Hysteresis loop of the LuGre Model ⋆ Mohammad Fuad Mohammad Naser a , Fay¸ cal Ikhouane b a mohammad.fuad@upc.edu b faycal.ikhouane@upc.edu Universitat Polit` ecnica de Catalunya, Escola Universit` aria d´Enginyeria, T` ecnica Industrial de Barcelona. Departament de Matem` atica Aplicada III, Comte d´Urgell, 187, 08036, Barcelona, Spain. Abstract The LuGre friction model is used in the current literature to describe the friction phenomenon for mechanical systems. In this paper, we focus on the hysteresis behaviour of the model. More precisely, we describe analytically the hysteresis loop of the model through the concepts of consistency and strong consistency. The description is illustrated by numerical simulations. Key words: LuGre model; hysteresis; (strong) consistency. 1 Introduction Friction is a nonlinear phenomenon that originates from the contact of two bodies. It has two types of charac- teristics, static and dynamic. The static characteristics of friction include the stiction friction, the kinetic force (the Coulomb force), the viscous force, and the Stribeck effect which are functions of steady state velocity. The static friction models give the friction force as a function of velocity and only describe the steady-state behaviour between velocity and friction force. Static friction mod- els are discontinuous at zero velocity with a dependence on the sign of velocity [1]. This discontinuity does not reflect accurately the real friction behaviour and may cause errors in numerical simulations, or even instability in the algorithms de- signed to compensate friction [1]. Dynamic friction models capture properties that can- not be captured by typical static friction models; for in- stance, presliding displacement related to the elastic and plastic deformations of asperities, frictional lag, that is the delay in the change of friction force as a function of a change of velocity, and stick-slip motion. These mod- els do not present a discontinuity at zero velocity which makes them more suitable for numerical simulations and friction compensation [1]. ⋆ Supported by grant DPI2011-25822 of the Spanish Min- istry of Economy and Competitiveness and by FI-DGR-grant IUE/2365/2009 of the Agency for Management of University and Research Grants (AGAUR). Dahl friction model is a dynamic model whose steady– state is the Coulomb friction [5]. The main contribution of the model is that it takes into account the existence of hysteresis between the presliding friction force input and the displacement output that is observed experimentally [1]. However, Dahl model does not capture the Stribeck effect. An improvement of this model is implemented in the LuGre model [4] which captures some essential prop- erties of friction such as hysteresis and Stribeck effect and thus can describe stick-slip motion [2]. Therefore, it has been used to describe the friction phenomenon for mechanical systems [10,2]. Necessary and sufficient conditions for the dissipativity of the LuGre model are given in [3]. Also, the model has been used for friction compensation [7,12,13]. In this paper, we focus on the hysteresis behaviour of the LuGre model. More precisely, we investigate the an- alytical expression of the hysteresis loop of the model through the concepts of consistency and strong consis- tency [8]. These concepts are particularly useful when dealing with rate–dependent hysteresis as is the case of the LuGre model. The reader is referred to [8] for a more detailed explanation and motivation of the concepts of consistency and strong consistency. The paper is organized as follows. Section 2 presents the needed background from [8]. The problem statement is formulated in Section 3. The main results of this paper are presented in Section 4. These results are commented upon is Section 5, and a simulation example is provided in Section 6. The conclusion is given in Section 7. Preprint submitted to Automatica February 4, 2015