AIM2005 - TA2-03 Proceedings of the 2005 IEEE/ASME...

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A New Continuously Differentiable Friction Model for Control Systems Design C. Makkar, W. E. Dixon, W. G. Sawyer, and G.Hu Mechanical and Aerospace Engineering University of Florida, Gainesville, FL 32608 Email:cmakkar, wdixon, wgsawyer, [email protected] fl .edu. Abstract — For high-performance engineering systems, model- based controllers are typically required to accommodate for the system nonlinearities. Unfortunately, developing accurate models for friction has been historically challenging. Typical models are either discontinuous and many other models are only piecewise continuous. Motivated by the fact that discontinuous and piecewise continuous friction models are problematic for the development of high-performance continuous controllers, a new model for friction is proposed in this paper. This simple continuously differentiable model represents a foundation that captures the major effects reported and discussed in friction modeling and experimentation. The proposed model is generic enough that other subtleties such as frictional anisotropy with sliding direction can be addressed by mathematically distorting this model without compromising the continuous differentiability. I. I NTRODUCTION General Euler-Lagrange systems can be described by the following nonlinear dynamic model: P ( t t + Y p ( t> ˙ t ) ˙ t + J ( t ) + i ( ˙ t ) = ? ( w ) = (1) In (1), P ( t ) 5 R q × q denotes the inertia matrix, Y p ( t> ˙ t ) 5 R q × q denotes the centripetal-Coriolis matrix, J ( t ) 5 R q denotes the gravity vector, i ( ˙ t ) 5 R q denotes a friction vector, ? ( w ) 5 R q represents the torque input control vector, and t ( w ) , ˙ t ( w ) , ¨ t ( w ) 5 R q denote the link position, velocity, and acceleration vectors, respectively. For high-performance engineering systems, model-based controllers [6] are typically required to accommodate for the system nonlinearities. In general, either accurate models of the inertial effects can be developed or numerous continuous adaptive and robust control methods can be applied to mitigate the effects of any potential mismatch in the inertial parameters. Unfortunately, developing accurate models for friction has been historically problematic. In fact, after decades of theoretical and experimental investi- gation, a general model for friction has not been universally accepted, especially at low speeds where friction effects are exaggerated. In fact, [1] examined the destabilizing effects of certain friction phenomena (i.e., the Stribeck effect) at low speeds. To further complicate the development of model-based controllers for high-performance systems, friction is often modeled as discontinuous; thus, requiring a discontinuous controller to compensate for the effects. Motivated by the desire to develop an accurate representa- tion of friction in systems, various control researchers have developed different analytical models, estimation methods to identity friction effects, and adaptive and robust methods to compensate for or reject the friction effects. In general, the dominant friction components that have been modeled include: Static friction (i.e., the torque that opposes the motion at
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