Energy-Based Modeling of Tangential Compliance in 3-Dimensional Impact Studies

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Explore the dynamics of impact and manipulation in the realm of robotics, focusing on tangential compliance modeling. Learn about the interplay between normal and tangential impulses, stiffnesses, and the two phases of impact. Discover the critical aspects of compliance models and their applications in analyzing force interactions during impacts.

  • Robotics
  • Compliance Modeling
  • Impact Studies
  • Tangential Compliance
  • Stiffness Ratios
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  1. Energy-based Modeling of Tangential Compliance in 3-Dimensional Impact Yan-Bin Jia Department of Computer Science Iowa State University Ames, IA 50010 Dec 14, 2010 Department of Computer Science, Iowa State University

  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) t 0 Linear relationships during impact ( ) = = Ii m V = fi m V + = Q r f Q Q r I i i i i Department of Computer Science, Iowa State University

  3. Impact with Compliance t = = + I fdt I I tI n t 0 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

  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: I = ( , , ) I I I u w n F tangential impulse opposing initial tangential contact velocity massless particle Department of Computer Science, Iowa State University

  5. Two Phases of Impact Compression E The normal spring (n-spring) stores energy . n Ends when the spring length stops decreasing: 0 = E En= : 1 0 e /e k k nv nv p 2E e max max energy coefficient of restitution 2 2 E F e E 0 0 nv 2 F n e n Restitution = 0 E Ends when n Department of Computer Science, Iowa State University

  6. Normal vs Tangential Stiffnesses : k tk stiffness of n-spring (value depending on impact phase) : stiffness of tangential u- and v-springs (value invariant) 2 0= Stiffness ratio: k / 0 tk Depends on Young s moduli and Poisson s ratios of materials. 2= k / tk = = (compression) (restitution) 0 e / 0 Department of Computer Science, Iowa State University

  7. Normal Impulse as Sole Variable Idea: describe the impact system in terms of normal impulse. Key fact: I = = = / 2 dI dt F kE n n n n Derivative well-defined at the impact phase transition. E I I = = = = ' / / / E dE dI kn n v n n n n n n E E = ' = ' u I w I u w E E n n = = 1 1 (signs of length changes of u- and w-springs) Department of Computer Science, Iowa State University

  8. System Overview tv Impact Dynamics Contact Mode Analysis ' E integrate n E I nI n integrate , ' ' uE E , I u I uI I , v v v Department of Computer Science, Iowa State University

  9. Sliding Velocity : tv tangential contact velocity from kinematics tv : sv velocity of particle p representing sliding velocity. sv = ( , ) 0 , w v v u s t = 0 sv Sticking contact if . Department of Computer Science, Iowa State University

  10. Stick or Slip? Energy-based Criteria = 0 sv By Coulomb s law, the contact sticks , i.e., if + I I I 2 2 + 2 2 F F F u w n u w n + 2 2 E E E u w n ratio of normal stiffness to tangential stiffness + = 2 2 E E E Slips if u w n Department of Computer Science, Iowa State University

  11. Sticking Contact Change rates of the lengths of the tangential u- and w-springs. = = 0 ( , ) 0 , w = v u v v u s t ) 0 , 0 , 1 ( = v ) 0 , 1 , 0 ( w 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 Department of Computer Science, Iowa State University

  12. Sticking Contact (contd) nI Keep track of as functions of . u k 0 2 , 2 k w 0 tE v , evaluating an integral involving D = ( , ) uD D w n Tangential elastic strain energies are determined as well. 2 w 2 u D D E = E = u 4 u 2 0 4 2 0 Department of Computer Science, Iowa State University

  13. Sliding Contact u , w can also be solved (via involved steps). Keep track of in impulse space. u k 0 , 2 2 k w 0 , , uG G , u w E Evaluating two integrals that depend on . (to keep track of whether the springs are being compressed or stretched). n w Tangential elastic strain energies: 2 u G 2 w G E = E = u 4 2 0 u 4 2 0 Department of Computer Science, Iowa State University

  14. Contact Mode Transitions Stick to slip when + = 2 2 E E E u w n Initialize integrals for sliding mode based on energy. w G , uG Slip to stick when = = ( , ) 0 , w vt u 0 sv i.e, Initialize integral for sliding mode. D Department of Computer Science, Iowa State University

  15. Start of Impact Initial contact velocity v = ( , 0 , u ) v v 0 0 0 n = ) 0 ( ' 0 w I Under Coulomb s law, we can show that + 2 2 0 2 0 4 0 2 0 1 v v v v = sticks if 0 ) 0 ( ' u I u w n u 2 0 v 0 n + 2 2 0 2 0 4 0 2 0 v v v = ) 0 ( ' uI slips if u w n = ) 0 ( ' E v 0 n n Department of Computer Science, Iowa State University

  16. Bouncing Ball Integration with Dynamics Velocity equations: V = = 0+ / V I 5 m z (Dynamics) I 0 2 mr Contact kinematics = ) 1 , 0 , 0 ( z 7 I = + + z v v z I 0 t 2 m m v0 tI Theorem During collision, is collinear with . t Impulse curve lies in a vertical plane. Department of Computer Science, Iowa State University

  17. Instance z Physical parameters: = = = 1 1 4 . 0 r m = 5 . 0 e = ) 3 . 0 ) 3 . 0 2 . 1 2 0 2 ( /( 2 2 Before 1st impact: = , 0 , 1 ) 0 , 2 , 0 ( ) 5 ( V 0 0= After 1st impact: = V = x 570982 . 0 ( . 1 , 0 ( ) 5 . 2 , 0 , ) 0 , 92746 Department of Computer Science, Iowa State University

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

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

  20. Bouncing Pencil = 1 5 . 0 r h = 1 3 m h 2= 1= = = 5 . 0 e 8 . 0 2 0 2 . 1 3 Department of Computer Science, Iowa State University

  21. Video Pre-impact: . 3 ( 962 (cos 5 . 0 , 3908 . 5 , , 0 , 5302 sin ) = ) V end of compression , 5 . 0 , 1 ( 0 6 6 = ) 5 . 0 0 Post-impact: = = . 0 ( 3681 . 0 , 3908 . 3 , 0302 ) V 0 ( 2362 . 0 , 8021 . 1 , ) 5 . 0 0 stick slip slip Slipping direction varies. Department of Computer Science, Iowa State University

  22. 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

  23. Simultaneous Collisions with Compliance Predicted post-hit velocities: Estimates of post-hit velocities: = = . 1 15 ( . 0 52 , = = . 1 24 ( . 0 80 , ( 658 938 . , 244 988 . . 0 , , 733 676 . 3 ) v ( 654 278 . , 361 537 . ) ) v ) Predicted trajectory Department of Computer Science, Iowa State University

  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). Department of Computer Science, Iowa State University

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

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