To improve the trajectory tracking performance and robustness for uncertain robot manipulators, a generalized sliding mode controller (GSMC) including an ideal controller and a continuous sliding mode controller (SMC) is proposed from the standpoint of motion constraints. First, the trajectory tracking requirements are formulated as the motion constraints, based on which an ideal controller is proposed to satisfy the motion constraints for robot manipulators whose dynamics are precisely known. Second, an additional continuous SMC is presented to compensate for the effects of uncertainty, and the chattering phenomenon that commonly exists in the SMC can be avoided by the introduction of a smoothing function. Third, Lyapunov analysis is conducted to verify that the proposed GSMC enables the tracking error restricted to a small region around zero. Finally, the numerical simulation and experiment are performed to verify the effectiveness and superiority of the proposed GSMC.

Recently, with the development of technology, robot manipulators have been widely applied in industrial manufacturing, aerospace, medical treatment, military, and other fields. Despite the different application scenarios, achieving fast and high-accuracy trajectory tracking is one of the common requirements for these tasks. However, acquiring a desirable trajectory tracking performance for robot manipulators remains a challenge, since robot manipulators have multiple variables, strong coupling, and nonlinear systems

Owing to its strong robustness, the SMC has attracted much attention

To realize the precision control of robot manipulators with uncertainty, a two-step control scheme is proposed. In the first step, consider the control of the “ideal” robot manipulator system whose dynamics model is precisely known. An exact closed-form ideal controller is designed from a novel standpoint, where the control requirements are taken as a set of constraints exerted on the controlled system, and based on the analytical dynamics theory

The main contributions are summarized below.

Based on the analytical dynamics theory, an ideal controller that can enable the ideal system to meet the desired control requirements is presented in closed form. Moreover, the ideal control input is optimal since it renders the quadratic control cost minimized at each moment.

By the introduction of the notion of the generalized sliding surface, an additional controller is designed to augment the ideal controller. The proposed controller can drive the system to meet the expected control requirements in the presence of initial condition deviations and uncertainty while avoiding the chattering problem.

Consider an

Since uncertainty inevitably exists in the robot manipulator, we assume that

The control objective of this paper is to design a suitable controller that causes the tracking error to converge to zero as

In this paper, the control design is split into two steps. The first step is to design an ideal controller for the ideal system (i.e., whose dynamic model is precisely known and has no external disturbances imposed on it). The second step is to design a GSMC to enhance the robustness.

Firstly, we suppose that the robot manipulator dynamic model parameters are all known and without load disturbances imposed on the robot manipulator. Then, we can obtain the ideal dynamic model of the robot manipulator as

In this paper, the control objective is reformatted as the form of motion constraint

Generally, the motion constraint (Eq.

Differentiating Eq. (

Rewriting Eqs. (

According to

In the previous section, the control input that drives the motion constraints satisfied is deduced based on the condition that the model parameters of robot manipulator are precisely known. However, precise knowledge of the system is difficult or even impossible to achieve. Therefore, in this section, we will explore the control of robot manipulator whose parameters are not perfectly known.

Define the auxiliary error variable as

According to Eq. (

The generalized sliding surface is defined as

Notice that the generalized displacement and velocity at the initial time are the same for the ideal system and the real system (i.e.,

Define

Select the Lyapunov candidate as

Differentiating Eq. (

Differentiating Eq. (

By Eqs. (

Substituting Eq. (

Then, one has

Substituting Eq. (

Moreover, when

To illustrate the effectiveness of the proposed controller, some simulations and experiments are conducted with a two-link robot manipulator. The dynamics of the manipulator is given as

To assess the superiority of the proposed controller, the comparison with a conventional SMC and robust control

The expression of the robust control is given as

The simulations are carried out in the MATLAB environment, employing a variable time step ode15i solver. The simulation results are depicted in Figs. 1–6. Figures 1 and 2 present the trajectory tracking response of joints 1 and 2 under three different controls. As seen in Figs. 1 and 2, both the three controls can track the reference trajectory. To compare the tracking performance of three algorithms, the trajectory tracking errors of joints 1 and 2 are depicted in Figs. 3 and 4. It can be observed that both the transient response and steady-state tracking performance of the generalized SMC are superior to those of the conventional SMC and robust control. In addition, the control input of joints 1 and 2 under the three algorithms is illustrated in Figs. 5 and 6; one can see that the control inputs of the conventional SMC and robust control are higher than the generalized SMC for both joint 1 and joint 2.

Simulation: trajectory tracking response of joint 1 under sinusoidal signal.

Simulation: trajectory tracking response of joint 2 under sinusoidal signal.

Simulation: trajectory tracking error of joint 1 under sinusoidal signal.

Simulation: trajectory tracking error of joint 2 under sinusoidal signal.

Simulation: control input of joint 1 under sinusoidal signal.

Simulation: control input of joint 2 under sinusoidal signal.

To further validate the proposed control algorithm, the experimental validation on the robot manipulator experimental platform (see Fig. 7) is conducted. The experimental platform adopts the rapid control prototyping method, which enables users to establish the control algorithm model in the MATLAB/Simulink environment and then use the automatic code generation tool to automatically generate the C code, thus facilitating users to complete the algorithm validation in a fast and efficient way.

Figures 8 and 9 show the trajectory tracking responses of the conventional SMC algorithm, robust control algorithm, and generalized SMC algorithm, respectively. One can see that the generalized SMC algorithm provides better tracking performance than the conventional SMC algorithm and robust control algorithm. Figures 10 and 11 compare the tracking errors of the two joints under three different algorithms. The tracking errors of the two joints under the generalized SMC algorithm are below 0.2

Robot manipulator experimental platform.

Experiment: trajectory tracking response of joint 1 under sinusoidal signal.

Experiment: trajectory tracking response of joint 2 under sinusoidal signal.

Experiment: trajectory tracking error of joint 1 under sinusoidal signal.

Experiment: trajectory tracking error of joint 2 under sinusoidal signal.

Experiment: control input of joint 1 under sinusoidal signal.

Experiment: control input of joint 2 under sinusoidal signal.

In this study, to improve the trajectory tracking control performance for robot manipulators subject to uncertainty, a generalized SMC is designed. The design procedure of the proposed generalized SMC contains two steps. In the first step, it is assumed that the dynamic model of robot manipulators is precisely known, and there are no external disturbances; an ideal control is designed based on analytical dynamics by reformulating the trajectory tracking as a problem of constrained motion. In the second step, a smooth-function-based SMC is designed to prevent the chattering phenomenon caused by discontinuous function and further enhance the robustness performance. Numerical simulations and experimental results simultaneously validate the effectiveness of the proposed generalized SMC algorithm. In the future, this method will be modified and applied to robotic manipulators with actuator saturation and output constraints.

All of the code used in this paper can be obtained from the corresponding author upon request.

No data sets were used in this article.

ZW designed the study and wrote the paper, LM provided guidance on the writing, and XM conducted the simulation research.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors acknowledge the financial support of the Jiangxi Provincial Department of Education.

This research has been supported by the Science and Technology Project of the Jiangxi Provincial Department of Education (grant no. GJJ2202403).

This paper was edited by Daniel Condurache and reviewed by four anonymous referees.