spacetime constraints revisited j. thomas ngo graduate biophysics program harvard university joe...

Spacetime Constraints Revisited

J. Thomas NgoGraduate Biophysics

ProgramHarvard University

Joe marksCambridge Research Lab

Digital Equipment Corporation

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

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

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

The Algorithm

• Dynamics module creates physically correct environment.

• Behavior module generates creature behaviors

• Search module finds values for stimulus-response parameters.

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.

Dynamics Module

• Friction is static when one joint touches floor

• Slippage proportional to contact force when two joints touchFriction

?

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

Behavior Module - Responses

• Response – direction for changing creature shape smoothly.– Time Constant– Set of Target values for

creature’s internal angles

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.

0)(2 02 iiii

0i

i

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

Behavior Module - Stimulus Functions• Stimulus functions

– Scalar function defined over sense space

V

jjjj vvW

1

20 )(max1

0jv

j

Sense Variables

Determined by search module

Determined by search module

Weight

V

j j

jW1

minlog

},...,,{ 21 Vvvv

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

V2,...,2,2

21

002

01 ,...,, Vvvv

Behavior Module - Stimulus Response Functions• Set of SR parameters consists of array of SR

pairs• To generate behavior:

Initialize creature state from SC problem descriptionActivate response 0for t = 1 to T

Determine deformation for time t from active responseSimulate resulting dynamics for time tMeasure sense variables from the environmentIdentify highest-valued stimulus functionActivate 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

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

Search Module

• Q: How to find values for stimulus and response parameter?

• A: Trial and Error. (Genetic Algorithm)

Search Module – Genetic Algorithm• Pseudocode:do parallel

Randomize genomeend dofor generation = 1 to number_of_generations

do parallelEvaluate genomeSelect mate from another processorCross genome with mateMutate genome

end do end for

Search Module – Genetic Algorithm

• Parameters of one SR pair in genome

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

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

Search Module – Genetic Algorithm• Crossover

– Typical linear layout for crossover not meaningful

– Crossover must be tailored to problem to get good performance

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

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…


After 64 generations…

Curling leap



Curling leap w/ a roll


• 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

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



Sideways cantering



Ghetto tripping cartwheel



Sideways shuffling

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



Skipping, back leg never gets In front of forward leg


Another trial…

Skipping walk


Another trial…

Weird walk

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?

???

spacetime constraints revisited j. thomas ngo graduate biophysics program harvard university joe...

Documents