EXISTENCE AND UNIQUENESS OF RIGIDBODY

DYNAMICS WITH COULOMB FRICTION

Pierre E Dup ont

Aerospace and Mechanical Engineering

Boston University

Boston MA USA

ABSTRACT

For motion planning and verication as well as for mo delbased control it is imp ortanttohavean

accurate dynamic system mo del In many cases friction is a signicant comp onent of the mo del When

considering the solution of the rigidb o dy forward dynamics problem in systems such as rob ots wecan

often app eal to the fundamental theorem of dierential equations to prove the existence and uniqueness

of the solution It is known however that when Coulomb friction is added to the dynamic equations

the forward dynamic solution may not exist and if it exists it is not necessarily unique In these

situations the inverse dynamic problem is illp osed as well Thus there is the need to understand the

nature of the existence and uniqueness problems and to know under what conditions these problems

arise In this pap er we show that even single degree of freedom systems can exhibit existence and

uniqueness problems Next weintro duce compliance in the otherwise rigidb o dy mo del This has two

eects First the forward and inverse problems b ecome wellp osed Thus we are guaranteed unique

solutions Secondlyweshow that the extra dynamic solutions asso ciated with the rigid mo del are

unstable

INTRODUCTION

To plan motions one needs to know what dynamic b ehavior will o ccur as the result of applying given

forces or torques to the system In addition the simulation of motions is useful to verify motion

planning op erations Alternatively one may wish to design a feedforward controller which applies the

forces or torques necessary to pro duce a desired output motion Rigidb o dy mo dels are often used for

these tasks

While frequently neglected friction has b een shown to b e very signicantinmany systems This

applies b oth to friction b etween the rigid b o dies making up the system and to friction b etween

the system and its environment In the context of this pap er the latter may constitute the most

imp ortant case since the friction co ecientbetween a mechanism and its environment can b e quite

high Rob otic examples include the insertion of a p eg into a hole and a nger sliding on an ob ject

surface prior to or during grasping

Beginning at the turn of the centuryanumb er of researchers haveshown that when the simple

Coulomb friction mo del is combined with the rigidb o dy forward dynamic equations there are cases

when no solutions exist and also cases when multiple solutions o ccur

Trans of the Canadian So cietyofMechanical Engineers Vol No A pp

In a recentpaperwehave demonstrated that systems p ossessing a single degree of freedom can

exhibit this b ehavior In these systems existence and uniqueness problems arise only when the

eective inertia of the system can take on a negativevalue In this case the value of input force or

torque determines the numb er of solutions for a single degree of freedom

The question to b e resolved is to determine if a unique motion can b e asso ciated with eachvalue

of input force or torque The case of zero solutions has b een successfully addressed in In the case

of multiple solutions it is easy to show that the static solution can b e separated from the dynamic

solutions What remains is to determine if the multiple dynamic solutions are all valid and if so

which will o ccur This problem has not b een successfully addressed in the literature and is the topic

of this pap er

In the next section background material is presented and we describ e the class of problems to b e

considered In Section the existence and uniqueness problem for systems with one degree of freedom

is reviewed In Section weintro duce compliant mo dels and compare their dynamic equations with

those of rigidb o dy mo dels A screw drive is used as an example A stability analysis is given in

Section to show that the extra solutions from the scalar rigidb o dy mo del are unstable Conclusions

are presented in the nal section of the pap er

BACKGROUND

The rigidb o dy dynamic equations for a mechanical system such as an op enkinematicchain rob ot are

of the form

Aq q bq qf q q q

The nvectors of generalized co ordinates suchasjoint p ositions and asso ciated input forces or torques

are q and resp ectively where n is the numb er of degrees of freedom of the system The conguration

nn

dep endent inertia matrix is represented by A R It is b oth symmetric and p ositive denite The

n n

vector b R consists of centrifugal Coriolis and gravity terms The vector f R includes all

friction terms and is a function of the generalized co ordinates and their rst and second derivatives

The Coulomb friction force is directed so as to opp ose relative motion and is prop ortional to the

normal force of contact during motion For unilateral constraints the normal force F must b e p ositive

n

and Coulomb friction can b e expressed as

F

n

v jF j F vF

f n f

v jF j F

f n

where F is friction force is the co ecient of friction and v is the velo city of relative motion

f

These equations dene what is commonly called the friction cone During motion the friction force

must lie on the friction cone while during static contact it may also lie inside the cone

The forward dynamics problem is to solve for the p ositions velo cities and accelerations given the

input torques or forces and the initial conditions This is the problem to b e solved for simulation At

each time step the known torques p ositions and velo cities are used to compute the accelerations In

the absence of friction this involves solving a set of linear algebraic equations for the accelerations

Using the values of acceleration and velo citynumerical integration yields the velo city and p osition at

the next time step The inverse dynamics problem is simpler Given the desired p ositions velo cities

and accelerations the system equation is solved for the forces or torques at each instant of time

This is the approachtaken in mo delbased control

Trans of the Canadian So cietyofMechanical Engineers Vol No A pp

Friction can arise due to relativemotionbetween the rigid b o dies making up the mechanism or

due to contact b etween one or more of these b o dies and the environment We will refer to these two

typ es as internal and external friction resp ectivelyInternal friction is due to such elements as the

transmissions and b earings External friction acts at contacts b etween the rob ot and its environment

Thus it is imp ortant in grasping and assembly op erations

In the case of external friction must b e mo died to include the Jacobian matrix relating

innitesimal joint and contact p oint displacements If the generalized co ordinate directions can b e

chosen so as to coincide with the directions of the external normal and friction forces then the Jacobian

reduces to the identity matrix and the form of the expression reduces to This is true in many

imp ortantcasesWe will only discuss problems of this typ e

In general the inclusion of loaddep endent friction in the dynamic equations renders them implicit

in the accelerations The cause of the implicitness is the dep endence of friction on the magnitude of

the normal force which is itself a function of the resultant force at the frictional contact Expressed in

a lo cal co ordinate frame the comp onents of the resultant force can b e formulated in terms of p ositions

velo cities and accelerations Just as with f in these comp onents will b e ane transformations

of the accelerations

If the direction of a normal force is constant in the lo cal co ordinate frame the normal force can b e

expressed as an ane transformation of the joint accelerations For the case of a single source of

friction asso ciated with each degree of freedom the friction vector f becomes

f q q q M jC q q dq qj

nn

Here M R is a diagonal matrix with diagonal elements sgnq and are the co ecients

i i i

nn n

of friction C R takes the form of an inertia matrix and d R The expression in the absolute

value is the normal force an ane transformation of acceleration vector

This expression is true for example of friction in transmissions p eginhole insertion and sliding

ngers Since the sign of the normal force at a bilateral constraintcanchange its absolute value must

b e used to obtain its magnitude Only friction of this form will b e considered here

EXISTENCE AND UNIQUENESS FOR THE SCALAR CASE

The scalar form of the dynamic equation with friction of the form is given by

aq q bq q jcq q dq qjsgnq

Note that the single normal force F isgiven by F cq dThus wehavetwo equations

n n

a cq b d sgnq sgnF

n

a cq b d sgnq sgnF

n

These equations are intersecting lines in the space formed by andq From it is clear that

these lines dep end on q andq For clarity of exp osition this dep endency is neglected in the following discussion

Trans of the Canadian So cietyofMechanical Engineers Vol No A pp

τ Fn <0 Fn >0 (5) . q>0 . .A’ (6) A . b+ µ d . q<0 q=0 .. -d/c B q .B’. .B’’ b- µ d b-ad/c C. .

C’

Figure Scalar Case of Versusq for acand d Equations and form twoVs

The upp er dotdashed V corresp onds toq The lower dotted V corresp onds toq The dark

line segment on the axis is the static region asso ciated withq The horizontal arrows indicate

discontinuities in acceleration which o ccur whenq

SOLUTION EXISTS AND IS UNIQUE

The case of ac and d isshown in Figure Note that a and c representing inertias must b e

p ositive Since these equations represent bilateral constraints normal force can b e of either sign and

F along the vertical line throughq dcThus forq the system lies on the upp er V and

n

forq on the lower V If the system is initially static it lies on the axis b etween the two lines

From this static region if the torque is increased ab ove b d motion withq ensues Similarlyif

the torque is decreased b elow b d motion withq ensues At b d the friction force lies on

the friction cone withq

Consider the b ehavior of various p oints on the graph At A b oth the velo city and acceleration are

p ositive At B acceleration is negative while the velo city is p ositive but decreasing Since p oint B is

within the static band on the axis the system will stick when the velo city reaches zero As shown

by the arrow the system jumps to B and the acceleration discontinuously jumps to zero Point C

however is outside the static region Whenq is reached at C the system jumps to C on the

negativevelo cityVForq p oints B and A lead to sticking and velo city reversal resp ectively

For almost all values of input torque Figure together with the currentvalue of velo cityprovides

a unique value of acceleration For example consider that the three values ofq at B B and B

are asso ciated with a single value of torque These solutions however corresp ond toq q

andq resp ectively The twoambiguous p oints o ccur for b d Note that a static friction

co ecient would resolve this ambiguity Let us now consider the conditions under which

s

solution existence and uniqueness fail for most values of input torque

SOLUTION NONEXISTENCE AND MULTIPLICITY

Consider the graph of and in Figure Here acand so has a negative slop e This

corresp onds to a negative eective inertia Once again the upp er V is asso ciated with p ositivevelo cities

and the lower with negativevelo cities Starting from rest the input torque must leave the static region

to initiate motion For example if the torque is increased ab ove b d motion in the p ositive direction

Trans of the Canadian So cietyofMechanical Engineers Vol No A pp

τ Fn <0 Fn >0 (5) . q>0 .A’ A. b+ µ d . .. B’’ q=0 q . -d/c B’. .B b-ad/c . C. D b- µ d .E’ .E q<0.

(6)

Figure Scalar Case of Versusq for ac In this case there can b e multiple solutions or none

will ensue Now however there are two p ossible solutions given for example by A and A Ifthe

torque is decreased to the value asso ciated with B there are three p ossible values of acceleration given

by B B and B B corresp onds to the static case B and B b oth corresp ond toq but with

dierentvalues of accelerationq

If the torque is decreased slightly b elow b adc the only valid solution is the static case C but

as wesaw with the previous gure the system would normally only jump like this when the velo city

reached zero Thus there app ears to b e no solution provided by the graph forq andb adc

We can make the same arguments for negativevelo cities According to the value of input torque and

velo citywecanhave either zero one two or three feasible solutions

In summary a necessary and sucient condition for solution existence and uniqueness for anyvalue

of input torque is given by

ac

When ac there exists a critical value of torque b adc for which yields no

cr cr

solutions and yields two or three solutions or vice versa dep ending on sgnq

cr

Several investigators have considered the case whenq and is decreased b elow b adc

cr

They have concluded that velo city jumps discontinuously to zero For an explanation the reader is

referred to and Recently the dynamics of this transition have b een studied using compliant

mo dels

In cases of multiple solutions suchasgiven by B B and B in Figure a metho d is needed to

select the actual solution from the set of feasible solutions Wehave already stated that the static case

can only b e escap ed by applying a torque outside of the static region b ounded by b dThus ifq

is initially zero B is the only p ossible solution Solution pairs suchasA A and B B however all

corresp ond toq and so velo city cannot b e used to distinguish b etween them This issue is resolved

in the following sections by adding compliance to the mo del

COMPLIANTBODYMODELS

Recall that the denition of system state is such that if the state is known at time t the b ehavior

of the system for t t can b e computed given the inputs for t t For those cases when is

Trans of the Canadian So cietyofMechanical Engineers Vol No A pp

not satised however sp ecifying the p osition and velo city of the rigid b o dies q q do es not givea

unique vector of accelerationsq and thus the problem is illp osed The rigidb o dy formulation do es

not provide an adequate denition of system state

To make the problem wellp osed wewould like to add the minimum numb er of degrees of freedom

to our state vector Solution multiplicity arises due to the unknown sign of the normal force at the

friction interface Thus our wellp osed mo del must allow us to compute normal force as a function of

system state

Todosoweintro duce nonzero compliance in the mo del This is done by adding a spring with

a comp onent of its displacement along the direction of the normal force to the otherwiserigid mo del

Any spring lo cation is acceptable as long as the normal force can b e expressed as a linear function of

spring displacement As a result the sign of the normal force equals the sign of the spring displacement

In this wayweintro duce one new degree of freedom two new state variables eg spring displacement

and velo city for each indep endent normal force

DERIVATION OF RIGID AND COMPLIANT MODELS

To explain the equations asso ciated with the compliant mo del it is worthwhile to rst explain in detail

the derivation of the rigid mo del The equations for the rigidb o dy mo del given by

aq b jF jsgnq

n

F cq d

n

are actually a simplied form If one was to write the equations for the scalar system from a freeb o dy

analysis two equations would b e obtained

a q b c F d jF jsgnq

n n r

a q b c F d jF jsgnq

n n r

In this form the normal force can app ear outside the absolute value andq is the relativevelo cityat

r

the friction interface

RIGIDBODYANALYSIS

In this case the displacements q and q are related by a p ossibly nonlinear function

q g q

Cho osing q as the dep endentvariable we can use and its rst twoderivatives to solve for

F in terms of q and its derivatives This gives an expression of the form of

n

COMPLIANTBODY ANALYSIS

In this case a spring is added to the system such that is no longer satised Instead we can write

an expression for normal force of the form

F k q g q

n ef f

in which k is the eective spring constant ef f

Trans of the Canadian So cietyofMechanical Engineers Vol No A pp

y = lq MASS α m

θ τ

MOTOR

Figure Screwdriven Mass The motor applies torque to the screw The mass misattached to

the nut In the rigid case its displacement y is related to the screw displacement q by the screw

lead l The helix angle and thread angle are also shown

With no longer satised q and q are now indep endentvariables We can substitute our

expression for F into and to obtain the nal form of these equations Clearly the sign of

n

normal force is known as is the relativevelo city and so the solution of these equations is a wellp osed

problem

EXAMPLE SCREW DRIVE

Screws are sometimes used as transmission elements in rob ots See for example Consider the

case of a screw moving a mass m with gravityactingdownward as shown in Figure The following

denitions will b e used

q screw displacement angle y mass displacement

l screw lead distanceangle y lq

screw helix angle screw thread angle

I rotational screw inertia tan cos

n

Summing forces on the screw and nut we can obtain equations of the form given by and

These are written in terms of the screw and mass displacements q and y resp ectively

l

I q F cos sin jF j cos sgnq

n n

tan

my F cos cos jF j sin sgnq mg

n n

RIGIDBODYANALYSIS

If the screw and nut are rigid the displacements q and y are related by the screw lead l Corresp onding

to wehave

y lq

Selecting y as the dep endentvariable we wish to obtain an expression for F like Dening as

n

cos

Trans of the Canadian So cietyofMechanical Engineers Vol No A pp

reduces to

sgnF sgnq

n

my F cos mg

n

sgnF sgnq

n

Rearranging to get the form of

 

mg ml

q F

n

cos cos

Note that the expression cos will b e p ositive for friction co ecients satisfying

cos

tan

Since and are small angles can b e true for very large co ecients of friction

To obtain the nal form of the rigidb o dy screw equations is substituted into After

simplication the dynamic equation is found to b e

 

mg l tan ml tan

q I

tan tan

Intro ducing the standard expression for screw eciency

i



tan

sgnF sgnq



n



tan

i



tan

sgnF sgnq

n

tan

we can also write this as

 

mg l ml

q I

i i

COMPLIANTBODY ANALYSIS

In this case we need to add compliance to the screwnut system such that a comp onent of the spring

displacement is parallel to the normal force It is p erhaps most appropriate to mo del the screw with

b oth torsional and axial springs to represent screw twisting and stretching resp ectively Since the

threads and hence normal force are inclined by the helix angle b oth torsional and axial springs

provide a comp onent of displacement in the normalforce direction For simplicity let us consider only

an axial spring as depicted in Figure

Clearly the springs impart a vertical force F on the nut given by

v

F k lq y

v

We can also express F as

v

F my mg

v

Using we can obtain an expression corresp onding to

 

k

F lq y

n

cos

Trans of the Canadian So cietyofMechanical Engineers Vol No A pp

y

SCREW

NUT

SCREW

q

Figure Planar Representation of Axiallycompliant Screw Mo del The springs are considered embed

ded in the screw adjacent to the friction interface Physically this might b e construed as the stiness

of the screw and nut threads

The variable sign is chosen as in the rigidb o dy case but note that the eective stiness is p ositiveif

is satised The sign of the normal force then dep ends on the sign of the spring displacement

sgnF sgnlq y

n

The forward and inverse dynamics of and together with are well p osed We can combine

these three equations to obtain

kl

I q lq y

i

my k lq y mg

These two equations can b e compared with the rigidb o dy equation In this case is chosen

i

according to sgnlq y sgnq

STABILITY ANALYSIS

As discussed in Section there are cases when the rigidb o dy formulation of the forward dynamics

problem yields several p ossible values of acceleration for a xed value of input torque Examples are

the pairs of p oints A A B B and E E in Figure These p oints are distinguished from each

other bysgnF

n

Since the compliant mo del provides a unique value of sgnF we can now p ose the following

n

questions Do corresp onding multiple solutions exist for the compliant mo del If so are they

stable

As the spring stiness b ecomes very large the compliant mo del approaches the rigid mo del By

considering this limiting case it is p ossible to p erform an approximate analysis We can cho ose the

stiness large enough so that the dynamics aecting normal force will b e fast and the spring displace

ment small compared to those aecting the co ecients a q b q q c q and d q in equations

i i i i

and Thus we can treat these co ecients as constants when considering the stability of the normal

force dynamics Put another waywe are considering the class of systems for which in the time scale

of interest xed values of input torque or force pro duce constant accelerations

Trans of the Canadian So cietyofMechanical Engineers Vol No A pp

To answer question consider that at each solution of the rigidb o dy equations F is constant

n

For the compliant mo del constant F implies that

n

q g q constant

and therefore

q g q

Under the assumption that the co ecients a b c and d are constantover the time scale of interest

i i i i

the rigidb o dy analysis which reduces and to and involves only the preceding equation

Therefore the xed p oints of the normal force dynamics for the compliant mo del corresp ond to the

rigidb o dy solutions

To answer question regarding the stability of these xed p oints let us reformulate the problem

Recalling that expresses the normal force of the compliantmodelwe can dene a variable v

expressing the very small spring displacement

v q g q

Dierentiating this equation twice we see that v evolves according to

v q g q q g q q

As b efore we assume that spring stiness is large enough and thus v is small enough that wecan

evaluate g and g at the op erating p oint of the slow dynamics

Substituting and into and expressing F in terms of v yields

n

h sgnv

v kh v u h

i i

h sgnv

in which h h and u are constants and k is spring stiness

The xed p oints of v corresp ond to the xed p oints of normal force for the compliant mo del Their

stability dep ends up on the eigenvalues which satisfy the characteristic equation

kh

i

Solutions are of the form

p p

v c sin kh tc cos kh t uk h h

i i i i

p p

kh t kh t

i i

c e uk h h v c e

i i

The former is the oscillatory solution exp ected from a compliant mo del without damping The latter

equation shows that if h the normalforce xed p oint is unstable with the spring displacement

i

v either increasing or decreasing exp onentially according to initial conditions An exp onentially

increasing normal force would rapidly bring the system to rest under impactlike conditions An

exp onentiallydecreasin g normal force would move the system towards the stable xed p oint

Wehave b een considering the limiting case when k and the compliant mo del approaches the

rigid mo del From it is clear that in the limit the particular solution for spring displacement

go es to zero as exp ected Now consider the addition of damping prop ortional tov This is actually

necessary to make the normalforce dynamics fast when h For h the system will b e stable

i i

and exhibit damp ed oscillations For h however no nite amount of damping will stabilize the

i

system indicating that this case is not a valid solution of the rigidb o dy mo del

Trans of the Canadian So cietyofMechanical Engineers Vol No A pp

EXAMPLE SCREW DRIVE

Let us reconsider the screw example The co ecients in and are in fact indep endentofq y

and their velo cities In this case it is clear without assumption that the normalforce xed p oints of

the compliant mo del corresp ond to the rigidb o dy solutions The spring displacement from is

v lq y

and we can use the compliantmo del equations and to solve

v l q y

These substitutions yield an expression of the form of

 

I ml

i

v k v lI g

mI

To determine stabilitywe are interested in the sign of h which for this case is

i

 

I ml

i

h

i

mI

The sign of h dep ends on the sign of its numerator which is the eective inertia of the rigidb o dy

i

mo del Recall from Section that questions of existence and uniqueness arise when one solution

gives a negativeeective inertia Therefore we can asso ciate h with normalforce xed p oints cor

i

resp onding to the p ositivelyslop ed line in Figure and h with unstable xed p oints corresp onding

i

to the negativelyslop ed line

Consideringq in Figure this analysis tells us that for the pairs of solutions A A and

B B the counterintuitive solutions A and B are unstable This analysis also suggests that when

initiating motion withq q must jump discontinuously to the p ositivelyslop ed branchofE

CONCLUSIONS

In this pap er wehave examined the eect of Coulomb friction on the forward and inverse dynamics of

rigidb o dy systems For the scalar case we presented a necessary and sucient condition for forward

solution existence and uniqueness whichwas rst derived in Violating this condition involves

friction co ecients which exceed typical magnitudes of internal friction This is an imp ortant result

b ecause it indicates that in many cases solution existence and uniqueness of b oth the forward and

inverse dynamics is not problematic

Co ecients of friction of this magnitude however may b e encountered in certain loweciency

transmissions and in external friction applications involving static or sliding contact with the envi

ronment For these cases it is imp ortant to understand the limitations of rigidb o dy mo dels and to

characterize the actual system motion Toward this end wehaveintro duced a compliantb o dy for

mulation for which the normal force at the friction interface can b e expressed as a function of system

state

The compliantb o dy mo del while p ossessing an additional degree of freedom leads to two results First this mo del do es not exhibit the existence and uniqueness problems of the rigidb o dy mo del

Trans of the Canadian So cietyofMechanical Engineers Vol No A pp

Secondly the compliantbody model was used to analyze the multiple forward dynamic solutions of

the rigidb o dy formulation It was shown that the extra solution of the rigidb o dy mo del is unstable

By extending this technique to higher dimensional systems it may b e p ossible even for high friction

co ecients to solve b oth forward and inverse dynamics problems using rigidb o dy mo dels together

with a set of stability rules for resolving the existence and uniqueness issues

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