Aaron Greenfield

1
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)
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Control Tasks with Frictional Contact
RHex Flipping Task
Snake Climbing Task
Presumption The physics of contact is critical
to the robots performance
Approach ? Utilize a model of robot dynamics
under contact constraints ? Solve for
behavior as a function of control input
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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

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Dynamic Response Function
where
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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-BodiesModeling 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)

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Dynamics Equations
(Pfeiffer and Glocker)
Two coordinate systems (1) Generalized
Coordinates (2) Contact Coordinates
Related by
Dynamic Equations on Generalized Coordinates
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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
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(Pictures adapted from Pfeiffer,Glocker 1996)
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Complete Dynamics Model
?
Dynamics Model
Desired Solution
AND
Normal Constraints
Tangential Constraints
Consider Branches Separately
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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
Mode Equality Constraints Inequality Constraints
S
L
R
F
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Form of Dynamic Response
Contact Mode Specific Dynamics Model
Contact Mode Solution
AND
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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
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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
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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
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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
Normal Vel. Tangential Vel. Modes
---- S
S,R
S,L
F,S,L,R
(Graphical Methods. Mason 2001)
Denote Reduced number of Modes
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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
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Solution Ambiguity
Ambiguity Definition
Two Ambiguity Types (Pfeiffer, Glocker 1996)
(1) Between Modes
(2) Within Mode
Multiple Domains contain same
Single Function has
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Solution Ambiguity Between Modes
Characterization Domain Intersection
Unambiguous Set
Example
Fall (SS) or Stick (FF)
(Brogliato)
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Solution Ambiguity Within Mode
Characterization Response Function
Ambiguity Variable
Examples
Unknown Rotational Deceleration
Unknown Tangential Forces
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Application to RHex Flip Task
Task Description
Initial Configuration
Final Configuration
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RHex Model Details
Generalized Coordinates
Contact Coordinates
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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)
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Application to Snake Climbing Task
Task Description
Initial Configuration
Final Configuration
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Snake Model Details
Generalized Coordinates
Contact Coordinates

• Other Model Details
• Single friction coefficient
• Point masses at each joint

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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
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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

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END TALK
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Movie, Rhex Flip
(Pure Animation)
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4 Cases
when
when
otherwise no solution
otherwise no solution