A Finite-Time Sliding Mode Control Approach for Constrained Euler–Lagrange System

Abstract

This paper investigates a general control strategy to track the reference trajectory for the constrained Euler–Lagrange system with model uncertainties and unknown external disturbances. Unlike the disturbances assumed to be upper-bounded by a constant in other papers, we consider the disturbances to be bounded by a function of the system states, which are more realistic. First, the nominal controller was designed based on the nonsingular fast terminal sliding mode control, and global fast finite-time convergence to the sliding surface was guaranteed. As the system is state-constrained in this paper, we introduce the control barrier function approach to formulate the constraints and ensure the system does not break the restrictions. The proposed control strategy was numerically assessed on a two-link robot manipulator system, and the simulation results illustrate the effectiveness of the proposed control strategy.

1. Introduction

The Euler–Lagrange system, as a mechanical system that is widely used in robotics, such as unmanned ground vehicles [1,2], unmanned air vehicles [3,4], robot manipulators [5,6], satellites [7], and so on, has been of broad interest in recent years. The system model is derived from the kinetic equations and the Euler–Lagrange equation. Consequently, it is worthy of further research, especially in practical applications. Moreover, as a typical motion control system, controller design based on the Euler–Lagrange system model is more versatile and dumbs down the designing process. Nonetheless, the sophistication of the dynamics makes it challenging to identify the system parameters for a specific robot. Given the vital importance of the linear in parameters (LIP) property [8] of the Euler–Lagrange system, estimating the uncertainties is achievable via an appropriate observer [1,7]. In addition, the Euler–Lagrange system is highly nonlinear and constantly affected by various external unknown disturbances. Hence, for the trajectory tracking control of the Euler–Lagrange system, researchers have introduced various advanced control strategies to solve the above issues in the actual control scenario, such as model predictive control [9,10] and sliding mode control [11].

As more attention has been paid to swarm behavior in insects and birds, approaches involving individual behavior have certain limitations, especially in complex task scenarios. Through the cooperation between multiple agents, work efficiency and reliability can be improved. Even if a single or part of the agents fails, the system can remain functional through appropriate reactions. Most agents can be modeled as a Euler–Lagrange system, even for a heterogeneous multiagent system. Recently, researchers have increasingly focused on multiagent cooperative control based on the Euler–Lagrange system [12,13].

Unlike classical control theory, which is drawn from engineering practice and focuses on the system’s dynamic performance, the modern control approach is based on stability analysis theory, i.e., designing the controller to guarantee the system converges to the equilibrium point. In practice, the rapid convergence of the system has a very significant impact on the control performance, especially in environments that require high dynamics.

Sliding mode control (SMC) is one of the most effective control approaches as a result of its insensitivity to certain external disturbances and model uncertainties. The basic idea of SMC is to steer the trajectory of the system to the sliding surface and converge to the equilibrium point. However, as a linear sliding surface is adopted, it can only guarantee the asymptotic stability of the system. Compared with the conventional sliding mode control, terminal sliding mode control (TSMC) can drive the system to the equilibrium point in finite time via a nonlinear function in the sliding surface. In [14], a continuous TSMC was tested for mechanical servo systems, which achieved robustness and finite-time convergence. Although TSMC could significantly improve the dynamic performance of the system, the singularity problem remained. A new terminal sliding surface [15] was proposed to eliminate the singularity problem. For further improvement on the convergence rate of the system, considering the different behavior of linear and nonlinear functions in different intervals, researchers proposed many improvement methods for the sliding mode surface. A robust ATNTSM control approach was introduced in [16] for robotic manipulators with finite-time convergence guaranteed based on the arctan function. It was shown that the convergence rate is apparently faster than the traditional TSMC and NTSMC. An adaptive nonsingular fast TSMC was proposed in [17] to control both the position and attitude of a quadrotor subject to model uncertainties and external disturbances.

In addition, there always exist various constraints for the agent to achieve complex tasks working in a restricted and practical environment, such as adaptive cruise control [18] and collision avoidance [19]. Model predictive control [20] is widely used for systems with constraints. However, model predictive control is a model-based optimal control approach, which means the model accuracy will affect the control performance directly. Due to the high calculation burden, more computing resources are required to solve the nonlinear optimization problem, especially in high dynamics environments. Another control approach is based on the barrier Lyapunov function. Unlike quadratic functions, the barrier will force the system to back away as it approaches the state’s boundary. In [21], an adaptive fuzzy controller was proposed based on a tan-type barrier Lyapunov function to achieve trajectory tracking with output constraints. Similarly, a tan-type barrier Lyapunov-based optimized backstepping controller was proposed in [22]. While the desired signal does not break the constraints, it is hard to say whether the state constraints can be satisfied in practice. A mixed $H_{-}/H_{\infty }$ fault-detection filtering was proposed in [23] to improve the safety and reliability of industrial processes. An observer-based fault estimation and fault-tolerant control of the linear discrete time-varying stochastic system was presented in [24].

In recent years, a safety-critical control strategy was proposed in [25] based on the set invariance theory to drive the system states into a predefined safe set. This work unifies the control Lyapunov function (CLF) with the control barrier function (CBF) through an online quadratic programming (QP) process for adaptive cruise controller design. It guarantees that the car can always go forward with the desired velocity unless the front vehicle is too close to maintain a safe distance. It has been shown that the online QP can be performed with sample rates of 200 Hz to 1 kHz in real time, which means the CLF-CBF-QP framework is an effective way to stabilize the system with additional constraints, such as state and output constraints. Similar frameworks have also been extended to various attractive applications in robotics, such as collision-free behaviors in multirobot systems [26,27], dynamic locomotion over rough terrain for quadruped robots [28], and bipedal robotic walking [29].

In this paper, we tackle the tracking problem of the constrained Euler–Lagrange system with model uncertainties and unknown external disturbances. The main contributions of this paper are summed up as follows: (i) To improve the control performance of the conventional NTSMC, we propose a novel nonsingular fast terminal sliding mode controller (NFTSMC). Compared with the algorithm in [16], the NFTSMC shows a faster convergence rate and better robustness; (ii) the state constraints are formulated as a function-defined safe set. The tracking problem is solved using a CBF-QP, and the simulation results illustrate the effectiveness of tracking a reference signal that sometimes breaks the limits on states.

The rest of this paper is organized as follows. The Euler–Lagrange system model and some preliminaries are given in Section 2. The NFTSMC and CBF-QP control strategy is presented in Section 3, and simulation results illustrate the effectiveness of the proposed control strategy in Section 4. The paper is concluded in Section 5.

2. Dynamic Model and Preliminaries

2.1. Dynamic Model

Consider a typical Euler–Lagrange system

$$\mathit{M}\left(\ddot{\mathit{q}}\right)+\mathit{C}(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}+\mathit{G}\left(\mathit{q}\right)=\mathit{\tau }+{\mathit{\tau }}_d,$$

(1)

where $\mathit{q}$, $\dot{\mathit{q}}$, $\ddot{\mathit{q}}\in {\mathbb{R}}^n$ denote the generalized position, velocity, and acceleration of the system, respectively. $\mathit{M}\left(\mathit{q}\right)\in {\mathbb{R}}^{n\times n}$, which is symmetric positive definite denotes the inertia matrix and $\mathit{C}(\mathit{q},\dot{\mathit{q}})\in {\mathbb{R}}^{n\times n}$, is the centrifugal and Coriolis matrix. $\mathit{G}\left(\mathit{q}\right)\in {\mathbb{R}}^n$ is a generalized force caused by the potential energy. $\mathit{\tau }$, ${\mathit{\tau }}_d\in {\mathbb{R}}^n$ are generalized control input and external disturbances applied to the system, respectively. For the convenience, the matrices are abbreviated as $\mathit{M}$, $\mathit{C}$, $\mathit{G}$. In fact, we cannot obtain the system parameters precisely, i.e., there exists uncertain terms $\delta \mathit{M}$, $\delta \mathit{C}$, and $\delta \mathit{G}$, which can be expressed as

$$\begin{array}{cc}\hfill \mathit{M}& ={\mathit{M}}_0+\delta \mathit{M},\hfill \\ \hfill \mathit{C}& ={\mathit{C}}_0+\delta \mathit{C},\hfill \\ \hfill \mathit{G}& ={\mathit{G}}_0+\delta \mathit{G},\hfill \end{array}$$

where ${\mathit{M}}_0$, ${\mathit{C}}_0$, and ${\mathit{G}}_0$ are estimated system parameters. Then, system (1) can be reformed as

$${\mathit{M}}_0\ddot{\mathit{q}}+{\mathit{C}}_0\dot{\mathit{q}}+{\mathit{G}}_0=\mathit{\tau }+\mathit{f},$$

(2)

where $\mathit{f}={\mathit{\tau }}_d-\delta \mathit{M}\ddot{\mathit{q}}-\delta \mathit{C}\dot{\mathit{q}}-\delta \mathit{G}$ is the total disturbances including model uncertainties and external disturbances.

Let ${\mathit{x}}_1=\mathit{q},{\mathit{x}}_2=\dot{\mathit{q}}$, then system (2) can be rewritten as

$$\left\{\begin{array}{cc}\hfill & {\dot{\mathit{x}}}_1={\mathit{x}}_2,\hfill \\ \hfill & {\dot{\mathit{x}}}_2=\mathit{F}({\mathit{x}}_1,{\mathit{x}}_2)+{\mathit{M}}_0^{-1}\left(\mathit{x}\right)\mathit{\tau }+\mathit{d}(t,\mathit{x}),\hfill \end{array}\right.$$

(3)

where

$$\begin{array}{cc}\hfill \mathit{F}({\mathit{x}}_1,{\mathit{x}}_2)& =-{\mathit{M}}_0^{-1}({\mathit{C}}_0\dot{\mathit{q}}+{\mathit{G}}_0),\hfill \\ \hfill \mathit{d}(t,\mathit{x})& ={\mathit{M}}_0^{-1}\mathit{f},\hfill \end{array}$$

The external disturbances such as friction and wind drag are bounded [15] and can be expressed as a function of the states. Then, we have the following assumption:

Assumption 1.

The disturbance $\mathit{d}(t,\mathit{x})$ is bounded by

$$\parallel \mathit{d}(t,\mathit{x})\parallel \le \parallel {\mathit{M}}_0^{-1}\parallel \left(b_0+b_1\parallel \mathit{q}\parallel +b_2\parallel \dot{\mathit{q}}\parallel \right)$$

(4)

where $b_0$, $b_1$, and $b_2$ are positive constants. Let $\rho =\parallel {\mathit{M}}_0^{-1}\parallel \left(b_0+b_1\parallel \mathit{q}\parallel +b_2\parallel \dot{\mathit{q}}\parallel \right)$. It is trivial that $\rho >0$.

The following lemma is given to analyze the stability of the system:

Lemma 1.

The states of the system will converge to the origin within finite time if there exists a continuous positive function $V\left(\mathit{x}\right):{\mathbb{R}}^n\to \mathbb{R}$ for any given initial condition $x\left(0\right)=x_0$, such that [30]

$$\dot{V}\left(\mathit{x}\right)\le -{\alpha }_{1}V\left(\mathit{x}\right)-{\alpha }_2{V}^{\alpha }\left(\mathit{x}\right),$$

(5)

where ${\alpha }_1,{\alpha }_2>0$, $0<\alpha <1$.

2.2. Conventional Nonsingular TSMC

Consider the following second-order nonlinear system with uncertainties:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2,\hfill \\ {\dot{x}}_2=f\left(x\right)+g\left(x\right)u+d(x,t),\hfill \end{array}\right.$$

(6)

where $d(x,t)$ denotes the uncertainties and external disturbances, which is bounded by $\parallel d(x,t)\parallel \le D$, $g\left(x\right)\ne 0$, and D is a positive constant. Let $\mathit{x}={[x_1,x_2]}^{\mathrm{T}}$ be the state vector with $x_1\in \mathbb{R}$ and $x_2\in \mathbb{R}$. The nonsingular terminal sliding surface is chosen as

$$s=x_1+\frac{1}{\beta }{x}_2^{\frac{p}{q}},$$

(7)

where $\beta >0$ and p, q are positive odd integers satisfying $p>q$. For system (6), the nonsingular terminal sliding mode controller is designed as

$$u=-g^{-1}\left(x\right)\left(f\left(x\right)+\beta \frac{q}{p}{x}_2^{2-\frac{p}{q}}+(D+\eta )sign\left(s\right)\right),$$

(8)

where $1<p/q<2$, $\eta >0$. For convenience, let $\lambda =1/\beta$, $\gamma =q/p$. Then, when the sliding mode motion is achieved, the states will converge to the equilibrium point within

$$T_n={\int }_0^{|x_1\left(0\right)|}\frac{{\lambda }^{1/\gamma }}{x_1^{1/\gamma }}d{x}_1=\frac{{\lambda }^{1/\gamma }·{\left|x_1\left(0\right)\right|}^{1-1/\gamma }}{1-1/\gamma }.$$

(9)

2.3. Control Barrier Function

The basic idea of the control barrier function method is to design a barrier function that satisfies some specific conditions and keeps the safe set forward invariant.

Definition 1.

The set $\mathcal{C}$ is the aforementioned forward invariant for system (10) if, from any initial state $x\left(t_0\right)\in \mathcal{C}$, the solutions $x\left(t\right)$ remain in that set for all $t\ge t_0$.

Consider a general control affine system,

$$\dot{x}=f\left(x\right)+g\left(x\right)u$$

(10)

with f and g being locally Lipschitz, and $x\in \mathcal{X}\subseteq {\mathbb{R}}^n$ being the state and $u\in \mathcal{U}\subseteq {\mathbb{R}}^m$ the control input. A safe set $\mathcal{C}$ is defined as

$$\begin{array}{c}\hfill \mathcal{C}=\{x\in {\mathbb{R}}^n:h\left(x\right)\ge 0\},\end{array}$$

(11)

where $h\left(x\right):{\mathbb{R}}^n\to \mathbb{R}$ is a continuously differentiable function and will be defined later. The safe set $\mathcal{C}$ is forward invariant if and only if there exists a control input u, such that

$$\dot{h}(x,u)\ge -\alpha \left(h\left(x\right)\right),$$

(12)

where $\alpha \left(·\right)$ is an extended class $\mathcal{K}$ function that is strictly increasing and $\alpha \left(0\right)=0$. There are many other equivalent conditions; for more details see [31] and the references therein. Then, we can have the following definition of control barrier function:

Definition 2.

The function $h\left(x\right)$ is said to be a zero control barrier function if there exists an extended class $\mathcal{K}$ function α such that [32]

$$\underset{u\in {\mathbb{R}}^m}{sup}(L_{f}h\left(x\right)+L_{g}h\left(x\right)u+\alpha \left(h\left(x\right)\right))\ge 0,$$

(13)

where $L_{f}h\left(x\right)$ and $L_{g}h\left(x\right)$ denote the Lie derivatives of $h\left(x\right)$ as

$$L_{f}h\left(x\right)=\frac{\partial h\left(x\right)}{\partial x}f\left(x\right),L_{g}h\left(x\right)=\frac{\partial h\left(x\right)}{\partial x}g\left(x\right).$$

One of the benefits of CBF is that we can design the controller separately. Given the nominal input $u_{\mathrm{norm}}$, which is intended to stabilize or tracks a reference signal for system (10), a CBF-QP problem can be expressed as

$$\begin{array}{cc}\hfill & u^{*}\left(x\right)=\underset{u\in {\mathbb{R}}^m}{\arg \min }\frac{1}{2}{\parallel u-u_{\mathrm{norm}}\parallel }^2\hfill \\ \hfill & \mathrm{s}.\mathrm{t}.L_{f}h\left(x\right)+L_{g}h\left(x\right)u+\alpha \left(h\left(x\right)\right)\ge 0.\hfill \end{array}$$

(14)

The applied control input u is equivalent to the nominal one if the safe condition is satisfied. Once the system become potentially unsafe, the CBF-QP guarantees that the system will not break the safe set and that the applied control input is as close as possible to the nominal control.

3. Controller Design

In this section, an NFTSMC controller design method is given such that the Euler–Lagrange system can track a given reference trajectory within finite time with a strong antidisturbance capacity. To overcome the singularity problem in conventional TSMC design and further improve the convergence rate, inspired by [33], for the cases of relative degree higher than 1, an exponential CBF (ECBF) is used to ensure the safe set $\mathcal{C}$ forward invariant.

3.1. NFTSM Controller Design

To achieve fast convergence and disturbance rejection, the nonsingular terminal sliding surface was designed as

$$\mathit{s}={\mathit{x}}_1+{\mathsf{\Lambda }}_1{sign}^{{\mathsf{\Gamma }}_1}\left({\mathit{x}}_1\right)+{\mathsf{\Lambda }}_2{sign}^{{\mathsf{\Gamma }}_2}\left({\mathit{x}}_2\right),$$

(15)

where

$$\begin{array}{cc}\hfill & {\mathsf{\Gamma }}_1=diag({\gamma }_{11},{\gamma }_{12},\dots ,{\gamma }_{1n}),{\mathsf{\Gamma }}_2=diag({\gamma }_{21},{\gamma }_{22},\dots ,{\gamma }_{2n})\hfill \\ \hfill & {\mathsf{\Lambda }}_1=diag({\mathsf{\lambda }}_{11},{\mathsf{\lambda }}_{12},\dots ,{\mathsf{\lambda }}_{1n}),{\mathsf{\Lambda }}_2=diag({\mathsf{\lambda }}_{21},{\mathsf{\lambda }}_{22},\dots ,{\mathsf{\lambda }}_{2n})\hfill \end{array}$$

with ${\gamma }_{1i}>{\gamma }_{2i},1<{\gamma }_{2i}<2$ and ${\mathsf{\lambda }}_{1i},{\mathsf{\lambda }}_{2i}>0$ for $i=1,2,\dots ,n$. Note that ${sign}^{\mathsf{\Gamma }}\left(\mathit{x}\right)$ denotes

$$\begin{array}{cc}\hfill {sign}^{\mathsf{\Gamma }}\left(\mathit{x}\right)& ={diag(sign\left(\mathit{x}\right))·\left|\mathit{x}\right|}^{\mathsf{\Gamma }}\hfill \\ \hfill & =\left[\right|x_1{|}^{{\gamma }_1}sign\left(x_1\right)\dots |x_n{{|}^{{\gamma }_n}sign\left(x_n\right)]}^{\mathrm{T}},\hfill \end{array}$$

where $sign\left(\mathit{x}\right)={[sign\left(x_1\right)\dots sign\left(x_n\right)]}^{\mathrm{T}}$, ${\left|\mathit{x}\right|}^{\mathsf{\Gamma }}=\left[\right|x_1{|}^{{\gamma }_1}\dots |x_n{{|}^{{\gamma }_n}]}^{\mathrm{T}}$ and it is trivial that

$$\frac{\mathrm{d}}{\mathrm{d}t}{sign}^{\mathsf{\Gamma }}\left(\mathit{x}\right)={\mathsf{\Gamma }diag\left(\right|\mathit{x}|}^{\mathsf{\Gamma }-{\mathit{I}}_n})·\dot{\mathit{x}},$$

where ${\mathit{I}}_n$ is an n-dimensional identity matrix. Then, we have the time derivative of (15) as

$$\begin{array}{cc}\hfill \dot{\mathit{s}}=& {\mathit{x}}_2+{\mathsf{\Lambda }}_1{\mathsf{\Gamma }}_{1}diag\left(\right|{\mathit{x}}_1{|}^{{\mathsf{\Gamma }}_1-{\mathit{I}}_n}){\mathit{x}}_2+{\mathsf{\Lambda }}_2{\mathsf{\Gamma }}_{2}diag\left(\right|{\mathit{x}}_2{|}^{{\mathsf{\Gamma }}_2-{\mathit{I}}_n}){\dot{\mathit{x}}}_2.\hfill \end{array}$$

(16)

Theorem 1.

Consider the rewritten Euler–Lagrange system (3) with uncertainties and disturbances satisfying Assumption 1, the system state will converge to the origin in finite time via the following control law:

$$\begin{array}{cc}\hfill \mathit{\tau }=& -{\mathit{M}}_0(M_2\mathit{s}+(\rho +M_1)\frac{\mathit{s}}{\parallel \mathit{s}\parallel }-{\mathit{M}}_0^{-1}({\mathit{C}}_0{\mathit{x}}_2+{\mathit{G}}_0)+\hfill \\ \hfill & {\mathsf{\Lambda }}_2^{-1}{\mathsf{\Gamma }}_2^{-1}({\mathit{I}}_n+{\mathsf{\Lambda }}_1{\mathsf{\Gamma }}_{1}diag\left(\right|{\mathit{x}}_1{|}^{{\mathsf{\Gamma }}_1-{\mathit{I}}_n}\left)\right){sign}^{2{\mathit{I}}_n-{\mathsf{\Gamma }}_2}\left({\mathit{x}}_2\right)),\hfill \end{array}$$

(17)

where $M_1$ and $M_2$ are positive constants.

Proof. 

Consider the following Lyapunov function

$$V=\frac{1}{2}{\mathit{s}}^{\mathrm{T}}\mathit{s}.$$

(18)

Then, the derivative of V becomes

$$\begin{array}{cc}\hfill \dot{V}=& {\mathit{s}}^{\mathrm{T}}\mathit{s}\hfill \\ \hfill =& {\mathit{s}}^{\mathrm{T}}({\mathit{x}}_2+{\mathsf{\Lambda }}_1{\mathsf{\Gamma }}_{1}diag\left(\right|{\mathit{x}}_1{|}^{{\mathsf{\Gamma }}_1-{\mathit{I}}_n}){\mathit{x}}_2+{\mathsf{\Lambda }}_2{\mathsf{\Gamma }}_{2}diag\left(\right|{\mathit{x}}_2{|}^{{\mathsf{\Gamma }}_2-{\mathit{I}}_n}\left)\right(\mathit{F}({\mathit{x}}_1,{\mathit{x}}_2)+\hfill \\ \hfill & \mathit{g}\left(\mathit{x}\right)\mathit{\tau }+\mathit{d}(t,\mathit{x})\left)\right).\hfill \end{array}$$

(19)

Substituting (17) into (19), it follows that

$$\begin{array}{cc}\hfill \dot{V}=& {\mathit{s}}^{\mathrm{T}}({\dot{\mathit{x}}}_1+{\mathsf{\Lambda }}_1{\mathsf{\Gamma }}_{1}diag\left(\right|{\mathit{x}}_1{|}^{{\mathsf{\Gamma }}_1-{\mathit{I}}_n}){\dot{\mathit{x}}}_1+{\mathsf{\Lambda }}_2{\mathsf{\Gamma }}_{2}diag\left(\right|{\mathit{x}}_2{|}^{{\mathsf{\Gamma }}_2-{\mathit{I}}_n})(\mathit{d}-M_2\mathit{s}-\hfill \\ \hfill & (\rho +M_1)\frac{\mathit{s}}{\parallel \mathit{s}\parallel }-{\mathsf{\Lambda }}_2^{-1}{\mathsf{\Gamma }}_2^{-1}\left({\mathit{I}}_n+{\mathsf{\Lambda }}_1{\mathsf{\Gamma }}_{1}diag\left(\right|{\mathit{x}}_1{|}^{{\mathsf{\Gamma }}_1-{\mathit{I}}_n})\right){sign}^{2{\mathit{I}}_n-{\mathsf{\Gamma }}_2}\left({\mathit{x}}_2\right)\left)\right)\hfill \\ \hfill =& {\mathit{s}}^{\mathrm{T}}({\dot{\mathit{x}}}_1+{\mathsf{\Lambda }}_1{\mathsf{\Gamma }}_{1}diag\left(\right|{\mathit{x}}_1{|}^{{\mathsf{\Gamma }}_1-{\mathit{I}}_n}){\dot{\mathit{x}}}_1+{\mathsf{\Lambda }}_2{\mathsf{\Gamma }}_{2}diag\left(\right|{\mathit{x}}_2{|}^{{\mathsf{\Gamma }}_2-{\mathit{I}}_n})(\mathit{d}-M_2\mathit{s}-\hfill \\ \hfill & (\rho +M_1)\frac{\mathit{s}}{\parallel \mathit{s}\parallel })-\left({\mathit{I}}_n+{\mathsf{\Lambda }}_1{\mathsf{\Gamma }}_{1}diag\left(\right|{\mathit{x}}_1{|}^{{\mathsf{\Gamma }}_1-{\mathit{I}}_n})\right){\dot{\mathit{x}}}_1)\hfill \\ \hfill =& -{\mathit{s}}^{\mathrm{T}}{\mathsf{\Lambda }}_2{\mathsf{\Gamma }}_{2}diag\left(\right|{\mathit{x}}_2{|}^{{\mathsf{\Gamma }}_2-{\mathit{I}}_n})\left((\rho +M_1)\frac{\mathit{s}}{\parallel \mathit{s}\parallel }+M_2\mathit{s}-\mathit{d}\right).\hfill \end{array}$$

Notice that

$$\begin{array}{cc}\hfill -({\mathit{s}}^{\mathrm{T}}\rho \frac{\mathit{s}}{\parallel \mathit{s}\parallel }-{\mathit{s}}^{\mathrm{T}}\mathit{d})& \le -\parallel \mathit{s}\parallel \rho +\parallel \mathit{s}\parallel ·\parallel d\parallel \hfill \\ \hfill & \le -\parallel \mathit{s}\parallel \rho +\parallel \mathit{s}\parallel ·\rho =0.\hfill \end{array}$$

Therefore, it is obviously that

$$\begin{array}{cc}\hfill \dot{V}& \le -{\mathit{s}}^{\mathrm{T}}{\mathsf{\Lambda }}_2{\mathsf{\Gamma }}_{2}diag\left(\right|{\mathit{x}}_2{|}^{{\mathsf{\Gamma }}_2-{\mathit{I}}_n})\left(\frac{M_1}{\parallel \mathit{s}\parallel }+M_2\right)\mathit{s}\hfill \\ \hfill & \le -min\left({\lambda }_{2i}{\gamma }_{2i}{\left|x_{2i}\right|}^{{\gamma }_{2i}-1}\right)\left(M_1\parallel \mathit{s}\parallel +M_2{\mathit{s}}^{\mathrm{T}}\mathit{s}\right).\hfill \end{array}$$

Let

$$\begin{array}{cc}\hfill & {\rho }_1\left({\mathit{x}}_2\right)=min\left({\lambda }_{2i}{\gamma }_{2i}{\left|x_{2i}\right|}^{{\gamma }_{2i}-1}\right)·2M_2,\hfill \\ \hfill & {\rho }_2\left({\mathit{x}}_2\right)=min\left({\lambda }_{2i}{\gamma }_{2i}{\left|x_{2i}\right|}^{{\gamma }_{2i}-1}\right)·\sqrt{2}{M}_1.\hfill \end{array}$$

Then, it follows that

$$\begin{array}{c}\hfill \dot{V}\le -{\rho }_{1}V-{\rho }_2\sqrt{V}.,\end{array}$$

(20)

When $x_{2i}\ne 0$, it is trivial that ${\rho }_1>0$ and ${\rho }_2>0$. By Lemma 1, it can be confirmed that the states of the system will converge to the sliding surface $\mathit{s}=\mathbf{0}$ within finite time. Then, we consider the case that $x_{2i}=0$. Substituting (17) into (3), we have

$$\begin{array}{cc}\hfill {\dot{\mathit{x}}}_2=& -({\mathit{C}}_0\dot{\mathit{q}}+{\mathit{G}}_0)-{\mathit{M}}_0(M_2\mathit{s}+(\rho +M_1)\frac{\mathit{s}}{\parallel \mathit{s}\parallel }-{\mathit{M}}_0^{-1}({\mathit{C}}_0{\mathit{x}}_2+{\mathit{G}}_0)+\hfill \\ \hfill & {\mathsf{\Lambda }}_2^{-1}{\mathsf{\Gamma }}_2^{-1}({\mathit{I}}_n+{\mathsf{\Lambda }}_1{\mathsf{\Gamma }}_{1}diag\left(\right|{\mathit{x}}_1{|}^{{\mathsf{\Gamma }}_1-{\mathit{I}}_n}\left)\right){sign}^{2{\mathit{I}}_n-{\mathsf{\Gamma }}_2}\left({\mathit{x}}_2\right))+d(t,\mathit{x})\hfill \\ \hfill =& -M_2\mathit{s}-M_1\frac{\mathit{s}}{\parallel \mathit{s}\parallel }+\rho \frac{\mathit{s}}{\parallel \mathit{s}\parallel }+d(t,\mathit{x})\hfill \\ \hfill \le & -M_2\mathit{s}-M_1\frac{\mathit{s}}{\parallel \mathit{s}\parallel }\hfill \end{array}$$

which suggests that ${\dot{x}}_{2i}<-M_1$ and ${\dot{x}}_{2i}>M_1$ for the cases $s_i>0$ and $s_i<0$, respectively. This indicates that the states of the system will not remain at the points $x_{2i}=0$. It is shown in Figure 1 that the controller will drive the states to the sliding surface $\mathit{s}=0$ when ${\mathit{x}}_2=0$. □

Remark 1.

Both NFTSM and conventional TSM share a nonlinear term of tracking error to achieve fast convergence when the states are far from the origin. However, the convergence rate changes slowly when the system approaches the sliding surface. Unlike TSM, a linear term of ${\mathit{x}}_1$ is introduced into the sliding surface to ensure a fast rate while approaching the nearby origin. See [15,16] for more details on how the states converge to zero when the sliding surface is reached.

Proposition 1.

If the following conditions are satisfied:

$${\lambda }_i={\left(\frac{{\lambda }_{2i}}{{\lambda }_{1i}}\right)}^{\frac{{\gamma }_i}{{\gamma }_{2i}}},{\gamma }_i=\frac{{\gamma }_{2i}}{{\gamma }_{1i}},$$

then the states converge to the equilibrium point controlled using the NFTSMC faster than with the conventional nonsingular TSMC for any nonzero initial state ${\mathit{x}}_1\left(0\right)$. ${\lambda }_i$ and ${\gamma }_i$ are parameters of the conventional nonsingular TSMC given by (6).

Proof. 

It is trivial that

$${(x_{1i}+{\lambda }_{1i}{x}_{1i}^{{\gamma }_{1i}})}^{\frac{1}{{\gamma }_{2i}}}>{\lambda }_{1i}^{\frac{1}{{\gamma }_{2i}}}{x}_{1i}^{\frac{{\gamma }_{1i}}{{\gamma }_{2i}}},$$

(21)

which follows that

$$\begin{array}{cc}\hfill T_{nf}=& {\int }_0^{|x_{1i}\left(0\right)|}\frac{{\lambda }_{2i}^{\frac{1}{{\gamma }_{2i}}}}{{(x_{1i}+{\lambda }_{1i}{x}_{1i}^{{\gamma }_{1i}})}^{\frac{1}{{\gamma }_{2i}}}}d{x}_{1i}\hfill \\ \hfill <& {\int }_0^{|x_{1i}\left(0\right)|}\frac{{({\lambda }_{2i}/{\lambda }_{1i})}^{1/{\gamma }_{2i}}}{x_{1i}^{{\gamma }_{1i}/{\gamma }_{2i}}}d{x}_{1i}=T_n,\hfill \end{array}$$

(22)

where $T_{nf}$ is the convergence time of the proposed controller. Therefore, the NFTSMC converges to the equilibrium faster than the conventional nonsingular TSMC. This completes the proof. □

Remark 2.

To reduce the chattering caused by switching, we use a continuous function in control law (17) to replace $\mathit{s}/\parallel \mathit{s}\parallel$,

$$\mathsf{\Theta }=\frac{\mathit{s}}{\parallel \mathit{s}\parallel +\delta },$$

(23)

where δ is a positive small constant. As δ is selected to be small enough, $\mathsf{\Theta }\approx \mathit{s}/\parallel \mathit{s}\parallel$.

3.2. High-Order CBF Design

Without loss of generality, consider the control affine system (10). Assume that the barrier function $h\left(x\right)$ is a continuously differentiable function and satisfies

$$h^{\left(r\right)}+k_{r-1}{h}^{(r-1)}+\cdots +k_1\dot{h}+k_{0}h\ge 0,$$

(24)

where $k_0,\dots ,k_{r-1}$ are real constants such that the roots of the following polynomial:

$$p\left(\lambda \right)={\lambda }^r+k_{r-1}{\lambda }^{r-1}+\cdots +k_1\lambda +k_0$$

(25)

are all negative real, i.e., $-{\lambda }_1,\dots ,-{\lambda }_r$ (${\lambda }_i>0,1\le i\le r$). We define

$$s_0(t,x)=h\left(x\right),s_k=(\frac{\mathrm{d}}{\mathrm{d}t}+{\lambda }_k)s_{k-1},1\le k\le r.$$

(26)

It is obvious from (26) that (24) is equivalent to $s_r(t,x)\ge 0$. Denote $s_k(0,x\left(0\right))$ as $s_k\left(0\right)$, then we have the following lemma:

Lemma 2.

Given a continuously differentiable function $h\left(x\right)$ and a set of real constant $k_0,\dots ,k_{r-1}$ such that (24) is satisfied and the roots of polynomial (25) are all negative real [34]. If $s_i\left(0\right)\ge 0,i=0,1,\dots ,r-1$, then $h\left(x\right)\ge 0$ for any $t\ge 0$.

Proof. 

As $s_r(t,x)\ge 0$ and (26), we have

$$\frac{d}{dt}\left(e^{{\lambda }_{r}t}{s}_{r-1}(t,x\left(t\right))\right)\ge 0.$$

Integrate both sides on $[0,t]$ and we have

$$s_{r-1}(t,x\left(t\right))\ge s_{r-1}\left(0\right)e^{-{\lambda }_{r}t}.$$

Similarly, multiply both sides by $e^{{\lambda }_{r-1}t}$ and from (26), we have

$$\frac{d}{dt}\left(e^{{\lambda }_{r-1}t}{s}_{r-2}(t,x\left(t\right))\right)\ge s_{r-1}\left(0\right)e^{({\lambda }_{r-1}-{\lambda }_r)t}.$$

After integrating, we have

$$\begin{array}{cc}\hfill s_{r-2}(t,x\left(t\right))\ge & s_{r-1}\left(0\right)e^{-{\lambda }_{r-1}t}{\int }_0^t{e}^{({\lambda }_{r-1}-{\lambda }_r){\tau }_1}d{\tau }_1+\hfill \\ \hfill & s_{r-2}\left(0\right)e^{-{\lambda }_{r-1}t}.\hfill \end{array}$$

Repeat the above iterative process and it becomes

$$\begin{array}{cc}\hfill s_0(t,x\left(t\right))\ge & s_0\left(0\right)e^{-{\lambda }_{1}t}+\sum _{k=1}^{r-1}\left[s_k\left(0\right)e^{-{\lambda }_{1}t}{\int }_0^t{e}^{\left({\lambda }_1-{\lambda }_2\right){\tau }_k}\right.\hfill \\ \hfill & \left.{\int }_0^{{\tau }_k}{e}^{\left({\lambda }_2-{\lambda }_3\right){\tau }_{k-1}}\dots {\int }_0^{{\tau }_2}{e}^{\left({\lambda }_k-{\lambda }_{k+1}\right){\tau }_1}d{\tau }_1\dots d{\tau }_{k-1}d{\tau }_k\right].\hfill \end{array}$$

It is easy to check that the integral item of the above inequality is positive and, as $t\to \infty$, it converges to zero. Since $s_i\left(0\right)\ge 0$, it is trivial that $s_0(t,x\left(t\right))\ge 0$. Therefore, $h\left(x\right)\ge 0$ for any $t\ge 0$. □

Then, we can have the following definition:

Definition 3.

Given control affine system (10), the smooth function $h\left(x\right):\mathcal{X}\to \mathbb{R}$ with relative degree r is defined as an exponential control barrier function if there exists $\mathcal{K}\in {\mathbb{R}}^r$, such that $\forall x\in \mathcal{C}$

$$\underset{u\in \mathcal{U}}{sup}\left(L_f^{r}h\left(x\right)+L_g{L}_f^{r-1}h\left(x\right)u+{\mathcal{K}}^{\mathrm{T}}\mathcal{H}\right)\ge 0,$$

where $\mathcal{K}={[k_0{k}_1\dots k_{r-1}]}^{\mathrm{T}}$ can be decided by the pole placement approach and

$$\mathcal{H}=\left[\begin{array}{c}h\left(x\right)\\ L_{f}h\left(x\right)\\ ⋮\\ L_f^{r-1}h\left(x\right)\end{array}\right].$$

(27)

For all $x\in \mathcal{X}$, define the set

$$K_{\mathrm{ecbf}}\left(x\right)=\{u\in \mathcal{U}|L_f^{r}h\left(x\right)+L_g{L}_f^{r-1}h\left(x\right)u+{\mathcal{K}}^{\mathrm{T}}\mathcal{H}\ge 0\}.$$

According to Lemma 2, we can have the following proposition:

Proposition 2.

Given control affine system (10), $h\left(x\right)$ is an ECBF with relative degree r. Then, any controller $u\in K_{\mathit{ecbf}}$ will render the safe set $\mathcal{C}$ defined by the (11) forward invariant.

Proposition 2 can be easily proved by Lemma 2. As defined above, the set $K_{\mathrm{ecbf}}$ provides a family of applicable control inputs that render the system inside the safe set. Therefore, we only need to find a proper control input inside $K_{\mathrm{ecbf}}$ to guarantee that the tracking performance is as accurate as possible.

We consider the state constraints as safety conditions to ensure that the Euler–Lagrange system can track the reference trajectory within state limits. Inspired by the work in [27], we use a super-ellipsoid safety region to ensure the safety of the system. We only consider the position constraints applied to the systems, and the safe set is defined as

$$\begin{array}{cc}\hfill & \mathcal{C}=\{\mathit{x}\in {\mathbb{R}}^{2n}:h\left(\mathit{x}\right)\ge 0\},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & h\left(\mathit{x}\right)=1-{\left(\frac{x_{11}-c_1}{p_1}\right)}^4-\cdots -{\left(\frac{x_{1n}-c_n}{p_n}\right)}^4,\hfill \end{array}$$

(29)

where $x_{ij}$ is jth component of ${\mathit{x}}_i$, $c_i$ is the center of the ellipse, and $p_i$ is the bound on the state. By taking the derivatives of $h\left(\mathit{x}\right)$ along $\mathit{x}$, we have

$$\begin{array}{cc}\hfill \frac{\partial h\left(\mathit{x}\right)}{\partial \mathit{x}}& =-4·\left[{\left(\frac{x_{11}-c_1}{p_1}\right)}^3\cdots {\left(\frac{x_{1n}-c_n}{p_n}\right)}^3{\mathbf{0}}_{1\times n}\right],\hfill \\ \hfill L_{f}h\left(\mathit{x}\right)& =\frac{\partial h\left(\mathit{x}\right)}{\partial \mathit{x}}{\left[{\mathit{x}}_2^{\mathrm{T}}\mathit{F}{({\mathit{x}}_1,{\mathit{x}}_2)}^{\mathrm{T}}\right]}^{\mathrm{T}}\hfill \\ & =-4{\left(\frac{x_{11}-c_1}{p_1}\right)}^3{x}_{21}-4{\left(\frac{x_{12}-c_1}{p_1}\right)}^3{x}_{22}-\cdots -4{\left(\frac{x_{1n}-c_1}{p_1}\right)}^3{x}_{2n},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill L_{g}h\left(\mathit{x}\right)& =\frac{\partial h\left(\mathit{x}\right)}{\partial \mathit{x}}{\left[{\mathbf{0}}_{n\times n}{\left({\mathit{M}}_0^{-1}\right)}^{\mathrm{T}}\right]}^{\mathrm{T}}={\mathbf{0}}_{1\times n}.\hfill \end{array}$$

(31)

Apparently, function $h\left(x\right)$ has relative degree two, which means the control input cannot be expressed explicitly, and the CBF-QP (14) of relative degree one is no longer applicable. Continue to take the derivative of $h\left(x\right)$ and we have

$$\begin{array}{cc}\hfill \nabla L_{f}h\left(\mathit{x}\right)& =\frac{\partial L_{f}h\left(\mathit{x}\right)}{\partial \mathit{x}},\hfill \\ \hfill L_f^{2}h\left(\mathit{x}\right)& =\nabla L_{f}h\left(\mathit{x}\right)·{\left[{\mathit{x}}_2^{\mathrm{T}}\mathit{F}{({\mathit{x}}_1,{\mathit{x}}_2)}^{\mathrm{T}}\right]}^{\mathrm{T}},\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill L_g{L}_{f}h\left(\mathit{x}\right)& =\nabla L_{f}h\left(\mathit{x}\right)·{\left[{\mathit{0}}_{n\times n}{\left({\mathit{M}}_0^{-1}\right)}^{\mathrm{T}}\right]}^{\mathrm{T}}.\hfill \end{array}$$

(33)

3.3. ECBF-Based QP Design

We can formulate the ECBF condition into a quadratic program to refine the nominal control law (17) in real time. To minimize the control deviation from the nominal one, we take the control input error as

$$\begin{array}{cc}\hfill \parallel {\mathit{e}}_u{\parallel }^2& ={(\mathit{u}-\mathit{\tau })}^{\mathrm{T}}(\mathit{u}-\mathit{\tau })\hfill \\ \hfill & ={\mathit{u}}^{\mathrm{T}}\mathit{u}-2{\mathit{\tau }}^{\mathrm{T}}\mathit{u}+{\mathit{\tau }}^{\mathrm{T}}\mathit{\tau }.\hfill \end{array}$$

Then, we can reformulate the CBF-QP problem (14) as

$$\begin{array}{cc}\hfill {\mathit{u}}^{*}\left(\mathit{x}\right)=& \underset{\mathit{u}\in {\mathbb{R}}^m}{\arg \min }{\mathit{u}}^{\mathrm{T}}\mathit{u}-2{\mathit{\tau }}^{\mathrm{T}}\mathit{u}\hfill \\ \hfill & \mathrm{s}.\mathrm{t}.L_f^{2}h\left(x\right)+L_g{L}_{f}h\left(x\right)u+{\mathcal{K}}^{\mathrm{T}}\mathcal{H}\ge 0,\hfill \end{array}$$

(34)

where $\mathcal{K}={\left[k_0{k}_1\right]}^{\mathrm{T}}$, $\mathcal{H}={\left[h\left(x\right)L_{f}h\left(x\right)\right]}^{\mathrm{T}}$ and $L_{f}h\left(x\right)$, $L_f^{2}h\left(x\right)$, $L_g{L}_{f}h\left(x\right)$ are defined by (30)–(33), respectively.

Remark 3.

Safety is always required to be satisfied even though the states of the system break the limit. When the constraints hold, the tracking error cannot converge to the origin, which means the system is unstable from this perspective. Therefore, this is a compromise between safety and performance.

4. Numerical Simulation

In this section, numerical simulations exemplify the validity and performance of the proposed control strategy. Consider a two-link robotic manipulator system [15], as shown in Figure 2, whose dynamics equation is described as an Euler–Lagrange system:

$$\mathit{M}\left(\ddot{\mathit{q}}\right)+\mathit{C}(\mathit{q},\dot{\mathit{q}})\dot{\mathit{q}}+\mathit{G}\left(\mathit{q}\right)=\mathit{\tau }+{\mathit{\tau }}_d.$$

In this case, $q_i$ denotes the angular position of the ith arm, $r_i$ the length of the arm, $m_i$ the mass, and $J_i$ the moment of inertial. Denote $\mathit{q}={\left[q_1{q}_2\right]}^{\mathrm{T}}$ as the state of the system and the relative matrices are defined as

$$\mathit{M}\left(\ddot{\mathit{q}}\right)=\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{21}& M_{22}\end{array}\right],$$

$$\mathit{C}(\mathit{q},\dot{\mathit{q}})=\left[\begin{array}{c}-{\beta }_{12}\left(q_2\right){\dot{q}}_1^2-2{\beta }_{12}\left(q_2\right){\dot{q}}_1{\dot{q}}_2\\ {\beta }_{12}\left(q_2\right){\dot{q}}_2^2\end{array}\right],$$

$$\mathit{G}\left(\mathit{q}\right)=\left[\begin{array}{c}\left(m_1+m_2\right)g{r}_{1}cos{q}_2+m_{2}g{r}_{2}cos\left(q_1+q_2\right)\\ m_{2}g{r}_{2}cos\left(q_1+q_2\right)\end{array}\right],$$

where

$$\begin{array}{cc}\hfill M_{11}& =\left(m_1+m_2\right)r_1^2+m_2{r}_2^2+2m_2{r}_1{r}_{2}cos{q}_2+J_1,\hfill \\ \hfill M_{12}& =m_2{r}_2^2+m_2{r}_1{r}_{2}cos{q}_2,\hfill \\ \hfill M_{21}& =m_2{r}_2^2+m_2{r}_1{r}_{2}cos{q}_2,\hfill \\ \hfill M_{22}& =m_2{r}_2^2+J_2,\hfill \\ \hfill {\beta }_{12}\left(q_2\right)& =m_2{r}_1{r}_{2}sin{q}_2\hfill \end{array}$$

with parameters given as $r_1=1 \mathrm{m}$, $r_2=0.8 \mathrm{m}$, $J_1=5 \mathrm{k}\mathrm{g} {\mathrm{m}}^2$, $J_2=5 \mathrm{k}\mathrm{g} {\mathrm{m}}^2$, $m_1= 0.5 \mathrm{k}\mathrm{g}$, $m_2=1.5 \mathrm{k}\mathrm{g}$, and $g=9.81 \mathrm{m}/{\mathrm{s}}^{-2}$. As mentioned above, the system parameters sometimes cannot be acquired accurately and we assume that the mass is uncertain. Instead, we assume that the nominal values of $m_1$ and $m_2$ are ${\widehat{m}}_1=0.4 \mathrm{k}\mathrm{g}$ and ${\widehat{m}}_2=1.2 \mathrm{k}\mathrm{g}$, respectively. The external disturbances are given by ${\mathit{\tau }}_d={[2sin\left(t\right)+0.5sin\left(200\pi t\right),cos\left(2t\right)+0.5sin\left(200\pi t\right)]}^{\mathrm{T}}$. Select the following initial conditions: $q_{10}=1,q_{20}=0$, and the desired trajectory is given by ${\mathit{q}}_{\mathrm{d}}={[-1.5sin\left(0.8t\right),2.5cos\left(t\right)]}^{\mathrm{T}}$. The uncertainty term is assumed to satisfy Assumption 1 where the parameters are given by $b_0=12$, $b_1=2.2$, $b_2=2.8$.

Define the tracking error ${\mathit{\epsilon }}_1=\mathit{q}-{\mathit{q}}_{\mathrm{d}}$ and ${\mathit{\epsilon }}_2=\dot{\mathit{q}}-{\dot{\mathit{q}}}_{\mathrm{d}}$. Therefore, the tracking error model of system (3) can be expressed as

$$\left\{\begin{array}{c}{\dot{\mathit{\epsilon }}}_1={\mathit{\epsilon }}_2,\hfill \\ {\dot{\mathit{\epsilon }}}_2={\mathit{F}}_e\left(\mathit{\epsilon }\right)+{\mathit{M}}_0^{-1}\left(\mathit{q}\right)\mathit{\tau }+\mathit{d}(t,\mathit{x}),\hfill \end{array}\right.$$

(35)

where ${\mathit{F}}_e\left(\mathit{\epsilon }\right)=\mathit{F}({\mathit{x}}_1,{\mathit{x}}_2)-{\ddot{\mathit{q}}}_{\mathrm{d}}$.

4.1. Controllers for Comparison

To demonstrate the effectiveness and robustness of the proposed NFTSM controller, we used the ATNTSM control scheme proposed in [16] with the same system parameters for comparison. In addition, the ATNTSM surface was chosen as

$$\mathit{s}=\mathit{\sigma }+{\mathit{C}}_a{\mathit{\epsilon }}_2^{\alpha /\beta },$$

and the control input $\mathit{\tau }$ can be expressed as

$$\mathit{\tau }={\mathit{\tau }}_{\mathrm{eq}}+{\mathit{\tau }}_{\mathrm{sw}},$$

where

$$\begin{array}{cc}\hfill {\mathit{\tau }}_{eq}& =-{\mathit{M}}_0\left(\mathit{q}\right)(\Pi +F\left(\epsilon \right)),\hfill \\ \hfill {\mathit{\tau }}_{sw}& =-{\mathit{M}}_0\left(\mathit{q}\right)\left(a_0+a_1\parallel \mathit{q}\parallel +a_2{\parallel \dot{\mathit{q}}\parallel }^2\right)sign\left(\mathsf{\Lambda }\mathit{s}\right)-k{\mathit{M}}_0\left(q\right)\mathit{s},\hfill \end{array}$$

and $k>0$ is an appropriately chosen constant. The remaining parameters of the controllers are given in

Table 1. Controller parameters.

Variables	Values	Variables	Values
$M_1$	1	$k_0$	600
$M_2$	2	$k_1$	50
${\mathsf{\Gamma }}_1$	$diag\left(\right[22\left]\right)$	$p_1$	1
${\mathsf{\Gamma }}_2$	$diag\left(\right[5/35/3\left]\right)$	$p_2$	2
${\mathsf{\Lambda }}_1$	$diag\left(\right[22\left]\right)$	$\alpha$	5
${\mathsf{\Lambda }}_2$	$diag\left(\right[11\left]\right)$	$\beta$	3
${\mathbf{C}}_a$	$diag\left(\right[0.30.4\left]\right)$	k	2.5
${\delta }_1,{\delta }_2$	$3.5\times {10}^{-3}$

.

The tracking performance of the given two-link robotic manipulator system is illustrated in Figure 3 and Figure 4. As shown in Figure 3, both controllers were able to track the reference trajectory, and the NFTSM controller performed better when the system was disturbed with a relatively large amplitude signal. To further illustrate the angular position tracking performance of the controllers, the position tracking error is shown in Figure 5. The NFTSM controller ensured that the position tracking error converged to zero, while the error caused by the ANTSM controller fluctuated in a small range. As can be seen in Figure 4, the ANTSM controller also fluctuated to a certain extent during speed tracking. Although the saturation function was utilized instead of the sign function, the ANTSM control input still changed relatively sharply. Figure 6 illustrates the control signal of each joint, and it shows that no singularity indeed occurred. Figure 7 shows that the sliding surfaces converged to zero within $0.5 \mathrm{s}$, much faster than the compared ANTSMC algorithm, and finite-time convergence was achieved. A comparison of the two controllers revealed that the NFTSM controller guarantees faster convergence and has stronger robustness.

4.2. States Constrained

Considering the states were constrained, we formulated the constraints on the angular position of each joint as safe constraints, we defined the safe set by (28), and $h\left(\mathit{x}\right)$ was defined as

$$h\left(\mathit{x}\right)=1-{\left(\frac{q_1}{p_1}\right)}^4-{\left(\frac{q_2}{p_2}\right)}^4.$$

(36)

To illustrate the effectiveness of our proposed approach, we assumed the constraints on the angular position of each joint to be $-1\le q_1\le 1$ and $-2\le q_2\le 2$, i.e., $p_1=1$, $p_2=2$. We still took ${\mathit{q}}_{\mathrm{d}}={[-1.5sin\left(0.8t\right),2.5cos\left(t\right)]}^{\mathrm{T}}$ as the reference trajectory, which sometimes broke the state constraints. As shown in Figure 8, when $q_i$ was about to break the limits, the controller responded quickly through the CBF-QP process to guarantee the states remained in the predefined safe set. When the system was operating safely, the zero tracking error with high robustness and finite-time convergence was guaranteed. The control inputs are shown in Figure 9.

4.3. Safety Behavior

We considered physical limitations on the robot manipulator system for more practical scenarios. We assumed that there was a wall above, and the joints should not hit the wall. The control purpose was to enforce the joints to the desired place without collisions. The initial position was ${\mathit{q}}_0={[0,0]}^{\mathrm{T}}$ and desired position was ${\mathit{q}}_{\mathrm{d}}={[\pi ,\pi ]}^{\mathrm{T}}$. The wall was about $1.5$ $\mathrm{m}$ high, i.e.,

$$r_{1}sin{q}_1+r_{2}sin{q}_2\le 1.5.$$

Thus, we defined the control barrier function as

$$h\left(\mathit{x}\right)=1.5-r_{1}sin{q}_1-r_{2}sin{q}_2.$$

(37)

We applied the CBF-QP controller to this two-link robot manipulator with the aforementioned safe constraints. Figure 10 and Figure 11 show the trajectory of the angular position and velocity of each joint. It is shown that the robot reached the desired position in about $3 \mathrm{s}$. For a more precise illustration, the time evolution of the robot manipulator is provided in Figure 12. We can see that the joints were controlled correctly to avoid collision, while the joints controlled by the NFTSM-only controller hit the wall.

5. Conclusions

In this paper, a novel terminal sliding surface is introduced to improve the convergence rate of the conventional nonsingular terminal sliding mode control. We illustrate the effect of model uncertainties and external disturbances of the Euler–Lagrange system, and the high control performance and robustness of the proposed controller are guaranteed. Furthermore, the CBF-QP control strategy ensures that the system remains safe with the formulated state constraints. The simulation results illustrate the control performance and effectiveness of the proposed control strategy. This control strategy provides a general and practical approach for any mechanical system formulated as a Euler–Lagrange system.