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Advanced Physics-Based Simulation in Robotics and Graphics

Explore forward dynamic simulation challenges, biomechanics integration, and efficient control in robotics and graphics applications. Learn about optimal system constraints and algorithms for accurate modeling. Visit links for in-depth resources.

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Advanced Physics-Based Simulation in Robotics and Graphics

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  1. Physics-Based Simulation: Graphics and Robotics Chand T. John

  2. Forward Dynamic Simulation Problem: Determine the motion of a mechanical system generated by a set of forces or control values. • Challenges: • Contact/collision • Large number of bodies • Drift • Control of end result • Creating natural motion • High-level motion control • Being efficient and accurate Update State Time Step

  3. Forward Dynamic Simulation • Forward dynamic simulation is needed in: • Computer graphics • Robotics • Biomechanics

  4. Bridging Biomechanics & Graphics • Biomechanics has: • A need for physically realistic simulations • Real science to back up dynamic models • Graphics has: • Fast algorithms for making visually • realistic movies of complicated scenes • Biomechanics goals: • Physically realistic • Make clinically relevant conclusions • Meaningfulness > accuracy > efficiency • Graphics goals: • Visually realistic • Make cool SIGGRAPH movies • Accuracy > efficiency > meaningfulness

  5. Constraining Mechanical Systems • Use fewer coordinates • Add constraint forces • Enforce constraints after each time step • Optimization • Control

  6. Robot Controller Simulation Maximal/redundant/operational-space/task-space/absolute coordinates (x2, y2, z2) (x3, y3, z3) (x1, y1, z1) 2 3 Reduced/generalized/joint-space/link coordinates

  7. Recursive Newton-Euler Algorithm Propagate accelerations outward a2 a1 a3 v2 v1 a0 v3 v0 Propagate velocities outward

  8. Recursive Newton-Euler Algorithm O(N) algorithm for inverse dynamics! f2 f1 f3 f0 Propagate forces inward

  9. Composite Rigid-Body Algorithm This part of the robot is in static equilibrium. When all velocity and acceleration-independent forces are zero, Mi is the force causing unit acceleration i to the robot. Link i Ci This part of the robot is a (composite) rigid body.

  10. Composite Rigid-Body Algorithm • If the robot is given an acceleration of i: • all of Ci will act like a single • rigid body with acceleration hi, • and the rest of the robot stays in • static equilibrium. • Let fiC = force needed to induce acceleration hi on Ci. Ci

  11. Composite Rigid-Body Algorithm Mji = hjTfiC for parents j of i Mji = 0 for other j not in Ci Fill in symmetric values Mij Inertia IiC of Ci = sum of inertias Ij fiC = IiChj fiC(j) = fiC for all j in Ci fiC = 0 for all other j Implement in joint space IiC Each parent of link i has force fiC hjTIiChi if i in Cj hjTIjChi if j in Ci 0 otherwise Mji = O(N3), fast for small N

  12. Articulated-Body Algorithm a = f + b Invert f = IAa + pA IA = -1 pA = -IAb Invertible if body unconstrained Key observation: Acceleration a of a body is always an affine function of applied force f.

  13. Articulated-Body Algorithm Propagate IiA, piA inward. 3 equations, 3 unknowns: ai = a(i) + hi’qi’ + hiqi’’ fiJ = IiAai + piA i = hiTfiJ Solve for qi’’ Evaluate ai Propagate ai outward. ai IiA piA O(N), faster than CRBA for N > 9

  14. Adaptive Dynamics Simplification • Based on: • Featherstone’s O(log N) divide-and-conquer algorithm • For forward dynamics on O(N) parallel processors • Closed loops not supported yet • Algorithm: • User picks number of DOFs desired • In each time step: • Algorithm picks active joints • Forward dynamics steps forward in time http://gamma.cs.unc.edu/AD/

  15. Dealing with Closed Loops Closed loop!

  16. Dealing with Closed Loops Compute spanning tree Compute dynamics for tree Mimic loop with constraint forces Equation of motion M LT q’’  – C + a = L 0 - c Constraint forces

  17. Lagrange Multiplier Constraints • Reduced-coordinate methods are fast • Multiplier methods (with maximal coordinates): • Have a drift problem • Involve solving a matrix equation • But multiplier methods: • Do not require parameterization • Allow use of nonholonomic constraints • Limited branching  sparse matrix • Baraff gives O(N) multiplier method for computing constraint forces for dynamics

  18. Lagrange Multiplier Constraints This is a system with 127 constraints. Each sphere is a 3 DOF constraint between two rigid bodies. There are 381 Lagrange multipliers.

  19. Post-Stabilization with DAEs State at time t + 1 State at time t After projection Constraint manifold http://www.cs.ubc.ca/nest/scv/demos/Slider/slider.html

  20. Spacetime Constraints • Given: • Equation of motion • Objective function to minimize • Spatial constraints to meet • Computes: • Actuation force function by constrained optimization • Position function by integrating equation of motion • High-energy motions modeled well • Low-energy motions require finer physics model http://www.cs.cmu.edu/~aw/

  21. Motion Transformation • First physics-based mocap editing method • Steps: • Map input motion onto simplified character • Find spacetime optimization problem most closely matching simplified character motion • Alter parameters, constraints, etc. • Map new motion onto original to get final animation • High-energy works well; low-energy not so well • No comparison to real actor’s motion physics http://www.cs.washington.edu/homes/zoran/sigg99/

  22. Physics-Based Motion Style • Tasks represented as constraints C • E(X; ) = total torque from muscle forces • X describes a motion sequence •  is a vector of style parameters • Given motion capture XT and C, minimize E(XT; ) to compute style  • With new constraints C’, minimize E(X; ) to compute new motion X http://grail.cs.washington.edu/projects/charanim/phys-style.html

  23. Neuromuscular Perturbation

  24. Proportional-Derivative Control kv mx’’ + kvx’ + kpx = 0 m Derivative Proportional kp Examples: Cruise control, CMC Rachel will talk about her work on PD control and human motion next week.

  25. Task-Level Control Vince will talk about task-level control of shoulder models two weeks from now.

  26. Summary • Reduced coordinates • Composite rigid-body algorithm, O(N3) • Articulated-body algorithm, O(N) • Adaptive dynamics simplification • Constraint force application • Closed-loop systems • Lagrange multiplier constraints • Projection onto constraint manifold • Post-stabilization with DAEs • Optimization • Spacetime constraints • Motion transformation • Physics-based motion style • Control • Neuromuscular perturbation • Proportional-derivative control • Task-level control

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