yan bin jia department of computer science iowa state university ames ia 50010 dec 14 2010 n.
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  1. Yan-Bin Jia Department of Computer Science Iowa State University Ames, IA 50010 Dec 14, 2010 Energy-based Modeling of Tangential Compliance in 3-Dimensional Impact

  2. Impact and Manipulation Impulse-based Manipulation  Potential for task efficiency and minimalism Foundation of impact not fully laid out Underdeveloped research area in robotics Huang & Mason (2000); Tagawa, Hirota & Hirose (2010) Linear relationships during impact ( )

  3. Impact with Compliance Normal impulse: 1. accumulates during impact (compression + restitution) 2. Poisson’s hypothesis. 3. variable for impact analysis. Tangential impulse: 1. due to friction & compliance 2. dependent on contact modes 3. driven by normal impulse 2D Impact: Routh’s graphical method (1913) Han & Gilmore (1989); Wang & Mason (1991); Ahmed, Lankarani & Pereira (1999) 3D Impact: Darboux (1880) Keller (1986); Stewart & Trinkle (1996) Tangential compliance and impulse: Brach (1989); Smith (1991); Stronge’s 2D lumped parameter model (2000); Zhao, Liu & Brogliato (2009); Hien (2010)

  4. Compliance Model  Gravity ignored compared to impulsive force – horizontal contact plane. Extension of Stronge’s contact structure to 3D.   Analyze impulse in contact frame: tangential impulse opposing initial tangential contact velocity massless particle

  5. Two Phases of Impact Compression The normal spring (n-spring) stores energy . Ends when the spring length stops decreasing: p energy coefficient of restitution  Restitution Ends when

  6. Normal vs Tangential Stiffnesses stiffness of n-spring (value depending on impact phase) stiffness of tangential u- and v-springs (value invariant) Stiffness ratio: Depends on Young’s moduli and Poisson’s ratios of materials. (compression) (restitution)

  7. Normal Impulse as Sole Variable Idea: describe the impact system in terms of normal impulse. Key fact: Derivative well-defined at the impact phase transition.  (signs of length changes of u- and w-springs)

  8. System Overview Impact Dynamics Contact Mode Analysis integrate integrate

  9. Sliding Velocity tangential contact velocity from kinematics velocity of particle p representing sliding velocity. Sticking contact if .

  10. Stick or Slip? Energy-based Criteria By Coulomb’s law, the contact sticks , i.e., if ratio of normal stiffness to tangential stiffness Slips if

  11. Sticking Contact Change rates of the lengths of the tangential u- and w-springs. Particle p in simple harmonic motion like a spring-mass system. Only signs of u and w are needed to compute tangential impulses. Impossible to keep track of u and w in time space. infinitesimal duration of impact unknown stiffness

  12. Sticking Contact (cont’d) evaluating an integral involving Tangential elastic strain energies are determined as well.  Keep track of as functions of .

  13. Sliding Contact can also be solved (via involved steps). Evaluating two integrals that depend on . (to keep track of whether the springs are being compressed or stretched). Tangential elastic strain energies: Keep track of in impulse space.

  14. Contact Mode Transitions Stick to slip when Initialize integrals for sliding mode based on energy. Slip to stick when i.e, Initialize integral for sliding mode.

  15. Start of Impact Initial contact velocity  Under Coulomb’s law, we can show that sticks if … … slips if 

  16. Bouncing Ball – Integration with Dynamics Velocity equations: (Dynamics) Contact kinematics TheoremDuring collision, is collinear with . Impulse curve lies in a vertical plane.

  17. Instance Physical parameters: Before 1st impact: After 1st impact:

  18. Impulse Curve (1st Bounce) Tangential contact velocity vs. spring velocity contact mode switch

  19. Non-collinear Bouncing Points Projection of trajectory onto xy-plane

  20. Bouncing Pencil

  21. Video end of compression stick slip slip Pre-impact: Post-impact: Slipping direction varies.

  22. Simultaneous Collisions with Compliance Combine with WAFR ‘08 paper (with M. Mason & M. Erdmann) to model a billiard masse shot. Trajectory fit

  23. Simultaneous Collisions with Compliance  Predicted trajectory Predicted post-hit velocities: Estimates of post-hit velocities:

  24. Conclusion • 3D impact modeling with compliance extending Stronge’s spring-based contact structure. • Impulse-based not time-based (Stronge) and hence ready for impact analysis (quantitative) and computation. • elastic spring energies • contact mode analysis • sliding velocity computable • friction • Physical experiment. • Further integration of two impact models (for compliance and simultaneous impact).

  25. Acknowledgement Matt Mason (CMU) Rex Fernando (ISU sophomore)