CONTROL OF APERIODIC WALKING AND THE ENERGETIC EFFECTS OF PARALLEL JOINT COMPLIANCE OF PLANAR BIPEDAL ROBOTS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Tao Yang, B.S., M.S. * * * * * The Ohio State University 27 Dissertation Committee: Approved by Eric R. Westervelt, Adviser James P. Schmiedeler Andrea Serrani David Orin Adviser Graduate Program in Mechanical Engineering
ABSTRACT In this dissertation, two problems related to bipedal robot walking are presented. The first problem is the influence of parallel knee joint compliance on the average power cost of walking in an underactuated planar bipedal robot, ERNIE. The second problem is the design of walking controllers that induce aperiodic bipedal robot walking. It has been found that compliance plays important roles in walking and running in animals. Compliance has been used in robotic bipedal machines to improve energetic efficiency or reduce the peak power demand on the robot s actuators. This dissertation presents numerical and experimental studies of the influence of parallel knee joint compliance on the average power cost of walking in an underactuated planar bipedal robot, ERNIE. The use of parallel compliance does not increase the control design complexity, as would the addition of series compliance. Four scenarios were studied: one without springs and three with springs of different stiffnesses and preloads. Optimal gaits in terms of average power cost for various speeds were designed for each scenario. It was found that for low-speed walking, soft springs are helpful to reduce power cost, while stiffer springs increase power cost. For high-speed walking, it was found that both soft and stiff springs reduce the average power cost of walking, but stiffer springs reduce the cost more than do softer springs. ii
The second problem addressed in this dissertation is aperiodic walking controller design. Along with the energetic efficiency of bipedal walking, the ability to walk stably in varying environments or with different tasks, such as stepping over stones, is another major concern in bipedal walking. In these scenarios, the walking is not periodic. This dissertation presents a new definition of stable walking that is not necessarily periodic for a class of biped robots. The inspiration for the definition is the commonly-held notion of stable walking: the biped does not fall. To make the definition useful, an algorithm is given to verify if a given controller induces stable walking. Also given is a framework to synthesize controllers that induce stable walking. The results are illustrated with numerical simulation and experiments. This dissertation also presents details of a modeling procedure for the experimental bipedal robot, ERNIE, and explores the possibility to apply iterative learning control to bipedal walking. iii
To my family iv
ACKNOWLEDGMENTS The work in this dissertation has been supported by many people, and I would like to take this opportunity to express my sincere gratitude to them for their support and encouragement. Foremost, I would like to thank my advisor, Dr. Eric Westervelt. I was guided by Dr. Westervelt into the field of robotics. He was patient with me when I was new to this field. He has always been enthusiastic to help me through my work. Without his motivation, generous support, and inspiring conversation, it was not possible for me to finish the work in my dissertation. I would like to thank Dr. James P. Schmiedeler. Beside having Dr. Schmiedeler on my committee, I had opportunities to get his comments and feedback frequently as a member of Locomotion And Biomechanics Lab (Labˆ2). He has been generous to provide me assistance with experimental equipment used in this study. I would like to extend my thanks to Dr. Andrea Serrani. I have been enjoying his lectures when I took ECE 751 and ECE 852 from him. In the development of the aperiodic walking design framework, Dr. Serrani has inspired me with his comments. I am grateful to Dr. David Orin for serving on my committee. I thank him for his advice in the course of my study. v
I have special thanks to many Labˆ2 members for their warm support and lasting friendship, particularly, Ryan Bockbrader, Jeff Wensink, Kenton Williams and Ben McCandless for their help during the experiments. Thanks to my family. They are always supporting me in my academic studies. vi
VITA May 6, 1975... Born - Minquan, Henan, China June, 1998...B.S., Mechanical Engineering, Zhejiang University, Hangzhou, China September, 1998 - December, 21...System Engineer, Shanghai Institute of Satellite Engineering, Shanghai, China January, 22 - August, 24... Graduate Research Assistant, Kansas State University, Manhattan, KS August, 24... M.S., Mechanical Engineering, Kansas State University, Manhattan, KS August, 24 - December, 27... Graduate Research Associate, The Ohio State University, Columbus, OH vii
PUBLICATIONS T. Yang, E. R. Westervelt, J. P. Schmiedeler, and R. A. Bockbrader. Design and Control of a Planar Bipedal Robot ERNIE with Parallel Knee Compliance. Submitted to Autonomous Robots, October 8, 27 T. Yang, E. R. Westervelt, and J. P. Schmiedeler. Using parallel joint compliance to reduce the cost of walking in a planar bipedal robot. In Proceedings of ASME International Mechanical Engineering Congress and Exposition, November 27. T. Yang, E. R. Westervelt, J. P. Schmiedeler, and R. A. Bockbrader. Design and control of the planar bipedal robot ERNIE. In Proceedings of ASME International Mechanical Engineering Congress and Exposition, November 27. T. Yang, E. R. Westervelt, and A. Serrani. A framework for the control of stable aperiodic walking in underactuated planar bipeds. In Proc. of the 27 IEEE International Conference on Robotics and Automation, Roma, Italy, 27. D. E. Schinstock, Z. Wei, and T. Yang. Loop shaping design for tracking performance in machine axes. ISA transactions, 45(1):55 66, January 26. Tao Yang. Current control of permanent magnet synchronous motors using DSP technologies. Master s thesis, Kansas State University, 24. viii
FIELDS OF STUDY Major Field: Mechanical Engineering Studies in: Robotics Control theory Dynamical systems Mathematics Professor E. R. Westervelt Professors A. Serrani, V. I. Utkin Professors C.H. Menq, R. G. Parker Professors F.R. Tian, J. Humphries ix
TABLE OF CONTENTS Page Abstract....................................... Dedication...................................... Acknowledgments.................................. Vita......................................... List of Tables.................................... List of Figures................................... ii iv v vii xiii xv Chapters: 1. Introduction.................................. 1 1.1 Background of Bipedal Robot Walking............... 2 1.1.1 An Overview of Bipedal Robots................ 2 1.1.2 Walking Stability........................ 6 1.1.3 Approaches to Bipedal Robot Locomotion.......... 7 1.1.4 Compliance in Legged Locomotion.............. 11 1.1.5 Tracking Control in Biped Walking.............. 11 1.2 Contributions.............................. 13 1.3 Dissertation Organization....................... 15 2. The Bipedal Robot ERNIE......................... 16 2.1 Bipedal Walking Modeling Overview................. 17 2.2 Swing Phase Model........................... 21 2.2.1 Development of the Swing Phase Model using the DH Convention............................. 22 x
2.2.2 Development of the Swing Phase Model using Screw Theory 29 2.3 Double Support Model......................... 33 2.4 Hybrid Model of Walking....................... 38 2.5 Mechanical Design........................... 39 2.5.1 Features of the Mechanical Design.............. 4 2.5.2 Limb and Torso Design.................... 41 2.5.3 Actuation and Transmission.................. 42 2.5.4 ERNIE s Experimental Setup................. 46 2.6 System Integration........................... 46 2.6.1 Sensors and Computation................... 46 2.6.2 Robot-Treadmill Interaction.................. 47 3. Control Philosophy Hybrid Zero Dynamics and its Application in ERNIE s Walking Gait Design............................. 49 3.1 Zero Dynamics............................. 5 3.1.1 An Example Illustrating Zero Dynamics: the Cart-Pendulum Problem............................. 5 3.1.2 Swing Phase Zero Dynamics.................. 54 3.1.3 Hybrid Zero Dynamics of Walking.............. 57 3.1.4 Ground Contact........................ 59 3.2 ERNIE s Walking Gait Design Using Parametric Optimization... 6 3.2.1 Choosing the Output Functions................ 61 3.2.2 Gait Design Using Parametric Optimization......... 63 3.3 Control System Implementation.................... 65 3.3.1 ERNIE s Control Program................... 66 3.3.2 System Tuning......................... 76 4. Using Parallel Joint Compliance to Reduce the Cost of Walking in ERNIE 82 4.1 ERNIE s Model Including Parallel Knee Compliance........ 83 4.2 Numerical Study of the Influence of Knee Springs on the Energetic Efficiency of Walking.......................... 84 4.2.1 Case 1: Gaits Designed without Knee Springs........ 85 4.2.2 Case 2: Gaits Designed with Knee Springs.......... 86 4.2.3 Effect of Spring Parameter Variation on Average Power Cost 89 4.3 Experimental Verification....................... 91 4.3.1 Experiment Procedure..................... 91 4.3.2 Experiment Results...................... 92 4.3.3 Walking Energy Economy of ERNIE............. 95 4.4 Conclusion............................... 96 xi
5. A Framework for the Control of Stable Aperiodic Walking in Underactuated Planar Bipeds.............................. 16 5.1 Introduction.............................. 16 5.2 A New Definition of Stable Bipedal Walking............. 17 5.3 Controller Design............................ 19 5.3.1 Basic Facts and Gait Stability................. 11 5.3.2 Switching Controller Analysis................. 112 5.3.3 Two Applications of Theorem 1................ 116 5.4 Examples of the Walking Controller Design............. 119 5.4.1 Example One: Stable Random Walking........... 12 5.4.2 Example Two: Speed Tracking................ 124 5.5 Experiment Verification........................ 128 5.5.1 Experiment Design....................... 128 5.5.2 Experiment Results...................... 129 5.6 Conclusions............................... 131 6. Conclusion and Future Work......................... 145 6.1 Conclusion............................... 145 6.2 Future Work.............................. 146 Appendices: A. Iterative Learning Control A Possible Way to Induce Better Walking. 149 A.1 Review of ILC algorithms....................... 149 A.2 Example................................. 154 B. Walking Experiment Guideline....................... 158 B.1 Adding a New Gait to ERNIE s Control Program.......... 158 B.2 ERNIE s Experiment Procedure.................... 161 C. Additional Experimental Plots for the Studies of the Effects of the Parallel Knee Compliance on Walking Energetic Efficiency............. 163 Bibliography.................................... 184 xii
LIST OF TABLES Table Page 2.1 Link parameters for ERNIE s five-link model.............. 24 2.2 Link parameters for ERNIE s seven-link full model.......... 36 2.3 Identified link parameters for ERNIE.................. 4 4.1 Optimization design cases........................ 84 4.2 Optimization constraints used in gait design.............. 85 5.1 Random stable walking: domain of definition S D,i j of individual controller Γ αi j for the example....................... 12 5.2 Random stable walking: δ zero,i j of individual controller Γ αi j for the example.................................. 12 5.3 Random stable walking: V zero,i j of individual controller Γ αi j for the example.................................. 121 5.4 Random stable walking: image S I,i j of the domain of definition S D,i j for the example.............................. 121 5.5 Random stable walking: self-transition gait fixed points........ 122 5.6 Random stable walking: individual controller switching sequence for twenty-step aperiodic walking for the example............. 123 5.7 Speed-tracking walking: domain of definition S D,i j of individual controller Γ αi j for the example....................... 124 xiii
5.8 Speed-tracking walking: δ zero,i j of individual controller Γ αi j for the example.................................. 125 5.9 Speed-tracking walking: V zero,i j of individual controller Γ αi j for the example.................................. 125 5.1 Speed-tracking walking: image S I,i j of the domain of definition S D,i j for the example.............................. 126 5.11 Speed-tracking walking: self-transition gait fixed points....... 126 5.12 Speed-tracking walking: individual controller switching sequence for twenty-step aperiodic walking for the example............. 126 A.1 Parameters of the two-link model.................... 154 xiv
LIST OF FIGURES Figure Page 1.1 Planes of the human body........................ 3 1.2 WAP-3 and ASIMO........................... 4 2.1 Frame assignment for ERNIE s five-link model............. 23 2.2 ERNIE s joint variable definition in the planar model......... 27 2.3 ERNIE s measurement conventions as depicted from the boom-side of the robot.................................. 28 2.4 ERNIE in its reference configuration.................. 3 2.5 ERNIE s full model frame assignment including two virtual prismatic links.................................... 35 2.6 The biped prototype: ERNIE s experimental setup.......... 39 2.7 Springs in parallel with the knee joints................. 44 2.8 Two transmission configurations..................... 45 2.9 Top view of ERNIE s experimental setup................ 48 3.1 A cart-pendulum model......................... 5 3.2 The monotonically increasing variable θ................. 61 3.3 ERNIE s controller Simulink diagram.................. 67 xv
3.4 ERNIE s observer and controller Simulink diagram.......... 68 3.5 State machine state flow......................... 7 3.6 State diagram of ERNIE s leg configuration logic........... 71 3.7 ERNIE s observer and controller Simulink diagram.......... 72 3.8 ERNIE s motion tracking Simulink diagram.............. 73 3.9 Collocated high gain PD control for joint trajectory tracking..... 75 3.1 Force sensitive resistor circuit scheme.................. 76 3.11 Example of the force sensitive resistor threshold tuning........ 79 4.1 Knee joint torque profiles for the.65 m/s gait............. 87 4.2 Average power cost over one step calculated from simulation..... 88 4.3 Average power with respect to spring stiffness and preload...... 9 4.4 Average power over one step calculated from experiment....... 97 4.5 Hip joint trajectories and tracking errors of ERNIE walking at.5 m/s with springs of spring constant 16 Nm/rad............ 98 4.6 Knee joint trajectories and tracking errors of ERNIE walking at.5 m/s with springs of spring constant 16 Nm/rad............ 99 4.7 Torso trajectory of ERNIE walking at.5 m/s with springs of spring constant 16 Nm/rad........................... 1 4.8 Hip and knee joint control input of ERNIE walking at.5 m/s with springs of spring constant 16 Nm/rad.................. 11 4.9 Hip horizontal displacement of ERNIE walking at.5 m/s with springs of spring constant 16 Nm/rad...................... 12 4.1 Statistics of ERNIE walking at.5 m/s with springs of spring constant 16 Nm/rad................................. 13 xvi
4.11 Cost of transport (COT) as function of the average walking rate in the four scenarios............................... 14 4.12 Comparison of the minimum cost of transport as a function of body mass for a variety of robots, animals, and vehicles........... 15 5.1 Example transitions in the lower layer of a walking controller..... 115 5.2 Joint angle trajectories for twenty steps of aperiodic walking..... 132 5.3 Stance leg hip joint angle q 1 versus swing leg joint angular velocity q 4 for twenty steps of aperiodic walking.................. 133 5.4 Ground reaction for twenty steps of aperiodic walking........ 134 5.5 Joint torque trajectories for twenty steps of aperiodic walking.... 135 5.6 Step-wise average walking rate and controller indices for 2 steps of walking.................................. 136 5.7 Joint angle trajectories for 2 steps of walking............ 137 5.8 Stance leg hip joint angle q 1 versus swing leg joint angular velocity q 4 for 2-steps of aperiodic walking.................... 138 5.9 Ground reaction for 2 steps of walking................ 139 5.1 Statistics of ERNIE s aperiodic walking (I)............... 14 5.11 Statistics of ERNIE s aperiodic walking (II).............. 141 5.12 Hip joint trajectories and tracking errors of ERNIE s aperiodic walking 142 5.13 Knee joint trajectories and tracking errors of ERNIE s aperiodic walking143 5.14 Torso trajectory of ERNIE s aperiodic walking............. 144 A.1 Two-link manipulator........................... 155 A.2 Two-link manipulator control scheme.................. 155 xvii
A.3 Tracking error on the second joint of the two-link manipulator.... 156 A.4 Joint trajectories at iteration 21.................... 156 C.1 Hip joint trajectories and tracking errors of ERNIE walking at.65 m/s without springs........................... 164 C.2 Knee joint trajectories and tracking errors of ERNIE walking at.65 m/s without springs........................... 165 C.3 Torso trajectory of ERNIE walking at.65 m/s without springs... 166 C.4 Hip horizontal displacement of ERNIE walking at.65 m/s without springs................................... 166 C.5 Hip and knee joint control input of ERNIE walking at.65 m/s without springs................................... 167 C.6 Statistics of ERNIE walking at.65 m/s without springs....... 168 C.7 Hip joint trajectories and tracking errors of ERNIE walking at.65 m/s with springs of spring constant 6 Nm/rad............. 169 C.8 Torso trajectory of ERNIE walking at.65 m/s with springs of spring constant 6 Nm/rad............................ 17 C.9 Hip horizontal displacement of ERNIE walking at.65 m/s with springs of spring constant 6 Nm/rad....................... 17 C.1 Knee joint trajectories and tracking errors of ERNIE walking at.65 m/s with springs of spring constant 6 Nm/rad............. 171 C.11 Hip and knee joint control input of ERNIE walking at.65 m/s with springs of spring constant 6 Nm/rad.................. 172 C.12 Statistics of ERNIE walking at.65 m/s with springs of spring constant 6 Nm/rad................................. 173 C.13 Hip joint trajectories and tracking errors of ERNIE walking at.65 m/s with springs of spring constant 16 Nm/rad............ 174 xviii
C.14 Torso trajectory of ERNIE walking at.65 m/s with springs of spring constant 16 Nm/rad........................... 175 C.15 Hip horizontal displacement of ERNIE walking at.65 m/s with springs of spring constant 16 Nm/rad...................... 175 C.16 Knee joint trajectories and tracking errors of ERNIE walking at.65 m/s with springs of spring constant 16 Nm/rad............ 176 C.17 Hip and knee joint control input of ERNIE walking at.65 m/s with springs of spring constant 16 Nm/rad.................. 177 C.18 Statistics of ERNIE walking at.65 m/s with springs of spring constant 16 Nm/rad................................. 178 C.19 Hip joint trajectories and tracking errors of ERNIE walking at.65 m/s with springs of spring constant 2 Nm/rad............ 179 C.2 Torso trajectory of ERNIE walking at.65 m/s with springs of spring constant 2 Nm/rad........................... 18 C.21 Hip horizontal displacement of ERNIE walking at.65 m/s with springs of spring constant 2 Nm/rad...................... 18 C.22 Knee joint trajectories and tracking errors of ERNIE walking at.65 m/s with springs of spring constant 2 Nm/rad............ 181 C.23 Hip and knee joint control input of ERNIE walking at.65 m/s with springs of spring constant 2 Nm/rad.................. 182 C.24 Statistics of ERNIE walking at.65 m/s with springs of spring constant 2 Nm/rad................................. 183 xix
CHAPTER 1 INTRODUCTION Locomotion is defined as an act or the power of moving from place to place [131]. In a natural or human-made setting, locomotion takes astonishing varieties. On a normal day, we can see birds fly, fish swim, and car pass by. We cannot forget the locomotion we perform ourselves everyday-we walk and run. Legged locomotion is one kind of locomotion among many locomotion forms, and it is of great interest in science and engineering nowadays for its advantages over other kinds of locomotion. Legged locomotion does not require a continuous supporting surface like wheeled and tracked locomotion does, which greatly extends the terrain range that can be accessed. According to a US Army investigation [2], half the earth s surface is inaccessible to wheeled and tracking vehicles, and such terrain is mainly exploited by legged animals. Legged locomotion is also superior to wheeled or tracked vehicles when crossing over obstacles. Bipedal locomotion is conducted by many two-leg animals such as birds and humans, as well as human-made machines bipedal robots. 1
1.1 Background of Bipedal Robot Walking 1.1.1 An Overview of Bipedal Robots Although the term bipedal robot does not exclusively refer to a human-like robot, most of the existing bipedal robots have a morphology similar to a human s. This dissertation will continue to use relevant terminologies presented in [125], in which a biped is assumed to be a kinematic chain consisting of two sub-chains called legs and, often, a sub-chain called the torso, all connected at a common point called the hip. The two legs may contact the supporting surface at the leg end. When only one leg is in contact with the supporting surface, the contacting leg is the stance leg, and the other leg is the swing leg. The end of the leg, whether or not it has links constituting a foot, is often called a foot. Figure 1.1 gives the definitions of the sagittal plane, frontal plane, and transverse plane. The interest in building humanoid robots that can emulate human motions has been ongoing for a long time. As early as in 14-th century, Leonardo da Vinci designed a humanoid robot having the appearance of a knight [118] that was able to emulate some human-like motions. The research and construction of bipedal robots proceeded slowly until the late 2-th century. With the rapid advance in computer technology, electronics, and high performance actuators, the research and construction of bipedal robots gained momentum in 196 s. Around the world, many bipedal robot prototypes have been built since then. Around 197, Vukobratovic proposed the concept of the Zero Moment Point (ZMP) to explain the stability of flat-footed bipedal walking [121, 122]. This criterion provides the theoretical foundation for a class of bipedal walking robots. In the same period, the bipedal robot WAP-3 shown in Fig. 1.2(a), which was able to 2
transverse frontal sagittal Figure 1.1: Planes of the human body. The sagittal plane is the longitudinal plane that divides the body into right and left sections. The frontal plane is the plane parallel to the long axis of the body and perpendicular to the sagittal plane that separates the body into front and back portions. A transverse plane is a plane perpendicular to sagittal and frontal plane [123] 3
(a) WAP-3 (1972) [71] (b) ASIMO [6] Figure 1.2: WAP-3 and ASIMO perform three-dimensional automatic bipedal walking for the first time [71], was built at Waseda University, Japan. Following WAP-3, a series of bipedal robots have been built at Waseda University with increasing complexity and the ability to perform walking or other human-like activities. The walking control approaches applied in this series of bipedal robots have relied heavily on the concepts of center of gravity (CoG) and ZMP. The bipedal robot development in Japan has gone beyond academia, and some commercial companies, such as Honda and Sony, have initiated research projects in this field. One remarkable success is ASIMO, shown in Fig. 1.2(b), from Honda [12]. ASIMO is able to walk on flat ground and climb stairs. In Europe, several projects are active. At the Technical University Munich, Germany, JOHNNIE [72,73,86] was designed and built for fast walking (up to 2.4 km/h) with a dynamically stable gait pattern. JOHNNIE has a complicated mechanical structure with 17 joints and is equipped with precise sensors and a high performance 4
control system to enable the fast walking motions. JOHNNIE is 1.8 meters high and is relatively light with a total weight of approximate 4 kg. At the Delft Biorobotics Laboratory (DBL) [3,39,13] located in Delft, Netherlands, a series of bipedal robots were built. Stappo is the first biped built at DBL that was able to walk on a flat surface without controls. After Stappo, research in bipedal robots at DBL was continued with the construction of Mike (fall 22), Denise (fall 24), and Meta (fall 25). The bipedal robots at DBL are mostly based on passive walking principles [75]. At the Laboratoire d Automatique de Grenoble, France, a robot named RABBIT [34] was constructed and served as the testbed on which the hybrid zero dynamics (HZD) approach to control biped walking [124] was first experimentally validated. In the United States, there have been comparatively fewer projects: At the MIT Leg Lab, extensive research in legged robots has been conducted since it was founded by Marc Raibert in 198 [89]. The walking bipedal robots built in this lab include the notable Spring Turkey and Spring Flamingo [91]. Both were planar walking bipeds walking with booms. The strong tradition of the MIT Leg Lab has been continued by the Robot Locomotion Group [39]. Since 1996 at Cornell, several passive biped projects [1,39,4] have been constructed since 1996 from Tinker Toy Walkers to the Cornell Ranger. In the development of bipedal walking robots, there are two problems of paramount importance. The first one is walking stability. The second one is energetic efficiency. Regarding these two problems, extensive research has been conducted, and many approaches have been proposed and tested. 5
1.1.2 Walking Stability One fundamental problem associated with bipedal walking is to ensure the stability of walking. Classic stability criteria such as Lyapunov stability are not applicable to this kind of problem. As a result, other stability criteria have been proposed. One of the common definitions of biped walking stability is that the robot does not fall and keeps walking. This definition is intuitive, but it is difficult to apply this definition in a rigorous way. As a result, more appropriate walking stability criteria are needed. For bipeds that walk flat-footed, the most popular methods to ensure gait stability is to ensure that the ground projection of center of mass (GCoM), the Zero-Moment Point (ZMP) [12, 119, 12], or the foot-rotation indicator (FRI) [5] lies in the supporting polygon. Since the GCoM only considers statics, it can only be applied when the walking motion is slow enough so that the dynamic forces can be ignored. The ZMP is defined as the point on the ground where the total moment generated due to gravity and inertia equals zero [113]. During stable walking, the supporting foot remains firmly on the supporting ground, i.e., without rotation and slippage on the ground. The ZMP coincides with the center of pressure (CoP) where the pressure between the ground and the foot can be replaced by single force [13]. The FRI [5] is a point on the foot/ground-contact surface where the net ground reaction force would have to act to keep the foot stationary. The ZMP, CoP and FRI are closely related, and interested readers can find detailed discussion in [5,88,13,14,119,12]. A recent definition of gait stability proposed by Pratt and Tedrake [92] is the velocity-based stability margin. The velocity-based stability margin provides a sufficient condition for stability by considering the biped s center of mass position and 6
velocity, the reachable region of its swing leg, the time required to swing its swing leg, and the amount of internal angular momentum available for capturing balance. The method of Poincaré is a useful tool for studying the stability of periodic walking. Considering a hyperplane intersecting the orbit that corresponds to the walking gait, the point of intersection is the fixed point of the orbit s return map with the surface, which is also known as the Poincaré map. The hyperplane is known as the Poincaré section. The periodic orbit s stability is the same as the fixed point s stability. If the moduli of the all eigenvalues of the linearized Poincaré map are smaller than 1, then the fixed point is stable; otherwise, it is unstable. Use of a Poincaré map enables the stability of the orbit corresponding to a periodic gait to be determined by examination of an associated discrete-time system. Cheng and Lin [32] derived the linearization of the Poincaré map for the gait of a 5-link planar biped at the fixed point. The eigenvalues of the linearized return map were then used to design controllers that induced periodic gaits. In [7, 51, 52, 76, 129], the eigenvalues of the Poincaré maps for the gaits of several different bipeds were obtained numerically. In the work that provides the foundations for the developments of this thesis, Grizzle et al. [56] and Westervelt et al. [125,126], the controller design was used to reduce the dimension of the Poincaré map to one. In [126], the Poincaré map was shown to be linear, thus allowing gait stability to be derived using simple stability metrics. 1.1.3 Approaches to Bipedal Robot Locomotion With the walking stability criteria and stability analysis tools reviewed previously, various approaches have been taken to realize bipedal walking, and many of them have concerns of energetic efficiency. 7
One design-oriented approach is passive-dynamic walking wherein the robot s dynamics are designed such that the robot is able to walk stably down shallow slopes without the need for control or energy input aside from that coming from gravity [48, 75]. Inspired by Mochon and McMahon s work [79], McGeer pioneered the study of passive walking [75]. McGeer built a two-dimensional biped walker without knees and demonstrated that the passive walker was able to walk down a small slope and obtain stable, periodic motion. Goswami [52, 54] studied the stability of symmetric and asymmetric passive gaits of a compass-like, planar biped. Garicia [48,49] studied passive walking stability and complexity, and developed a scaling law of planar two-link models. Adolfsson [4] showed the existence of passive walking numerically based on a 3-D, five-link biped model. Collins et al. [4] built the first three-dimensional, kneed, two-legged passive-dynamic biped. Wisse et al. [128] showed that a straight-leg walker with an upper body may exhibit successful passive walking with disturbance rejection if a kinematic constraint is applied on the upper body. Borzova et al. [25] found three different passive gaits based on a five-link robot with knees and an upper body. In the studies of passive walking, the method of Poincaré was used extensively [47, 49, 52, 53, 76]. There are three primary drawbacks of passive-dynamic walkers [69]: they are only able to walk down slopes, their gaits are restricted to the few admitted by their dynamics, and they are sensitive to perturbations. Realizing these limitations, researchers have sought to improve passive-dynamic walkers by adding actuation [39]. With the exploration of passive walking, active control approaches have been developed to make a biped mimic the passive walking motion. Asano et al. [16,17] introduced a virtual gravity field in which virtual gravity forces drive the biped to 8
walk forward as in the case with gravity for passive biped walkers. In [15,18], Asano et al. formulate the biped gait generation problem by investigating the gait generation mechanism in passive dynamic walking considering mechanical energy and proposed a robust control design using mechanical energy behavior characteristics. Collins et al. [39] reported on three robots based on passive-dynamics, with small active power sources substituted for gravity, which can walk on level ground with high efficiency. In Anderson et al. [1], three bipeds were reported whose design and control are based on principles learned from the gaits of the passive dynamic walking bipeds. Spong et al. [19,11] showed the existence of a nonlinear control law that reproduces so-called passive gaits independent of the particular ground slope. Another way to realize biped walking is the model-based optimization approach. In this approach, the joint trajectory references are optimized with respect to a chosen cost function while ensuring the stability criterion is satisfied. Active control is then applied to control the joints to follow the trajectory references. The most common cost functions used are the average energy cost and the average quadratic norm of torques. To define and find the joint trajectory references that minimize the chosen cost function, the most common means is to use parametric optimization to choose the parameters that specify the joint trajectory references. For example, Chevallereau et al. [35] used parametric optimization to design fourth-degree polynomial functions that gave the joint motions over a step as functions of time, and the chosen cost function, average energy, was minimized. In another example using average energy as the cost function, Channon et al. [3] used parametric optimization to design third-degree polynomial functions that gave the motions of the hip and swing foot of the robot as functions of time, and the corresponding joint trajectory references 9
were determined by an inverse kinematic model of the robot. Djoudi et al. [43] used fourth-degree polynomials to describe the joint trajectory references. The polynomial coefficients were found by minimizing the average norm of applied torque. Unlike the previous examples in which the trajectory reference for each joint was described by a single polynomial function of time, in [11] cubic splines connected at points uniformly distributed along the motion time were used to generate complete optimal steps, including a double-support phase. Another means of designing gaits that minimize the chosen cost function is by application of Pontryagin s Maximum Principle (PMP). The first application of PMP to gait generation can be found in [37] in which the average energy cost was minimized. More recently, Rosatami et al. [96,97] used PMP to design impact-less gaits by casting the constrained gait optimization problem as a two-point boundary value problem to minimize the average energy cost. In [22], Bessonnet et al. carried out dynamicsbased optimization of sagittal gait of a planar seven-link biped using PMP to minimize the average norm of the torques. The third means to find optimal trajectory references is to use genetic algorithms and evolutionary programming. In [57], Hasegawa et al. used a hierarchical evolutionary algorithm to generate natural walking motions via energy optimization considering the ZMP stability condition. Different from these methods based on off-line model-based optimization is gait generation based on learning. Capi et al. [29] enabled real-time gait generation with a neural network designed using learning. Tedrake [39, 114] used online learning to continuously adapt the gait. 1
1.1.4 Compliance in Legged Locomotion Compliance is able to save and release energy by elastic deformation. This ability has resulted in compliance playing an important role in legged locomotion, including both animals and humans, as well as human-made machines. Alexander [6] identified three uses of the springs in legged locomotion: pogo stick-like springs and return springs can save energy and reduce fuel consumption and unwanted heat production; foot pads can reduce foot impact forces and improve road holding by preventing chatter. Compliance was used in the legged robots of Raibert s pioneering work [93, 94]. Compliance continues to be utilized in the design of hopping and running robots, recently by Ahmadi et al. [5] and Hurst et al. [62]. Mechanical compliance has also been used in walking bipedal robots. In [1, 38, 46], springs were added at the passive ankles to improve the energetic efficiency of walking. In [63], springs were added across the hip and shank, and thigh and heel simultaneously. Series-elastic actuation was implemented on Spring Flamingo to enable control of the ground reaction forces in walking, thus providing an active suspension [9]. Vanderborght et al. [116] used McKibben s muscles as a means to add controllable joint compliance. 1.1.5 Tracking Control in Biped Walking For biped walking, after the reference trajectories are defined, controlling joints to follow the references is classically solved as a manipulator trajectory tracking problem. Most of the tracking control schemes fall into one of the following three categories: 11
The first approach to the trajectory tracking problem is based on the classic servomechanism, where the joint controllers could be simple proportionalintegral-derivative (PID) controllers, and the dynamics of the manipulator is ignored. This approach is model-independent and robust with respect to the model variation. This approach is commonly used in applications that only require low-to-moderate performance. Another approach is the inverse dynamics control approach. To apply this approach, the manipulator dynamics need to be known accurately, and the resulting model is explicitly used in the feedback control design. With this approach, high performance can be expected when the model is accurate. However, accurately modeling the plant is challenging for high-dimensional systems such as multiple-degree-of-freedom manipulators. This difficulty greatly limits the application of this approach. The third approach is the adaptive control approach [21,42,66,67,99,1,16 18, 127]. With the adaptive control approach, the system s mathematical description needs to be known precisely, but not all parameters need to be known exactly. These three control approaches are generally applicable to the robot trajectory tracking control problem, but they do not fully utilize the desired motion characteristics such as periodicity. On the other hand, iterative learning control (ILC) and repetitive control (RC) are control approaches that are particularly useful for tracking repetitive motion. 12
ILC and RC are different from the three approaches given above in several ways. Foremost, ILC uses previous trial information to improve future trial performance, and it can be expected that the performance is improved over trials. The classic servomechanism and inverse dynamics approaches have fixed feedback controller structure and parameters, and if each trial has the same initial conditions and random disturbances are ignored, then the performance will be the same for each trial. ILC and RC are also different from conventional adaptive control. In most adaptive control schemes, the unknown plant parameters are identified online, and essentially a closed-loop control is formed with the identified plant parameters, while ILC and RC could have an open-loop control structure, i.e., the control applied in one trial is not based on the current trial feedback. Instead, it is learned from previous trials. Another difference between ILC and RC and adaptive control lies in that ILC and RC adjust controller output or control reference at the end of each trial, while in adaptive control schemes the controller parameters are adjusted continuously. 1.2 Contributions The dissertation continues the work of Westervelt [123,126] with two contributions. The first contribution is the study of the influence of parallel knee compliance on the energetic efficiency of walking. The second contribution is a new definition of stable biped walking and a framework for designing controllers that induce aperiodic walking that is stable in the sense of the given definition. In the broader spectrum of bipedal robot walking, walking with great energetic efficiency and dexterity are two major goals. One of the most promising uses of bipedal robots is as service machines that assist humans in obstacle-ridden environments, such 13
as the home. In such applications, where a tether is impractical, a bipedal robot will most likely rely on its onboard power supply, such as batteries. Therefore, energy efficiency is of crucial importance for bipedal robots. The bipedal robot in this kind of application also requires great dexterity like a human to be able to avoid obstacles. The two contributions of this dissertation address these two goals respectively. Compliance has been used by others to save energy in legged locomotion, but using compliance in parallel with knee actuators for bipedal walking has not been reported. This dissertation gives a thorough examination of the influence of parallel knee compliance on the energetic efficiency of walking. The study was verified with simulation and experiment. It was found that the addition of springs in parallel with the knee actuators can improve the energetic efficiency of walking, with higher stiffness providing greater benefit at higher speeds and lower stiffness providing benefit at lower speeds. The second contribution addresses the walking dexterity problem. The proposed definition subsumes the common notion of gait stability: the robot does not fall. The new definition of gait stability makes it possible to prove rigorously that walking is stable. The definition is independent of the controller used. Using the proposed definition, a framework for the design of walking controllers is given. This framework extends the work of Westervelt et al. [14]. Unlike [14], which enabled the design of controllers that only induce stable periodic walking, the proposed framework allows the induced gaits to be aperiodic. Moreover, the gait associated with an individual controller of the framework is not necessarily asymptotically stable. 14
1.3 Dissertation Organization Chapter 2 first introduces the modeling procedures for the class of bipedal robots studied and then presents ERNIE s mechanical design and system integration. Chapter 3 first reviews the hybrid zero dynamics technique applied in bipedal walking and then presents ERNIE s control program implementation. Chapter 4 presents numerical and experimental studies of the influence of parallel knee joint compliance on the average power cost of walking in ERNIE. Chapter 5 presents a framework in which aperiodic bipedal walking can be introduced. The walking controller design procedure is illustrated with simulation and with ERNIE in experiment. Chapter 6 concludes the dissertation and gives several suggestions for future works. 15
CHAPTER 2 THE BIPEDAL ROBOT ERNIE ERNIE is a bipedal robot constructed at the Locomotion and Biomechanics Lab of The Ohio State University. The primary motivation for the design and construction of ERNIE was to provide a scientific and educational platform for the development of novel control strategies for bipedal walking and running. In this dissertation, ERNIE is used as a testbed for the studies of the influence of parallel knee compliance on walking energetic efficiency and aperiodic walking. The general morphology of ERNIE was inspired by that of RABBIT [34]. It has two symmetric legs with a torso. The two legs are connected with the torso at the hip joint. ERNIE has an approximate anthropomorphic structure. The motion of ERNIE is considered to occur only in its sagittal plane. It is assumed that a normal step consists of a double support phase and a single support phase. Walking is assumed to consist of consecutive steps. There are a number of unique features in the mechanical design of ERNIE that impact the range of experiments that can be carried out as well as the controller design and implementation. ERNIE can be configured to walk on a treadmill to make it possible to walk infinitely in a confined space. 16
This chapter presents the details of the development of ERNIE s hybrid walking model as well as ERNIE s mechanical design and electrical and computer system integration. 2.1 Bipedal Walking Modeling Overview Walking involves alternation of leg configuration. To facilitate modeling of walking, it is common to divide a step into different parts according to the leg configuration. One part is the swing phase in which one leg is in contact with the supporting surface and the other leg swings in the air. Generally speaking, the swing phase can be modeled as an open chain multi-link manipulator described by continuous differential equations. Normally, the swing phase comprises the most significant portion of a step. Another part is the double support phase in which the legs contact the supporting surface at the same time. The double support phase can be modeled as a closed-chain manipulator described by continuous differential equations. For actuated bipedal robots, because of holomonic constraints imposed on both leg ends during the double support phase, the degrees of the freedom are reduced, which could result in the model being over-actuated. The third part is the impact phase that connects the swing phase with the double support phase. The impact phase could be modeled with good accuracy as an instantaneous rigid impact when both the supporting surface and the foot are sufficiently stiff and the impact duration is sufficiently short. When the impact phase is modeled as an instantaneous rigid impact, it could be described by an algebraic 17
map that calculates the post-impact state variables from the pre-impact state variables. These state variables include the joint angles and joint velocities. Conversely, when the supporting surface and the foot are not stiff enough, a compliant model will be required to model the impact. The compliant model provides the possibility to model the impact more precisely, but it introduces additional computation complexity. Specifically, the compliant model could result in stiff differential equations [44], which require special integration methods to ensure the accuracy of the simulation. Due to these additional complexities, the compliant model is not desirable for the design of controllers for bipedal walking. The presence of all three parts in a walking model for bipedal robots is not required and depends on the physical properties of the bipedal robots, such as the stiffness of the foot covering material and the supporting surface material, as well as the control strategies used in the walking. Considering the presence of these three parts in the walking model, the walking model can be categorized into three cases. In the first case, such as [31,77,78,117], all three parts appear in the model to constitute a complete step. The swing phase and the double support phase are modeled as constrained Lagrangian systems, and the impact is modeled as an instantaneous rigid impact to provide transition from the swing phase to the double support phase. In the second case, a complete step of the bipedal walking model includes only the swing phase and the impact, such as the models used to study dynamically stable walking [11,3,33,35,43,45,55,87,98,15,126]. In this case, the double support phase is absent, or the impact and double support phase are merged together. The impact provides transition from one single support phase to successive single support phases. 18
In the third case, the impact is avoided by carefully designing and controlling the motion such that the end of the swing leg has zero velocity when it touches the supporting surface [22, 23, 82]. As a result of this, the walking model consists of a single support phase and a continuous double support phase. The model of a biped walking depends on the physical properties of the bipedal robots under study as well as the control strategies being considered. For example, the instantaneous rigid impact model is accurate enough only when the foot material and supporting surface material are sufficiently stiff. The control strategy to be applied also affects the modeling. Like the third case, the impact phase disappears because the swing leg velocity is controlled to be zero when it touches the supporting surface. ERNIE has aluminum hemispherical feet covered by half of a racket ball and walks on a concrete surface or on treadmill belts that are supported by a stiff plastic layer. With this configuration, the impact between the swing leg and the supporting surface can be approximately modeled as an instantaneous rigid impact. In ERNIE s walking, the double support phase is merged with the rigid impact to form an instantaneous double support phase. With this modeling assumption, ERNIE s walking model falls into the second case. The modeling of bipedal robot walking has been presented in previous studies, but the current literature seems to lack a thorough presentation of the modeling procedure. In this chapter, two methods for deriving a model of bipedal walking are presented. The first method uses the Denavit-Hartenberg (DH) convention, and the second method uses screw theory. These two methods are demonstrated with the development of ERNIE s model. 19
In the development of the biped walking model, the robot hypotheses and gait hypotheses used in [126] are assumed here, and copied below for convenience. Robot hypotheses The robot is assumed to be: RH 1. comprised of 5 rigid links with mass, connected by revolute joints with no closed kinematic chains; RH 2. planar, with motion constrained to the sagittal plane; RH 3. bipedal, with symmetric legs connected at a common point called the hip; RH 4. actuated at each joint; and RH 5. unactuated at the point of contact between the stance leg and ground. Gait hypotheses From the intuition and observation of simple walking, the following hypotheses are assumed for the bipeal robot gaits. GH 1. there are alternating phases of single support and double support; GH 2. during the single support phase, the stance leg acts as a pivot joint. That is, throughout the contact, it can be guaranteed that the vertical component of the ground reaction force is positive and that the ratio of the horizontal component to the vertical component does not exceed the coefficient of static friction; GH 3. the double support phase is instantaneous and can be modeled as a rigid contact[hm94]; GH 4. at impact, the swing leg neither slips nor rebounds; 2
GH 5. in steady state, successive phases of single support are symmetric with respect to the two legs; GH 6. walking is from left to right, so that the swing leg starts from behind the stance leg and is placed strictly in front of the stance leg at impact. 2.2 Swing Phase Model During the swing phase, ERNIE has one leg contacting the supporting surface and the other leg swinging in the air. ERNIE is modeled as an open-chain manipulator. The dynamical modeling of ERNIE can be accomplished in a variety of ways. The most common approach is to use Lagrange s equations. The second most common approach is to use the Newton-Euler formulation. In this dissertation, the first approach is taken to derive the equations of motion of the swing phase. The application of the Newton-Euler formulation to open-chain manipulator modeling can be found in [111]. To apply Lagrange s formalism, the Lagrangian L is defined as the difference between the kinetic energy K and the potential energy V of the system as L := K V. (2.1) Lagrange s equations are given as d L L = u i, (2.2) dt q i q i where q i and u i are the i-th generalized coordinate and the i-th generalized force, respectively, of the system of degree of freedom N, and i = 1,...,N. The degree of freedom (DOF) N of a system is the minimum number of the coordinates required to describe the system configuration. 21
In this dissertation, the kinetic energy K and the potential energy V of ERNIE are derived using two different methods. The first method follows the de facto standard method, the DH convention, and the second method uses screw theory. 2.2.1 Development of the Swing Phase Model using the DH Convention In this section, the swing phase model of ERNIE is developed using the DH convention and Lagrange s equations. In the following development, the coordinate frames are chosen first. In the second step, the forward kinematics and differential kinematics are found such that the link positions, orientation and velocities can be expressed by the joint variables and their derivatives. In the third step, the kinetic and potential energies are expressed as functions of the joint variables and their derivatives. Then, the equations of motion are obtained by applying Lagrange s equations. Finding the forward kinematics is the first step to derive ERNIE s equations of motion during the swing phase. The forward kinematics gives the link positions and orientations as functions of the joint variables. In this development, the position of the center of mass and orientation of each link are of interest. The five-link model of ERNIE during the swing phase is depicted in Fig. 2.1. As an open-chain manipulator, the stance leg tibia of ERNIE is numbered as the first link, and the other links are numbered from 2 to 5 successively from the stance leg femur to the swing leg tibia, as shown in Fig. 2.1. The base frame o x y z is established as shown. Its origin coincides with the contact point of the stance leg and the supporting surface; its x axis points in the direction of walking; its y axis points 22
L b L H y 2 x 2 L bcom hip z 3 y 3 o 3 = o 3 = o 4 o 2 L fcom hip L fcom hip z 2 x 3 L f z 4 y 1 L f x 1 o 4 = o 5 y 4 L tcom knee x 4 o 1 o 1 = o 2 z 1 L t L tcom knee z 5 y o 5 L t y 5 x 5 z o = o 1 x Figure 2.1: Frame assignment for ERNIE s five-link model 23
upward; and its z axis is determined by the right-hand rule. After the base frame is chosen, other frames are chosen following the DH convention [111] given in Fig. 2.1. Link a i α i d i θ i 1 L t θ 1 2 L f θ 2 3 π L H θ 3 4 L f θ 4 5 L t θ 5 Table 2.1: Link parameters for ERNIE s five-link model The DH parameters are given in Table 2.1. The parameter a i is the distance between o i and o i. The parameter d i is the coordinate of o i along z i 1. The parameter α i is the angle between axes z i 1 and z i about axis x i. The parameter α i is zero or π due to the fact that all joint motions happen in the sagittal plane. The parameter θ i is the angle between axes x i 1 and x i about axis z i 1 to be taken positive when rotation is made counter-clockwise. The parameter θ i represents the relative angular motion between the two consecutive link frames, and they are the joint variables of ERNIE s model. The homogeneous transformation matrix A i that expresses the position and orientation of the frame o i x i y i z i with respect to the frame o i 1 x i 1 y i 1 z i 1 has the general form A i 1 i = c θi s θi c αi s θi s αi a i c θi s θi c θi c αi c θi s αi a i s θi s αi c αi d i 1, (2.3) 24
where s θi stands for sin(θ i ) and c θi stands for cos(θ i ). The homogeneous transformation matrix T i that expresses the position and orientation of the frame o i x i y i z i with respect to the base frame o x y z is given by T i = A 1A 1 2...A i 1 i. (2.4) The position of the center of mass of the i-th link in the inertial frame is given by p com,i := x com,i y com,i z com,i 1 = T i x i com,i y i com,i z i com,i 1, (2.5) where x i com,i, y i com,i, and z i com,i are the coordinates of the i-th link COM in the associated link frame. To find the kinetic energy expressed as a function of the joint variables and their derivatives, it is convenient to express the COM linear velocities and the angular velocity of the each link as functions of the joint variables and their derivatives first. Let v com,i be the linear velocity vector of the i-th link COM in the inertial frame and ω i the angular velocity vector of the i-th link in the inertial frame. The body velocity vector ξ i is given as which can be computed by ξ i := [ vcom,i ω i ], (2.6) ξ i = J i θ, (2.7) where J i is Jacobian matrix associated with the i-th link and θ = [θ 1, θ 2, θ 3, θ 4, θ 5 ] T. The Jacobian matrix J i has the form J i = [ J i 1 J i 2 J i 3 J i 4 J i 5], (2.8) 25
where and J i j = [ zj 1 (p com,i o j 1 ) z j 1 ], j i, (2.9) Ji+1 i = = Ji 5 = [ ]T, (2.1) since the i-th link position and velocity are determined by only the motions of the links 1,...,i, and the motion of the joint after the i-th joint has no effect on the i-th link s motion. The i-th link s kinetic energy is given by K i (θ, θ) = 1 2 m iv T com,i v com,i + 1 2 ωt i (R i) T I i R i ω i, (2.11) where R i is the orientation transformation between the i-th body attached frame and the inertial frame, which is a submatrix of T i given by T i [1 : 3; 1 : 3], and I is the inertia tensor of the i-th link expressed in the body frame. The i-th link s potential energy is given by V i (θ) = m i gz com,i, (2.12) where m i is the mass of the i-th link, g is the acceleration due to gravity, and z com,i is the height of the i-th link COM in the base frame. The total kinetic energy K θ (θ, θ) of ERNIE in the swing phase is given in the generalized coordinates θ 1,...,θ 5 by K θ (θ, θ) = 5 K i (θ, θ), (2.13) i=1 and the total potential energy is given by V θ (θ) = 5 V i (θ). (2.14) i=1 26
θ 3 θ 4 θ 2 θ 5 y o θ1 x Figure 2.2: ERNIE s joint variable definition in the planar model Since ERNIE was designed to walk in its sagittal plane, only dynamics in this plane are of concern. The third link s offset a 3, whose value equals ERNIE s hip width L H, is set to zero. The planar model of ERNIE is given in Fig. 2.2, which illustrates the joint variable definitions corresponding to Fig. 2.1. In Fig. 2.3, a different coordinate choice for ERNIE is depicted in which the torso orientation is an absolute angle with respect to the vertical direction, and all other angles are relative angles. This coordinate system is used for convenience in the rest of the dissertation. The two coordinate systems given in Fig. 2.2 and Fig. 2.3 are related by a coordinate transformation given by 1 1 1 1 θ = 1 q + 1 1 1 π 2 2π π =: Θ 1 (q), (2.15) 27
q 5 p v H q 1 q 2 p H p h H p v 2 q 4 q 3 p 2 p h 2 A Figure 2.3: ERNIE s measurement conventions as depicted from the boom-side of the robot and θ = Θ 1 q q = 1 1 1 1 1 1 1 q, (2.16) where q := [q 1, q 2, q 3, q 4, q 5 ] T Q, Q is a simply-connected, open subset of [, 2π) 5, and q := [ q 1, q 2, q 3, q 4, q 5 ] T R 5. This is a valid coordinate transformation since the determinant of the Jacobian matrix Θ 1 q is nonzero. By replacing θ and θ by q and q in K θ (θ, θ) and V θ (θ), the kinetic and potential energies K and V are given in the new coordinates q as and K q (q, q) := K θ ( Θ1 (q), Θ 1(q) q q ) (2.17) V q (q) := V θ (Θ 1 (q)). (2.18) 28
By using Lagrange s equations d dt ( L q ) L q = u, (2.19) where u is the general force applied with the associated coordinate, and regrouping terms, the resulting equations of motion have the form D(q) q + C(q, q) q + G(q) = Bu, (2.2) where u R 4. Since the dimension of u is smaller than that of q, the robot is underactuated. The matrix D(q) is the mass-inertia matrix, C(q, q) is the matrix of centripetal and Coriolis terms, G(q) is the gravity vector, and B is the input matrix given as B = 1 1 1 1. (2.21) Defining the state vector x := [q q; ] T, the model written in state space is [ ẋ = ] [ q + D 1 ( C q G) ] u, (2.22a) D 1 B =: f(x) + g(x)u (2.22b) with state space T Q := { (q; q) q Q, q R 5 }. 2.2.2 Development of the Swing Phase Model using Screw Theory In the previous section, the swing phase equations of motion of ERNIE were developed using the DH convention and Lagrange s equations. In that procedure, the kinetic and potential energies were represented as functions of the joint variables 29
by choosing the coordinate frames according to the DH convention. In this section, screw theory is used to find the kinetic and potential energies represented as functions of the joint variables. L H L 3 L bcom hip L b θ 3 θ 4 p 3 = p 4 L fcom hip L 2 L fcom hip L 4 L f L f p 2 = p 5 θ 2 L tcom knee L 1 θ 5 L tcom knee L 5 L t L t y θ 1 z o p 1 x Figure 2.4: ERNIE in its reference configuration Assume that in Fig. 2.4 ERNIE is in its reference configuration, where its hip and knee joints are straight, and the body is vertical. The twists ξ i, where i = 1,..., 5, 3
for joints 1 to 5 are given by ξ 1 = 1, ξ 2 = L t 1, ξ 3 = L t + L f 1, ξ 4 = L t + L f 1, ξ 5 = L t 1. (2.23) Attach to the center of mass of each link a frame L i aligned with the principle inertia axes of the link as shown Fig. 2.4. Then, the configuration of the frame L i represented in the chosen inertial frame is given by g sl1 () = I L t L tcom knee 1, (2.24) g sl2 () = I L t + L f L fcom hip 1, (2.25) g sl3 () = I L t + L f + L hcom hip L H 2 1, (2.26) g sl4 () = I L t + L f L fcom hip L H 1, (2.27) and g sl5 () = I L t L tcom knee L H 1, (2.28) where I is the 3 3 identity matrix. 31
With the choice of the link frames, the links inertia matrices have the following form: M i = m i m i m i, (2.29) I zi where m i is the i-th link mass and I zi is the i-th link inertia along its attached body frame z axis. The body Jacobian J b sl i () corresponding to the i-th link frame can be calculated following Eq. 3.55 in [83]. The kinetic energy is given by K θ = 5 i=1 1 2 θ T Jsli() b T Mi Jsli() b θ, (2.3) and the potential energy is given by V θ = 5 m i gh i (θ), (2.31) i=1 where θ = [θ 1,...,θ 5 ] T, and h i is the height of the i-th link COM given by the its y axis coordinate in the inertial frame, and can be found via the forward kinematics map g sli (θ) = eˆξ 1 θ1 eˆξ i θ i g sli (). With a coordinate transformation θ = Θ 2 (q) and θ = Θ 2(q) q q, where q is defined in Fig. 2.3, and Θ 2 (q) := 1 1 1 1 1 1 1 q + π π π, (2.32) 32
and following the same procedure given by (2.17) (2.2), the same equations of motion can be obtained as given by (2.2) with the coordinates q by setting the hip width L H to zero. 2.3 Double Support Model At the end of each swing phase, when the swing leg contacts the supporting surface, an impact occurs. The impacts are assumed to be instantaneous rigid-body impacts. Various rigid impact models have been studied in [19,26,27,61,85]. This dissertation follows [125], which uses the model of Hürmüzlü et al. [61] with the following impact hypotheses copied below for convenience: Impact model hypotheses: The impact model of [61] is used under the following assumptions: IH 1. the contact of the swing leg with the ground results in no rebound and no slipping of the swing leg; IH 2. at the moment of impact, the stance leg lifts from the ground without interaction; IH 3. the impact is instantaneous; IH 4. the external forces during the impact can be represented by impulses; IH 5. the impulsive forces may result in an instantaneous change in the velocities, but there is no instantaneous change in the configuration; and IH 6. the actuators cannot generate impulses and hence can be ignored during impact. 33
The development of the impact model of ERNIE requires a 7-DOF model of the biped. In order to develop the 7-DOF full model of ERNIE, two virtual prismatic joints are added to the end of the stance leg as illustrated in Fig. 2.5. The development of the full model uses the DH convention, and the coordinate frames are chosen as shown in Fig. 2.5. The first and second joints are the two virtual links added to the tip of the stance leg. The corresponding joint variables X and Y are the horizontal and vertical positions of the stance leg tip. With this coordinate frame assignment, the DH parameters for the 7-DOF full model are given in Table 2.2. Following a procedure similar to the one given in Section 2.2, the equations of motion of the full model may be derived, which have the following form. ( ) T D e (θ e ) θ e + C e (θ e, θ E(θe ) e ) θ e + G e (θ e ) = B e u + δf ext, (2.33) θ e where θ e = [X; Y ; θ 1 ; θ 2 ; θ 3 ; θ 4 ; θ 5 ], δf ext are impulses acting on the swing leg end during the impact, E(θ e ) = ( p h 2 (θ e); p v 2 (θ e) ), and p h 2 (θ e) and p v 2 (θ e) are the swing leg end horizontal and vertical coordinates, respectively, as given in Fig. 2.5 in the coordinates θ e. For convenience, a new coordinate system is chosen. Define the new general coordinates q e := [ q 1 ; q 2 ; q 3 ; q 4 ; q 5 ; p h H ; pv H], where p h H and p v H are the hip horizontal and vertical positions with respect to frame. The general coordinates θ e and the general coordinates q e are related by a coordinate transformation Θ e : q e θ e. 34
L b L H y 4 z 4 x 4 L bcom hip z 5 y 5 o 5 = o 5 = o 6 o 4 L fcom hip L fcom hip x5 L f z 6 L f x 3 o 6 = o 7 L tcom knee x 6 y 3 y 6 o 3 = o 4 z 3 o 1 L t L tcom knee L t y 2 y 7 z 7 o 7 p 2 (p h 2, p v 2, p z 2) o o = o b y b z 2 X z x z 1 x b y 1 x 1 o 1 = o 1 z b y x 2 o 2 = o 2 = o 3 Y x 7 Figure 2.5: ERNIE s full model frame assignment including two virtual prismatic links 35
Link a i α i d i θ i 1 π/2 X 2 π/2 Y π/2 3 L t θ 1 4 L f θ 2 5 π L H θ 3 6 L f θ 4 7 L t θ 5 Table 2.2: Link parameters for ERNIE s seven-link full model θ e = p h H L t cos(q 1 + q 3 + q 5 1 2 π) L f cos(q 1 + q 5 1 2 π) p v H L t sin(q 1 + q 3 + q 5 1 2 π) L f sin(q 1 + q 5 1 2 π) q 1 + q 3 + q 5 1/2π q 3 q 1 + 2π q 2 + π q 4 =: Θ e (q e ). (2.34) The determinant of the Jacobian Θe q e of the coordinate transformation is found to be nonzero globally, and thus this coordinate transformation is a valid coordinate transformation. With a coordinate transformation, the equations of motion of the full model are given in the new general coordinate q e by ( ) T E(qe ) D e (q e ) q e + C e (q e, q e ) q e + G e (q e ) = B e u + δf ext, (2.35) q e where E(q e ) = ( p h 2 (q e); p v 2 (q e) ) and p h 2 (q e) and p v 2 (q e) are the swing leg end horizontal and vertical coordinates, respectively, as given in Fig. 2.5 in the coordinates q e. 36
After the equations of motion with the generalized coordinate q e are obtained, the calculations to obtain the impact map are the same as presented in [123]. These calculation are reproduced below to complete the model development. Let q e and q+ e be the generalized velocities just before the impact and just after the impact. Then, by applying the angular momentum conversation, the generalized angular momenta just before and after the impact are related by ( ) [ T D e (qe ) q e = E(qe ) ˆF T 2 qe=q q e e ˆF 2 N ] + D e (q e ) q+ e, (2.36) where ˆF T 2 and ˆF N 2 are the tangential and normal impulses acting on the swing leg tip. With the assumption that the stance leg end has zero velocity after the impact, E(q e ) qe=q q e e By combining equations (2.36) and (2.37), q e +, T ˆF 2 and ˆF 2 N q + [ ] e ˆF 2 T De (q e ) ( E(qe) 1 [ q = e ) T De (qe ) ] q e ˆF N 2 E(q e) q e q e =. (2.37) can be computed as =: Π(q e ) [ De (q e ) q e ], (2.38) and q + e, ˆF T 2 and ˆF N 2 can be expressed more explicitly by partitioning Π(q e ) as q + e = Π 11(q e )D e(q e ) q e, (2.39) and [ ˆF T 2 ˆF N 2 ] = Π 21 (q e )D e (q e ) q e. (2.4) According to the impact hypothesis IH 1, the extended coordinate q e and their velocities are related to q and q by q e = π 1 (q), (2.41) 37
and q e = π 1 (q) q q, (2.42) where π 1 := [ q T, p h H, T H] pv. Since the impact causes the stance leg to change, it is necessary to relabel the coordinates. This can be done by a circular matrix R defined as: 1 1 R = 1. (2.43) 1 1 In the end, the algebraic map that relates the post-impact state with the preimpact state is written as (x ) := [ Rq q ( q ) ], (2.44) where q ( q ) := [R ] Π 11 π 1 (q )D e π 1 (q 1) π 1 (q) q q=q. (2.45) 2.4 Hybrid Model of Walking By combining the swing phase model given by (2.22) and the impact given by (2.44), a hybrid model of walking can be written. { ẋ = f(x) + g(x)u, x / S x + = (x ), x S, (2.46) where the switching set S is defined as S := { (q, q) T Q p v 1 (q) =, p v 2 (q) =, ph 2 (q) > }. (2.47) In the switching set S, ERNIE has a configuration such that both legs contact the supporting surface, and the swing leg is in front of the stance leg. 38
Figure 2.6: The biped prototype: ERNIE s experimental setup With this model, the state vector composed of the joint variables and their derivatives evolves in the state space T Q. Once the state enters the switching set S, an impact event occurs, and the post-impact state x + and pre-impact state x are related by the impact map. 2.5 Mechanical Design To restrict ERNIE s walking motion to its sagittal plane, a boom is attached to the torso via an unactuated revolute joint coaxial with the hips. The revolute connection allows for the body to pitch relative to the boom. Fig. 2.6 is a photograph of ERNIE s experimental setup when ERNIE is on a treadmill. Table 2.3 gives ERNIE s geometric and inertial parameters as determined from a solid model assembly composed of the individual parts from which all of the specialized components were manufactured. 39
Model Parameter Units Link Value torso 13.6 Mass kg femur 1.5 tibia 1. torso.28 Length m femur.36 tibia.36 torso.14 Mass center a m femur.13 tibia.12 torso.9 Inertia b kg m 2 femur.2 tibia.2 Motor rotor inertia kg m 2-2.9 1 5 Gearhead ratio - - 91 Gearhead inertia kg m 2-1. 1 6 a The mass center of each link is measured along link axis from the nearest joint. b The link inertia is measured with respect to its center of mass. Table 2.3: Identified link parameters for ERNIE 2.5.1 Features of the Mechanical Design The following are the key features of ERNIE s mechanical design. Parallel compliance at knees: With a simple cable-mounting assembly at the knee joints, extension springs can be easily added across the knee joints in parallel with the actuators. The addition of compliance has the potential to improve the energetic efficiency of walking. This design feature was inspired by Alexander s [6] idea that it is possible to reduce the work done by the actuators 4
by using return springs to decelerate the leg at the end of each forward or backward swing and accelerate the leg in the other direction. Modular legs: The mechanical couplings at the knee joints and hip joints are designed to make the femurs and tibias independent modules. With this modular design, the leg lengths and leg ends may be changed with minimal redesign. In this way, modularity facilitates future studies involving mechanical changes such as walking with feet. Low-mass links: ERNIE s boom and legs are made primarily of carbon fiber to reduce the total mass without compromising structural rigidity. Actuators in the torso: Locating all of the actuators in the torso reduces the mass that is distal to the robot s center of mass. The result is lighter legs, thus enabling smaller motors to be used. 2.5.2 Limb and Torso Design Femur and Tibia Design: To minimize high-order modes of the robot s mechanics, the robot s legs were designed to be stiff. Carbon fiber tubing was used for the tibias and femurs because of its high rigidity and strength-to-weight ratio. Each limb segment is composed of two tubes. The tube ends are bonded to aluminum plugs that fit inside the tubes. The two carbon fiber tubes of each femur(28.7 mm OD and 1.6 mm wall thickness) are spread apart from the neutral bending axis, while the two carbon fiber tubes of each tibia (28.7 mm and 32.1 mm OD s and 1.6 mm wall thickness) are aligned concentrically. The bending moment on the femur is larger than that on the tibia during stance, 41
so greater strength is needed in the femur. The concentric arrangement of the tubes in the tibia modestly increases its bending resistance without increasing its width in the sagittal plane. A wide tibia would create ground interference problems because the foot, described below, is directly attached to the end of the tibia. Torso Design: The torso was designed to house the motors, connect to the boom, and generally provide structure for the robot as a whole. The torso is made of 6.35 mm (.25 inch) aluminum plates because of its combination of machinability, structural stiffness, and light weight. Foot Design: ERNIE s feet are hemispherical and designed to provide an unactuated degree of freedom in the sagittal plane between the tibia and the ground. Each aluminum hemisphere is covered with half of a racquet ball to provide cushioning at ground impact as well as to increase the coefficient of friction between the foot and the walking surface. With this covering, the foot radius is about 3 mm. A force sensitive resistor is placed between the foot and tibia and is used to detect foot-ground contact. 2.5.3 Actuation and Transmission Actuation: Since the weight of actuators accounts for a significant portion of the total weight of the biped, it is important to choose actuators with a high power-to-weight ratio. As a result, brushless DC motors were chosen for ERNIE, with the size determined from simulations of a detailed model of the robot walking under closed-loop feedback control. Using these simulations, the design of ERNIE was iterated until the needed components specifications 42
matched those that were available off the shelf. The following motors from Maxon Precision Motors Inc. were chosen: brushless DC motor EC45-136212 combined with motor gearhead GP42C-23125 and the incremental encoder HEDL 914. The assigned power rating of the brushless DC motor is 25 Watts, and the gear ratio of the gearhead is 91:1. The motors are powered by brushless servo amplifier B6A4AC from Advanced Motion Controls. Transmission: Since all of the motors are located in the torso, transmissions are needed to transmit power from the motors to the joints. Wire rope cabling (7X19 with uncoated OD of 3.2 mm and vinyl coated for a total OD of 4.8 mm) is combined with pulleys (63.5 mm diameter) to give a simple, compact, and light transmission. For the knee joint transmissions, additional idler pulleys are used at the hips. The pulleys feature slip rings that have a running fit about the pulley body packed with grease. This design allows both halves of the cables to have relative motion during cable tensioning. Parallel Energy Storage: Extension springs can be mounted across the knee joints in parallel with the actuators as shown in Fig. 2.7. The design allows for two extension springs to be attached in parallel across each knee joint to achieve a relatively high effective stiffness without employing an unduly large spring. One end of each extension spring is attached at the top of the femur assembly, and the other end is attached to a wire rope using a wire rope thimble. The wire rope wraps around a circular cam of radius r that is rigidly fixed to the tibia. Since the cam is circular, the torque applied to the knee joint is linearly related to the knee s angular displacement. With this design, the knee springs 43
Femur Extension spring r Cam Tibia (a) Scheme of springs in parallel with knee joint (b) Actual assembly of springs in parallel with the knee joints Figure 2.7: Springs in parallel with the knee joints engage only when the knee joints flex to a certain angle, which is termed the knee spring offset. The overall wire rope length can be adjusted with an in-line turnbuckle or a cable stop as shown in Fig. 2.7(b). Hence, the knee spring offset can be adjusted by changing the wire rope length. When the selected extension springs have a nonzero preloaded F,l and constant spring stiffness Ksp, l the equivalent torsional spring stiffness K sp and the equivalent torsional spring preload τ are K sp := 2K l sp r2, τ := 2F,l r. (2.48a) (2.48b) Comparison between ERNIE s and RABBIT s transmissions The difference between ERNIE and RABBIT s transmission designs lies in the locations of the gear reducers. RABBIT s gear reducers are mounted at the hip and 44
knee joints, timing belts are used to connect the motors output shafts to the gear reducers input shafts to drive the actuated joints as illustrated in Fig. 2.8(a). ERNIE s gearheads are attached to the motors output shafts directly, and the gearhead outputs are connected to the actuated joints shafts by steal cables via pulleys as illustrated by Fig. 2.8(b). Since ERNIE s motors and gearheads are assembled together and mounted in the upper body, ERNIE s hip and knee joints have simpler mechanical structures than RABBIT s, and, consequently, ERNIE s legs are less heavy. Although this transmission design gives ERNIE light legs, but the transmission stiffness is not as high as RABBIT s design. (a) RABBIT s Transmission Configuration (b) ERNIE s Transmission Configuration Figure 2.8: Two transmission configurations The transmission compliance can be modeled as follows. Assume that the motor shaft is fixed and that the torque τ is applied on the output shaft. Then, the transmission stiffness k T is defined by k T = τ θ (2.49) where θ is the output shaft angular displacement under the applied torque τ. 45