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Two adaptive switching control strategies are proposed for the trajectory tracking problem of robotic manipulator in this paper. The first scheme is designed for the supremum of the bounded disturbance for robot manipulator being known; while the supremum is not known, the second scheme is proposed. Each proposed scheme consists of an adaptive switching law and a PD controller. Based on the Lyapunov stability theorem, it is shown that two new schemes can guarantee tracking performance of the robotic manipulator and be adapted to the alternating unknown loads. Simulations for two-link robotic manipulator are carried out and show that the two schemes can avoid the overlarge input torque, and the feasibility and validity of the proposed control schemes are proved.

With the increasing number of robotic manipulators used in industry, it has become an important research field for the control of manipulators with unknown or changing dynamics. The excellent performance of tracking can be considered as one of the most important requirements of robotic manipulator because of its highly nonlinear, multivariable, strongly coupling, and time-varying systems. Many schemes were presented in the past years [

For compensation of those uncertainties stemming from inaccurate estimation of inertial parameters of the load mass changes and so on, many methods are proposed to overcome these uncertainties, such as adaptive control [

The adaptive control is very popular to cope with the parameter uncertainty of robotic system [

Iterative learning control requires that the reference trajectory is periodic with known period [

Neural networks (NNs) and fuzzy systems are two typical tools to parameterize the nonlinear systems with unknown nonlinearities [

In fact, from an implementation point of view, neural networks (NNs) and fuzzy systems are much more complicated. It is well known that the linear PD control is one of the most simple and effective control methods, which has been widely used in the field of industrial robots [

Practically, robots often must pick up or lay down some objects and the load for manipulator is not constant. Therefore, parameter jumping exists in this system. So it is difficult for the traditional adaptive control to solve the above problem. It is well known that a system with a jumping parameter can be viewed as a switched system whose subsystems differ from each other only by parameters [

The purpose of this paper is to provide an efficient solution. In this paper, two adaptive switching controllers with PD parameters for a serial

According to whether the supremum of bounded disturbance for robot manipulator is known, two difficult disturbance compensation algorithms are designed. Those strategies are all composed of a nonlinear PD and compensated controller. The portion of nonlinear PD can avoid the overlarge output of initial torque, and the adaptive controller including a regression matrix can compensate the dynamic uncertainty of robot manipulator.

Advantages of methods are as follows: when the initial error is bigger, the nonlinear PD feedback plays a main role, which can avoid the overlarge output of initial torque; when the error is smaller, the adaptive controller plays a main role, which can obtain the good dynamic performance.

Based on switched common Lyapunov function method, the adaptive updated laws and the switching signals have been developed to guarantee that the resulting closed-loop system is asymptotically Lyapunov stable and the position of manipulator’s joint can follow any given bounded desired output signal. Finally, a simulation example of robotic manipulator is given to illustrate the proposed methods.

The organization of this paper is as follows. In Section

Considering an

Some properties and assumptions of robot (

The disturbance input and errors

If

Considering the payload variation, the following switched model of robotic manipulator for subsystems has been used:

For model (

The following adaptive switching controllers will be classified as two sections: the first is designed under the condition of knowing the supremum of bounded disturbance for robot manipulator, and the other is done without knowing it.

This section introduces the adaptive switching controller applied to robotic manipulator. Our purpose is to design a robustly stable controller to ensure the system stability and improve the robot tracking performance in the case of the variational payload for robotic manipulator. For the dynamic model of robot (

The adaptive laws (

For robotic system (

Combining (

Taking the time of derivative of

In this section, a sufficient condition is proposed to ensure the system is stable without knowing the supremum of bounded disturbance.

For robotic system (

Choosing a Lyapunov function candidate

Similar to Theorem

From the process of controller design, it can be seen that it is easier to obtain the parameters of proportional and differential. And controller can be working on the state of arbitrary switching.

In this section, the above proposed adaptive switching strategies are employed to control the robotic manipulator to illustrate the feasibility and effectiveness. Simulations are carried out for a two-DOF planar manipulator whose load is persistently changing. The model of robotic dynamics is

The linkage is composed of two rigid beams with actuators mounted at the joints. The load can be considered as a part of the second link. Parameters of dynamic model (

The length and mass of robot are

The given reference trajectory and initial state of system are

Control parameters are

It is switching adaptive control when the supremum of bounded disturbance for robot manipulator is known.

Choose disturbance input and errors as

Simulation results are shown in Figures

The switching signal.

Tracking error for Case

Torque for link 1 (Case

Torque for link 2 (Case

It is switching adaptive control when the supremum of bounded disturbance for robot manipulator is not known.

Choose

In Case

Tracking error for Case

Tracking error (PID).

Torque for link 1 (Case

Torque for link 2 (Case

In this paper, two adaptive switching control schemes have been investigated for robotic manipulator with changing loads. The first scheme is designed for the supremum of bounded disturbance for robot manipulator being known, and the other is contrary. When the corresponding subsystem is activated, the proposed adaptive update law works. Based on Lyapunov stability theorem, it is shown that the proposed control scheme can guarantee the tracking performance of robotic manipulator system. Simulations show that the satisfactory tracking performance can be obtained and the adaptive switching controller is simple to realize for engineering applications. In our further work, we will try to extend the proposed results to the case of force tracking for the end effector of robotic manipulators.

The authors declare no conflict of interests.

Shumin Fei managed the overall progress and gave some useful advises on this project. Zhen Yang carried out the theoretical analysis and numerical calculations for the results. Both authors have read and approved the final paper.

This work was supported by National Natural Science Foundations of China (61273119, 61374038, and 61473079).