Control of Systems with Friction

Friction appears in all mechanical systems and has a significant impact on control. Successful design of mechatronic systems requires an understanding of the effects of friction as well as techniques to compensation. Friction phenomena are complicated because they are caused by many different physical mechanisms. Friction can cause a substantial deterioration of the performance of a control system. Typical effects are steady state errors and oscillations. Many attempts have been made to compensate for friction. Early efforts include introduction of dither signals. Other ideas are to based on model based control. This paper reviews several models for friction that have been useful to model friction in motion control systems and to compensate for effects of friction. INTRODUCTION Friction is important in all motion control systems. It leads to deterioration in precisions and can generate limit cycles. It is therefore important to understand friction phenomena to understand and improve the behavior of the systems. Friction has been investigated for a long time. Leonardo da Vinci investigated the motion of rectangular blocks sliding over flat surfaces. The French physicist Amonton [1] made similar investigations and concluded that the friction force at a sliding interface is proportional to the normal load. He also discovered that the the friction force does not depend on the apparent area of contact. A small block sliding on a surface thus experiences the same friction as a large block if the normal forces are the same. The French physicist Coulomb [15] found that once the motion starts the friction is independent of the velocity, resulting in the famous Coulomb’s friction law. Friction phenomena have been extensively studied in tribology see [11], [12] and [19]. Lately there have been ∗ This work has been partially supported by the Swedish Research Council for Engineering Science, contract 95-759. a resurgence of interest in friction in many scientific communities [42], [26]. This has been driven by scientific curiosity, engineering needs and improved instrumentation such as laser interferometry and atomic force microscopes. Physicists have investigated friction on an atomic scale [6]. Geologists have been investigating friction to better understand earth quakes [40], [25]. Friction plays an important role in micro-mechanical systems. Control engineers have tried to come up with better ways of dealing with frictions in mechatronic systems see [32], [29] and [35]. Even if the phenomenological properties of friction are reasonably well understood the fundamental physical mechanism that cause friction are not well known. For example there are no strong correlation between surface roughness and friction. There are cases where friction is less if a surface is rougher. Experiments with mica surfaces have shown that there is friction even if there are no asperities in the surfaces. In this case the friction is explained in terms of complicated molecular adhesion. The issue of lubrication is also poorly understood. Experiments on some surfaces have indicated that friction is less when surfaces are dry. FRICTION MODELS Before going presenting detailed mathematical model we give a phenomenological description of sliding friction. For this purpose we will consider a rectangular block over a flat surface. Friction is the tangential reaction force between the surfaces. If a small tangential force is applied there will initially be a small deflection of the block but the block will return to its original position if the force is reset to zero. There may be a small residual off set. The residual off-set increases with increasing force. If the force increases further the block starts to move. The force required to do this is called the break-away force. Its value depends on how long the block has been at rest. The force is also varying depending on many factors. The force required to keep the block moving depends on the velocity and