Adaptive super-twisting global nonsingular terminal sliding mode control for robotic manipulators

This paper develops a novel global nonsingular terminal sliding mode control (GNTSMC) strategy based on an adaptive super-twisting algo-rithm (STA) for tracking control of robotic manipulators with uncertain perturbations. A novel global nonsingular terminal sliding manifold is designed to steer the system trajectory to reach the switching surface at the beginning, thereby removing the reaching stage and achieving strong robustness throughout the entire response. Moreover, the proposed sliding manifold can ensure the finite time convergence of the trajectory error to the origin. Then, an adaptive STA, which does not require the boundary information of the perturbations, is devised not only to attenuate the chattering effect without degrading the tracking precision, but also to guarantee the finite time stability of the system. Finally, the superiority of the adopted GNTSMC is validated by comparative studies.


Introduction
The robotic manipulator has been widely utilized in industrial systems to realize mechanical automation and ensure the safety of operators.Due to the complexity of working environments and nonlinear and highly coupled dynamics of robot systems, it is unable to establish relatively accurate manipulator models.In order to achieve high accuracy tracking of robotic manipulators and reject various disturbances, effective control strategies independent of the mathematical models of manipulators are required.Sliding mode control (SMC) is one of the most popular nonlinear control schemes thanks to its strong robustness against external interference, invariance to system variations, fast response, and flexible design.However, conventional SMC approach has some defects: (1) discontinuous switching control results in chattering phenomena; (2) prior information of perturbations is required in the design process; (3) only asymptotic convergence of trajectory errors is guaranteed due to the adopted linear hyperplane; (4) system performance is susceptible to perturbations during the reaching stage.
Over the past decades, numerous studies have been proposed to tackle these problems.A general means of chattering attenuation is to replace the discontinuous switching function by a continuous one, such as saturation, hyperbolic tangent, sigmoid, and various adjustable reaching laws.However, this method degrades the tracking precision and the robustness property of the system.Higher order SMC technique is another effective way to alleviate chattering by transferring the switching function into the derivative of control input, while system control accuracy can be improved [1][2][3][4][5][6][7].The super-twisting algorithm (STA), one kind of second-order SMC, has been investigated recently.It can realize chattering-free control with high tracking performance and does not require the derivatives of the sliding manifold and control input [8][9][10][11][12][13][14].However, aforementioned works require the upper bounds of system perturbations, which are impossible to be accurately acquired in applications.In order to estimate the unknown perturbations, researches integrating STA with various control schemes, such as adaptive control [15][16][17][18][19][20][21][22], observer techniques [23,24], fuzzy systems [25], and neural networks [26,27], have been presented and successfully applied to industrial systems.
To realize the finite time convergence of trajectory errors, terminal sliding mode control was put forward for nonlinear systems, such as robotic manipulators and permanent magnet synchronous motors.Then, nonsingular terminal sliding mode control and nonsingular fast terminal sliding mode control (NFTSMC) emerged to address the singularity problem in terminal sliding mode control and implement high accuracy tracking control for practical plants [7,14,28,29].To eliminate the reaching stage in the sliding motion and improve robustness throughout the entire response, global SMC techniques have been developed generally by introducing the exponential function into linear hyperplanes [30][31][32][33][34][35].Chu et al. proposed dynamic global SMC to realize global robustness and chattering-free control for micro electromechanical systems gyroscopes [36].Researchers combined global SMC with terminal sliding mode control to achieve superior tracking performance and chattering suppression for nonlinear systems [37][38][39].Liu and Sun presented a novel form of global SMC by introducing a generalized terminal function [40].Subsequently, it was integrated into second-order SMC approaches, which can enhance the tracking property and robustness of the controlled systems and attenuate chattering phenomena [41,42].Besides, integral SMC strategies have been utilized to omit the reaching stage [43,44].However, these methods only guarantee the asymptotic convergence of trajectory tracking errors.
In order to achieve fascinating control performance for robotic manipulators and deal with the presented defects of SMC simultaneously, a novel global nonsingular terminal sliding mode control (GNTSMC) strategy based on an adaptive STA is proposed in this work.The main contributions are: (1) A novel global nonsingular terminal sliding manifold is designed to eliminate the reaching stage, that is, the system state can be drove to the switching surface at the beginning, so that strong robustness can be guaranteed during the whole response and the tracking performance can be improved.Subsequently, the proposed GNTSMC scheme derived from the adopted sliding manifold can not only ensure the finite time convergence of the state trajectory to the sliding manifold, but also drive the trajectory error to the origin within limited time.This greatly differs from the aforementioned global SMC methods.(2) An adaptive STA is developed to suppress chattering effect without degrading the tracking accuracy.Moreover, the designed STA does not require the prior information of perturbations and can avoid overestimating the adaptive gain.Finally, the finite time stability is analyzed.
The remainder of this paper is arranged as follows.The mathematical mode of a rigid robotic manipulator with uncertain disturbances is described in Section 2. The GNTSMC based on an adaptive STA is presented in Section 3, and the finite time stability of the controlled system is proven in this section as well.Then, Section 4 provides the simulation results.Section 5 concludes the paper.

Problem Formulation
Consider the dynamic mode of a robotic manipulator with n degree of freedom as where θ ∈ R n is the joint position; the symmetric positive definite matrix F m (θ) ∈ R n×n denotes the joint inertia; F c (θ, θ) ∈ R n×n , F g (θ) ∈ R n , and F f (θ, θ) ∈ R n represent the coriolis and centrifugal force, gravity, and friction, respectively; τ ∈ R n and τ d ∈ R n denote the joint control torque and external perturbation, respectively.As mentioned above, there exist parameter variations and un-modeled dynamics, thus (1) is reformulated as where Fm (θ), Fc (θ, θ), and Fg (θ) represent the nominal parameter matrices; Γ (θ, θ) denotes the lumped uncertainties bounded by ∥Γ (θ, θ)∥ < D, and D > 0; ∆F m (θ), ∆F c (θ, θ), and ∆F g (θ) stand for the system perturbations.Let e = θ − θ d be the trajectory tracking error, where θ d ∈ R n is the twice continuously differentiable desired trajectory.The objective is to devise a robust controller to perform rapid and accurate tracking of the manipulator joint position and effectively eliminate chattering effect in the presence of time-varying perturbations.
Lemma 1 Assume that the function V (t) is positive definite and its derivative satisfies, where µ > 0 and 0 < ρ < 1.Then, V converges to zero from any initial state within finite time, which can be represented by The following notations are required in the next section to formulate the control law and analyze the finite time stability.For a n-dimensional column vector x and a n-dimensional diagonal matrix α, we define 3 The design of GNTSMC

Global nonsingular terminal sliding manifold design
In order to realize the finite time convergence of the state trajectory to the switching surface and the trajectory error to the origin, and ensure robustness during the entire response, a global nonsingular terminal sliding manifold is designed as where k 1 , k 2 , α, and β are coefficient matrices expressed by Φ is an auxiliary function to be determined according to the following principles: the function Φ should drive the state trajectory to the manifold s = 0 at the beginning to remove the reaching stage; for another, it cannot degenerate the finite time convergence of the trajectory error after the manifold is reached; besides, the continuity of the sliding dynamic should be guaranteed.Consequently, the continuous function Φ can de devised as where t n is a time constant, and Then, the sliding dynamic is derived as According to ( 6) -( 8), s(t) and ṡ(t) are continuous and s(0) = ṡ(0) = 0, the manifold can be reached at the initial time, so as to eliminate the reaching stage and realize strong robustness throughout the entire response.When t > t n , the sliding manifold ( 6) is similar to that of NFTSMC, which can steer the trajectory error to the origin in limited time.
Consider the reaching law ṡ = −η 1 s − η 2 sign(s), the robust control law is given as where ) are gain matrices with η 1i > 0 and η 2i > ∥ F −1 m (θ)∥D.Obviously, the matrix β(t) is crucial to the control law (9) and will decide whether there exists a singularity.In order to achieve singularity-free control, the following design is necessary, where 0 < ξ < 1 is a constant.It allows the state trajectory to move continuously while avoiding the singularity, and guarantees the finite time convergence of the trajectory error.
Define the following Lyapunov function to analyze the stability of the proposed controller, Differentiating (11) yields where ) > 0 for ėi ̸ = 0. Substituting ( 9) into (2) in the case of s ̸ = 0 and ė = 0 yields Therefore, ė = 0 is not an attractor before the state trajectory reaches the sliding manifold s = 0, and the function V (t) can monotonously decrease to zero within finite time.Since V (0) = 0, we obtain V ≡ 0, which means that s ≡ 0, i.e. the state trajectory can reach the manifold and stay on it from the beginning, achieving global robustness and high tracking precision.When t ≥ t n , the sliding manifold is presented as Then, the trajectory error moves along it and converges to the origin in limited time given by Evidently, the value of t n will affect the convergence rate of the state trajectory error.The smaller the value of t n , the faster the convergence rate.

Adaptive STA
Although the proposed sliding manifold can remove the reaching stage and achieve global robustness, the discontinuous control law still leads to unexpected chattering phenomena.In this subsection, a STA is employed to attenuate the chattering effect and enhance tracking performance.Nevertheless, selecting appropriate gains for STA is complicated.This requires not only knowing the boundaries of perturbations, but also adjusting the gains based on experience.To tackle this problem, an adaptive STA is developed as where η1 = diag(η 11 , • • • , η1n ) and η2 = diag(η 21 , • • • , η2n ) denote the adaptive gain matrices.
To ensure the finite time stability of the controlled system and avoid overestimating the control gains, the gain matrices and an adaptive law are defined as follows [18,19], where Consequently, the GNTSMC law based on the adaptive STA can be given as 2 )sign(s) + Substituting ( 19) into (8) yields where . Assumption 1 Ξ is supposed to be bounded by ∥Ξ∥ ≤ δ, where δ > 0.

Finite time stability analysis
In what follows, we will take the ith joint of the manipulator to briefly analyze the system stability based on the analysis method in [20].For the ith link, the sliding dynamic of the system is prescribed by Define a state vector z = [|s i | 1 2 sign(s i ), ϑ i ] T and consider the following Lyapunov functions where εi = ε i − ε * ≤ 0 and P is a positive definite matrix.According to Rayleigh theorem, one has λ min (P )∥z∥ 2 ≤ V 2 ≤ λ max (P )∥z∥ 2 .
It is worth noting that the function V 2 is not locally Lipschitz continuous; even so, the system stability can still be proven in the light of Zubov's theorem if the functions V 1 and V 2 are continuous and positive definite.To analyze the finite time stability of the proposed controller, we first differentiate (23) where ) .
If the matrix R is positive definite, then z T Rz ≥ λ min (R)∥z∥ 2 .Note min (P ) and ∥z∥ 2 ≥ V 2 /λ max (P ), consequently, the following inequality can be derived where ς = λ min (P )λ min (R)/λ max (P ) > 0. To ensure the positive definiteness of R, the adaptive gains should satisfy the following inequality Consider ( 17) and ( 18) and let λ 2i = 1 4 λ 1i ω i , then ( 26) can be reformulated as where Obviously, the positive definiteness of R can be guaranteed if For ė = 0 and s ̸ = 0, substituting (19) into (2) yields we have ë ̸ = 0, which implies that ė = 0 is not an attractor before the sliding manifold s = 0 is reached, therefore, it will not disturb the convergence of the function V 2 .
Then, differentiating (22) yields According to the inequality ( where r = min(ς, 1).It follows that Assume the state trajectory reaches the subset Ω 1 \ {(0, 0)} at the moment t 0 , such that s(t 0 ) = 0 and ṡ(t 0 ) = ωϑ(t 0 ) ̸ = 0.This indicates that at least one entry s i will monotonically pass through the manifold s i = 0 at some moment.Consequently, the absolutely continuous dynamic trajectory derived from (16) will not stay on this subset, and (32) holds almost everywhere, making the function V 1 monotonically decreasing.In the light of Lemma 1, the dynamic trajectory can reach the origin (s, ϑ) = (0, 0) in finite time.Further, e = ė = 0 can be derived in limited time.Moreover, η1 and η2 are bounded, which avoids overestimating the adaptive gains.

Simulation results
To investigate the validity of the adopted robust controller, simulation analysis and comparisons with the adaptive NFTSMC presented in [28] are conducted on a two-link manipulator system with the following parameter matrices, ] , ] , where m i , l i , and J i denote the mass, length, and inertia of the link i (i = 1, 2), respectively, whose nominal values are chosen as m 1 = 0.5 kg, m 2 = 1.5 kg, l 1 = 1 m, l 2 = 0.8 m, and J 1 = J 2 = 5 kg•m 2 .Furthermore, consider the parametric fluctuations with 20% of the nominal values and sudden load variation that m 2 increases to 2.5 kg at t = 5 s, which means the manipulator picks up an object weighting 1 kg at that moment.For better comparison, the sliding manifold and the adaptive controller of NFTSMC are rewritten as follows, ḃ0 =a 0 ∥s∥∥ ė∥ ϱ−I , ḃ1 =a 1 ∥s∥∥θ∥∥ ė∥ ϱ−I , ḃ2 =a 2 ∥s∥∥ θ∥ 2 ∥ ė∥ ϱ−I ,  Figs. 1 and 2 depict the sliding manifold curves of the manipulator.It can be found that the designed controller can make the state trajectory reach the sliding manifold s = 0 at the beginning and then converge to the manifold rapidly after slight oscillation.The tracking performance comparisons are illustrated in Figs.3-6.We can observe that the proposed controller exhibits faster response than the adaptive NFTSMC.Moreover, it can remarkably attenuate the chattering effect, while the NFTSMC only alleviates chattering to some extent.Fig. 7 describes the control torque of the manipulator, which also verifies the ability of the GNTSMC to attenuate chattering.Furthermore, Figs. 8 and 9 show the influences of the coefficient t n on the sliding manifold and the tracking performance, respectively.It can be seen that a larger t n results in smoother sliding dynamic, yet slower response rate, which is in accordance with the above analysis.

Conclusion
In this work, a GNTSMC scheme based on an adaptive STA has been developed for the tracking problem of robotic manipulators suffering uncertain disturbances.By integrating a novel global nonsingular terminal sliding manifold with an adaptive STA, the presented controller performs excellent tracking control without singularity.It can not only eliminate the reaching stage and realize global robustness, but also guarantee the finite time convergence of system states and trajectory errors.Further, the chattering phenomena can be significantly attenuated without deteriorating the tracking accuracy, and the prior information of perturbations is unnecessary in the design process.The Data sharing not applicable to this article as no datasets were generated or analysed during the current study.