Adaptive observer designs for friction estimation in position control of simple mechanical systems with time delay
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Friction force/torque is a well known natural eﬀect that can cause performance degradation or even instability in mechanical systems, although it sometimes can be disregarded in closed loop feedback design phase. Hence, friction modeling and cancellation methods can be vital to achieve desired robustness and performance criteria in position control problems. Basically, the topic of friction cancellation is divided into two main categories named model based and non-model based methods. Friction modeling is a broad area of research and there are lots of diﬀerent modeling approaches in various complexities. Among these approaches, Coulomb Model is one the simplest yet fundamental models. Nevertheless, in some cases, being a classical static model, it is inadequate to exhibit the dominant friction components occurring at diﬀerent motion stages such as break-away force, stick-slip motion, pre-sliding behavior or friction lag. Generally, dynamical models, i.e. LuGre Model, are more advanced as a result, they are better to describe such friction eﬀects. Unfortunately, for these cases, the number of friction parameters are increased. In fact, there is a trade-oﬀ between model complexity and parameter identiﬁcation. A desired system response may not be achieved when model parameters do not coincide with the existing friction coeﬃcients. In this manner, precise identiﬁcation of each parameter can be challenging when there are many of them. Besides, some of these parameters might be time varying due to environment, temperature, material properties, position, etc. Therefore, non-model based adaptive schemes are prevalent in the literature since these methods do not require any parameter identiﬁcation. In this study, we focus on adaptive observer based friction compensation techniques and provide some stability conditions. First, we consider simple second order mechanical systems with or without time delay under Coulomb friction. To estimate the Coulomb friction, we ﬁrst consider Friedland-Park observer. Then, some necessary conditions are stated to extend the estimation function in the observer structure to a larger class of functions. Especially measurement delay can be signiﬁcant since observers estimate friction based on the velocity measurements. Therefore, it is proposed to employ a velocity predictor either based on numerical diﬀerential equation solvers or inverse Pade approximant when the existing time delay is large. What is more, a new observer design that considers friction and velocity error dynamics together is proposed as a novel contribution. Extensive MATLAB simulations are conducted to investigate the performances of proposed observers in a closed loop position control system with and without delay. To this end, Smith predictor and ITAE index-based designs are considered to utilize a position controller. In some of these simulations, LuGre model is preferred to mimic the actual friction instead of Coulomb friction in order to observe the eﬀects of dynamic parameters. Moreover, some experiments are performed on DC motor platform driven by Arduino Uno microcontroller. Under the light of acquired results, observer based friction compensation improves the system performance even existing friction cannot be conﬁned to Coulomb coeﬃcient, especially when the implemented controller has low bandwidth. Also, in terms of practicability, it is an advantage that these observer structures do not require any parameter identiﬁcation.
Nu-merical diﬀerential equation solvers