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Michael Posa Joint work with Mark Tobenkin and Russ Tedrake

Exploiting the complementarity s tructure: 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. TexPoint fonts used in EMF.

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Michael Posa Joint work with Mark Tobenkin and Russ Tedrake

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  1. 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 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA

  2. Stability Analysis and Contact FastRunner [IHMC 2013] Atlas [Boston Dynamics, MIT 2014] Domo [Edsinger 2007]

  3. Lyapunov Functions Lyapunov Function Capture stability properties of dynamic systems

  4. Sums-of-Squares [Parrilo 2000, Lasserre 2001] For polynomials, non-negativity is NP-hard Replace with sufficient condition Convex constraint in a Semidefinite Program

  5. Regional Stability Rarely have global stability Instead, show

  6. S-Procedure Sufficient condition: Positivity over a basic semi-algebraic set:

  7. Hybrid Barrier Certificates For valid a hybrid jumps: . . . [Prajna, Jadbabaie, and Pappas 2007]

  8. Hybrid Systems Approach Number of hybrid modes exponential in number of contact points

  9. 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]

  10. Measure Differential Inclusions • Dynamics from set-valued functions • v(t) is of locally bounded variationand has no singular part Lyapunov Condition Alternative framework for describing solutions [Moreau, Brogliato, Stewart, Leine, …]

  11. Lyapunov Conditions How to efficiently express ? Contact forces λ(q,v) are discontinuous Easy to write

  12. Leveraging Structure Robot kinematics are algebraic… Semialgebraic conditions in statesand forces Contact model constrains λ

  13. Lyapunov Conditions In the air: Impacts: Admissible Set Non-penetration Normal force Dissipation Friction Cone Complementarity over admissible statesand forces

  14. Convexity and Connected Components v

  15. A Sufficient Condition Verify that V decreases along a path from (q,v-) to (q,v+) Not Verified Not Verified Verified

  16. 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

  17. [Desbiens, Asbeck, Cutkosky 2010]

  18. Perching Glider Feet attached to wall Tail can collide with wall [Desbiens et al.] 4 state model of glider after perching Modifying a previous example from Glassman Find largest set of safe initial conditions

  19. Scaling Contact conditions and constraints are separable By continuity, sufficient to write For nstate variables and mcontact points, size of SDP is

  20. Control Design (work in progress) Find u(x) that maximizes the verified region SOS problem is bilinear in u and V

  21. Conclusion Exploit algebraic structure of contact models Scalable framework for automated stability analysis Numerical conditioning still an issue

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