presented together in Fig. 3.18. These 5 surfaces contain all the hysteresis model parameters under 5 dierent constant loads. Noticeable incongruency can be observed between each surface, which veries the SMA wire hysteresis' stress-dependency and thus validates the necessity to address the time-varying load's impact on SMA's deformation con- trol. Similarly, the complete set of parameters that fully characterizes the SMA wire's hysteresis under stress in the range [105 : 145] MPa is then formulated by linearly interpolating between the 5 constant-stress FORC increment surfaces along the direction, creating an FORC strain increment hypersurface. It is known that cyclic thermomechanical loading would induce irreversible changes in SMA wire's property and hysteresis characteristics [93, 94, 95]. To investigate this phenomenon and 63 Fig. 3.18: Set of experimentally identied FORC strain increment surfaces. F ( 0 ; 0 ; m ), for m = 105; 115; 125; 135; 145 MPa. Note that the results shown here are collected from the 127 C thick SMA wire with a nominal phase transition temperature of 90 C. estimate its impact on the accuracy of identied hysteresis model parameters, we performed a long-duration test on a brand new SMA wire with the same diameter and nominal phase transition temperature. The results show that the hysteresis behavior undergoes marked changes during the earlier cycles of thermomechanical loading before settling into a rather steady state, which is in agreement with ndings in [94]. This implies that if the model parameters are identied after the SMA wire's property has mostly stabilized, they could possibly remain accurate in describing the wire's true dynamics for extended cycles of thermomechanical loading. Based on the runtime of temperature and stress controller tuning tests we had performed on the SMA wire, according to our long-duration test results, the wire's property had likely already evolved into a steady condition by the time we initialized the system identication process. Consequently, the procured FORC strain increment hypersurface could still remain accurate even after repeated position control experiments. Details of the SMA wire long-duration experiment can be found in the Appendix A.3. 64 3.5 Preisach Model Inversion 3.5.1 Method Derivation From a system control perspective, SMA wires' stress-dependent hysteresis needs to be compen- sated for in order for the wire to provide eective and precise actuation. Intuitively, an inverse of the identied Preisach model , 1 , can be possibly exploited to mitigate the hysteresis eect. Upon receiving a set of reference strain r and stress r signals, the inversion algorithm should be capable of generating a reference temperature prole T out , that once applied to the SMA wire together with r , would lead to a strain output that closely tracks r . Thus, the Preisach model inversion also presents a MISO system, with two time-varying inputs r and r , and an output T out : T out (t) = 1 ( r ; r )(t): (3.13) Among the well-studied operator-based hysteresis models, only the Prandtl-Ishlinskii model is analytically invertible [83, 84]. In comparison, numerical methods are usually used to solve the Preisach model inversions [44, 52, 96]. In some cases, this type of approach relies on computationally-intensive iterative searching process. The inverse scheme introduced in this study exploits the transformation of FORC strain increment hypersurface derived in Section 3.4 and avoids complex iterative calculations. In the derivation of the inverse mapping, similar to the Preisach model development described by (3.10) and (3.11), we need to consider the two cases depending on whether SMA wire is subject to heating or cooling, separately. In agreement with the convention dened in Section 3.4, heating caused contraction in the SMA wire is considered increasing in strain while cooling initiated elongation is treated as reduction in strain. Although change in r can also cause wire deformation, within the range of stress we have modeled the SMA wire's dynamics, its overall impact is less 65 Fig. 3.19: Inversion bases. (a) -inverse basis and (b) -inverse basis for m = 145 MPa. (c) The set of discrete -inverse bases and (d) -inverse bases formulated at m = 105; 115; 125; 135; 145 MPa. pronounced than that from temperature uctuation. Therefore in this study, we are assessing the movement ofT out (t) by evaluating _ r (t) alone. A part of the ongoing research objective is focusing on integrating variations in both r and r to determine the temperature trend. That being said, here, _ r (t)6 0 implies monotonically decreasing temperature at t; then by referring to (3.11), we can express T out (t) as T out (t) = 1 ( n ; r (t) +S cooling (t); r (t)); (3.14) where S cooling (t) = n(t)1 X k=1 [F ( k ; k1 ; r )F ( k ; k ; r )] +F ( n ; n1 ; r ): (3.15) 1 is the inverse function introduced to solve for T out when a decreasing r is detected; n is the most recent local maximum recorded. Evidently, 1 bears the past temperature and strain 66 histories. From a purely geometric perspective, nding a solution to 1 involves rst locating a specic surface from the FORC strain increment hypersurface created in Section 3.4 according to r , before identifying a unique FORC based on n . From there, solving for T out is equivalent to searching for the end value of the identied FORC whose peak (reversal) point temperature value is n , so that the corresponding strain increment equals to r (t) +S cooling (t). Following the same approach, _ (t) > 0 indicates monotonically increasing temperature at t; thus by invoking (3.10), T out (t) can be formulated as T out (t) = 1 ( n1 ; r (t)S heating (t); r (t)); (3.16) where S heating (t) = n(t)1 X k=1 [F ( k ; k1 ; r )F ( k ; k ; r )]: (3.17) Analogous to the previous case, 1 denotes the inverse procedure when an increasing r is detected and n1 is the most recent local minimum recorded. 1 also bears the past temperature and strain histories. Similarly, solving 1 requires rst identifying a specic surface from the FORC strain increment hypersurface according to r ; afterward, a unique FORC can be selected based on n1 . From there, calculating T out is equivalent to searching for the peak (reversal) point temperature of the identied FORC whose end point temperature value is n1 , so that the resulted strain increment equals to r (t)S heating (t). In summary, the Preisach model inversion process can be described as the following. First, based on the sign of _ (t), either 1 or 1 is selected to solve for a monotonically increasing or decreasing T out , respectively. At each sampling step, the history of temperature extrema is updated with any newly detected extremal values. This process is also closely examined to monitor the occurrence of the wiping-out eect. Meanwhile,S cooling (t) andS heating (t) are computed based 67 on the history of temperature extrema recorded and r (t) using the FORC strain increment hypersurface. From there,T out (t) can be calculated using either (3.14) or (3.16) and registered to the temperature history which is required to continue the inversion process for the next sampling moment. 3.5.2 Inversion Method Realization Real-time implementation of the described Preisach model inversion process relies on a method capable of solving (3.14) and (3.16) within each sampling period. In [51, 52, 76], this process involves rst locating a region on the Preisach plane where the solution may lie, before invoking computationally-heavy iterative interpolations to nd out the specic value. Here, we introduce a method that provides almost-exact numerical solutions to (3.14) and (3.16) without involving computationally-heavy iterative interpolations. Recall from Section 3.5.1, solving 1 and 1 requires locating a specic curve from the FORC strain increment hypersurface, based on the instantaneous stress value r (t) and the most recent temperature extremum recorded; T out (t) is either the reversal or end point temperature of the identied FORC. Evidently, the complexity of computation involved can be reduced if the FORC hypersurface is inverted to form 2 additional hypersurfaces serving as look-up tables, one with entriesf 0 ;F;g and output 0 , referred to as the inverse base, and the other one with entriesf 0 ;F;g and output 0 , denoted as the inverse base, respectively. The process to formulate these two inverse bases is described in the following paragraph. In Section 3.4, the FORC strain increment hypersurface is created from linearly interpolating between 5 experimentally-obtained constant-stress FORC increment surfaces along the direc- tion. Consistently, each of the ve surfaces is individually inverted and arranged before being interpolated along the direction to create the two inverse hypersurface bases. Additionally, a data-density-enhancement procedure is performed prior to the inversion process in order to prevent potential distortion in the inverted surfaces. Recall that each of the original discretized 68 FORC increment surfaces consists of 231 empirically determined strain increments evenly spaced over the Preisach plane according to a 21 21 square mesh grid. The data-density-enhancement process thus involves interpolating within each existing grid to construct a new 201 201 grid evenly spaced across thef 0 ; 0 g plane, which corresponds to 20301 experimentally-obtained and interpolated data points per surface. From there, each density-enhanced FORC strain increment surface is rotated 90 clockwise about the line 0 = 130 C rst, and linearly-tted (using MAT- LAB function sftool) with respect to thef 0 ;Fg plane. The outcome is a set of ve inverse surfaces evenly discretized along the 0 and F axes. Similarly, to create the set of inverse sur- faces, these density-enhanced surfaces are rotated 90 counterclockwise about the line 0 = 30 C and linearly-tted with respect to thef 0 ;Fg surface to ensure even discretization along the 0 and F axes. As an example, the and inverse surfaces corresponding to m = 145 MPa are shown in Fig. 3.19(a) and (b) while the ve discretized alpha and ve discretized beta inverse surfaces are demonstrated in Fig. 3.19(c) and (d), respectively. The 5 constant-stress and inverse surfaces are then linearly interpolated along the direction to formulate the and hypersurface inverse bases. Therefore, in the process of solving 1 or 1 , r (t) is examined rst within each sampling time to pinpoint the proper or inverse surface from the two hypersurface inverse bases. From there, with the most recently recorded temperature extremum n1 (or n ) and the updated r (t)S heating (t) (or r (t) +S cooling (t)),T out (t) can be quickly located from the identied inverse base using a simple bilinear-interpolation algorithm like the one used in [97]. Subsequently, the time required to compute each T out (t) by interpolating the inverse hypersurfaces is found out to be as low as 8:59 10 6 s, using MATLAB function timeit performed on a commercial laptop simultaneously engaged in other tasks. 69 Fig. 3.20: Block diagram illustrating the feedforward position control structure used in real-time simulation. is the Preiscah hysteresis model identied in Section 3.4. The reference temperature T r obtained through the inverse process together with the time-varying reference stress r are the input signals evaluated by to generate the simulated strain output s . 3.6 Position Control Implementation 3.6.1 Position Control Simulations and Results The proposed Preisach-model-inversion-based position control algorithm is rst tested through a series of real-time simulations following the scheme demonstrated in block diagram shown in Fig. 3.20. Accordingly, the control objective is to have the SMA wire track a reference strain sequence r under non-constant external loading. Specically, r and a time-varying sequence of r are introduced as inputs to the Preisach model inversion algorithm 1 derived in Section 3.5.1. The temperature proleT r generated by 1 , together with r are then evaluated by the Preisach hysteresis model identied in Section 3.4 to compute the simulated strain output s . In all simulations performed, r is bounded within 0 and the empirically-identied maximum attainable strain under = 105 MPa. Moreover, r is conned within the spectrum of stresses the wire's dynamics was modeled with, [105 : 145] MPa. Consequently, any strain and stress input sequences within the selected range can be mapped to a bounded and unique reference temperature through the inversion algorithm. Furthermore, all real-time simulations are performed at a sampling rate of 2:5 kHz using the same xPC target setup described in Section 3.3.1. Here, 2 sample results are presented in Fig. 3.21(a) and (b), respectively. In the rst case, reference strain (RS), shown in blue in Fig. 3.21(a) Part I is a sinusoidal signal; reference loading (RL) shown in black in Fig. 3.21(a) Part II is also a sinusoidal signal oscillating at a dierent 70 Fig. 3.21: Real-time simulation results of purposed SMA position control scheme. (a) Simulation results for tracking a sinusoidal-shaped reference strain under a sinusoidal-shaped stress prole. (b) Simulation results for tracking a strain sequence that oscillates between ran- domly selected peaks and troughs while subject to a random stress prole. In both cases, reference strain (RS) is shown in blue and simulated strain (SS) is shown in red in Part I, while the reference loading (RL) and simulated temperature (ST) are presented in black and magenta in Part II and III, respectively. frequency; in Part III, the inverse algorithm generated simulated temperature (ST) is presented in magenta. The simulated strain (SS) is plotted together with RS in red in Part I. For the second set of simulation results presented in Fig. 3.21(b), RS and RL are selected to be oscillating between two sets of randomly selected peak and trough values. In both cases, the matching between the SS and RS signals is almost perfect, providing convincing evidence for the eectiveness of position control method. More importantly, the fact that all simulations are performed in real-time at a sampling period of 0:0004 s demonstrates its computational eciency which makes possible of the method's implementation in real-time experiments. 71 Fig. 3.22: Real-time simulation results of purposed SMA position control scheme under constant stress. (a) Simulation results for a sinusoidal-shaped reference strain signal. (b) Simulation results for a randomly generated reference strain signal. In both (a) and (b), the reference and simulated strain signals are shown in blue, and the simulated temperature signal is shown in red. Note that the identied hysteresis model of a dierent wire (same dimension and nominal phase transition temperature) was used in these simulations. An additional set of position control simulation results based on the identied characteriza- tion of another wire (same dimension and nominal phase transition temperature) is presented in Fig. 3.22(a) and (b). Here, the simulated stress is kept to be constant. In Fig. 3.22(a), the RS is a sinusoidal signal while in Fig. 3.22(b), the RS is colored noise randomly generated using Matlab. Consistent with the results observed in Fig. 3.21, the SS signals match almost perfectly with the RS signals. 3.6.2 Position Control Experiments and Results In real-time position control experiments, the Preisach model inversion algorithm is employed as a feedforward controller in a scheme similar to that described in Fig. 3.20. However, T r and r are instead imposed onto the SMA wire through the experimental setup and regulated using the temperature and stress feedback controllers described in Section 3.3. Since the output strain in this case cannot be directly measured, it is calculated using the deformation measured by the laser sensor. Consistently, all the experiments are also run at a sampling rate of 2:5 kHz. To allow for sucient computation time within each sampling step, a second target computer is utilized to 72 Table 3.2: Summary of position control results under constant stresses. RL MPa SPI SPII SPIII SPIV RMSE %of RMSE %of RMSE %of RMSE %of (×10 −3 ) max. ε r (×10 −3 ) max. ε r (×10 −3 ) max. ε r (×10 −3 ) max. ε r 105 2.60 6.89 2.22 6.19 2.42 6.44 2.81 7.46 115 2.48 6.59 1.92 5.10 1.90 5.04 1.69 4.50 125 2.98 7.91 2.96 7.88 1.65 4.38 1.68 4.47 135 2.31 6.15 2.24 5.96 1.51 4.02 2.30 6.11 145 1.96 5.21 1.83 4.87 1.52 4.04 1.93 5.14 operate the inversion algorithm while the other target computer performs data collections as well as temperature and stress regulations. Data are transmitted between the two target computers through raw Ethernet packets using dedicated Ethernet cards. Prior to incorporating time-varying stress into to the position control task, the inverse algo- rithm is rst tested to track a series of dierent reference strain sequences under the ve constant stresses m = 105; 115; 125; 135; 145 MPa. Specically, four sinusoidal-based reference strain proles are employed in this group of test: a sinusoidal signal oscillating at 1=17 Hz, a damped sinusoidal signal, an amplifying sinusoidal signal and a signal oscillating within a set of randomly selected peak and trough points, denoted as strain prole (SP) I, II, III and IV, respectively. Consistent with the simulation process, magnitude of SPs is limited within [0 : 0:0376], the empirically-identied maximum strain attainable under 105 MPa. An exemplary graphic presen- tation of the strain, loading and temperature tracking performances corresponding to SP III at m = 135 MPa is shown in Part I, II and III, respectively, in Fig. 3.23. The root mean square error (RMSE) between RS and measured strain (MS) and such values as percentages of the maximum attainable strain (% of max. r ) collected from this series of tests are presented in Table 3.2. As can be seen in Fig. 3.23 Part II and III, RL shown in green, ST shown in cyan, are tracked almost perfectly by the measured loading (ML) shown in black, and measured temperature (MT) shown in magenta, respectively. Comparable stress and temperature tracking performances are observed at all the experiments conducted, and are presented in Appendix A.2. The near-perfect 73 Fig. 3.23: Sample position control under constant stress results. Here, SP III (amplifying sinusoidal signal) is tracked as the reference strain when a constant stress of 135 MPa is maintained on the wire. The reference and measured strains, RS and MS, shown in blue and red, are presented in Part I; the reference and measured loadings, RL and ML, shown in green and black, are demonstrated in Part II; and the simulated and measured temperatures, ST and MT, shown in cyan and magenta, are displayed in Part III. matching between RS and MS, shown in blue and red, can also be observed from Fig. 3.23 Part I. Quantitatively, the RMSE between RS and MS in Fig. 3.23 equals to 1:51 10 3 , which translates to approximately 4:02% of the maximum r . Since both the temperature and stress feedback controllers performed well, the minor discrepancy between RS and MS is most likely caused by the mismatch between the true dynamics and the Preisach modeling of the SMA wire hysteresis. Overall, comparable strain tracking results are recorded across all the constant stress position control experiments, as evidenced by the summary listed in Table 3.2. The average RMSE between RSs and MSs is 2:15 10 3 with a standard deviation of 0:46 10 3 , which translates to an average error within 5:71 1:22% of the peak r value tested. Conclusively, the Preisach-model-inversion-based feedforward controller is proven to be feasible and eective in real-time experiments. Furthermore, each of the ve FORC strain increment surfaces and their corresponding inverse bases can in general accurately capture the SMA wire's true hysteresis characteristics at their respective loading stresses. Since the strain increment and inverse bases hypersurfaces are constructed based on these individual surfaces, their accuracy is fundamental to the successful realization of the actuator's position control at the presence of time-varying stress. 74 Naturally, the results of these constant-stress experiments also serve as benchmarks to evaluate the controller performance when dynamic loading is introduced. Results from an example series of SMA wire position control experiments under time-varying stress are presented in Fig. 3.24(a)-(f). In these cases, the RS to be tracked is SP IV and six types of dierent loading proles (LP) are applied onto the SMA wire: two sinusoidal-shaped signals oscillating between 105 and 145 MPa at 1=17 Hz and 1=47 Hz, a pair of monotonically increasing and decreasing linear signals, and two signals oscillating between two dierent sets of peak and trough values bounded within 105 and 145 MPa, denoted as LP I, II, III IV, V and VI, respectively. Consistent with Fig. 3.23, the strain, stress and temperature tracking results are presented in Part I, II and III in each subgure, shown in red and labeled as MS1, ML1, ST1 and MT1, respectively, while RS and RL are shown in black. To demonstrate the necessity of addressing the time-varying stress in regulating the deformation of the SMA wire as well as to validate the ecacy of the proposed control scheme, we also performed a series of experiments in which the dynamic loading condition was neglected. Specically, instead of adjusting for the time- varying loading proles by invoking the FORC strain increment hypersurface and its hypersurface inverse bases, only a single FORC strain increment surface and inverse bases formulated at a constant stress was utilized to compute the temperature output solely based on the RS signal. As comparison, example results from this series of experiments, shown in blue and labeled as MS2, ML2, ST2 and MT2, are also presented in Fig. 3.24(a)-(f). Consistently, SP IV and LP I through VI are the strain and loading references, while the hysteresis parameters collected at 145 MPa are employed in the example experiments presented here. The Preisach inversion algorithm is observed to have generated dierent ST1s to track the same RS in order to adjust for the six dierent stress signals applied. Again, ST1 in all tests are tracked almost perfectly by MT1; the matching between RL1 and ML1 is also satisfying, despite minor incongruencies. Most importantly, MS1 can be seen to match well with RS across all six cases, although clear but minor discrepancies can be detected. Quantitatively, the six tests shown here 75 Fig. 3.24: Sample position control under time-varying stresses results. Here, SP IV is selected as the reference strain to be tracked under 6 dierent LPs. Experimental results corre- sponding to each of the 6 LPs are displayed in (a) through (f). Consistent with Fig. 3.23, the strain, stress and temperature tracking performances are presented in Part I, II and III, respec- tively in each subgure. Similarly, RS and RL, marked using black circles, are the reference strain and loading signals; MS1, ML1, ST1 and MT1, shown in red, are the measured strain and loading, simulated and measured temperature recorded in the experiments. To further demonstrate the necessity and ecacy of the proposed position control method in adjusting for the dynamic stress, we performed comparative tests in which we attempted to track the same SP while overlooking the loading condition; specically in these cases, temperature output was computed solely based on RS. Corresponding results are shown in blue and labeled as MS2, ML2, ST2 and MT2 in each subgure. Note that in these example experiments, the FORC strain increment surface and inverse bases extracted at the constant stress 145 MPa were utilized to compute temperature output, ST2. 76 Table 3.3: Position control results under time-varying stress. Loading Profile SP I SP II SP III SP IV RMSE %of RMSE %of RMSE %of RMSE %of (×10 −3 ) max. ε r (×10 −3 ) max. ε r (×10 −3 ) max. ε r (×10 −3 ) max. ε r LP I 1.82 4.82 2.21 5.87 2.21 5.89 2.12 5.64 LP II 2.41 6.40 2.37 6.29 1.84 4.89 2.27 6.02 LP III 1.91 5.07 2.16 5.73 1.95 5.19 2.21 5.89 LP IV 1.99 5.28 1.53 4.07 1.70 4.51 2.05 5.44 LP V 2.18 5.79 1.87 4.98 2.33 6.20 2.16 5.74 LP VI 1.78 4.73 1.61 4.28 1.74 4.63 1.95 5.19 recorded an average RMSE between RS and MS1 of 2:1310 3 , which translates to approximately 5:65% of the maximum r value. Experimental results of tracking all four SPs under the six LPs are summarized in Table 3.3. Clearly, comparable position control performance is achieved across all the strain and loading proles tested, registering an average RMSE of 2:0210 3 with a standard deviation of 0:2510 3 , which equals to 5:360:65% of the maximum r value. The comprehensive graphic presentation of these experimental results are also included in Appendix A.2. In comparison, in cases where time-varying stress is overlooked, the strain tracking results deteriorated substantially. Since only a constant stress FORC strain increment surface and inverse bases were used in computing the temperature output, given the same RS signal (SP IV), ST2s are identical across Fig. 3.24(a)-(f), Part III, despite dierent loading conditions applied, as expected. Temperature and stress are also closely tracked in this series of experiments, evidenced by the close matching between ST2 and MT2, as well as between RL and ML2. However, signicant discrepancies between RS and MS2 can be observed across all the subgures. Numerically, the average RMSE between RS and MS2 among these six cases equal to 4:8510 3 , or approximately 12:9% of the maximum r value, which is over 127% higher than the value recorded when dynamic loading condition is adjusted for. Table 3.4 summarizes the strain tracking performances of SP IV where the dynamic loading conditions were neglected. As the header of each column suggests, the constant stress FORC strain increment surface and inverse bases formulated at 105, 115, 125, 135 and 145 MPa were individually employed and tested in the feedforward position control scheme 77 Table 3.4: Sample position control results neglecting time-varying stress (SP IV). Loading Profile 105 MPa 115 MPa 125 MPa 135 MPa 145 MPa Base Base Base Base Base RMSE %of RMSE %of RMSE %of RMSE %of RMSE %of (× max. (× max. (× max. (× max. (× max. 10 −3 ) ε r 10 −3 ) ε r 10 −3 ) ε r 10 −3 ) ε r 10 −3 ) ε r LP I 5.11 13.59 4.70 12.51 4.37 11.64 3.02 8.28 3.97 10.57 LP II 4.93 13.12 5.08 12.52 4.14 11.00 3.08 8.19 4.67 12.41 LP III 3.73 9.92 4.40 11.71 3.29 8.75 2.48 6.59 4.18 11.11 LP IV 3.74 9.94 3.54 9.42 3.01 7.99 3.20 8.52 5.48 14.58 LP V 3.53 9.38 3.35 8.91 2.22 5.89 3.06 8.13 6.12 16.27 LP VI 3.02 8.04 3.18 8.46 2.06 5.48 2.69 7.15 4.68 12.44 under all six LPs. The complete graphic presentation of these experimental results can be found in Appendix A.2. Overall, these twenty-four tests recorded an average RMSE between RS and MS2 of 3:80 10 3 with a standard deviation of 1:00 10 3 , which translates to 10:10 2:66% of the maximum r value. Clearly, the position control performance worsened when time-varying loading condition is neglected, as the tracking errors (RMSE) recorded in Table 3.4 have all exceeded the values registered in the 7 th column in Table 3.3, especially when FORC strain increment surface and inverse bases extracted at 105 and 145 MPa were exploited in the position controller. Similar outcomes have been observed when dierent SPs are tracked. In conclusion, the impact on SMA wire's strain output from time-varying stress cannot be overlooked and the proposed position control method has been proven to be highly eective in mitigating SMA wire's hysteresis under dynamic loading condition during actuation. Since the Preisach-model-inversion-based algorithm is realized in a feedforward controller, given that both the temperature and stress signals were well tracked during all the experiments, the position con- trol errors are likely caused by the mismatch between the identied model and the true dynamics of the SMA wire. This is a plausible explanation because commonly, the performance of a feed- forward controller relies on the accuracy of the system modeling. The magnitude of mismatch can 78 be potentially reduced if a larger number of experimentally obtained FORC strain increment sur- faces identied at a denser selection of stresses are employed in the inversion algorithm. Although repeated thermomechanical loading is known to cause hysteresis property change in SMA wires and can potentially aggravates model mismatch in the proposed control scheme, the position con- trol performances remained consistent after over 220 experiments with an accumulated runtime exceeding 26; 000 s. This could possibly be explained by ndings from the long-duration test we performed. As mentioned in Section 3.4, the property of the SMA wire had likely already settled into a stable state before we started characterizing the hysteresis model parameters, due to the temperature and stress controllers tuning process. Therefore, the identied Preisach model was able to remain accurate in describing the wire's hysteresis dynamics even after repeated thermo- mechanical loading cycles. However, our long-duration test has its limit. To formulate a detailed understanding of SMA wire's hysteresis property change due to repeated thermomechanical load- ing will require a more comprehensive and longer experiments carried out with a larger sample size. Such test could potentially generate a fatigue factor that can be exploited to reduce the mismatch between the hysteresis model and the true wire dynamics as more tests accumulate. This is a matter of current and future research. Across all the position control experiments performed, we logged an average task-execution- time (TET ) of 2:8610 4 s and a maximum TET of 2:9310 4 s on the xPC Target running the inverse algorithm, which remained well within the 410 4 s step time under the 2:5 kHz sampling rate we imposed. The other xPC Target tasked with data collection and temperature and stress control recorded an average and maximum TET of 5:57 10 5 s and 7:20 10 5 s, respectively. Evidently, the proposed feedfoward hysteresis compensation method requires minimal computa- tion time and is therefore well suited for real-time robotic applications. In case for autonomous robots with limited computation capacities, the sampling rate can be adjusted accordingly. 79 3.7 Summary In Chapter 3, we rst introduced a method based on a modied Preisach model to characterize the hysteresis of SMA wire based actuators. Specically, we constructed a hypersurface consists of FORC strain increment surfaces obtained at a set of discretized stresses. To account for the im- pact of time-varying stress on SMA wire's displacement, the parameters within the hypersurface are weighted by the instantaneous stress value. To create and experimentally validate the pro- posed model, we developed a reliable apparatus and control algorithm, achieving highly accurate temperature and stress regulation. We also proposed and tested a Preisach-model-inversion-based position control method for SMA wire actuators subject to time-varying stress. The previously identied hysteresis model was processed to formulate the mapping bases for the inversion algo- rithm which is implemented in a feedforward control scheme. The suitability and eectiveness of proposed position regulation method were demonstrated through a series of real-time simulations and experiments. Note that throughout this study, we emphasized on using temperature and stress instead of current or voltage as both the parameters to formulate the hysteresis model and control input that modulates the deformation of SMA wires. Consequently, the modeling and displacement control strategy can be readily employed to monitor the deformation of SMA wires driven by means other than joule heating, for instance, catalytic reaction. Furthermore, the only sensing apparatuses required by the proposed position control method are lightweight and compact ther- mocouples and load cell, which can be easily modied to t onto cm-scale autonomous robots. In addition, the hysteresis identication study has provided invaluable insights into SMA-based actuator design. Instead of reaching the maximum deformation possible by heating the SMA wire between its lower and higher saturation temperature, signicantly swifter actuation at the cost of reduced displacement magnitude can be realized if the SMA wire is heated to follow a minor 80 loop temperature-strain hysteresis prole. Such feature is explored with corresponding ndings presented in Chapter 4. 81 Chapter 4 Fast Actuation of SMA-Based Articial Muscles Our research on characterizing SMA wire based actuators' dynamics and designing ecient con- troller to regulate their deformation originates from the aspiration to replace conventional battery- powered actuation mechanism with SMA-based articial muscles driven by catalytic reaction and therefore realize full autonomy on mm-cm scale robots. Despite their unmatched power-to-weight ratio, SMA-based actuators are known to have relatively low actuation frequency, which limits their applications in areas where speed is prioritized [6, 72]. Specically, SMA actuators' actua- tion rate is hindered by its cooling speed, as the energy removal rate is limited by the process of heat conduction and convection. Figure 4.1 illustrates such behavior observed from experiments we conducted. In this case, an SMA wire is heated between a prescribed temperature bandwidth while experiencing only natural cooling. Clearly, heating appears to be substantially faster than cooling. To improve cooling rate and thus the overall actuation frequency, researchers have exper- imented with active cooling methods including forced air [98, 99, 100], liquid cooling [101, 102], as well as heat sinks and conductive materials [103, 104, 105]. To tackle this problem, we took on an approach focusing on exploiting the characteristics of SMA wire's hysteresis behavior. The related ndings are detailed in this chapter. 82 Fig. 4.1: From top to bottom, the temperature and displacement histories of an electrically-heated SMA wire led to cool naturally, respectively. A bang-bang controller was implemented to oscillate the temperature of the wire between 77 C and 80 C. 4.1 Major and Minor Hysteresis Loop The hysteretic temperature-strain relationship displayed by SMAs are commonly described as hysteresis loops [43]. Under a constant stress, the largest hysteresis loop bounded by the SMA's lower and upper saturation temperatures is known as the major loop, which corresponds to largest deformation (strain) attainable from the wire. In Fig. 1.3, an experimentally measured major hys- teresis loop is shown in red. Naturally, under the identical loading condition, as the temperature bandwidth shrinks, the strain output of the SMA wire is expected to decrease accordingly. The reduced hysteresis loop is referred to as the minor loop. Two minor loops of dierent sizes are shown in blue in Fig. 1.3. Note that shape of the SMA major hysteresis loop is characterized with gradual rise/fall near its saturation temperatures but steep slopes neighboring its nominal phase transition temperature. This implies that within certain range, a decrease in the heating bandwidth would lead to only 83 disproportionally minor reduction in the strain output. For example as can be observed from Fig. 1.3, reducing the heating temperature bandwidth from [30 : 130] C by 50%, to [50 : 100] C would only lead to an approximately 13% decrease in strain output, from 4% to 3:3%. Further reducing the bandwidth to [60 : 85] C will lead to a strain output of 2:1%, indicating that a 75% temperature range reduction (from the saturation temperature bandwidth) would only cause a less than 50% decrease in displacement. Reducing the heating temperature bandwidth will naturally cause the corresponding cyclic heating/cooling time to fall. Hence, the envisioned high-frequency SMA wire actuation strategy stems from the possibility of exploiting the minor hysteresis loop. Namely, signicantly faster cyclic deformation can be achieved at SMA-based actuators controlled to operate within properly selected minor hysteresis loops. 4.2 Multiplier Mechanism 4.2.1 Displacement and Frequency Multiplication Since SMA wires of the same diameter and identical chemical composition are expected to display comparable if not the same temperature-strain hysteresis relationship, a longer SMA wire is theoretically capable of generating greater absolute displacement than the shorter one when both are heated with the same temperature prole. Consequently, cyclic displacement attainable by a shorter wire heated between the saturation temperatures can be achieved on a longer wire heated within a much narrower bandwidth of temperature and cycling in a minor hysteresis loop. This concept is further illustrated in Fig. 4.2. Here, l 0 ‡ is the contracted length of a heated SMA wire under a constant stress produced by the hanging weight; 1;l0 (t) is the instantaneous extension of the wire associated with the ‡ Denotations of variables employed in Chapter 3 do not carry over to Chapter 4. 84 Fig. 4.2: Idealized illustration of the proposed the multiplier mechanism. (a) A SMA wire measures l 0 in length in the fully autenitic state when heated. As it is cooled to room temperature, it recovers to the fully martensitic state, producing a maximum instantaneous displacement of 1 (t). (b) Compound SMA mechanism with an actuation wire of length nl 0 , which is capable of producing a displacement of n 1 (t) or nl 0 1 (t) when heated within its major hysteresis loop, multiplying the amplitude of displacement attainable in (a) by a factor of n. instantaneous strain of the wire, 1 (t), resulting from the cooling of the wire; and l 1;l0 (t) is the instantaneous length of the wire, which is related to the other variables according to 1 (t) =l 1 0 1;l0 (t): (4.1) Here, the subscript \1" denotes that these variables are associated with the hysteresis major loop. As described in Fig. 4.2, for a given constant stress, two wires of identical diameter and com- position but with dierent full austenitic state length, l 0 and nl 0 , respectively, will theoretically follow the same hysteretic trajectory at the same rate provided that they are excited to follow the same temperature signal T (t) over a constant amount of time 1 . Therefore, during a major loop, two wires of lengthsl 0 andnl 0 will both achieve the same maximum strain 1 , which implies 85 that the maximum displacement produced by the wire of length nl 0 will theoretically be n times longer than that produced by the wire of length l 0 . Here, we specically show that the multiplier system illustrated in Fig. 4.2 can be utilized to multiply the output frequency exploiting the concept of the minor loop. As we have mentioned before, under the same heating/cooling rate, the time required to complete a minor loop 2 is always smaller than the 1 on a same wire. In fact, as explained in [47, 48], we can always choose a minor loop small enough (therefore fast enough) such that 2 can be made arbitrarily small. Subsequently, in order to achieve the same output displacement than that obtained with the wire of lengthl 0 operating in major loop, but with a signicantly higher frequency f 2 , we can potentially adopt the setup depicted in Fig. 4.2(b) while satisfying the relationships f 1 f 2 = 1 2 and 2;nl0 = 1;l0 ; (4.2) where 2;nl0 is the displacement of the nl 0 long wire cycling in the chosen minor loop. Note that ultimately, the multiplier mechanism shown in Fig. 4.2 is envisioned to be installed onto mm-cm scale robots and the wire winding design is conceived to be a possible solution to the limited space available on these robots. The preliminary design utilizes the capstan conguration, illustrated using solid red circles. 4.2.2 Preliminary Multiplier Design To validate the proposed displacement and frequency multiplier concept, we designed and built the testing apparatus presented in Fig. 4.3. Here Fig. 4.3(a) shows the CAD rendering of overall setup. As pointed out in Section 4.2.1, at this stage, the multiplier draws inspiration from capstan. Note that capstan-based devices are normally utilized to balance a large loading force using a 86 signicantly smaller holding force by exploiting the interaction of frictional forces and tension, according to the Capstan equation i = 0 e i f ; (4.3) where is the angle swept by the wire on the capstan's surface, f is the friction coecient, i is tension in each segmenti required to generate the desired tension 0 . This implies that for system of this type to optimize its performance, the heating of each segment has to be `synchronized' so that the deformation of the segments are compatible with each other at the presence of frictional forces. Thus ultimately, individualizes heating for each segment is desired, which is a focus of future research. Additionally according to (4.3), reducing the coecient of friction between the SMA wire and capstans can minimize the frictional force and thus mitigate the impact from inconsistent tensions between each segment. To this end, we chose to fabricate the capstan using polytetra uoroethylene (PTFE), which was experimentally determined to have a f of 0:0237 0:0037 (in contact with the Nitinol SMA wire we use), leading to a tension variation factor ranging within approximately [1:060 : 1:085] between each segment between capstans. Figure 4.3(b) and (c) provide the magnied view of the CAD rendering and actual image of the capstan-based multiplier. Specically, a series of PTFE tubes of outer-diameter 6:35 mm are secured onto an aluminum peg board using shoulder screws so that these tubes cannot spin. In addition, a thin groove as means to hold the SMA wire between each capstan in the same plane was lathed onto each PTFE tube. Employing the same techniques described in Section 3.3.1, we wrapped one type K bare-wire thermocouples (50m in diameter) at each of the three segments on the SMA wire for the purpose of monitoring and regulating temperature, as can be seen in Fig. 4.3(c). A layer of Kapton tape is added between the thermocouples and aluminum peg board to provide electric insulation. Loading was realized using a calibration weight of 150 g. The power electronics created to operate this 87 Fig. 4.3: (a) The CAD drawing of over experimental setup. The multiplier design is highlighted in a red dashed line box and can be better observed in closed-up views in (b) and (c). (b) is a CAD rendering while (c) shows the actual apparatus. (d) oers an overview of the complete setup together with associated power electronics. apparatus as well as the Keyence LK-081 laser sensor used to measure SMA wire displacement are accordingly shown in Fig. 4.3(d). Consistently, sensor readings and output signals are collected and communicated through a DAQ board installed on a target computer running the xPC Target system. A PID controller is implemented and tuned to regulate temperature in this study. 4.2.3 Multiplier Experimental Results and Discussion The SMA wire tested on the multiplier setup measures 214 mm in length, 127 m in diame- ter, and has a nominal phase transition temperature of 90 C. When heated between its lower (30 C) and upper (130 C) saturation temperatures, the wire generates a maximum major hys- teresis displacement of 8:0 mm, as shown in red in Fig. 4.4(a) Part II, which is equivalent to a 3:7% strain. As a comparison, under the same constant stress, a shorter standalone wire 88 (56 mm) with the same diameter and nominal phase transition temperature is also heated fol- lowing the exact same temperature prole shown in blue in Fig. 4.4(a) Part I. Accordingly, the shorter wire's displacement output shown in blue in Fig. 4.4(a) Part II, reached a maximum value of 2:2 mm. Therefore, the experimentally calculated displacement magnication factor equals to 8 2:2 3:64, which is close to the nominal magnication factor following the displace- ment multiplication relationship dened in Section 4.2.1, n = 214 56 3:8. This result can also be veried by the corresponding temperature-strain hysteresis curves presented in Fig. 4.4(a) Part III. Since the same temperature prole was well tracked by both wires under the same loading condition, the incongruency between the hysteresis loops can be explained by the possible incon- sistency in the wires' composition and impact of friction between the capstans and SMA wire. Video recording of this experiment can be found in the supplementary movie multiplier.mp4 (http://www.uscamsl.com/resources/gethesis2020/multiplier.mp4). Similar heating experiments adopting minor loop temperature proles are repeated on both wires. The example results of the wire heated between 50 C and 100 C as well as between 60 C and 90 C are presented in Fig. 4.4(b) and (c), respectively. The displacement multiplication eect can be well observed. The peak-to-peak minor loop displacements measured in these two cases for both the multiplier and the short SMA wire are [6:92; 1:82] mm and [5:18; 1:56] mm, respectively. To enable fast actuation, we initially adopted a bang-bang controller that initiates and termi- nates heating of the SMA wire upon reaching prescribed temperature values. The temperature thresholds here, are determined from examining the minor loop strain output. Figure 4.5(a) demonstrates the fast actuation experimental results obtained from the multiplier setup employ- ing this control scheme. In this case, heating is turned `on' and `o' at 75 and 90 C, respectively; an average deformation of approximately 500 m was attained at around 2 Hz. In comparison, when a similar heating pattern was applied to the shorter wire without the multiplier mecha- nism, an average deformation of approximately 300m at around 2 Hz was recorded, as shown in Fig. 4.5(b). The results implied that multiplier is still capable of magnifying deformation at higher 89 Fig. 4.4: (a), (b) & (c) Displacement magnication verication of the multiplier design. (a) The temperature tracking (Part I), displacement (Part II) as well as the temperature- strain hysteresis loop (Part III) results of the multiplier (shown in red) and the short wire (shown in blue) undergoing the exact same major loop heating sequence (between 30 and 130 C). (b) & (c) The displacement, temperature tracking, as well as the temperature-strain hysteresis loop results of the multiplier and the short wire undergoing the same major and minor loop heating sequences. The minor loop temperature bandwidths are chosen to be [50 : 100] C and [60 : 90] C for cases shown in (b) and (c), respectively. actuation rate, though the magnication factor appeared to be reduced. A plausible explanation to this phenomenon is that at higher actuation frequency, the rapid cyclic deformation of wire will also lead to the frequent acceleration and de-acceleration of the calibration weight connected to the wire. This induced dynamics can potentially intensify incongruency in tensions between each segment of SMA wire and therefore limiting the displacement output. To quantify the dynamics 90 Fig. 4.5: (a) & (b) Temperature and deformation measurements of the multiplier and the short wire when heated between 75 and 90 C adopting the bang-bang control scheme. The switching temperatures are marked in black and blue in the temperature prole plot and the actual measurement is shown in red. Additionally, zoomed-in views of segments enclosed within dashed lines are displayed accordingly. associated with high frequency actuation and reduce frictional forces of the apparatus are of the ongoing and future research interests. Considering the response time of the thermocouple and temperature overshoot observed when using the temperature-driven bang-bang controller, we later switched to pulse heating. In addition, we replaced the free-hanging calibration weight to a tensioned elastic spring anchored to the optical table. The spring has a nominal stiness constant of 2:8 lbs=in (490 N=m) and is preloaded to exert a baseline stress of approximately 125 MPa on the SMA wire. The amplitude, pulse period as well as duty cycle of heating signal were experimentally tested and tuned to maximize the actuation frequency while maintaining an eective displacement. Sample experimental results are presented in Fig. 4.6. Evidently, the setup and control method updates are seen to have drastically improved the wire actuation outputs. Under a 20 Hz, 13:125 V pulse signal with a duty cycle of 15%, an average displacement of approximately 220 m was achieved on the same SMA wire (multiplier). This outcome further demonstrates the potential of multiplier conguration in enhancing SMA wire's actuation rate and deformation magnitude. 91 Fig. 4.6: SMA wire displacement, temperature measurements and their respective zoomed-in views when the heating pulse period is chosen to be 0:05 s (20 Hz). In this case, the heating pulse has an amplitude of 13:125 V and duty cycle of 15%. 4.3 SMA Wire Based Bending Actuator 4.3.1 Actuator Design and Fabrication The multiplier mechanism is developed aiming to extract large displacement at higher frequency directly from SMA wires. Yet such displacement can enable even larger deformation at an actu- ator's end eector when SMA wires are jointed with additional structures. For instances, SMA wires have been employed to produce bending motion of elastic beam structures. In these cases, the linear contraction of SMA wires are shown to be capable of achieving large maximum beam tip de ections [106, 107, 108]. However, these studies have focused on improving the magnitude of the bending de ection rather than achieving fast actuation. Here, we discuss the potential of incorporating our ndings in Section 4.2 into the development of an SMA wire driven bending actuator capable of fast actuations. The proposed bending actuator is illustrated in Fig. 4.7. A pair of SMA wires are placed at equal distance on the opposite two sides of two coplanar elastic beams. The eccentric placement of the SMA wires is adopted to increase the bending moment exerted at the tips of the beams according to [58]. In theory, the SMA wire contraction induced bending of the elastic beam can be described as buckling-due-to-tilting-of-forces [59], or tilt buckling. The bending actuator therefore 92 Fig. 4.7: (a) & (b) CAD rendering of the bending actuators. A still actuator is shown in (a) and a bent actuator is shown in (b). Note that the magnitude of beam de ection shown in (b) is for illustration purpose only. (c) & (d) Image of a fabricated actuator prototype. The caps and central beam are made of FR4 and carbon ber, respectively. In this iteration, the distance between the two caps is set to be 10 mm and the two wires are spaced 1:7 mm apart from the central beam. exploits the range of tip de ection before buckling occurs. Figure 4.7(b) presents the actuator's bending conguration when the SMA wire on one side is heated. The elastic beams also serve as a pair of leaf springs that facilitate the SMA wire's recovery from austenite to martensite after heating is terminated. In addition, under alternating heating and cooling, the contraction of the SMA wire on one side could further accelerate the relaxation of the wire on the opposite side. A prototype of the described bending actuator is presented in Fig. 4.7(c) and (d). The caps are made of FR4 while the central beams are composed of carbon ber. Both parts are manufactured using a laser micro-cutting system with a 10 m spot size. The distance between the two caps in this prototype is set to be 10 mm. The SMA wire employed in this case has a diameter of 50 m and a nominal transition temperature of 90 C. In order to achieve balanced de ection in both directions during actuation, two wire segments of equal length (at room temperature) are rst preloaded equally by an elastic spring before being crimped using a hydraulic crimper between terminals made of miniature stainless steel tubing. 93 Fig. 4.8: A quadruped crawling robot employing the described bending actuator. (a) & (b) CAD rendering of the proposed robot shown in stationary and motion, respectively. (c) CAD rendering of the foot design stemming from the top of the bending actuator. A slanted hook is introduced at the tip of foot to induce anisotropic friction. (d) The leg in bending conguration. Consistently, the magnitude of beam de ection shown is for illustration purpose only. (e) The image of a manufactured robot prototype before wiring. 4.3.2 Actuator Testing and Application Here, we discuss a possible application of the bending actuator in realizing robotic locomotion. Specically in this case, the bidirectional de ection of the actuator induced by alternately heating two SMA wires is employed to approximate the femur sweeping motion of reptiles during crawling. The proposed quadruped crawling robot is presented in Fig. 4.8. A `foot' stems from the top of each actuator and can further amplify the bending motion in order to increase each stride length. In addition, a slanted hook is introduced at the tip of foot to induce frictional anisotropy that enables locomotion. At this stage, the crawler is powered electrically. Figure 4.8(e) shows an assembled robot prototype void of wirings. The body frame is made of FR4 and it serves to anchor the base of each leg. Both the foot and hook are made of carbon ber. Consistently, all pieces are manufactured using the same in-house laser cutter. 94 Fig. 4.9: Testing setup for the bending-actuator-based leg. The laser measures the de ec- tion of a spot 17 mm from the base of actuator, which is held stationary onto a 3D-printed frame. Video recording of this test series can be found in the supplementary movie bending actuator.mp4 (http://www.uscamsl.com/resources/gethesis2020/bending actuator.mp4). The performance of the leg design is then experimentally tested using the setup displayed in Fig. 4.9. A laser sensor (Keyence LK 031) measures the de ection of a spot measured 17 mm apart from the base of the actuator, which is held stationary to a 3D-printed testing rig. In addition, a type K thermocouple (50 m in diameter) is wrapped onto the SMA wire to provide real-time temperature measurement. Pulse heating is employed to drive the deformation of the SMA wires. Figure 4.10 presents a series of example test results. The tip de ection measured when only either one SMA wire (denoted as `A' or `B' only) was heated can be seen in Fig. 4.10(a) and (b). Evidently, an average tip-to-tip displacement of 400 m was registered in both cases, under a heating signal with a frequency, amplitude and duty cycle of 4 Hz, 5 V, and 5%, respec- tively. Such balanced de ection in both directions validates the eectiveness of the pretension- ing process. The same pulse signal was then employed to alternately excite both wires (set apart with a 0:125 s time dierence) to realize bidirectional de ection. The corresponding re- sult is shown in Fig. 4.10(c). An approximately 800 m tip-to-tip displacement is recorded, almost exactly two times the displacement obtained when only a single wire is excited. The corresponding temperature measurement of SMA wire A is depicted in Fig. 4.10(d). It can be observed that the wire was cycling within a minor loop temperature bandwidth of approx- imately 10 C during actuation. Overall, at 4 Hz, the 800 m displacement is equivalent to a 95 Fig. 4.10: (a) & (b) Tip de ection recorded when only either one of of two SMA wires was excited. (c) Tip de ection recorded when both SMA wire were heated alternately. (d) Temperature measurement of one SMA wire during the test when both SMA wires are excited alternately. In all cases presented, the excitation pulse signal employed had a frequency, amplitude and duty cycle of 4 Hz, 5 V and 5%, respectively. When alternating heating was enabled, the excitation signal was set apart with a 0:125 s delay. 8% strain of the SMA wire segment bounded between the caps, a value exceeding the maxi- mum recoverable strain attainable from this type of SMA wires. Video recording of one ac- tuator's characterization test can be found in the supplementary movie bending actuator.mp4 (http://www.uscamsl.com/resources/gethesis2020/bending actuator.mp4). 96 Fig. 4.11: Movie stills of a testing gait realized on the crawler prototype. Cartoons demonstrating the desired gait are also presented. Footage of the test can be found in the sup- plementary movie crawler.mp4 (http://www.uscamsl.com/resources/gethesis2020/crawler.mp4). Four of these mechanisms were then installed onto the body frame and wired. Fig- ure 4.11 presents the photographic sequence composed of movie stills showing a crawl- ing gait achieved by the quadruped crawler held in a third hand. The desired gait described by the cartoons in Fig. 4.11 in general can be followed by this prototype. Footage of the preliminary gait test is recorded in the supplementary movie crawler.mp4 (http://www.uscamsl.com/resources/gethesis2020/crawler.mp4). Improving the assembling and wiring quality as well as enabling the robot to crawl following a prescribed gait is of ongoing research interests. In the future, characterizing de ection and optimizing the actuator design to enhance crawling performance deserve further research attention. 97 Chapter 5 Conclusion and Direction of Future Work 5.1 Conclusion As the title of this dissertation suggests, my graduate research has been focusing on creating and controlling biologically-inspired robots and actuation mechanisms. Accordingly, we presented our progresses made thus far in the areas of creating and modeling an earthworm-inspired robot, characterizing and controlling of SMA wire actuators as well as performance improvements of SMA-wire-based articial muscles in Chapter 2, 3 and 4, respectively. Drawing inspiration from earthworms, we developed a soft crawling robot capable of bidirec- tional locomotion on both at and inclined surface. The robot is composed of pneumatic actuators made of silicone rubber and 3D-printed casings, designed to jointly emulate the functionalities of an earthworm's longitudinal and circular muscles as well as the bristle-like setae structures em- ployed to modulate traction. Locomotion is generated and controlled through the active switching of the robot's surfaces in contact with the supporting ground, and thus the corresponding coe- cients of friction, while a main central actuator provides mechanical power periodically. For the purposes of analysis and control, we modeled the robot as a double-mass-spring-damper system and described its crawling dynamics using a reduced-complexity LTI state-space representation. Employing this model, we proved mathematically that frictional forces can be used as inputs 98 to generate and control locomotion. This nding was thoroughly validated through numerical simulations and real-time control experiments. Driven by the aspiration to create small-size (mm-to-cm scale) autonomous robots, we then delved into the eld of modeling and control the dynamics shape memory alloy wires because of SMA's unmatched power-to-weight ratio. The nonlinear hysteresis inducible by both loading and temperature variations makes precisely regulating SMA actuator's deformation a challenging problem. To rst characterize the hysteresis, employing a modied classical Preisach model, we constructed a hypersurface consists of rst-order-reversal-curve strain increments collected at a set of discrete stress values. A mapping algorithm was developed based on this hypersurface that computes the strain output given the time-varying temperature and stress input signals. To realize position control (or deformation regulation), we then inverted the identied Preisach model and incorporated the inversion in a feedforward control structure. Upon receiving the reference strain and dynamic stress signals, the inversed Preisach model is capable of generating a temperature prole that will allow the SMA wire's displacement output to track the desired strain prole. The proposed method is computationally ecient and requires only light weight and compact sensing apparatuses, making it suitable for cm-scale autonomous micro robots. Based on our nding in SMA wires' hysteresis, we concluded that high-frequency actuation can be realized if we can modulate the SMA actuator to cycle within its hysteresis minor loop. Adopting this concept, periodic deformation of equal magnitude can be achieved on longer SMA wires at a much higher frequency when arranged in proper congurations. To this end, we proposed and tested two methods: the SMA multiplier and an SMA-driven bending actuator. Preliminary tests results have proven their eectiveness. To further demonstrate the concept's applicability in microrobotic design, we designed and fabricated a quadruped crawling robot employing the proposed bending actuators. 99 5.2 Direction of Future Work Our research interest in SMA-based articial muscles originates from SMA wire's high power- to-weight ratio and subsequently the promising potential of creating fully autonomous cm-to- mm scale robots driven by such mechanism. In the aspect of realizing power autonomy, our lab has successfully demonstrated controlled catalytic reaction on platinum-coated SMA wires, proving that hydrocarbon fuels (and hydrogen) can possibly replace batteries as means to power microrobotic movements. Intuitively, the next step is to fuse together the ndings presented in this dissertation and catalytic-combustion-powered SMA articial muscles. Specically, the hysteresis of SMA wire subject to catalytic-combustion has to be characterized and the deformation control strategy described in this dissertation needs to be tested accordingly. Along this process, designing ecient and accurate temperature regulation methods for catalytic reaction should be another research focus. In addition, the design of the SMA wire driven bending actuators could be optimized for the purpose of achieving even faster and more reliable locomotion for the quadruped crawling robot. 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The tasks of data acquisition and signal processing are performed using a National Instruments PCI-6229 data acquisition (DAQ) board mounted on a target PC which communi- cates with a host PC as part of a xPC Target 5.5 conguration setup (MATLAB R2013b). In this scheme, three digital proportional-integral-derivative (PID) controllers, ^ K i (z), for i = 1; 2; 3, control the internal pressure of each actuator with associated discrete-time dynamics P i (z), for i = 1; 2; 3, according to the feedback loop in Fig. A.1. During actuation, both the in ation and exhaust pumps are maintained at constant ow rates and output pressures; for each actuator, the net air ow and internal pressure are manipulated by the fast solenoid valves using pulse width modulation (PWM). Accordingly, each plant P i (z) represents the entire combined SISO dynamics of the actuator i, including those associated with 114 ˆ K i (z) P i (z) p r,i p m,i - Fig. A.1: LTI scheme used to independently control each actuator (i = 1; 2; 3) of the soft robot. Here, the discrete-time plant P i (z) represents the entire SISO dynamics of the actuatori, including those associated with the corresponding in atable silicone-rubber structure, pressure sensor, solenoid valve and DAQ board. The input toP i (z) is the duty cycle of the PWM signal that opens and closes the corresponding solenoid valve; p m;i is the measured internal air pressure; p r;i is the reference pressure; and ^ K i (z) is the PID controller that controls the internal air pressure of the corresponding actuator i. the corresponding in atable structure, solenoid valve, pressure sensor and DAQ board. The input toP i (z) is the duty cycle of the PWM that opens and closes the corresponding solenoid valve, and the output from P i (z) is the measured pressure value p m;i . The dierence between the reference pressure p r;i and p m;i is the input to the corresponding controller ^ K i (z), as shown in Fig. A.1. The experimental characterization process of the robot is based on the procedure introduced in [5]. Consistently, to identify the relationship between the internal pressure and elongation of the central longitudinal actuator, a set of in ation{de ation experiments were conducted in order to measure relevant pressure{elongation data points in a predened set, without causing signicant radial expansion of the test actuator. In the case of the extremal actuators, the minimum pressure thresholds required to establish a rm contact with the supporting surface were found through visual observation. The information obtained from the system characterization experiments is employed to determine the actuator pressure references p r;i required to implement the four-phase actuation locomotion sequence in Fig. 2.6 (see Table. 2.1), employing the simple PID control scheme in Fig. A.1. As discussed in Section 2.6, this approach was demonstrated to be greatly eective to generate friction-controlled bidirectional locomotion. 115 A.2 Additional SMA Wire Position Control Results Figure 3.23 in Section 3.6.2 presents the detailed SMA strain tracking result of strain prole (SP) III, which is a amplifying sinusoidal the signal, under the constant stress 135 MPa. Meanwhile, Table 3.2 summarizes the root-mean-square-errors (RMSEs) between the reference strain (RS) and measured strain (MS) obtained for 4 dierent SPs when 5 dierent constant stresses were applied. Aside from SP III, the other 3 LPs include a sinusoidal signal oscillating at 1=17 Hz (SP I, a damped sinusoidal signal (II) and a signal oscillating between with a set of randomly selected peak and trough points (IV). In addition of SPs is limited within [0 : 0:0376], the empirically- identied maximum strain attainable under 105 MPa. Here, Fig A.2, A.3, A.4, A.5 and A.6 demonstrate the complete SMA position control experimental results of the 4 SPs under 105, 115, 125, 135 and 145 MPa, respectively. Note that the variable labeling and the notation here follow the same convention used in Fig. 3.23. Figure 3.24 in Section 3.6.2 presents the detailed SMA strain tracking result of the SP IV under 6 dierent time-varying loading proles (LPs): two sinusoidal signals oscillating between 105 and 145 MPa at 1=17 and 1=47 Hz (LP I and II) , a pair of monotonically increasing and decreasing linear signals (LP III and IV) and two signals oscillating between two sets of dierent randomly selected peak and trough points (LP V and VI). Meanwhile, Table 3.3 summarizes the RMSEs between RSs and MSs obtained for the same 4 SPs when the 6 LPs were applied. Here, Fig. A.7, A.8 and A.9 demonstrate the complete SMA position control experiments results of the rest 3 SPs when subject to the 6 dierent LPs. Note that the variable labeling and the notation here follow the same convention used in Fig. 3.23. Figure 3.24 in Section 3.6.2 also demonstrates a series of SMA strain tracking results when the time-varying stress was overlooked. Specically in the presented case, when tracking SP IV under 6 dierent LPs, RS was the only input utilized to compute the temperature output, using the rst order reversal curves (FORCs) strain increment surface and inverse bases formulated at 145 MPa. 116 Fig. A.2: Results of SMA wire position control experiments under a constant stress of 105 MPa. In (a), (b), (c), and (d), the reference strain (RS) to be tracked are SPI, II, III and IV, respectively. Meanwhile, Table 3.4 summarizes the RMSEs between RSs and MSs (SP IV) obtained when dynamic stress was neglected, and the single FORC strain increment surface and inverse bases collected at 105, 115, 125, 135 and 145 MPa were invoked to calculate the heating prole. Here, Fig. A.10, A.11, A.12 and A.13 demonstrate the complete experimental results corresponding to the RMSE values listed in Table 3.4. Note that the variable labeling and the notation here follow the same convention used in Fig. 3.24. Namely, MS1, ML1, ST1 and MT1, shown in red are the measured temperature, loading, simulated and measured temperatures obtained when the time-varying stress was adjusted for using the proposed position control method; and MS2, ML2, ST2 and MT2, shown in blue are the results attained when dynamic loading was ignored. 117 Fig. A.3: Results of SMA wire position control experiments under a constant stress of 115 MPa. In (a), (b), (c), and (d), the RS to be tracked are SPI, II, III and IV, respectively. Fig. A.4: Results of SMA wire position control experiments under a constant stress of 125 MPa. In (a), (b), (c), and (d), the RS to be tracked are SPI, II, III and IV, respectively. 118 Fig. A.5: Results of SMA wire position control experiments under a constant stress of 135 MPa. In (a), (b), (c), and (d), the RS to be tracked are SPI, II, III and IV, respectively. Fig. A.6: Results of SMA wire position control experiments under a constant stress of 145 MPa. In (a), (b), (c), and (d), the RS to be tracked are SPI, II, III and IV, respectively. 119 Fig. A.7: Results of SMA wire position control experiments under time-varying stress and SP I is the RS. In (a), (b), (c), (d), (e), and (f), the reference loadings (RLs) introduced are LP I, II, III, IV, V and VI, respectively. 120 Fig. A.8: Results of SMA wire position control experiments under time-varying stress and SP II is the RS. In (a), (b), (c), (d), (e), and (f), the RLs introduced are LP I, II, III, IV, V and VI, respectively. 121 Fig. A.9: Results of SMA wire position control experiments under time-varying stress and SP III is the RS. In (a), (b), (c), (d), (e), and (f), the RLs introduced are LP I, II, III, IV, V and VI, respectively. 122 Fig. A.10: SMA wire position control results when time-varying stress is neglected I. SP IV is the sole input evaluated to calculate the simulated temperature 2 (ST2), utiliz- ing the FORC strain increment and inverse bases procured at 105 MPa. Consistently, In (a), (b), (c), (d), (e), and (f), the RLs introduced are LP I, II, III, IV, V and VI, respec- tively. measured strain 2 (MS2), measured loading 2 (ML2) and measured temperature 2 (MT2), shown in blue, are the strain, loading and temperature measured in each case. Moreover, following the same convention of Fig. 3.24, as a comparison, the results from experiments where dynamic loading is adjusted for are also exhibited here, labeled as MS1, ML1, ST1 and MT1, shown in red. 123 Fig. A.11: SMA wire position control results when time-varying stress is neglected II. SP IV is the sole input evaluated to calculate ST2, utilizing the FORC strain increment and inverse bases procured at 115 MPa. Consistently, In (a), (b), (c), (d), (e), and (f), the RLs introduced are LP I, II, III, IV, V and VI, respectively. MS2, ML2 and MT2, shown in blue, are the strain, loading and temperature measured in each case. Moreover, following the same convention of Fig. 3.24, as a comparison, the results from experiments where dynamic loading is adjusted for are also exhibited here, labeled as MS1, ML1, ST1 and MT1, shown in red. 124 Fig. A.12: SMA wire position control results when time-varying stress is neglected III. SP IV is the sole input evaluated to calculate ST2, utilizing the FORC strain increment and inverse bases procured at 125 MPa. Consistently, In (a), (b), (c), (d), (e), and (f), the RLs introduced are LP I, II, III, IV, V and VI, respectively. MS2, ML2 and MT2, shown in blue, are the strain, loading and temperature measured in each case. Moreover, following the same convention of Fig. 3.24, as a comparison, the results from experiments where dynamic loading is adjusted for are also exhibited here, labeled as MS1, ML1, ST1 and MT1, shown in red. 125 Fig. A.13: SMA wire position control results when time-varying stress is neglected IV. SP IV is the sole input evaluated to calculate ST2, utilizing the FORC strain increment and inverse bases procured at 135 MPa. Consistently, In (a), (b), (c), (d), (e), and (f), the RLs introduced are LP I, II, III, IV, V and VI, respectively. MS2, ML2 and MT2, shown in blue, are the strain, loading and temperature measured in each case. Moreover, following the same convention of Fig. 3.24, as a comparison, the results from experiments where dynamic loading is adjusted for are also exhibited here, labeled as MS1, ML1, ST1 and MT1, shown in red. 126 A.3 SMA Wire Long-Duration Test In the proposed SMA wire position control method, a priori identied mapping between tem- perature, stress and strain is formulated to compensate for an SMA wire's hysteresis behavior in a feedforward position controller. Performance of controllers of this type relies on the accu- racy of the system modeling. However, it is known that cyclic thermomechanical loading could induce irreversible changes in the wire's property, which could aggregate the mismatch between the identied Preisach model and the true hysteresis dynamics, and subsequently rendering the controller less eective. To investigate the change in SMA wire's hysteresis characteristics caused by repeated thermomechanical cycles, we performed a long-duration test descried in the following. A brand new SMA wire measures 51:59 mm is the subject of this study. It shares the same diameter and nominal transition temperature, 127m and 90 C, respectively with the wire stud- ied in Section 3.6.2. In each test, a constant stress was maintained, while the SMA wire was heated between 30 C and 130 C following in a sinusoidal-shaped temperature prole at a fre- quency of 0:05 Hz. The prescribed heating signal was repeated 20 times in each experiment, leading to a total of 400 s runtime. The experiment was performed at 5 constant stresses: 105; 115; 125; 135; 145 MPa. Overall, we performed 25 tests at each stress value, leading to a total runtime in excess of 50; 000 s. Note that here, we reused the previously identied controller parameters and omitted majority of the tuning process. Figure A.14(a)-(e) summarize the long-duration tests results obtained at the 5 constant stress values. Specically, following the same convention adopted in this dissertation, strain is rst calculated by dividing the measured displacement by the room-temperature length of the wire. Afterwards, the temperature and strain measurements from the 20 heating cycles of each test were averaged rst before being plotted against each other to generate each temperature-strain major hysteresis loop presented in Fig. A.14. The test results from the rst 13 sets of tests conducted at each constant stress, labeled as T1 through T13, are presented on the left in each subgure; while 127 the major hysteresis loops obtained from the second 13 sets of tests, including results from T13, labeled as T13 through T25, are shown on the right. A red-to-blue color gradient is employed in each subgure to illustrate the gradual impact on SMA's hysteresis behavior from the increased number of thermomechanical loading cycles. Note that we performed these tests by applying stress following the order [125 135 115 145 105] MPa in each round for 25 times so the wire is worn to a comparable degree under dierent stresses in all the experiments. Following this experiment scheme, the rst 13 tests at each of the 5 constant stresses accounted for approximately 26; 000 s of runtime. According to Fig. A.14, during this period of time, the SMA property has gone through appreciable changes. Quantitatively, the RMSEs between strains obtained from T1 and T13 equal to [2:2 3:4 4:2 3:7 3:0] 10 3 , which translate to approximately [5:23 7:55 9:30 8:15 6:67]% of maximum strain obtained under 105, 115, 125, 135 and 145 MPa tensile stresses, respectively. In comparison, the hysteresis loops attained from T13 through T25 can be observed to be almost congruent across all ve loading conditions. Numerically, the RMSEs between strain measured from T13 and T25 equal to [1:4 1:5 1:2 1:2 1:0] 10 3 , or [3:19 3:43 2:73 2:74 2:18]% of maximum strain obtained under 105, 115, 125, 135 and 145 MPa tensile stresses, respectively. The long-duration test has shown that the hysteresis behavior of a brand new SMA wire can undergo relatively substantial change within the earlier series of thermomechanical loading cycles. However, the wire's property is observed to be gradually settling into a rather stable state as total runtime accumulated. This nding implies that possibly, as long as the hysteresis model parameters are identied after the wire's property has evolved into a relatively steady state (or suciently worn), the accuracy of the Preisach model and its inverse will remain robust for an extended number of thermomechanical loading cycles. We kept record of the comprehensive data collected from all tests conducted on the specic SMA wire on which we ran the position control experiments and presented the results in Fig. 3.23 and 3.24. According to the record, we had already performed 25; 360 s of experiments on the wire for the purposes of tuning the 128 electronic hardwares as well as the temperature and stress controllers, before we started the system identication process. That value is close to the total runtime of 65 rounds of the long- duration tests (26; 000 s), which suggests that we procured the hysteresis model parameters after the SMA wire's property had approached a relatively steady condition. Consequently, the position control performances remained consistent after over 220 experiments with an accumulated runtime exceeding 26; 000 s. 129 Fig. A.14: Preliminary SMA wire long-duration test results. The temperature-strain major hysteresis loops obtained at constant stresses 105; 115; 125; 135; 145 MPa, are presented in (a), (b), (c), (d) and (e), respectively. In all subgures, the results of the rst 13 sets of tests, T1 through T13 are shown on the left side while the results of the second 13 sets of tests (including T13), T13 through T25, are presented on the right side. 130

Abstract (if available)

Abstract For decades, researchers have been creating robots that can mimic the locomotion patterns or functional movements of animals. However, most of these biologically-inspired robots are driven by conventional actuation technologies including combustion engines and electric motors, which usually require additional complex transmission systems. Such deficiencies are further exposed in the development of small size (mm to cm scale) robots of which the spacing and loading capacities are strictly limited. Additionally, these mechanisms can potentially be hazardous in applications where human-machine interactions are involved. Intuitively, ‘muscle-like’ actuation technologies are highly desirable in these situations and have garnered immense research attentions. ❧ In this work, we present our research progresses made on the study on two types of artificial muscle designs. We first created a pneumatically-driven earthworm-inspired soft robot capable of bidirectional locomotion on both horizontal and inclined platforms. In this approach, the locomotion patterns are controlled by actively varying the coefficients of friction between the contacting surfaces of the robot and the supporting platform, thus emulating the limbless locomotion of earthworms at a conceptual level. Using the notion of controllable subspace, we show that friction plays an indispensable role in the generation and control of locomotion in robots of this type. Based on this analysis, we introduce a simulation-based method for synthesizing and implementing feedback control schemes that enable the robot to generate forward and backward locomotion. From the set of feasible control strategies studied in simulation, we adopt a friction-modulation-based feedback control algorithm which is implementable in real time and compatible with the hardware limitations of the robotic system. ❧ Another type of artificial muscle of interest is shape-memory-alloy (SMA)-based actuators. SMA wires are known for their unmatched power-to-weight ratio. Under tension, they can generate repeatable extension and contraction characterized by nonlinear hysteresis when subject to cyclic heatings according to the shape-memory effect (SME). Empirical evidence has also shown that such hysteretic deformation is also a function of the loading condition. Achieving accurate position control of an SMA wire thus requires the hysteresis phenomenon in the presence of time-varying stress to be modeled and compensated for. To this end, we create a Preisach-model-based a priori forward mapping from time varying temperature and stress inputs to the strain output through a series of system identification processes. The identified parameters are then used in a Preisach-model-inversion algorithm which is integrated into a feedforward controller designed to mitigate the hysteresis effect. The proposed deformation tracking approach employs light-weight sensing apparatuses and has proven to be computationally efficient

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Asset Metadata

Creator Ge, Zaoyuan (Joey) (author)

Core Title Development and control of biologically-inspired robots driven by artificial muscles

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 05/13/2020

Defense Date 05/15/2020

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag artificial muscle,biologically-inspired,Control,earthworm-inspired,fast SMA actuators,hysteresis,OAI-PMH Harvest,Preisach model,shape-memory-alloy (SMA),soft robot

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Perez-Arancibia, Nestor O. (committee chair), Culbertson, Heather (committee member), Flashner, Henryk (committee member)

Creator Email gezaoyuan@gmail.com,zaoyuang@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-305959

Unique identifier UC11663134

Identifier etd-GeZaoyuanJ-8502.pdf (filename),usctheses-c89-305959 (legacy record id)

Legacy Identifier etd-GeZaoyuanJ-8502.pdf

Dmrecord 305959

Document Type Dissertation

Rights Ge, Zaoyuan (Joey)

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions 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...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA