Bipedal Locomotion on Small Feet. Bipedal Locomotion on Small Feet. Pop Quiz for Tony 6/26/2015. Jessy Grizzle. Jessy Grizzle.

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1 Bipedal Locomotion on Small Feet Jessy Grizzle Elmer G. Gilbert Distinguished University Professor Levin Professor of Engineering ECE and ME Departments Pop Quiz for Tony Can you give the first name of either D Alembert or Lagrange? Jean le Rond d'alembert and Joseph Louis Lagrange What is the air-speed velocity of an unladen swallow? What do you mean? An African or European swallow? Whose birthday am I unable to celebrate today? My wife s birthday! Bipedal Locomotion on Small Feet Jessy Grizzle Elmer G. Gilbert Distinguished University Professor Levin Professor of Engineering ECE and ME Departments Major Themes Walking may be easy for people, still challenging for robots! Control of balance on small feet enables agility and robustness on normal feet. The unreasonable effectiveness of geometry + nonlinear control + parameter optimization. From Simple More Complex From Not So Simple Very Complex Bipedal Aaron Ames 1

2 MABEL: Planar & Underactuated Control Algorithms Jonathan Hurst Does the bar hold up the robot? Large Springs K. Sreenath H. W. Park 1 m 65 Kg Ultimate Goal: Bipedal robots outdoors 11 J. Hurst: 3 ATRIAS robots built & delivered MARLO Lateral leg motion MARLO Carnegie Mellon Oregon State Michigan 2

3 13 DOF in SS and 6 Actuators MARLO Look ma, no bar Brian Buss A. Ramezani K. Akbari Hamed Brent Griffin Models are Hybrid Models Lagrangian Dynamics Hybrid SS Single Support DS Double Support Underactuated Walking gait: alternating phases: SS, DS, SS, DS, Models are Hybrid Models are Hybrid Impact Dynamics DS Double Support DS Double Support Walking gait: alternating phases: SS, DS, SS, DS, Walking gait: alternating phases: SS, DS, SS, DS, 3

4 Models are Hybrid Models are Hybrid Walking gait: alternating phases: SS, DS, SS, DS, SS SS DS Models are Hybrid Models are Hybrid Hybrid System: May have multiple phases Degree of Actuation Degree of Actuation joint angles & velocities motor torques joint angles motor torques & velocities No longer fully actuated! Fully actuated: dim u = dim q Fully actuated: dim u = dim q Common assumption, but dangerous. severe limitations on ankle torque (to prevent rotation of foot) 4

5 Degree of Actuation State of the Art: DRC 6 & 7 June 2015 The bipeds in the following videos are trying to stay fully actuated! joint angles motor torques & velocities No longer fully actuated! Fully actuated: dim u = dim q severe limitations on ankle torque (to prevent rotation of foot) ZMP, M. Vukobratovic (1968) Capture Point, Pratt et a. (2006) State of the Art: DRC 6 & 7 June 2015 State of the Art: DRC 6 & 7 June 2015 Degree of Actuation Degree of Actuation joint angles & velocities motor torques series elastic actuators Fully actuated: dim u = dim q Underactuated: dim u < dim q Passive Pivot Fully actuated: dim u = dim q Underactuated: dim u < dim q 5

6 Periodic Orbit Poincaré map ( ) Steady-state Walking Periodic Solution How to find a periodic solution? How to check for stability? Trajectory Tracking vs. Virtual Constraints Feedback Design Virtual constraints Design via parameter optimization Initial 3D experiments Trajector y Gait Generato r NL PID Cont. Physical Virtual Constraints Virtual constraints y = (variables to be controlled) (desired evolution) Gait phase or timing variable Feedback controller: y(t) 0 6

7 Virtual constraints Virtual constraints holonomic constraint Gait phase or timing variable Feedback controller: y(t) 0 Virtual constraints Virtual constraints Virtual Constraints in ODE model Switching Surface Virtual constraints Virtual constraints Virtual Constraints in ODE model Virtual Constraints in ODE model Byrnes-Isidori Zero Dynamics ( 88) 7

8 Geometric Interpretation Geometric Interpretation In general Hybrid Zero Dynamics Theorem: [TAC 2001, TAC 2009, Book] Can design surface such that Hybrid zero dynamics Hybrid zero dynamics dim 2(N-k) dim 2k Rendering Z hybrid invariant Rendering Z sufficiently rapidly attractive. Hybrid zero dynamics From invariance reset map is often expansive Design of the Periodic Orbit Computation and Orbit Design Adjust feedback gains to achieve attractivity of Z 8

9 Desired Evolution Constrained Optimization (standard) y = (variables to be controlled) (desired evolution) Design desired evolution Cost Function Free parameters to be chosen Integrate over Hybrid Zero Dynamics 2(N-k) dim Constrained Optimization (standard) Nonlinear-Hybrid Feedback Control Design desired evolution Cost Function Equality Constraints Swing leg impact at end of step Walking speed Periodicity or not Inequality Constraints Ground reaction force positive Friction coefficient < 0.6 Swing foot clearance Torque bounds Stability??? Choosing What to Control? Free Pivot For 1 DUA, STABILITY is INDEPENDENT of CHOICE of outputs! angular vs. Cartesian Otherwise (such as in 3D), choice matters. 9

10 3D: Output Choice Matters 3D: Output Choice Matters Swing leg Stance leg Robot Falls Sagittal Plane Frontal Plane 3D: Output Choice Matters 3D: Output Choice Matters Swing leg Stance leg Eigenvalues now have magnitude less than one Robot Walks Sagittal Plane Frontal Plane 3D: Output Choice Matters Choosing What to Control Eigenvalues have magnitude greater than one Systematic Selection of Outputs Robot Falls Kaveh Akbari Hamed Brian Buss CDC 2014; ADHS 2015; IJRR [to appear] 10

11 Systematic Selection of Outputs Parameterized family of controllers Periodic orbit: independent of parameters Seek parameter values giving (exp.) stability Periodic Orbit Orbit independent of parameter choice but changes the zero dynamics! Applies Much More Broadly Periodic Orbit Feedforward term so that orbit independent of parameter choice 11

12 Seek parameter vector so that Jacobian is Hurwitz Taylor Series Expansion Taylor Series Expansion Nominal parameter vector Objective: Choose such that sum of matrices is Hurwitz Choosing What to Control Bilinear Matrix Inequality Seek Lyapunov function and satisfying BMI Optimization Problem Can be handled by the solver PENBMI. 12

13 Giving new outputs that couple pitch and roll in ways that we would not have found through intuition Giving new outputs that couple pitch and roll in ways that we would not have found through intuition standard new Robot falls when we cut the power. Can Include Disturbance Rejection Goal Outdoors IJRR (to appear) & ADHS (submitted) Rough Terrain 13

14 Virtual Constraints and HZD Virtual Constraints and HZD Westervelt et al Martin et al. Sreenath et al Chicago Rehabilitation Institute and UT Dallas Robert D. Gregg et al., ICORR 2013 Leg placement? No! ZMP? No! Ames et al Gregg et al Buss et al Vanderbilt leg being controlled with methods conceived at Michigan for bipedal robots. Virtual Constraints and HZD Virtual Constraints and HZD Robert Gregg Aaron Ames (Ga Tech & TAMU) Oregon State: Mikhail Jones and J. Hurst Acknowledgments University of Michigan Brian Buss, Brent Griffin, Kaveh Akbari Hamed, Ali Ramezani, Kevin Galloway, Dennis Da, Ross Hartley, Omar Harib Oregon State University Jonathan Hurst, Dynamic Robotics Laboratory, Jesse Grimes, Ross McCullough, Soo-Hyun Yoos 14

15 Concluding Remarks Great area for feedback control There is a lot going on and much more to do 15