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A discrete-time return map analysis and prediction of gait-modulated robot dynamic under repeated obstacle collisions

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A discrete-time return map analysis and prediction of gait-modulated robot dynamic under repeated obstacle collisions

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Content ADiscrete-timeReturnMapAnalysisandPredictionof Gait-modulatedrobotdynamicunderrepeatedobstacle collisions by Haodi Hu A Thesis Presented to the Faculty of the VITERBI SCHOOL OF ENGINEERING UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (Electrical Engineering) May 2021 Copyright 2021 Haodi Hu Acknowledgements I would first like to thank my thesis advisor Prof. Feifei Qian for her guidance, motivation, patience and immense knowledge. I would also like to thank my colleague Elijah Yap for his help in building robot and Hanyu She for his help in collecting experiment data. A special thanks to my family and friends for their endless motivation. ii TableofContents Acknowledgements ii ListofFigures v Abstract viii Chapter1: Introduction 1 Chapter2: EXPERIMENT 2 2.1 Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Obstacle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Gaits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.5 Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.6 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6.1 Robot Trajectoreis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.6.2 Basin of Attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter3: Modeling 12 3.1 Bounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.1 Mode 1 - Leg pair 1 slipping phase . . . . . . . . . . . . . . . . . . . . . 13 3.1.2 Mode 2 - Leg pair 1 advancing phase . . . . . . . . . . . . . . . . . . . . 15 3.1.3 Mode 3 - Leg pair 2 slipping phase . . . . . . . . . . . . . . . . . . . . . 15 3.1.4 Mode 4 - Leg pair 2 advancing phase . . . . . . . . . . . . . . . . . . . . 16 3.1.5 Exception cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Trotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.1 Mode 1 - Leg pair 1 slipping phase . . . . . . . . . . . . . . . . . . . . . 18 3.2.2 Mode 2 - Leg pair 1 advancing phase . . . . . . . . . . . . . . . . . . . . 20 3.2.3 Mode 3 - Leg pair 2 slipping phase . . . . . . . . . . . . . . . . . . . . . 20 3.2.4 Mode 4 - Leg pair 2 advancing phase . . . . . . . . . . . . . . . . . . . . 22 3.2.5 Exception cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Pacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3.1 Mode 1 - Leg pair 1 slipping phase . . . . . . . . . . . . . . . . . . . . . 24 3.3.2 Mode 2 - Leg pair 1 advancing phase . . . . . . . . . . . . . . . . . . . . 25 3.3.3 Mode 3 - Leg pair 2 slipping phase . . . . . . . . . . . . . . . . . . . . . 26 iii 3.3.4 Mode 4 - Leg pair 2 advancing phase . . . . . . . . . . . . . . . . . . . . 27 3.3.5 Exception cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Chapter4: SimulationResultsandComparisonWithExperiments 29 4.1 Robot Trajectories and Robot Orientation VS. Time . . . . . . . . . . . . . . . . . 29 4.2 Basin of Attraction Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3 Mechanisms Leads to Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . 31 4.3.1 Mechanism 1: Disturbance-balance . . . . . . . . . . . . . . . . . . . . . 31 4.3.2 Mechanism 2: Collision-free Steady . . . . . . . . . . . . . . . . . . . . . 32 4.4 Mechanism Leads to Limit Cycle States . . . . . . . . . . . . . . . . . . . . . . . 33 4.5 Ongoing and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.5.1 Obstacle-aided navigation through sequential selection of gaits . . . . . . . 36 4.5.2 Beyond Virtual Bipedal Gaits . . . . . . . . . . . . . . . . . . . . . . . . 38 4.5.3 Converge Process of Steady State . . . . . . . . . . . . . . . . . . . . . . 40 Chapter5: Conclusion 42 References 43 iv ListofFigures 2.1 Robot and Gaits. Left figure shows the robot body, (a) is bio-inspired C-shaped legs, (b) is customized Arduino Uno Rev3 board function as CPG, (c) is 2600mAh Lithium-powered battery, (d) is Lynxmotion Smart Servo (LSS). . . . . . . . . . . 2 2.2 Obstacle structure and distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 Gaits and Beulher Clock. (a) is bounding gait, (b) is trotting gait, (c) is pacing gait, read color represents leg pair 1, green color represents leg pair 2; right figure is Beulher clock which used to implement the three virtual bipedal gaits, green line representsj 1 , red line representsj 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Motion capture system. We used average position of five reflection marker to rep- resent robot CoM position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.5 Experiment environment. Left figure initial obstacle fields setup and robot initial state; right figure shows robot coupling with obstacles. . . . . . . . . . . . . . . . 5 2.6 Sequence of images from a bounding experiment showing the robot orientation was locked to 0 degree under periodic obstacle modulation. In this trial, initial orientation is 50 degrees, initial position is -27cm. . . . . . . . . . . . . . . . . . . 7 2.7 Sequence of images from a trotting experiment showing the robot orientation was locked to 48 degrees under periodic obstacle modulation. In this trial, initial orien- tation is 50 degrees, initial position is -27cm. . . . . . . . . . . . . . . . . . . . . 8 2.8 Sequence of images from a pacing experiment showing the robot orientation was locked to 60 degrees under periodic obstacle modulation. In this trial, initial orien- tation is 40 degrees, initial position is -30cm. . . . . . . . . . . . . . . . . . . . . 8 2.9 Robot experiment orientation and trajectories. From top to bottom are bounding, trotting and pacing respectively. Left side is robot trajectories of CoM in repeated obstacle field, right side is robot orientation vs. experiment time, color bar repre- sents value of robot initial orientation. . . . . . . . . . . . . . . . . . . . . . . . . 9 v 2.10 Initial positions and initial orientations affects on steady orientations. From top to bottom are bounding, trotting and pacing respectively; different marker represents different initial positions, markers belong to two dash line are correspond to same initial orientation, different marker color represents the marker corresponded initial orientation, error bar on each marker represent the standard deviation for the steady orientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.11 Experiment attraction basin. Each marker represents a steady state orientation, each shaded area corresponding to marker represents initial orientation that can converge the marker represented steady orientation, error bar represents standard deviation for each steady orientation. . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1 Robot in model. Left is robot with bounding gait, middle is robot with trotting gait, right is robot with pacing gait. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1 Simulation results. Left side from top to bottom are robot simulation trajectories of bounding, trotting and pacing gait based on robot CoM position respectively; right side from top to bottom are robot simulation orientation of bounding, trotting and pacing gait respectively, different color represents simulation results with different initial orientation corresponding to the color bar. . . . . . . . . . . . . . . . . . . . 30 4.2 Steady states attraction basin in simulation and experiment. Every color marker represents one steady state orientation, the color represents value of orientation corresponding to right color bar. Color marker with error bar represents equilib- rium states, color marker without error bar represents limit cycle states. Color shaded area means the attraction basin for each steady state, each attraction basin correspond to the steady states markers with same horizontal position. . . . . . . . 31 4.3 Steady mechanism 1 schematic. Black rectangle represents robot body before sides, green and red filled circle represents active leg pair 1 and 2. . . . . . . . . . 32 4.4 Steady mechanism 2 schematic. Black rectangle represents robot body before sides, green and red filled circle represents active leg pair 1 and 2. . . . . . . . . . 33 4.5 Sequential gait composition to reach different steady states orientation. Steady states orientation of one gait treated as initial orientation as another gait finally converged to a new steady state orientation. . . . . . . . . . . . . . . . . . . . . . 37 4.6 Trajectory of sequential gait combination. Left figure show the trajectory of one trail from pacing gait shifted to bounding gait, right figure show the trajectory of one trail from bounding to trotting gait. . . . . . . . . . . . . . . . . . . . . . . . 37 4.7 Gait space. Each point in gait space represents a specific gait, j LFRF represents phase difference of LF leg and RF leg, j RBRF represents phase difference of RB leg and RF leg, j LBRF represents phase difference of LB leg and RF leg; TP0.5, BT0.5, PB0.5 represents intermediate gait between trotting and pacing, bounding and trotting, pacing and bounding respectively. . . . . . . . . . . . . . . . . . . . 38 vi 4.8 Trajectories of intermediate gaits. From top to bottom are BT0.5, TP0.5 and PB0.5 respectively, a steady state orientation 56 1:4 was observed in BT0.5, a steady state orientation 62 0:9 was observed in TP0.5, a steady state orientation 90 3:2 was observed in PB0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.9 Basin formation process for bounding gait. Red and green lines represents obstacle in terms of robot CoM,X k and robot orientation,q k ; each point in black point cloud represent a robot state (X k ,q k ), left top number represents sequential step number in simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.10 Basin formation process for trotting gait. Red and green lines represents obstacle in terms of robot CoM,X k and robot orientation,q k ; each point in black point cloud represent a robot state (X k ,q k ), left top number represents sequential step number in simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.11 Basin formation process for pacing gait. Red and green lines represents obstacle in terms of robot CoM, X k and robot orientation, q k ; each point in black point cloud represent a robot state (X k ,q k ), left top number represents sequential step number in simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 vii Abstract Animals are able to flexibly adapt their gaits to cope with a wide variety of challenging environ- ments. Understanding the fundamental principles of such adaptation can guide the creation of control and planning strategies to improve the locomotion performance of legged robots. How- ever, one of the key challenges is the limited understanding of the complex interactions between legged locomotors and perturbation-rich terrains, due to the wide variety in terrain parameters, and the large disturbances from dynamic collisions. The objective of this study is to understand how gaits influence the interactions between legged robots and obstacle-cluttered terrains and resulting in different obstacle-modulated orientation dynamics. We study the horizontal-plane dynamics of a quadrupedal robot as it traverses an array of evenly-distributed obstacles, and discovered that the robot can passively converge to different steady-state (SS) orientations with three virtual-bipedal gaits: bound, trot, pace. In order to investigate the interaction mechanism behind the passive SS, we developed a discrete-time model map model that can predict robot state transition dynamics in the obstacle field for given gaits and step lengths. Comparison between experiment, simulation, and analytical results suggest that the highly-simplified model are capable of predicting the robot steady-state orientations and converging dynamics for the three virtual bipedal gaits, facilitating the discovery of the governing mechanism behind such passive convergence. Building upon the mode-map model, we are developing a gait transit method to utilize the existing passive steady states to navigate robot in the obstacles fields. Going forward, we envision these interaction mod- els, combined with the exploration of complete gait space, will enable even low-cost robots to flexibly exploit complex environments and achieve robust navigation. viii Chapter1 Introduction Obstacles and disturbances are frequently encountered in natural environments that terrestrial robots must cope with. Animals are observed to exhibit a wide variety of gait adjustments, such as relative phase, duty cycle, and step length, to flexibly move across obstacle terrains [1, 2], but the inherent complexity in neural control and sensory feedback can obscure the discovery of phys- ical mechanisms governing their strategy adaptations. Recent robotics studies have revealed that leg-obstacle interaction forces depend primarily on obstacle inclination at the contact position [3]. By adapting different leg-obstacle contact position patterns [4], the repeated leg-obstacle collisions will allow legged robot to robustly converge to stable orientations without active steering effort [5]. To enable analytical prediction of how these stable orientations depend on gait parameters and further explore the possibility of assisted navigation based on obstacles, in this study we construct a highly-simplified “mode map” representation that can allow better understanding and analytical prediction of robot state transition dynamics in the obstacle field for given gaits and step lengths. These robotics results are beginning to reveal how robot dynamics is mechanically governed by gait and terrain parameters, and lead towards development of new navigation method for locomotion robot with different combination of gaits to approach desired behavior in obstacle fields. 1 Chapter2 EXPERIMENT Experiments were performed to track robot dynamics under repeated obstacle contacts and obtain an initial understanding of the effect of gait and environment parameters. In the first half of the chapter, robot parameters, data collection protocol and environment setup are discussed. In the latter half of the chapter, different data analysis procedures based on collected experiment raw data are discussed. 2.1 Robot Figure 2.1: Robot and Gaits. Left figure shows the robot body, (a) is bio-inspired C-shaped legs, (b) is customized Arduino Uno Rev3 board function as CPG, (c) is 2600mAh Lithium-powered battery, (d) is Lynxmotion Smart Servo (LSS). 2 In order to investigate the interaction between legged robot and obstacles, a quadrupedal robot as shows in figure 2.1 was used. Robot body width, 2W, is 15.4 cm, robot body length, 2L, is 16.0 cm. 2.2 Obstacle Figure 2.2: Obstacle structure and distribution. In order to investigate robot’s behavior under repeated interaction with obstacles, we evenly- distribute obstacles to form a repeated obstacles field. Half-cylinder is used as the shape of obsta- cles to represent highly simplified complex terrain. Diameter of obstacle , D, is 4.8cm, length of obstacle is 300cm, spacing between each obstacle,P, is 3.5cm. 2.3 Gaits As shown in figure 2.3. Three different gaits were used in the experiment to explore the effect of gaits on robot interaction with obstacles. In order to achieve better robot stability, for each stride, the instantaneous angular position of each leg was specified by the “Buehler clock” parameteriza- tion of stance and flight phasing as described in Saranli et al. [6]. We divided four individual legs into two group O 1 and O 2 owing to the gait symmetry. For bounding gait, O 1 =f LF, RFg, O 2 = 3 Figure 2.3: Gaits and Beulher Clock. (a) is bounding gait, (b) is trotting gait, (c) is pacing gait, read color represents leg pair 1, green color represents leg pair 2; right figure is Beulher clock which used to implement the three virtual bipedal gaits, green line representsj 1 , red line representsj 2 . f LB, RBg; for trotting gait, O 1 =f LB, RFg, O 2 =f LF, RBg; for pacing gait O 1 =f LF, LB g, O 2 =f RF, RBg. j 1 and j 2 represents phases of O 1 and O 2 respectively. After implemented Beulher clock with two phases, slower phase started from 0 to 120 and time period 1.5s, faster phase started from 120 to 360 and time period 1.5s. Step length of bounding, trotting and pacing gaits are 9.0cm, 12.5cm and 8.7cm indicated byS B ,S T andS P respectively. 2.4 DataCollection Reflection marker on four servos and top of robot are used for motion capture system(MCS) to track robot center of mass (CoM) position and body rotation angle with frame rate of 120Hz. We defined CoM as average point of four markers on servos. Four LED tracking cameras and two video cameras are included in MCS so as to obtain color video during each tracking. The data collected from MCS include frame sequence number N, the positions of CoM indicated by X CoM , y CoM , z CoM , roll angle a, pitch angle b, and yaw angle g. We used X CoM and z CoM to calculate robot CoM position andg to calculate robot orientation during each trial. 2.5 Experimentsetup In order to investigate the effect of gaits, initial orientation, q 0 and initial position, X 0 affects on robot behaviors during repeated interactions with obstacles, we performed 45 experiments we 4 Figure 2.4: Motion capture system. We used average position of five reflection marker to represent robot CoM position. Figure 2.5: Experiment environment. Left figure initial obstacle fields setup and robot initial state; right figure shows robot coupling with obstacles. performed 45 experiments with systematic variations in q 0 and x 0 . For each gait, q 0 was varied from 0 to 70 with an increment of 5 , whereas x 0 was tested at 24cm, 27cm, and 30cm. Here x 0 is measured from the robot CoM to the near edge of the first obstacle. For initial orientation larger than 70 , the robot deflected off the first obstacle and failed to enter the obstacle field in most of 5 the trials. We only performed trials with positive initial orientation angle owing to the symmetry. For each trial, the robot was set to a fixed stride frequency 0.33Hz. 2.6 ExperimentResults 2.6.1 RobotTrajectoreis In experiments, passively SS were found in repeated obstacles fields as shown in Figure 2.6, Fig- ure 2.7, Figure 2.8. We observed that with a bounding gait there exists three steady orientation, 0 1:1 , 33 2:1 , 33 2:6 . With the trotting gait, however, the stabilized orientations were significantly different from those with the bounding gait, four steady orientation, 4:8 5:4 , 12:3 3:8 , 47:4 5:3 , 68:1 4:1 were observed. With the trotting gait, three steady ori- entation 5:6 10:3 , 60:0 4:8 were observed. Robot trajectories measured from experiments are plotted in Figure 2.9 shows the overall experiment trajectory for bounding, trotting and pacing respectively. From Figure 2.9 we found there are passive SS exists under repeat obstacle modulation. With further process of experiment data, we observed there wsisting different steady states orientation like bounding steady orientation at 0 1:1 ; the other is limit-cycle steady(LCS) state like pacing steady orientation at 5:6 9:8 . In order to enable analytical analysis, we define SS as a trajectory satisfy following conditions: s(q)h 0 (2.1) jarctan( h(T)h(0:6T) g(T)g(0:6T) ) tan(q)jh 0 (2.2) Where h 0 is tolerable experiment noise in term of orientation, which set to 10 ; T represents the time robot stay in obstacle field in one trajectory; f is a function of the Angle of the robot in terms of time;g is a function of cumulative change ofz CoM regardless of slipping in terms of time;h is a 6 Figure 2.6: Sequence of images from a bounding experiment showing the robot orientation was locked to 0 degree under periodic obstacle modulation. In this trial, initial orientation is 50 degrees, initial position is -27cm. function of cumulative change of x CoM regardless of slipping in terms of time; q is average value of robot angle during last 40 percent ofT . 2.6.2 BasinofAttraction In order to distinguish initial orientation and initial position affects on steady states orientation we further processed robot orientation with definition of ES and LCS. As show in Figure 2.10, both initial position and initial orientation can affect steady states orientation, this study focus on 7 Figure 2.7: Sequence of images from a trotting experiment showing the robot orientation was locked to 48 degrees under periodic obstacle modulation. In this trial, initial orientation is 50 degrees, initial position is -27cm. Figure 2.8: Sequence of images from a pacing experiment showing the robot orientation was locked to 60 degrees under periodic obstacle modulation. In this trial, initial orientation is 40 degrees, initial position is -30cm. 8 Figure 2.9: Robot experiment orientation and trajectories. From top to bottom are bounding, trot- ting and pacing respectively. Left side is robot trajectories of CoM in repeated obstacle field, right side is robot orientation vs. experiment time, color bar represents value of robot initial orientation. the relationship between initial orientation and steady orientation with regarding initial position as a unknown variable. As a result, given a steady orientation, we can form an attraction basin in terms of initial orientation that including all initial orientations that can lead to the given steady orientation. 9 Figure 2.10: Initial positions and initial orientations affects on steady orientations. From top to bottom are bounding, trotting and pacing respectively; different marker represents different ini- tial positions, markers belong to two dash line are correspond to same initial orientation, different marker color represents the marker corresponded initial orientation, error bar on each marker rep- resent the standard deviation for the steady orientation. 10 Figure 2.11: Experiment attraction basin. Each marker represents a steady state orientation, each shaded area corresponding to marker represents initial orientation that can converge the marker represented steady orientation, error bar represents standard deviation for each steady orientation. 11 Chapter3 Modeling In order to enable analytical prediction, a highly simplified model named mode-map based on discrete dynamic was developed to predict robot’s CoM state at step k+1, (X k+1 ,q k+1 ) given robot’s CoM state at step k, (X k , q k ), as show in figure 3.1. We define the robot CoM state in the obstacle field as (X, q) , where X is the relative CoM fore-aft position on the periodic obstacle orbit, [0;P+D), andq is the CoM orientation in world frame. In the simplified model we consider two virtual “phases” for each step: a “slipping phase” and a “advancing phase”. Each stride-based return map, therefore, is a composition of the four mode maps: leg pair 1 slipping phase! leg pair 1 advancing phase! leg pair 2 slipping phase! leg pair 2 advancing phase. In the following analysis we use the subscript, k, to denote the step number, whereas superscripts denote the 1 st and 2 nd half-stride events, respectively. There are three basic assumption of this model. First, we assume the obstacle surface is very smooth and there is no friction on the obstacle surface. Second, gait frequency is low to ensure enough time for active leg pair to slip if they contact on obstacles. Third, robot legs movement is represented as a point attached at the hip joint. 3.1 Bounding The following discussion focuses on a virtual bipedal gait, bounding, where two front legs of a quadrupedal robot move synchronously, and alternate with the hind leg pair. Each stride consists 12 Figure 3.1: Robot in model. Left is robot with bounding gait, middle is robot with trotting gait, right is robot with pacing gait. two steps: 1 st half-stride (i.e., step) by leg pair O 1 =fLF, RFg (red, Fig. 3.1), and 2 nd half-stride by leg pairO 2 =fRB, LBg (green, Fig. 3.1). 3.1.1 Mode1-Legpair1slippingphase Mode map: (X 1 k ,q 1 k )! (X 1 k + ,q 1 k + ) During the slipping phase, the change in the robot orientation state, q, is rigidly related to the difference in fore-aft positions of the “active” leg pair: DX 1 k = 2W sin(dq 1 k ); (3.1) where DX 1 =X LF X RF (3.2) We use subscript to denote the time right before leg-obstacle or leg-ground contact, and + to denote the time right after the leg-obstacle interaction. DX 1 k + =DX 1 k +DS 1 k (3.3) 13 DS represents the difference of slipping distance between the two active legs: DS 1 k =S LF S RF (3.4) where the slipping distance of each leg depends solely on the leg’s fore-aft position in relative to the near edge of the obstacle in contact ([3]): S i = 8 > > > > > > < > > > > > > : N(P+D)X i ; X i 2[N(P+D);N(P+D)+D=2) N(P+D)+DX i ; X i 2(N(P+D)+D=2;N(P+D)+D] 0; (X i =N(P+D)+D=2)_(X i 2(N(P+D)+D;(N+ 1)(P+D))) (3.5) where X i ;i2fLF;RF;LB;RBg represents the relative fore-aft position of each leg on the peri- odic obstacle orbit,N is natural number. After calculating the slip distance for each active leg, their fore-aft position can be calculated: X LF =X 1 k +Ccos(dq 1 k )+S LF (3.6) X RF =X 1 k +Ccos(d+q 1 k )+S RF (3.7) The change in robot orientation, q, therefore can be computed by substitutingDX in Eqn. 3.1 with Eqn. 3.3: 2W sinq 1 k + = 2W sinq 1 k +DS 1 k (3.8) We can compute the CoM fore-aft position based on middle fore-aft position of two active legs: X 1 k + = X LF +X RF 2 Lcosq 1 k + (3.9) 14 3.1.2 Mode2-Legpair1advancingphase Mode map: (X 1 k + ,q 1 k + )! (X 2 k ,q 2 k ). The orientation does not change during the advancing phase: q 2 k =q 1 k + (3.10) The CoM fore-aft position advances by a constant step length, S B , along the robot current orientation,q 1 k + : X 2 k =X 1 k + +S B cos(q 1 k + ) (3.11) Eqn. 3.10 and Eqn. 3.11 specifies the mode map for leg pair 1 advancing phase. 3.1.3 Mode3-Legpair2slippingphase Mode map: (X 2 k ,q 2 k )! (X 2 k + ,q 2 k + ) Similar to mode1: DX 2 k = 2W sin(dq 2 k ); (3.12) where DX 2 k =X LB X RB (3.13) Subscript denote the time right before leg-obstacle or leg-ground contact,+ denote the time right after the leg-obstacle interaction. DX 2 k + =DX 2 k +DS 2 k (3.14) 15 DS represents the difference of slipping distance between the two active legs: DS 2 k =S LB S RB (3.15) After calculating the slip distance for each active leg, their fore-aft position can be calculated: X LB =X 2 k Ccos(dq 2 k )+S LB (3.16) X RB =X 2 k Ccos(d+q 2 k )+S RB (3.17) The change in robot orientation,q, therefore can be computed by substitutingDX in Eqn. 3.12 with Eqn. 3.14: 2W sinq 2 k + = 2W sinq 2 k +DS 2 k (3.18) We can compute the CoM fore-aft position based on middle fore-aft position of two active legs: X 2 k + = X LB +X RB 2 +Lcosq 2 k + (3.19) 3.1.4 Mode4-Legpair2advancingphase Mode map: (X 2 k + ,q 2 k + )! (X 1 (k+1) ,q 1 (k+1) ) Similar to mode 2, The orientation does not change during the advancing phase: q 1 (k+1) =q 2 k + (3.20) The CoM fore-aft position advances by a constant step length, S B , along the robot current orientation,q 2 k + : X 1 (k+1) =X 2 k + +S B cos(q 2 k + ) (3.21) Eqn. 3.20 and Eqn. 3.21 specifies the mode map for leg pair 1 advancing phase. 16 3.1.5 Exceptioncases Cases where the mode map cannot be directly applied: DX 1 k + > 2W orDX 2 k + > 2W. Since the distance between robot’s two active legs cannot excess 2W, we assume there is a dominated leg with stronger slide force and the other one follow its step. For the caseDX 1 k + > 2W: q 1 k + = p 2 sgn(q 1 k ) (3.22) CoM fore-aft position can be computed as follows: X 1 k + = 8 > > > > > > < > > > > > > : X 1 LF Wsgn(q 1 k + ); S RF = 0_jS RF j>jS LF j X 1 k ; S LF =S RF 6= 0 X 1 RF +Wsgn(q 1 k + ); S LF = 0_jS LF j>jS RF j (3.23) For the caseDX 2 k + > 2W: q 2 k + = p 2 sgn(q 2 k ) (3.24) CoM fore-aft position can be computed as follows: X 2 k + = 8 > > > > > > < > > > > > > : X 2 LB Wsgn(q 2 k + ); S RB = 0_jS RB j>jS LB j X 2 k ; S LF =S RF 6= 0 X 2 RB +Wsgn(q 2 k + ); S LB = 0_jS LB j>jS RB j (3.25) 3.2 Trotting The following discussion focuses on a virtual bipedal gait, trotting, where two diagonal legs of a quadrupedal robot move synchronously, and alternate with the other diagonal leg pair. Each stride consists two steps: 1 st half-stride (i.e., step) by leg pair O 1 =fLF, RBg (red, Fig. 3.1), and 2 nd half-stride by leg pairO 2 =fRF, LBg (green, Fig. 3.1). 17 3.2.1 Mode1-Legpair1slippingphase Mode map: (X 1 k ,q 1 k )! (X 1 k + ,q 1 k + ) Before the slipping phase, the fore-aft positions of the two legsfLF;RBg that belong to the “active” leg pair,O 1 , are given by: X LF =X 1 k +C sin( p 2 +dq 1 k ) X RB =X 1 k C sin( p 2 +dq 1 k ) (3.26) During the slipping phase, the change in the robot orientation state, q, is rigidly related to the difference in fore-aft positions of the active leg pair indicated byDX 1 k : DX 1 k =X LF X RB = 2C sin( p 2 +dq 1 k ) (3.27) We use subscript to denote the time right before leg-obstacle or leg-ground contact, and + to denote the time right after the leg-obstacle interaction. DX 1 k + =DX 1 k +DS 1 k (3.28) DS represents the difference of slipping distance between the two active legs: DS 1 k =S LF S RB (3.29) 18 where the slipping distance of each leg depends solely on the leg’s fore-aft position in relative to the near edge of the obstacle in contact ([3]): S i = 8 > > > > > > < > > > > > > : N(D+P)X i ; X i 2[N(D+P);N(D+P)+D=2) N(D+P)+DX i ; X i 2(N(D+P)+D=2;N(D+P)+D] 0; X i =N(D+P)+D=2 _ X i 2 N(D+P)+D;(N+ 1)(D+P) ; (3.30) whereN= 0;1;2;::: is the index of the contacting obstacle, andX i ;i2fLF;RF;LB;RBg represents the fore-aft position of each leg. The change in robot orientation, q, therefore can be computed by substitutingDX in Eqn. ?? with Eqn. 3.27: 2C sin( p 2 +dq 1 k + )= 2C sin( p 2 +dq 1 k )+DS 1 k (3.31) Since cos is not invertible on[p=2;p=2], here we specify how to choose from the two possible solutionsq 1 ,q 2 : q 1 = p 2 +d arcsin(sin( p 2 +dq 1 k )+ DS 1 k 2C ) (3.32) q 2 = p 2 +d+ arcsin(sin( p 2 +dq 1 k )+ DS 1 k 2C ) (3.33) Since the robot will always select the angle closer to original angle: q 1 k + = 8 > > > > > > < > > > > > > : q 1 ;jq 1 q 1 k j<jq 2 q 1 k j q 2 ;jq 1 q 1 k j>jq 2 q 1 k j q 1 =q 2 ;jq 1 q 1 k j=jq 2 q 1 k j (3.34) 19 Now we compute the mode map for the CoM fore-aft position, X. Since X is the midpoint between legLF and legRB, the change inX during the slipping phase is given by: X 1 k + =X 1 k +(S LF +S RB )=2 (3.35) Eqn. 3.43 and Eqn. 3.35 specifies the mode map for leg pair 1 slipping phase. 3.2.2 Mode2-Legpair1advancingphase Mode map: (X 1 k + ,q 1 k + )! (X 2 k ,q 2 k ). The orientation does not change during the advancing phase: q 2 k =q 1 k + (3.36) The CoM fore-aft position advances by a constant step length, S T , along the robot current orientation,q 1 k + : X 2 k =X 1 k + +S T cos(q 1 k + ) (3.37) Eqn. 3.36 and Eqn. 3.37 specifies the mode map for leg pair 1 advancing phase. 3.2.3 Mode3-Legpair2slippingphase Mode map: (X 2 k ,q 2 k )! (X 2 k + ,q 2 k + ) 20 We first update the positions of all legs,X i , based onX 2 k . Before leg pair 2 slipping phase, the fore-aft positions of the two legs that belong toO 2 ,fRF;LBg, are given by: X RF =X 2 k +C sin( p 2 q 2 k d) X LB =X 2 k C sin( p 2 q 2 k d) (3.38) This will allow us to computeDX 2 k andDS 2 k : DX 2 k =X RF X LB = 2C sin( p 2 q 2 k d) (3.39) DS 2 k =S RF S LB : (3.40) During leg pair 2 slipping, we have DX 2 k + =DX 2 k +DS 2 k : (3.41) The change in robot orientation,q, therefore can be computed by substitutingDX in Eqn. 3.41 with Eqn. ??: 2C sin( p 2 dq 2 k + )= 2C sin( p 2 dq 2 k )+DS 2 k (3.42) Similar to Mode 1, the change in robot orientation, q, is rigidly related to the dference in leg pair 2 fore-aft positions,DX 2 . By substitutingDX 2 in Eqn. 3.41 with Eqn. ??, we get: q 2 k + = 8 > > > > > > < > > > > > > : q 1 ; jq 1 q 2 k j<jq 2 q 2 k j q 2 ; jq 1 q 2 k j>jq 2 q 2 k j q 1 =q 2 ; jq 1 q 2 k j=jq 2 q 2 k j (3.43) 21 Where: q 1 = p 2 d arcsin(sin( p 2 dq 2 k )+ DS 2 k 2C ) (3.44) q 2 = p 2 d+ arcsin(sin( p 2 dq 2 k )+ DS 2 k 2C ) (3.45) Since the CoM fore-aft position, X, is the midpoint between leg RF and leg LB, the change in X during leg pair 2 slipping phase can be computed as: X 2 k + =X 2 k +(S RF +S LB )=2; (3.46) whereS i can be computed using Eqn. 3.30. Eqn. 3.43 and Eqn. 3.46 specifies the mode map for leg pair 2 slipping phase. 3.2.4 Mode4-Legpair2advancingphase Mode map: (X 2 k + ,q 2 k + )! (X 1 (k+1) ,q 1 (k+1) ) Similar to mode 2, The orientation does not change during the advancing phase: q 1 (k+1) =q 2 k + (3.47) The CoM fore-aft position advances by a constant step length, S T , along the robot current orientation,q 2 k + : X 1 (k+1) =X 2 k + +S T cos(q 2 k + ) (3.48) Eqn. 3.47 and Eqn. 3.48 specifies the mode map for leg pair 1 advancing phase. 22 3.2.5 Exceptioncases For exception conditions DX 1 > 2C and DX 2 > 2C, the desired differential fore-aft positions of the active leg pair exceed the physical limit, and the mode maps cannot be directly applied. In which case we modify mode maps 1 (for leg pair O 1 exception) and 3 (for leg pair O 2 exception) as follows: First, the distance between the two active legs will continuously increase up to its maximum, 2C. At which point we will haveDY 1 =Y LF Y RB = 0 forO 1 , orDY 2 =Y RF Y LB = 0 forO 2 . The post-slipping robot orientation will beq 1 + =d forO 1 , andq 2 + =d forO 2 . Once DY 1 = 0 or DY 2 = 0 is reached, the robot orientation will remain atd, but the CoM fore-aft position can continue to change. Based on experiment measurements, we expect that the leg with smallerS i corresponds to larger slipping force and dominates the slipping, while the other leg passively follows. The post-slipping CoM position, X + , can thus be computed based on the post-slipping fore-aft position of the dominant leg: X 1 + = 8 > > > > > > < > > > > > > : X RB +S RB +C; jS LF j>jS RB j_S LF = 0 X LF +S LF C; jS LF j<jS RB j_S RB = 0 X 1 ; jS LF j=jS RB j X 2 + = 8 > > > > > > < > > > > > > : X LB +S LB +C; jS RF j>jS LB j_S RF = 0 X RF +S RF C; jS RF j<jS LB j_S LB = 0 X 2 ; jS RF j=jS LB j 23 3.3 Pacing The following discussion focuses on a virtual bipedal gait, pacing, where two left legs of a quadrupedal robot move synchronously, and alternate with right leg pair. Each stride consists two steps: 1 st half- stride (i.e., step) by leg pair O 1 =fLF, LBg (red, Fig. 3.1), and 2 nd half-stride by leg pair O 2 = fRF, RBg (green, Fig. 3.1). 3.3.1 Mode1-Legpair1slippingphase Mode map: (X 1 k ,q 1 k )! (X 1 k + ,q 1 k + ) During the slipping phase, the change in the robot orientation state, q, is rigidly related to the difference in fore-aft positions of the “active” leg pair: DX 1 k = 2Lsin( p 2 q 1 k ); (3.49) where DX 1 k =X LF X LB (3.50) We use subscript to denote the time right before leg-obstacle or leg-ground contact, and + to denote the time right after the leg-obstacle interaction. DX 1 k + =DX 1 k +DS 1 k (3.51) DS represents the difference of slipping distance between the two active legs: DS 1 k =S LF S LB (3.52) 24 where the slipping distance of each leg depends solely on the leg’s fore-aft position in relative to the near edge of the obstacle in contact ([3]): S i = 8 > > > > > > < > > > > > > : N(P+D)X i ; X i 2[N(P+D);N(P+D)+D=2) N(P+D)+DX i ; X i 2(N(P+D)+D=2;N(P+D)+D] 0; (X i =N(P+D)+D=2)_(X i 2(N(P+D)+D;(N+ 1)(P+D))) (3.53) where X i ;i2fLF;RF;LB;RBg represents the relative fore-aft position of each leg on the peri- odic obstacle orbit, N is natural number. After calculating the slip distance for each active leg, their fore-aft position can be calculated: X LF =X 1 k +Ccos(dq 1 k )+S LF (3.54) X LB =X 1 k Ccos(d+q 1 k )+S RF (3.55) The change in robot orientation,q, therefore can be computed by substitutingDX in Eqn. 3.49 with Eqn. 3.51: 2Lsin( p 2 q 1 k + )= 2Lsin( p 2 q 1 k )+DS 1 k (3.56) We can compute the CoM fore-aft position based onX LF andX LB : X 1 k + = X LF +X LB 2 Wsinq 1 k + (3.57) 3.3.2 Mode2-Legpair1advancingphase Mode map: (X 1 k + ,q 1 k + )! (X 2 k ,q 2 k ). 25 The orientation does not change during the advancing phase: q 2 k =q 1 k + (3.58) The CoM fore-aft position advances by a constant step length, S P , along the robot current orientation,q 1 k + : X 2 k =X 1 k + +S P cos(q 1 k + ) (3.59) Eqn. 3.58 and Eqn. 3.59 specifies the mode map for leg pair 1 advancing phase. 3.3.3 Mode3-Legpair2slippingphase Mode map: (X 2 k ,q 2 k )! (X 2 k + ,q 2 k + ) Similar to mode1: DX 2 k = 2Lsin( p 2 q 2 k ); (3.60) where DX 2 =X RF X RB (3.61) We use subscript to denote the time right before leg-obstacle or leg-ground contact, and + to denote the time right after the leg-obstacle interaction. DX 2 k + =DX 2 k +DS 2 k (3.62) DS represents the difference of slipping distance between the two active legs: DS 2 k =S RF S RB (3.63) 26 After calculating the slip distance for each active leg, their fore-aft position can be calculated: X RF =X 2 k +Ccos(d+q 2 k )+S RF (3.64) X RB =X 2 k Ccos(dq 2 k )+S RB (3.65) The change in robot orientation,q, therefore can be computed by substitutingDX in Eqn. 3.60 with Eqn. 3.62 : 2Lsin( p 2 q 2 k + )= 2Lsin( p 2 q 2 k )+DS 2 k (3.66) We can compute the CoM fore-aft position based onX RF andX RB : X 2 k + = X RF +X RB 2 +Wsinq 2 k + (3.67) 3.3.4 Mode4-Legpair2advancingphase Mode map: (X 2 k + ,q 2 k + )! (X 1 (k+1) ,q 1 (k+1) ) Similar to mode 2, The orientation does not change during the advancing phase: q 1 (k+1) =q 2 k + (3.68) The CoM fore-aft position advances by a constant step length, S P , along the robot current orientation,q 2 k + : X 1 (k+1) =X 2 k + +S P cos(q 2 k + ) (3.69) Eqn. 3.68 and Eqn. 3.69 specifies the mode map for leg pair 1 advancing phase. 27 3.3.5 Exceptioncases Cases where the mode map cannot be directly applied: DX 1 k + > 2L orDX 2 k + > 2L. Since the distance between robot’s two active legs(O 1 ,O 2 ) cannot excess 2L, we assume there is a dominated leg with stronger slide force and the other one follow its step. For the caseDX 1 k + > 2L: q 1 k + = p 2 + p 2 sgn(q 1 k p 2 ) (3.70) CoM fore-aft position can be computed as follows: X 1 k + = 8 > > > > > > < > > > > > > : X 1 LF +Lsgn(q 1 k + p 2 ); S LB = 0_jS LB j>jS LF j X 1 k ; S LF =S LB 6= 0 X 1 LB Lsgn(q 1 k + p 2 ); S LF = 0_jS LF j>jS LB j (3.71) For the caseDX 2 k + > 2W: q 2 k + = p 2 + p 2 sgn(q 2 k p 2 ) (3.72) CoM fore-aft position can be computed as follows: X 2 k + = 8 > > > > > > < > > > > > > : X 2 RF +Lsgn(q 2 k + p 2 ); S RB = 0_jS RB j>jS RF j X 2 k ; S RF =S RB 6= 0 X 2 RB Lsgn(q 2 k + p 2 ); S RF = 0_jS RF j>jS RB j (3.73) 28 Chapter4 SimulationResultsandComparisonWithExperiments Here we analyze the obstacle-modulated robot state dynamics predicted by the mode map model for given gaits and initial conditions. In order to verify the validity of the model, same parameters were used in simulation to compare with experiments. In simulation, q 0 was varied from -90 degree to 90 degrees with 0.1 degree increment whereasX 0 varied from -30cm to -21.7cm(-30 +P + D) with 0.1cm increment for each gait, simulation step number is set to 20 for robot to generate complete trajectory in obstacle field. 4.1 RobotTrajectoriesandRobotOrientationVS.Time Figure 4.1 shows robot trajectories and orientation dynamics for the three virtual biped gaits, bound, trot, and pace. Similar to observations from experiments (Figure 2.9), robot trajectories converge to different steady state orientations under repeated obstacle interactions, where the po- sitions of steady-state orientations depend sensitively on the gait used. 4.2 BasinofAttractionComparison With model predicted orientation, a simulation attraction basin can be calculated. Figure 4.2 shows the model predicted steady states orientation and attraction basin has a good agreement between experiment and simulation. 29 Figure 4.1: Simulation results. Left side from top to bottom are robot simulation trajectories of bounding, trotting and pacing gait based on robot CoM position respectively; right side from top to bottom are robot simulation orientation of bounding, trotting and pacing gait respectively, different color represents simulation results with different initial orientation corresponding to the color bar. The discovery of these gait-dependent basins of attractions suggest that by sequentially com- posing different gaits, a multi-legged robot can potentially take advantage of obstacle disturbances to robustly achieve desired trajectories in obstacle-cluttered environments. Furthermore, in the fol- lowing sections, we discuss how we can analytically predict the steady states and associated basins of attraction using the mode map model. We discovered two simple mechanisms for the robot to passively converge to a steady state orientation:i) alternating leg pairs passively select opposing obstacle disturbances; and ii) projected step length allows periodic sampling of the environment while robot legs step between obstacles. The discovery of these mechanisms have enabled analyt- ical prediction of equilibrium orientations for given legged robot and terrain parameters. 30 Figure 4.2: Steady states attraction basin in simulation and experiment. Every color marker repre- sents one steady state orientation, the color represents value of orientation corresponding to right color bar. Color marker with error bar represents equilibrium states, color marker without error bar represents limit cycle states. Color shaded area means the attraction basin for each steady state, each attraction basin correspond to the steady states markers with same horizontal position. 4.3 MechanismsLeadstoEquilibriumStates 4.3.1 Mechanism1: Disturbance-balance As discussed in chapter 3, the sliding of active legs is what causes robot orientation changes and based on the modeling assumption the sliding is depends on the contact position. As a result, if the two legs in one pair of active leg pair have same relative contact position after one certain step k,DS 1 k andDS 2 k in equations 3.4, 3.29, 3.52 and equations 3.15, 3.40, 3.63 are equals to 0. So that q 1 k + equals to q 1 k in equations 3.8, 3.31, 3.56 and q 2 k + equals to q 2 k in equations 3.18, 3.42, 3.56. Since Mode 1 and Mode3 doesn’t change the robot orientation, the robot orientation keep the same during Mode1 to Mode4, which satisfy the definition of equilibrium definition 2.1 and 2.2. The mechanism is shows in figure 4.3. The gait shows in figure 4.3 is bounding, but trotting and pacing 31 gait has the same mechanism behavior. Therefore, given 2W, 2L and S B , S T , S P , this mechanism can be described as Equation 4.1, 4.2, 4.3 for bounding, trotting and pacing respectively. 2NMsinq =M(P+D) (4.1) 2NLcosq =M(P+D) (4.2) 2NLsinq =M(P+D) (4.3) WhereN;M2N,q2 [ p 2 , p 2 ]. Figure 4.3: Steady mechanism 1 schematic. Black rectangle represents robot body before sides, green and red filled circle represents active leg pair 1 and 2. 4.3.2 Mechanism2: Collision-freeSteady The other mechanism is intuitive, since the robot orientation changes relay onDS 1 k andDS 2 k in equa- tions 3.4, 3.29, 3.52 and equations 3.15, 3.40, 3.63, if there is no active leg contact with obstacle after one certain stepk,DS 1 k andDS 2 k equals to 0. As a result,q 2 k + equals toq 1 k after applying Mode 1 to 4, which also satisfy the equilibrium states definition 2.1 and 2.2. The mechanism is shows in figure 4.4. The gait shows in figure 4.4 is bounding, but trotting and pacing gait has the same 32 mechanism behavior. This mechanism can be described as Equation 4.4, 4.4, 4.4 for bounding, trotting and pacing respectively. NS B cosq =M(P+D);X i 2[D;D+P] (4.4) NS T cosq =M(P+D);X i 2[D;D+P] (4.5) NS P cosq =M(P+D);X i 2[D;D+P] (4.6) WhereX i X i (mod P+D),i2f LF, RF, RB, LBg,N;M2N,q2 [ p 2 , p 2 ]. Figure 4.4: Steady mechanism 2 schematic. Black rectangle represents robot body before sides, green and red filled circle represents active leg pair 1 and 2. 4.4 MechanismLeadstoLimitCycleStates Limit cycle states has a distinguish characteristic that robot orientation will oscillate around a virtual average angle in orientation VS. times figure, for instance, wave line in figure 4.1, which can be also derived from the definition. In order to get more intuitive thought about the mechanism 33 behind LCS, we define the LCS more specifically with a repeat period indicated by N. A N- period LCS means that under the constraints?? and??, another constraint equation should also be satisfied. q k+N =q k (4.7) Where k represents k-th step after robot entering LCS. Since simulation is based on discrete time point, equation?? can be transferred to equation 4.8. N1 å n=0 q k+n = 0 (4.8) We can analyze simple LCS cases with characteristic in equation 4.8. However, for complicate cases it’s hard to analyze by only using the characteristic. To show how complicate the cases would be, here we start from a 2-period LCS with trotting gait. It can be derived from definition 4.7 that there are only two different robot angle exists after robot entering LCS, let’s indicate them by q 1 and q 2 respectively. Since q 1 +q 2 = 0, and based on the LCS definition, q 1 6=q 2 , which means when robot angle transfer from q 1 to q 2 and from q 2 to q 1 there would be one increasing and one decreasing. With out of loss of generality, q 1 is output of Mode 1 and q 2 is output of Mode3. DS can be decided increasing or decreasing given increasing or decreasing of q 1 and q 2 based on equations 3.31 and 3.29, 3.42 and 3.40. S R F and S L B has 5 different combinations with given increasing or decreasing ofDS. For instance, given theDS increasing, S R F and S L B can be increasing, increasing, increasing, no change, increasing, decreasing, no change, decreasing and decreasing, decreasing. Thus, we have 2 5= 10 kinds of position changes of active legs to form a 2-period LCS. As a result, a formula aim to describe the complexity of N-period LCS based on the position change tendency of active legs is showed as 4.9. S LCS = 5(2 N 2) (4.9) 34 Where S LCS represents the number of kinds of position changes tendency of active legs for a N- period LCS. From equation 4.9, we know that without given parameters of robot, there are many possible LCS exists in the modeling world, refer to Section 4.5.3 we will construct a state-space directed graph to analyze these more complex dynamics in ongoing work. 35 4.5 Ongoingandfuturework 4.5.1 Obstacle-aidednavigationthroughsequentialselectionofgaits Different SS of each gait provide possibility to combine them to robustly achieve desired trajecto- ries under repeated obstacle interactions. As a result, the question becomes how can we use gait combination to access previous unreachable area. In order to specify what gait combination we could have to achieve this target, a transfer matrixX which is an 3 size matrix is defined, where the n represents we take n different gaits to achieve the goal. G a is defined to describe gait space. At previous section we discussed the model can be used to describe robot states transition and position transition. F defined as a function to represents robot states transition by using proposed mode map model. The problem can be described as find a transfer matrix X that have the highest possibility to make robot transit from given stateR s to desired stateR d . G a = Bounding;Trotting;Pacing T (4.10) In this study, we assume that when we switch robot gait from one to other, the switching process happens very quickly and during the process robot has no additional interaction with obstacles. Moreover, we set a gait switching constrain that before entering SS robot cannot make another gait switching. A possibility functionP r is introduced to describe the uncertainty effects on robot state during gait transition. Given a input set of robot statesS I ,P r can be described as Equation 4.11. P r (R d ;S I )= j S R s 2S I fR d jk ! R sd k<h R gj j S R s 2S I S R i 2R fR i jk ! R si k<h R gj (4.11) Where ! R si =R i R s , ! R sd =R d R s . As described in Equation 4.11, with increasing number of sequential gait changes, gait transfer 36 Figure 4.5: Sequential gait composition to reach different steady states orientation. Steady states orientation of one gait treated as initial orientation as another gait finally converged to a new steady state orientation. noise will be accumulated. In this study, we only consider one time gait change. State transit success rateP f can be described as Equation 4.12. P f = å F(R m )=R d ;k ! R si k<h R P r (R m ;R i ) (4.12) In order to investigate our proposed sequential gait combination potential for developing future obstacle-aided navigation. Two preliminary experiments were performed as show in Figure 4.6. Figure 4.6: Trajectory of sequential gait combination. Left figure show the trajectory of one trail from pacing gait shifted to bounding gait, right figure show the trajectory of one trail from bound- ing to trotting gait. 37 4.5.2 BeyondVirtualBipedalGaits From Figure 2.11 we observed a total of 10 different steady state orientations for the three virtual bipedal gaits. As a result, we can expand available steady orientations available since there are numerous gaits in gait space as shown in Figure 4.7 [7]. In order to investigate others gait than bounding, trotting and pacing, experiments with three intermediate gaits, BT0.5, TP0.5 and PB0.5 were performed. As show in Figure 4.8, a new steady orientation 56 1:4 appeared in experiment with BT0.5 gait, another new steady orientation 62 0:9 appeared in experiment with TP0.5 gait. These additional new steady states orientations prove a wide range of dynamic composition opportunities in the gait space that can facilities flexible obstacle-aided [8] [9]. In the future, I plan to adapt the obstacle disturbance framework and the mode map model to understand and predict the obstacle-modulated steady states for these non-virtual-bipedal gaits. Figure 4.7: Gait space. Each point in gait space represents a specific gait,j LFRF represents phase difference of LF leg and RF leg,j RBRF represents phase difference of RB leg and RF leg,j LBRF represents phase difference of LB leg and RF leg; TP0.5, BT0.5, PB0.5 represents intermediate gait between trotting and pacing, bounding and trotting, pacing and bounding respectively. 38 Figure 4.8: Trajectories of intermediate gaits. From top to bottom are BT0.5, TP0.5 and PB0.5 respectively, a steady state orientation 56 1:4 was observed in BT0.5, a steady state orientation 62 0:9 was observed in TP0.5, a steady state orientation 90 3:2 was observed in PB0.5. 39 4.5.3 ConvergeProcessofSteadyState We discussed in previous sections about mechanisms that leads to occurrence of ES, however, it is hard to describe LCS mechanism due to its complexity. The mode map model allows construction of directed graphs in the robot state space. These directed graphs are beginning to form energy landscapes and facilitating the understanding of the convergence of gait-dependent steady states. Such understandings will guide the development of more ”reactive” strategies to cope with less structured environments. In order to better describe ES and LCS and SS attraction ability which pave the way to enable the combination of different SS, a visualize method based on mode map model is introduced in this study as shows in Figure 4.9, Figure 4.10 and Figure 4.11. (1) (2) (5) (6) (3) (4) (7) (8) (9) (10) (13) (14) (11) (12) (15) (16) Figure 4.9: Basin formation process for bounding gait. Red and green lines represents obstacle in terms of robot CoM,X k and robot orientation,q k ; each point in black point cloud represent a robot state (X k ,q k ), left top number represents sequential step number in simulation. 40 (1) (2) (5) (6) (3) (4) (7) (8) (9) (10) (13) (14) (11) (12) (15) (16) Figure 4.10: Basin formation process for trotting gait. Red and green lines represents obstacle in terms of robot CoM,X k and robot orientation,q k ; each point in black point cloud represent a robot state (X k ,q k ), left top number represents sequential step number in simulation. (1) (2) (5) (6) (3) (4) (7) (8) (9) (10) (13) (14) (11) (12) (15) (16) Figure 4.11: Basin formation process for pacing gait. Red and green lines represents obstacle in terms of robot CoM,X k and robot orientation,q k ; each point in black point cloud represent a robot state (X k ,q k ), left top number represents sequential step number in simulation. 41 Chapter5 Conclusion We experimentally characterized passive steady state orientations for three virtual bipedal gaits: bound, trot, and pace. A discrete-time mode-map model was developed to predict steady states ori- entations, which agreed well with experiment data. Moreover, underlying mechanisms governing the coupling between gaits and periodic obstacles that produced the steady states were uncovered by our proposed model. We show that these gait-dependent steady states have begun to enable obstacle-aided navigation through sequential composition of gait selections. In on-going and fu- ture work, we will explore the complete gait space to enable additional steady-state orientations. We will also build upon the mode map model to understand the transitional dynamics, to create reactive gait adaptation strategies for less structured environments. 42 References 1. Wilshin, S., Reeve, M. A., Haynes, G. C., Revzen, S., Koditschek, D. E. & Spence, A. J. Longitudinal quasi-static stability predicts changes in dog gait on rough terrain. Journal of ExperimentalBiology220, 1864–1874 (2017). 2. Wilshin, S., Haynes, G. C., Porteous, J., Koditschek, D., Revzen, S. & Spence, A. J. Mor- phology and the gradient of a symmetric potential predict gait transitions of dogs. Biological cybernetics111, 269–277 (2017). 3. Qian, F. & Goldman, D. I. The dynamics of legged locomotion in heterogeneous terrain: uni- versality in scattering and sensitivity to initial conditions. in Robotics: Science and Systems (2015), 1–9. 4. Ramesh, D., Kathail, A., Koditschek, D. E. & Qian, F. Modulation of Robot Orientation Via Leg-Obstacle Contact Positions.IEEERoboticsandAutomationLetters5, 2054–2061 (2020). 5. Qian, F. & Koditschek, D. E. An obstacle disturbance selection framework: emergent robot steady states under repeated collisions. The International Journal of Robotics Research 39, 1549–1566 (2020). 6. Saranli U Buehler M, K. D. RHex: A Simple and Highly Mobile Hexapod Robot. The Inter- nationalJournalofRoboticsResearch20, 616–631 (2001). 7. Haynes G Rizzi A, K. D. M. Phase regulation for robust steady and transitional legged gaits. TheInternationalJournalofRoboticsResearch31, 1712–1738 (2012). 8. Revzen S Koditschek DE, F. R. Towards testable neuromechanical control architectures for running.AdvExpMedBiol, 629:25–55 (2009). 9. Wilshin, S., Reeve, M. A., Haynes, G. C., Revzen, S., Koditschek, D. E. & Spence, A. J. Longitudinal quasi-static stability predicts changes in dog gait on rough terrain. Journal of ExperimentalBiology (2017). 43

Abstract (if available)

Abstract Animals are able to flexibly adapt their gaits to cope with a wide variety of challenging environments. Understanding the fundamental principles of such adaptation can guide the creation of control and planning strategies to improve the locomotion performance of legged robots. However, one of the key challenges is the limited understanding of the complex interactions between legged locomotors and perturbation-rich terrains, due to the wide variety in terrain parameters, and the large disturbances from dynamic collisions. The objective of this study is to understand how gaits influence the interactions between legged robots and obstacle-cluttered terrains and resulting in different obstacle-modulated orientation dynamics. We study the horizontal-plane dynamics of a quadrupedal robot as it traverses an array of evenly distributed obstacles, and discovered that the robot can passively converge to different steady-state (SS) orientations with three virtual-bipedal gaits: bound, trot, pace. In order to investigate the interaction mechanism behind the passive SS, we developed a discrete-time model map model that can predict robot state transition dynamics in the obstacle field for given gaits and step lengths. Comparison between experiment, simulation, and analytical results suggest that the highly-simplified model is capable of predicting the robot steady-state orientations and converging dynamics for the three virtual bipedal gaits, facilitating the discovery of the governing mechanism behind such passive convergence. Building upon the mode-map model, we are developing a gait transit method to utilize the existing passive steady states to navigate robots in the obstacles fields. Going forward, we envision these interaction models, combined with the exploration of complete gait space, will enable even low-cost robots to flexibly exploit complex environments and achieve robust navigation.

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Asset Metadata

Creator Hu, Haodi (author)

Core Title A discrete-time return map analysis and prediction of gait-modulated robot dynamic under repeated obstacle collisions

School Viterbi School of Engineering

Degree Master of Science

Degree Program Electrical Engineering

Publication Date 04/27/2021

Defense Date 03/12/2021

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

Tag discrete-time dynamic,gaits,legged locomotion,mode map,OAI-PMH Harvest,repeated obstacle collisions

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Qian, Feifei (committee chair), Nguyen, Quan (committee member), Nuzzo, Pierluigi (committee member)

Creator Email haodihu@usc.edu,myddhhhd@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-456486

Unique identifier UC11668246

Identifier etd-HuHaodi-9551.pdf (filename),usctheses-c89-456486 (legacy record id)

Legacy Identifier etd-HuHaodi-9551.pdf

Dmrecord 456486

Document Type Thesis

Rights Hu, Haodi

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