An Approach to Model-Based Control of Frictionally Constrained Robots

1. Biorobotics Lab An Approach to Model-Based Control of Frictionally Constrained Robots Aaron Greenfield CFR Talk 02-22-05

2. Talk Outline 1. Control Under Frictional Contact 2. Planar Dynamics Model - Multi-Rigid-Body - Coulomb Friction 3. Dynamic Response Calculation 4. Applications MOVIE: Real Rhex Flipping MOVIE: Real Snake Climbing (Borer) (Saranli) Slide 2 / 26

3. Control Tasks with Frictional Contact RHex Flipping Task Snake Climbing Task Presumption: The physics of contact is critical to the robot’s performance Approach: ●Utilize a model of robot dynamics under contact constraints ● Solve for behavior as a function of control input Slide 3 / 26

4. Advantages • ● Small number of coordinates • ● Simple Contact Model (1 parameter) Disadvantages ● No Body is perfectly rigid ● Coefficient of friction can be hard to determine, non-static ● Solution Ambiguities and Inconsistenciescan exist Dynamics Model: Multi-Rigid-Bodies and Coulomb Friction Dynamic Equations: 2nd order ODE relates coordinates, forces Rigid Body Model: No penetration, Compressive Normal Force Friction Model: Tangential Force Opposing Slip Slide 4 / 26

5. Accelerations Reaction Forces Control Inputs Ambiguity Variables System State Relates instantaneous behavior to controls and ambiguity for a particular What is this function? To select control inputs which achieve desired instantaneous behavior Why compute it? By solving a series of linear systems of equalities and inequalities consisting of: 1) Lagrange’s equation 2)Contact constraints How do we compute it? Non-linearity, Solution Ambiguity, Inconsistency, Inequality Constraints Why is it hard to compute? Dynamic Response Function where Slide 5 / 26

6. Related Research • Single Rigid Bodies • Ambiguities with Rigid Object, Two Walls. (Rajan, Burridge, Schwartz 1987) • Configuration Space Friction Cone. (Erdmann 1994) • Graphical Methods. (Mason 2001) • Multi-Rigid-Bodies:Modeling and Simulation • Early Application of LCP. (Lostedt 1982) • Lagrangian dynamics and Corner Characteristic. (Pfeiffer and Glocker 1996) • 3D Case, Existence and Uniqueness Extensions. (Trinkle et al. 1997) • Framework for dynamics with shocks (J.J. Moreau 1988) • Early Application of Time Sweeping. (Monteiro Marques 1993) • Formulation Guarentees Existence. (Anitescu and Potra 1997) • Review of Current Work. (Stewart 2000) Multi-Rigid-Bodies: Control Computing Wrench Cones. (Balkcom and Trinkle 2002) (MPCC) Mathematical Program with Complementarity Constraint. (Anitescu 2000) Application of MPCC to Multi-Robot Coordination. (Peng, Anitescu, Akella 2003) Stability, Controllability, of Manipulation Systems. (Prattichizzo and Bicchi 1998) Open Questions for Control of Complementary Systems. (Brogliato 2003) Slide 6 / 26

7. Joint Actuations Reaction Forces Dynamics Equations (Pfeiffer and Glocker) Two coordinate systems (1)Generalized Coordinates (2) Contact Coordinates Related by Dynamic Equations on Generalized Coordinates: Slide 7 / 26

8. Contact Force Constraints Key Points on Contact Model (1) Reaction Forces are NOT an explicit function of state (2) Reaction Forces ARE constrained by state, acceleration Normal Force-Acceleration (Rigid Body) Tangential Force: Acceleration (Coulomb Friction) Contact Point Slide 8 / 26 (Pictures adapted from Pfeiffer,Glocker 1996)

9. Complete Dynamics Model ? Dynamics Model Desired Solution AND Normal Constraints Tangential Constraints Consider Branches Separately Slide 9 / 26

10. Contact Modes Contact Modes: Separate (S), Slide Right (R) and Left (L), Fixed (F) Tangential Direction Normal Direction (S) (R) (L,R,F) (F) (L) Constraints in Matrix Form Slide 10 / 26

11. Form of Total Solution ● Linear function from equality constraints ● Polytope domain from inequality constraints Form of Dynamic Response Contact Mode Specific Dynamics Model Contact Mode Solution AND Slide 11 / 26

12. is f.r.r. but not f.c.r. is f.r.r. and f.c.r Solving for Response Function Consider equality constraints only ● Contact Mode Acceleration Constraints ● Contact Mode Force Constraints ● Dynamical Constraints (Group terms) Solve constraints based on rank- 4 cases Slide 12 / 26

13. Solving for Response Domain Now consider inequality constraints ● Contact Mode Acceleration Constraints ● Contact Mode Force Constraints Substitute to eliminate acceleration, forces Reduce inequality constraints Non-Supporting Supporting Use Linear Programs to generate minimal representation: Slide 13 / 26

14. Response Domain on Control Input Description ● Domain of on BOTH control inputs, ambiguity variables Description ● Domain of on ONLY control inputs Computation ● Polytope Projection by Fourier-Motzkin. ● Reduce by Linear Program Slide 14 / 26

15. Mode Enumeration Do we need to repeat this process for all ? Not necessarily. Two pruning techniques (1) Contact point velocity: Necessary (2) System Freedoms: Computational Normal Velocity Tangential Velocity Opposite Accelerations Existence of Solution to: (Graphical Methods. Mason 2001) Denote Reduced number of Modes: Slide 15 / 26

16. Algorithm Summary Goal: Characterize system dynamics as a function of control input Approach: Break up by contact mode, solve each mode Algorithm Steps: (1) Computed Mode Response (2) Computed Mode Response Domain (3) Computed Modes we need to consider Slide 16 / 26

17. Multiple solutions exist for a particular Solution Ambiguity Ambiguity Definition: Two Ambiguity Types: (Pfeiffer, Glocker 1996) (1) Between Modes (2) Within Mode Multiple Domains contain same Single Function has Slide 17 / 26

18. Solution Ambiguity: Between Modes Characterization: Domain Intersection Unambiguous Set Example Fall (SS) or Stick (FF) (Brogliato) Slide 18 / 26

19. Solution Ambiguity: Within Mode Characterization: Response Function Ambiguity Variable Examples Unknown Rotational Deceleration Unknown Tangential Forces Slide 19 / 26

20. High-level Task Description: • Flip RHex Over • No Body Separation until past vertical (Saranli) • Technical Task Description: • Maximize pitch acceleration • No separating contact mode Application to RHex Flip Task Task Description Initial Configuration Final Configuration Slide 20 / 26

21. Other Model Details • Legs Massless • Body Mass Distribution: C.O.M at center, Inertia • Body Friction Toe Friction RHex Model Details Generalized Coordinates Contact Coordinates Slide 21 / 26

22. Algorithm Outline and Simulation Input: Output: • Dynamic Response • Calculate Possible Modes • Compute Response • Compute Domains • Ambiguities • Compute Unambiguous Regions • Optimize • Optimize over • Subject to no body separation (Saranli) Slide 22 / 26

23. High-level Task Description: • Immobilize Lower ‘V’-Brace • Disregard Controls for Remainder-Free • Technical Task Description: • Ensure ‘FFF’ contact mode • Reduce to a Disturbance Application to Snake Climbing Task Task Description Initial Configuration Final Configuration Slide 23 / 26

24. Brace Free Brace Dynamics where Snake Model Details Generalized Coordinates Contact Coordinates • Other Model Details • Single friction coefficient • Point masses at each joint Slide 24 / 26

25. Algorithm Outline Input: Output: • Parameterize Disturbance Forces • Calculate disturbance set • Dynamic Response Function • Calculate Possible Modes • Compute Response • Compute Domains • Robust Ambiguities • Compute Unambiguous Region • for all disturbances (Pure Animation) Disturbance Forces Slide 25 / 26

26. Conclusion • ● Objective: An approach to model-based control of frictionally • constrained robots • ● Dynamics Model: Multi-Rigid-Body with Coulomb Friction • ● Model Prediction: Generate the dynamics response function • ● Application: RHex flipping and Snake Climbing Slide 26 / 26

27. END TALK

28. Movie, Rhex Flip (Pure Animation) Slide 24 / 25

29. is f.r.r. but not f.c.r. is not f.r.r. and not f.c.r. is f.r.r. and f.c.r is not f.r.r. and f.c.r 4 Cases when when otherwise no solution otherwise no solution