# Combined effects of sampling and dry friction on position control

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## Abstract

This paper studies the combined effect of sampling and dry friction on the dynamics of mechanical systems subjected to position control using discrete-time state feedback. This paper aims to highlight that even in case of a single degree-of-freedom mechanical subsystem, the controlled system cannot be characterized by a model with a single dominant frequency and some of the upper harmonics become relevant resulting multi-frequency vibrations. The presented stability analysis of the frictionless system gives insight into this intricate dynamics of sampled-data systems. The numerical simulation results show that the presence of dry friction can stabilize an otherwise unstable motion due to the effect of sampling. The effect of dry friction results that decreasing vibrations occur with concave envelope, but it makes the dynamics dependent on the initial conditions.

## Keywords

Motion control Positioning Digital effects Higher harmonics Coulomb friction## 1 Introduction

Position control is a fundamental task in mechatronic systems. In such tasks, the aim of the control system is to maintain a desired position of the mechanical subsystem. The desired position can include a specified position or tracking a trajectory defined as a series of positions in time. Many applications of position control are typically related to robotic systems where state feedback is used in the control algorithm. For example, in case of industrial robots steady-state position control and trajectory tracking are very common. In those applications, the main objectives are high accuracy and high speed. The gains are usually tuned to values that guarantee best performance in terms of the primary objectives.

For such applications, comprehensive studies [1, 2, 3] are available on digital control theory. In classic control theory, phase and/or gain margins are used to design the controllers, while another basic design method is the so-called pole placement. This method determines the control gains of the feedback matrix in order to modify the characteristic roots or poles of the uncontrolled system to achieve the desired motion. In case of discrete-time control [4, 5, 6], the characteristic roots are described in continuous time, and they are transformed back to discrete-time domain with a given sampling time in order to get the control gains for the discrete-time case.

On the other hand, in force-feedback haptic systems [7], position control serves a different purpose. In such systems, the difference between desired and actual positions is seen as the deflection of a virtual spring, and the proportional control gain is the stiffness coefficient of the virtual spring. These generate the force/torque values that are fed back to the human user from the virtual environment through the haptic device. Therefore, the value of proportional gain is very important in such applications since it determines how soft or hard the user feels the virtual environment. For these systems, the sampling frequency often cannot be increased to high enough values [8], and the time discretization effects become very visible leading to stability problems.

A discrete-time position control with a proportional gain only is always unstable [9, 10], and the level of instability, the rate of divergence depends on the sampling rate; the higher it is, the slower the system position diverges away from the desired value. To stabilize a discrete-time position controlled system, some physical or virtual dissipation is needed. For high sampling rates, often the inherent physical dissipation of the mechanical system can be sufficient. But, such dissipation can also be added via derivative feedback that acts as a virtual viscous damping element. The other source of physical dissipation can come through Coulomb friction.

Little is known about the dynamic effects of Coulomb friction in haptic systems. It is typically neglected in order to achieve a linearized, worst-case system model for stability analysis [11]. Considering time discretization, delay, quantization, and Coulomb effects, a passivity condition is given in [8]. But, other papers mostly consider viscous damping that is necessary to achieve/maintain passivity [11, 12].

The analysis of the effect of sampling requires the handling of infinite dimensional mathematical models. The complexity of the mathematical treatment is further increased by the non-smooth property of dry friction [13, 14]. In order to simplify the analysis and/or the control design, the effect of dry friction is often neglected. This can be justified by considering friction compensation [15], or aiming for designing a robust controller with conservative stability conditions [16].

Although the above simplifications can be reasonable from the engineering point of view, precise control and the explanation of some intricate vibration phenomena require the use of more accurate models. As it was demonstrated in [17, 18], when Coulomb friction compensates for the possible instability caused by the sampling, decreasing vibrations occur with concave envelope. In these two papers, the only considered source of physical dissipation was Coulomb friction. As was demonstrated in a sampled-data setting, Coulomb friction can act as an important dissipative phenomenon; however, it can cause intricate dynamic effect.

In [19], the effect of the additional viscous damping was investigated while this current paper focuses on how the effect of virtual damping can modify the dynamic behavior of sampled-data systems. The main objective of this paper is not related to traditional controller design approaches, i.e., how to tune the gains to achieve best performance. We rather intend to show the complex dynamic effects that can arise due to virtual damping and Coulomb friction. These can be important in applications such as haptic systems where sampling rates often cannot be increased to high values, and the control gains themselves can carry physical meaning.

The paper is organized as follows. In Sect. 2, the mathematical model of a sampled-data system is introduced where the effect of dry friction is considered as the main source of physical dissipation. As a reference, stability and dynamic analysis of the corresponding sampled-data system with negligible dry friction is presented in Sects. 3 and 4 . The nonlinear vibrations induced by sampling and friction effects are presented in Sect. 5. Section 6 concludes the paper.

## 2 Model of position control

*q*(

*t*) represents the generalized coordinate as a function of time

*t*, which corresponds to the modeled degree of freedom,

*m*denotes the generalized mass corresponding to the generalized coordinate

*q*, and dot refers to differentiation with respect to time. In addition, \(f_{\mathrm {u}}\) is the generalized control force, \(f_{\mathrm {fr}}\) is the generalized friction force, and \(t_j = j \tau \) denotes the

*j*th sampling instant, where \(\tau \) is the sampling time.

By assuming that the direction of motion does not change between two consecutive sampling instants, the piecewise linear system presented in Eq. (5) can be solved analytically for the discrete state variables collected in \(\mathbf {x}_j = [q_j \, v_j]^\mathrm {T}\), where the discrete states are \(q(T_j) = q_j\) and \(q'(T_j) = v_j\).

## 3 Stability analysis with negligible friction

Using this linear map, the time series of the discrete states can be determined for any initial conditions \(\mathbf {x}_0\) which forms a generalized, or multidimensional geometric series, called Neumann series [16]. The convergence of this series is equivalent to \(\rho (\mathbf {A}) < 1\) where \(\rho \) denotes spectral radius. Thus, the eigenvalues of matrix \(\mathbf {A}\) determine the asymptotic stability of the system, which is stable if, and only if, the eigenvalues are inside the complex unit disk [4], i.e., \(|z_n| < 1\) with \(n = 1,2\).

*w*, the stability condition yields

*p*and

*d*as a shaded triangular area in the left panel of Fig. 1.

Since the possible maximum steady-state error is \(\Delta = \sigma / p\), the proportional gain needs to be selected as high as possible [16] to improve the precision of positioning. As an example, consider the case of \(p=3.5\) and \(d=1.935\), which selection results in somewhat unexpected multi-frequency oscillations in the frictionless case. Such oscillations are plotted in black in both panels of Fig. 2.

## 4 Dynamic analysis with negligible friction

*z*. It follows that these conditions result in three separation curves in the stable domain by

*z*. It is important to note that

*z*is also a rotating vector which means \(\vartheta \) is periodic and \(\vartheta + 2k \pi \), \(k \in {\mathbb {Z}}\). If the transformation is applied, then the resulting infinite number of characteristic roots in the domain of

*s*are called characteristic exponents.

*x*(

*T*) can be given as

*x*(

*T*) can be characterized as

Accordingly, without considering the effect of higher harmonics corresponding to \(k=-1\), the dominant motions cannot be characterized correctly. This consequence is also valid for the characterization of the typical motions in sub-domains \(\#\)2, \(\#\)3 and \(\#\)4 where the common property is that one of the characteristic multipliers lies in the left half of the complex plane *z*.

## 5 Nonlinear vibrations induced by sampling and friction effects

When Coulomb friction can stabilize an otherwise unstable motion, the resulting motions may have concave envelope [17, 18]. If the control gains are selected from the lower region of the stability map, the motion is exponentially unstable with one dominant frequency. It is presented as black curve in the left panel of Fig. 5. When the friction stabilizes the motion, the predicted concave envelope is presented as red curve in the right panel of Fig. 5.

This concave envelope is also observed when the effect of friction stabilizes such kind of unstable motions that can be characterized by two dominant vibration frequencies. This is presented in the left panel of Fig. 6. In these cases, due to the effects of two dominant frequencies, it results in the vibrations exhibit double concave envelope as shown in the right panel of Fig. 6.

As it was mentioned above, the stabilization effect of Coulomb friction is sensitive to the initial conditions resulting unstable limit cycle around the zero desired position. In case of the presented examples, when the motions are initiated with zero velocity at \(x_0 = 5.8\) instead of \(x_0 = 5\), or at \(x_0 = 9.8\) instead of \(x_0 = 9.4\) the motions become unstable due to the presence of an unstable oscillation.

## 6 Conclusions

In this paper, the main characteristics of position controlled mechanical systems were investigated by considering dry friction as the primary source of physical dissipation. For the analysis of the effect of sampling with dry friction, a detailed analysis was presented for the frictionless system. It was demonstrated that even a single degrees-of-freedom mechanical system can have multi-frequency vibrations when it is positioned by a discrete-time state-feedback controller. The observed vibrations can be explained by the higher harmonics in the solution which correspond to the periodicity of the eigenvalues of the transition matrix. When the effect of dry friction is also taken into account, it was also shown that the position controlled systems with discrete-time state feedback can become sensitive to the initial conditions, and unstable oscillations may appear around the desired position.

## Notes

### Acknowledgements

Open access funding provided by Budapest University of Technology and Economics (BME). The research reported in this paper was supported by the National Research, Development and Innovation Fund of Hungary under Grant No. PD 128398, and by TUDFO/51757/2019-ITM, Thematic Excellence Program. The work was also supported by the Higher Education Excellence Program of the Ministry of Human Capacities of Hungary (BME FIKP-MI/FM), and by the Natural Sciences and Engineering Research Council of Canada (NSERC).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Ethical approval

This paper does not contain any studies with human participants or animals performed by any of the authors.

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