SECOND-ORDER LUGRE FRICTION MODEL Acho L., Moreno J., and Guerra R. CITEDI – IPN Av. del Parque No. 131º, Mesa de Otay, Tijuana, Baja California, México C.P. 22510 leonardo@citedi.mx rguerra@citedi.mx ABSTRACT A second-order LuGre friction model is presented which can be viewed as an extension of the well known LuGre friction model. This model is based on a dynamic extension, which can be seen as an extra dynamic to capture some kind of periodic motion produced by the bristles in motion. The additional dynamic can be viewed as an internal disturbance due to the vibration associated with the use of motors. Our model can capture the friction phenomena of the original LuGre friction model and presents two new behaviors, one is the multi-loop behavior in the hysteresis curve when velocity is varied during unidirectional motion, and the other, in the pre- sliding motion curve of friction force versus displacement in the spring regime, where two jumps appear. KEY WORDS Friction, friction model, hysteresis. 1.Introduction Friction modeling is an important issue in control theory because, in most cases, the design of a control law is based on the model of the plant to be controlled. In this context, a congruent friction model that captures all the phenomena that the friction produces in a mechanical system, such as: stribeck effect, pre-sliding motion, hysteresis, etc., is highly important. So far, there exist three important friction models: 1) The Coulomb model, 2) The Dahl model, and 3) The LuGre model. The first one is a static model whereas the others are dynamical. It is well know that dynamical models can produce most of the phenomena produced by friction [1]. There have been some techniques to compensate friction forces, see for example [2]-[4]. For instance, in [2] the Coulomb friction model is used for the compensator design; however, and as it is pointed out in [2], if this model differs from reality, this techniques is not effective. In [5] a chattering control design is presented to uncouple the effects of the friction forces on the mechanical system, but, it requires a control law that commutes very fast, which is hard to produce because of the high band required to drive the control-law’s signal. The demonstration of this important fact was possible using the LuGre friction model. In this work (in [5]), it is also shown that the chattering control law is robust against variation in the friction model used; in other words, in the simulation experiments shown in [5], with Dahl model or LuGre model, the chattering control law could uncouple the effects of the friction forces on the system. One important observation on friction model is that this is not unique, that is, there can exist other friction models that can captures most of the phenomenon produced by friction force. The main objective of the present paper is to show that there exists a modification of the LuGre friction model (a second order model) that can still capture the main effects of friction force; however two other effects appears, for instance, the Hysteresis effect which is produced when velocity is varied during unidirectional motion, in our model, presents multi-loop variation which is not captured by the LuGre model. Also, the pre-sliding motion curve of friction force versus displacement in the spring regime, presents a jump, which is not captured by LuGre model. This jump appears in experimental and simulations results shown in [6]. Other friction models have been reported recently (see [6], [7], and [8]), our model differs from them by the fact that we are using an extra nonlinear dynamic to capture some kind of periodic motion produced the average deflection of the moving elastic bristles, this additional dynamic can be attributed to an internal disturbance, such as the vibration generated by motors intended to produce motion. II. LuGre Friction Model The LuGre friction model is given by [1]: () () ( ) ( ) 2 / 0 2 1 0 S v v C S C e F F F v g v z z F z v g v v dt dz z - - + = + + = - = = σ σ σ σ (1)