Trajectory Tracking of Delta Parallel Robot via Adaptive Backstepping Fractional-Order Non-Singular Sliding Mode Control

Abstract

The utilization of the Delta parallel robot in high-speed and high-precision applications has been extensive, with motion stability being a critical performance measure. To address the inherent inaccuracies of the model and minimize the impact of external disturbances on motion stability, we propose an adaptive backstepping fractional-order non-singular terminal sliding mode control (ABF-NTSMC). Initially, by employing a backstepping algorithm, we select the virtual control for subsystems as the state variable function in joint space while incorporating a calculus operator to enhance fractional-order sliding mode control (SMC). Subsequently, we describe factors such as model uncertainty and external disturbance using a lumped uncertainty function and estimate its upper bound through an adaptive control law. Ultimately, we demonstrate system stability for our proposed control approach and provide an analysis of finite convergence time. The effectiveness of this presented scheme is demonstrated through simulation and experimental research.

1. Introduction

The Delta parallel robot is a highly efficient robotic system with extensive assembly, spraying, and welding applications due to its exceptional speed, precision, and load capacity [1]. Researchers have proposed various methods for achieving precise trajectory tracking in the context of smooth trajectory planning for the Delta parallel robot. These methods include B-spline curve fitting [2], optimization-based approaches [3], and machine learning techniques [4] aimed at reducing vibrations and improving trajectory accuracy. However, the complex mapping relationship between the pose and joint position of the end effector introduced by the nonlinear kinematics/dynamics model of the Delta parallel robot poses a challenge [5]. Therefore, it is essential to consider the dynamic characteristics of the Delta parallel robot when designing a trajectory planning algorithm that meets requirements while maximizing motion speed and accuracy [6]. To effectively address this challenge, we propose a joint trajectory planning approach that considers joint and Cartesian space parameters [7]. We validate our proposal through experimental simulations and testing procedures. The resulting trajectories generated by our algorithm typically exhibit continuous smooth curves, which need to be converted into position and velocity instructions for accurate tracking by the control system of the robot joints [8]. Additionally, it is crucial to select appropriate control algorithms that enable accurate trajectory tracking for the Delta parallel robot while incorporating effective techniques for smoothing out trajectories [9,10].

The conventional PID control, a traditional control method, computes the joint motion command by considering the current position error and velocity error. It aims to track the desired trajectory of the Delta parallel robot’s end effector by adjusting parameters related to proportionality, integration, and differentiation [11]. However, due to its linear nature, the traditional PID control fails to adapt well to nonlinear variations and the inherently dynamic characteristics exhibited by robots. On the contrary, model predictive control establishes a dynamics model for robots while considering constraints on control inputs and state variables [12]. In one study, the driving torque was realized to change in the direction of the joint movement on any trajectory, and the accurate tracking ability of the robot trajectory was improved [13]. This approach predicts robot trajectories within a finite time frame in order to determine optimal joint motion instructions that exhibit exceptional tracking performance and robustness even under nonlinear and constrained conditions [14]. Nevertheless, limitations arise when employing mode predictive control for trajectory tracking in Delta parallel robots due to uncertain factors like unmodeled errors, joint clearance issues, and frictional effects [15]. To effectively cope with real-time changes and uncertainties present in the system, we propose an adaptive algorithm capable of automatically adjusting control parameters based on dynamic characteristics. This algorithm ensures optimal tracking performance even in uncertain environments [16]. Researchers have widely utilized sliding mode control as a nonlinear control method for addressing nonlinearity issues and uncertainties and disturbances encountered during Delta parallel robot control [17,18]. While sliding mode control offers benefits such as its ability to handle nonlinear models, resistance against disturbances, quick response times, and limitations on input controls, it also poses challenges in terms of selecting parameters and designing the sliding mode surface [19,20]. By incorporating an adaptive algorithm into the sling mode control approach, real-time adjustments can be made to both the slope of the sliding mode surface and control law parameters based on system dynamics and variations in external disturbances. This enhances system robustness and improves trajectory tracking performance [21].

In addition, researchers have suggested the integration of nonlinear functions into the sliding mode surface design to improve system performance in terms of tracking and stability [22]. Approaches such as boundary layer technology and inverse function-based sliding mode control are utilized [23,24]. Furthermore, robust sliding mode surfaces and control laws are developed using robust control theory and H-finite control techniques to address parameter uncertainty and external disturbances. These measures ensure system stability and trajectory tracking performance even in the presence of these factors [25,26]. Integrating adaptive inversion control, fractional-order control, and non-singular terminal sliding mode control effectively addresses uncertainties, disturbances, and external perturbations in nonlinear systems [27,28]. This approach accurately models the system using fractional calculus and ensures convergence to the desired state within a finite time frame. It enhances the capability to handle complex nonlinear systems with uncertainty. However, designing the ABGONSTSMC system presents challenges due to the computational complexity associated with fractional-order calculus and determining positive definite control parameters [29]. To address this issue, we propose a precise trajectory tracking control method for the Delta parallel robot based on adaptive backstepping fractional-order non-singular sliding mode control (ABFONSTSMC). The main contributions of this paper are as follows:

• By incorporating an adaptive algorithm and backstepping control, the trajectory-tracking performance of the Delta parallel robot is significantly improved by dynamically adjusting the control law to effectively compensate for errors arising from system dynamics and parameter estimation.
• The integration of an adaptive algorithm and backstepping control enhances the trajectory-tracking performance of the Delta parallel robot by dynamically adjusting the control law to mitigate errors caused by system dynamics and parameter estimation.
• The Delta parallel robot is effectively controlled using a combination consisting of adaptive backstepping control, fractional-order control, and non-singular terminal sliding mode control, enabling rapid response, minimizing steady-state error, compensating for uncertainty factors, and achieving the desired state within a finite time.

The rest of this paper is structured as follows: In Section 3, a dynamics model of the robot manipulator is described. The proposed control method is derived, and the stability of the control system is proved in Section 4. The corresponding simulations and experiments are described in Section 5. Finally, conclusions are drawn in Section 6.

2. Dynamic Model of the Delta Parallel Robot

Considering modeling errors and external disturbances, a dynamic model of an $n-$ degree-of-freedom (DoF) robot can be expressed by the Newton-Euler formula as follows:

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

(1)

where $q,\dot{q},\ddot{q}\in R^{n\times 1}$ are the vectors of position, velocity, and acceleration in the joint space, respectively. $M\left(q\right)\in R^{n\times n}$ is the positive definite inertial matrix, $C(q,\dot{q})\in R^{n\times n}$ is the Coriolis and centripetal matrix, and $G\left(q\right)\in R^{n\times 1}$ is the matrix of gravity, while $\tau \in R^{n\times 1}$ is the vector of the input torque, and $\overline{\mathbf{F}}\in R^{n\times 1}$ is the lumped uncertainty, which is satisfied by the following:

$$\overline{F}=d-(\Delta M\left(q\right)\ddot{q}+\Delta C(q,\dot{q})\dot{q}+\Delta G\left(q\right))$$

(2)

where $\mathbf{d}\in R^{n\times 1}$ is the external disturbances, while $\Delta M\left(q\right)$, $\Delta C(q,\dot{q})$, and $\Delta G\left(q\right)$ are the modeling errors.

The dynamics model of the robot manipulator has the following properties:

Property 1.

The inertial matrix $M\left(\mathrm{q}\right)$ is a symmetric positive definite matrix and is bounded:

$$mI<M\left(\mathrm{q}\right)=M^T\left(\mathrm{q}\right)\le \overline{m}I$$

(3)

where m and $\overline{m}$ are positive constant parameters, and $0<m<\overline{m}$.

Property 2.

$\dot{M\left(\mathrm{q}\right)}-2C(\mathrm{q},\dot{\mathrm{q}})$ is skew-symmetric (the elements of the matrix on the main diagonal are all zeros, while the elements located symmetrically on either side of the main diagonal are inversely signed) and can be satisfied by the following:

$$D^T(\dot{M\left(\mathrm{q}\right)}-2C(\mathrm{q},\dot{\mathrm{q}}))D=0$$

(4)

where D is any non-singular matrix.

Property 3.

$G\left(q\right)$ is bounded by the following:

$$\parallel G\left(q\right)\parallel \le G_k$$

(5)

where $G_k$ is a positive constant.

3. Improved Integral Non-Singular Terminal Sliding Mode Control

3.1. Design of Control System

The position tracking error in joint space $e\left(t\right)\in R^{n\times 1}$ is defined as follows:

$$e=q-q_d$$

(6)

where $q_d\in R^{n\times 1}$ is the desired trajectory in joint space. The non-singular terminal sliding mode surface s can be defined as follows:

$$s=e+{\int }_0^t\left({\lambda }_{1}sign\left(\dot{e}\right)|\dot{e}{|}^{{\omega }_1}+{\lambda }_{2}sign\left(e\right){\left|e\right|}^{{\omega }_2}\right)dt$$

(7)

where ${\lambda }_1$ and ${\lambda }_2$ are positive constants $1<{\omega }_1<2$, and ${\omega }_2={\omega }_1/(2-{\omega }_1)$.

To ensure the convergence of the sliding mode surface s in finite time and suppress the vibration of the system, a new sliding mode surface related to the initial condition is constructed, yielding the following:

$$\begin{array}{c}\sigma =\dot{s}+cs=\dot{e}+{\lambda }_{1}sgn\left(\dot{e}\right)|\dot{e}{|}^{{\omega }_1}+{\lambda }_{2}sgn\left(e\right){\left|e\right|}^{{\omega }_1}+\\ c\left({\int }_0^t\left({\lambda }_{1}sgn\left(\dot{e}\right)|\dot{e}{|}^{{\omega }_1}+{\lambda }_{2}sgn\left(e\right){\left|e\right|}^{{\omega }_2}\right)dt\right)\end{array}$$

(8)

Assume that the upper-bound estimate of lumped uncertainty of the robot system is ${\parallel \mathit{K}\parallel }_{max}$, and the estimated error is $\widehat{K}=K-\overline{F}$.

Deriving (8) from the time, the control law of integral non-singular terminal sliding mode control is derived as follows:

$$\begin{array}{cc}\hfill u=& G+C\dot{q}-\widehat{K}-M\left({\displaystyle \frac{{\lambda }_2{\omega }_2\dot{e}{\left|e\right|}^{{\omega }_2-1}}{1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{{\omega }_1-1}}}-{\ddot{q}}_d+\epsilon sign\left(\sigma \right)+\right.\hfill \\ \hfill & \left.k_v\sigma \right)-M\left({\displaystyle \frac{c\left(\dot{e}+{\lambda }_{1}sign\left(\dot{e}\right)|\dot{e}{|}^{{\omega }_1}+{\lambda }_{2}sign\left(e\right){\left|e\right|}^{{\omega }_2}\right)}{1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{{\omega }_1-1}}}\right)\hfill \end{array}$$

(9)

where $\epsilon >0$ is a design parameter, $k_v={\parallel \mathit{K}\parallel }_{max}+\zeta$, $\zeta$ is the switching control gain, and $\zeta \ge 0$.

3.2. Stability Analysis in Finite Time

The Lyapunov function is constructed as follows:

$$V={\displaystyle \frac{1}{2}}{\sigma }^T\sigma$$

(10)

Considering Derivative (8) with respect to the time, we obtain the following:

$$\begin{array}{cc}\hfill \dot{\sigma }=& \ddot{s}+c\dot{s}\hfill \\ \hfill & =\ddot{e}+{\lambda }_1{\omega }_1|\dot{e}{|}^{{\omega }_1-1}\ddot{e}+{\lambda }_2{\omega }_2{\left|e\right|}^{{\omega }_2-1}\dot{e}+\hfill \\ \hfill & c\left(\dot{e}+{\lambda }_{1}sign\left(\dot{e}\right)|\dot{e}{|}^{{\omega }_1}+{\lambda }_{2}sign\left(e\right){\left|e\right|}^{{\omega }_2}\right)\hfill \\ \hfill & =\left(1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{{\omega }_1-1}\right)\ddot{e}+{\lambda }_2{\omega }_2{\left|e\right|}^{{\omega }_2-1}\dot{e}+\hfill \\ \hfill & c\left(\dot{e}+{\lambda }_{1}sign\left(\dot{e}\right)|\dot{e}{|}^{{\omega }_1}+{\lambda }_{2}sign\left(e\right){\left|e\right|}^{{\omega }_2}\right)\hfill \end{array}$$

(11)

Considering Derivative (6) with respect to the time, combined with (1), there is

$$\ddot{e}=M{\left(q\right)}^{-1}(\tau +\overline{F}-C(q,\dot{q})\dot{q}-G\left(q\right))-{\ddot{q}}_d$$

(12)

Substituting (9) into (12), we obtain

$$\begin{array}{cc}\hfill \ddot{e}& =M^{-1}(-K+\overline{F})-\left({\displaystyle \frac{{\lambda }_2{\omega }_2\dot{e}{\left|e\right|}^{{\omega }_2-1}}{1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{{\omega }_1-1}}}+\epsilon sign\left(\sigma \right)-{\ddot{q}}_d+\right.\hfill \\ \hfill & \left.k_v\sigma \right)-\left({\displaystyle \frac{c\left(\dot{e}+{\lambda }_{1}sign\left(\dot{e}\right)|\dot{e}{|}^{{\omega }_1}+{\lambda }_{2}sign\left(e\right){\left|e\right|}^{{\omega }_2}\right)}{1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{{\omega }_1-1}}}\right)-{\ddot{q}}_d\hfill \end{array}$$

(13)

Substituting (13) into (11), we obtain

$$\begin{array}{cc}\hfill \dot{\sigma }& =\left(1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{{\omega }_1-1}\right)\left(-M^{-1}\widehat{K}-\left(-{\ddot{q}}_d+k_v\sigma +\epsilon sign\left(\sigma \right)\right)-{\ddot{q}}_d\right)\hfill \\ \hfill & -{\lambda }_2{\omega }_2\dot{e}{\left|e\right|}^{{\omega }_2-1}-c\left(\dot{e}+{\lambda }_{1}sign\left(\dot{e}\right)|\dot{e}{|}^{{\omega }_1}+{\lambda }_{2}sign\left(e\right){\left|e\right|}^{{\omega }_2}\right)\hfill \\ \hfill & +{\lambda }_2{\omega }_2{\left|e\right|}^{{\omega }_2-1}\dot{e}+c\left(\dot{e}+{\lambda }_{1}sign\left(\dot{e}\right)|\dot{e}{|}^{{\omega }_1}+{\lambda }_{2}sign\left(e\right){\left|e\right|}^{{\omega }_2}\right)\hfill \end{array}$$

(14)

Simplifying (14) yields the following:

$$\dot{\sigma }=\left(1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{{\omega }_1-1}\right)\left(-M^{-1}\widehat{K}-k_v\sigma -\epsilon sign\left(\sigma \right)\right)$$

(15)

Considering Derivative (10) with respect to the time, combined with (1) and (8), we obtain the following:

$$\dot{V}={\sigma }^T\dot{\sigma }=\left(1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{{\omega }_1-1}\right){\sigma }^T\left(-M^{-1}\widehat{K}-k_v\sigma -\epsilon sign\left(\sigma \right)\right)$$

(16)

where $1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{{\omega }_1-1}$ is a positive constant value.

As ${\sigma }^T\sigma ={\parallel \sigma |}^2$ and ${\sigma }^{T}sign\left(\sigma \right)=\parallel \sigma |$, (16) can be rewritten as

$$\dot{V}=\left(1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{{\omega }_1-1}\right)\left(-M^{-1}\parallel \sigma \parallel \parallel \widehat{K}\parallel -k_v{\parallel \sigma \parallel }^2-\epsilon \parallel \sigma \parallel \right)$$

(17)

Considering the properties of the dynamic model of the robot manipulator, when $\epsilon >0$ and $k_v>0$, there is

$$\dot{V}<0$$

(18)

As $\parallel \widehat{K}\parallel$ is bounded and the product of $\parallel \widehat{K}\parallel$ and $\parallel \sigma \parallel$ is very small, (12) can be simplified as follows:

$$\dot{V}\le \left(-{\sigma }^T{k}_v\sigma -{\sigma }^T\epsilon sign{\left(\sigma \right)}^{\mu }\right)\left(1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{{\omega }_1-1}\right)$$

(19)

where $\mu \approx 1$.

Also,

$${\sigma }^T{k}_v{\sigma }^{\xi }=\sum _{i=1}^n{k}_{vi}{\sigma }_i^{\xi +1}\ge \psi {\left(\sum _{i=1}^n{\displaystyle \frac{1}{2}}\overline{h}{\sigma }_i^2\right)}^{\sigma }\ge \psi {\left({\displaystyle \frac{1}{2}}{\sigma }^T\sigma \right)}^{\sigma }$$

(20)

where $\varpi \approx (1+\xi )/2$, $\psi \approx k_{min}{(2/\overline{h})}^{\sigma }$, and $k_{min}\approx min\left(k_{vi}\right)$. Furthermore, (19) can be rewritten as

$$\dot{V}\le \left({\left(k_v\right)}_{min}\left({\sigma }^T\sigma \right)-{\epsilon }_{min}{\left({\sigma }^T\sigma \right)}^{(\mu +1)/2}\right)\left(1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{({\omega }_1-1}\right)$$

(21)

Theorem 1.

For finite-time stability, the Lyapunov function $V\left(t\right)$ is satisfied by [30]

$$\dot{V}\left(t\right)\le -bV\left(t\right)-\vartheta V^{\eta }\left(t\right),\forall t\ge t_0,V\left(t_0\right)\ge 0$$

(22)

where $b>0$, $\vartheta >0$ and $0<\eta <1$. The corresponding arrival time $T_r$ can be calculated by the following:

$$T_r⩽t_0+{\displaystyle \frac{1}{b(1-\eta )}}ln{\displaystyle \frac{b{V}^{1-\eta }\left(t_0\right)+\vartheta }{\vartheta }}$$

(23)

According to Theorem 1, the proposed control system can be stable in a finite time. When the initial state is $x_0=0$, its stability time is given by the following:

$$t_s⩽{\displaystyle \frac{1}{b(1-\eta )}}ln\left(1+{\displaystyle \frac{b{V}^{1-\eta }\left(t_0\right)}{\vartheta }}\right)$$

(24)

where $\vartheta =\left(1+{\lambda }_1{\omega }_1{\left|\dot{e}\right|}^{{\omega }_1-1}\right)2^{(\mu +1)/2}{\epsilon }_{min}$, $\eta =(1+\mu )/2$, and $b=2\left(1+{\lambda }_1{\omega }_1\left|\dot{e}\right|\right){\left(k_v\right)}_{min}$.

4. Adaptive Backstepping Fractional-Order Non-Singular Terminal SMC

In practical scenarios, it can be challenging to accurately predict and determine the lumped uncertainty $\overline{F}$ beforehand. Poor selection of $\overline{F}$ may worsen chattering in robotic systems and compromise their motion stability. Moreover, during switching processes, there exists a gradual energy transfer within fractional-order sliding surfaces that effectively suppresses chattering.

With the definition of $x_1=q$, $x_2=\dot{q}$, the dynamic mode of (1) in state space can be expressed as follows:

$$\begin{array}{cc}\hfill & {\dot{x}}_1=x_2\hfill \\ \hfill & {\dot{x}}_2=M^{-1}\left(-C{x}_2-G+\tau +\overline{F}\right)\hfill \\ \hfill & y=x_1\hfill \end{array}$$

(25)

Step 1. Assuming that $e_1=q-q_d$, the Lyapunov function of the first step is selected as follows:

$$V_1={\displaystyle \frac{1}{2}}{e}_1^T{e}_1$$

(26)

Considering Derivative (26) with respect to the time, we obtain the following:

$${\dot{V}}_1=e_1^T{\dot{e}}_1=e_1^T\left(x_2-{\dot{q}}_d\right)$$

(27)

Through setting the virtual control $x_{2d}=-K_1{e}_1+{\dot{q}}_d$ to ensure the asymptotic stability of the system and substituting it into (26), we obtain the following:

$${\dot{V}}_1=e_1^T\left(e_2-K_1{e}_1\right)$$

(28)

where $e_2=x_2-x_{2d}$, $K_1>0$ is a design parameter.

Step 2. The Lyapunov function of the second step is defined as follows:

$$V_2=V_1+{\displaystyle \frac{1}{2}}{s}^{T}s$$

(29)

where s is the fractional-order sliding mode surface, satisfied by the following:

$$s=e_1+e_2+D^{\alpha -1}{\chi }_1{e}_1^{\gamma /\beta }$$

(30)

where $0<\alpha <1$, $\gamma$, and $\beta$ are odd, and they can be satisfied by $1<\gamma /\beta <2$, ${\chi }_1\in R^{n\times n}$, a positive definite diagonal matrix.

The differential and integral operator factors of the fractional order can be defined as follows:

$$t_0{D}_t^{\alpha }=\left\{\begin{array}{c}{d}^{\alpha }/d{t}^{\alpha }Re\left(\alpha \right)>0\hfill \\ 1Re\left(\alpha \right)=0\hfill \\ {\int }_{t_0}^t{\left(d\tau \right)}^{-\alpha }Re\left(\alpha \right)<0\hfill \end{array}\right.$$

(31)

Considering Derivative (29) with respect to the time, combined with (25) and (30), we obtain the following:

$$\begin{array}{cc}\hfill {\dot{V}}_2& ={\dot{V}}_1+s^T\left({\dot{e}}_1+D^{\alpha }{\chi }_1{e}_1^{\gamma /\beta }\right.\hfill \\ \hfill & +M^{-1}(-C\dot{q}-G+\tau +\overline{F})+K_1{\dot{e}}_1-{\ddot{q}}_d\hfill \end{array}$$

(32)

Assume that $\Vert \overline{F}\Vert <B_0+B_1\parallel q\parallel$, $B_0$ and $B_1$ are positive constant. The control system is stable when the control law ${\tau }_{eq}$ is satisfied by the following:

$${\tau }_{eq}=C\dot{q}+G-\left({\tilde{B}}_0+{\tilde{B}}_1\parallel q\parallel \right)sign\left(s\right)-M\left(hs+{h\left|s\right|}^{\mu }sign\left(s\right)\right)$$

(33)

Assuming that ${\widehat{B}}_0$ and ${\widehat{B}}_1$ are the estimate values of $B_0$ and $B_1$, the adaptive error is defined as ${\tilde{B}}_0={\widehat{B}}_0-B_0$, and ${\tilde{B}}_1={\widehat{B}}_1-B_1$, the adaptive law is given by the following:

$${\dot{\widehat{B}}}_0={\delta }_0^{-1}\parallel s\parallel ,{\dot{\widehat{B}}}_1={\delta }_1^{-1}\parallel s\parallel \parallel q\parallel$$

(34)

where ${\delta }_0$ and ${\delta }_1$ are the adaptive tuning parameters, and ${\delta }_0>0$, ${\delta }_1>0$.

Step 3. Combining (26) and (29) with adaptive law, the Lyapunov function of the third step is given by the following:

$$V_3={\displaystyle \frac{1}{2}}{e}_1^T{e}_1+{\displaystyle \frac{1}{2}}{s}^{T}s+{\displaystyle \frac{1}{2}}\left({\zeta }_0{\tilde{B}}_0^2+{\zeta }_1{\tilde{B}}_1^2\right)$$

(35)

Considering Derivative (35) relative to the time, we obtain the following:

$$\begin{array}{c}{\dot{V}}_3=e_1^T{\dot{e}}_1+s^T\left({\dot{e}}_1+D^{\alpha }{\chi }_1{e}_1^{\gamma /\beta }+M^{-1}(-C\dot{q}-G+\tau +\overline{F})+\right.\\ \hspace{1em}\left.K_1{\dot{e}}_1-{\ddot{q}}_d\right)+{\zeta }_0\left({\widehat{B}}_0-B_0\right){\dot{\widehat{B}}}_0+{\zeta }_1\left({\widehat{B}}_1-B_1\right){\dot{\widehat{B}}}_1\end{array}$$

(36)

Substituting (33) and (34) into (36), we obtain the following:

$$\begin{array}{cc}\hfill & {\dot{V}}_3=-s^T{M}^{-1}\left({\widehat{B}}_1\parallel q\parallel +{\widehat{B}}_0\right)sign\left(s\right)-h{s}^T{\left|s\right|}^{{\mu }_1}sign\left(s\right)\hfill \\ \hfill & -K_1{e}_1^T{e}_1-h{s}^{T}s+s^T{M}^{-1}\overline{F}+{\zeta }_0{\delta }_0^{-1}{\tilde{B}}_0\parallel s\parallel +{\zeta }_1{\delta }_1^{-1}{\tilde{B}}_1\parallel s\parallel \parallel q\parallel \hfill \end{array}$$

(37)

As $s^{T}s={\parallel s\parallel }^2$, $-\left(h{{\Vert }_S\Vert }^2+h{{\Vert }_S\Vert }^{{\mu }_1}\right)\le -h{{\Vert }_S\Vert }^2$, (37) can be rewritten as follows:

$$\begin{array}{cc}\hfill & {\dot{V}}_3\le -\left(h\parallel S\parallel -M^{-1}\parallel \overline{F}\parallel +M^{-1}\left({\widehat{B}}_1\parallel q\parallel +{\widehat{B}}_0\right)\right)\parallel S\parallel -K_1{∥e_1∥}^2\hfill \\ \hfill & -\left(M^{-1}-{\zeta }_0{\delta }_0^{-1}\right)\parallel S\parallel ∥{\tilde{B}}_0∥-\left(M^{-1}-{\zeta }_1{\delta }_1^{-1}\right)\parallel S\parallel ∥{\tilde{B}}_1∥\parallel q\parallel \hfill \end{array}$$

(38)

Theorem 2.

Consider a fractional non-autonomous system $D^{\alpha }x\left(t\right)=f(x,t)$, where $f(x,t)$ satisfies the Lipschitz condition, and $0<\alpha <1$. Assuming that $x=0$ is the equilibrium point, the system is stable, while a Lyapunov function $V(t,x)$ can be found, satisfying the conditions as follows [31]:

$$k_1\parallel x\parallel \le V(t,x)\le k_2\parallel x\parallel ,\dot{V}(t,x)\le -k_3\parallel x\parallel$$

(39)

where $k_1$, $k_2$, and $k_3$ are constants greater than zero.

Theorem 3.

It is assumed that the Lyapunov function $V(t,x)$ satisfies the following inequality [32]:

$$\dot{V}(t,x)\le -\vartheta V^{\eta }(t,x),\forall t\ge t_0,V\left(t_0\right)\ge 0$$

(40)

where $\vartheta >0$, $0<\eta <1$.

Then, the system can converge in a finite time $T_r$, and

$$T_r\le t_0+{\displaystyle \frac{V^{1-\eta }\left(t_0\right)}{\vartheta (1-\eta )}}$$

(41)

Based on Theorems 2 and 3, (38) can be rewritten as follows:

$${\dot{V}}_3\le -{\vartheta }_{min}{V}_3^{1/2},\mathrm{and}{t}_r={\displaystyle \frac{2V^{1/2}\left(0\right)}{{\vartheta }_{min}}}$$

(42)

where

${\vartheta }_{min}=min\left\{{\vartheta }_0\sqrt{2},{\vartheta }_1\sqrt{2},{\vartheta }_2\sqrt{{\displaystyle \frac{2}{{\zeta }_0}}},{\vartheta }_3\sqrt{{\displaystyle \frac{2}{{\zeta }_1}}}\right\}$

${\vartheta }_0=K_1∥e_1∥$

${\vartheta }_2=h{\Vert }_S\Vert -M^{-1}\left(\parallel \overline{F}\parallel -∥B_1∥q∥+B_0∥\right)$

${\vartheta }_1=\left(M^{-1}-{\zeta }_0{\delta }_0^{-1}\right)\parallel S\parallel ,{\vartheta }_3=\left(M^{-1}-{\zeta }_1{\delta }_1^{-1}\right)\parallel S\parallel \parallel q\parallel$

A diagram of adaptive backstepping fractional-order sliding mode control is shown in Figure 1.

5. Simulations and Experiments

The robustness of integral order sliding mode control is evident; however, the presence of nonlinear factors in the dynamic model exacerbates process vibrations and affects accurate tracking performance when converting the end effector trajectory from Cartesian space to joint space. On the other hand, fractional-order sliding mode control integrates differential and integral processes, allowing for a wider range of controller adjustments. This facilitates smoother transitions between sliding mode surfaces and effectively reduces system chattering.

5.1. Simulations

By integrating the interface file within the Adams/Controls module and synergizing the kinematics analysis capability of Adams with the control functionality of Matlab, we successfully achieved joint simulation for the multi-software system of the Delta parallel robot. The steps required for this are as follows:

(1) A 3D graph of the Delta parallel robot is saved in $\ast .X_T$ format and inputted into ADMAS 2020.

(2) Redefine the constraint relationship between each part, give each part’s material and quality properties, and add drivers.

(3) Complete the data interface setting, generate the $.m$ files required for data exchange, and import the defined Delta model into Matlab 2020a through the Adams/Controls extension module.

The Delta parallel robot model established is shown in Figure 2a, and a block diagram of the co-simulation system is shown in Figure 2b.

The external disturbance is given by the following:

$$d=\left[\begin{array}{c}0.002cos\left(\pi t\right)\\ 0.002cos\left(0.9\pi t\right)\\ 0.002cos\left(0.8\pi t\right)\end{array}\right]$$

(43)

Set the simulation time to 10 s and the simulation step size to 0.0005 s. The Parameters of the traditional PID controller and fractional-order PID used for simulation comparison are given in

Table 1. The parameters of the comparison controller and the controller proposed in this paper.

Controller	Parameters
PID	$k_P=diag(400,400,400),k_I=diag(10,10,10),k_D=diag(15,15,15)$
INSTSMC	${\chi }_1=$1.200$,{\chi }_2=1.300,{\omega }_1=5/3,\lambda =15$
	$v=diag(0.913,0.913,0.913),\epsilon =diag(0.351,0.351,0.531)$
FOPID	$K_p=diag(600,600,600),K_I=diag(10,20,10),{\mu }_{FO}=0.700$
	${\lambda }_{FO}=0.800,K_D=diag(15,10,15)$
ABFONSTSMC	${\eta }_1=diag(4,3,5),K_1=diag(25,25,50),h=diag(45,45,50),\alpha =0.800$

.

Trajectory tracking of the three drive joints of the Delta parallel robot is shown in Figure 3.

Figure 3 presents a compelling demonstration of the precise trajectory-tracking capabilities exhibited by all four controllers in stable conditions. However, when faced with uncertain factors such as external disturbances, it is observed that the PID controller displays a noticeable overshoot during the initial phase. In contrast, due to the integration of fractional calculus operators, the FOPID controller demonstrates a reduced adjustment time and overshoot compared to its PID counterpart. Furthermore, both steady-state error and overshoot are found to be superior for INSTSMC in comparison to both PID and FOPID controllers. Importantly, our proposed controller effectively eliminates any initial-stage overshoot and achieves a lower steady-state error than the other three controllers after 0.05 s, instilling confidence in its performance.

The joint driving torque obtained using POPID, INSTSMC, and the control method proposed in this paper is shown in Figure 4.

Figure 4 shows that due to the existence of discontinuous terms such as ${\chi }_{1}sign\left(\dot{e}\right){\left|\dot{e}\right|}^{{\omega }_1}$, ${\chi }_{2}sign\left(e\right){\left|e\right|}^{{\omega }_2}$, and $\epsilon sign\left(\sigma \right)$ in INSTSMC, the control torque is kept within a certain range, and the chattering phenomenon is obvious. The Proposed ABFONSTSMC controller has the advantages of fractional-order and sliding mode control, which can effectively suppress the control torque chattering and improve the trajectory tracking and motion stability of the Delta parallel robot.

5.2. Experiments

The Delta parallel robot’s experimental platform was constructed utilizing the EtherCAT communication mode. The hardware components encompass a Delta parallel robot, motion control card, and sensor, among others. To establish communication, an RJ45 industrial Ethernet cable is connected to the workstation JSDG2S-15A-E Solid High technology Co. Ltd, China, Guangdong via the communication port CN5. MotionStudio software 2020 is employed to configure the motion control settings, while Microsoft Visual Studio is utilized to develop the control system application. Figure 5 illustrates the test platform of the Delta parallel robot.

The parameters of each controller are still set according to Table 1, and the trajectory tracking of the end effector of the Delta parallel robot is shown in Figure 6. Joint trajectory tracking and control torque are shown in Figure 7 and Figure 8, respectively.

Figure 6 shows that all controllers exhibit accurate trajectory tracking in Cartesian space. However, notable disparities arise in their performance at the transition rounded corner, where INSTSMC exhibits the poorest performance, FOPID displays oscillatory behavior, and PID demonstrates the largest steady-state error. In contrast, our proposed controller, ABFONSTSMC, showcases superior trajectory tracking capabilities, specifically at the transition rounded corner.

Figure 7 shows the effective reduction in tracking error achieved by the proposed ABFONSTSMC controller while maintaining a root mean square error within $1\times {10}^{-3}$ rad even in the presence of uncertain factors such as unmodeled errors and external disturbances. Additionally, Figure 8 shows the effective suppression of control torque chattering by the controller proposed in this paper (

Table 2. The root-mean-square tracking error and control torque in joint space of the Delta parallel robot.

Controller	Tracking Error of Joints ${∥{\mathit{e}}_{\mathit{i}}∥}_{\mathit{R}\mathit{M}\mathit{S}}/\left(\mathbf{m}\right)$			Control Torque of Joints ${∥{\mathit{e}}_{\mathit{i}}∥}_{\mathit{R}\mathit{M}\mathit{S}}/(\mathit{N}·\mathit{m}\mid )$
	${\mathbf{1}}^{\#}$	${\mathbf{2}}^{\#}$	${\mathbf{3}}^{\#}$	${\mathbf{1}}^{\#}$	${\mathbf{2}}^{\#}$	${\mathbf{3}}^{\#}$
PID	0.0043	0.0010	0.0032	0.0098	0.0085	0.0063
INSTSMC	0.0025	0.0015	0.0021	0.0093	0.0053	0.0083
FOPID	0.0016	0.0018	0.0013	0.0123	0.0054	0.0120
ABFONSTSMC	0.0013	0.0010	0.0005	0.0039	0.0020	0.0036

).

6. Conclusions

The utilization of the Delta parallel robot in high-speed and high-precision applications has been extensive, with motion stability being a critical performance measure. To tackle the inherent model inaccuracies and minimize the impact of external disturbances on motion stability, we propose an adaptive control approach known as ABF-NTSMC, which combines fractional-order non-singular terminal sliding mode control with a backstepping control strategy. This combination offers several advantages: adaptability to changing conditions, the utilization of fractional-order dynamics, the implementation of a backstepping control technique, and the incorporation of non-singular terminal sliding mode control.

(1) Based on adaptive backstepping control, a feedback controller was designed to compensate for the nonlinear term in the system and adjust the control law based on actual dynamic and parameter errors to enhance system robustness and tracking performance.

(2) By incorporating fractional-order calculus, it accurately describes the complex nonlinear nature of the Delta parallel robot system.

(3) In conjunction with an adaptive algorithm, it estimates and compensates for system uncertainty factors using system response and error information. Combined with non-singular terminal sliding mode control, it ensures finite-time convergence to desired states.

The simulation and experimental results show the effectiveness of the proposed method. The future application of a data-driven approach is anticipated to enhance the dynamic modeling accuracy and trajectory tracking precision of parallel robots.