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)