Aaron Greenfield

CFR Talk

02-22-05

An Approach to Model-Based Control of Frictionally Constrained Robots

Biorobotics LabM ic ro d ynam icSystem sLa b o ra to ry

Talk Outline1. 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

(Saranli) (Borer)Slide 2 / 26

Control Tasks with Frictional ContactRHex 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

Dynamics Model: Multi-Rigid-Bodies and Coulomb Friction

Dynamic Equations: 2nd order ODE relates coordinates, forcesRigid Body Model: No penetration, Compressive Normal ForceFriction Model: Tangential Force Opposing Slip

● Small number of coordinates ● Simple Contact Model (1 parameter)

Advantages Disadvantages ● No Body is perfectly rigid ● Coefficient of friction can be hard to determine, non-static ● Solution Ambiguities and Inconsistencies can exist

Slide 4 / 26

Dynamic Response Function

where

Accelerations

Reaction ForcesControl Inputs

System StateAmbiguity Variables

Why compute it? To select control inputs which achieve desired instantaneous behavior

How do we compute it? By solving a series of linear systems of equalities and inequalities consisting of:1) Lagrange’s equation 2)Contact constraints

Why is it hard to compute? Non-linearity, Solution Ambiguity, Inconsistency,Inequality Constraints

What is this function? Relates instantaneous behavior tocontrols and ambiguity for a particular

Slide 5 / 26

Related ResearchSingle 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 SimulationEarly 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: ControlComputing 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

Dynamics Equations

Dynamic Equations on Generalized Coordinates:

Two coordinate systems (1) Generalized Coordinates (2) Contact Coordinates

Related by

(Pfeiffer and Glocker)

Joint Actuations

Reaction Forces

Slide 7 / 26

Contact Force Constraints

(Pictures adapted from Pfeiffer,Glocker 1996)

Normal Force-Acceleration(Rigid Body)

Tangential Force: Acceleration (Coulomb Friction)

Key Points on Contact Model(1) Reaction Forces are NOT an explicit function of state(2) Reaction Forces ARE constrained by state, acceleration

ContactPoint

Slide 8 / 26

Complete Dynamics Model

Normal Constraints Tangential Constraints

Dynamics Model ? Desired Solution

Consider Branches Separately

AND

Slide 9 / 26

Contact ModesContact Modes: Separate (S), Slide Right (R) and Left (L), Fixed (F)

(S)

(L,R,F)

(R)

(F)

(L)

Mode Equality Constraints

Inequality Constraints

S

L

R

F

Normal Direction Tangential Direction

Constraints in Matrix Form

Slide 10 / 26

Form of Dynamic Response

● Linear function from equality constraints ● Polytope domain from inequality constraints

Form of Total Solution

Contact Mode Specific Dynamics Model

Contact Mode Solution

AND

Slide 11 / 26

Solving for Response Function

● Contact Mode Acceleration Constraints● Contact Mode Force Constraints● Dynamical Constraints

Consider equality constraints only

(Group terms)

is f.r.r. and f.c.r is f.r.r. but not f.c.r.

Solve constraints based on rank- 4 cases

Slide 12 / 26

Solving for Response Domain

● Contact Mode Acceleration Constraints● Contact Mode Force Constraints

Substitute to eliminate acceleration, forces

Now consider inequality constraints

Supporting

Non-SupportingReduce inequality constraints

Use Linear Programs to generate minimal representation:

Slide 13 / 26

Response Domain on Control Input

● Polytope Projection by Fourier-Motzkin. ● Reduce by Linear Program

Description● Domain of on BOTH control inputs, ambiguity variables

Description● Domain of on ONLY control inputs

Computation

Slide 14 / 26

Do we need to repeat this process for all ? Not necessarily.

Mode Enumeration

(1) Contact point velocity: Necessary (2) System Freedoms: Computational

Two pruning techniques

Normal Vel.

Tangential Vel.

Modes

---- S

S,R

S,L

F,S,L,R

Existence of Solution to:

Denote Reduced number of Modes:

(Graphical Methods. Mason 2001)

Normal Velocity Tangential Velocity Opposite Accelerations

Slide 15 / 26

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

Algorithm Summary

Slide 16 / 26

Solution Ambiguity

Two Ambiguity Types: (Pfeiffer, Glocker 1996)

Multiple solutions exist for a particular Ambiguity Definition:

Multiple Domains contain same Single Function has

(1) Between Modes (2) Within Mode

Slide 17 / 26

Solution Ambiguity: Between Modes

Fall (SS) or Stick (FF)

(Brogliato)

Characterization: Domain Intersection

Unambiguous Set

Example

Slide 18 / 26

Solution Ambiguity: Within Mode

Unknown Tangential ForcesUnknown Rotational Deceleration

Characterization: Response Function

Ambiguity Variable

Examples

Slide 19 / 26

Application to RHex Flip TaskTask Description

Initial Configuration Final Configuration

High-level Task Description:1) Flip RHex Over2) No Body Separation until past vertical (Saranli)

Technical Task Description:1) Maximize pitch acceleration2) No separating contact mode Slide 20 / 26

RHex Model Details

Generalized Coordinates

Contact Coordinates

Other Model Details1) Legs Massless2) Body Mass Distribution: C.O.M at center, Inertia 3) Body Friction Toe Friction

Slide 21 / 26

Algorithm Outline and Simulation

(Saranli)

Input:Output:

(1) Dynamic Responsea) Calculate Possible Modesb) Compute Responsec) Compute Domains

(2) AmbiguitiesCompute Unambiguous Regions

(3) OptimizeOptimize over Subject to no body separation

Slide 22 / 26

Application to Snake Climbing TaskTask Description

Initial Configuration Final Configuration

High-level Task Description:1) Immobilize Lower ‘V’-Brace2) Disregard Controls for Remainder-Free

Technical Task Description:1) Ensure ‘FFF’ contact mode2) Reduce to a Disturbance Slide 23 / 26

Snake Model DetailsGeneralized Coordinates

Contact Coordinates

Other Model Details1) Single friction coefficient2) Point masses at each joint

Brace Free

where

Brace Dynamics

Slide 24 / 26

(1) Parameterize Disturbance Forcesa) Calculate disturbance set

(2) Dynamic Response Functiona) Calculate Possible Modesb) Compute Responsec) Compute Domains

(3) Robust AmbiguitiesCompute Unambiguous Regionfor all disturbances

Algorithm Outline

(Pure Animation)

Input:Output:

0.5 1 1.5 2

-0.05

0.05

0.1

0.5 1 1.5 2

4.9

5.1

5.2

5.3

5.4

0.5 1 1.5 2

-0.25

-0.2

-0.15

-0.1

-0.05

0.5 1 1.5 2

0.125

0.15

0.175

0.225

0.25

0.275

0.3

Disturbance Forces

Slide 25 / 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

END TALK

Movie, Rhex Flip

(Pure Animation)

Slide 24 / 25

4 Casesis f.r.r. and f.c.r is f.r.r. but not f.c.r.

is not f.r.r. and f.c.r

when

otherwise no solution

is not f.r.r. and not f.c.r.

when

otherwise no solution