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Energy-based Modeling of Tangential Compliance in 3-Dimensional Impact

Energy-based Modeling of Tangential Compliance in 3-Dimensional Impact

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## Energy-based Modeling of Tangential Compliance in 3-Dimensional Impact

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**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**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 ( )**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)**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**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**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)**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)**System Overview**Impact Dynamics Contact Mode Analysis integrate integrate**Sliding Velocity**tangential contact velocity from kinematics velocity of particle p representing sliding velocity. Sticking contact if .**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**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**Sticking Contact (cont’d)**evaluating an integral involving Tangential elastic strain energies are determined as well. Keep track of as functions of .**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.**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.**Start of Impact**Initial contact velocity Under Coulomb’s law, we can show that sticks if … … slips if **Bouncing Ball – Integration with Dynamics**Velocity equations: (Dynamics) Contact kinematics TheoremDuring collision, is collinear with . Impulse curve lies in a vertical plane.**Instance**Physical parameters: Before 1st impact: After 1st impact:**Impulse Curve (1st Bounce)**Tangential contact velocity vs. spring velocity contact mode switch**Non-collinear Bouncing Points**Projection of trajectory onto xy-plane**Video**end of compression stick slip slip Pre-impact: Post-impact: Slipping direction varies.**Simultaneous Collisions with Compliance**Combine with WAFR ‘08 paper (with M. Mason & M. Erdmann) to model a billiard masse shot. Trajectory fit**Simultaneous Collisions with Compliance** Predicted trajectory Predicted post-hit velocities: Estimates of post-hit velocities:**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).**Acknowledgement**Matt Mason (CMU) Rex Fernando (ISU sophomore)