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Exploiting the complementarity structure: stability analysis of contact dynamics via sums-of-squaresMichael PosaJoint work with Mark Tobenkin and Russ TedrakeMassachusetts Institute of TechnologyBIRS Workshop on Computational Contact Mechanics2/17/2014TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAANumerical methods for lyapunov analysis of robots1Stability Analysis and Contact2/17/2014

Domo [Edsinger 2007]Atlas [Boston Dynamics, MIT 2014]

FastRunner [IHMC 2013]

Continuity of solutions, zeno, uniqueness.Make optimization difficult, like for simulation. Recent work in planning in control, but stability analysis tools still use hybrid models.Stability important to prove the robot will do what you want it to, influence control design, planning, etc.2

Lyapunov FunctionsCapture stability properties of dynamic systems

Lyapunov Function2/17/2014Introduce how we think about stability.Stability is an old problem in nonlinear control. For linear systems, simply an eigenvalue problem.Positivity of these functions.3Sums-of-SquaresFor polynomials, non-negativity is NP-hard

Replace with sufficient condition

Convex constraint in a Semidefinite Program2/17/2014

[Parrilo 2000, Lasserre 2001]

SOS gap4Regional StabilityRarely have global stabilityInstead, show

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S-ProcedurePositivity over a basic semi-algebraic set:

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Sufficient condition:Hybrid Barrier Certificates2/17/2014

For valid a hybrid jumps:

...[Prajna, Jadbabaie, and Pappas 2007]72/17/2014Hybrid Systems Approach

Number of hybrid modes exponentialin number of contact pointsCompare to planning and simulation.8ObjectiveGiven a system of rigid bodies with:Inelastic impacts and Coulomb friction

Automated numerical analysis ofEquilibrium stability in the sense of Lyapunov Positive invarianceUnsafe region avoidance

Algorithms polynomial in number of contacts[Posa, Tobenkin, and Tedrake. HSCC 2013]2/17/20149Measure Differential InclusionsAlternative framework for describing solutions[Moreau, Brogliato, Stewart, Leine, ]

2/17/2014

Dynamics from set-valued functions v(t) is of locally bounded variation and has no singular part

Lyapunov ConditionInstead of differential equations.LBV implies accumulation points, V is also LBV, tie into hybrid guards/reset10Lyapunov Conditions2/17/2014How to efficiently express ?

Contact forces (q,v) are discontinuous

Easy to write

Verify over states, contact forces.Emphasize seperability of impact equation.11Leveraging StructureContact model constrains

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Robot kinematics are algebraicSemialgebraic conditions in states and forces

Observation: impacts and friction can fit naturally in this framework12Lyapunov Conditions2/17/2014ComplementarityDissipationFriction Cone

Non-penetrationNormal forceAdmissible Set

over admissible states and forces

In the air:

Impacts:

Verify over states, contact forces.Emphasize seperability of impact equation.13

Convexity and Connected Components2/17/2014

vTechnical hurdles with discontinuous systems. Energy as a natural lyapunov candidate.A lot of previous analysis restricted to convex/quadratic V. A hard problem, in general14A Sufficient Condition2/17/2014

Not VerifiedNot VerifiedVerifiedVerify that V decreases along a path from (q,v-) to (q,v+)

Parameterize path by incremental impulses15Rimless Wheel2/17/20145 state model with two contact pointsExhibits ZenoBilinear alternation searching over quartic Lyapunov functionsVerify stability and region of invariance about equilibrium

Dyn invariant under lateral motion162/17/2014

Discuss energy, double single support17[Desbiens, Asbeck, Cutkosky 2010]

2/17/2014Perching Glider2/17/2014

Feet attachedto wallTail can collidewith wall4 state model of glider after perchingModifying a previous example from GlassmanFind largest set of safe initial conditions

[Desbiens et al.]192/17/2014

Over tens of degrees20ScalingContact 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 regionSOS problem is bilinear in u and V 2/17/2014

ConclusionExploit algebraic structure of contact models

Scalable framework for automated stability analysis

Numerical conditioning still an issue

2/17/2014Alternative to discrete abstractionsdiscrete modes to continuous framework.Semialgebraic opt. techniques, complementarity, MDI frameworks. dV < 0, LaSalle23