1. Spacetime Constraints Chris Moore CS 552

2. Purpose Physically Accurate Realistic Motion

3. Components

4. Bodies

5. Components Bodies Inputs

6. Components Bodies Inputs Constraints

7. Components Bodies Inputs Constraints Objective Function

8. Bodies Single Particle / Rigid Body Articulated Bodies Multiple Bodies

9. Inputs Denoted Force Vector Components Spring Variables –Linear and Angular –Rest Length, Damping Coefficient, etc Object Positions

10. Constraints Function of Inputs:

11. Constraints Function of Inputs: Examples: –Newton’s 2 nd :

12. Constraints Function of Inputs: Examples: –Newton’s 2 nd : –Desired Keyframes

13. Constraints Function of Inputs: Examples: –Newton’s 2 nd : –Desired Keyframes –Maximum Force Magnitude

14. Constraints Function of Inputs: Examples: –Newton’s 2 nd : –Desired Keyframes –Maximum Force Magnitude –Collision Surfaces

15. Objective Function Function of Inputs: Optimization Problem

16. Objective Function Function of Inputs: Optimization Problem Examples

17. Objective Function Function of Inputs: Optimization Problem Examples –Minimize Force or Torque Where or is derived from

18. Objective Function Function of Inputs: Optimization Problem Examples –Minimize Force or Torque –Minimize Power Used

19. Objective Function Function of Inputs: Optimization Problem Examples –Minimize Force or Torque –Minimize Power Used –Minimize Distance to Keyframe

20. Objective Function Function of Inputs: Optimization Problem Examples –Minimize Force or Torque –Minimize Power Used –Minimize Distance to Keyframe –Maximize Jumping Height / Walking Distance

21. Process Start With Initial State Until a Desired Result is Found: –Evaluate Objective Function –Optimize Result –Constrain Solution

22. Optimization

23. Broyden-Fletcher-Goldfarb-Shanno (BFGS) –Quasi-Newton Method –Gradient Descent

24. Optimization Broyden-Fletcher-Goldfarb-Shanno (BFGS) Sequential Quadratic Programming (SQP)

25. Optimization Broyden-Fletcher-Goldfarb-Shanno (BFGS) Sequential Quadratic Programming (SQP) Lagrangians

26. Optimization Broyden-Fletcher-Goldfarb-Shanno (BFGS) Sequential Quadratic Programming (SQP) Lagrangians Monte Carlo Markov Chains (MCMC)

27. Optimization Broyden-Fletcher-Goldfarb-Shanno (BFGS) Sequential Quadratic Programming (SQP) Lagrangians Monte Carlo Markov Chains (MCMC) Genetic Algorithms (GA)

28. Applications Luxo - Generating realistic movement Generating intermediate frames between BVH sequences Creating realistic walking actions Causing the desired dice roll to occur

29. My Project

30. Bodies Inputs Constraints Objective Function

31. Implementation - Bodies Current Simulation: –Single Sphere

32. Implementation - Bodies Current Simulation: –Single Sphere –Potential for Complex Bodies

33. Implementation - Bodies Current Simulation: –Single Sphere –Potential for Complex Bodies –Represented within ODE

34. Implementation - Bodies Current Simulation: –Single Sphere –Potential for Complex Bodies –Represented within ODE –ODE handles system integration and collision detection

35. Implementation - Inputs Force Vector Components

36. Implementation - Inputs Force Vector Components –Inputs Represent Keyframes of

37. Implementation - Inputs Force Vector Components –Inputs Represent Keyframes of –Keyframes are linearly interpolated

38. Implementation - Inputs Force Vector Components –Inputs Represent Keyframes of –Keyframes are linearly interpolated –Applied at Center of Mass

39. Implementation - Constraints None Implemented So Far

40. Implementation - Constraints None Implemented So Far Challenge Mixing Newton’s 2 nd With Collision Response

41. Implementation - Objective Objective Function:

42. Implementation - Objective Objective Function: Influence Coefficients –Force Minimization

43. Implementation - Objective Objective Function: Influence Coefficients –Force Minimization –Velocity Error

44. Implementation - Objective Objective Function: Influence Coefficients –Force Minimization –Velocity Error –Position Error

45. Implementation – Class Layout Class Descriptions –STCSoln Represents one possible solution Contains real-valued input variables representing the components to force vectors Force vectors represent keyframes of the force function Discretize solution into N time steps: 3N variables per STCSoln

46. Implementation – Class Layout Class Descriptions –STCSolnPop Represents a population of STCSoln solutions Stored as a square grid –display purposes –compatibility with the GA parent selection

47. Implementation – Class Layout Class Descriptions –STCSolnPopEvaluator Interface class Initializes solutions Evaluates solution fitness Unique per problem

48. Implementation – Class Layout Class Descriptions –STCSolnPopEvaluator Interface class Initializes solutions Evaluates solution fitness Unique per problem –Problem Instance of STCSolnPopEvaluator Unique methods: –Initialize / Shutdown Physical System –Initializes & Updates Solution Playback –Evaluates STCSoln Objective Values

49. Implementation – Class Layout Class Descriptions –Solver Abstract Class for Optimizing Populations Uses the Problem Class to Calculate Fitness Optimizes Solutions based upon fitness value Subclasses: –GASolver –NewtonSolver

50. Implementation - Solvers GASolver –Genetic Algorithm based

51. Implementation - Solvers GASolver –Genetic Algorithm based –Population Specifics Size of Population Grid Layout –Choosing Parents –Maintaining Multiple Minima

52. Implementation - Solvers GASolver –Genetic Algorithm based –Gene Encoding Genes represented by STCSoln class Chromosomes represented by inputs

53. Implementation - Solvers GASolver –Genetic Algorithm based –Algorithm Order of Operations Within Iterations –Order of Operations –Succession of Children –Elitism Mutation Operator Crossover Operator –Choosing Parent –Random or Pivot

54. Implementation - Solvers NewtonSolver –Quasi-Newton Algorithm (Gradient Descent)

55. Implementation - Solvers NewtonSolver –Quasi-Newton Algorithm (Gradient Descent) –Potentially BFGS / SQP

56. Implementation - Solvers NewtonSolver –Quasi-Newton Algorithm (Gradient Descent) –Potentially BFGS / SQP –Requires Gradient Calculation of the Objective Function

57. Implementation - Solvers NewtonSolver –Quasi-Newton Algorithm (Gradient Descent) –Potentially BFGS / SQP –Requires Gradient Calculation of the Objective Function Jacobian

58. Implementation - Solvers NewtonSolver –Quasi-Newton Algorithm (Gradient Descent) –Potentially BFGS / SQP –Requires Gradient Calculation of the Objective Function Jacobian Hessian

59. Implementation - Solvers NewtonSolver –Quasi-Newton Algorithm (Gradient Descent) –Potentially BFGS / SQP –Requires Gradient Calculation of the Objective Function Jacobian Hessian –Still to be implemented

60. Implementation - Interface How it all works:

61. Implementation - Interface Components: –Scene Components: Green / Red / White Model

62. Implementation - Interface Components: –VCR-Style Playback Play / Pause / Stop

63. Implementation - Interface Components: –File Menu New / Load / Save

64. Implementation - Interface Components: –Population Graph Global Fitness Shading Local Fitness Shading Displays Selected Solution

65. Implementation - Interface Components: –GA Solver Button Starts / Stops the GA Solver

66. Implementation - Interface

67. Components: –Future Implementation: More Solvers Newton Solver Button Force Function Graph User Manipulation of Inputs

68. Conclusion Demo 1 Demo 2 Demo 3

69. Sources Spacetime Constraints, Andrew Witkins, Michael Kass Spacetime Constraints Revisited, J. Thomas Ngo, Joe Marks Motion from Spacetime Constraints, Matt Ahrens, Brett Levin © 2001 Spacetime Constraints Attempted, Taylor Shaw © 2001 Spacetime Constraints for Biomechanical Movements, David C. Brogan, Kevin P. Granata, Pradip N. Sheth Automatic Control for Animation, Ron Metoyer © 2004 Efficient Generation of Motion Transitions Using Spacetime Constraints, Charles Rose, Brian Guenter, Bobby Bodenheimer, Michael F. Cohen N-Body Spacetime Constraints, Diane Tang, J. Thomas Ngo, Joe Marks © 1994 Sampling Plausible Solutions to Multi-body Constraint Problems, Stephen Chenney, D. A. Forsyth