MovementControl Metho ds

for Complex Dynamically Simulated Agents

Adonis Dances the Macarena

Ma ja J Mataric Victor B Zordan Zachary Mason

Computer Science Department College of Computing Computer Science Department

University of Southern California Georgia Institute of Technology Brandeis University

Los Angeles CA Atlanta GA Waltham MA

tel tel tel

matariccsuscedu victorccgatechedu zmasoncsbrandeisedu

area in realistic physicallybased animation where Abstract

the control of dynamic simulations of human charac

We describ e and compare two implemented controllers

ters involving realistic physical mo dels matches the

for Adonis a physically simulated humanoid torso

complexity of the rob otics problem Pai Ho d

one based on jointspace torques and the other on

gins Wo oten Brogan OBrien Van de Panne

convergent forceelds applied to the hands The two

Lamouret

come from dierent application domains the former

Weintro duce a novel approach to manipulator

is a common approach in manipulator rob otics and

control based on mo dels from neuroscience which

graphics while the latter is inspired by biological limb

employconvergent forceelds at the endp oints of

control Both avoid explicit inverse kinematic calcu

a manipulator We explore the feasibilityofsucha

lations found in standard Cartesian control trading

mo del for complex animation and agent control in

generality of motion for programming eciencyThe

general and discuss how it can b e generalized to

two approaches are compared on a common sequen

highlevel motor tasks and incorp orated into a gen

tial task the familiar dance Macarena and evalu

eral control framework F urthermore we compare

ated based on ease of generating new b ehaviors exi

the metho dology to a standard rob otics approach

bility and naturalness of movement w e also compare

which has also b een adopted in animation employ

them against human p erformance on the same task

ing jointspace control with torque actuators Both

Finallywe discuss the tradeos and presentamore

approaches are app ealing b ecause they avoid explicit

general framework for addressing complex motor con

computation of inverse kinematics IK found in stan

trol of simulated agents

dard Cartesian control The inputs of each controller

are used explicitly as either p ositions or orientations

without IK solvers converting the input data How

Intro duction

ever this presents a tradeo b etween generalityof

Control of humanoid agents dynamically simulated

motion and programming eciencyTo compare the

or physical is an extremely dicult problem due to

two approaches we implemented them on a common

the high dimensionality of the control space ie the

motor task a continuous sequence of smo oth move

many degrees of freedom and the redundancy of the

ments For the purp ose of evaluation a well known

system In rob otics standard metho ds havebeende

goaldriven sequence was chosen the p opular dance

velop ed for simpler manipulators and have b een grad

Macarena The dance presents a nontrivial well

ually scaled up to more complex arms Paul

dened task that can b e precisely sp ecied and eval

Brady Hollerbach Johnson LozanoPerez Mason

uated b oth relative to the quantitative sp ecication

and recently to physical humanlikearmsSchaal

and to qualitativehuman p erformance

Williamson The problem of anthrop o

morphic control has also found a new application

Background

Control in Rob otics and Computer Animation

Computer animation and rob otics are two primary

areas of researchinto motion for articial agents D

animated character motion has traditionally b een cre

Our work is inspired by a sp ecic principle derived ated by hand through a timeconsuming pro cess Re

from evidence in neuroscience MussaIvaldi Giszter centlyphysical mo deling has b een used to generate

Giszter MussaIvaldi Bizzi and re motion by minimizing usersp ecied constraints while

lated work on spinalized frogs and rats suggest the ex allowing the mo del constraints to add physical real

istence of forceeld motor primitives that converge ism Witkin Kass pursue physical mo deling

to single equilibrium p oints and pro duce highlevel through such a constraintbased approach bycho os

behaviors suchasreaching and wiping When a par ing start and end conditions they generate anticipa

ticular eld is activated the frogs leg executes a b e tion and determination in the action Cohen

havior and comes to rest at a p osition that corre extended this approach with higher DOF systems and

sp onds to the equilibrium p oint when two or more more complex constraints Ngo Marks in

elds are activated either a linear sup erp osition of tro duce a constraint approach of creating b ehaviors

the elds of tested cases or a winnertakeall automatically using genetic algorithms

resp onse of remaining cases results Mussa Dynamic simulation has b een used to generate

Ivaldi Giszter Bizzi This suggests an el graphical motion by applying dynamics to physically

egant organizational principle for motor control in based mo dels and using forward integration Simu

tire b ehaviors are co ded with lowlevel force whichen lation ensures physically plausible motion by enforc

elds and maybecombined into higherlevel more ing the laws of physics Pai simulates walk

complex b ehaviors The idea of supplying an agent ing gaits drawing strongly from rob otics work His

with a collection of basis behaviors or primitives rep torso and legs use a controller based on highlevel

resenting forceelds and combining those into a gen timevarying constraints Handtuned control of sim

eral rep ertoire for complex motion is very app ealing ulations has b een applied successfully to more com

Our previous work Mataric Mataric in plex systems such as full articulated human gures

spired by the same biological results has already suc Raib ert Ho dgins demonstrate rigid b o dy

cessfully applied the idea of basis behaviors to control dynamic simulations of legged creatures Their hand

of planar mobile agentsrob ots This pap er extends tuned controllers consist of state machines that cy

the notion to agents with more DOFs cle through rulebased constraints to p erform dier

Another inspiration comes from psychophysical data ent gaits Ho dgins et al extend this work

describing what p eople xate on when observing hu to human characters suggesting a to olb ox of tech

man movement MataricPomplun demon niques for controlling articulated humanlike systems

strate that when presented with videos of human n to generate athletic b ehaviors such asDrunning

ger hand and arm movements observers fo cus on diving and bicycling Van de Panne Lamouret

the hand yet when asked to imitate the movements use searchtechniques to nd balancing con

sub jects are able to reconstruct complete tra jecto trollers for humanlikecharacter lo comotion aiming

ries even for unnatural movements involving mul at more automatic control of simulated agents

tiple DOFs in spite of having attended to the end In rob otics manipulator control has b een largely

point This could suggest some form of internal mo d addressed for p ointtop ointreaching typically by

els of complete b ehaviors or primitives for movement sp ecifying Cartesian D goals and explicitly solving

which eectively encapsulate the details of lowlevel the IK for the manipulators joint angles Paul

control Given an appropriately designed motor con Brady et al Various neural network approaches

troller tasks could b e sp ecied largely by endp oint to learning IK for simple manipulators havebeenex

p ositions and a few additional constraints and the plored and more sophisticated learning metho ds for

controller could generate the appropriate corresp ond dynamic tasks and higher DOF systems are b eing

ing p ostures and tra jectories The forceeld approach develop ed Atkeson Schaal Atkeson

we describ e is a small step toward such an approach The work most similar to the forceeld approachwe

To compare it with an alternative we implemented describ e was p erformed by Williamson who

b oth on a common testb ed describ ed next presented a controller for a DOF rob ot arm based

on the same biological evidence we describ e next It

consists of four b ehaviors three reaching and one

The Dynamic Anthrop omorphic Simulation Adonis

resting p osture intermediate targets are achieved by

linear interp olation

Adonis is a rigidb o dy simulation of a human torso

with static graphical legs Figure consisting of

eight rigid links connected with revolute joints of one

Biological Inspiration

and three DOFs totaling DOFs The dynamic

The exibility and eciency of biological motion pro

mo del for Adonis was created by metho ds describ ed

vides a desirable mo del for complex agentcontrol

meaningful instructions as opp osed to fully describ

ing the tasks at the simulation level The Macarena

sp ecication omitting the hip and wholeb o dy sub

tasks at the end is given b elow

Extend Right Arm straight out palm down

Extend Left Arm straight out palm down

Rotate Right Hand palm up

Rotate Left Hand palm up

TouchRight Hand to top of your left shoulder

Touch Left Hand to top of your right shoulder

TouchRight Hand to the backofyour head

Touch Left Hand to the backofyour head

TouchRight Hand to the left side of your ribs

Touch Left Hand to the right side of your ribs

Move Right Hand to your right hip

Move Left Hand to your left hip

No taskplanning was necessary b ecause the se

quence is provided by the dance sp ecication How

Figure The Adonis dynamic simulation testb ed

ever the individual subtasks are not sp ecied in the

same frame of reference The rst four deal with ex

tending the arms b est expressed in joint angles while

in Ho dgins et al Mass and momentofinertia

the rest are b etter describ ed in an egocentric Carte

information is generated from the graphical b o dy parts

sian reference frame This typ e of heterogeneous task

and equations of motion are calculated using a com

sp ecication is common in natural language descrip

mercial solver SDFast SDFast Users Manual

tions and control systems must satisfy eachofthe

The simulation acts under gravity accepts other ex

dierent goals regardless of the underlying represen

ternal forces from the environment and applies low

tation To address the issue of controller representa

level PDservo control see Section to keep bal

tion we implemented twovery dieren t alternatives

ance at the waist and neck No collision detection

and compared them based on common p erformance

with itself or its environment is used in the describ ed

metrics to demonstrate the tradeos involved

implementation we are currently implementing this

extension

Adonis is particularly well suited for testing and

The ForceField Approach

comparing dierent motor control strategies the sim

ulation allows us to apply forces to the endp oints Weintro duce a biologicallyinspired control approach

based on applying convergent forceelds to end ef

while a physical rob ot implementation would require

explicit calculation of the IK of the arms to solve fectors Adonis hands Inspired by the neuroscience

for actuator torques from the desired forces This work on motor control in frogs describ ed in Section

in turn enables us to implement exp erimental con this approach aords a more intuitive user inter

trollers for humanlikemovement more easilywhile face at the exp ense of motion generalityAnumber of

force elds are used as a set of basis functions for con

having the simulation software handle the issue of IK

and dynamics trolling Adonis desired Cartesian goals are reached

by applying forces to the hands to move them to the

desired destinations Each eld has a single equilib

Task Sp ecication

rium p oint EP at a target p osition where the eld is

zero From anypoint in Adonis reachable workspace

Natural goaldriven movement relies on precise sp ec

the hand will stably movetoward the EP excluding

ication co ordination and constraints imp osed by

singularities in the kinematics EPs are chosen em

implementation and evaluation A test task should b e

pirically as target p ositions for the hands in the

challenging to control but familiar enough to evalu

subtasks of the Macarena Some intermediate p os

ate The Macarena is a p opular dance whichinvolves

tures were used as describ ed in Section

a sequence of co ordinated movements that consti

The force elds used as basis functions dep end on

tute natural subtasks We used a verbal description

the p osition and velo city of the hand b eing controlled

aimed at teac hing p eople the dance found on the

The exerted force is prop ortional to the dierence

web at httpwwwradioprocommacarenahtml to

between the current and desired velo city calculated

implementcontrollers according to a common set of

where and corresp ond to the actual as a function of the dierence b etween the current

actual desir ed

and desired jointvelo cities and and p osition and the equilibrium p oint Formally

actual desir ed

corresp ond to the actual and desired joint angles

F c v v jx x j

actual desir ed EP

To generate the Macarena controller the desired

angles used for the feedback error are interp olated

where v is the desired velo city v is the

desir ed actual

from handpicked key p ostures The p ostures were

actual velo cityand c is the gain constant The magni

derived directly from the task sp ecication eachcor

tude of c determines the sp eed of the complete move

resp onding to one of the subtasks enumerated in

ment For wellchosen values of c this mo del makes

Section Intermediate p ostures b etween subtasks

the hand p osition converge to the EPfollowing a

help guide the joint tra jectories through dicult tran

roughly b ellshap ed velo city prole consistent with

sitions For example an intermediate p osture was

human data on regularsp eed reaching motions Flash

needed for swinging the hands around the head to

Hogan Linear combinations of the force

prevent a direct yet unacceptable path through the

elds can pro duce new elds with convergence p oints

head The incremental desired angles use a spline

in the convex hull of the EPs of the bases Bases

to smo othly interp olate b etween the p ostures Gains

can b e combined using vector summation as in our

for the PDservowere chosen by hand and remained

approach or using a winnertakeall framework as

constant throughout the b ehavior The joint torque

used by Williamson for rob ot reaching Our

approachallows direct control of each actuated join t

tra jectories are generated bymoving the equilibrium

in the system giving the user lo cal control of the

p oint through a set of subgoals or control p oints The

details of eachbehavior However the controller in

EP is computed from a linear weighted comp osition

turn requires a complete set of desired angles at all

of the bases functions We use feedbac ktomove

times Sp ecifying all of this information can b e te

between control p oints transitioning when the cur

dious esp ecially for joints that are less imp ortant for

rent control p oint is reached This works well since

the b ehavior b eing generated Interp olating b etween

the Macarena is a relatively slow task highly dy

key p ostures is a reasonable metho d for reducing the

namic b ehaviors eg ball throwing are b etter and

required amount of information

more robustly handled with feedforward control but

The control of actuated joints may b e individually

nontrivial arm dynamics make it dicult to predict

mo died using their resp ective desired angles giving

prop er transition timing

lo calized control over the generated motion All de

The forceeld approach presents a convenient con

sired key p ostures are sp ecied as a set of angles in

trol interface b ecause it eectively lowers the dimen

joint space In b ehaviors such as the Macarena p osi

sionality of the system This makes the control task

tion constraints like hands b ehind the head can b e

easier although the current implementation limits

satised with userlevel feedback However precise

exibility Several of the joints including the waist

Cartesianspace constraints like nger on the tip

neck and wrists use PDservocontrollers describ ed

of the nose would b e dicult to control by hand

in Section to maintain fairly rigid b ehavior Be

tuning using jointspace errors directlyAninverse

cause these joints are not explicitly controlled bythe

kinematics solver could b e used to generate desired

user the range of controllable motion is limited and

angles from p osition constraints although the con

the naturalness of the resulting b ehavior can b e re

troller has no direct measure of errors in Cartesian

duced as discussed in Section

space currently

The Joint Torque Approach

Performance Analysis and Comparisons

Jointspace controllers with torque actuators have b een

Analysis and evaluation of complex b ehavior is an

used successfully to generate b ehaviors for a varietyof

op en researchchallenge As synthetic b ehavior b e

systems Pai Raib ert Ho dgins Ho dgins

comes more complex the issue b ecomes increasingly

et al Van de Panne Lamouret In gen

acute in animation rob otics and AI in general We

eral these controllers cho ose a set of torques for all

explored several evaluation criteria b efore deciding on

actuated joints We designed a handtuned PDservo

those elab orated b elow

feedback controller for p erforming the Macarena on

Adonis In this approach torques are calculated for

t as a function of angular p osition and ve each join

Controller Flexibility

lo city errors b etween the feedback state and desired

Behavior constraints come in various forms Com

state

plex movements can b e brok en down into subtasks

T kd k with Cartesianspace or jointspace constraints For

desir ed actual desir ed actual

Figure Adonis p erforming the Macarena using the jointspace controller

example placing hands b ehind the head is b est rep straints such as gesturing and freeform movement

resented by a Cartesian constraint while turning the are more easily controlled with this approach and

hand is b etter treated as a joint constraint The issue subtleties such as head motion and elb ow p osition

of co ordinate and representation transformation has ing can b e controlled Figure The richness of the

b een addressed extensively in manipulator rob otics control however requires the user to sp ecify joint an

Paul Brady et al The twocontrollers gle information for each DOF in each subtask In this

implemented favor very dierent representations and approach the representational transformation from

thus the evaluation of their ability to satisfy all of the Cartesian to joint space is done implicitly by the pro

various subgoals serves as an eective p erformance grammer Consequently tight Cartesian constraints

metric are dicult to achieve with the current implementa

The forceeld controller employs a D equilib tion and may require an IK solver

rium p oint as a control handle This representation Acontrol structure should easily accommo date a

is well suited to subtasks that describ e the lo cation variety of tasks and constituent b ehaviors and gener

of the endp ointshands such as p ointing reaching ating new b ehaviors should require a minimal amount

and basic interaction with a D environment How of user input The jointtorque approachtocontrol

ever a Cartesian equilibrium p oint is an indirect and is not easily generalizable b etween b ehaviors as lit

at times unusable control handle in that complex tle or no knowledge can b e transferred timing and

p ostures like folding the arms or turning the palms p ostures are usually sp ecic and need to b e regener

cannot b e naturally achieved Akey advantage of ated with eachnewcontroller In contrast the force

this approach is that it limits the amount of infor eld approachmay b e used to generate new b ehaviors

mation required for describing a b ehavior reducing more easily due to its reduced control dimensionality

the dimensionality of the problem However it also Also using its linear additive prop erties new b ehav

removes access to the individual degrees of freedom iors may b e generated by combining existing ones

for the agent While sp ecializations of the forceeld

approach are p ossible they compromise its simplic

Naturalness of Movement Qualitative

ity

Judging the naturalness of movement is imp ortant

The jointspace controller uses desired angles as

but aesthetic judgmentisdiculttoquantify Dy

control handles allowing more complete control of

namic simulation constrains motion to b e physically

the agents motion Behaviors that include joint con

controllers against the motion of a human p erforming

Human Data Torquedriven Forcedriven

the Macarena Wechose this metric over other alter

T jerk time jerk time jerk time

natives such as minimum torque change suggested

byUnoKawato Suzuki for its simplicity

of computation in our domain the hand p osition was

easily measured for b oth the simulation and the hu

man movement The comparison is p erformed using

the metric of total square jerk in D Cartesian space

we added a z p osition term to the calculation

Motion for D hand p ositions of a human sub

ject p erforming the Macarena was recorded using a

commercial Flo ck of Birds electromagnetic tracking

system Each subtask was isolated and calculated

separately for a nergrain evaluation Corresp ond

ing data were collected for the twosimulations the

results are presented in Figure The values in the

table corresp ond to the square jerk over the length

Figure A comparison of minimum jerk values

of the subtask for the ma jor moving hand in that

subtask eg in subtask one the right arm in sub

task two the left arm and so on The units for

plausible but it is not necessarily natural Viewer



squared jerk are m sec and the duration of each

based evaluation is one qualitative approach to mea

subtask lab eled time is given in seconds Subtasks

suring naturalness Realtime playbacks of b oth con

and could not b e implemented with the forceeld

trollers we implemented are available from http

approach since it lacked the ability to control hand

wwwrob oticsuscedu agentsmacarenahtml

orientation these entries are indicated with Jerk

Rigid b o dy simulation imp oses limitations that

is a sensitive measure susceptible to p osition error

cannot b e overcome by control For instance its

unavoidable in motion capture data collected with

unactuated back will necessarily app ear sti Fur

electromagnetic trackers Furthermore timing has a

thermore the controllers havenoknowledge of the

signicant eect on the jerk slower movements imply

b o dy p osition and avoiding selfcollisions was done by

lessjerkasisshown in the case of the forcecontrol

hand This results in conservative unnatural tra jec

approach Therefore the exact values ab ove are less

tories whichwould b e improved with a controller ca

signicant than the general trends they indicate

pable of collision prediction and avoidance Similarly

In this evaluation there is no corresp ondence b e

joint limits would contribute to more natural b ehav

tween the goal p ositions and timing for the human

ior To improve app earance the forceeld metho d

and the simulated agent motions Therefore wedo

uses strong PDservos to clamp some joints avoiding

not exp ect corresp ondence b etween the controllers

marionettelike unconstrained movement The joint

and the human jerk values but instead in trends

torque metho d interp olated p ostures with splines to

across each subtask and b etween the three sets of

smo oth the resulting motion and includes small head

data Rather than make a onetoone comparison we

and hand movements that result in richer motion

consider where the data are signicantly consistentor

Head gaze can add much in terms of naturalness of

inconsistent and suggest some reasons for these pat

motion and its implementation is straightforward

terns The hand jerk calculation is notably noisy and

varies widely from task to task so a qualitative analy

Naturalness of Movement Quantitative

sis is more b enecial than an actual numeric compar

Minimal jerk of hand p osition has b een prop osed by

ison It should b e noted that the forceeld controller

Flash Hogan as a metric for describing hu

explicitly applies forces to the hand and thus directly

man arm mo vements in the plane by minimization

aects the jerk Therefore the magnitude of the jerk

of the following index

is a contrived metric for the forcecontrolled mo ve

R

ment as a whole However the relative jerk within

t

f inal

jerk jerk dt C

j

x y

each of the three sets of controllers provides valuable

insight

where jerk and jerk corresp ond to the third

x y

An imp ortant consideration in this comparison is

derivativeofthe x and y p ositions with resp ect to

the amountofenvironment knowledge available to

time We used jerk as the evaluation metric for com

the controller human or simulated Human access

paring the simulated motion for the two implemented

to knowledge ab out b o dy lo cation and eective means implementation as a metric Both of the describ ed

of avoiding selfcollision allow for generating smo oth metho ds avoid explicit IK computations in favor of

movements of arbitrary complexity In contrast sim simple but less general alternatives Their relative

ulations haveminimal environment knowledge or tra eciency is dicult to compare however due to the

tegral role of the designer in b oth approaches jectory planning capabilities Consequently subtasks in

involving complex selfcollision avoidance result in

less ecient and less natural b ehavior For example

Continuing Work and Conclusion

subtasks and exhibit these characteristic

tra jectory planning diculties

Wehave compared two approaches to anthrop omor

Interestingly while tasks and are similar in

phic agentcontrol b oth implemented on a dynamic

nature and in fact app ear most complex in terms of

torso simulation Adonis and tested on a subgoal

selfcollision avoidance their corresp onding jerk val

Macarena task Weintro duced a biologicallymotivated

ues do not show the same pattern One interpreta

forceeld approach whichsetsupconvergent force

tion of this result stems from the subgoals involved

elds that op erate as a function of the velo city and

in generating the complex movement While wehave

p osition of the hands We then compared it to a

little understanding of the strategies p eople use in

jointspace controller which manipulates the individ

reaching this subgoal we can infer from the b ehavior

ual torques for each DOF of the system allowing com

that co ordinated movement of the head and the arm

plete control over the agent but requiring more intu

and subsequent contact with the head is involved In

ition on the b ehalf of the user Both approaches avoid

contrast the simulated b ehavior pro ceeds through a

explicit IK computation by feeding unconverted user

set of intermediate p ositions whichavoid selfcollision

inputs to the simulation making tasksp ecication

and result in the arm b ehind but not in contact with

ecient but not fully general We compare the con

the head Not surprisingly the resulting jerk pattern

trollers against eachotherandtohuman data on the

is quite dierent To study this discrepancy further

same task using a minimum jerk computation as a

a less familiar and intuitive set of subtasks could b e

common metric The fundamental tradeo b etween

used that mayprovide a stronger comparison b etween

b elievability and control eort still remains as the

human and simulated motion

twoapproaches pro duce dierent results dep ending

To facilitate an honest comparison sim ulated and

on subtask sp ecication

human motion were generated indep endentlyHow

The describ ed work is a part of a pro ject aimed

ever various techniques can b e implemented to gen

at developing a biologicallyinspired b ehaviorbased

erate a closer t b etween the data if that is desired

approach to motor control We presented implemen

Sp ecicallyhuman hand p ositions could b e used to

tations that employ a single representational metho d

select goal p ositions for the forceeld equilibrium

ology which are a part of an eort toward a system

p oints Similarly an IK solver could nd joint p os

that encapsulates the lowlevel control details within

tures for the jointspace controller that achieved these

primitive parametric b ehaviors that satisfy various

hand p ositions In addition timing considerations in

motor tasks including movement to p oint using a

uence the comparison As seen from the data the

sp ecic achievementgoal such as reaching and most

timing of subtasks was not correlated Timing taken

of the Macarena subtasks and rep etitive and oscil

from human motion could b e used to generate simu

latory movements using maintenancegoals suchas

lated motion that more closely t the human p erfor

b ouncing waving swinging etc Individual b ehav

mance Lastlyminimization techniques could b e ap

iors may rely on dierent representations but their

plied to the controller parameters to nd movements

use and p erformance can b e seamlessly integrated

that minimize jerk or other p erformance metrics

by sequencing and coactivation as in the biologi

cal mo del In addition to its biological motivation

and p otential dimensionality reducing prop erties the

Other Evaluation Criteria

behaviorbased mo del for complex motor control pro

Other criteria were also considered We explored

vides a simple and scalable interface for graphical

evaluating the correctness of the resulting b ehavior

agents One of our goals is to develop a control archi

but without a set of analytical denitions of each

tecture that allows for combining various movement

subtask this proved of little use Such sp ecic con

primitives p ossibly from dierent users into a ver

straints could b e computed but were rather inconsis

satile and general system As complex articulated

tent with the graphical environment where the def

agents b ecome more prevalent such a mo dular ap

inition of achieving the task did not rest on precise

proachtocontrol would use its op en architecture

placementasmuchasontheoverall impression of the

to combine the advantages of various approaches by

complete movement We also considered eciency of

MussaIvaldi F A Giszter S F Vector encapsulating them into primitives

eld approximations a computational paradigm

for motor control and learning Biological Cy

Acknowledgments

bernetics 

This work is supp orted by the NSF Career Grant IRI

MussaIvaldi F A Giszter S F Bizzi E

to M Mataric The authors thank Nancy

Linear combinations of primitives in ver

Pollard for help with the jerk calculations Len Nor

tebrate motor control Proceedings of the Na

ton for help with human motion data and Stefan

tional Academy of Sciences 

Schaal and Jessica Ho dgins for sharing exp ertise and

providing many insightful comments The Adonis

Ngo J T Marks J Spacetime Constraints

simulation was develop ed by Jessica Ho dgins at Geor

Revisited in J T Ka jiya ed Computer

gia Institute of Technology

Graphics Pro ceedings SIGGRAPH

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