Decomposition-based friction compensation of mechanical systems

doi:10.1016/S0957-4158(01)00010-1

Mechatronics

Volume 12, Issue 5, June 2002, Pages 755-769

https://doi.org/10.1016/S0957-4158(01)00010-1 Get rights and content

Abstract

In this paper, a linear parametric friction model is formulated by linearizing a nonlinear empirical friction model in parameters. A proposed decomposition-based control design framework is applied to synthesize the friction compensation scheme. A separate compensator is designed for each type of friction utilizing the most suitable control technique. The nominal friction is compensated by feed forward. An adaptive compensator is derived to compensate for parametric unmodeled friction with unknown but constant parameters, and a robust compensator is used to deal with friction model parameter variations, as well as non-parametric unmodeled friction. The combination of the compensators yields the overall compensation scheme. Adaptive and robust compensators complement each other in compensating the effects of model uncertainties. The analytical and simulation studies have confirmed the efficiency of the proposed friction compensation method.

Introduction

Precise control of mechanical systems such as robot manipulators requires joint friction compensation, especially at low velocities. Extensive research has been conducted in the literature to friction modeling and compensation as reviewed in [1]. Recently, a dynamic state variable friction model is presented in [2], and several adaptive friction compensation methods have been developed based on this state variable model [11]. In [3], a reduced-order version of the friction model [2] is used to design an adaptive controller to compensate linear parametric friction for tracking control, and the Stribeck parameter is assumed to be accurately known. While the advantage of adaptive friction compensation can be readily understood, it may encounter difficulties from lack of excitation and even instability due to incompleteness of friction model. The problem becomes worse if the position dependency of friction is involved. In theory, a robust controller can easily compensate for low velocity friction, since the unmodeled low velocity friction is obviously bounded [4], [5], [7]. However, high feedback gain is applied in order to achieve high accuracy, which is always limited by hardware issues [5], [7], [8]. The key in robust control design is to achieve desired performance with minimum feedback gains.

In this paper, a friction compensation scheme is synthesized by applying the decomposition-based control design approach proposed in [5], [6], [7]. The fundamental strategy of the decomposition-based system modeling and control approach is to distinguish between uncertain parameters and variables of different physical types, and to design a separate compensator for each of them, while taking into account each specific physical feature. This approach advocates treating each type of model uncertainty with the most suitable and efficient means, including PID, robust, adaptive, and sensor-based control methods. The overall controller is generated by synergetic integration of these compensators. In the proposed friction compensation scheme, adaptive control and robust control techniques are both applied, and they complement each other in dealing with model uncertainties. To apply adaptive and robust control techniques in the friction compensation, the friction model presented [1] is linearized at the nominal values of the static and Stribeck friction model parameters, so that all friction model parameters appear linearly in the linearized model. We assume that the deviation of friction model parameters due to off-line estimation error, position dependency, as well as temperature and lubrication changes, is relatively small compared to their nominal values. In practice, it is always encouraged to estimate the friction model parameters off-line to obtain the nominal values, so that the controller can be initialized with the best available knowledge of the plant.

It is well understood that the friction model parameters may vary due to changes in temperature or in contact forces. Slow variations can be caused by temperature and lubrication changes, as well as component wear. Contact force change can cause fast friction variations. Two control schemes are derived in this paper as a result of considering both situations. The first scheme is developed based on the assumption that the parameters of the friction model are not accurately known but constant. In the second scheme, variable parametric friction model uncertainty is considered. An adaptive compensator is designed with respect to the constant portion of the variable parametric friction model uncertainty while a robust compensator deals with the variable portion and guarantees the uniform ultimate boundedness of the system error. The first scheme is a special case of the second, which can be reduced to the first when the friction parameters are constant or change only slowly.

Section snippets

Nonlinear model

Since the focus of the present work is friction compensation, we consider a single joint mechanical system as studied in [1], [3]. The mathematical model is formulated to be of the form: $M q ̈ +B q ̇ +(F_{c} +F_{s} exp (−F_{τ} q ̇^{2})) sat (q ̇)+F_{q} (q, q ̇)=τ,$ where $q ̈ (t), q ̇ (t), q(t)$ represent the acceleration, velocity and position, respectively. M denotes the constant inertia, and B denotes the viscous friction coefficient. F_c denotes the Coulomb friction-related parameter. F_s denotes the static friction-related parameter.

Constant parametric uncertainty compensation

In this section, the parametric model uncertainty F̃ is assumed unknown but constant. The decomposition-based control design framework proposed in [5], [6] is applied to derive the friction compensation scheme. The two uncertainty groups are distinguished from each other, and a compensator is designed with respect to each of them. Then the combination of the compensators, together with the nominal control, yields the overall control law. Such an approach allows us to treat the parametric and

Variable parametric friction compensation

In Section 3, it has been assumed that the parametric model uncertainty defined by (8) is unknown, but constant. Actually, in the proof of Theorem 1, it is required that the differential of F̃ equals to zero, i.e. $d F ̃ / d t=0$ .

In practice, the parametric model uncertainty F̃ may not always be constant, due to temperature and lubrication changes. Also, in the case of robot manipulators, the friction parameters vary as the manipulator moves to different positions. In certain circumstances, the

Dynamic model

The simulation studies in this section are to investigate the effectiveness of the proposed decomposition-based friction compensation schemes derived using the linearized model , , . For the simulations, the plant is represented by the original nonlinear model (1), with the following parameters: $M=0.15 kg m^{2}, B=1.5 Nm s / rad,$ $F_{s} =5.0 Nm, F_{c} =3.5 Nm, F_{τ} =100 s^{2} / rad^{2} .$

The inertia parameter M is assumed to be exactly known, and the nominal parameters of the friction model are assumed as in the following table. The

Concluding remarks

A linear parametric friction model is formulated by linearizing a nonlinear empirical friction model in parameters. A proposed decomposition-based control design framework is applied to synthesize the friction compensation scheme. A separate compensator is designed for each type of friction utilizing the most suitable control technique. An adaptive compensator is derived to compensate for parametric unmodeled friction with unknown but constant parameters, and a robust compensator is used to

Acknowledgements

This work is supported by NSERC, Canada, through a research grant.

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Technical NoteDecomposition-based friction compensation of mechanical systems

Abstract

Introduction

Section snippets

Nonlinear model

Constant parametric uncertainty compensation

Variable parametric friction compensation

Dynamic model

Concluding remarks

Acknowledgements

A survey of models, analysis tools and compensation methods for the control of machines with friction

Automatica

A new model for control of systems with friction

IEEE Trans Automat Control

An experimental comparison of robust control algorithms on a direct drive manipulator

IEEE Trans Control Syst Technol

Robust control of robot manipulators based on dynamics decomposition

IEEE Trans Robotics Automation

Technical Note
Decomposition-based friction compensation of mechanical systems