1. Multi-Policy Control of Biped Walking
Eric Whitman
7/20/09

2. Outline Existing controller Evaluation
Resistance to perturbations
Speed Control
Proposed controller and work towards it

3. 3D Biped (Complex System)
5 Rigid Links
24 dimensional state space
12 control dimensions
Torque-controlled joints
Static friction with ground
μ=1.0

4. Dividing the System
Simplified Coronal Model
Simplified Sagittal Model

5. Sagittal Plane Dynamics
Newton’s Second Law
Non-rotating Torso
For Leg
Substituting:
Forward Kinematics
Simplifying:

6. Sagittal Plane Dynamics (Touch Down)
Commanded as an action at any time

7. Sagittal Plane Policy Generation
Ankle Torque Policy
Touchdown Policy

8. Full System Sagittal Control

9. Sagittal System Comparison
Test test test
Major Discrepancies:
Swing Leg Acceleration – Negative initial acceleration
Torso Bob – High late-step acceleration

10. Touchdown Prediction Ankle Torque Policy Touchdown Policy
Duration of Simulation:

11. Time Until Touch Down

12. Predicted Angle and Angular Velocity at Touch Down

13. Swing Leg Requirements
Must reach touchdown angle at touchdown time
Must touch down without slipping (velocity matching)
Must not accelerate rapidly
Must not touch ground during swing

14. Trajectory Generation
Cubic Spline
Start at current leg angle and angular velocity
Go to predicted touch down angle and angular velocity

15. Trajectory Update Angle Time
Start at currently desired angle and angular velocity
End at new estimate of touch down angle and angular velocity
Angle
Time

16. Swing Leg Trajectory Tracking

17. Swing Knee Use knee to control swing leg length
Must not touch down during swing
Command foot to 5 cm above ground
Must touch down at appointed time with a straight leg
Command knee angle to straighten linearly:

18. Swing Foot Height and Knee Angle

19. Coronal Plane Control Similar dynamics to sagittal plane
Touchdown is different
Switch sides
Pick angle of touch down
Small angles
State not action
Simplified system – counts down and reset to nominal period
Full system – provided by sagittal policy

20. Full and Simple Coronal Comparison

21. Coronal Swing Leg Simulate forward to get desired touch down angle
Track with PD servo

22. Yaw Control
PD controller on the ankle rotation axis – axis parallel to shin
Does not work well due to large coupling
Must use low gains

23. Perturbation Resistance by Time

24. Failure Modes

25. Speed Control by Leaning

26. Speed Control By Policies

27. Parameters Sagittal Stance Coronal Plane Sagittal Swing PD servos
Cost function weights – 2
Cost function weights – 3
Nominal Speed
Nominal leg angle
Speed lost at touchdown
Sagittal Swing
PD servos
Foot height
Stance hip – 2
Knee extension rate
Swing hip – 2
Swing knee
Yaw ankle

28. Planned Improvements Collaborative Coordination Swing leg policy
More sophisticated sagittal and coronal models
Arms
Better Yaw Control

29. Collaborative Coordination
Each policy supplies V(ttd)
Select ttd to minimize sum(V(ttd)) at each time step
Each policy must then know how to act optimally given that ttd

30. Sagittal Policy with ttd

31. Sagittal Policy with ttd

32. Swing Hip Policy Dynamic programming Cost Model for Trajectories
5D state
Angle
Angular velocity
Target Angle
Target Angular Velocity
Target Time
Cost Model for Trajectories

33. Upgraded Simple Models
Unrestrict the torso angle
Adds 2 more states
Adds an action
Better touch down model
Effect on velocity depends on angle

34. Arms Add an arm swing policy
Probably does not need to participate in collaborative coordination – can be subordinate
Add an estimate of arm swing effect to sagittal policy

35. Improved Yaw Control
Use adjusted pitch and roll ankle torques to cancel the coupled portion of the rotation torques
Change gain with step
High gain mid-step
Low gain at touchdown
Coordinate with leaning or foot placement?

36. Coronal Swing
Can cause trouble if it does not reach its target by touch down
Make a model of tracking error
PD rise time or slew speed
Report expected error to coronal controller
Coronal controller knows cost of this error

37. Robust DP
Normally:
Use:
Or: