Exploiting the complementarity
structure: stability analysis of contact
dynamics via sums-of-squares
Michael Posa
Joint work with Mark Tobenkin and Russ Tedrake
Massachusetts Institute of Technology
BIRS Workshop on Computational Contact Mechanics
2/17/2014
Stability Analysis and Contact
FastRunner [IHMC 2013]
Atlas [Boston Dynamics, MIT 2014]
Domo [Edsinger 2007]
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Lyapunov Functions
Capture stability properties of dynamic systems
Lyapunov Function
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Sums-of-Squares
For polynomials, non-negativity is NP-hard
Replace with sufficient condition
Convex constraint in a Semidefinite Program
[Parrilo 2000, Lasserre 2001]
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Regional Stability
Rarely have global stability
Instead, show
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S-Procedure
Positivity over a basic semi-algebraic set:
Sufficient condition:
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Hybrid Barrier Certificates
For valid a hybrid jumps:
.
.
.
[Prajna, Jadbabaie, and Pappas 2007]
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Hybrid Systems Approach
Number of hybrid modes exponential
in number of contact points
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Objective
Given a system of rigid bodies with:
Inelastic impacts and Coulomb friction
Automated numerical analysis of
• Equilibrium stability in the sense of Lyapunov
• Positive invariance
• Unsafe region avoidance
Algorithms polynomial in number of contacts
[Posa, Tobenkin, and Tedrake. HSCC 2013]
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Measure Differential Inclusions
Alternative framework for describing solutions
[Moreau, Brogliato, Stewart, Leine, …]
• Dynamics from set-valued functions
• v(t) is of locally bounded variation and has no singular part
Lyapunov Condition
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Lyapunov Conditions
How to efficiently express
Contact forces λ(q,v) are discontinuous
Easy to write
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?
Leveraging Structure
Contact model constrains λ
Robot kinematics are algebraic…
Semialgebraic conditions in states and forces
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Lyapunov Conditions
over admissible states and forces
In the air:
Impacts:
Admissible Set
Non-penetration
Normal force
Dissipation
Friction Cone
Complementarity
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Convexity and Connected
Components
dV · 0; V · 10
v¡
v+
v
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A Sufficient Condition
Verify that V decreases along a path from (q,v-) to (q,v+)
Not Verified
v¡ (t)
Verified
Not Verified
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Rimless Wheel
• 5 state model with
two contact points
• Exhibits Zeno
• Bilinear alternation
searching over
quartic Lyapunov
functions
• Verify stability and
region of invariance
about equilibrium
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[Desbiens, Asbeck, Cutkosky 2010]
Perching Glider
• 4 state model of glider after
perching
• Modifying a previous example
from Glassman
• Find largest set of safe initial
conditions
Feet attached
to wall
Tail can collide
with wall
[Desbiens et al.]
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Scaling
Contact conditions and constraints are separable
By continuity, sufficient to write
For n state variables and m contact points,
size of SDP is
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Control Design (work in progress)
Find u(x) that maximizes the verified region
SOS problem is bilinear in u and V
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Conclusion
Exploit algebraic structure of contact models
Scalable framework for automated stability
analysis
Numerical conditioning still an issue
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