Abstract
In order to make the manipulator track the desired trajectory in a finite time, a new control method based on fast nonsingular terminal sliding mode has been designed. This method combines the traditional fast terminal sliding mode with the nonsingular terminal sliding mode, has rapidity, nonsingularity, finite-time convergence, and strong robustness, and can effectively suppress the inherent chattering phenomenon of the sliding mode controller. The nonsingular fast terminal sliding mode surface is used to accelerate the convergence speed of manipulator trajectory tracking error, and the singularity problem in terminal sliding mode is solved. The finite-time convergence of the algorithm is proved by the Lyapunov function. The simulation results demonstrate that the proposed method can achieve accurate finite-time trajectory tracking characteristics and has robustness against external disturbances.
1. Introduction
Realizing the fast and high-precision tracking of the desired trajectory has always been one of the key issues in the research of the trajectory tracking control of the manipulator. Considering that the dynamic model of the manipulator is a strongly coupled nonlinear system with time-varying parameters, and under the influence of uncertain factors such as external disturbances and friction, many traditional control methods, such as PD control, adaptive control, optimal control, model predictive control, etc., have limited tracking performance, making it difficult to meet the control requirements of the actual manipulator [1–7].
The sliding mode control method is a nonlinear control algorithm with strong robustness that can eliminate internal coupling and realize decoupling. It is widely used in the control of manipulators [8–10]. However, the traditional sliding mode control can only ensure the progressive convergence of the system state and cannot guarantee that the system state error converges to zero within a finite time, which limits its application in actual industrial production. In order to overcome the above shortcomings, the terminal sliding mode control method has been proposed and developed in recent years to realize high-precision trajectory tracking in finite time and suppress uncertainty [11–17].
At present, in the process of practical application, there are still some problems to be solved in the terminal sliding mode control method. First of all, terminal sliding mode control often brings chattering problems [18] and singular problems [19]. Chattering problem refers to the high-frequency chattering phenomenon that occurs when the controller passes through the sliding plane. This is due to the noncontinuous switching items in the sliding mode control signal. In practice, the switching operation time cannot be ignored and is easily affected by model parameters. Uncertainty and the influence of external interference. Chattering will excite the unmodeled high-frequency characteristics of the system, which will lead to a decrease in control performance; the singular problem means that the designed control signal will become infinite at certain moments, which is due to the fact that the terminal sliding mode control often contains negative fractional power. Since the actual driving function provided by the motor is bounded, the design of the control signal with singular problems is obviously inappropriate, which will not only deteriorate the control performance of the system but also seriously damage the driving equipment. In addition, the traditional terminal sliding mode control has a slower convergence speed when the system state is far from the equilibrium point.
In response to the above problems, a fast nonsingular terminal sliding mode control strategy for manipulators is proposed in this paper. Firstly, an improved fast terminal sliding surface is designed to improve the convergence speed of the system state away from the equilibrium point based on the traditional terminal sliding surface. Secondly, considering the chattering phenomenon in the design of the sliding mode controller, an adaptive chattering suppression algorithm is proposed. Then, the sliding mode control law is designed based on the equivalent control theory, and the negative fractional power term in the control law is suppressed by the saturation function to avoid singularity. Finally, the simulation results verify the effectiveness of the designed control method.
2. Problem Formulation
A simplified dynamic model of the n-joint manipulator considering the existence of external disturbances and inertia uncertainties can be expressed aswhere
and are the estimation terms of the manipulator dynamics model, is the uncertainty term of the manipulator model, and denotes the bounded unknown disturbances.
Equation (1) can be rewritten aswhere denotes the external disturbance and modeling error of the system. Assuming is a bounded function, andwhere and are constants. Let be the desired trajectory, then the error function can be written as
The state equation of the system can be written aswhere
The state equation of the system iswhere
Problem 1. To find a feedback control law for the system (8) to guarantee that all the system states of the closed-loop system are uniformly ultimately bounded, and the tracking errors converge into small regions in finite time.
3. Design of Trajectory Tracking Controller
3.1. Analysis and Design of Sliding Surface
The sliding surfaces of terminal sliding mode control (TSM) and nonsingular terminal sliding mode control (NTSM) are designed respectively as follows:where, , , .
If and , the above two sliding surfaces are equal. Then it can be concluded that the equilibrium point (9a) and (9b) is globally finite-time stable. For any given initial state , the system state converges to 0 in finite timeand when or , .
The sliding surface of fast terminal sliding mode control (FTSM) is described by the following differential equation:where , . , and is odd and .
By solving the differential equation, it can be seen that the system state reaches in finite timewhere is a Gaussian hypergeometric function. The parameters , , , and can make converge.
If the system state is far from the equilibrium point, has a greater impact on the system than . At this time, equation (11) can be written as to ensure that the sliding surface can converge quickly.
When the system state approaches the equilibrium point, has a greater impact on the system than . The term determines the finite-time convergence. Therefore, the fast sliding surface will converge at a faster speed.
Equation (11) can be written as follows:
If the system state is far from the equilibrium point, equation (13) can be approximately written aswhere, .
If the system state approaches the equilibrium point, equation (13) can be approximately written aswhich is the same form as (9a).
Equation (13) reaches the equilibrium point within the time
Using the sliding surface designed above can ensure faster convergence speed, but due to the negative exponential term in the derivation of , it will lead to singular problems. A nonsingular fast terminal sliding mode surface is designed as follows:
The nonsingular fast terminal sliding mode surface (NFTSM) is constructed aswhere , , and are constants.
By a direct calculation, the time derivative of s is
According to the above analysis, the nonsingular fast terminal sliding mode surface (NFTSM) (17) and the derivative are continuous.
Remark 1. If , , the sliding surface is transformed into a structure similar to equation (11) and has similar convergence characteristics. When the system states are far from the equilibrium point, the term in equation (17) plays a leading role to ensure rapid convergence. When the system states approach the equilibrium point, the term in equation (17) plays a major role and the system is stable in a finite time. The system maintains a fast convergence rate throughout the dynamic process.
Remark 2. In a small neighborhood of the origin , the sliding surface and the derivative of s do not contain negative exponential power terms, which avoids singularity.
3.2. Design and Stability Analysis of Tracking Controller
For Problem 1, a nonsingular fast terminal sliding mode surface is constructed as follows:where , , , , . is the switching function and ,
Letwhere, is the identity matrix.
Using the sliding mode surface (17) of nonsingular fast terminal control, the control law is designed as follows:where , , .
Theorem 1. For system (8), using the finite time nonsingular terminal sliding mode (14) designed in this paper and selecting the control law (20), the system state can reach the equilibrium point in finite time , which is related to the design parameters.
Proof. The stability analysis is performed below. Firstly, the Lyapunov function is selected asDerivation of equation (22) can be obtained can be expressed asThen equation (23) can be expressed asSubstituting the controller into equation (25), there isLetThen equation (26) can be rewritten asIt can be seen from equation (29), . The closed-loop system is stable.
Find the exponential reaching law satisfying the following conditions from the control lawwhereIn order to suppress chattering, the following function can be used to replace the switching function in the control law:where is a small constant.
4. Simulation
In this paper, sliding modes NFTSM and FTSM are simulated under the same controller parameters, initial conditions, and external disturbance conditions. A simplified dynamic model of the manipulator is expressed aswhereand .
The desired tracking trajectory is , and the initial states are .
The sliding mode parameters and controller parameters are shown in Table 1.
Based on the proposed nonsingular fast terminal sliding mode control law (16), the parameters of the controller are selected as follows:
The block diagram of the control system is shown in Figure 1.
The simulation results are shown in Figures 2–4 by using the NFTSM method.
The position tracking of joint 1 and joint 2 are shown in Figure 2. It can be observed that it takes 0.5 s to track the desired signal on each joint angle. The tracking error and the control input of joint 1 and joint 2 are shown in Figures 3 and 4, respectively. It can be seen that the manipulator can track the reference trajectory well, and the error signals converge to zero rapidly. The control torques are exhibited in Figure 4. From the simulation results, it can be concluded that the proposed method has robustness against external disturbances.
The simulation results are shown in Figures 5–7 by using the FTSM method.
The position tracking and the tracking error of joint 1 and joint 2 are shown in Figures 5 and 6. It can be seen that it takes 2.1 s to track the desired signal on each joint angle. The time of tracking the desired trajectory in the FTSM method is longer than that in the NFTSM method. The control inputs of joint 1 and joint 2 are shown in Figure 7.
Compared with the two control methods, although the manipulator can track the desired trajectory in a limited time, the convergence speed of NFTSM is obviously better than FTSM. It can also be seen that the proposed method can achieve better control performance with a lower control energy cost.
5. Conclusion
In this paper, a fast-nonsingular terminal sliding mode control method is designed to meet the requirements of manipulator trajectory tracking. The new sliding mode controller has the following advantages:(1)Compared with the traditional sliding mode, it has fast convergence and solves the control singularity problem well.(2)The control system can converge to the expected value in a finite time and has robustness.(3)The new sliding mode controller can effectively suppress the inherent chattering phenomenon of the traditional sliding mode controller.
The nonsingular finite-time sliding mode control method established in this paper has fast convergence speed, low energy consumption, and no chattering. It has good control quality and is convenient for practical application.
Data Availability
The [.m ] data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Science and Technology Development Plan Project of Henan Province (no. 212102210512), the Research and Innovation Team of Henan University of Animal Husbandry and Economy (no. 2018KYTD19), and the Research and Innovation Fund Project of Henan University of Animal Husbandry and Economics (no. XKYCXJJ2020010).