Mechanism and Machine Theory 105 (2016) 568–582 Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt A linear complementarity formulation for contact problems with regularized friction Farnood Gholami a, * , Mostafa Nasri a, 1 , József Kövecses a , Marek Teichmann b a Department of Mechanical Engineering and Centre for Intelligent Machines, McGill University, 817 Sherbrooke Street West, Montreal, Quebec H3A 2K6, Canada b CM Labs Simulations Inc., 645 Wellington Street, Montreal, Quebec H3C 1T2, Canada ARTICLE INFO Article history: Received 7 April 2016 Received in revised form 18 July 2016 Accepted 20 July 2016 Available online xxxx Keywords: Multibody systems Contact dynamics Linear complementarity problems Regularized friction ABSTRACT A mathematical formulation in terms of a linear complementarity problem with regularized friction is introduced for multibody contact problems. In this approach, contacts are character- ized based on kinematic constraints while the friction forces are simultaneously regularized and incorporated into the formulation. The variables of the resulting linear complementarity problem are only the normal forces. The main advantage of this formulation is that the dimen- sion of the resulting linear complementarity problem is significantly less than its counterpart formulations in the literature, and hence, faster simulations can be achieved. The proposed formulation is examined for a set of benchmark examples yielding promising results. © 2016 Elsevier Ltd. All rights reserved. 1. Introduction Contact modelling, simulation, and analysis of mechanical systems need proper treatments due to the non-smooth nature of the problem. Constraint and explicit force representation formulations are two main approaches to deal with the contact problems [11,16,17,28,36]. In constraint based approaches, the contact problem is formulated by imposing the kinematic unilateral constraints between bodies in contact. Such a representation was first formulated in Ref. [30], which was further expanded using methods of modern mathematics such as measure differential equations that may directly lead to effective numerical algorithms for contact problems [29]. For frictionless sustained contacts, a complementarity problem can be formed using the fact that either normal contact velocity or force exists at each time instance. In this case the resulting mathematical model is a linear complementarity problem (LCP) and relatively straightforward to solve, because it can be proven that such an LCP always has a solution [27]. By adding the Coulomb friction, as one of the most common friction models, this phenomenon can be still represented by a complementarity problem but the resulting cone is not a positive orthant and hence, non-linear optimization techniques needs to be considered, such as the Gauss–Seidel or fixed-point iteration methods [5,22,23,31]. Another possibility is to derive a linear complementarity problem based on a polyhedral approximation of the actual friction cone [4,17,43]. In this case, the resulting LCP formulation can be stated at the acceleration [17], velocity [4] or position [43] levels. Acceleration level formulations are not guaranteed to always have a solution. This was one of the motivations to propose an LCP formulation at the velocity-level [4]. The formulation * Corresponding author at: Department of Mechanical Engineering and Centre for Intelligent Machines, McGill University, 817 Sherbrooke Street West, Montreal, Quebec817 Sherbrooke Street West, Montreal, Quebec, H3A 2K6, Canada. E-mail addresses: farnood.gholami@mail.mcgill.ca (F. Gholami), m.nasri@uwinnipeg.ca (M. Nasri), jozsef.kovecses@mcgill.ca (J. Kövecses), marek@cm-labs.com (M. Teichmann). 1 M. Nasri is currently with the Department of Mathematics and Statistics, University of Winnipeg, 515 Portage Ave, Winnipeg, MB, R3B 2E9, Canada. http://dx.doi.org/10.1016/j.mechmachtheory.2016.07.016 0094-114X/© 2016 Elsevier Ltd. All rights reserved.