Font Llagunes, Josep Maria

Kövecses, József

Peiret Giménez, Albert

UPCommons

The discretized coulomb friction model in a non-singular complementarity formulation for multibody systems with contacts

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ENOC 2017, June 25-30, 2017, Budapest, HungaryThe Discretized Coulomb Friction Model in a Non-Singular ComplementarityFormulation for Multibody Systems with ContactsAlbert Peiret, József Kövecses and Josep M. Font-LlagunesDepartment of Mechanical Engineering, McGill University, Montréal, CanadaDepartment of Mechanical Engineering, Universitat Politècnica de Catalunya, Barcelona, SpainSummary. The dynamics of a multibody system with contacts and friction can be formulated as a linear complementarity problem(LCP) via the discretization of the friction cone. Several discrete friction models are available in the literature, and all of them producerank-deficient LCPs due to the use of multiple directions to discretize the tangent plane. In this work, a new discrete friction model isproposed which couples the equations in these directions and allows the dynamic formulation to take the form of a full-rank LCP.AbstractThe dynamics of multibody systems with contacts presents some well-known challenges, especially when it involvesfrictional contacts. If friction is neglected, the representation takes the form of a linear complementarity problem (LCP),for which the existance of solution is guaranteed, and several direct or iterative solution algorithms are available in theliterature. However, considering friction in contacts using the Coulomb model turns the formulation into a nonlinearcomplementarity problem (NCP), for which the existence of solution is not guaranteed.Several friction models have been proposed to approximate the Coulomb model so that the model can take the form of anLCP [1, 2, 3]. These models are based on the discretization of the friction cone, which turns the non-linear inequalitiesof the original model into linear ones by choosing a finite number of directions in the tangent plane. As a result, thisdiscretized model approximates the friction cone with a pyramid. It is worth mentioning that this issue is only related tothe contact points that are not sliding, because the static friction force is confined to the friction cone; whereas the kineticfriction force is defined by a constitutive relation.Generally, there are two ways to discretize the Coulomb friction model. One approach is based on a discrete force, inwhich the directions are used to define non-negative components of the friction force t =åb je j, being e j the unit vectorsassociated with each direction of the tangent plane [2, 3]. The second approach is based on a discrete acceleration, inwhich the directions are used to define non-negative components of the pre-sliding tangential acceleration u˙t =åk je j [1].The size of the problem to solve is very similar in both cases, and they can behave the same for certain sets of chosendirections.A common characteristic of these models in multibody system formulations is that the lead matrix of the LCP is rankdeficient, even though there is no redundancy in the contact forces of the system. This is due to the use of more than twonon-idependent directions in the tangent plane, and so it is an artifact of the model. Nevertheless, it is possible to define afull-rank LCP formulation using the discretized friction model proposed here.Let us define k different directions in the tangent plane, so that the relative tangent acceleration can be discretized andparameterized as followsu˙t =kåj=1(k j+e j k j e j) (1)where e j are the unit vectors of these directions, and k j+ > 0 and k j  > 0 are the positive acceleration parameters pointingto the positive and negative directions of e j, respectively. On the other hand, two friction saturations are defined in eachdirection to enforce the limits of the friction force in the positive and negative directionss j+ = mln  eTjt+rk j+ > 0 (2)s j  = mln+ eTjt+rk j  > 0 (3)where m is the static friction coefficient, ln is the normal force, and r is the phantom inertia. The friction saturations sneed to be complementary to the parameters k , so that sliding only starts when the friction force reaches the bounds at theopposite direction. This is enforced via the complementarity conditions s j+k j  = 0 and s j k j+ = 0.The novelty of this model is the addition of the phantom inertia r into the equations, which does not change the dynamicsof the contact model, but it adds some extra information into the equations. This makes the lead matrix of the LCP fullrank, in spite of using a discrete friction model. The complementarity conditions make sure that force and accelerationwill never point to the same direction, that is, adding that term does not change this. It can be shown that a solution whereboth parameters k j+ and k j  are positive is not possible, no matter how big r is. When the friction force reaches a bound,only the acceleration pointing to the opposite direction will be positive; and when the friction force is within the boundsof that direction, i.e., jeTjtj< mln, the two parameters k j+ = k j  = 0. The inertia r only makes the friction saturation ofone bound larger, in case the sliding starts in the oposite direction, and thus it does not influence the stick-slip transition,see Figure 1.This model can be introduced in a formulation for multibody systems, where the dynamic representation takes the formof an LCP. Let q be the minimal set of n generalized coordinates of a multibody system, so that any bilateral constraint isENOC 2017, June 25-30, 2017, Budapest, Hungarye jjσ j–μλnCoulomb model e jjσ j–= 0Discretized model σ j+ρκj+κj+e jT μλn=–ttte jT tx xyyFigure 1: Discretized friction model, and model variables for the j-th direction. Static friction (left), and stick-slip transition (right).embedded. A minimal set of n generalized velocities v is chosen, such that q˙ =  v, where  (q; t) is the transformationmatrix. With this representation, the dynamic equations can be written asMv˙+ c= fa+ ft+ fn (4)where M(q) is the n n mass matrix, c(q;v) is the n 1 array containing the Coriolis and centrifugal terms, fa is thegeneralized applied force, and ft and fn are the generalized tangential and normal contact forces, respectively.The gap functions that define the distance between the r contacting surface pairs are arranged into the array (q)  0.Therefore, the normal acceleration is ¨= u˙n = Jnv˙+ J˙nv> 0, where Jn(q) is the rn contact Jacobian matrix. Therefore,the generalized normal contact forces are fn = JTnn, where the r normal contact forces aren > 0 complementary to u˙n > 0 (5)The tangent plane of each contact point is characterized by two orthogonal directions, in which the components of thesliding acceleration can be expressed as u˙t = Jtv˙+ J˙tv, where Jt(q) is the 2rn friction Jacobian matrix. The generalizedfriction force is then ft = JTt t, where t contains the 2r friction contact force components. The discrete friction modelrequires a set of directions in the tangent plane, so that the components of their ki unit vectors of the i-th contact point canbe arranged as columns of the matrix Ei =e1    eki, and therefore, u˙ti = Eii+ Eii . Equations (1), (2) and (3)can be arranged in a matrix form for each contact, so that the 2 arrays of ki friction saturations arei+ = lni  ETi ti +ii+ > 0 complementary to i  > 0 (6)i  = lni +ETi ti +ii  > 0 complementary to i+ > 0 (7)where i = [ mi    mi ]T, and i = diag(ri; : : : ; ri). Then, the system dynamic equations can be writen as266664M  JTt  JTn 0 0Jt 0 0  E EJn 0 0 0 00 ET  0 0  ET   0377775266664v˙tn+ 377775+266664c  faJ˙tvJ˙nv00377775=26666400u˙n +377775 (8)where E = diag(E1; : : : ; Er),  = diag(1; : : : ; r), and  = diag(1; : : : ; r). The variables v˙ and t, which do nothave any complementarity condition, can be eliminated from the system via the Schur complement, so that an LCP withthe variables n, + and  , and the complementarity conditions in Eqn. (5), (6) and (7) can be formulated.The role of the square diagonal phantom inertia matrix  is clearly shown in equation (8). Due to the use of dependentdirections to discretize the friction cone, the two last block columns of the lead matrix would be linearly dependent.However, the discrete friction model proposed here gives rise to a full-rank LCP formulation, by coupling the frictionsaturations without changing the dynamics of the system. If the contact forces are not redundant, the contact normaland friction Jacobian matrices are full-rank, and so is the lead matrix. Nevertheless, if the system presents redundancy,some physics-based techniques (e.g., constraint relaxation, minimum-norm solution, etc.) would be required to solve thedynamic equations and determine the contact forces.References[1] Glocker C. (2001) Set-Valued Force Laws. Troy, New York, USA: Springer.[2] Stewart D.E., Trinkle J.C. (1996) An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and coulomb friction. Inter-national Journal for Numerical Methods in Engineering, vol. 39, pp. 2673–2691.[3] Anitescu M., Potra F.A. (1997) Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems.Nonlinear Dynamics, vol. 14, pp. 231–247.

The discretized coulomb friction model in a non-singular complementarity formulation for multibody systems with contacts

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