Department of Computer Science, Iowa State University Energy-based Modeling of Tangential Compliance in 3-Dimensional Impact Yan-Bin Jia Department of Computer Science Iowa State University Ames, IA Dec 14, 2010

Department of Computer Science, Iowa State University 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 ( )

Department of Computer Science, Iowa State University 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)

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

Department of Computer Science, Iowa State University Two Phases of Impact  Ends when the spring length stops decreasing:  The normal spring (n-spring) stores energy. Compression  Ends when Restitution p energy coefficient of restitution 

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

Department of Computer Science, Iowa State University 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) 

Department of Computer Science, Iowa State University System Overview Impact Dynamics Contact Mode Analysis integrate

Department of Computer Science, Iowa State University Sliding Velocity tangential contact velocity from kinematics velocity of particle p representing sliding velocity. Sticking contact if.

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

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

Department of Computer Science, Iowa State University Sticking Contact (cont’d)  Tangential elastic strain energies are determined as well. evaluating an integral involving  Keep track of as functions of.

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

Department of Computer Science, Iowa State University 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.

Department of Computer Science, Iowa State University Start of Impact  Initial contact velocity sticks if slips if … …   Under Coulomb’s law, we can show that

Department of Computer Science, Iowa State University Bouncing Ball – Integration with Dynamics Contact kinematics Theorem During collision, is collinear with. Velocity equations: (Dynamics) Impulse curve lies in a vertical plane.

Department of Computer Science, Iowa State University Instance Physical parameters: Before 1 st impact: After 1 st impact:

Department of Computer Science, Iowa State University Impulse Curve (1 st Bounce) Tangential contact velocity vs. spring velocity contact mode switch

Department of Computer Science, Iowa State University Non-collinear Bouncing Points Projection of trajectory onto xy-plane

Department of Computer Science, Iowa State University Bouncing Pencil

Department of Computer Science, Iowa State University Video Slipping direction varies. end of compression slip stick slip Pre-impact: Post-impact:

Department of Computer Science, Iowa State University Simultaneous Collisions with Compliance  Combine with WAFR ‘08 paper (with M. Mason & M. Erdmann) to model a billiard masse shot.  Trajectory fit

Department of Computer Science, Iowa State University Simultaneous Collisions with Compliance Estimates of post-hit velocities:  Predicted trajectory Predicted post-hit velocities: 

Department of Computer Science, Iowa State University 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).

Department of Computer Science, Iowa State University Acknowledgement Matt Mason (CMU) Rex Fernando (ISU sophomore)