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Existence and Uniqueness of Rigid-Body Dynamics With 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) .
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