Mathematical Problems in Engineering

Volume 2017, Article ID 4275201, 19 pages

https://doi.org/10.1155/2017/4275201

## Task Space Trajectory Tracking Control of Robot Manipulators with Uncertain Kinematics and Dynamics

Key Laboratory of High Efficiency and Clean Mechanical Manufacture, Ministry of Education, School of Mechanical Engineering, Shandong University, 17923 Jingshi Road, Jinan City, Shandong, China

Correspondence should be addressed to Yi Wan; nc.ude.uds@iynaw

Received 19 July 2017; Revised 13 October 2017; Accepted 18 October 2017; Published 11 December 2017

Academic Editor: Jean Jacques Loiseau

Copyright © 2017 Xichang Liang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To improve the tracking precision of robot manipulators’ end-effector with uncertain kinematics and dynamics in the task space, a new control method is proposed. The controller is based on time delay estimation and combines with the nonsingular terminal sliding mode (NTSM) and adaptive fuzzy logic control scheme. Kinematic parameters are not exactly required with the consideration of kinematic uncertainties in the controller. No dynamic models or numerous parameters of the robot manipulator system are required with the use of TDE. Thus, the controller is simple structure and suitable for practical applications. Furthermore, errors caused by time delay estimation are compensated by the adaptive fuzzy nonsingular terminal sliding mode scheme. The simulation is performed on a 2-DOF robot manipulator with three cases in the task space. The results show that the proposed controller provides faster convergence rate and higher tracking precision than TDE based NTSM and improved TDE based NTSM controller.

#### 1. Introduction

Robot manipulators play an important role in the industry automation field in recent years [1–3]. One of the irreplaceable capabilities of robot manipulations is the high accuracy and high-speed performance of trajectory tracking. Therefore, engineers have sought to realize their automatic control by various control methods, such as sliding mode control (SMC) [4–7], adaptive control [8], neural network methods [9], and fuzzy logic control [10]. The automatic control of robot manipulators presents a unique challenge from the control aspect, which is caused by inherent large nonlinearities and external uncertainties in system dynamics. Generally, to obtain a satisfactory control performance, the above control methods are mostly either model-free types introducing numerous parameters or model-based types requiring nominal model [11]. However, robot manipulators, similar to many other mechanical systems, are complex and nonlinear. It is difficult to establish an accurate dynamic model of a robot manipulator system, which limits the practical applications of the control methods.

It is noteworthy that time delay estimation (TDE) provides a simple way to solve the above problems. The main idea of the TDE is to estimate unknown dynamics and disturbances by intentionally using time delayed information [12]. The main advantage of the TDE is the mitigation of tedious modeling burden of complex system [13]. Due to its advantages, TDE has been widely used in the design process of various controllers and provides satisfactory results [14–16]. Lee et al. proposed an adaptive robust controller using TDE and adaptive integral sliding mode control, which was proved to be robust, chattering-free, and highly accurate [17]. Roy et al. proposed a new adaptive robust control strategy with time delay control to remove prerequisite of system model and to alleviate the over- or underestimation problems of the switching gain [13]. A systematic method was proposed using time delay estimation to simplify the tune process of fuzzy PID controller by Kim et al. [18].

Controllers designed based on TDE are typically composed of two elements. One element is the TDE element, which cancels nonlinear dynamics. The other element is an injecting element, which endows desired error dynamics. Linear error dynamics are widely used as the desired error dynamics in TDE based controllers, such as time delay control (TDC) [19]. Through extensive research, controllers based on TDC have been developed to improve the control performance. To suppress estimation error in TDC, fuzzy logic system was introduced in TDC as the third element by Bae et al., and satisfactory results were obtained [20]. To realize the automatic tuning of TDC parameters, adaptive time delay control was proposed by Jin et al. [21] and Cho et al. [22], respectively. To improve the convergence rate and tracking precision, the nonlinear error dynamics such as nonsingular terminal sliding mode (NTSM) were used together with TDE by recent work [23, 24]. Jin et al. employed this control method in the trajectory tracking of robot manipulators by simulations and experiments. The tracking results showed that the controller is highly accurate, model-free, simple to implement, and robust. However, there are two aspects to be improved. Kinematic uncertainties are not considered in the controller design process. When the desired trajectory is planned in the task space such as Cartesian space, the Jacobian matrix from joint space to Cartesian space and the kinematic parameters of the robot manipulator are assumed exactly to be known. Satisfactory performance can be obtained in the task space without the feedback of the end-effector position with the assumption. But in most practical applications, the kinematics parameters may not be exactly known, which is caused by the interaction between manipulator and different environments and the imprecise measurements of physical parameters. TDE can not eliminate the nonlinearities such as Coulomb friction perfectly. Those nonlinearities may cause TDE error, which reduces the precision of trajectory tracking [25–27].

In order to solve the above problems, a novel control method is proposed for trajectory tracking of robot manipulator’s end-effector in the task space. The controller is based on TDE and combines with the NTSM and adaptive fuzzy logic control scheme. The contributions of this controller are listed as follows. (1) The kinematic uncertainties of the manipulator are taken into consideration without the prior knowledge of system. (2) No dynamic models of the robot manipulator system are required. (3) The adaptive fuzzy logic scheme is implemented as the third element to compensate the TDE error and eliminate the undesired chattering in NTSM. (4) The controller is easy to implement with simple structure, and it is suitable for practical applications.

#### 2. TDE Based NTSM Control with Kinematic and Dynamic Uncertainties

In this section, the conventional TDE based NTSM controller is improved with the consideration of kinematic uncertainties. The control objective of this controller is to make the end-effector of manipulator follow a desired trajectory in the task space with unknown dynamics and uncertain kinematics.

The dynamical equation of -link robot manipulator can be shown as where are vector of position, velocity, and acceleration of the joints respectively; denotes the actuator torque; and represents the inertia matrix; stands for the Coriolis and centrifugal matrix; is the gravitational vector; is the friction term; and denotes the disturbance torques.

Equation (1) can be rewritten as follows after defining a constant diagonal matrix :where .

The trajectory of manipulator end-effector in the task space can be expressed as where is the transformation relationship between the task space and the joint space; is the trajectory of end-effector in the task space, which can be measured by laser sensor or vision sensors. is the velocity in task space and it is related to in the joint space, which can be expressed aswhere is the Jacobian matrix.

The differential of (4) can be expressed as

It should be noted that the inverse of Jacobin matrix will be an ill conditioned matrix when the manipulator is at the vicinity of a singular configuration, and the singularity problem exists in the control of robot manipulators. Some methods have been proposed to solve this problem [28]. In this paper, it is assumed that the Jacobian matrix is nonsingular, and then (5) can be rewritten as

Substituting (6) into (2), it can be obtained as

Because the kinematic and dynamic parameters of robot manipulators are not exactly known in the practical applications, the parameters in (8) are substituted by the estimated value.where and are estimated parameters of** H** and .

Replacing by the acceleration of the desired trajectory , the equivalent control input can be expressed as

To compensate the kinematic and dynamic uncertainties and guarantee the stability, the NTSM scheme is implemented as

The NTSM scheme is designed by two steps. The first step is the design of a sliding surface, and the second step is the design of a control law to guarantee the existence of sliding mode.

Then, sliding mode surface is chosen aswhere is the tracking error between the desired position and actual position of the manipulator end-effector in the task space, and ; represents an vector;* K* denotes an defined positive constant matrix; and are positive odd integers and .

To guarantee the existence of sliding mode, the NTSM scheme is designed as where is a constant matrix to be designed.

Then, the control law can be expressed as

In this control law, is the estimation of all the nonlinearities kinematic and dynamic uncertainties, which makes it difficult to establish model and identify parameters. TDE is used to estimate in this controller, which can be expressed as where denotes the time delayed value of , and if the time delay* L* is set as infinitesimally small, an estimation of would be possible by TDE.

Then the output of the controller can be expressed as

Substituting controller equation (15) into dynamics equation (2), it can be obtained as

In (6), the Jacobin matrix is substituted by the estimated value. It can be expressed as

Substituting (17) into (16), it can be obtained as where and it is defined as TDE error. The TDE error is mainly caused by the finite time delay* L* under nonlinearities such as Coulomb friction.

Lyapunov function is chosen to prove the stability of the system, which is expressed asThen, the derivative of* V *is expressed as

According to the study of Jin et al. [24], is proved to be bounded. In (20), and are positive odd integers and ; then for . To guarantee the stability of the system, should be chosen as Then, and .

Thus, the controller considering the kinematic and dynamic uncertainties is proposed as (15). However, it still has two drawbacks. One is the chattering problem caused by sign function. Although the replacement of the sign function by saturation function can eliminate the chattering, it reduces the tracking precision [29, 30]. The other problem is the TDE errors, which is shown in (18). TDE errors may cause the tracking error of the system.

#### 3. TDE Based AFNTSM Control with Kinematic and Dynamic Uncertainties

In order to solve the above problems, adaptive fuzzy logic control scheme is used to eliminate the chattering and improve the tracking precision. In this section, the controller is presented and the stability is proved based on the Lyapunov method. The new controller structure is designed as

Figure 1 demonstrates the block diagram of the proposed controller. It is composed of three terms, including the equivalent control term, the adaptive fuzzy nonsingular terminal sliding mode term, and time delay estimation term. The main difference of controller equation (22) and equation (15) is the replacement of sign function by an adaptive fuzzy logic control scheme , where is the adaptive fuzzy logic controller and is its compensator. is designed as a diagonal positive definite matrix and , where is a positive constant and is a positive value.