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Simultaneous Three-Dimensional Impacts with Friction and Compliance Yan-Bin Jia Department of Computer Science Iowa State University Ames, IA 50010, USA Sep 27, 2012 Department of Computer Science, Iowa State University Why Impact? Because impact is everywhere … Accomplishing tasks otherwise very difficult. Reduction of harmful impulsive forces Efficiency over static and dynamic forces Because it is relevant to robotics… Collision between robots and environments, walking robots … Impulsive manipulation being underdeveloped in robotics Higuchi (1985); Izumi and Kitaka (1993); Hirai et al. (1999) Huang & Mason (2000); Han & Park (2001), Tagawa et al. (2010) Foundation of impact not fully laid out Department of Computer Science, Iowa State University Describing Impact by Impulse “Infinitesimal” duration before after “Infinite” contact force Finite change in momentum m(v1 v0 ) t 0 I v0 v1 Fdt impulse Department of Computer Science, Iowa State University Impact Phase 1 – Compression Contact point virtual spring. The spring changes its length by storing elastic energy E . x, v p Compression ends when the spring length stops decreasing: v0 E Emax Department of Computer Science, Iowa State University Transition at End of Phase 1 Loss of energy at transition to restitution: E e Emax 2 [0,1] : energy coefficient of restitution v p Increase in stiffness at transition to restitution: k0 k0 / e 2 F kx F x e2 x 1 2 E kx e 2 E 2 change in spring length Department of Computer Science, Iowa State University Impact Phase 2 – Restitution During restitution the normal spring (n-spring) releases the remaining e 2 E . Restitution ends when E 0 v Department of Computer Science, Iowa State University Impulse-Energy Relationship dE I x v v0 dI m Energy is a piecewise quadratic function of impulse. One-to-one correspondence between impulse and time. Describe the process of impact in terms of impulse. Department of Computer Science, Iowa State University Example of General 3D Impact – Billiard Shooting Two contacts: - cue-ball - ball-table normal impulses At each contact: - normal impulse - tangential impulse I1 I2 tangential impulses How are they related at the same contact? (compliance & friction) At different contacts: How are the normal impulses related? (simultaneous impacts) Department of Computer Science, Iowa State University Talk Outline I. Impact with Compliance Relationship of tangential impulse to normal impact at single contact (also with friction) II. Simultaneous Impacts Relationships among normal impacts at different contacts (no friction or compliance) III. Model Integration (Billiard Shooting) Simultaneous impacts with friction and compliance Department of Computer Science, Iowa State University Related Work on Impact Newton’s law (kinematic coefficient of restitution) Maclaurin (1742); Bernoulli (1969); Ivanov (1995) Energy increase Wang et al. (1992); Wang & Mason (1992) No post-impact motion of a still object Brogliato (1999); Liu, Zhao & Brogliato (2009) Poisson’s hypothesis (kinetic coefficient of restitution) Routh (1905); Han & Gilmore (1989); Ahmed et al. (1999); Lankarani (2000) Darboux (1880); Keller (1986); Bhatt & Koechling (1994); Glocker & Pfeiffer (1995); Stewart & Trinkle (1996); Anitescu & Portta (1997) Energy increase Department of Computer Science, Iowa State University Related Work (cont’d) Energy-Based Restitution (energetic coefficient of restitution) Stronge (1990); Wang et al. (1992); Energy conservation Simultaneous Collisions Chatterjee & Ruina (1998); Ceanga & Hurmulu (2001); Seghete & Murphey (2010) Liu, Zhao & Brogliato (2008, 2009); Jia, Mason & Erdmann (2008) Tangential Impulse & Compliance Mindlin (1949); Maw et al. (1976) Smith (1991); Bilbao et al. (1989); Brach (1989) Stronge (1994; 2000); Zhao et al. (2009); Hien (2010) ; Jia (2010) Department of Computer Science, Iowa State University Impact with Compliance t I fdt I n I 0 Normal impulse: 1. accumulates during impact (compression + restitution) 2. energy-based restitution 3. variable for impact analysis Tangential impulse: 1. due to friction & compliance 2. dependent on contact modes 3. driven by normal impulse Department of Computer Science, Iowa State University Compliance Model Gravity ignored in comparison with impulsive force. Extension of Stronge’s contact structure to 3D. Analyze impulse in contact frame: I (Iu , I w , I n ) F tangential impulse opposing initial tangential contact velocity massless particle Department of Computer Science, Iowa State University Normal vs Tangential Stiffnesses k : stiffness of normal spring (value varying with impact phase) k : stiffness of tangential u- and v-springs (value invariant) Stiffness ratio: k0 / k 2 0 Depends on Young’s moduli and Poisson’s ratios of materials. k / k 2 0 0 / e (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: In dI n / dt Fn 2kEn Derivative well-defined at the impact phase transition. En ' dEn / dI n vn Iu ' 1 1 Eu En Iw ' Ew En (1 if extension of tangential u- and w-springs –1 if compression) Department of Computer Science, Iowa State University Tangential Springs Cannot determine changes u and v in length without knowing stiffness. Can keep track of integrals of u , En v En Gu , Gv over In Elastic strain energies: 2 u 2 0 G Eu 4 2 w 2 0 G Eu 4 Department of Computer Science, Iowa State University System Overview v Impact Dynamics & Contact Kinematics In integrate Contact Mode Analysis En I Iu , Iv vn integrate Iu ' , Iv ' Eu , Ev Department of Computer Science, Iowa State University Sliding Velocity v : tangential contact velocity from contact kinematics vs : sliding velocity represented by velocity of particle p v vs vs v (u, w ,0) Sticking contact if vs 0 . Department of Computer Science, Iowa State University Stick or Slip? Energy-Based Criteria By Coulomb’s law, the contact sticks , i.e., v s F F Fn 2 u 0 if Iu2 Iw2 In 2 w Eu Ew En 2 Slips if 2 Eu Ew En 2 2 Department of Computer Science, Iowa State University Contact Mode Transitions Stick to slip when Eu Ew En 2 Slip 2 to stick when vs 0 i.e, ,0) v (u, w Department of Computer Science, Iowa State University Sticking Contact Rates of change in spring length. 0 vs v (u, w ,0) u v (1,0,0) w v (0,1,0) Particle p in simple harmonic motion like a spring-mass system. (No energy dissipation tangentially) Department of Computer Science, Iowa State University Sliding Contact u, w can also be solved (via involved steps). Energy dissipation rate (tangentially): E' I n vs Department of Computer Science, Iowa State University System of Impact with Compliance Differential equations with five functions and one variable In : I 'u f1 ( En , Gu ) Tangential impulses Length (scaled) of tangential springs Energy stored by the normal spring I 'w f 2 ( En , Gu ) G'u f 3 ( I n , En , Gu , Gw ) G'w f 4 ( I n , En , Gu , Gw ) E 'n f 5 ( I n , I u , I w ) Department of Computer Science, Iowa State University Bouncing Ball z Physical parameters: m 1 r 1 0.4 e 0.5 02 (2 0.3) /( 2 2 0.3) 1.2 Before 1st impact: V0 (1,0,5) 0 (0,2,0) After 1st impact: V (0.570982,0,2.5) x v0 0 (0,1.92746,0) All bounces are in one vertical plane Department of Computer Science, Iowa State University Impulse Curve (1st Bounce) contact mode switch Department of Computer Science, Iowa State University Non-collinear Bouncing Points V0 (1,0,5) 0 (6,6,0) v0 0 Department of Computer Science, Iowa State University Trajectory Projection onto Table Department of Computer Science, Iowa State University Bounce of a Pencil Pre-impact: V0 5(cos 6 ,0, sin 6 ) 0 (1,0.5,0.5) Post-impact: V0 (0.3681,0.3908,3.0302) 0 (0.2362,1.8021,0.5) z x Department of Computer Science, Iowa State University Impulse Curve (3.962,0.3908,5.5302) end of compression z stick y slip slip Slipping direction varies. Department of Computer Science, Iowa State University Where Are We? I. Impact with Compliance Relationship of tangential impulse to normal impact at single contact (also with friction) II. Simultaneous Impacts Relationships among normal impacts at different contacts (no friction or compliance) III. Model Integration (Billiard Shooting) Simultaneous impacts with friction and compliance Department of Computer Science, Iowa State University Simultaneous Collisions in 3D High-speed photographs shows >2 objects simultaneously in contact during collision. Lack of a continuous impact law. Our model Collision as a state sequence. Within each state, a subset of impacts are “active”. Energy-based restitution law. Department of Computer Science, Iowa State University Two-Ball Collision Problem: One rigid ball impacts another resting on the table. Question: Ball velocities after the impact? virtual springs x1 v1 v2 x2 v2 m1v1 F1 k1x1 m2 v2 F2 F1 k1 x1 k 2 x2 (kinematics) (dynamics) Department of Computer Science, Iowa State University Impulses, Velocities & Stain Energies Ball-ball impulse: I 1 F 1dt Ball-table impulse: I 2 F 2 dt Velocities: 1 v1 v0 I 1 m1 v2 1 ( I 2 I 1) m2 Rates of change in contact strain energies: 1 dE1 1 1 I1 v0 I2 dI1 m2 m1 m2 dE2 1 I1 I 2 dI 2 m2 Department of Computer Science, Iowa State University State Transition Diagram The two impacts almost never start or end restitution at the same time. An impact may be reactivated after restitution. spring 2 ends restitution first spring 1 ends restitution first e2 0 and v2 0 before spring 1 ends restitution e1 0 and v1 v2 before spring 2 ends restitution both springs end restitution together s4 otherwise, spring 2 ends restitution otherwise, spring 1 ends restitution Department of Computer Science, Iowa State University Some Facts A change of state happens when a contact disappears or a disappeared contact reappears. Impulse & strain energy for one impact also depend on those for the other (correlation). Compression may restart from restitution within an impact state due to the coupling of impulses at different contacts. Department of Computer Science, Iowa State University Assumptions on Simultaneous Impacts Every end of compression of single impact: increase in stiffness invariance of contact force No change in stiffness when restitution switches back to compression, or when a contact is reactivated. Department of Computer Science, Iowa State University Stiffness, Mass, and Velocity Ratios Theorem 2 Collision outcome depends on the contact stiffness ratio but not on individual stiffness. The outcome does not change if the ball masses scale by the same factor. Output/input velocity ratio is constant (linearity). Consider upper ball with unit mass and unit downward velocity. Department of Computer Science, Iowa State University Impulse Curve I2 Theorem 3 During the collision, the impulses I 1,I 2 accumulate along a curve that is first order continuous and bounded within an ellipse. 1 2 1 I1 ( I1 I 2 ) 2 v0 I1 0 2m1 2m2 lines of compression S4 v1 0.94 v2 0.30 S1 S2 S1 2 m1 1 m2 v0 1 3 k2 1 e1 e2 1 k1 I1 Department of Computer Science, Iowa State University Example with Energy Loss I2 E2 energy loss I1 2 m m1 1 2 3 k2 1 k1 e1 0.9 e2 0.7 energy loss E1 Department of Computer Science, Iowa State University Convergence I (i ) 1 , I 2(i ) : impulses at the end of the ith state in the sequence. Sequence {( II(1(ii)),, II 2((ii)) )}: 1 2 monotone non-decreasing bounded within an ellipse Theorem 4 : The state transition will either terminate with v1 v 2 0 or the sequence will converge with either v1 v 2 0 or v1 v 2 0 . Department of Computer Science, Iowa State University Ping Pong Experiment Department of Computer Science, Iowa State University Experiment m 0.00023 kg (m / s) v1 Measured values: e1 0.950043 (ball-ball) e2 0.846529 (ball-table) v2 Guessed value: k2 3 k1 same trial v0 (m / s) Department of Computer Science, Iowa State University Where Are We? I. Impact with Compliance Relationship of tangential impulse to normal impact at single contact (also with friction) II. Simultaneous Impacts Relationships among normal impacts at different contacts (no friction or compliance) III. Model Integration (Billiard Shooting) Simultaneous impacts with friction and compliance Department of Computer Science, Iowa State University Billiard Shooting c I1 n z I2 Simultaneous impacts: cue-ball and ball-table! Department of Computer Science, Iowa State University Contact Structures Cue-ball contact Ball-table contact Normal impulses at the two contacts are described by the simultaneous impact model. At each contact, normal impulse drives tangential impulse as described by the compliance model. Department of Computer Science, Iowa State University Combing the Two Impact Models normal CB impulse vc , normal BT impulse The two normal impulses take turns to drive the system. Department of Computer Science, Iowa State University Mechanical Cue Stick Department of Computer Science, Iowa State University A Masse Shot Department of Computer Science, Iowa State University reconstructed trajectory from video Shot video if not ended by cushion Trajectory fitting Post-shot ball velocities Impact model predicted trajectory by model Predicted post-shot ball velocities increasing cue-ball compliance Department of Computer Science, Iowa State University Conclusion • 3D impact modeling with compliance and friction • • • • elastic spring energies impulse-based not time-based contact mode analysis (stick / slip) sliding velocity computation • Multiple impacts • • • • • state transition diagram impulse curve stiffness ratio scalability convergence • Physical experiment. • Integration of two impact models Department of Computer Science, Iowa State University Extensions of Collision Model Rigid bodies with arbitrary geometry General simultaneous multibody collision ≥3 contact points State transition templates Measurement Robot of relative contact stiffness pool player! Department of Computer Science, Iowa State University Acknowledgement HR0011-07-1-0002 Matt Mason, Michael Erdmann, Ben Brown (CMU) Amir Degani (Israel Institute of Technology) Rex Fernando, Feng Guo (ISU students) Department of Computer Science, Iowa State University Online Papers International Journal of Robotics Research, 2012: 1. Yan-Bin Jia. Three-dimensional impact: energy-based modeling of tangential compliance. DOI: 10.1177/0278364912457832. http://www.cs.iastate.edu/~jia/papers/IJRR11a-submit.pdf 2. Yan-Bin Jia, Matthew T. Mason, and Michael A. Erdmann. Multiple impacts: a state transition diagram approach. DOI: 10.1177/0278364912461539. http://www.cs.iastate.edu/~jia/papers/IJRR11b-submit.pdf Department of Computer Science, Iowa State University Appendix 1: Start of Impact Initial contact velocity v0 (v0u ,0, v0 n ) I w ' (0) 0 Under Coulomb’s law, we can show that sticks if v v slips if v v 2 0u 2 0u 2 0w 2 0w v I u ' (0) v I u ' (0) 2 2 4 2 0 0n 4 2 0 0n 1 02 v0u v0 n … … En ' (0) v0 n Department of Computer Science, Iowa State University Bouncing Ball – Integration with Dynamics Velocity equations: V V0 I / m 5 0 zI 2mr (Dynamics) Contact kinematics z (0,0,1) Iz 7 v v0 z I m 2m Theorem 1 During collision, I is collinear with v0 . Impulse curve lies in a vertical plane. Department of Computer Science, Iowa State University Impulse Curve (1st Bounce) Tangential contact velocity vs. spring velocity v contact mode switch Department of Computer Science, Iowa State University Bouncing Pencil 3 m 1 r 1 h1 3 h2 0.5 0 .8 e 0.5 02 1.2 Department of Computer Science, Iowa State University