Discrete-time second order sliding mode with time delay control for uncertain robot manipulators

doi:10.1016/j.robot.2017.04.010

Robotics and Autonomous Systems

Volume 94, August 2017, Pages 53-60

https://doi.org/10.1016/j.robot.2017.04.010 Get rights and content

Highlights

•
The discrete-time dynamic model of robot manipulators is introduced.
•
A novel method which consists on a combination of second order sliding mode and time delay estimation is proposed.
•
Design, stability and convergence analysis are addressed.
•
Experimental results prove the effectiveness of the proposed controller.

Abstract

Control of uncertain nonlinear systems is one of the main topics in automation, control problem. However, uncertainties caused by uncertain parameters, load variations, unmodeled dynamics and external disturbances affect the control performance. This paper investigates the problem of high accuracy joint space trajectory tracking. A discrete-time second order sliding mode combined with time delay estimation is designed to make the joint positions track a desired trajectory with high accuracy. In addition, sufficient condition assuring convergence of the error to zero in finite time based on Lyapunov theory is presented. Experiments on a 7-DOF ANAT robot arm are presented to verify the performance of the proposed controller.

Introduction

Nowadays, robot manipulators are present in different areas. This is why the design of robust controller to ensure high accuracy positioning and repeatability is very important. However, robot manipulators suffer from uncertainties caused by uncertain parameters, unmodeled dynamics, load variations and external disturbances, which make the design of accurate control more challenging. Sliding Mode Control (SMC) due to its robustness to a significant class of uncertainties is one of the powerful nonlinear control techniques [1], [2], [3]. However, to ensure a good performance, the switching gains are chosen larger than the overestimated bound of these uncertainties, which makes the values of the switching gains excessive. Then, the chattering phenomenon [4], [5] which is the major drawback of SMC will be important and might not be accepted by the robot actuators and might deteriorate the controlled system if the control has any physical sense.

One of the first proposed solution to reduce the chattering phenomenon is the use of continuous functions as saturation function or hyperbolic tangent instead of the sign function [1]. These propositions reduce the problem of chattering. However, the steady state error does not converge to zero in a finite-time. Furthermore, the observer-based sliding mode has been proposed in [6], [7]. This method allows exact and robust estimation. However, the control performance can be affected once the estimation is not exact. Otherwise, a combination of sliding mode with intelligent controller as neural-network and fuzzy logic has been proposed [8], [9], [10], [11]. Even if these intelligent controllers provide a good approximation of unknown dynamics and disturbances, however, their real-time implementation is still difficult due to the large number of parameters or fuzzy rules.

In [12], a Higher Order Sliding Mode (HOSM) has been proposed to attenuate chattering. The order is generally chosen to be equal to the relative degree $r$ of the controlled system (the relative degree is the number of time for which the output is differentiated to obtain an explicit expression in function of the input). Since introduced, many algorithms have been proposed to improve the HOSMC such as twisting algorithm, sub-optimal algorithm, global algorithm, Super-Twisting Algorithm and others [13], [14], [15]. The basic idea of HOSMC is to make the signum function acting on the higher time derivative of the switching function while keeping the main advantage of standard sliding mode control, the chattering effect is eliminated and higher order precision is provided. However, in all these algorithms, the knowledge of the bound of the uncertainties and disturbances is needed.

In order to propose a solution to the aforementioned problem (the chattering reduction and the non-requirement of the uncertainties and disturbances bound), this paper introduces a new nonlinear control strategy based on Second Order Sliding Mode (SOSM) [16] with Time Delay Estimation (TDE) [17] for high accuracy tracking trajectory of robot manipulators with unknown dynamics and unexpected disturbances. The basic idea of this method can be summarized in two points as:

$•$
The TDE is used to estimate uncertainties and external disturbances simply and effectively using time-delayed information of the control torque inputs and state derivatives.
$•$
The SOSM is used to provide higher precision, to reduce chattering and to eliminate the effect of TDE error.

In addition, the implementation in real time is done through discrete systems [18]. For this reason, it will be better to develop the controller in discrete time. Therefore, it is more suitable to design the controller using the robot model in discrete time representation because the inherent properties of the proposed method might not be maintained after discretization. Therefore, this work deals with a discrete time second order sliding mode with time delay estimation of uncertain robot manipulators and in presence of unexpected disturbances for high accuracy trajectory tracking. This method is meant to force the sliding surface to converge to zero with chattering reduction. Sufficient condition for the stability of the resulting closed-loop dynamics is established to that end.

This paper is organized as follows. Section 1 introduces the paper. The dynamic equation of $n$ -DOF robot manipulators and control objective are introduced in Section 2. In Section 3, the discrete model of the robot is given and the proposed controller is designed such as the stability and the finite time convergence are proved. In Section 4, experiments are performed on a 7-DOF robot arm and compared with two other controllers to illustrate the effectiveness of the proposed controller. Section 5 concludes the paper.

Section snippets

Robot dynamics

The equations of motion of an $n$ DOF robot manipulators are described according to the Euler–Lagrange theory [19] by: $M (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) + F (\dot{q}) + H (q, \dot{q}, \ddot{q}) = τ$ where $q, \dot{q}, \ddot{q} \in R^{n}$ are the joint position, velocity and acceleration vectors, respectively, $M (q) \in R^{n \times n}$ is the inertia matrix (symmetrical definite positive, thus, $M {(q)}^{- 1}$ always exists), $C (q, \dot{q}) \in R^{n \times n}$ is the centrifugal and Coriolis matrix, $G (q) \in R^{n}$ is the gravitational vector, $F (\dot{q}) \in R^{n}$ is the vector of viscous friction torque at the joints, $H (q, q$

Controller design

In this section, the development of discrete-time second order sliding mode with time delay controller will be described. Considering the nonlinear continuous system given in Eq. (3) and using the Euler approximation method under the assumption of a sufficiently small sampling period $T_{s}$ , the discrete model of an $n$ -DOF robot manipulator can be written as: $\{\begin{matrix} x_{1} (k + 1) = x_{1} (k) + T_{s} x_{2} (k) \\ x_{2} (k + 1) = x_{2} (k) + T_{s} f (k) + T_{s} w (k) + T_{s} g (k) u (k) \end{matrix}$ with some abuse of notation, notice that $f (k) = f (x (k T_{s}))$ , $g (k) = g (x (k T_{s}))$ , $u (k) = u (k T_{s})$

System description

In this section, the proposed controller is implemented on the 7-DOF ANAT robot shown in Fig. 1 using the Simulink with Real-TimeWorkshop (RTW) of Mathworks. ANAT is built by Robotics Design inc. (Canada) and stands for Articulated Nimble Adaptable Trunk, this robot consists of a combination of H-shaped and U-shaped motorized modules, variably interconnected. In our case, the robot consists of seven degrees of freedom, the first joint is prismatic, followed by six rotary joints [25].

The

Conclusion

For an $n$ -DOF robot manipulators with uncertain dynamics and external disturbances, a discrete-time second order sliding mode with time delay estimation is presented in order to achieve our control objective. Sufficient condition of convergence is established. Experimental results on the ANAT robot arm showed the effectiveness of the proposed controller, improved tracking performance even in presence of uncertain dynamics and unexpected disturbances are provided. Further research should be

Acknowledgments

This work was supported in part by Automatic and Industrial Informatics Laboratory, Ecole Mohammadia d’Ingénieurs, Mohammed V University, Rabat, Morocco and Ecole de Technologie Supérieure , Montreal, Canada.

Yassine Kali was born in Fez, Morocco, in 1989. He received the Diploma degree in electrical engineering from the Faculté des Sciences et Techniques, Sidi Mohamed Ben Abdellah University, Fez, Morocco in 2013. He is currently Ph.D. student with A2I Laboratory at Ecole Mohammadia d’Ingénieurs, Rabat, Morocco. His main research interests include nonlinear and intelligent control, robotics.

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Maarouf Saad received the B.S. and M.S. degrees in electrical engineering from Ecole Polytechnique of Montreal, Montreal, QC, Canada, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from McGill University, Montreal, in 1988.

In 1987, he joined Ecole de Technologie Supérieure, Montreal, where he is currently teaching control theory and robotics courses. His research is mainly in nonlinear control and optimization applied to robotics and flight control system.

Khalid Benjelloun received the degree in electrical engineering from Ecole Mohammadia d’Ingénieurs in 1987, M.Sc.A and Ph.D. degrees in mechanical engineering from Ecole Polytechnique of Montreal, QC, Canada, in 1993 and 1996, respectively. In 1989, he joined Ecole Mohammadia d’Ingénieurs, Rabat, where he is currently teaching control theory, nonlinear control systems and robotics courses. His research is mainly in nonlinear control and jump systems, optimization applied to robotics and stochastic control system.

Abdelilah Fatemi received the degree in electrical engineering from Ecole Mohammadia d’Ingénieurs in 1980. In This year, he joined Ecole Mohammadia d’Ingénieurs, Rabat, Morocco, where he is currently teaching Industrial Automation, Programmable Industrial Logic Controllers, Industrial Local Area Networks and robotics courses. His research is mainly in autonomous mobile robotics, Instrumentation control, technical supervision systems and control/monitoring based on web technology.

^☆: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

View full text

Discrete-time second order sliding mode with time delay control for uncertain robot manipulators☆

Highlights

Abstract

Introduction

Section snippets

Robot dynamics

Controller design

System description

Conclusion

Acknowledgments

Robot. Auton. Syst.

Neurocomputing

Robot. Auton. Syst.

Robot. Auton. Syst.

Robot. Auton. Syst.

Control Eng. Pract.

Control Eng. Pract.

Applied Nonlinear Control

Sliding Mode in Control and Optimization

Sliding Mode Control in Electromechanical Systems

An averaging approach to chattering

IEEE Trans. Automat. Control

Analysis of chattering in continuous sliding-mode controllers

IEEE Trans. Automat. Control

Disturbance-observer-based sliding-mode control for a 3-dof nanopositioning stage

IEEE/ASME Trans. Mechatronics