TRITA-MMK 2003:19 ISSN 1400-1179 ISRN KTH/MMK–03/19–SE Doctoral thesis Legged locomotion: Balance, control and tools — from equation to action Christian Ridderström Stockholm Department of Machine Design Royal Institute of Technology 2003 S-100 44 Stockholm, Sweden Akademisk avhandling som med tillstånd från Kungliga Tekniska Högskolan i Stockholm framläggs till offentlig granskning för avläggande av teknologie dok- torsexamen, den 12:e maj 10:00 i Kollegiesalen, vid Institutionen för Maskinkon- struktion, Kungliga Tekniska Högskolan, Stockholm. c Christian Ridderström 2003 Stockholm 2003, Universitetsservice US AB Till familj och vänner Abstract This thesis is about control and balance stability of legged locomotion. It also presents a combination of tools that makes it easier to design controllers for large and complicated robot systems. The thesis is divided into four parts. The first part studies and analyzes how walking machines are controlled, ex- amining the literature of over twenty machines briefly, and six machines in detail. The goal is to understand how the controllers work on a level below task and path planning, but above actuator control. Analysis and comparison is done in terms of: i) generation of trunk motion; ii) maintaining balance; iii) generation of leg sequence and support patterns; and iv) reflexes. The next part describes WARP1, a four-legged walking robot platform that has been built with the long term goal of walking in rough terrain. First its modular structure (mechanics, electronics and control) is described, followed by some ex- periments demonstrating basic performance. Finally the mathematical modeling of the robot’s rigid body model is described. This model is derived symbolically and is general, i.e. not restricted to WARP1. It is easily modified in case of a different number of legs or joints. During the work with WARP1, tools for model derivation, control design and control implementation have been combined, interfaced and augmented in order to better support design and analysis. These tools and methods are described in the third part. The tools used to be difficult to combine, especially for a large and com- plicated system with many signals and parameters such as WARP1. Now, models derived symbolically in one tool are easy to use in another tool for control design, simulation and finally implementation, as well as for visualization and evaluation — thus going from equation to action. In the last part we go back to “equation” where these tools aid the study of balance stability when compliance is considered. It is shown that a legged robot in a “statically balanced” stance may actually be unstable. Furthermore, a criterion is derived that shows when a radially symmetric “statically balanced” stance on a compliant surface is stable. Similar analyses are performed for two controllers of legged robots, where it is the controller that cause the compliance. Keywords legged locomotion, control, balance, legged machines, legged robots, walking robots, walking machines, compliance, platform stability, symbolic modeling 5 Preface There is a really long list of people to whom I wish to express my gratitude — the serious reader may skip this section. In short, I would like to thank the entire DAMEK group, as well as some people no longer working there. I’ve had two supervisors, no three actually. my main supervisors have been Professor Jan Wikander and Doctor Tom Wadden, but I’m also grateful for being able to talk things over with Bengt Eriksson in the beginning. We did a project involving a hydraulic leg and managed to spray oil on Tom’s jacket. Tom was also the project leader for WARP1 for a while and he put the fear of red markings in us PhD. students. When you got something you’d written back from Tom, you were sure to get it back slaughtered. He definitively had an impact on my English. Speaking of the WARP1 project, I want to thank all the other PhD. students in it for a great time, especially Freyr who finished last summer and Johan who’s still “playing” with me in the lab on Tuesdays. Johan and I are almost too creative when working together — we can get so many ideas that we don’t know which to choose. Then there is also Henrik, who I wrote my first article with. The research engineers, Micke, Johan and Andreas also deserve a special thank you, as well as “our” Japanese visitors Yoshi and Taka. Peter and Payam should not be forgotten either — the department’s SYS.ADM. — both very fun guys. The social aspect of a PhD. student’s life is very important, and Martin Grimhe- den earns a special thanks for his fredagsrus1 which I believe were introduced by Micke. Most of the other PhD. students at the department have also been social. Ola got us to play basketball once a week for a season. in retrospect its amazing that so many of us actually were there playing at 08.00 in the morning. Thank you guys! Finally I want to thank the developers of LYX [1] for a great program, and also everybody on the User’s List for their help in answering questions. I’ve used LYX to write this thesis and all my articles, and not regretted it once! Acknowledgments This work was supported by the Foundation for Strategic Research through its Cen- tre of Autonomous Systems at the Royal Institute of Technology. Scientific and technical contributions are acknowledged in each part respectively. 1A Swedish word local to DAMEK, the etymology involves weekday, running and beer. 6 Notation See chapter 6 for details on coordinate systems, reference frames, triads, points, vectors, dyads and general vector algebra notation. Table 6.2 on page 152 lists points and bodies in the WARP1 rigid body model (rbm). Vectors fixed in bodies of the WARP1 rbm are listed in table 6.3 on page 152. Bodies, points and matrices are denoted using captial letters, e.g. A. Vectors are written in bold and the column matrix with components of a vector r is written as r. Similarly, dyads are written in bold A and the components of a dyad A, i.e. its matrix representation, is written as A. Note that a left superscript indicates reference body (or triad) for vectors and matrices (see 144). T b1, b2, b3, b triad fixed relative to B, b = [b1, b2, b3] T n1, n2, n3, n triad fixed relative to N, n = [n1, n2, n3] N,B,Tl,Sl,Hl, Kl, Pl points in WARP1 rbm (table 6.2). rBHl , rHTl , rHKl , rKSl , rKPl body fixed vectors of the WARP1 rbm (table 6.3). β duty factor l or l leg l, index indicating leg l L total number legs (L=4 for WARP1) t time T period or cycle time ∂x x, x˙ state and state derivative, i.e. x˙ = ∂t f, F force m mass g constant of gravity q, q,˙ q¨ generalized coordinates, velocities and accelerations w, w˙ generalized speeds and corresponding derivative Ix,Iy,Iz inertia parameters of a body λ,λi,j eigenvalue, λi,j = c means that λi = c and λj = c ± − 7 CM centre of mass of robot (or trunk) PCM projection of CM onto a horizontal plane PCP centre of pressure Asup support area or region σi φiτi instance of a controller from a control basis Jl Jacobian for leg leg τ l joint torques for leg l P1, P2 foot points for leg 1 and 2 C1,C2 ground contact point for leg 1 and 2 φl relative phase of leg l k stiffness coefficient d damping coefficient Kl, BKl stiffness coefficients for leg l Dl, BDl damping coefficients for leg l k1, k2, k3 stiffness coefficients see (11.2) on page 221. kB, kα, kL stiffness coefficients (table 11.1) dB, dα, dL stiffness coefficients (table 11.1) α rotation angle planar rigid body (part IV) r radius (in part IV) h height (in part IV) γ asymmetry parameter (part IV) R asymmetry parameter, R = γr (part IV) a, a2, a3, a4, aL bifurcation parameters (part IV) 8 Contents Abstract ................................... 5 Keywords .................................. 5 Preface.................................... 6 Acknowledgments.............................. 6 Notation................................... 7 List of Figures 15 List of Tables 19 1. Introduction 21 1.1. Thesis contributions . 22 I. A study of legged locomotion control 25 Introduction to part I . 27 2.Introductiontoleggedlocomotion 29 2.1. Basicwalking............................. 31 2.2. Terminology.............................. 31 2.3. Definition of gait and gaits . 33 2.4. Staticbalance ............................. 38 2.5. Dynamicbalance ........................... 40 2.5.1. Center of pressure and dynamic stability margin . 42 2.5.2. ZeroMomentPoint. 43 2.5.3. ZMPandstability. 44 2.6. Miscellaneous joint and leg controller types . ..... 44 9 Contents 3. Controller examples 45 3.1. Deliberative controllers I . 47 3.1.1. Statically balancing controller . 49 3.1.2. Dynamically balancing controller — The expanded trot gait . 52 3.1.3. The Sky-hook Suspension . 56 3.1.4. Dynamically balancing controller — The intermittent trot gait . 57 3.1.5. Summary and discussion . 60 3.2. Deliberative controllers II . 62 3.2.1. The Adaptive Suspension Vehicle . 63 3.2.2. TheASVcontroller. 64 3.2.3. RALPHY,SAPandBIPMAN . 69 3.2.4. Thecontrol .......................... 70 3.2.5. Summary and discussion . 75 3.3. A hybrid DEDS controller . 76 3.3.1. The control architecture . 77 3.3.2. The control basis . 78 3.3.3. Learning Thing to turn . 87 3.3.4. Summary and discussion . 89 3.4. A biologically inspired controller . 91 3.4.1. The Walknet controller . 92 3.4.2. Summary and discussion . 97 4. Analysis 99 4.1. Problem analysis . 100 4.1.1. Goals ............................. 100 4.1.2.