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.: AAA A A AAA A

2. Stability Analysis and Contact 2/17/2014 Domo [Edsinger 2007] Atlas [Boston Dynamics, MIT 2014] FastRunner [IHMC 2013]

3. Lyapunov Functions Capture stability properties of dynamic systems Lyapunov Function 2/17/2014

4. Sums-of-Squares For polynomials, non-negativity is NP-hard Replace with sufficient condition Convex constraint in a Semidefinite Program 2/17/2014 [Parrilo 2000, Lasserre 2001]

5. Regional Stability Rarely have global stability Instead, show 2/17/2014

6. S-Procedure Positivity over a basic semi-algebraic set: 2/17/2014 Sufficient condition:

7. Hybrid Barrier Certificates 2/17/2014 For valid a hybrid jumps:...... [Prajna, Jadbabaie, and Pappas 2007]

8. 2/17/2014 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] 2/17/2014

10. Measure Differential Inclusions Alternative 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 Condition

11. Lyapunov Conditions 2/17/2014 How to efficiently express ? Contact forces λ(q,v) are discontinuous Easy to write

12. Leveraging Structure Contact model constrains λ 2/17/2014 Robot kinematics are algebraic… Semialgebraic conditions in states and forces

13. Lyapunov Conditions 2/17/2014 Complementarity Dissipation Friction Cone Non-penetration Normal force Admissible Set over admissible states and forces In the air: Impacts:

14. Convexity and Connected Components 2/17/2014 v

15. A Sufficient Condition 2/17/2014 Not Verified Verified Verify that V decreases along a path from (q,v - ) to (q,v + )

16. Rimless Wheel 2/17/2014 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. 2/17/2014

18. [Desbiens, Asbeck, Cutkosky 2010] 2/17/2014

19. Perching Glider 2/17/2014 Feet attached to wall Tail can collide with wall 4 state model of glider after perching Modifying a previous example from Glassman Find largest set of safe initial conditions [Desbiens et al.]

20. 2/17/2014

21. Scaling Contact conditions and constraints are separable By continuity, sufficient to write For n state variables and m contact points, size of SDP is 2/17/2014

22. Control Design (work in progress) Find u(x) that maximizes the verified region SOS problem is bilinear in u and V 2/17/2014

23. Conclusion Exploit algebraic structure of contact models Scalable framework for automated stability analysis Numerical conditioning still an issue 2/17/2014