A Novel Fast Terminal Sliding Mode Tracking Control Methodology for Robot Manipulators

This paper comes up with a novel Fast Terminal Sliding Mode Control (FTSMC) for robot manipulators. First, to enhance the response, fast convergence time, against uncertainties, and accuracy of the tracking position, the novel Fast Terminal Sliding Mode Manifold (FTSMM) is developed. Then, a Supper-Twisting Control Law (STCL) is applied to combat the unknown nonlinear functions in the control system. By using this technique, the exterior disturbances and uncertain dynamics are compensated more rapidly and more correctly with the smooth control torque. Finally, the proposed controller is launched from the proposed sliding mode manifold and the STCL to provide the desired performance. Consequently, the stabilization and robustness criteria are guaranteed in the designed system with high-performance and limited chattering. The proposed controller runs without a precise dynamic model, even in the presence of uncertain components. The numerical examples are simulated to evaluate the effectiveness of the proposed control method for trajectory tracking control of a 3-Degrees of Freedom (DOF) robotic manipulator.


Introduction
Robots are gradually replacing people in the fields of social life, manufacturing, exploring, and performing complex tasks. In order to improve productivity, product quality, system reliability, electronics, measurement, and mechanical systems of robotic systems, more advanced designs are required. Therefore, this leads to an increase in the complexity of the structural and mathematical model when there is an additional occurrence of uncertain components.
Sliding Mode Control (SMC) [1][2][3][4][5][6][7][8][9][10][11][12][13][14] is capable of handling high non-linearity and external noise when it possesses outstanding features such as fast response, and robustness towards the existing uncertainties. However, the chattering problem in the SMC causes oscillations in the control input system leading to vibrations in the mechanical system, heat, and even causing instability. Furthermore, the SMC does not yield convergence in a defined period and provides a slow convergence time θ +Ĝ(θ) , Φ(x, t) =M −1 (θ), and D(θ, ∆U) = −M −1 (θ)∆u. The control target of the system is to further increase the response speed and accuracy of the trajectory tracking control for robot manipulators, even if the effects of uncertain dynamics and external perturbations are valid. First, to enhance the response, fast convergence time against uncertainties, and accuracy of the tracking position, the novel FTSMM is developed. Then, STCL is applied to combat the unknown nonlinear functions in the control system. By using this combined technique, the exterior disturbances and uncertain dynamics will be compensated more rapidly and more correctly with the smooth control torque. Finally, the designed controller is launched from the proposed sliding mode manifold and the STCL to obtain the control efficiency.

Main Results
The position control error and the velocity control error on each joint are, respectively, defined as follows: where x d ∈ R n×1 represents the angle of the expected position.

The Designed FTSMM
To enhance the response, fast convergence time, and accuracy of the tracking position, the novel FTSMM is developed as follows: where s i is the proposed sliding mode manifold, γ 1 , γ 2 , µ 1 , µ 2 are the positive constants, 0 < α < 1, and . Based on the SMC, when the control errors operate in the sliding mode, the following constrain is satisfied [1]: Appl. Sci. 2020, 10, 3010 4 of 16 From condition in Equation (9), it is pointed out that: Remark 1. When the control error of |x ei | is much greater than φ, the first component of Equation (10) offers the role of providing a quick convergence rate and the second component has a smaller role. Contrariwise, when the control error of |x ei | is much smaller than φ, the second component of Equation (10) offers a greater role than the first one.
The following theorem is launched to guarantee that convergence takes place within the defined time.
Theorem 1. Let us consider dynamic of Equation (10). x ei = 0 is defined as the equilibrium point and the state variables of the dynamic of Equation (10), including x ei and x dei stabilize to zero in finite-time.

Proof.
To validate the correctness of Theorem 1, the Lyapunov function candidate is proposed as follows: and its time derivative is It is shown that . L 1 < 0, hence, x ei and x dei concentrate on the equilibrium state in finite time. When x ei (0) > φ, the sliding motion consists of two phases: The first phase: x ei (0) → |x ei | = φ , the first component of Equation (10) offers the role of providing a quick convergence rate and the second component has a smaller role.
The second phase: |x ei | = φ → x ei = 0 , the second component of Equation (10) offers a role greater than the first one.
The total time of the sliding motion phase is defined as: Appl. Sci. 2020, 10, 3010

of 16
The state variable of the dynamic in Equation (10) converges to sliding manifold ( s(0) → 0 ) within the defined time T r , which was pointed out in [8]. Therefore, the total time for stability on the sliding manifold is computed as: T ≤ T r + T s (16)

The Designed Control Methodology
Let us take the time derivative of Equation (8): With .
x d , the time derivation of Equation (17) gets along with the system in Equation (5) as follows: In order to facilitate controller design, there is the following assumption: , need to satisfy the following standard condition: where K i > 0.
In order to achieve the stabilization target of the robot system, the following control action is proposed: Here, it should be noted that the u eq is designed as: and u r is designed as: where . Σ 1i and Σ 2i are assigned to satisfy the following relationship [8]: Based on those above statements, the following theorems are written to prove the stability problem.
Theorem 2. Consider the robot system in Equation (1). If the designed torque actions are proposed for system in Equation (1) as Equations (20)- (22), then x ei and x dei stabilize to zero in finite time. That means that robot system in Equation (1) runs in a stable mode. (20)- (22) to Equation (19) gains:

Proof. Applying control torque in Equation
Based on the assumption in Equation (19), and the selection condition of the sliding gains in Equation (23), it can be verified that the sliding manifold and its time derivative will converge to zero in finite time. Now, considering one of the elements in Equation (24) as follows: The following Lyapunov function is defined for the system in Equation (25): Here Based on the assumption in Equation (19), we can gain: .
According to [8], s i and . s i are equal to zero in finite-time (t ri = 2L 1/2 2 (t = 0)/υ). Therefore, s and . s equal to zero in finite time (T r = max i=1,...,n {t ri }) and both x ei and x dei also stabilize to equilibrium in finite time (T ≤ T r + T s ) under the control action in Equation (20)- (22).

Numerical Simulation Studies
In this numerical example, a 3-DOF PUMA560 robot manipulator (with the first three joints and the last three joints blocked) is adopted. The MATLAB/SIMULINK software (2019a MATLAB Version Appl. Sci. 2020, 10, 3010 7 of 16 of The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760 USA) was used for all computation, the sampling time was set to 10 −3 s, and the solver ode3 was used. The kinematic description for the robot system is displayed in Figure 1. The design parameters and dynamic models of the robot system are referenced from the document [39]. There are many essential parameters of a robot that need to be presented. Therefore, to present briefly, the design parameters and dynamic models of the robot system are reported in [39].

Numerical Simulation Studies
In this numerical example, a 3-DOF PUMA560 robot manipulator (with the first three joints and the last three joints blocked) is adopted. The MATLAB/SIMULINK software (2019a MATLAB Version of The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760 USA) was used for all computation, the sampling time was set to 10 −3 s, and the solver ode3 was used. The kinematic description for the robot system is displayed in Figure 1. The design parameters and dynamic models of the robot system are referenced from the document [39]. There are many essential parameters of a robot that need to be presented. Therefore, to present briefly, the design parameters and dynamic models of the robot system are reported in [39]. To explore the potential of our designed approach, the robot is controlled to follow the designated trajectory configuration at first. Later, its control performance is then evaluated and compared with the performance of different control algorithms, including SMC and NFTSMC. These control methods for comparison are briefly explained as follows: The normal SMC [14] has the following control torque:  To explore the potential of our designed approach, the robot is controlled to follow the designated trajectory configuration at first. Later, its control performance is then evaluated and compared with the performance of different control algorithms, including SMC and NFTSMC. These control methods for comparison are briefly explained as follows: The normal SMC [14] has the following control torque: where s = x de + cx e is the linear sliding manifold, c is a positive constant. Further, the NFTSMC [40] has the following control torque: where s = x e + −1 x l/q de is a nonlinear sliding manifold. The designed parameters of three control methodologies are given in Table 1. The designated trajectory configuration for position tracking when the robot manipulator operates: Friction and disturbance models are hypothesized to analyze the strong capability of the designed FTSMC. It is not amenable to accurately calculate these friction and disturbance terms; therefore, the physical values of frictions and disturbances are not measured. Therefore, the following friction forces and disturbances were modeled, respectively: To clearly present the results within the simulation period and to facilitate easier comparison, the averaged tracking error i: where Z is the number of simulation steps.
To demonstrate the superiority of the designed controller, the average control error is calculated over two different simulation periods (10 s and 30 s).
The averaged tracking errors are reported in Table 2. The designated trajectory configuration and real trajectory under three control methods at the first three joints are displayed in Figure 2. It can be seen from Figure 2 that all three controllers appear to have a similar tracking control performance. However, they have different convergence times in the following order: the designed controller has the fastest convergence time among all three control methods, and NFTSMC has faster convergence time than the normal SMC. Figures 3 and 4 show the position control errors and the velocity control errors, respectively. It can be seen from Figure 3 and Table 2, the position control errors of the designed control scheme are relatively small compared to those of the other control methods, in the order of 10 −7 rad. The position control errors of the NFTSMC are in the order of 10 −6 rad. SMC provides the largest position control errors of the three control methods, in the order of 10 −4 rad.
From Figure 4, it is seen that the designed control method also has the smallest velocity control errors among all the three control methods.
The control torque for all three control manners, including SMC, NFTSMC, and the designed FTSMC, are displayed in Figure 5. It can be recognized from Figure 5, SMC and NFTSMC have discontinuous control torque because of using the high-frequency control law. Meanwhile, the designed system has smooth control torque with a significant elimination of the chattering phenomena. To achieve this goal, the suggested controller applies STCL to substitute the high-frequency control law in removing chattering behavior. The designated trajectory configuration and real trajectory under three control methods at the first three joints are displayed in Figure 2. It can be seen from Figure 2 that all three controllers appear to have a similar tracking control performance. However, they have different convergence times in the following order: the designed controller has the fastest convergence time among all three control methods, and NFTSMC has faster convergence time than the normal SMC.    From Figure 4, it is seen that the designed control method also has the smallest velocity control errors among all the three control methods.  The control torque for all three control manners, including SMC, NFTSMC, and the designed FTSMC, are displayed in Figure 5. It can be recognized from Figure 5, SMC and NFTSMC have discontinuous control torque because of using the high-frequency control law. Meanwhile, the designed system has smooth control torque with a significant elimination of the chattering phenomena. To achieve this goal, the suggested controller applies STCL to substitute the highfrequency control law in removing chattering behavior. Response time of the sliding mode manifolds, including SMC, NFTSMC, and designed FTSMC, are shown in Figure 6. Response time of the sliding mode manifolds, including SMC, NFTSMC, and designed FTSMC, are shown in Figure 6.

Conclusions
This paper focuses on designing a novel FTSMC for robot manipulators. In the first step, the novel FTSMM is developed to enhance response capability, fast convergence time, uncertainties opposition, and especially, improve the accuracy of the tracking position. To alleviate unknown nonlinear parameters in the control system, STCL is then applied. Thanks to this valuable technique, exterior disturbances and nonlinear elements are compensated more rapidly and more correctly with the smooth control torque. Finally, combining STCL and our proposed sliding mode manifold, under the flexible controller, the stability and robustness of the control system are guaranteed with highperformance and limited chattering. To evaluate the efficiency, a simulation example is performed for the trajectory tracking control of a 3-DOF robotic manipulator.

Conclusions
This paper focuses on designing a novel FTSMC for robot manipulators. In the first step, the novel FTSMM is developed to enhance response capability, fast convergence time, uncertainties opposition, and especially, improve the accuracy of the tracking position. To alleviate unknown nonlinear parameters in the control system, STCL is then applied. Thanks to this valuable technique, exterior disturbances and nonlinear elements are compensated more rapidly and more correctly with the smooth control torque. Finally, combining STCL and our proposed sliding mode manifold, under the flexible controller, the stability and robustness of the control system are guaranteed with high-performance and limited chattering. To evaluate the efficiency, a simulation example is performed for the trajectory tracking control of a 3-DOF robotic manipulator.
From theoretical evidence, simulation results, and a comparison with SMC and NFTSMC, our proposed controller has some of the following contributions: (1) the proposed controller provides finite-time convergence and faster transient performance without singularity problem in controlling; (2) the proposed controller inherits the benefits of the FTSMC and CRCL in the characteristics of robustness towards the existing uncertainties; (3) a new FTSMM was proposed, and evidence of finite-time convergence was sufficiently proved; (4) the precision of the proposed controller was further improved in the trajectory tracking control; (5) the proposed controller shows the smoother control torque commands with lesser oscillation.

Conflicts of Interest:
The authors declare no conflict of interest.