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Spacetime Constraints Revisited

Spacetime Constraints Revisited. J. Thomas Ngo Graduate Biophysics Program Harvard University. Joe marks Cambridge Research Lab Digital Equipment Corporation. Spacetime Constraints . Indirectly controlling physically realistic motion of articulated figures Animator Defines:

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Spacetime Constraints Revisited

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  1. Spacetime Constraints Revisited J. Thomas Ngo Graduate Biophysics Program Harvard University Joe marks Cambridge Research Lab Digital Equipment Corporation

  2. Spacetime Constraints • Indirectly controlling physically realistic motion of articulated figures • Animator Defines: • Physical structure • Actuators • Criteria for evaluating motion • Computer computes optimal trajectory according to specified criteria

  3. Globally Optimal Solutions • Current SC techniques are local in nature • Finding Globally Optimal Solutions to SC Problems is tough • Multimodality • Exponential number of possible trajectories • Many may be locally optimal or near optimal • Search-space discontinuities • Small change in actuators causes large change in trajectory

  4. A Way to Find Globally Optimal Solutions: • Global search algorithm • Generates multiple near-optimal trajectories • Encode trajectories as sets of stimulus-response behavior rules • Genetic algorithm for choosing behavior parameters

  5. The Algorithm • Dynamics module creates physically correct environment. • Behavior module generates creature behaviors • Search module finds values for stimulus-response parameters.

  6. Dynamics Module • Uses forward dynamics • Find motions from forces • Articulated figure treated as autonomously deforming object with no DOF • Deformations produced kinematically by stimulus-response control algorithm. • Save CPU cycles.

  7. Dynamics Module • Friction is static when one joint touches floor • Slippage proportional to contact force when two joints touch Friction?

  8. Behavior Module • Uses parameterized algorithm based on stimulus and response • Stimulus-Response Control Algorithm • Reflexes triggered by conditions sensed in environment • Conditions – stimulus functions • Reflexes - responses

  9. Behavior Module - Responses • Response – direction for changing creature shape smoothly. • Time Constant • Set of Target values for creature’s internal angles

  10. Behavior Module - Responses • Throw values into critically damped equation of motion for each angle • time constant • target value of angle • actual value of angle • Creature approaches target shape smoothly.

  11. Behavior Module - Stimulus Functions • Sense variable • Real-valued function of environment • Proprioceptive senses • Um, joint angles • Tactile senses • Force exerted by endpoint on floor • Kinestheitic sense • Vertical velocity of center of mass • Position sense • Vertical position of center of mass

  12. Behavior Module - Stimulus Functions • Stimulus functions • Scalar function defined over sense space

  13. Behavior Module - Stimulus Functions • Constants normalized so sense variables fall between 0 and 1 • Sensitive Region • Locus of points in sense space where stimulus-function is positive • Forms a hyper-rectangle • Dimensions • Centered at

  14. Behavior Module - Stimulus Response Functions • Set of SR parameters consists of array of SR pairs • To generate behavior: Initialize creature state from SC problem description Activate response 0 for t = 1 to T Determine deformation for time t from active response Simulate resulting dynamics for time t Measure sense variables from the environment Identify highest-valued stimulus function Activate corresponding response if stimulus positive end for • Active response changed only if highest-valued stimulus function is positive • Causes response to be active for several time step • Coherent motion

  15. Search Module – Genetic Algorithm • Parallel genetic algorithm • Written in C* on Thinking Machines CM-2 with 4096 processors • Each processor evaluates one genome per generation

  16. Search Module • Q: How to find values for stimulus and response parameter? • A: Trial and Error. (Genetic Algorithm)

  17. Search Module – Genetic Algorithm • Pseudocode: do parallel Randomize genome end do for generation = 1 to number_of_generations do parallel Evaluate genome Select mate from another processor Cross genome with mate Mutate genome end do end for

  18. Search Module – Genetic Algorithm • Parameters of one SR pair in genome

  19. Search Module – Genetic Algorithm • Randomization • All parameters initially chosen at random • Uses probability distribution • Hill climbing to enrich initial gene pool • Evaluate initial gene pool • Mutate and re-evaluate solution 4 times • On each processor choose best out of five • Makes population skewed in favor of multi-step behaviors

  20. Search Module – Genetic Algorithm • Mate Selection • Processors laid on in imaginary 64x64 grid • Each processor does 10 step random walk • If best solution is itself, no mating • Else mates with best encountered genome • Good genes diffuse through population • Spreading is not instantaneous so inferior solutions get some processing time • Population converged when one population dominates

  21. Search Module – Genetic Algorithm • Crossover • Typical linear layout for crossover not meaningful • Crossover must be tailored to problem to get good performance

  22. Search Module – Genetic Algorithm • Mutation • Tailored specifically for the SR representation. • One SR Pair creeps • One SR Pair randomized form scratch • But at least one corner of sensitive region of new stimulus function must coincide with original sense-space trajectory. • Else new stimulus functions dominate trajectory or don’t do anything

  23. Results – Five-Rod Fred • Five-Rod Fred • 5 consecutively linked rods • Middle rods have same mass • Terminal rods five times heavier • Evaluation Function – Move COM as far right as possible • Expected inch-worm behavior • But…

  24. Results – Five-Rod Fred After 64 generations… Curling leap

  25. Results – Five-Rod Fred After 100 generations… Curling leap w/ a roll

  26. Results – Five-Rod Fred • Final behavior generated by 5 of 10 SR pairs • Two pairs produce initial curling motion • Two pairs produce leaping motion • One pair produces final curled shape

  27. Results – Mr. Star-Man • Mr. Star-Man • Five rods in star shape • All rods have equal length, mass • Evaluation Function – Move COM as far right as possible

  28. Results – Mr. Star-Man After 20 generations… Sideways cantering

  29. Results – Mr. Star-Man After 37 generations… Ghetto tripping cartwheel

  30. Results – Mr. Star-Man After 94 generations… Sideways shuffling

  31. Results – Beryl Biped • Beryl Biped • Headless 2D humanoid • Rigid torso • Jointed legs • Point feet • Rod masses of human proportion • Evaluation Function – Move point between feet as far right as possible • If used COM, just fell down

  32. Results – Beryl Biped After 100 generations… Skipping, back leg never gets In front of forward leg

  33. Results – Beryl Biped Another trial… Skipping walk

  34. Results – Beryl Biped Another trial… Weird walk

  35. Results • GA will fail to find good optima of evaluation function is no good • Small changes can restore expected behavior • How to change evaluation function to reward grace? ???

  36. Questions?

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