INVERSE AND GEOMETRIC CONSTRAINTS

FOR ARTICULATED FIGURE MANIPULATION

by

Chris Welman

BSc Simon Fraser University

a thesis submitted in partial fulfillment

of the requirements for the degree of

Master of Science

in the Scho ol

of

Computing Science

c

Chris Welman

SIMON FRASER UNIVERSITY

Septemb er

All rights reserved This work may not b e

repro duced in whole or in part by photo copy or other means without the p ermission of the author

APPROVAL

Name Chris Welman

Degree Master of Science

Title of thesis and Geometric Constraints for Articulated Fig

ure Manipulation

Examining Committee Dr L Hafer

Chair

Dr TW Calvert

Senior Sup ervisor

Dr J Dill

Dr D Forsey

External Examiner

Date Approved ii

Abstract

Computer of articulated gures can b e tedious largely due to the amount of data which

must b e sp ecied at each frame Animation techniques range from simple interp olation b etween

keyframed gure p oses to higherlevel algorithmic mo dels of sp ecic movement patterns The former

provides the animator with complete control over the movement whereas the latter may provide

only limited control via some highlevel parameters incorp orated into the mo del Inverse kinematic

techniques adopted from the rob otics literature have the p otential to relieve the animator of detailed

sp ecication of every motion parameter within a gure while retaining complete control over the

movement if desired

This work investigates the use of inverse kinematics and simple geometric constraints as to ols

for the animator Previous applications of inverse kinematic algorithms to are

reviewed A pair of alternative algorithms suitable for a direct manipulationinterface are presented

and qualitatively compared Application of these algorithms to enforce simple geometric constraints

on a gure during interactive manipulation is discussed An implementation of one of these algo

rithms within an existing gure animation editor is describ ed whichprovides constrained inverse

kinematic gure manipulation for the creation of keyframes

Contents

Intro duction

Organization

Approaches to Figure Animation

Bo dy Mo dels

Scop e

Skeleton Mo delling

Kinematic Metho ds

Inverse Kinematics

Dynamic Metho ds

Forward Dynamics

Inverse Dynamics

Control Issues

Summary

Inverse Kinematics

The Inverse Kinematic Problem

Resolved Motion Rate Control

Redundancy

Singularities

OptimizationBased Metho ds

Evaluation

Applications to Computer Graphics

Ecient Algorithms for Direct Manipulation

A Simplied Dynamic Mo del

The Jacobian Transp ose Metho d i

CONTENTS ii

Implementation Details

Computing the Jacobian

Scaling Considerations

Integration

Joint Limits

A Complementary Heuristic Approach

The CyclicCo ordinate Descent Metho d

Overview

Comparison

Incorp orating Constraints

Constraint Satisfaction

Maintaining Constraints

The Constraint Condition

Computing the Constraint Jacobian Matrix

Computing the ConstraintForce

Solving for Lagrange Multipliers

Feedback

Overview

Implementation Issues

Skeletons as Ob jects

Handles on Skeletons

Constraints on Handles

The Global Picture

Summary

A CCDbased Penalty Metho d

An Interactive Editor

A System Overview

Skeletons

The Sequence

The Editor

Direct Manipulation

Constraints

Conclusion

Summary

Results

CONTENTS iii

Comments ab out Constraints

Directions

Bibliography

List of Figures

a ob jects dened in lo cal co ordinate systems b Lo cal rotations applied cd

Lo cal translations applied

Three congurations of a D redundant manipulator

A manipulator in a singular conguration

Interactive control lo op mo del for Jacobian transp ose metho d

A case not handled well with the Jacobian transp ose metho d Pulling inwards on the

tip of the manipulator on the left will not pro duce an exp ected conguration like the

one shown on the right

Example CCD iteration step for rotation jointi

A force applied to a p oint constrained to lie within a plane A constraining force

normal to the plane is added to the applied force to obtain a legal force tangential to

the plane

Iteration steps for maintaining constraints

A generic function blo ck

Example network

A branching chain with two endeector constraints

Sample skeleton description

Sequence editor screen

Plan view of goal determination in a an orthographic view and b a p ersp ective

view

A gure b eing p ositioned by rst tilting then twisting the p elvis

a First keyframe p ose b Interp olated p ose c Second keyframe p ose iv

Chapter

Intro duction

Computer graphics has advanced to a p oint where generating images of striking realism and com

plexity has b ecome almost commonplace However making ob jects moveconvincingly within these

pictures remains dicult particularly as ob ject mo dels grow increasingly complex The sp ecication

and control of motion for computer animation has emerged as one of the principal areas of research

within the computer graphics community

One area in particular which continues to receiveattention is that of gure animation The goal

of work in this area is to provide a means of generating lifelike p ossibly humanlike articulated

gures and to design and control their actions within simulated environments Animated human

gures could for example b e placed in simulated environments for ergonomic evaluation or simply

to provide some aesthetic qualities to a presentation In the arts and entertainment area the concept

of computergenerated characters roaming through articial worlds seems universally app ealing

Although gure animation raises technical challenges in b oth mo delling and rendering the fun

damental problem of designing and controlling movement for these gures remains a particularly

dicult one Part of the problem lies in deciding from whichlevel of detail to approach the task

At one end of the scale the movements of individual parts of the b o dy must b e known for each

instant in time At the other end of the scale co ordinating movements and handling interaction

between gures and with the environmentmay require algorithms based on b ehavioural rules and

knowledge bases Many of the most impressive examples of gure animation bycomputerhavebeen

the result of algorithms implementing highlevel b ehavioural and motor control mo dels However

these algorithms are often limited to generating sp ecic usually rep etitive movement patterns such

as walking and running For the animator who wishes to create new movements there is little al

ternative to painstakingly constructing the movementby hand Given the complex structure of a

typical articulated gure this can involve an inordinate amountofwork

CHAPTER INTRODUCTION

The motivation b ehind this work is a desire to improve the animation capabilities of an existing

interactive articulated gure animation package which is currently used to create movements for

b oth dance and animation It is shown how inverse kinematic techniques for controlling rob otic

manipulators can b e adopted to relieve the animator of some of the more tedious asp ects of creating

new movements by hand After reviewing the inverse kinematic problem and solutions that have

previously b een applied to gure animation a pair of alternative solution algorithms are presented

and qualitatively compared These algorithms are simple yet eective and can supp ort b oth direct

manipulation of articulated gures as well as the imp osition of simple geometric constraints up on a

gure Implementations of these algorithms are presented and are applied to develop a basic set of

interactive to ols for gure manipulation and animation

Organization

Chapter reviews computer animation techniques in general and discusses their applicabilityin

the context of gure animation In Chapter the inverse kinematic problem is stated and com

mon approaches to solving the problem are reviewed In Chapter a pair of fast reliable inverse

kinematic algorithms are describ ed suitable for interactive manipulation tasks and diering from

previous algorithms adopted for computer graphics In Chapter pro cedures for satisfying simple

geometric constraints using these algorithms are considered Chapter intro duces an interactive

gure animation editor and discusses implementation of the algorithms as p ositioning aids

Chapter

Approaches to Figure Animation

Placing this work in context requires some understanding of computer animation techniques in

general and of how they may b e applied to gure animation in particular This chapter provides an

overview of the advantages and disadvantages of basic motion control techniques for gure animation

The emphasis here is on metho ds to create and control the movements of articulated gures

rather than simply replaying digitized movement It is fair to say that for many pro ductions dig

itizing or rotoscoping themovements of real sub jects remains the metho d of choice for obtaining

convincing lifelike motion Rotoscoping can refer to techniques ranging from visually matching

graphic images to prerecorded video fo otage to attaching some sort of sensors to a p erformers

body whose p ositions can b e tracked by computer and stored for later playback Neither of these

are particularly attractive options the former b eing quite tedious and the latter relying on the

availability of reliable unobtrusive instrumentation for the b o dy and sophisticated software to re

construct the original motion from the sensor data neither of which are readily available yet A

further limitation of rotoscoping is that a gure animated in this way is limited to those movements

actually p erformed by a live sub ject Computer animation techniques can b e applied to animate

gures in situations for which rotoscoping is neither a viable nor practical solution

Bo dy Mo dels

Scop e

First wemust decide exactly what we are trying to animate Although the ideal computergenerated

character would include muscle and tissue that deforms during movement skin and clothing that

wrinkles and stretches hair that ows and expressive facial features the accurate mo delling ani

mation and rendering of these attributes are research topics in their own right and work in these

CHAPTER APPROACHES TO FIGURE ANIMATION

areas is still at the exp erimental stage For the time b eing wewillhave to restrict our attention to

animating simple approximations to real b o dies It is useful to think of these simple approximations

as a skeletal layer up on whichmuscle tissue and skin can later b e layered The imp ortant p oint here

is that any b o dy mo del can b e animated bymoving an underlying skeletal approximation which

need not b ear any resemblance to the nal rendered app earance of the character Thus the motion

control problem for gures reduces to that of controlling the movement of an abstract articulated

skeleton

Skeleton Mo delling

Askeleton can b e represented by a collection of simple rigid ob jects connected together byjoints

The joints are usually rotational but may also b e sliding or prismatic Each rotary jointmay

allow rotation in or orthogonal directions these are the degrees of freedom DOF of the

joint A detailed approximation to the human skeleton mayhaveasmany as degrees of freedom

although often fewer suce Restrictions on the allowable range of movements for a jointcanbe

approximated by limiting the rotation angle in each of the rotation directions at each joint

The individual ob jects comprising the skeleton are each dened in their own lo cal co ordinate

systems and are assembled into a recognizable gure in a global world co ordinate system bya

nested series of transformations In Figure a simple articulated limb is built up by applying lo cal

rotations and translations to blo cks dened in their own lo cal co ordinate systems

More complex skeletons can b e built up by arranging the segments in a treestructured hierarchy

Each no de in the tree maintains the rotations currently in eect at the corresp onding joint these

joint rotations are osets from the orientation of the parent segment in the tree These nested

transformations in the hierarchy ensure that segments inherit the rotations applied to joints higher

in the tree a rotation applied at say the shoulder joint causes the entire arm to rotate and not

just the upp er arm segment One jointintheskeleton needs to b e sp ecied as the ro ot of the tree

transformations applied to this jointmove the entire skeleton in the world co ordinate system The

choice of which jointistoserve as the ro ot is irrelevant and it is convenient to b e able to restructure

an existing hierarchy around a new ro ot jointatany time The global transformations applied to

any particular ob ject within the skeleton can b e computed bytraversing the hierarchy from the ro ot

to the segment and concatenating the lo cal transformations at each joint visited by the traversal

Most animation systems provide a means of building up the transformation hierarchy needed to

dene a skeleton and it is easy enough to dene a simple grammar for sp ecifying skeletons Zelb

Sims SZ has describ ed an interactive editor for designing new skeletons which applies some

simple heuristics to streamline the pro cess Regardless of how it is created a skeleton denition will

CHAPTER APPROACHES TO FIGURE ANIMATION

(a) (b)

(c)

(d)

Figure a ob jects dened in lo cal co ordinate systems b Lo cal rotations applied cd Lo cal

translations applied

minimally sp ecify the individual b o dy segment lengths the joint degrees of freedom and the overall

hierarchy of the structure

Askeleton can b e animated byvarying the lo cal rotations applied at eachjointover time as well

as the global translation applied at the ro ot joint The motion sp ecication and control problem

is that of managing the wayinwhich these transformations change over time In general there

are two fundamental approaches to this problem kinematic and dynamic The following sections

review b oth kinematic and dynamic metho ds for motion sp ecication in general the typ es of control

available for each and applications of these to gure animation in particular

Kinematic Metho ds

Forward Kinematics

Forward kinematics involves explicitly setting the p osition and orientation of ob jects at sp ecic frame

times For skeletons this means directly setting the rotations at selected joints and p ossibly the

global translation applied to the ro ot joint to create a poseToavoid doing this for each frame

of an animation a series of keyframe p oses can b e sp ecied at dierent frames with intermediate

CHAPTER APPROACHES TO FIGURE ANIMATION

p oses calculated byinterp olating the joint parameters b etween the keyframes The gure can then

b e animated by displaying eachintermediate p ose

While linear interp olation b etween keyframes is the simplest metho d for generating these inter

mediate p oses the resulting motion is usually unsatisfactory Discontinuous rst derivatives in the

interp olated joint angles at the keyframes lend a jerky rob otic quality to the motion The use of

higherorder interp olation metho ds such as piecewise splines can provide continuous velo city and

acceleration and hence smo other transitions b etween and through the keyframes Keyframe inter

p olation is well established Ste GomHS ShoStu and is invariably provided in

commercial animation systems

Controlling interp olation

Interp olation often pro duces intermediate values that do not quite meet the animators requirements

some control over the interp olation pro cess is crucial Las The interp olated values for a single

DOF over the course of an animation form a trajectory curve which usually passes through the

keyframe values The shap e of the tra jectory and hence the motion of the ob ject is dep endanton

b oth the keyframed values and the typ e of interp olating spline used An interactive editor which

allows the animator to view and mo dify the shap e of a tra jectory can b e a useful to ol Once a

tra jectory is dened the quality of the movement can b e further mo died byvarying the rate at

which the tra jectory is traversed Anumb er of parameterized interp olation metho ds havebeen

prop osed which provide varying degrees of control over b oth the shap e of a tra jectory and variations

in sp eed along the tra jectory

Ko chanek KB describ es an interp olation technique based on a generalized form of piecewise

cubic Hermite splines Three parameters continuity tension andbias are provided to control the

length and direction of vectors tangent to the tra jectory at the keyframes Mo difying the direction

of the tangentvectors gives lo cal control over the shap e of the curve as it passes through a keyframe

Changing the length of the tangentvectors aects the rate of change of the interp olated value around

the keyframe and thus provides some control over sp eed Some traditional animation eects such

as action fol lowthrough and exaggeration Las can b e achieved with appropriate settings of these

parameters Unfortunately since all three parameters in the spline formulation aect the shap e of

the curve the metho d provides no means for mo difying sp eed along a tra jectory without mo difying

the tra jectory itself

Steketee and Badler SB advo cate a doubleinterp olant metho d which do es separate timing

control from the tra jectory itself As b efore a tra jectory is dened as a piecewise cubic spline passing

through a series of keyframed values An additional spline curve isalsointro duced to control the

CHAPTER APPROACHES TO FIGURE ANIMATION

parameter with which the tra jectory curve is sampled This provides control over the parametric

sp eed at which the tra jectory curveistraversed However there is often no meaningful relationship

between parametric sp eed and actual sp eed in the geometric sense sampling a curve at uniformly

spaced parametric p oints will not necessarily yield uniformly spaced p oints in space This approach

to timing adjustment therefore is somewhat ad hoc and nonintuitive requiring a trialanderror

pro cess on the part of the animator to achieve the desired velo city prole along the tra jectory

More intuitivecontrol over sp eed along a tra jectory can b e obtained by reparameterizing the

tra jectory curveby arclength Arclength parameterization provides a direct relationship b etween

parametric sp eed and geometric sp eed along a tra jectory the distance d travelled along a tra jectory

is prop ortional to the increments of the tra jectorys arclength parameter sAllowing the animator

to sketchacurve representing s over time provides an intuitivemechanism for varying sp eed along

the tra jectory BH However although theoretically a reparameterization by arclength exists for

any curve it is often not p ossible to nd an analytic solution for arbitrary curves and one must

resort to numerical approximation metho ds Gir GP

Evaluation

Keyframebased computer animation has a direct analogy in traditional animation where key ani

mation cels are drawn by senior animators while less exp erienced animators draw the action in the

intermediate cels Computerbased keyframing is intuitive and the interp olation can usually b e p er

formed fast enough to provide near realtime feedback For skeleton animation however keyframe

interp olation do es not work well the few go o d examples of keyframed gure animation are more a

tribute to the skill and patience of the animator than to the techniques suitability for the task

One ma jor diculty can b e lab elled the degrees of freedom problem for interesting skeletons

there are simply to o many DOFs for whichvalues must b e provided the level of detail required

from the animator to sp ecify even a single key p ose is excessive Trying to control the interp olated

motion by manually mo difying p ossibly hundreds of tra jectory curves can b e tedious frustrating

and errorprone While it is essential that the animator have some control at the joint level higher

levels of control are desirable for sp ecifying the co ordinated movements of groups of joints

Even supp osing that the numb er of degrees of freedom within a gure is manageable the common



practice of displaying interp olated joint angles as a set of three splined tra jectory curves is rarely

helpful Unlike translations an ordered series of rotations do not combine intuitively making it

dicult to predict the consequences of editing a single rotation tra jectory and almost imp ossible to

decide on the appropriate changes to all three curves which will pro duce a desired change in a single



one each for the XY and Z rotation directions at each joint

CHAPTER APPROACHES TO FIGURE ANIMATION

b o dy segments motion

The hierarchical structure of the skeleton also causes problems The only jointwhich an animator

can explicitly p osition is the ro ot joint in the hierarchy the p ositions of all other ob jects in the

skeleton dep end on the rotations at ancestor joints This makes it dicult to enforce p ositional

constraints when creating a keyframe p ose For example if the hierarchy ro ot for a bip ed is at the

p elvis then placing a fo ot on the o or and keeping it there is troublesome if the fo ot is already in

place then a b end at the knee will move the fo ot whichmust then b e rep ositioned by mo difying

the rotation at the hip joint The ability to rearrange the hierarchy ab out a new ro ot joint is only

marginally useful In our example making the supp ort fo ot the new ro ot of the hierarchywould

allow a knee b end which leaves the fo ot in place However this will also move the rest of the b o dy

whichmaymove another previously p ositioned b o dy segment such as the other fo ot This makes

enforcing multiple p ositional constraints a frustrating pro cess

The same problem crops up during interp olation Even if an animator has made sure that

b oth feet are p ositioned correctly in a series of keyframe p oses there is no guarantee that simply

interp olating joint rotations will maintain the correct fo ot p ositions at the intermediate frames

It is quite common to see interp olated keyframed sequences for gures in which the feet seem to

p enetrate through or slide around on the o or While this can b e remedied by sp ecifying additional

keyframes as the keyframe spacing b ecomes smaller the animation pro cess b egins to resemble the

framebyframe p ositioning of traditional stopaction animation claymation for example This

defeats the whole purp ose of interp olation whichisintended to relieve the animator from the tedium

of sp ecifying the motion on a framebyframe basis

While forward kinematics combined with a simple interp olation scheme may suce for animating

simple ob jects it is not really up to the task of animating articulated gures

Inverse Kinematics

Using forward kinematics the p osition of any ob ject within a skeleton can only b e indirectly con

trolled by sp ecifying rotations at the joints b etween the ro ot and the ob ject itself In contrast

inverse kinematic techniques provide direct control over the placementofan endeector ob ject at

theendofakinematic chain of joints solving for the joint rotations which place the ob ject at the

desired lo cation In light of the preceeding discussion it should b e apparentthatinverse kinematics

oers an attractive alternative to explicitly rotating individual joints within a skeleton An animator

can instead directly sp ecify the p osition of an endeector while the system automatically computes

the joint angles needed to place the part Not surprisingly the inverse kinematic problem has b een

studied extensively in the rob otics eld although it is only fairly recently that the techniques have

CHAPTER APPROACHES TO FIGURE ANIMATION

b een adopted for computer animation

Chadwicks Critter system p ermits inverse kinematic manipulation of a skeleton for creating

keyframes CHP Badler has prop osed an inverse kinematic algorithm to enforce p ositional con

straints on multiple b o dy parts during skeleton manipulation BMW and has incorp orated joint

range limits into the inverse kinematic solution CP ZB Both Girards PODA system GM

and Sims gait controller SZ provided highlevel lo comotion mo dels for skeletons using inverse

kinematics to generate the leg motion In these systems a planning stage determines fo ot place

ments and tra jectories while the inverse kinematic algorithm is resp onsible for generating the leg

joint angles as the feet are moved along tra jectories b etween each fo othold

Inverse kinematics provides higherlevel control over joint hierarchies than simple forward kine

matics moving the limbsofaskeleton b ecomes much more manageable However often the un

derlying metho d for generating motion still relies on strictly kinematic metho ds Unfortunately

kinematic metho ds do not pro duce convincing movement without a considerable amount of eort

on the animators part Often the motion exhibits a weightless qualitywhich is dicult to disp el

by editing the tra jectories and timing for individual degrees of freedom Kinematic metho ds b oth

forward and inverse do not pro duce movement with the sort of dynamic integrity wehave come to

exp ect from our exp erience with the physical laws of the real world

Dynamic Metho ds

Animation based on dynamic simulation is attractive b ecause the generated motion adheres to phys

ical laws providing a level of realism that is extremely dicult to duplicate with kinematic metho ds

For dynamic analysis ob ject descriptions must include suchphysical attributes as the center of mass

the total mass and the moments and pro ducts of inertia Although there are manyformulations for

the equations of motion they are all essentially equivalent to the familiar F mawhich relates the

acceleration a an ob ject of mass m undergo es in resp onse to a force F applied at the ob jects center



of mass The motion generated byphysical simulation is controlled by the application of forces

and torques whichmayvary over time Techniques for dynamic motion control can b e categorized

as either forward dynamic metho ds or inverse dynamic metho ds The essential distinction b etween

the twoisintheway that the basic forces and torques driving the motion are arrived at

Forward Dynamics

Forward dynamics involves explicit application of timevarying forces and torques to ob jects Some

forces such as those due to gravity and collisions b etween ob jects may b e handled automatically

 a similar equation relates angular acceleration to applied torques

CHAPTER APPROACHES TO FIGURE ANIMATION

by the animation system other forces are applied directly by the animator to ob jects in the scene

The motion is approximated by taking a series of discrete steps in time and at each step solving the

equations of motion for the acceleration an ob ject undergo es in resp onse to the applied forces Given

the p osition and velo city of an ob ject from the previous time step the acceleration a can b e twice

integrated to determine a new velo city and p osition resp ectively for the current time step A go o d

intro duction and overview of the basics of forward dynamic simulation for animating rigid b o dies

can b e found in Wil A comprehensiveapproach to simulating the motion of rigid p olyhedral

ob jects accounting for collisions is presented by Hahn Hah

Extending this approach to the simulation of articulated skeletons is challenging In general

there will b e one equation of motion for each degree of freedom in the skeleton This leads to a large

system of equations whichmust b e solved bynumerical metho ds at considerable computational

exp ense The formulation adopted to represent the equations of motion signicantly aects the cost

of the solution metho d A solution for the matrixbased GibbsApp ell formulation for example



has On complexityfor n degrees of freedom Wil Armstrong has prop osed an alternative

recursive formulation which reduces the complexityto O nAG enabling dynamic simulations

of simple articulated structures to b e p erformed in close to realtime But dynamic simulation of

reasonably complex articulated skeletons cannot in general b e p erformed at interactivespeedson

singlepro cessor machines although the recursiveformulation may b e fast enough to b e tolerable

Complicating matters is the fact that the equations of motion for articulated skeletons are con

siderably more complex than those for simple ob jects since they must include terms to mo del the

interactions b etween connected b o dy parts This coupling of the dynamics equations makes control

extrememly dicult since movement of one segmentoftheskeleton will exert forces and torques

on adjacent segments the notion that the motion of the skeleton can b e adequately controlled by

applying joint torques individually is incorrect Wil Eorts to counteract this unwanted prop

agation of torques usually involve placing springs and damp ers at eachjoint to maintain a desired

orientation Unfortunately this typ e of control invariably leads to a sti set of equations which

causes severe instabilityinmostnumerical solution techniques A summary of numerical stability

and control issues that must b e addressed during dynamic simulation is presented in Gir

Comp ounding the problem of numerical instabilities is the fact that the equations of motion for

articulated skeletons are inherently illconditioned indep endent of their formulation Mac The

illconditioning arises when the skeleton assumes a p osture in which small incremental changes in

one degree of freedom pro duce large accelerations elsewhere almost all numerical solution techniques

have diculty handling such cases Maciejewski contends that these situations o ccur frequently for

articulated gures and are inherent in the structure of most skeletons The illconditioning of the

CHAPTER APPROACHES TO FIGURE ANIMATION

equations has implications not only for dynamic analysis but inverse kinematic algorithms as well

Mac gives a lucid description of the problem and discusses metho ds for detecting and handling

the illconditioning in b oth cases

One of the earliest attempts to control an articulated gure purely through forward dynamic

simulation was Wilhelms Virya system Wil Virya p ermitted the interactive design of force or

torque versus time functions for individual degrees of freedom Force and torque keyframes could b e

sp ecied at dierent times cubic splines were then used to construct the force and torque proles over

the course of the entire motion sequence During dynamic simulation these forcetorque proles were

sampled and combined with forces due to collisions and gravity to determine instantaneous force

and torque measurements for the current time step The use of interp olating curves is conceptually

similar to the direct kinematic keyframe interp olation approach describ ed previously The dierence

is that the motion is driven not directly bytheinterp olated curves but indirectly through the

equations of motion Virya exhibited most of the problems outlined ab ove In particular Wilhelms

rep orts that the coupling of the dynamic equations made control of the gure dicult and non

intuitive Other eorts to simulate skeleton motion using pure forward dynamics rep ort similar

problems AGWCH

Even supp osing that a reliable and fast numerical solution technique is available the lackof

intuitive control remains the principal problem in using forward dynamics for animation In fact

forward dynamic simulation is b est suited for tasks which can b e p osed as initialvalue problems

That is tasks for which initial p ositions and velo cities and forcetorque proles are known a priori

and the goal is to generate the resulting motion This formulation may b e satisfactory for animating

scenes of simple inanimate ob jects realistically tumbling and b ouncing through an environment but

do es not apply for the animation of sp ecic tasks For example simulating a ball b ouncing on a

o or is simple to do given an initial height and velo city the simulation need only consider the force

of gravity and reactions to collisions with the o or to generate convincing motion However if the

goal is to have the ball b ounce three times and land in a cup the problem is much more dicult

the exact initial p osition and release velo city of the ball which will land it in the cup is dicult to

determine Yet this is precisely the sort of problem that app ears in animation the animator knows

what motion should o ccur but do es not knowinadvance the initial conditions and forcetorque

proles needed to pro duce the desired result

Inverse Dynamics

Inverse dynamic metho ds automatically determine the force and torque functions needed to accom plish a stated goal In the degenerate case the stated goal is a complete description of the motion

CHAPTER APPROACHES TO FIGURE ANIMATION

and the aim is to determine the forces and torques which repro duce the motion under forward dy

namic simulation While this case is of interest in rob otics its application is of little use in an

animation system after all if the motion tra jectories and timing are known b eforehand the exp ense

of the physical simulation is unnecessaryMoreinteresting are recent metho ds whichallow rela

tively highlevel constraints or goals to b e sp ecied and which then compute the forces and torques

necessary to meet the goals

Geometric Constraints

Barzel and Barr BB made early use of inverse dynamics for mo delling A mo del was dened as

a collection of ob jects related by geometric constraints A numb er of useful simple constraints for

mo delling were presented including pointtopoint constraints for attaching two ob jects together

pointtopath constraints for moving an ob ject along a predened path and twist constraints to

control an ob jects orientation The constraints were used to intro duce forces and torques into a

forward dynamics simulation of the mo del These constraint forces and torques act in concert to

move the mo del towards a state in which all the constraints are satised This approach blurs

somewhat the distinction b etween mo delling and motion control as it allows for the animation of

selfassembling structures if the constituent parts of the mo del initially are in a state which violates

the geometric constraints turning on dynamic simulation results in the mo del assembling itself using

the laws of Newtonian mechanics

Forsey and Wilhelms haveusedinverse dynamics to manipulate an articulated skeleton into

keyframe p ositions for a traditional kinematic interp olation system FW The Manikin system

p erformed dynamic analysis during interaction with the gure using Armstrongs recursive formu

lation for the equations of motion A p ositional goal for a b o dy part could b e sp ecied interactively

with M anik in computing the forces to push the part towards the goal This allowed manipulation of

the gure in a manner similar to inverse kinematic manipulation The imp osition of p ositional con

straints up on b o dy parts was accomplished by articially increasing the mass of constrained parts

with the system constantly computing additional forces necessary to keep the part in place as other

parts were moved Motion sequences could b e generated by storing the state of the b o dy at dierent

p oints during the dynamic analysis and later using these stored states as keyframes for kinematic

interp olation

The penaltyforce approachtaken here of converting all constraints into forces and torques which

steer the motion during dynamic analysis has its limitations The p enalty forces are often mo delled

as simulated springs and damp ers whichdeliver a force prop ortional to the velo city of the motion

This metho d of control is vulnerable to stiness in the resulting system of equations and byunde

sirable oscillations ab out constraint satisfaction p oints Cho osing appropriate spring and damping

CHAPTER APPROACHES TO FIGURE ANIMATION

co ecients for the constraints is often a matter of trialanderror

In contrast to the p enaltyforce approach a numberofformulations for the dynamic equations

of motion can include explicit constraint equations Isaacs Dynamo system ICIC combines

keyframed kinematic constraints with inverse dynamics Rather than causing the intro duction of

additional forces into the simulation the kinematic constraints are instead used to remove degrees

of freedom from the system since they implicitly sp ecify some of the accelerations in the system

The remaining accelerations for unconstrained DOFs can then b e solved for The solution metho d

ensures that reactant forces due to the keyframe constraints are intro duced into the solution for

the unconstrained DOFs This allows the kinematic constraints to sp ecify motion for some parts

of a skeleton while the other unconstrained parts react realistically to the prescrib ed motion In

cases where all parts are constrained the technique reduces to a simple keyframing approach This

approach illustrates an interesting mixture of dynamic simulation with kinematic control However

Isaacs most ambitious attempt at skeleton animation is the simulation of a traditional marionette

controlled by ro ds and strings attached to the limbs While technically impressive this example

p oints out the need for b etter metho ds of control over dynamically simulated skeleton motion

NonGeometric Constraints

Consider the inverse dynamic problem of moving a p oint mass from p osition A to p osition B in a

given time interval t There is no unique force function over the interval t which will accomplish

this the system must cho ose b etween applying a large force for a short p erio d of time or applying

a smaller force over a longer p erio d b oth metho ds mayachieve the goal of reaching the keyframed

p osition B at time t This problem is one of determining not only what is to o ccur in this case

moving from A to B but also how the motion is to o ccur A numb er of metho ds have b een prop osed

which attempt to describ e the quality of motion by considering nongeometric constraints in the

inverse dynamic solution These approaches are based on wellestablished techniques for optimizing

functions sub ject to a set of constraints

Brotman and Netravali BN prop ose an inverse dynamic approach to motion interp olation

which uses p enalty forces to enforce keyframed kinematic constraints However the solution in

corp orates an additional constraint on the energy exerted by these p enalty forces The problem is

formulated as that of solving for the set of constraint forces whichminimizes the energy exp ended

in meeting the constraints imp osed by kinematic keyframe values

Girard Gir has applied constrained optimization techniques to determine sp eed distribution

along predened limb tra jectories for articulated gures Girard notes that the choice of optimization

criteria has a signicant eect on the p erceived quality of motion Solving for a velo city prole which

CHAPTER APPROACHES TO FIGURE ANIMATION

minimizes energy exp enditure yields a relaxed swinging motion for the limb while minimizing jerk

ab out the end of the limb yields movement suggestiveofsuch goaldirected tasks as reaching for an

ob ject The establishment of additional corresp ondence s b etween optimization criteria and expressive

qualities of movement remains an op en area of research

These constrained optimization metho ds assume that the complete or partial motion paths for

limbs are known in advance and attempt to derive the b est set of forces and torques which

move the limb along the path This sidesteps the fundamental problem of synthesizing the limb

tra jectories for co ordinated movement in the rst place Witkin and Kass WKhave prop osed an

intriguing metho d of motion synthesis they call Spacetime Constraints which they demonstrated to

b e capable of synthesizing b oth the tra jectories and the timing of movements for simple articulated

gures This use of constrained optimization seems particularly promising as it seems capable

of pro ducing complex co ordinated physicallycorrect motion with very little input from a user

However the approach results in very large systems of equations whichmust b e solved and cannot

b e considered useful for interactive gure animation Ongoing research is addressing the interactivity

limitations of the metho d Coh

Badler LWZB has used a form of constraintbased inverse dynamics to synthesize the tra

jectories of limbs charged with the task of moving a load b etween two dierent p ositions The

tra jectories are computed incrementally and are constrained by measures of strength comfort and

exertion The iterative nature of the algorithm diers fundamentally from the global solution found

by optimization metho ds Instead a set of biomechanical heuristics whichareintended to mimic the

pro cess bywhich p eople move loads are used to guide the solution pro cess The metho d successfully

pro duces feasible alb eit suboptimal limb tra jectories which accomplish the task

Control Issues

The research eorts outlined ab ove are attempts to provide higher levelsofcontrol over b oth kine

matic and dynamic motion The goal is to b e able to sp ecify movements at the task level and to

have the system take care of the underlying details of pro ducing the motion Given the current

state of these eorts it seems that it will b e some time b efore the emergence of systems capable of

synthesizing motion to accomplish arbitrary tasks However there has b een some success in devel

oping sp ecial purp ose control strategies for sp ecic typ es of movements The system is resp onsible

for decomp osing highlevel task descriptions suchaswalk to the do or or reach for the cup

into lowerlevel movement primitives and for the co ordination of these primitives The lowlevel

primitives may consist of keyframes for interp olation inverse kinematic goals forward dynamic

simulations constrained inverse dynamic goals or a mixture of all these approaches

CHAPTER APPROACHES TO FIGURE ANIMATION

Zeltzer was an early prop onent of the need for highlevel control over articulated gures Zela

He describ es a control strategy for synthesizing walking sequences for a skeleton Highlevel walking

instructions are decomp osed into a set of motor control programs MCP whichdrive the motions

of individual limbs or joints The control stategy is based on a nitestate machine resp onsible

for activating and deactivating the appropriate MCPs at the appropriate times Zeltzers MCPs

consisted of kinematic jointvalues obtained from rotoscop ed human walks and thus were purely

kinematic Nevertheless the system demonstrated the usefulness of the concept

Building on Zeltzers work Bruderlin BCdevelop ed a similar hierarchical control strategy

for generating bip ed walking sequences but incorp orated dynamic simulation to derive leg motion

rather than relying on rotoscop ed data The user is able to instruct a skeleton to walk at a particular

sp eed and is able to sp ecify b oth desired step frequency and step length These instructions are

decomp osed into dynamicallybased lowlevel MCPs whichdrive the motion of an abstract kneeless

pair of legs The MCPs essentially p erform dynamic interp olation of a set of kinematic keyframes

for the leg movements during the walk cycle The kinematic keyframe values and spacing are derived

from the input parameters combined with knowledge ab out human lo comotion patterns gleaned from

the biomechanics literature The forces and torques driving the motion of the simplied walking

mechanism are iteratively adjusted until the keyframed joint angles are achieved at the correct

times A purely kinematic overlayoftheskeletons jointed legs onto the underlying mechanism is

then p erformed The algorithm is able to pro duce a wide range of realistic walking sequences and

is a true hybrid of b oth dynamic and kinematic motion control The decision to use a simplied

dynamic mo del sp ecically tuned for walking seems sound the resulting system of equations is small

relatively stable and inexp ensivetosolve A similar approach has b een used by this author to build

a jumping algorithm based on the simulation of a simple underlying massandspring mo del

Unfortunately the highlevel control provided by algorithms of this nature come at the exp ense

of generality eachcontrol strategy must b e tuned for a sp ecic movement But developing sucha

control strategy is dicult Deriving the equations for simulating the dynamics of the underlying

mechanism requires some mathematical sophistication In the absence of an inverse kinematic algo

rithm Bruderlins metho d of mapping the motion of the underlying dynamic mo del to the motion

of the skeleton can p ose problems to the implementor To a large extent the success of the ab ove

control strategies is due to the predictable rep etitive nature of lo comotion Developing highlevel

control strategies for arbitrary movement sequences still seems a distant goal

CHAPTER APPROACHES TO FIGURE ANIMATION

Summary

What has hop efully emerged from the discussion so far is that no one technique has emerged as

a clear winner A successful gure animation system is likely to incorp orate all of the techniques

discussed ab ove to some degree Pure dynamics applied to gure animation seems to raise as many

problems as it addresses unless it is conned to sp ecic movementcontrol strategies The research

into automatic motion synthesis from highlevel constraints while promising is still at to o early a

stage to b e considered useful For the time b eing designing arbitrary movement sequences remains

in the hands of the animator

A simple interactivekeyframe editor in the hands of a user who understands howthebodymoves

and has some patience can pro duce some surprisingly go o d animation sequences even for gures as

complex as the human form Dynamic simulation or algorithmic motion mo dels while useful in some

contexts will only b e appreciated if they can alleviate some of the work involved in interactively

handcrafting new movement sequences and it can b e argued that given the current state of research

in these areas this is not generally the case The most promising interactive techniques reviewed in

this chapter are those based on the use of inverse kinematics whichprovide a level of control higher

than simple forward kinematics yet still leave the user with complete control over the animation

In the remaining chapters the use of inverse kinematics to complement an existing interactive

keyframe editor is explored The goal is to address the limitations of a simple forward kinematic

approach by providing a set of to ols which supp ort direct manipulation of kinematic chains within

a gure and the imp osition of simple geometric constraints which are maintained during keyframe

creation and interp olation Along the waywe identify twoinverse kinematic algorithms which dier

from those previously adopted for computer graphics and describ e their suitability to the problem

Chapter

Inverse Kinematics

The inverse kinematic problem has b een studied extensively in the rob otics literature which remains

the b est source of information on the sub ject In this chapter we formally state the problem and

review the most common approaches to solving it Previous applications of these approaches to

computer graphics are also describ ed

The Inverse KinematicProblem

Section showed that a skeleton can b e mo delled as a hierarchical collection of rigid ob jects

connected by joints We will refer to a of segments within a skeleton as a manipulator

and will assume that the joints connecting segments within this chain are revolute joints rotating

ab out a single axis One end of the manipulator the base is xed and cannot move the distal

end of the chain is free to move The endeector is emb edded in the co ordinate frame of the most

distal jointinthechain the endeector p osition is a p oint within this frame and the endeector

orientation refers to the orientation of the frame itself

At eachjointinthechain a jointvariable determines a transformation M between the two adjacent

co ordinate frames sharing the joint The transformation M at a rotation joint i is a concatenation

i

of a translation and a rotation b oth of which are relative to the co ordinate frame of joint is parent

That is

M Tx y z R

i i i i i

where Tx y z is the matrix that translates by the oset of joint i from its parent joint i and

i i i

R is the matrix that rotates by ab out joint is rotation axis

i i

CHAPTER INVERSE KINEMATICS

The relationship b etween anytwo co ordinate systems i and j in the chain is found by concate

nating the transformations at the joints encountered during a traversal from joint i to joint j

j

M M M M M

i i i j  j

So the p osition and orientation of the endeector with resp ect to the base frame is found by simply

concatenating the transformations at each joint in the manipulator

Given a vector q of known jointvariables then the forward kinematic problem of computing

the p osition and orientation vector x of the endeector is a simple matter of matrix concatenation

and has the form

x f q

But if the goal is to place the endeector at a sp ecied p osition and orientation x then determining

the appropriate jointvariable vector q to achieve the goal requires a solution to the inverse of



q f x

Solving this inverse kinematic problem is not so simple The function f is nonlinear and while there

is a unique mapping from q to x in equation the same cannot b e said for the inverse mapping

of there maybemany qs for a particular x The most direct approach for solving the problem

would b e to obtain a closedform solution to But closedform solutions can only b e derived

for a restricted set of manipulators with sp ecic characteristics and even these result in a set of

nonlinear equations to b e solvedPau A general analytic solution for arbitrary manipulators

do es not exist instead the problem must b e solved with numerical metho ds for solving systems of

nonlinear equations The most common solution metho ds are based on either matrix inversion or

optimization techniques

Resolved Motion Rate Control

Since the nonlinear nature of equation makes it dicult to solve a natural approachisto

linearize the problem ab out the current manipulator conguration then the relationship b etween

jointvelo cities and the velo city of the endeector is

x Jqq

The linear relationship is given bytheJacobian matrix

f

J

q

CHAPTER INVERSE KINEMATICS

whichmapschanges in the jointvariables q to changes in the endeector p osition and orientation

x J is an m n matrix where n is the numb er of jointvariables and m is the dimension of the

endeector vector xwhich is usually either for a simple p ositioning task or for a more general

p ositionandorientation task The ith column of J represents the incremental change in the p osition

and orientation of the endeector resulting from an incremental change in the jointvariable q

i

Inverting the relationship of provides the basis for resolved motion rate control



q J qx

If the inverse of J is known we can compute incremental changes in the jointvariables which pro duce

a desired incremental change in the endeector p osition and orientation

A simple iterativescheme for solving the inverse kinematic problem can b e based on equation

At each iteration a desired x can b e computed from the current and desired endeector

p ositions The jointvelo cities q can then b e computed using the Jacobian inverse and integrated

once to nd a new joint state vector q The pro cedure rep eats until the endeector has reached

the desired goal Note that since the linear relationship represented by J is only valid for small

p erturbations in the manipulator conguration Jqmust b e recomputed at each iteration A

pro cedure for eciently computing the Jacobian is presented in Section

Of course this scheme assumes that the Jacobian matrix is invertible that J is b oth square and

nonsingular This assumption is not in general a valid one Diculties arise when a manipulator

is redundant or when it passes through or near a singular conguration

Redundancy

A manipulator is considered kinematical ly redundant when it p ossesses more degrees of freedom than

are required to sp ecify a goal for the endeector For example consider the simple D case in

Figure The manipulator p ossesses degrees of freedom the rotation angles at each joint For

a simple p ositioning task the goal is to place the endeector the tip of the distal link of the chain

at some p ointx y As the gure shows for a given goal x y there is no unique solution each

of the congurations shown will place the tip at the goal p osition The manipulator is therefore

redundant for this D p ositioning task

In general p ositioning an ob ject in Cartesian space requires the sp ecication of six co ordinates

three for lo cation and three for orientation Therefore any manipulator p ossessing more than six

degrees of freedom is redundant for the general Dspace p ositioning task and there is no unique

CHAPTER INVERSE KINEMATICS

(x,y)

θ3

θ2

θ1

Figure Three congurations of a D redundant manipulator

set of jointvalues solving the inverse kinematic problem

For a redundant manipulator the Jacobian matrix has fewer rows than columns and cannot

be inverted In this case equation is underdetermined and there are an innite number of

 y

solutions from whichtocho ose If J in is replaced by some generalized inverse J then a

useful solution to the underdetermined problem can b e found One such generalized inverse is the

Mo orePenrose pseudoinverse Gre GM It can b e shown KH that this pseudoinverse is

optimal in the sense that it yields solutions with a minimum Euclidean norm for cases in which

is underdetermined mn and that in cases in which the system is overdetermined mna

leastsquares solution is obtained In practice these prop erties ensure that joints move as little as

p ossible to match the desired endeector velo city as closely as p ossible

Exploiting Redundancy

Since a redundant manipulator can satisfy a p ositioning task in anynumber of ways it is often useful

to consider exploiting the redundancy in an attempt to satisfy some secondary criteria This can b e

accomplished by incorp orating an additional term into equation

y y

q J x I J JrH q

The function H q is a measure of some criterion to b e minimized sub ject to satisfying the

y

primary p ositioning task The other comp onent I J J is a pro jection op erator which selects

those comp onents of the gradientvector rH q which lie in the set of homogeneous solutions to

A homogeneous solution to is a set of jointvelo cities q which do es not change the endeector

CHAPTER INVERSE KINEMATICS

y

singular direction

x

Figure A manipulator in a singular conguration

p osition ie for which x In eect then the rst term of the general equation selects a

jointvelo cityvector which pro duces the desired change in the endeector p osition while the second

term exploits the redundancy of the manipulator byvarying these jointvelo cities in suchawaythat

H q is minimized without disturbing the endeector p osition By exploiting redundancy in this

manner secondary goals have b een created to avoid collisions with obstacles Bai to exploit joint

range availability GM KH and even to maintain manipulator dexteritybyavoiding kinematic

singularities SS

Singularities

The pseudoinverse metho d outlined ab oveprovides useful solutions to when the Jacobian matrix

J is rectangular and therefore not invertible But wemust also consider the case where J is not

invertible b ecause it is singular A matrix is said to singular when twoormorerows are linearly

dep endent and a manipulator is said to b e in a singular conguration when the Jacobian b ecomes

singular Figure depicts a simple example of a jointed manipulator in a singular conguration

In this example an incremental change to any of the joint angles will result in approximately the

same movement of the endeector in the y direction no combination of jointvelo cities will pro duce



an endeector velo city in the singular ie x direction The Jacobian matrix computed for this

conguration will contain zero es in the rst row and is therefore singular and cannot b e inverted

The pseudoinverse can still b e applied to obtain a useful solution when J is singular However

as a manipulator passes through a singular conguration there are discontinuities in elements of



Although intuitively it might seem obvious that a rotation at anyjoint will result in at least some movementin

the x direction recall that equation deals with instantaneous quantities

CHAPTER INVERSE KINEMATICS

the computed pseudoinverse due to the change in rank of J at the singular conguration Mac

Furthermore as the manipulator approaches this conguration the pseudoinverse tends to pro duce

large jointvelo cities Numerical integration techniques typically do not handle suchderivativespikes

well The problem manifests itself as a tendency of the manipulator to oscillate wildly around the

singular conguration So while the pseudoinverse is able to provide a usable solution at a singular

conguration its principal drawback is that it do es not provide a continuous stable solution around

singularities While industrial rob otic manipulators may b e programmed to follow tra jectories which

explicitly avoid singular congurations for just this reason this is not really an option for an algorithm

to b e used for computer animation

The Singular Value Decomp osition

Numerical instabilities near singular congurations are a ma jor problem which raises the question

of whether there is a means of detecting and correcting the problem Probably the most useful to ol

for analyzing the Jacobian matrix is the singular value decomposition SVD PFTV The SVD

theorem states that any matrix can b e written as the pro duct of three nonunique matrices

T

J UDV

The pro cedure for computing the SVD of a matrix is b eyond the scop e of this discussion but is

well known and describ ed elsewhere PFTV MK The signicance of the SVD lies in the

interpretation of each of the three matrices U Dand V

For an m n matrix J D is an n n diagonal matrix with nonnegative diagonal elements known

as singular values If one or more of these diagonal elements is zero then the original matrix is itself

singular Even b etter the ratio of the largest singular value to the smallest one the condition number

of the matrix is a measure of how il lconditioned the matrix J is When the condition number is



to o large then the matrix is illconditioned It is this illconditioning that is resp onsible for the

large jointvelo cities generated by the pseudoinverse near a singular conguration Mac

The other matrices U and V are orthonormal bases for the range and nul l space resp ectivelyof

JFor any zero singular values in D the corresp onding columns in V form a set of orthogonal vectors

which span the space of homogeneous solutions to equation ie the set of jointvelo cities which

will not move the endeector Likewise the nonzero singular values have corresp onding columns

in U which span the space of solutions which wil l move the endeector We will refer to these basis

matrices again when discussing constraints in Chapter

While the SVD provides a means for detecting illconditioning in the Jacobian matrix it do es



ie its recipro cal approaches machine precision limits

CHAPTER INVERSE KINEMATICS

not in itself provide a way for dealing with the illconditioning Nevertheless it is useful as an

analytical to ol Klein and Huang KH have used singular value analysis to demonstrate the

optimal prop erties of the Mo orePenrose pseudoinverse Maciejewski Mac has used the SVD to

illustrate the discontinuity that o ccurs in the pseudoinverse at a singularity and to develop a strategy

for damping the high velo cities which o ccur near singular congurations But the cost of computing



the SVD O n logn for an n n matrix adds signicantly to the p eriteration cost of any control

algorithm so it is often not feasible to incorp orate it into online control schemes Maciejewski

MK do es describ e a metho d of incrementally up dating the SVD from one iteration to the next



which reduces the cost to O n p er iteration but this requires careful implementation to reduce

cumulative errors and the cost is still high enough to deter its use

OptimizationBased Metho ds

A fundamentally dierent approach to solving the inverse kinematic problem avoids the matrix

inversion step altogether The idea is to cast the basic problem of equation as a minimization

problem then apply standard iterative nonlinear optimization techniques to obtain a solution

As an example consider the problem of p ositioning the endeector x at a goal p osition p The

distance from the current p osition xq to the goal p osition p serves as an error measurement



E qp xq

By varying the jointanglevector q the endeector either moves awayfromp increasing the error

measure or towards p decreasing the error Clearly the intent is to nd a jointvector q which

minimizes the error measure Limits on the joint ranges of motion provide additional constraints on

the individual jointvalues q Formallywe need to nd a vector q whichsolves the problem

i

minimize E q

sub ject to l q u i n

i i i

where l and u are the lower and upp er b ounds resp ectivelyonthevalue of jointvariable q For

i i i

this example the error measure E is just the distance formula but the approach generalizes to more

complex goals for the endeector since E qcanbeany arbitrary function of the jointvector q

This formulation is a classic nonlinear constrained optimization problem which can b e solved by

anumberofstandardnumerical metho ds A go o d intro duction to the topic of optimization and the

issues to consider in selecting a solution metho d is presented by Press et al PFTV Gill et al

survey a numb er of practical optimization techniques GMW The eectiveness of any particular

metho d is usually determined bythecharacteristics of the ob jective function in this case E and

CHAPTER INVERSE KINEMATICS

of the constraints For our example a solver for minimizing smo oth quadratic functions sub ject to

linear inequality constraints would b e an appropriate choice

Atypical solver will iteratively converge from an initial state towards a solution state at each

step p erturbing the state variables slightly and reevaluating the ob jective function to evaluate its



progress Some solvers maymake use of the gradient of the ob jective function rE to suggest new

directions in which to p erturb the state vector This may increase the computation p er iteration

but pay o in an improved rate of convergence toward a solution

Once selected the optimizer can b e thoughtofasablackbox which is fed as inputs the

current jointvector q a function for evaluating the ob jective function E and p ossibly a function to

evaluate rE as well The output from the blackbox is a new jointvector which minimizes the

error measure E and therefore solves the inverse kinematic problem

Evaluation

While conceptually simple there are some practical diculties in implementing this approach Con

strained optimization of arbitrary nonlinear functions is still an op en research area whichhas

pro duced a collection of numerical metho ds whichmayormaynotwork for a particular problem

Selecting an appropriate solver and determining what the problem is when it fails to work can b e

dicult A solver maywork well for one particular typ e of problem but fail miserably on others

Furthermore there is no guaranteethatasolver will nd the true global minimum for a con

strained optimization problem Since most solvers converge on a solution by iteratively moving

downhill along the ob jective function surface they cannot distinguish b etween a true global min

imum and merely a lo cal minimum of the surface In practical terms this implies that the solver

may return a jointvector q whichdoesnotprovide the b est solution and the user mayhaveto

somehow suggest a more appropriate conguration from which to restart the iterative searchfora

b etter solution

For interactive computer graphics the demands of interactivity place some additional demands

on the solver Interactive dragging of a manipulator involves rep eatedly sampling the cursor lo cation

onscreen to determine a goal p osition for the endeector then invoking the solver with the current

manipulator state as the initial guess at a solution Tomaintain the illusion of continuous interactive



control during dragging the screen needs to b e up dated at a reasonable refresh rate If the solver

blackbox cannot pro duce a solution quickly enough to provide go o d feedback to a user dragging



each step should decrease or increase when maximizing the ob jective function



framessec for example is a minimal goal to aim for during interaction

CHAPTER INVERSE KINEMATICS

a manipulator onscreen then the interface will feel sluggish and unresp onsive A related problem is

that optimizers typically work b est when the initial state is close to the nal solution But during

interactive dragging the cursor may get ahead of the solver so that on the next invo cation of the

solver the state of the manipulator b eing dragged is not close to the next solution As a result

the solver may drastically alter the state while solving for the next solution the result is that the

manipulator will seem to suddenly jump to a completely dierent conguration in order to satisfy the

goal This is of course quite disconcerting to a user One option is to interrupt the solver and obtain

its current state in order to refresh the screen However since the solver is free to try dierent paths

as it feels its waytowards the closest minimum there is no guarantee that intermediate solutions

will b e suitable for refreshing the screen

Applications to Computer Graphics

Each of the approaches ab ovehave previously b een adopted for computer graphics The pseudoin

verse metho d was intro duced to the computer graphics communityby Girard in GM for his

PODA gait generator Girard exploited the redundancy of animal limbs in an attempt to minimize

joint limit violations using the pro jection op erator metho d of Section The inverse kinematic

capabilities of Sims gait controller SZ and Chadwicks Critter system are also based on the tech

nique None of these eorts suggest sp ecic solutions to the problems the metho d exhibits around

singularities so it seems reasonable to assume that each of these systems will not p erform well near

singular congurations

Badler and Zhao CP ZB have adopted the second approach for the Jack system applying

avariablemetric optimization pro cedure to provide interactivecontrol over an articulated gures

p osture Joint range limits are presented as constraints to the optimizer and a number of objective

functions for simple geometric constraints are develop ed These include for example p ointto

p oint constraints p ointtoplane constraints orientation constraints and others of a similar nature

Simultaneous constraints on multiple b o dy parts can b e imp osed by simply summing the individual

ob jective functions for each constrained part into an aggregate ob jective function to b e minimized

This p ermits inverse kinematic manipulation of a gure while maintaining a set of constraints on the

body which is a useful interactive to ol Phillips PB even describ es an ob jective function which

attempts to balance the Jack gure providing a go o d example of how the metho d can b e extended

to handle arbitrarily complex nongeometric goals While Jacks capabilities are impressive and do

p erform at interactive sp eeds it works b est on highp erformance graphics workstations and even

then the imp osition of just a few constraints noticeably degrades the resp onse time of the interface

Badler admits to p erio dically refreshing the screen with intermediate solutions from the optimizer

to retain interactivityeven after stating CP that only the nal solution is useful and that the

CHAPTER INVERSE KINEMATICS

intermediate solutions should not b e considered motion

A simpler conjugategradientminimization technique is also used by Alt and Nicolas AN

providing inverse kinematic manipulation and animation of limbs by animating p osition goals over

time In constrast to Badlers approach they p erform unconstrained minimization electing to enforce

joint limits by simply clamping solution values to the jointvariable b ounding values

It is also worth noting that some early attempts by Badler to solve the inverse kinematic problem

were based on simple heuristic algorithms KB BMW These metho ds do not app ear to have

gained wide acceptance Badler has since abandoned these approaches in favour of the optimization

metho d describ ed ab ove

Chapter

Ecient Algorithms for Direct

Manipulation

One of our goals is to provide direct manipulation of an articulated gure and for this the metho ds

of the previous chapter have some shortcomings The pseudoinverse is exp ensive to compute and is

sub ject to numerical instabilities around singular congurations Badlers optimization metho d is

considerably more p owerful but its p erformance leaves something to b e desired and it maybemore

suitable for solving problems oline than for interactive manipulation The continuing app earance

of inverse kinematic pap ers in the rob otics literature seems to suggest that neither of these metho ds

is entirely satisfactory

In this chapter a pair of simple inverse kinematic algorithms are presented as alternatives to

those adopted for computer graphics in the past Both are relatively simple to implement are

numerically stable and are ecient enough to provide reasonable interactive p erformance on even

lowp erformance machines Each metho d exhibits a dierent b ehaviour than the other and it is

suggested that they mightwork well together as complementary manipulation to ols

A Simplied Dynamic Mo del

The rst metho d b elongs to a class of solutions to the inverse kinematic problem that are based

on the transp ose of the Jacobian matrix Like the resolvedmotion rate control of section it

relies on the linear relationship b etween endeector and jointvelo cities Unlike this other approach

however no inversion of the Jacobian matrix is required which signicantly reduces the cost of each

iteration Consequently the metho d has b een advo cated for the online dynamic control of rob otic

manipulators

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

The metho d was intro duced in byWolovich and Elliot WE who describ ed a dynamic

control scheme based on the use of the Jacobian transp ose and showed that it could provide stable

tracking of an arbitrary endeector tra jectory Sciavicco and Siciliano SS applied the metho d to

redundant manipulators and showed that the redundant degrees of freedom could b e used to satisfy

b oth obstacle avoidance constraints and constraints on joint ranges of motion Das et al DSS

develop a more general technique for satisfying secondary criteria similar to the pseudoinversebased

metho d discussed in section and compare the metho d to a minimization algorithm based on

Newtons metho d Novakovic and Nemec NNshow that the metho d can b e used to generate

either jointvelo cities or joint accelerations We are interested here in pro ducing jointvelo cities to

drive the joint parameters

The Jacobian Transp ose Metho d

Consider a comp osite force F applied to the tip of a real manipulator consisting of b oth a pull f

in some direction and a twist torque m ab out some axis

T

F f f f m m m

x y z x y z

This external force F applied to the endeector will result in internal torques and forces at the

manipulator joints Under the simplifying assumption of the principle of virtual work Pau the

relationship b etween F and the vector of internal generalized forces is

T

J F

This suggests a rather simple iterative metho d for forcing an endeector to track a timevarying

tra jectory x t If the current endeector p osition is given by x t then the error measure

d c

etx t x t

d c

can b e thoughtofasaforce f pulling the endeector toward the desired tra jectory p oint x t

d

Equation can then b e used to transform this external force to a generalized force on eachof

the jointvariables If we are interested in the dynamic b ehaviour of the manipulator in reaction to

the applied force then can b e considered the vector of jointvariable accelerations q Since weare

not particularly interested in accurate dynamic simulation of the manipulator it suces to think of

as the vector of joint displacements velo cities so that

T

q J F

Once q is computed a single integration step yields a new vector q whichmoves the endeector

towards x t This pro cedure rep eats until the endeector reaches the desired p osition or some

d other stopping criterion is met

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

In eect at each iteration we treat the vector e as an elastic force pulling on the end of a dy

namically simple manipulator and adopt a heuristic simulation mo del in which force is prop ortional

to velo city rather than acceleration Using f mv as the governing equation of motion eliminates

the eects of inertia things stop moving as so on as the applied external force disapp ears Gleicher

GW has used the relationship combined with this simple equation of motion to provide

direct manipulationover a variety of geometric ob jects The term dierential manipulation has b een

coined to describ e this and the discussion ab oveshows that we can add articulated structures to

the set of ob jects amenable to direct manipulation of this typ e

Equation is in fact equivalenttothewellknown steepest descent metho d of minimization

which is generally regarded as having p o or convergence prop erties PFTV This raises the question

of why one would adopt this approachover other minimization metho ds with sup erior convergence

prop erties The intenthereistoprovide direct manipulation of articulated gures in an interac

tive setting rather than to solveinverse kinematic problems oline and for this typ e of online

application the Jacobian transp ose metho d has some app ealing characteristics

The rst of these is that no matrix inversion is required Using the transp ose of the Jacobian not

only avoids problems with matrix singularities but also means that at each iteration only forward

kinematic calculations are required Since these can b e computed quickly a tra jectory which is b eing

sp ecied interactively can b e sampled frequentlyproviding resp onsive feedback to the user

More imp ortantly since the solution is based on a physical mo del alb eit a simplied one the

solutions obtained in successive iterations are b oth predictable and intuitive That is the manipula

tor will resp ond as though the user were in fact pulling on the endeector with an elastic band In

contrast other minimization techniques may generate successive solutions which are quite dierent

from each other particularly when the manipulator is redundant and there are an innite number of

acceptable solutions When using a physicallybased mo del successive congurations are implicitly

constrained to b e close to each other the pulling metaphor uniquely sp ecies a path from one

solution to the next For interactive applications this advantage arguably outweighs the p otentially

poor convergence prop erties of the metho d

One drawback is that since the metho d is based on the Jacobian it is not entirely immune to

kinematic singularities since the matrix will b e inherently illconditioned Near a singularity high

jointvelo cities can result in oscillations ab out the singular conguration This typically o ccurs

only when trying to fully extend the manipulator The problem can b e overcome by either using a

smaller integration stepsize using an integration metho d with an adaptive stepsize or by clamping

jointvelo cities to some maximum b ounds b efore integrating Since the basic solution step is so quick

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

F q’ q x − T d J ∫

x c

f(q)

Figure Interactivecontrol lo op mo del for Jacobian transp ose metho d

to compute often it is p ossible to simply use a smaller integration stepsize while retaining go o d

interactive p erformance

Implementation Details

In comparison with the metho ds of the previous chapter implementation of the Jacobian transp ose

metho d is relatively simple Each iteration involves computing a force applied to the endeector

computing the Jacobian matrix for the current conguration and computing the resulting joint

velo cities The jointvelo cities are integrated once to obtain a new conguration for the manipulator

Figure depicts the basic control lo op

Forces applied to the endeector are a result of interaction with the user and will b e discussed

in Chapter For nowwe assume that the applied force is known and concentrate on the other

asp ects of the metho d

Computing the Jacobian

Ateach iteration we need to compute the Jacobian matrix J whose columns convey how the end

eector frame moves in the world frame as the individual jointvariables change Given a vector of

jointvariables q the endeector frame is sp ecied by a p osition Pq and orientation Oq The

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

Jacobian column entry for the ith jointis

P

x

P

y

P

z

J

i

O

x

O

y

O

z

where the derivative op erator is with resp ect to q These entries can either b e computed directly

i

in the world frame or may b e computed in the lo cal joint frame b efore b eing transformed to the

world frame Here we describ e b oth pro cedures for eciently computing the full Jacobian matrix

for a manipulator

Each joint i in the manipulator either translates along or rotates ab out one of the principal axes

u in the lo cal joint frame At eachjoint M is the matrix which transforms the lo cal joint frame

i i

to the world frame Supp ose that axis represents the normalized transformation of the lo cal joint

i

axis in the world co ordinate frame

axis u M

i i i

Then for a translating joint the corresp onding Jacobian entry is

T

axis

i

J

i

and for a rotating joint the Jacobian entry is

T

p j axis

i i

J

i

T

axis

i

where p denotes the p osition of the endeector and j is the p osition of joint i in the world

i

Lo cal vs World Frame

In practice it is exp ected that the manipulator itself will b e a subtree emb edded within a gen

eral transformation hierarchy within an animation system This subtree may contain arbitrary

transformation no des in addition to those corresp onding to the manipulator joints These additional

transformation no des do not constitute degrees of freedom and are not sub ject to up dates byinverse

kinematic manipulation But since they transform some of the lo cal joint frames of the manipulator

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

they must b e considered The signicance of this is that the typ es of no de present in the hierarchy

determine whether the Jacobian entries can b e computed in the global world frame or whether they

must b e computed in the lo cal joint frames at a slightly higher cost

If the transformation hierarchyinwhich the manipulator is emb edded consists only of trans

lations and rotations then the Jacobian entries can b e computed directly in the world frame A

concatenation of only rotations and translations guarantees that the upp erleft p ortion of the

transformation M at a joint will b e orthogonal This implies that the rst three rows of M are the

i i

transformed principal axes of the joints lo cal frame and so axis can b e extracted directly from M

i i

Similarly the joint p osition j can b e extracted from the fourth rowofM Since the transformation

i i

matrices M need to b e computed to display the structure caching them at each joint during display

reduces the calculation of each Jacobian entry J to just a vector subtraction and a cross pro duct

i

op eration



If scaling transformations are p ermitted in the hierarchy then there is no guarantee that M is

i

orthogonal Arbitrary nonuniform scaling transformations within the manipulator may result in an

upp erleft p ortion of M which do es not represent a pure rotation In this case axis cannot

i i

b e extracted from M Instead the vector quantities in and must b e computed in the

i

lo cal joint frame then transformed to the world frame by M To compute in the lo cal frame

i

the endeector p osition p needs to b e transformed from the world frame to the joint frame and so



the inverse transformation M is required

i



For eciency M can b e computed incrementally while traversing the hierarchy Using the

i

matrix identity

  

AB B A

and the fact that the inverse transformation for a single joint is simple to determine the matrix

inversion step at a joint can b e replaced with a cheap er matrix multiplication Further simplications

result from noting that the world frame joint p osition j corresp onds to the origin of the lo cal joint

i

frame and that the joint axis is simply one of the principal axes making the cross pro duct step

trivial

The Jacobian matrix for the endeector frame then can b e assembled byasingletraversal of

the hierarchyAteach joint i either or is computed to obtain the column J and these

i

calculations may b e p erformed either in the global frame or the lo cal joint frame



as they are likely to b e in any computer animation system

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

Scaling Considerations

In practice the simple mo del expressed in equation do es not work well for the general case One

problem is that the resp onse of the manipulator to an applied force dep ends on the choice of scale

for the jointvariables If rotation jointvalues represent degrees for example a given applied force

will result in a dierent resp onse than if the rotations were measured in radians The overall scale

of the world must also b e considered since this will govern the magnitude of the vector quantities

in These factors must b e taken into account if the technique is to provide uniform b ehaviour

over a range of scales

To provide stable scaleinvariant b ehaviour we can mo dify equation to comp ensate for these

factors

T

q KJ F

where K is simply a constant scaling matrix whose ith diagonal entry K actsasaweighting factor

i

for the computed jointvelo city q The following term is suggested

i

K

i

w

i i

where the w term is prop ortional to the length of link i and is intended to oset the eects of

i

the overall scale of the world The term is a weighting factor for joint i which can b e thoughtof

i

as a parameter controlling the resp onsiveness of the joint to an applied force This parameter might

b e made available to the user to allow editing of a joints p erceived stiness

Although K is sp ecied here to b e a constant gain matrix it is p ossible to compute a timevarying

gain matrix Ktsoastocontrol the rate of convergence of a tracking algorithm based on

WE NN However given that the additional computation required is considerable relativeto

the basic iteration cost of there is some incentive to just consider K constant

Integration

Once the jointvelo cities q are known a single integration step is required to generate the new state of

the manipulator q for the next iteration The simplest metho d is to take a single Euler integration

step

q q hq

for some integration stepsize h This metho d is notoriously inaccurate since it assumes constant

velo city across the step interval As the interval width h is made smaller the accuracy improves

but overall p erformance suers Nevertheless the Euler metho d with a small h may b e adequate for

manipulators with just a few joints

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

The Euler metho d can b e replaced with a more robust metho d An adaptive stepsize Runge

Kutta metho d can improve p erformance taking small steps when required but allowing the stepsize

h to increase when small steps are unnecessary Adaptiveintegration implementations are available

in the public domain PFTV and their use is recommended

Joint Limits

Since the formulation do es not explicitly include constraints on the jointvariable values joint limits

are enforced by clamping q to upp er and lower b ounds after the integration step Although this is

i not recommended in most optimization texts in practice it app ears to b e adequate

force

goal

Figure A case not handled well with the Jacobian transp ose metho d Pulling inwards on the

tip of the manipulator on the left will not pro duce an exp ected conguration like the one shown on

the right

A Complementary Heuristic Approach

The forcebased approach of the preceding metho d has some app ealing characteristics for interactive

manipulation There are some cases however for which the metho d do es not p erform well Figure

illustrates one such case On the left is a manipulator in a singular conguration with a new

desired p osition for the tip shown as a black dot The conguration on the rightshows a reasonable

solution to this inverse kinematic problem and probably reects what a user exp ects to get by

dragging the tip in towards the goal However as the user attempts to drag the tip inwards the

applied force exerted on the tip p oints straightdown the singular direction it is in eect cancelled

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

out and the tip will not move This b ehaviour is reasonable given the physical analogy of the

Jacobian transp ose metho d but may not really match the users exp ectation

An alternativeinverse kinematic algorithm is presented here suitable for interactive p ositioning

and capable of providing reasonable b ehaviour in cases such as that in Figure It is based on a

heuristic metho d which has b een prop osed to quickly nd an initial feasible solution for a standard

minimizationbased algorithm WC However it can stand on its own as an inverse kinematic

p ositioning to ol Like the Jacobian transp ose metho d the technique is ecient simple and immune

to problems with singularities However its b ehaviour is quite dierent from that exhibited by the

previous algorithm and it is suggested here as a complement to rather than a replacement for the

Jacobian transp ose metho d

The CyclicCo ordinate DescentMethod

The cycliccoordinate descent CCD metho d is an iterative heuristic searchtechnique which at

tempts to minimize p osition and orientation errors byvarying one jointvariable at a time Each

iteration involves a single traversal of the manipulator from the most distal link inward towards the

manipulator base Each jointvariable q is mo died in turn to minimize an ob jective function The

i

minimization problem at each joint is simple enough that an analytic solution can b e formulated so

each iteration can b e p erformed quickly

As a solution is obtained at eachjoint the endeector p osition and orientation are up dated

immediately to reect the change Thus the minimization problem to b e solved at any particular joint

incorp orates the changes made to more distal joints during the current iteration This diers from

the previously describ ed metho d which in eect determines the changes to eachjointsimultaneously



Supp ose that the current endeector p osition is

P x y z

c c c c

and that the currentorientation of the endeector is sp ecied by the three orthonormal rows of the

rotation matrix O

c

u

c

O

u

c

c

u

c



ie although the changes are computed sequentially the state of the manipulator remains constant during the iteration

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

u 2d P ic u 3d u φ P 1d d P j id

i

Figure Example CCD iteration step for rotation jointi

The endeector can b e placed as close as p ossible to some desired p osition P and orientation O

d d

by nding a jointvector q whichminimizes the error measure

E qE q E q

p o

which is just the sum of a p ositional error measure



E qkP P k

p d c

and an orientation error measure



X



E q u u

o jd jc

j 

The metho d pro ceeds by considering one joint at a time from the tip to the base Each jointvariable

q is mo died to minimize equation b efore pro ceeding to the next joint i At each joint

i

minimizing b ecomes a simple onedimensional optimization problem since only q is allowed

i

to change while the other elements of q are xed Since joint i is either a rotation or a translation

joint there are two cases to b e considered

Rotation Joint

Figure illustrates the situation for rotation joint i during an iteration The vector P is the

ic

vector from the joint p osition j to the current endeector p osition and P is the vector from j to

i id i

the desired endeector p osition We are free to rotate the vector P ab out the world space joint

ic

axis axis by some amount This rotated vector is

i



P R P

ic

ic

axis i

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION



As varies P sweeps out a circle centered at j The p oint on this circle closest to the desired

i

ic

p osition P is the p ointatwhich the circle would intersect the line along P so the b est wecando

d id



byvarying jointvariable j alone is to align the twovectors P andP This implies that we

i id

ic

seek a value for which maximizes the expression



g P P

 id

ic

Reasoning along similar lines an orientation error is b est corrected by making sure that also

maximizes the expression



X



g u u

 jd

jc

j 

Combining b oth and gives an aggregate ob jective function to b e maximized for joint i

g w g w g

p  o 

Here w and w are arbitrary p osition and orientation weighting factors resp ectively and are intro

p o

duced to play a role similar to that of the gain matrix K of the Jacobian transp ose metho d The

following ad hoc values for these factors are suggested WC

w

o

w

p

Here is a scaling factor inversely prop ortional to the overall world scale W

kW

and is required to make the algorithms b ehaviour scaleinvariant The factor is an arbitrary weight

which dep ends on the conguration of the manipulator Wang WC suggests without justication

that

minkP k kP k

id ic

maxkP k kP k

id ic

which seems to b e adequate in practice

With some algebraic manipulation the ob jective function to b e maximized at joint i can b e

reduced to

g k cosk cos k sin

  

with the constant co ecients k k and k given by

  



X

k w P axis P axis w u axis u axis

 p id i ic i o jd i jc i

j 

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION



X

k w P P w u u

 p id ic o jd jc

j 



X

k axis w P P w u u

 i p ic id o jc jd

j 

From elementary calculus we know that the ob jective function is maximized over the interval

when its rst derivative is zero and its second derivative is negative The rst condition

k k sin k cos

  

implies that

k





tan

k k

 



However since tan is p erio dic which determines a candidate value in the range

c c

 

there are p otentially two other candidate values to consider and Of these candidate

c c

values those which lie in the interval xand which pass the second derivative test are

maximizing values for the ob jective function If there is more than one of these the ob jective

function is evaluated with each to determine which yields the true maximumOnce has b een

uniquely determined in this way it is added to the currentjointvalue q At this p ointwecan

i

intro duce an arbitrary weighting factor w w whichcontrols the p erceived stiness of

i i

the joint so that the up date b ecomes

q q w

i i i

The endeector frame is then rotated to reect this change and the iteration continues on to the

next joint i using the up dated endeector

Translation Joint

If joint i is a translation joint then it can only reduce the p osition error It is not dicult to

showWC that the b est that can b e done to minimize the p osition error is to change the joint

displacementby

P P axis

id ic i

This is weighted by w as b efore and added to the currentvalue of jointvariable q The endeector

i i

p osition is up dated b efore continuing on to the next joint

Overview

A single iteration of the CCD metho d for an njointed manipulator visits joints n through in turn

Ateachjoint the original ndimensional optimization problem is reduced to a onedimensional

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

problem involving just the jointvariable q which admits to an analytic solution An incremental

i

change to q is computed with either if the joint rotates or with if the joint translates

i

The variable q is then incremented and clamp ed to upp er and lower b ounds The current endeector

i

frame P and O is up dated to reect the change b efore pro ceeding to the next joint

c c

The algorithm b ehaves well around singular congurations and since the value of the ob jective

function is reduced with each step the metho d is guaranteed to converge But the heuristic

nature of the metho d makes the rate of convergence somewhat dicult to quantify since it is de

p endent on the structure of the manipulator itself In practice most problems can b e solved with

only a few iterations although there are situations for which the metho d can converge very slowly

In terms of b ehaviour the heuristic implies that distal links move more readily than links closer to

the base if the endeector goal can b e reached bymoving only the nal link of the chain then only

that link will move

Comparison

Each of the metho ds describ ed ab ove is suitable for interactive direct manipulation with some

caveats The p eriteration cost for each is minimal so that b oth can provide go o d feedbackwhen

dragging reasonably complex manipulators Each can b e made numerically stable near kinematic

singularities although the CCD metho d has an edge in this regard since it is completely immune to

diculties near singularities Where neither metho d particularly excels is in the rate at whichthey

converge toward a solution b oth can exhibit p o or convergence rates particularly if high accuracy

is required

Figures and compare the p erformance of each algorithm for solving a pair of inverse

kinematic problems The manipulator has degrees of freedom ab out the same complexityas

a simple approximation to a limb although for this example there are no limits on the range of

movement for eachjoint In each case a p osition goal is sp ecied for the endeector and the

inverse kinematic problem is solved to varying degrees of accuracyEach gure shows the initial

conguration and the nal solution obtained with each metho d In addition the time to achieve the

solution is plotted with resp ect to the degree of accuracy requested In the rst case of Figure

the endeector goal is well within reach and b oth metho ds are able to solve the problem reasonably

quicklyHowever it is already apparant that the Jacobian tranp ose converges slowly when it is

close to a solution there is a marked decrease in p erformance for each additional digit of accuracy

requested

In the second case of Figure the goal is close to the edge of the reach space of the limb

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

whichmust approach a singular conguration to achieve the goal Although b oth metho ds are able

to solve the goal the CCD metho d clearly outp erforms the other particularly as the accuracy of

the solution increases As it approaches the goal the Jacobian transp ose metho d is hamp ered bya

small applied force which is pulling in a direction which do esnt aect most of the joints so the

progress towards the goal is very slow The heuristic approach of the CCD fares much b etter in this

example

The nal congurations shown in the gures also illustrate the CCD metho ds preference for

moving distal links in contrast with the other metho ds tendency to distribute jointchanges more

equally along the chain This dierence in b ehaviour is quite noticeable during interactive dragging

Manipulating a chain with the Jacobian transp ose metho d tends to feel likeplaying a exible elastic

ro d while the CCD metho d imparts a feel more akin to pulling on a chain of lo osely connected

links While the nal solutions for the CCD metho d might lo ok inferior to those obtained with the

Jacobian tranp ose keep in mind that these solutions were obtained noninteractively ie they were

computed oline When dragging interactively using the CCD metho d it is usually not dicult to

gain control over the nal p osition bymoving the cursor appropriately Also b oth metho ds supp ort

a means of controlling joint resp onsiveness by sp ecifying an appropriate weighting factor w at each

i

joint In particular this parameter can b e useful to oset the default b ehaviour of the CCD metho d

These examples illustrate the main advantages and disadvantagesofeach metho d Both are

quick and therefore worth considering if direct manipulation is required even on machines with

mo dest p erformance The Jacobian metho d has the advantage of resp onding in an intuitive fashion

to pushes and pulls on the endeector the CCD metho d is not as intuitive in this regard The CCD

metho d exhibits more stability around singular congurations and although its rate of convergence

slows it is not nearly to the extent that the Jacobians do es Moreover it can b e argued that

direct manipulation do es not necessarily require a high degree of accuracy Certainly while a user is

sweeping a cursor across the screen it is not critical that the endeector track it to six decimal places

of precision one or two decimal places of accuracy probably suces When accurate p ositioning is

required particularly near singularities the CCD metho d would b e the appropriate metho d to use

In general the two metho ds complementeach other nicelyproviding alternate interaction mo dels to oer the user

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

(a) Initial (b) Jacobian transpose (c) CCD

200

X

X Time (milliseconds)

100 X

X O X O O O X O O

X − Jacobian 1e−1 1e−2 1e−3 1e−4 1e−5 1e−6

O − CCD Convergence Criteria Figure

CHAPTER EFFICIENT ALGORITHMS FOR DIRECT MANIPULATION

(a) Initial (b) Jacobian transpose (c) CCD

2750 X

2000

X Time (milliseconds)

1000 X

O

X O

X O O OX O X − Jacobian 1e−1 1e−2 1e−3 1e−4 1e−5 1e−6

O − CCD Convergence Criteria Figure

Chapter

Incorp orating Constraints

Inverse kinematic manipulation is a useful shortcut but do es not in itself provide any more function

ality than simple forward kinematics Creating a p ose for a skeleton can b e less tedious but there

are still problems in trying to edit the p ose without moving previously p ositioned b o dy parts It

would b e useful for example to b e able to sp ecify that parts of the b o dy should not move no matter

howwe manipulate the skeleton Badlers recentwork CP has shown the usefulness of b eing able

to imp ose constraints on a gure during editing The constrained optimization metho d employed

with Jack p ermits any constraintwhich can b e expressed as a function of the skeleton state to b e

sp ecied However with the exception of the exp erimental balance constraint describ ed by Phillips

PB this capability has not b een exploited The basic set of constraints available within Jack are

almost all simple geometric constraints on p ositions and orientations of b o dy parts It would seem

that even this limited set of constraints is enough to greatly improveinteraction with a gure

In this chapter we consider howeach of the manipulationtechniques describ ed in the previous

chapter can b e used to satisfy multiple simple geometric constraints on the p ositions and orientations

of endeectors within a skeleton and discuss some of the p erformance and implementation issues

that arise

Constraint Satisfaction

A manipulator or a skeleton can b e considered a system describ ed by a set of state variables for

our purp oses joint rotations Solving the inverse kinematic problem is essentially one of solving a

nonlinear system of equations for these state variables and there are a numb er of approaches one

can take to incorp orate constraints in the solution pro cess Wehave already seen that constrained

optimization is one p ossibility but there are other p ossibilities for the manipulation metho ds of the

previous chapter

CHAPTER INCORPORATING CONSTRAINTS

The simplest approach is to use some variant of the penalty method This metho d can b e used

with any iterativetechnique and the essential idea is that when a constraint is violated a restoring

force is intro duced to push the system backinto a legal state in which all constraints are satised

A feedback control lo op monitors the state of the system and applies the appropriate p enalty forces

as it detects constraint violations An advantage of this approach is that if the constraints are not

initially met the restoring forces pull the system towards a legal state A disadvantage is that the

constraints cannot b e enforced exactly since the restoring force only comes into play after a constraint

has already b een violated Thus two p oints constrained to b e coincident will app ear to pull apart

slightly b efore the constraining force can pull them back together For metho ds based on physical

simulation the restoring forces are usually springbased If the springs are made suciently sti

then the constraint violations may b e small enough to b e unnoticeable But this causes problems

when trying to simulate the system requiring tinyintegration steps to retain numerical stability

PFTV

Both the Jacobian transp ose metho d and the CCD metho d could b e used within a control lo op

to provide this sort of constraint satisfaction In fact due to the heuristic nature of the CCD metho d

and its reliance on strictly geometric information this is the only option op en Before describing

a p enaltymetho d approach based on the CCD metho d we consider rst a more comprehensive

solution which is a generalization of the Jacobian transp ose metho d

Maintaining Constraints

The approachtaken is to consider the problem of maintaining a set of constraints which are already

satised rather than trying to satisfy constraints that have b een violated By assuming that the

system is already in a legal state the problem is reduced to one of ensuring that changes to the system

never violate the constraints The technique is based on wellknown metho ds for implementing

constrained dynamics within physical simulations Sur AW using the same sort of simplied

rstorder equations of motion describ ed in Section and by Gleicher GW The essential idea

is that geometric constraints on an ob ject are treated as mechanical connections whichintro duce

constraining forces into the system b eing simulated When an external force is applied to the system

some comp onents of the applied force maybecounteracted by these constraining forces so that the

net force acting on the system do es not violate the constraints The nub of the problem is to

determine what these constraining forces are given a set of geometric constraints whichmust b e

satised

Figure illustrates the concept for the simplest p ossible system a p oint The systems state q

is simply the p oints lo cation in space A single constraint is imp osed on the system stating that the

CHAPTER INCORPORATING CONSTRAINTS

Fa Fc

Ft

Figure A force applied to a p oint constrained to lie within a plane A constraining force normal

to the plane is added to the applied force to obtain a legal force tangential to the plane

p ointmust lie within a plane This constraint condition can b e written as a function of the system

state cqNow supp ose a force F is applied to the p oint If the p oint is to remain in the

a

plane then the constraintmust intro duce a counteracting force F which removes from the applied

c

force those comp onents whichwould movethepointaway from the plane Summing the applied

force with this constraint force yields a net force acting on the p ointwhich ensures that the p oint

only moves in a legal direction Although simple this example illustrates twoimportantpoints

which apply to more complicated systems the net force lies in a plane tangent to the constraint



surface cq while the constraining force p oints in a direction normal to this tangent plane

Wenow describ e a pro cedure for computing the appropriate constraint forces for arbitrarily complex

systems

The Constraint Condition

Supp ose we write the constraints on a system as a vector function of the system state

C f q

If the constraints are initially satised then C In order to maintain the constraints C must

not change so the derivatives of C must always b e zero In particular this imp oses the condition

C

C q

q

In Chapter equation dened the simplied equations of motion for a system which had the

form

q Kg



in fact for this trivial example the tangent plane is the constraint surface

CHAPTER INCORPORATING CONSTRAINTS

for some generalized force g acting on the system state Substituting this equation into and

c

denoting the constraint Jacobian matrix by J the constraint condition b ecomes

c

q

C J Kg

c

Any generalized force g acting on the system which satises this condition will not violate the

constraints

Computing the Constraint Jacobian Matrix

The constraint Jacobian matrix J describ es how the individual constraints change as the state

c

variables vary The section ab oveintro duced the constraintvector Cq which is made up of m

scalar constraint functions c q c q If there are n state variables q q thenJ is an

 m  n c

m n matrix Typically there are more state variables than there are constraints

In practice constraints are not sp ecied on q directly but rather on more meaningful geometric

entities suchaspoints or orientations which are themselves functions of the state qFor example

supp ose we wish to imp ose a constraint that the tip of a limb on an articulated skeleton denoted

Pq should remain xed at some lo cation R An appropriate constraint condition for this example

would b e



R Pq

In this case the constraintvector C consists of scalar constraints one on eachofthex y andz

comp onents of P

Cq c P q c P q c P q

 x  y  z

  

R P R P R P

x x y y z z

c

i

to the constraint Jacobian matrix Thus Each scalar constraint c contributes a single row

i

q

j

each geometric constraintcontributes a blo ckofr rows to the matrix where r is the dimension of

the geometric quantityeg r for a p oint constraint Since geometric constraints are expressed

in terms of functions of qthechain rule is applied to calculate the Jacobian entries In the case

ab ove for example

c c P

i i k

q P q

j k j

Typically a single constraint will dep end on only a few of the system state variables If the

left fo ot of the sample skeleton of Figure is constrained to some lo cation for example the arm

CHAPTER INCORPORATING CONSTRAINTS

jointvariables have no eect on this constraint Thus the ith rowofJ corresp onding to the

c

constraint c usually contains just a few nonzero elements It is imp ortant that this sparsityin

i

J b e exploited and preserved sparse matrix techniques oer substantial p erformance gains over a

c

naive implementation

Computing the ConstraintForce

In Chapter it was shown that an external force applied to a p oint on an articulated skeleton could b e

converted to a generalized force g on the skeleton state by the Jacobian transp ose metho d However

the g of equation is the net generalized force on the system As such it may include the sum

of multiple forces applied to various p oints on a skeleton More to the p oint it must also include

some unknown constraining forces which prevent the applied forces from violating the constraints

imp osed on the system In general the net force is considered to b e a sum of known applied forces

and of unknown constraint forces g g g Substituting this into the constraint condition

a c

yields the linear system of equations

J Kg J Kg

c c c a

in which only the constraining generalized force vector g is unknown

c

Equation simply states that an appropriate constraining force is one which when added

to the applied forces causes the constraint derivatives to b e zero But this is to o ambiguous since

the system is usually underconstrained and there maybemany constraining forces g which satisfy

c

equation One approach to removing this ambiguity is to insist that the constraint force must



lie in a direction in which the system maynotmove just as the constraining force of Figure

do es Recalling the discussion in Chapter ab out the Singular Value Decomp osition SVD this

implies that the constraint force must lie within the range of the constraint Jacobian matrix J

c

Thus for constrained systems in general the constraint force g is restricted to lie in the range of

c

J while the net force g must lie in its null space This is a generalization of the sp ecic example

c

presented in Figure

If the constraint force is in the range of J then g J for some vector Therefore equation

c c c

can b e written as

T

J Kg J KJ

c a c c

and solved for the vector of Lagrange multipliers Once the Lagrange multipliers are known the

constraint force which removes the illegal comp onents of the applied force is computed as

g J

c c



this restriction follows from the principle of virtual work Pau

CHAPTER INCORPORATING CONSTRAINTS

Note that computing g requires the solution to a linear system of equations even though the

c

constraints themselves may b e nonlinear

Solving for Lagrange Multipliers

The linear system must b e solved to nd the Lagrange multiplier vector

 m

Note that the numb er of equations in the system is equal to the dimension of the constraintvector

CAny metho d for solving sets of linear equations can b e applied here

A metho d recommended here to solve the system emphasizes robustness over sp eed Due to the

structure of most articulated skeletons it is imp ossible to avoid illconditioning in all of the

time Mac To cop e with this the system is solved using the truncated SVD of the lefthand

T

combined with a backsubstition algorithm PFTV The truncated SVD m m matrix J KJ

c c

is formed from the original SVD by zeroing any small singular values which eectively throws

awayany illconditioned comp onents of the matrix PFTV MK This solution metho d do es

not take advantage of any sparsity in the matrix but has the advantage of b eing robust in cases

where illconditioning o ccurs

Feedback

The discussion so far has b een based on the assumption that the constraints are initially satised

and that they remain so To handle cases in which the initial state violates some constraints or

where numerical inaccuracies cause constraint violations an additional feedback term is added to

the basic equation of motion

T

q Kg g k CJ

a c c

This term eectively adds a p enalty spring to the system the strength of which is prop ortional to

the magnitude of the constraint deviation C

Overview

Equation is the complete simplied equation of motion for a constrained system Constraints

are maintained byevaluating the right hand side of the equation using the metho ds outlined in the

sections ab ove to nd the state variable velo cities q A single integration step then yields a new state

q which resp ects any constraints imp osed on the system In an interactive setting the pro cedure

iterates as the user applies changes to the system state through the user interface Screen up dates

o ccur after each iteration to provide feedback Figure provides an overview of the pro cedure

CHAPTER INCORPORATING CONSTRAINTS

for a single iteration Note that equation is a generalization of the Jacobian transp ose metho d

of chapter in the absence of any constraints on the system the equation reduces to the simple

formula intro duced earlier

Implementation Issues

The basic constrained equation of motion has a regular structure which lends itself to an ob ject

oriented implementation A solver for the equation only needs to know a few things the length and

value of a global state vector qhowtoevaluate a numb er of functions of q and howtoevaluate a

matrix of partial derivatives with resp ect to q for each of these functions

Using software ob jects which resp ond to sp ecic requests to provide these pieces of information

a generic constraintsolver can b e written which is insulated from the sp ecic details of articulated

skeletons The solver queries these ob jects to obtain the information it needs to assemble a global

system of constrained equations of motion solves the system and then communicates the results

back to some of the ob jects Implementing this metho d of maintaining constraints requires three

dierenttyp es of these ob jects skeletons hand les and constraints

Skeletons as Ob jects

Eachskeleton is considered a single ob ject whichcontributes variables to the solvers global state

vector qEachskeleton must b e able to resp ond to requests for sp ecic pieces of information that

the solver requires In addition a metho d must b e provided bywhich the solver can communicate a

solution back to the skeleton If eachskeleton provides these capabilities the solver do es not need

to knowanything ab out the internal structure of the skeleton itself The solver must b e able to

query a skeleton for the length of its state vector

query a skeleton for its state variable vector

query a skeleton for its scaling matrix K

send a skeleton a new state vector reecting the solution

Handles on Skeletons

A hand le is an abstraction representing a geometric quantity asso ciated with a skeleton whose value

dep ends on the skeletons internal state A handle can refer to something as simple as the p osition

of a joint within the skeleton or as complex as the lo cation of the skeletons center of mass Handles

are the geometric entities up on which constraints may later b e imp osed and to which forces maybe applied

CHAPTER INCORPORATING CONSTRAINTS

Manipulate Specify skeleton constraints

Compute g a constraint Jacobian Jc

Solve for Lagrange multipliers λ

Compute Compute constraint force feedback term g c

+

q´

∫

q

Figure Iteration steps for maintaining constraints

CHAPTER INCORPORATING CONSTRAINTS

Each handle attached to a skeleton knows how to compute a function of the skeletons state

hq A numberofhandletyp es can b e dened each computing a dierentquantity These

sk eleton

might include

A point hand le which computes the global p osition of some p oint dened within a lo cal joint

co ordinate system eg a p oint on the left shin inches b elow the knee

An orientation hand le which computes the global orientation of a lo cal joint co ordinate system

A centerofmass hand le which computes the global p osition of a skeletons center of mass

computed as a weighted average of each b o dy parts center of mass

Usually a single handle will dep end on only a subset of the joints within a skeleton When queried

for its state vector a skeleton should return only those elements of q which are referenced

sk eleton

by a handle This will keep the total number of variables in the constrained system to a minimum

In addition to the function hq each handle must also knowhow to compute the Jacobian

sk eleton

h

matrix J q The constraintsolver will query each handle for this information

sk eleton

h q

skeleton

when assembling the global constraint Jacobian matrix J Implementing a new handle typ e is

c

straightforward requiring only handlesp ecic routines to b e co ded These routines are resp onsible

for initializing the handle computing the handle function h and computing the handles Jacobian

matrix J

h

Constraints on Handles

Constraints can b e applied to the quantities computed by a handle The simplest constraints are

those that constrain a handle to a geometric constant Useful examples of this include

constraining a p oint handle to a given lo cation

constraining a p oint handle to a plane

constraining a p oint handle to a line

constraining an orientation handle to a given orientation

These simple constraints would b e useful for sp ecifying interactions with a static environment For

example the feet of a gure can b e constrained to lie in the plane dened by the o or or the hand

can b e constrained to the rung of a ladder To sp ecify more complex b ehaviours constraints could

also b e created b etween two handles For example a gures hands could b e constrained to stay

clasp ed together by creating a p oint handle on each hand then sp ecifying an equality constraint

between these two handles

CHAPTER INCORPORATING CONSTRAINTS

Each constraint computes a function ch h where the input values are the outputs of

 n

one or more handles These constraint functions are usually quite simple often reducing to just a

variation of the distance formula In addition each constraintmust b e able to compute its Jacobian

c

matrix with resp ect to its inputs Adding a new constraint is just a matter of writing co de to

h

evaluate these two items Geometric constraints are for the most part simple enough that this is a

minor amountofwork

The Global Picture

An application sets up constraint problems by creating handles on skeletons imp osing constraints

on these handles and applying forces to them b efore invoking a solver to evaluate equation

But adding the constraint enforcementscheme of Section to an interactive program is not quite

straightforward A user should b e able to interactively imp ose temp orary constraints on a gure

while editing its p osture Over the course of a normal work session numerous constraints may

b e created and deleted as the work pro ceeds So the system b eing simulated by equation is

constantly changing each time a constraint is added new handles may b e referenced and new state

variables intro duced It is not p ossible to determine b eforehand the correct set of equations to b e

simulated and to simply hardwire the co de to do so Instead some sort of mechanism must exist

for assembling and evaluating the system of equations on the y based on the current set of active

constraints The constraint solver itself can b e made resp onsible for this It will maintain a data

ow network which represents the system b eing simulated the equations of motion for the system

are completely determined by the networks conguration The dataownetwork scheme has b een

describ ed by Witkin AW and explored more fully by Kass Kas

The solver is resp onsible for assembling and solving the global constrained system of equations

taking an integration step and communicating new state information backtoeachskeleton ob ject

It also handles any bookkeeping required to map lo cal skeleton variable indices to global indices As

an application adds constraints to the system the dataownetwork within the solver is up dated

The network consists of a set of function blocks with outputs of some blo cks feeding into the inputs

of others Each blo ck computes two items a function of the blo cks inputs f x and the lo cal

f

The blo ck outputs f x as well as Jacobian matrix of that function with resp ect to its inputs

x

the result of multiplying the lo cal Jacobian matrix by an incoming Jacobian matrix thus applying

the chain rule Figure depicts a typical function blo ck

Each constraint and each handle within the system contribute a single function blo ck to the

network At the top level the inputs of a constraintblock are wired to the outputs of one or more

handle blo cks or in the case of simple constraints directly to constantvalues At the b ottom level

the inputs to each handle blo ck are connected directly to elements in the global state vector qAs

CHAPTER INCORPORATING CONSTRAINTS

∂f f(x) ∂q

f

∂ x x

∂q

Figure A generic function blo ck

constraints are created and destroyed no des are added to and deleted from the network whichis

rewired to reect the change

When the solver is invoked it consults the top level of the network to determine the global

constraintvector C as well as the constraint Jacobian matrix J The constraintvector C is formed

c

by simply instructing each top level no de to evaluate its function and concatenating the results

Each no de recursively instructs its inputs to evaluate themselves b efore computing its own function

The recursion b ottoms out at the lowest level when it reaches the state variable values qEachnode

caches its last computed value in case it is needed again

A similar recursivescheme is used to compute the global constraint Jacobian matrix J Eachtop

c

level constraint no de computes its own lo cal Jacobian matrix with resp ect to its inputs then applies

the chain rule bymultiplying this against the Jacobian matrix of each of its inputs with resp ect to

the state variables see Figure The solvers recursion scheme takes care of the mapping b etween

lo cal and global indices as well as the allo cation of temp orary memory for intermediate matrices

Matrix sparsity is preserved and exploited to reduce computation and caching is again used to

avoid duplicate computations

The dataownetwork provides the exibility required to supp ort the interactive restructuring of

a simulated system One attractive feature of this organization is that implementing new classes of

handles and constraints is particularly simple All that is required is to implementthetwo routines

required by a function blo ck since this is all the solver needs as it traverses the network

CHAPTER INCORPORATING CONSTRAINTS

An Example

Consider the problem of constraining the tips of a gures hands to its hips This can b e accomplished

by rst creating four p oint handles on the gure

handle h on the tip of the right hand



handle h on the right hip



handle h on the tip of the left hand



handle h on the left hip



To x the hands to the hips two equality constraints on these handles are created



c h h

  



c h h

  

Figure shows the resulting dataownetwork constructed by the constraintsolver Each handle

references a few variables within the skeletons state vector these variables are concatenated to

form the global state vector q The handle function blo cks compute their resp ectivepoint lo cations

and feed the results to the constraint function blo cks These compute the distances b etween the



incoming p oints and the blo ck function outputs are concatenated to form the element global

constraintvector CEachblock also computes its own lo cal Jacobian matrix with resp ect to its

inputs multiplies it with an incoming matrix and passes the result along The matrix outputs of

the constraintblocks contribute sparse rectangular blo cks to the global constraint Jacobian J

c

Any time an elementofq changes value a disable cache signal is triggered on any handle blo ck

connected to the element This signal p ercolates up through the network forcing any blo ckthatis

encountered to discard its cached data The next time the solver evaluates the network only these

blo cks will recompute their outputs When the network has b een reevaluated the solver has all the

information it needs to evaluate equation

Summary

This constraint enforcement pro cedure has some attractivecharacteristics Most imp ortantly con

straints can theoretically b e maintained exactly since any forces acting on the system whichwould

cause a constraint violation are removed The pro cedure is quite general as well constraints and

handles can b e arbitrarily complex functions and are therefore not restricted to referring to geomet

ric quantities alone The dataownetwork architecture is extensible since adding new handle and

constrainttyp es is simply a matter of writing a handful of functions



for this example two sets of x y andz scalar constraints

CHAPTER INCORPORATING CONSTRAINTS

C Jc

c 1 c2

h1 h h 2 3 h4

q

Figure Example network

CHAPTER INCORPORATING CONSTRAINTS



The main drawback to the approach is that the solution cost is dominated by the O n cost

of solving a linear system for the Lagrange multipliers of which there is one for eachofn scalar

constraints in the system b eing simulated For even a few simple geometric constraints the solution

time at this b ottleneckquickly degrades to the p oint where interaction b ecomes dicult But Surles

Sur has shown that when constraints are applied to articulated structures with sp ecic charac

teristics the resulting system of equations can b e solved in linear time under certain conditions

This result is promising but is recent enough that it has not b een pursued for this thesis

A sample implementation of the metho d outlined ab ove b oth conrms its p otential as well as the

limitations imp osed by the b ottleneck Retaining interactive up date rates for any reasonably complex

skeleton with just a few p osition and orientation constraints applied is dicult given currentCPU

p erformance Simple Euler integration not surprisinglyintro duces numerical instabilities due to

cumulative errors when the integration stepsize is large enough to provide reasonable interactive

feedback Adopting more robust adaptiveintegration metho ds is not necessarily helpful either

These metho ds typically require multiple evaluations of equation p er iteration and since it

is so exp ensive to compute the refresh rate drops dramatically and the interface tends to feel

unresp onsive In addition adaptive stepsizing implies that some steps will b e thrown awayentirely

when the stepsize must b e reduced in order to retain accuracy As a result one iteration maytake

much longer to complete than another so screen up dates o ccur at irregular intervals and consequently

the interface is p erceived as b eing jerky and unresp onsive However these are problems which

will disapp ear as CPU p erformance improves to the p oint where accuracy in the solution can b e

maintained at interactive up date rates

Another source of p otential stability problems is the feedback term of equation This term

suers from the usual problems asso ciated with springbased constraint enforcement a high spring

constant k leads to a sti set of equations which are unstable while a lowvalue for k p ermits

noticeable constraint violations Cho osing a go o d spring constantvalue can b e dicult and is

currently done purely on a trialanderror basis Ideally the feedback spring should b e a lo ose one

ie k is small with the assumption that as CPU p erformance improves it will b e p ossible to

p erform enough iterations b etween screen up dates to eliminate any noticeable constraint violations

Trying to eliminate the inevitable drifting of a system away from its constraints a result of

numerical inaccuracies by using a high spring constant is almost sure to intro duce severe stability

problems

Given current CPU sp eeds this technique seems b etter suited to solving constraint problems

oline than in an interactive setting Our goal is to develop immediately useful interactivetools

for working with nontrivial skeletons and this metho d cannot b e considered practical yet for this

CHAPTER INCORPORATING CONSTRAINTS

purp ose on mo destly p owered workstations Instead we will consider a p enaltyforce metho d based

on the CCD algorithm as a substitute with whichwe can exp erimentinaninteractive editor

A CCDbased Penalty Metho d

The previous metho d is capable of handling multiple constraints imp osed on dierent parts of a

skeleton To accomplish the same with the CCD metho d requires some slight mo dications to the

basic algorithm presented in Chapter Whereas the basic algorithm considered just a serial chain

of links and a single goal solving for multiple constraints means wemust also consider branches in

the manipulator and more than one goal as shown in Figure

The scheme of Chapter solved the problem for a serial chain bytraversing from the endeector

in towards the base at eachjoint adjusting the joint parameter to minimize an error measure derived

from the current endeector p osition and a known goal In a branching chain with multiple end

eectors and multiple goals a jointmaycontribute to the p osition of more than one endeector

so p otentially more than one goal must b e considered when varying the jointvariable Also the

algorithm dep ends on the inward traversal scheme distal joints must b e solved rst b efore a more

proximal one can b e considered This precludes treating the multiplegoal problem as a series of

singlegoal problems which can b e solved sequentially

A recursivetraversal of the hierarchy from the leaves in towards the ro ot ensures that all sub

problems at a jointaresolved b efore the joint itself is considered Each joint instructs all of its

children to solve for any constraints that may b e applied to endeectors in their subtree Then the

jointvariable which minimizes the summed error measure from eachchild subtree can b e computed

The only change from the previous CCD algorithm is that the traversal must now consider branches

A

B

Figure A branching chain with two endeector constraints

CHAPTER INCORPORATING CONSTRAINTS

in a chain and that the error measure equation of Chapter b ecomes a summation

n

X

E q E qE q

p o

i

i

i

over the n constraints which are distal to the current joint The net eect of this change is that the

co ecients k k andk must b e computed by summing equations over each distal

  

endeector goal b efore a minimizing change in the currentjointvariable can b e computed The

pro cedure must iterate until either all endeector constraints have b een satised or until successive

iterations pro duce negligible changes indicating that the constraints are conicting and cannot b e

simultaneously satised

This recursivescheme resembles Badlers early heuristic approach BMW for p ositioning a

gure with multiple constraints although the CCD metho d for computing a minimizing change in a

jointvariable app ears to dier from the one used there

This p enalty metho d approach is arguably inferior to the more comprehensivescheme outlined

previously Constraints are only approximately enforced so we can exp ect to see some drift on

segments that are constrained to stay in place Also we are restricted to geometric constraints

due to the CCD metho ds reliance on geometric relationships alone Nevertheless the metho d is

usually stable enough to p erform adequately at interactive up date rates and gives us a means of

exp erimenting with constraints within our interactive editor In our prototyp e implementation

instabilities do o ccur when p osition and orientation constraints conict with each other but this can

b e alleviated with appropriate weighting factors on the individual constraints

Chapter

An Interactive Editor

To place the discussion of the preceding chapters in context an interactive program for animating

articulated gures is intro duced here The program is designed to create keyframed motion sequences

for arbitrary skeletons and has b een b oth a motivation and a testb ed for the ideas develop ed

in this thesis Both the Jacobian transp ose and the CCD metho ds have b een implemented and

the application interface mo died to accomo date direct inverse kinematic manipulation as well as

constrained manipulation This chapter briey describ es the program and the manipulation interface

A System Overview

The SequenceEditoris an interactive to ol for creating and editing keyframed movement sequences

for a single arbitrary skeletal gure Its primary function has b een to act as the movement creation

window of a motion planning package CWGLforchoreographers whose needs are quite dierent

from those of computer animators Toa typical user planning the movementofmultiple gures in a

work space a rough approximation of a movement that can b e created quickly can b e more valuable

than a more realistic animation that would take hours to p erfect Consequently the programs design

has favoured ease of use over functionality The hop e is that direct manipulation and constraint

satisfaction based on the metho ds of the previous chapters can improve the qualityofmovement

created with the editor while retaining a simple and intuitive user interface

Before describing the interface itself some of the terminology warrants a brief explanation

Skeletons

The program itself knows nothing ab out skeletons sp ecically a skeleton app ears as an abstract data

typ e supp orted by a to olkit library of routines The to olkit supp orts the creation manipulation and

CHAPTER AN INTERACTIVE EDITOR



animation of skeletons dened by a simple ASCI I le format Figure shows a sample description

for a simple approximation to a human gure Anyvalid skeleton description can b e read in to the

editor and animated

As the example shows eachskeleton is simply a collection of named joints arranged in a hierarchy

The joints are either rotary or prismatic each with a single degree of freedom Comp ound joints with

multiple degrees of freedom eg ballandso cket can b e mo delled as a series of these single DOF

joints sharing the same lo cation Asso ciated with each jointarea numb er of attributes including

the jointtyp e rotary or prismatic the lo cal axis axes of freedom and lo cal limits on the range

of movementofthejoint Polygonal ob jects maybeattached to anyjoint no de in the hierarchy

These rigid ob jects comprise the app earance of the skeleton and are drawn as they are encountered

during a displaytraversal of the hierarchy

Askeleton maintains an internal state vector q consisting of all the jointvariable values as well

as a translation vector for the skeleton as a whole This information is enough to completely dene

the lo cation and shap e of the skeleton for a single frame An application can animate a skeleton by

rep eatedly loading the skeleton state q and instructing the skeleton to display itself

The to olkit now supp orts inverse kinematic control over a skeleton by allowing the application

to identify b oth a base and an endeector within the joint hierarchy A desired p osition andor

orientation for the endeector may then b e sp ecied and the skeleton instructed to solve the

inverse kinematic problem using either of the metho ds of the previous chapter The internal state q

is automatically up dated to reect the solution to the problem This isolates the application from

the details of the inverse kinematic algorithms

The Sequence

A sequence is an abstract data typ e for storing skeleton animation data it represents the movements

ofaskeleton over some p erio d of time An application may query a sequence to determine the

skeleton state at any frame of the animation Although a sequence could encapsulate a pro cedural

motion mo del for this discussion a sequence will b e dened as a series of keyframed p oses eachof

which sp ecies the state vector q at a particular frame When a sequence is queried for the skeleton

state at a frame b etween key frames an interp olated state is computed onthey and returned For



each rotating joint either linear or splined quaternion interp olation Shomay b e p erformed

For each translating joint either linear interp olation or generalized CatmullRom spline interp olation

KB may b e used Currently no further control over the interp olation pro cess is provided other



courtesy of Phil Peterson



a quaternion is a compact representation for rotations with some app ealing prop erties

CHAPTER AN INTERACTIVE EDITOR

# Example .scr file for a stick figure.

define skeleton "stickman" joint "pelvis" group "Upper body" joint "torso" group "Right Arm" joint "right arm" joint "right hand" end group group "Left arm" joint "left arm" joint "left hand" end group joint "head" # "head" is a child of "torso" end group group "Right leg" joint "right leg" joint "right foot" end group group "Left leg" joint "left leg" joint "left foot" end group end skeleton

head

torso right arm left arm define joint "right leg" object "r_lleg.pol" offset 0 −0.32 0 hinge x 0 170 right hand left hand mass 1 orientation 0 0 0 pelvis mirror "left leg" end joint right leg left leg

right foot left foot

Figure Sample skeleton description

CHAPTER AN INTERACTIVE EDITOR

than that aorded bychanging the keyframe spacing or the key p oses

Anumb er of simple editing op erations are dened for keyframe sequences Ranges of keyframe

p oses may b e copied inserted andor deleted Keyframe spacing may also b e adjusted by stretching

shrinking or sliding moving a range within the sequence This provides some crude control over

timing

The Editor

An interactive editor for creating and editing animation sequences for arbitrary skeletons is shown

in Figure The currentskeleton is displayed in three orthogonal views and a single p ersp ective

view Each of these views mayoccupy the large main work area in whichtheskeleton maybe

manipulated Existing sequences are group ed into menus and displayed in iconic form to the right

Figure Sequence editor screen

of the screen Each sequence icon can cycle through all of the keyframe p oses on request generating

a simple ipb o ok preview of the movement as a memory aid A sequence can b e named and added

to one of these menus and a menu item can later b e reloaded for editing A small display b eneath

the main viewp ort shows the keyframe p oses within the loaded sequence These are laid out along

CHAPTER AN INTERACTIVE EDITOR

a timeline to indicate the keyframe spacing A set of VCRlike transp ort controls are provided for

playing back the animated sequence in the main display area

The small timeline provides a convenientinterface for copying cutting and pasting ranges of

keyframes within a sequence as well as mo difying the keyframe spacing But the key p oses them

selves must b e created or mo died by adjusting the skeleton displayed in the main window For

manipulating the skeleton forward kinematic controls are provided in the form of sliders for adjust

ing individual joints within the skeleton one at a time In addition the interface has b een mo died

to p ermit inverse kinematic manipulation of the skeleton In this mo de the user may use the metho ds

presented in the previous chapter to up date multiple joints by dragging one b o dy part around The

implementationofthisinterface for direct manipulation is briey describ ed b elow

Direct Manipulation

With direct manipulation the user adjusts the skeleton by selecting and dragging a b o dy part rather

than adjusting individual joints A dragging session b egins when a b o dy part is selected with the

mouse and ends when the mouse button is released Dragging is a p ositioning mechanism onlyit

do es not directly control the orientation of the selected part To supp ort dragging the interface

must provide a wayofidentifying a serial chain of joints within the skeleton which will b e considered

a manipulator as wellasawayofunambiguously sp ecifying endeector goals for the chain

Identifying the Chain

Achain of joints within the skeleton is dened when a b o dy part is rst selected at which time

bothabasejoint and an endeector must b e determined To determine a base joint the skeleton

hierarchyistraversed backwards from the selected part towards the ro ot Three separate interaction

mo des control how far up the tree this traversal pro ceeds b efore stopping at a jointwhichbecomes

the base of the chain The stopping criteria corresp onding to each of the three dragging mo des are

stop when the parentjoint has a b o dy part attached to it

stop when the parentjointhasmultiple children

stop when a named jointisreached

The rst of these results in only the selected b o dy part b eing dragged and is the direct manipulation

equivalent of p ositioning the part using the forward kinematic slider controls The second results in

achain which extends from the nearest branch in the hierarchy and is therefore a useful default to

use when dragging limbs Selecting and dragging a hand for example would movetheentire arm

without disturbing the torso The third interaction mo de p ermits the user to override these default

CHAPTER AN INTERACTIVE EDITOR

behaviours by explicitly naming a joint from whichthechain extends Thus the user can sp ecify a

torso joint as the chain base for example in which case pulling on the arm would result in a b end

of the spine

In addition to the base of the chain the endeector p osition must b e known since that is the

p oint b eing dragged There are two p ossible approaches to determing the endeector lo cation when

a user p oints at a b o dy segment and presses the mouse button The rst is to explicitly compute the

p oint on the b o dy segment underneath the cursor by casting a ray and intersecting it with the ob ject

The alternative is to predene lo cations within the skeleton whichmay b e selected for dragging the

lo cations of the joints themselves for example This requires either that the lo cations b e displayed

in some iconic form so the user can select one or that the cursor warp to the nearest one when the

mouse button is pressed The initial decision was to cho ose the former metho d explicitly computing

apoint on the surface of the ob ject and making that p oint the lo cation of the endeector frame

This gave mixed results On one hand it allowed the user to clickanywhere on the ob ject for

dragging without any disconcerting cursor warping On the other hand having the endeector

frame lo cated on the ob ject surface often resulted in unexp ected twisting of the segment esp ecially

with the Jacobian metho d As a compromise the current implementation places the endeector

halfway along the segments axis of selfrotation warping the cursor to the corresp onding onscreen

lo cation Lo cating the endeector along the segment axis eliminates the problem of unexp ected

twisting

Determining an EndEector Goal

To compute new goal p ositions for the endeector as the cursor moves around on the screen we

must resolvetheambiguity in mapping a D screen lo cation to a D lo cation The solution adopted

here is to cast a ray through the cursor into the world and to nd the p oint on this line which

is closest to the endeector This p oint b ecomes the goal p osition for the current iteration This

technique provides reasonable b ehaviour in b oth orthographic and p ersp ective views

With an orthographic pro jection the cast ray is parallel to the line of sight and the closest p oint

on the line will lie in the plane p erp endicular to the ray and containing the endeector In this case

the endeector is constrained to lie within the plane while it is dragged around In a p ersp ective

pro jection view the cast raymaydiverge from the line of sight and the shortest path from the

endeector to the ray is no longer constrained to lie parallel to the image plane The endeector

is therefore free to moveinany direction and mayeven b e pushed and pulled towards and

away from the viewer with a little practice Figure illustrates the case for b oth p ersp ective and orthographic pro jections

CHAPTER AN INTERACTIVE EDITOR

goal goal

(a) (b)

Figure Plan view of goal determination in a an orthographic view and b a p ersp ectiveview

Once the new goal p osition for the endeector has b een computed the skeleton is instructed

to solve the inverse kinematic problem the user can select the solution metho d to use by setting

a program option The screen is refreshed and the cursor lo cation resampled after each iteration

to provide resp onsive feedback Of course when a single iteration is insucienttosolve the most

recent problem the movementoftheskeleton lags b ehind the cursor In practice this small tracking

delay has not b een a problem

To provide some control over the resp onsive b ehaviour of the skeleton the user can interactively

mo dify weighting parameters at each individual joint This is particularly useful when the CCD

metho d is used allowing a the resp onsiveness of a chain to b e varied from that of a sti ro d to that

of a set of lo osely coupled links

Constraints

As an additional p ositioning aid a small set of constraints have b een dened which the user may

apply to any segmentofaskeleton The most useful of these are intended to lo ck limb extremities

in place while the p ose is b eing edited Currently constraints are enforced by iteratively solving a

set of p ersistentinverse kinematic goals using the CCD metho d This restricts us to a set of simple

geometric goals

Lo cking an endeector p osition

Lo cking an endeector orientation This can b e a partial lo ckofonlyoneortwo directions

eg lo ck Y orientation only

CHAPTER AN INTERACTIVE EDITOR

Aweighted combination of b oth of these

Constraining an endeector to lie within a plane

Toapplyalock the user selects a segment then makes a lo ck selection from a menu Once created

alock remains in place until it is explicitly deleted A lo ck can b e edited to change either its typ e its

imp ortance weighting or the joints which can contribute to maintaining the lo ck The lo ck lo cation

or orientation can also b e mo died by either dragging the lo ck or twisting the endeector through

a set of sliders

The intent is that lo cks represent constraints which are to b e maintained during any subsequent

p ositioning of the gure Toachieve this after each editing op eration of a gure whether through

direct manipulation or the forward kinematic slider controlsanumb er of iterations of the multiple

goal CCD solver are p erformed prior to each screen refresh When the numb er of iterations p erformed

is insucient to satisfy any constraints violated by the editing op eration some drifting from the lo ck

constraint p ositions is unavoidable But a button is provided for invoking the solver explicitly until

all constraints are met to the users satisfaction The user mayalsochange the numb er of iterations

p erformed b etween screen refreshes On a highp erformance machine this numb er can b e set high to

minimize visible constraint violations on a lowp erformance machine a smaller numb er can improve

the resp onsiveness of the interface at the cost of constraint violations b ecoming more noticeable

Lo cking constraints are particularly useful when applied to the supp orting limbs of a gure In

the example of Figure the gures feet are constrained to stay in place bylocking b oth their

p osition and orientation The rst image shows the p osition at which the lo cks are created In the

second image the p elvis has b een tilted to lean the gure and in the third image a twist has b een

applied to the p elvis In all three images the fo ot p ositions and orientations are maintained and the

CCD solver is fast enough that the p elvis can b e tilted and twisted interactively without noticeable

sliding of the feet For this example iterations b etween screen refreshes were sucienttomaintain

the constraints with negligible drift and provide an up date rate of frames p er second on a Silicon

Graphics R Entry Level Indigo workstation This up date rate is inadequate for animation but

quite sucientforinteractive p ositioning

While lo cking endeectors in p osition is useful for creating a series of consistentkeyframe p oses

there still remains the problem of maintaining the constraints at the interp olated inb etween frames

There is no guarantee that an endeector lo cked to some p osition in two adjacentkeyframes will

remain in that p osition if joint angles are merely interp olated b etween the p oses Figure illustrates

the problem The rst and last images are keyframes in which the feet have b een constrained to

the same p osition and orientation The image in the middle is the result of interp olating joint

CHAPTER AN INTERACTIVE EDITOR

Figure A gure b eing p ositioned by rst tilting then twisting the p elvis

angles b etween the two p oses the feet clearly move through the o or during the transition b etween

the keyframes The problem is that the gures hierarchy is ro oted at the p elvis and there is no

relationship b etween the rate at the which the p elvic translation is b eing interp olated and the rate

at which the leg joint angles need to change to maintain the fo ot p ositions Of course we could

rearrange the hierarchy to ro ot it at one of the feet but this would only alleviate the problem for

that fo ot rearranging the hierarchy is not helpful when multiple endeectors are constrained

In the current implementation the only attempt made to address this problem is to provide an

option to enable constraint satisfaction during interp olation Each frame is computed as usual by

interp olating joint angles then the CCD solver is invoked until any violated constraints havebeen

resatised In essence a series of singleframe constraint problems are solved This approachis

admittedly ad hoc and it is not clear yet howwell it will work it app ears adequate for the few

situations tested so far although interp olation can no longer b e p erformed onthey

Initial feedback suggests that even this simple approach to enforcing geometric constraints com

bined with direct inverse kinematic manipulation is a signicant improvement to the keyframe editor

CHAPTER AN INTERACTIVE EDITOR

Figure a First keyframe p ose b Interp olated p ose c Second keyframe p ose

Chapter

Conclusion

Summary

Wehave examined solutions to the inverse kinematic problem applied to manipulating articulated

skeletons in an interactivekeyframe animation editor As alternatives to previously published algo

rithms in the graphics literature a pair of simple algorithms have b een describ ed which can provide

relatively inexp ensive direct manipulation of a gure The rst of these is really just an application

of a simple minimization metho d but its application to inverse kinematics has not previously b een

made explicit The second is a new heuristic algorithm adopted and mo died from the rob otics liter

ature The advantages and disadvantages of eachhave b een discussed and their relative p erformances

gauged

Metho ds by whicheach algorithm can b e used to satisfy constraints during manipulation have

b een develop ed and their resp ectiveadvantages and disadvantages discussed An implementation

of the simpler approach has b een incorp orated into a keyframe editor and the interface briey

describ ed

As a bypro duct a to olkit library for dening and manipulating skeletons has b een implemented

which insulates an application from the details of creating editing drawing and animating a gure

Askeleton is also able to solveinverse kinematic problems p osed by the application allowing the

application to deal with interface issues rather than dealing explicitly with inverse kinematic algo

rithms This also allows new inverse kinematic solution metho ds to b e implemented and added to

the to olkit without requiring mo dications to the application

CHAPTER CONCLUSION

Results

After exp erimenting with several inverse kinematic metho ds in the course of this work some things

have b ecome apparant The rst is that inverse kinematic algorithms all exhibit problems of one

typ e or another no one approach seems uniformly sup erior to others The metho ds presented here

should not necessarily b e considered any b etter than other approaches but rather provide alternative

approaches whichmay b e suitable in some applications

The second p ointisthatinverse kinematics as a p ositioning to ol is a useful complement to but

not a replacement for simple forward kinematic p ositioning In particular for direct manipulation

inverse kinematics is really only useful for chains with relatively few degrees of freedom single

limbs for example In fact an interesting observation is that even with the ability to drag multiple

jointed chains around many users of the editor describ ed in Chapter seem quite content with

the default dragging mo de which drags just a single b o dy part Informal feedback from users

suggests that while direct manipulation is a huge improvementtotheinterface over the previous

sliderbased p ositioning the value of multiplejoint p ositioning is not quite so clear In fact trying

to p osition a chain with many degrees of freedom with direct manipulation is often likely to cause

unwanted changes to the skeleton For example trying to b end the spine of a gure by pulling on its

ngers is probably asking to o much in the absence of additional constraints sp ecifying howchanges

should b e distributed among the joints anymethodislikely to give unnatural results Joint limits

provide some constraints as do the weighting parameters at each joint but these do not seem to

b e adequate particularly for manipulating recognizable b o dies Inverse kinematic metho ds which

p erform quite acceptably for disembodied chains or mechanical rob ots do not necessarily translate

well to animate gures the results can often app ear unnatural This problem is that the term

unnatural is sub jective and dicult to quantify More p owerful p ositioning to ols are likely to

result from considering these factors more carefully

Either of the inverse kinematic algorithms presented in Chapter is adequate for interactive direct

manipulation They are b oth simple to implement and are fast enough to provide go o d feedback

for gross p ositioning tasks However if high accuracy is a requirement and interactive resp onse time

not so much of a concern then other solution metho ds may oer b etter p erformance

Comments ab out Constraints

In Chapter the choice of the CCDbased p enalty metho d for implementation within the gure

animation editor was largely based on p erformance issues the alternative Lagrange multiplierbased

approachwas deemed to o demanding for interactive use on a typical workstation This problem

is temp oraryhowever and it is worth reconsidering the decision in lightofcontinuing increases in

CHAPTER CONCLUSION

CPU p ower At the same time we can sp eculate a little ab out what typ es of constraints mightbe

useful to develop and how constraints might b e used for animation as well as manipulation

Given sucient CPU sp eed to retain numerical stability the Lagrange multiplier approachisthe

more general of the two The CCDbased metho d is fast b ecause it reduces a dicult problem to a

series of simpler ones which can b e solved analytically using only simple geometric quantities But

for problems more complicated than solving p osition and orientation constraints the reduction to

analytical form is not always p ossible so the approach is limited to these typ es of constraints In

contrast the Lagrange multiplier metho d could b e used to enforce many dierenttyp es of constraints

including nongeometric ones the only requirementisthattheconstraints b e some function of the

jointvariables within a gure

Lo oking Ahead

What sorts of constraints other than the obvious geometric ones might b e useful for gure ma

nipulationInitial exp erimen tation with the constraintsolver implemented in the animation editor

indicates that geometric constraints alone while useful are not always enough To o often a solution

which is correct in the sense that all geometric constraints are satised seems incorrect b ecause it

puts the gure into an awkward lo oking p ose Think for example of a p erson going from a standing

p osition into a crouch The natural tendency is for the knees to move apart as the p erson crouches

b ecause that is the comfortable way to crouch To manipulate a standing gure in the editor into a

crouch one mightlock the to es of each fo ot in place and then pull the p elvis down to make the gure

crouch But without any other information ab out how the legs should move one is just as likely to

see the knees move together as the gure crouches as to see them move together Perhaps this could

b e rectied by sp ecifying additional geometric constraints on the knees but cho osing and sp ecifying

these auxiliary constraints would probably b e dicult and timeconsuming A useful alternative

would b e a standard set of constraints which applied to the gure at all times These might include

constraints to avoid uncomfortable p ostures or p ostures whichwould place a p erson obalance

The latter can b e implemented as a constraint on a gures centerofmass a computable quantity

given a set of joint angles while the former would require some function which measured comfort

or naturalness given the same set of variables

In addition to manipulation constraints could also b e useful in animating a gure A natural

place to start would b e to consider animating the constraintvalues Animating a p osition constraint

for example might dene a tra jectory for some limb extremitytofollow A simultaneous animated

orientation constraintwould control the orientation of the extremityasittraversed the tra jectory

But for complex motions constraints will probably b e more helpful in describing a desired movement

with some sort of scripting approach For example the movements of the supp ort fo ot of a walking

CHAPTER CONCLUSION

bip ed might b e describ ed in terms similar to the following

After heel strike the heel of the fo ot maintains its p osition until the ball of the fo ot

touches the ground For a short p erio d the entire fo ot remains lo cked in place while the

body moves forward Toward the end of the step the heel of the fo ot breaks contact with

the ground rst followed shortly after by the ball of the fo ot

A description such as this identies a numb er of simple constraints which are in eect during the

movement If the description were more sp ecic ab out the time intervals and lo cations involved we

would have a fairly detailed script for animating the stance fo ot during a walk A simple scripting

language could b e implemented whichwould allowmovements such as this to b e describ ed textu

ally A script would need to sp ecify a set of constraints to b e imp osed and provide some wayof

activating and deactivating these constraints over the course of a movement An interpreter for such

a descriptive scripting language would b e a useful to ol for gure animation

Directions

There are plenty of problems still to b e addressed Atalowlevel of detail it is worth determining

whether the p erformance of the Lagrange multiplier constraint satiser can b e improved up on First

Surles has shown that some constraint problems for articulated gures can b e solved in linear time

Sur He lists a numb er of prerequisites for this but in the context of chemical protein mo delling

which is his application area An analysis of these prerequisites and what they mean in the context

of articulated skeleton manipulation would b e useful A second area which could b e explored to

sp eed up the solution pro cess is to consider Maciejewskis incremental algorithm for computing the

SVD MK the referenced pap ers on this topic are interesting reading and may suggest alternate

approaches to the inverse kinematic problem in general

Designing interactiveinterfaces whichmake eective use of constraints is challenging and deserves

some attention How should constraints b e representedSelectedEditedWhat should happ en

when constraints conict with each otherHo w should constraints b e animated and controlled over

timeWhat ab out constrain ts involving multiple gures

And nallydoinverse kinematics and constraints provide a convenientlayer up on which higher

level pro cedural motion mo dels can b e builtAn in teresting exercise would b e to extend Bruderlins

walking algorithm to handling uneven terrain making use of the basic techniques describ ed in this

thesis to achieve and maintain fo otholds

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