Uncertainty estimation in robust position control of robotic arms

Abstract

       This thesis deals with uncertainty estimation in robust control of robotic arms and presents new methods based on voltage control strategy for uncertainty estimation. The voltage control method is much simpler compared to the conventional torque control method, because it does not require a complex non-linear model of the robot. As a result, the volume of controller calculations to determine the applied voltage to the motors is reduced. According to the general approximation theorem, fuzzy systems and neural networks are able to approximate real continuous nonlinear functions with desired accuracy. It should be noted that in addition to fuzzy systems, there are other general approximations such as Fourier series, Legendre functions and Chebyshev polynomials. In this thesis, these approximations are used in the robust position control of robotic arms. The main advantage of using these approximations compared to fuzzy systems and neural networks is to reduce the feedback required by the control system. So far, some references have used Fourier series in robust control of robotic arms. We show that if the optimal paths of functions are periodic, the least common multiple (LCM) of their fundamental periodicity can be a suitable criterion for the fundamental periodicity of the Fourier series used to estimate uncertainties. Another innovation of this thesis is to present a stability proof based on Lyapunov for the control of first-order nonlinear systems using emotional controllers. For the first time, the proposed voltage control rules are implemented on a Scara robot.

Keywords: voltage control strategy, Fourier series, Legendre functions, emotional control, permanent magnet electric motor, skilled robotic arm.

Review of past works

1-1-1-Torque control strategy

Considering that improving the performance of robot control systems has a significant impact on the quality of industrial products and increasing efficiency production, the design of robot control systems has always been one of the most attractive research areas. Studying the historical progress of the presented control methods clarifies the progress made in this field. Robotic arms are complex multivariable nonlinear systems with many couplings. For this reason, researchers have provided many different methods to control them, the simplest of which are model-based methods. Feedback linearization [1-2] is the most popular and widely used technique for controlling non-linear systems, because by using it, the non-linear dynamics of the robot complex can be easily converted into second-order linear equations.  This method is known as computational torque, inverse dynamics or torque control in robotics. But the success of model-based methods depends on having an accurate model of the system. Unfortunately, obtaining the exact mathematical model of robotic systems is very difficult, time-consuming and sometimes impossible. Because some system dynamics such as friction may not be repeatable or an accurate model cannot be proposed for them. In addition, the parameters of the system model may change over time or under the influence of certain conditions. For example, when the robot lifts objects with different masses, the center of mass of the last link, which is one of the dynamic parameters of the robot, changes. For this reason, the model we propose for the system (nominal model) is different from the real system model. Therefore, uncertainty has always been one of the most important challenges of designing control systems. It should be noted that the uncertainty in robotic systems is usually assumed to be non-random and it means the unknown parameters of the system, the presence of unknown or unmodeled dynamics, as well as external disturbance. To overcome the uncertainty caused by model mismatch, adaptive and robust control methods [3-7] have been presented. Adaptive control can compensate the effects of parameter uncertainty. In addition to parametric uncertainty, robust control is able to compensate for uncertainties caused by modeled dynamics and external disturbances. Extensive research has been done to design adaptive control systems of rigid robots in order to ensure the stability of the control system and to keep the internal signals limited. Spong has provided a comprehensive classification of adaptive methods [8] and divides them into two main groups of methods based on inverse dynamics and methods based on passivity. In all the above methods, only parametric uncertainty is considered.Another important point about the adaptive methods is the stable stimulation [1] of the stimulation signals [7]. Otherwise, the estimated parameters will not converge to the real parameters. In robust control methods, it is necessary to know the limits of uncertainty. Uncertainty limits are one of the most important challenges in these methods. If the uncertainty limits are greater than the actual value, the size of the control signal may be greater than its allowed value, in which case the saturation phenomenon will occur and the controller will not be able to control the system. In addition, if the amplitude of the control signal exceeds the permissible limit, it may damage the system, and the vibration phenomenon of the control signal is also strengthened. On the other hand, if the uncertainty limits are lower than the actual value, the tracking error increases and may lead to the instability of the control system [9-11]. Some robust control methods lead to discontinuous control rules. As an example, we can refer to the sliding mode control method [2]. These laws increase the possibility of high frequency fluctuations (vibration) in the control signal. Vibration of the control signal is an undesirable phenomenon that causes wear of parts and stimulation of unmodeled dynamics. With the emergence of fuzzy logic as a powerful tool in the control of uncertain and complex systems, a tremendous transformation occurred in control engineering. Fuzzy laws can be used to describe systems that do not have a precise mathematical model [12]. Indirect adaptive fuzzy method uses this idea [13-15]. Another feature of fuzzy logic is the modeling of human knowledge and ability to control complex systems, which the direct adaptive fuzzy method [16-17] provides this possibility. In addition, it is possible to combine direct and indirect adaptive fuzzy methods and obtain a method that performs better [18]. One of the most important features of fuzzy logic, which has led to their widespread use in control systems, is the general approximating feature of fuzzy systems [12]. For this reason, in recent years, researchers have focused more on fuzzy control and many efforts have been made for robust robot control using fuzzy control and neural networks [19-35], because the general approximation feature is also established for different types of neural networks such as multilayer perceptron and radial basis function networks [36-40].  In [19], adaptive fuzzy systems are presented to compensate uncertainties such as parametric uncertainty, external disturbance (such as the mass of the object that the robot moves), unmodeled dynamics (such as friction) and also the approximation error of the fuzzy system. In [20], a method to reduce the number of required fuzzy systems is presented. Also, it has been shown how the tracking error can be reduced by choosing the appropriate parameters of the control law. In [22], it is assumed that the velocity and acceleration feedbacks are not available and a nonlinear approach is proposed to estimate these signals. In [26], two-layer neural networks have been used to approximate robot dynamics, and new adaptation rules have been obtained to adjust the weights of both layers using Lyapunov's stability proof. But the number of inputs of designed neural networks are large. These inputs are the current of the motors, the position and speed of the joints, the desired path and its first and second derivatives. In these methods, a Lyapunov function is proposed for the stability of the control system, and the matching law of fuzzy system parameters or neural network weights is obtained from the negative condition of the Lyapunov function derivative being definite. Some authorities approximate the dynamics of the system by using fuzzy systems or neural networks and use this approximation in the design of the control law, and some others consider the controller as a fuzzy system or neural network and adjust its parameters using the adaptation rules obtained. In [41], a new and different adaptive fuzzy method from these two conventional methods is presented. In this method, a nominal model is considered for the system and the control law is designed based on this nominal model. Then, a fuzzy system is added to the control law to compensate for the uncertainty caused by the mismatch between the nominal model and the real model. To prove the stability of the system, the direct Lyapunov method is used, and the matching law of fuzzy system parameters is derived from the condition of the negative definiteness of the derivative of the Lyapunov function. In recent years, regressor-free and model-independent methods have been proposed in the control of uncertain nonlinear systems [42-51].