Transcript Presentation Slides (ppt 2.3Mb)

An Approach to Model-Based Control of
Frictionally Constrained Robots
Aaron Greenfield
CFR Talk
02-22-05
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Biorobotics Lab
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
(Saranli)
(Borer)
Slide 2 / 26
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
Dynamics Model: Multi-RigidBodies and Coulomb Friction
Dynamic Equations:
Rigid Body Model:
Friction Model:
2nd order ODE relates coordinates, forces
No penetration, Compressive Normal Force
Tangential Force Opposing Slip
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
Inconsistencies can exist
Slide 4 / 26
Dynamic Response Function
Accelerations
where
What is this function?
Why compute it?
How do we compute it?
Why is it hard to compute?
Reaction Forces
Control Inputs
Ambiguity Variables
System State
Relates instantaneous behavior
to
controls
and ambiguity
for a particular
To select control inputs which achieve
desired instantaneous behavior
By solving a series of linear systems of equalities
and inequalities consisting of:
1) Lagrange’s equation
2)Contact constraints
Non-linearity, Solution Ambiguity, Inconsistency,
Inequality Constraints
Slide 5 / 26
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
Dynamics Equations
(Pfeiffer and Glocker)
Two coordinate systems
(1) Generalized Coordinates
(2) Contact Coordinates
Related by
Dynamic Equations on Generalized Coordinates:
Joint Actuations
Reaction Forces
Slide 7 / 26
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
(Pictures adapted from Pfeiffer,Glocker 1996)
Slide 8 / 26
Complete Dynamics Model
Dynamics Model
?
Desired Solution
AND
Normal Constraints
Tangential Constraints
Consider Branches Separately
Slide 9 / 26
Contact Modes
Contact Modes: Separate (S), Slide Right (R) and Left (L), Fixed (F)
Normal Direction
Tangential Direction
(S)
(R)
(L,R,F)
(F)
(L)
Constraints in Matrix Form
Mode Equality
Constraints
S
L
Inequality
Constraints

R
F
Slide 10 / 26
Form of Dynamic Response
Contact Mode Specific
Dynamics Model
Contact Mode Solution
AND
Form of Total Solution
● Linear function
● Polytope domain
from equality constraints
from inequality constraints
Slide 11 / 26
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
is f.r.r. and f.c.r
is f.r.r. but not f.c.r.
Slide 12 / 26
Solving for Response Domain
Now consider inequality constraints
● Contact Mode Acceleration Constraints
● Contact Mode Force Constraints
Substitute
to eliminate acceleration, forces
Reduce inequality constraints
Use Linear Programs to generate
minimal representation:
Non-Supporting
Supporting
Slide 13 / 26
Response Domain on Control Input
Description
● Domain of
on BOTH
control inputs, ambiguity variables
Description
● Domain of
control inputs
on ONLY
Computation
● Polytope Projection by Fourier-Motzkin.
● Reduce by Linear Program
Slide 14 / 26
Mode Enumeration
Do we need to repeat this process for all
? Not necessarily.
Two pruning techniques
(1) Contact point velocity: Necessary
Normal Velocity
Tangential Velocity
(2) System Freedoms: Computational
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:
Slide 15 / 26
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
Solution Ambiguity
Ambiguity Definition: Multiple solutions
exist for a particular
Two Ambiguity Types: (Pfeiffer, Glocker 1996)
(1) Between Modes
(2) Within Mode
Multiple Domains contain same
Single Function has
Slide 17 / 26
Solution Ambiguity: Between Modes
Characterization: Domain Intersection
Unambiguous Set
Example
Fall (SS) or Stick (FF)
(Brogliato)
Slide 18 / 26
Solution Ambiguity: Within Mode
Characterization: Response Function
Ambiguity Variable
Examples
Unknown Tangential Forces
Unknown Rotational Deceleration
Slide 19 / 26
Application to RHex Flip Task
Task Description
Initial Configuration
Final Configuration
High-level Task Description:
1) Flip RHex Over
2) No Body Separation until past vertical (Saranli)
Technical Task Description:
1) Maximize pitch acceleration
2) No separating contact mode
Slide 20 / 26
RHex Model Details
Generalized Coordinates
Contact Coordinates
Other Model Details
1) Legs Massless
2) Body Mass Distribution: C.O.M at center, Inertia
3) Body Friction
Toe Friction
Slide 21 / 26
Algorithm Outline and Simulation
Input:
Output:
(1) Dynamic Response
a) Calculate Possible Modes
b) Compute Response
c) Compute Domains
(2) Ambiguities
Compute Unambiguous Regions
(3) Optimize
Optimize
over
Subject to no body separation
(Saranli)
Slide 22 / 26
Application to Snake Climbing Task
Task Description
Initial Configuration
Final Configuration
High-level Task Description:
1) Immobilize Lower ‘V’-Brace
2) Disregard Controls for Remainder-Free
Technical Task Description:
1) Ensure ‘FFF’ contact mode
2) Reduce to a Disturbance
Slide 23 / 26
Snake Model Details
Generalized Coordinates
Brace
Free
Contact Coordinates
Other Model Details
1) Single friction coefficient
2) Point masses at each joint
Brace Dynamics
where
Slide 24 / 26
Algorithm Outline
Input:
Output:
(1) Parameterize Disturbance Forces
a) Calculate disturbance set
(2) Dynamic Response Function
a) Calculate Possible Modes
b) Compute Response
c) Compute Domains
(3) Robust Ambiguities
Compute Unambiguous Region
for all disturbances
(Pure Animation)
Disturbance Forces
-0.05
5.4
0.3
-0.1
0.1
0.275
5.3
-0.15
0.25
5.2
0.05
0.225
-0.2
0.5
1
1.5
2
5.1
0.5
-0.25
1
1.5
2
0.175
0.5
-0.05
0.5
1
1.5
2
4.9
1
1.5
2
0.15
0.125
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 Cases
is f.r.r. and f.c.r
is not f.r.r. and f.c.r
is f.r.r. but not f.c.r.
is not f.r.r. and not f.c.r.
when
when
otherwise no solution
otherwise no solution