Trajectory Tracking Closed-Loop Cooperative Control of Manipulator Neural Network and Terminal Sliding Model

Abstract

To address the issue of low trajectory tracking accuracy in six-degree-of-freedom robotic arms, this study proposes a trajectory tracking control strategy that integrates a Radial Basis Function Neural Network (RBFNN) with non-singular fast terminal sliding mode (NFTSM) control. (1) The Lagrangian method is utilized to develop the dynamic model of the robotic arm. At the same time, a non-singular fast terminal sliding surface is designed to accelerate trajectory convergence and resolve the singularity problem commonly associated with traditional sliding mode control by integrating nonlinear and fast terminal terms. (2) The RBF neural network is employed to globally approximate and compensate for uncertainties in the model and variations in the parameters of the robotic arm. (3) To confirm the overall stability of the control system with the proposed NFTSM control strategy, the Lyapunov stability theory is applied to formulate a Lyapunov function. (4) The six-degree-of-freedom robotic manipulator is simulated in the MATLAB/Simulink environment to assess the effectiveness of the proposed control method. In addition, experimental validation is carried out on a real robotic manipulator to verify the effectiveness of the proposed method. The simulation and experimental results show that, compared with NFTSM and RBFNN-SMC, the proposed control strategy significantly enhances the trajectory tracking accuracy of the six-degree-of-freedom robotic manipulator, thereby offering an effective and practical solution for its trajectory tracking control.

1. Introduction

The six-degree-of-freedom manipulator is an advanced, automated mechanical device comprising a mechanical structure, a drive system, a control system, and sensors. Multiple inputs and outputs, high nonlinearity, and high coupling characterize this system. This technology has extensive applications across various fields, including industry, medical and healthcare, and aerospace. The manipulator can be used to track the desired trajectories with high accuracy. However, achieving this precision is challenging due to the dynamics of the manipulator, which are affected by model errors and parameter uncertainties.

To address the issue of slow trajectory tracking speed and low accuracy in the six-degree-of-freedom manipulator, various types of manipulator control methods have been proposed both domestically and internationally. These include fuzzy control [1,2,3], disturbance observer control [4,5,6], robust control [7], sliding mode control, and other methods. Wu et al. [8] proposed a control strategy that combines a Radial Basis Function Neural Network (RBFNN) with terminal sliding mode control. The method facilitates model-free control and achieves a maximum steady-state tracking error of merely 0.005 rad. However, its high computational complexity presents challenges for real-time implementation, limiting its practical applicability. Jia et al. [9] proposed a control approach that combines a block-wise approximated neural network with terminal sliding mode control based on the dynamic model of the manipulator. This technique enables model reconstruction and achieves a maximum average steady-state tracking error of 0.0073 rad. However, the high design complexity and insufficient coordination among subnetworks may result in systemic errors. Li et al. [10] proposed a novel adaptive neural network-based sliding mode control strategy. The method effectively mitigates excessive chattering while maintaining high tracking accuracy, achieving a maximum average steady-state error of 0.0372 rad. However, the inability to precisely identify the nominal model embedded in the controller may compromise overall system performance. Zhou et al. [11] proposed a fuzzy sliding mode control scheme with adaptive switching gains. While this approach significantly enhances tracking stability, its dependence on expert knowledge limits its robustness and generalizability. Jiang et al. [12] developed a fractional-order PID (FOPID) control strategy to enhance the tracking accuracy of manipulators. The proposed method significantly enhances tracking performance, achieving a maximum average steady-state error of 0.0088 rad. However, it is susceptible to being trapped in local optima, which could limit overall system performance.

In summary, researchers worldwide have proposed various control strategies to improve the trajectory tracking accuracy of manipulators, with most being validated on six-degree-of-freedom systems. However, these methods often involve high computational complexity and significant design challenges. To address this issue, a control strategy is developed for a six-degree-of-freedom manipulator that integrates a Radial Basis Function (RBF) neural network with non-singular fast terminal sliding mode control. The RBF network is utilized to globally approximate and compensate for nonlinear dynamics, thereby enhancing the accuracy of trajectory tracking. The main contributions are as follows:

• This paper proposes a symmetrically designed non-singular fast terminal sliding surface with a reduced number of parameters, ensuring consistent convergence rates for both positive and negative joint errors, thereby enhancing control accuracy.
• A Radial Basis Function Neural Network compensation mechanism with symmetrically distributed centers and a uniform width is introduced, which reduces the number of network parameters and enhances generalization capability, thereby lowering computational complexity.
• A Lyapunov function is constructed to verify the overall stability of the system.
• Simulations and experimental validation are performed, and the proposed method with the RBFNN-SMC and NFTSM control algorithms are compared.

2. Manipulator Control System

2.1. The Dynamic Model of the Manipulator Is Constructed

Taking the JAKA Zu 5 collaborative manipulator as our control object, this six-degree-of-freedom industrial robot is characterized by its high flexibility and strong adaptability.

The three-dimensional structure of the manipulator, along with its corresponding joint coordinates, is illustrated in Figure 1.

The definitions of the system parameters are presented in

Table 1. The definitions of the system parameters.

Symbol	Definition/Unit
q	Joint angle vector of the manipulator
$\dot{q}$	Joint angular velocity vector
$\ddot{q}$	Joint angular acceleration vector
$M\left(q\right)$	Symmetric positive definite inertia matrix
$C\left(q,\dot{q}\right)$	Coriolis and centrifugal force matrix
$G\left(q\right)$	Gravity vector
F	External disturbance torque
$\tau$	Control input torque vector
W	Weight vector of the RBF neural network
f	RBFNN approximation of the equivalent control term
V	Lyapunov function
$tanh\left(·\right)$	Hyperbolic tangent switching function for sliding mode compensation

.

The dynamic equation of the manipulator is as follows:

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

(1)

in Equation (1), $M\left(q\right)\in R^{6\times 6}$ is the positive definite symmetric inertia matrix of the manipulator; $q\in {\left[q_1,q_2,q_3,q_4,q_5,q_6\right]}^T$ is expressed as the joint angle of the manipulator. The unit is rad; $\dot{q}\in {\left[{\dot{q}}_1,{\dot{q}}_2,{\dot{q}}_3,{\dot{q}}_4,{\dot{q}}_5,{\dot{q}}_6\right]}^T$ represents the angular velocity of each joint of the manipulator. The unit is $\mathrm{rad}/\phantom{rads}\mathrm{s}$; $\ddot{q}\in {\left[{\ddot{q}}_1,{\ddot{q}}_2,{\ddot{q}}_3,{\ddot{q}}_4,{\ddot{q}}_5,{\ddot{q}}_6\right]}^T$ is expressed as the angular acceleration of each joint of the manipulator. The unit is ${\mathrm{rad}/\phantom{rads}\mathrm{s}}^2$; $C(q,\dot{q})\in R^{6\times 6}$ is expressed as the Coriolis force and centrifugal force matrix of the manipulator; $G\left(q\right)\in R^{6\times 1}$ is expressed as the gravity matrix of the manipulator; $F\in R^{6\times 1}$ is expressed as the sum of the model error and parameter uncertainty factors of the manipulator; and $\tau \in R^{6\times 1}$ is expressed as the control input of the manipulator.

The dynamic model of the manipulator exhibits the following two properties:

Property 1: The inertia matrix $M\left(q\right)$ is symmetric and positive definite. There exist positive constants $m1$ and $m2$ such that the following inequality is satisfied:

$$m1{∥x∥}^2\le x^{T}M\left(q\right)x\le m2{∥x∥}^2$$

(2)

Property 2: $\dot{M}\left(q\right)-2C(q,\dot{q})$ is a skew-symmetric matrix. There exists a real number x satisfying

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

(3)

2.2. Controller Design

2.2.1. Design of Non-Singular Fast Terminal Sliding Mode Controller

The expected angle of each joint of the manipulator is $q_d={\left[q_{d1},q_{d2},q_{d3},q_{d4},q_{d5},q_{d6}\right]}^T$; the actual joint angle is $q=\left[q_1,q_2,q_3,q_4,q_5,q_6\right]$; the joint angle error is e, the joint angular velocity error is $\dot{e}$, and the joint angular acceleration error is $\ddot{e}$.

The calculation method of joint error is as follows:

$$\left\{\begin{array}{c}e=q_d-q\\ \dot{e}={\dot{q}}_d-\dot{q}\\ \ddot{e}={\ddot{q}}_d-\ddot{q}\end{array}\right.$$

(4)

to enhance the tracking speed of the manipulator and to resolve the singularity issue in sliding model control, a non-singular fast terminal sliding mode surface [13] is designed as follows:

$$S=\nu e+c\dot{e}+\mu {\dot{e}}^{p/\phantom{pk}k}$$

(5)

in Equation (5), S denotes the function of the sliding surface $\nu <0;\mu >0;1<p/k<2$. Both p and k are positive odd numbers, and c is a constant.

When the joint error e is particularly large, the nonlinear term $\mu {\dot{e}}^{p/k}$ is the dominant term, and the system state quickly approaches the equilibrium state. When e is small, the nonlinear term $\mu {\dot{e}}^{p/k}$ can be ignored. At this time, the linear term $\upsilon e+c\dot{e}$ plays a leading role in ensuring that the system’s state converges quickly to the equilibrium point.

Based on the sliding surface, the control law is designed. The time derivative of the sliding surface S is given by

$$\dot{S}=\upsilon \dot{e}+c\ddot{e}+\mu (p/k)diag\left({\dot{e}}^{(p/k)-1}\right)\ddot{e}$$

(6)

in Equation (6), $diag\left({\dot{e}}^{\left(p/k\right)-1}\right)$ represents that the diagonal matrix is generated by taking the elements of the vector ${\dot{e}}^{(p/k)-1}$ as diagonal elements.

It can be concluded from Equation (2):

$$\ddot{q}=M{\left(q\right)}^{-1}\left[\tau -C\left(q,\dot{q}\right)\dot{q}-G\left(q\right)-F\right]$$

(7)

based on Equation (4), the expression of $\ddot{e}$ is obtained as

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

(8)

let $\dot{S}=0$, then the equivalent control rate ${\tau }_{eq}$ of the equivalent control term can be obtained as follows:

$${\tau }_{eq}=M\left(q\right){\ddot{q}}_d+\nu {\left[cI+\mu {\displaystyle \frac{p}{k}}diag\left({\dot{e}}^{\left(p/k\right)-1}\right)\right]}^{-1}M\left(q\right)\dot{e}+C\left(q,\dot{q}\right)\dot{q}+G+F$$

(9)

in Equation (9), I denotes the unit matrix of 6 × 6.

In a sliding mode control system, the switching control term is essential for ensuring the effectiveness of the control process. A crucial aspect of the switching control law is the reaching law. The typical standard approach rates are the constant approach rate, exponential approach rate, and power approach rate [14].

Constant approach rate:

$$\dot{S}=-\epsilon sign\left(S\right)$$

(10)

in Equation (10), $\epsilon >0$, whereby the greater the value of $\epsilon$, the faster the system reaches the switching surface.

Exponential approach rate:

$$\dot{S}=-\epsilon sign\left(S\right)-kS$$

(11)

in Equation (11), $\epsilon >0,k>0$; the exponential approach rate is suitable for solving the corresponding control problem of a large step.

Power approach rate:

$$\dot{S}=-k{\left|S\right|}^{\alpha }sign\left(S\right)$$

(12)

in Equation (12), $k>0,0<\alpha <1$; the system state can be guaranteed to approach at a larger speed when the system state is far away from the synovial surface by adjusting $\alpha$.

To accelerate convergence and enhance system robustness, a switching control law is designed:

$${\tau }_{sw}=k_{1}S+k_{2}tanh\left(S\right)$$

(13)

the $tanh\left(S\right)$ function in Equation (13) is the hyperbolic tangent function, which is continuous and differentiable. It helps reduce chattering in the system while maintaining a balance between control accuracy and chattering suppression $k_1>0,k_2>0$.

In summary, the control rate of the controller can be designed as

$$\tau ={\tau }_{eq}+{\tau }_{sw}$$

(14)

the Lyapunov function [15] is constructed as

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

(15)

derivation of time:

$$\dot{V}=S^T\dot{S}$$

(16)

from the above equations, we can obtain

$$\begin{array}{cc}\hfill \dot{V}=& -S^T\left[cI+\mu \left(p/\phantom{pk}k\right)diag\left({\dot{e}}^{p/k-1}\right)\right]M^{-1}{\tau }_{sw}\hfill \\ \hfill =& -S^T\left[cI+\mu \left(p/\phantom{pk}k\right)diag\left({\dot{e}}^{p/k-1}\right)\right]M^{-1}\left(k_{1}S+k_{2}tanh\left(S\right)\right)\hfill \end{array}$$

(17)

because c, $\mu$, and $p/\phantom{pk}k$ are positive scalars and M is a positive definite matrix, P is also positive definite. Let $P=\left[cI+\mu \left(p/\phantom{pk}k\right)diag\left({\dot{e}}^{p/k-1}\right)\right]M^{-1}$

$$\dot{V}=-S^{T}P\left(k_{1}S+k_{2}tanh\left(S\right)\right)<0$$

(18)

in Equation (18), $k_1>0,k_2>0$. It is guaranteed that $\dot{V}<0$, when $S=0$, $\dot{V}=0$.

According to the Lyapunov stability theory, the system is stable. However, in practice, the parameters required in the equivalent control law for a six-degree-of-freedom manipulator are challenging to obtain. Therefore, the control law requires further modification.

2.2.2. RBF Neural Network Controller Design

The Radial Basis Function Neural Network (RBFNN) is highly effective in approximating nonlinear functions. Therefore, it is employed to perform a global approximation of the system’s nonlinearities. The RBFNN consists of an input layer, a hidden layer, and an output layer [16], as illustrated in Figure 2.

The RBF neural network structure shown in Figure 2 consists of nnn input layer neurons, mmm hidden layer neurons, and a single output neuron. $X_1,X_2\dots X_n$ represent the output signals of the input layer. In practical applications, these input signals typically include joint angles, angular velocities, and joint angle errors. Let $h_1,h_2,h_3\dots h_m$ denote the outputs of the hidden layer, and let f represent the output of the output layer, which corresponds to the result of the overall approximation. The hidden layer outputs are determined using radial basis functions, described by the following equation:

$$h_m\left(t\right)=exp\left(-{\displaystyle \frac{{∥X-c_m∥}^2}{2{b_m}^2}}\right)$$

(19)

in Equation (18), $c_m$ denotes the Gaussian basis function center vector of the mth neuron node; $b_m$ denotes the width of the Gaussian basis function of the mth neuron node.

Let f denote the ideal output of the RBF neural network used for the global approximation of the equivalent control law, which can be expressed as

$${\tau }_{eq}=f=\left[\begin{array}{c}{f}_1\\ f_2\\ ⋮\\ f_m\end{array}\right]=\left[\begin{array}{c}{W_1}^{\mathrm{T}}{h}_1\\ {W_2}^{\mathrm{T}}{h}_2\\ ⋮\\ {W_m}^{\mathrm{T}}{h}_m\end{array}\right]$$

(20)

in Equation (20), $W={\left[W_1,W_2\dots Wm\right]}^{\mathrm{T}}$ is the weight value of the RBF neural network.

Set the adaptive rate of neural network weights as follows:

$$\dot{W}=\varphi h\left(t\right)S^{\mathrm{T}}$$

(21)

in Equation (21), $\varphi$ is a 6 × 6 positive definite diagonal learning rate matrix.

The modified total control law is

$$\tau =f+k_{1}S+k_{2}tanh\left(S\right)$$

(22)

in Equation (22), $k_{1}S$ can promote the system state of the manipulator to converge to the sliding mode surface quickly, thereby improving the convergence speed. The $k_{2}tanh\left(S\right)$ term can make the control input smoother and enhance the robustness and stability of the system.

2.2.3. Stability Proof of Control System

The Lyapunov function is constructed as

$$V={\displaystyle \frac{1}{2}}{S}^{\mathrm{T}}M\left(q\right)S+{\displaystyle \frac{1}{2}}tr\left({e_W}^{\mathrm{T}}{\varphi }^{-1}{e}_W\right)+{\displaystyle \frac{1}{2}}{\int }_0^t{S}^{\mathrm{T}}\mathrm{\Theta }Sd\tau$$

(23)

in Equation (23), $tr\left(·\right)$ denotes the trace of the matrix; $e_W=W-W^{*}$, where W represents the current weight and $W^{*}$ represents the ideal weight; $\varphi ,\mathrm{\Theta }$ denotes a positive definite diagonal matrix and a positive definite symmetric matrix, respectively; and in the function, ${\int }_0^t{S}^{\mathrm{T}}\mathrm{\Theta }Sd\tau$ is introduced to enhance the description of the dynamic characteristics of the system.

The time derivative of Equation (23) is obtained as

$$V=S^{\mathrm{T}}M\left(q\right)\dot{S}+{\displaystyle \frac{1}{2}}{S}^{\mathrm{T}}\dot{M}\left(q\right)S+tr\left(e_W{\varphi }^{-1}{\dot{e}}_W\right)+{\displaystyle \frac{1}{2}}{S}^{\mathrm{T}}\mathrm{\Theta }S$$

(24)

according to the RBF neural network framework, the dynamic terms of the manipulator are globally approximated to steer the system toward the ideal sliding mode, while ensuring overall stability.

$$M\dot{S}=M\nu \left({\dot{q}}_d-\dot{q}\right)+Mc\left({\ddot{q}}_d-\ddot{q}\right)+M\mu {\displaystyle \frac{p}{k}}\mathrm{diag}\left({\dot{e}}^{{\displaystyle \frac{p}{k}}-1}\right)\left({\ddot{q}}_d-\ddot{q}\right)$$

(25)

$$M{\ddot{q}}_d+C\dot{q}+G+F=W^{{*}^T}h$$

(26)

in Equation (26), the variable $W^{*}$ denotes the ideal weight.

$$W^{T}h+k_{1}S+k_{2}tanh\left(S\right)=\tau$$

(27)

substituting Equations (26) and (27) into Equation (24) yields

$$M\dot{S}=\underset{{\tau }_1}{\underbrace{M\nu \left({\dot{q}}_d-\dot{q}\right)+Mc{\ddot{q}}_d+M\mu {\displaystyle \frac{p}{k}}diag\left({\dot{e}}^{{\displaystyle \frac{p}{k}}-1}\right){\ddot{q}}_d}}-\underset{{\tau }_2}{\underbrace{\left(Mc+M\mu {\displaystyle \frac{p}{k}}diag\left({\dot{e}}^{{\displaystyle \frac{p}{k}}-1}\right)\right)\left(\tau -C\dot{q}-G-F\right)}}$$

(28)

$$\left(Mc+M\mu {\displaystyle \frac{p}{k}}diag\left({\dot{e}}^{{\displaystyle \frac{p}{k}}-1}\right)\right)\left(\left(W^{T}h-W^{{*}^T}h\right)+k_{1}S+k_{2}tanh\left(S\right)+M{\ddot{q}}_d\right)={\tau }_2$$

(29)

substituting Equation (29) into Equation (28), and with a suitable selection of the parameter $\nu ,c,\mu$, we can obtain

$$M\dot{S}=M\nu \left({\dot{q}}_d-\dot{q}\right)-\left(Mc+M\mu {\displaystyle \frac{p}{k}}diag\left({\dot{e}}^{{\displaystyle \frac{p}{k}}-1}\right)\right)\left({e_W}^{T}h+k_{1}S+k_{2}tanh\left(S\right)\right)$$

(30)

since $\nu \left({\dot{q}}_d-\dot{q}\right)$ in the sliding surface design corresponds to the linear feedback of the error derivative, according to the sliding surface reaching condition $S=0$, $\nu \left({\dot{q}}_d-\dot{q}\right)$ can be chosen to cancel out the error derivative term in the system. Consequently, we obtain

$$M\dot{S}=-\left(Mc+M\mu {\displaystyle \frac{p}{k}}diag\left({\dot{e}}^{(p/k)-1}\right)\right)\left({e_W}^{T}h+k_{1}S+k_{2}tanh\left(S\right)\right)$$

(31)

since M is a positive definite inertia matrix and $c,\nu$ consists of design parameters, $Mc+M\mu {\displaystyle \frac{p}{k}}diag\left({\dot{e}}^{(p/k)-1}\right)$ can be regarded as a positive definite gain. This does not affect the negative definiteness of the resulting equation.

In summary,

$$M\left(q\right)\dot{S}=-{e_W}^{\mathrm{T}}h-k_{1}S-k_{2}tanh\left(S\right)$$

(32)

according to the simplification of Equation (24) by $\dot{W}=\dot{e_W}$ and Equation (32),

$$\begin{array}{cc}\hfill \dot{V}=& -S^{\mathrm{T}}{e}_{W}h-k_1{S}^{\mathrm{T}}S-k_2{S}^{\mathrm{T}}tanh\left(S\right)+S^{\mathrm{T}}C\left(q,\dot{q}\right)S+tr\left(S^{T}h{e}_W\right)+{\displaystyle \frac{1}{2}}{S}^{\mathrm{T}}\mathrm{\Theta }S\hfill \\ \hfill =& -k_1{S}^{\mathrm{T}}S-k_{2}tanh\left(S\right)+{\displaystyle \frac{1}{2}}{S}^{\mathrm{T}}\mathrm{\Theta }S\hfill \\ \hfill \le & -k_1{S}^{\mathrm{T}}S-k_{2}tanh\left(S\right)+{\displaystyle \frac{1}{2}}\lambda min\left(\mathrm{\Theta }\right)S^{\mathrm{T}}S\hfill \\ \hfill \le & -\left(k_1-{\displaystyle \frac{1}{2}}\lambda min\left(\mathrm{\Theta }\right)\right)S^{\mathrm{T}}S-k_{2}tanh\left(S\right)\hfill \end{array}$$

(33)

in Equation (33), ${\lambda }_{min}\left(\mathrm{\Theta }\right)>0$ is the smallest eigenvalue of the positive definite symmetric matrix $\mathrm{\Theta }$; it can be concluded that the appropriate $k_1$ value can be selected to ensure $\dot{V}<0$ and make the system stable.

2.2.4. The Overall Schematic Design of Cooperative Controller

The block diagram illustrating the trajectory tracking principle of the manipulator, which is controlled by the RBF neural network and the non-singular fast terminal sliding mode control, is illustrated in Figure 3.

Figure 3 primarily consists of two components: the global approximation term and the switching control term, both of which are based on the RBF neural network. The expected and actual joint angles, angular velocities, and angular accelerations of the manipulator serve as input signals for training the RBF neural network. Furthermore, the non-singular fast terminal sliding mode surface is employed to constrain the error dynamics and enhance system robustness.

3. Simulation and Experimental Validation

3.1. Simulation

To verify the effectiveness and feasibility of the proposed control algorithm, a six-degree-of-freedom collaborative robot is selected as the control object. The trajectory tracking control model is developed within the MATLAB R2021b/Simulink environment, utilizing Simscape in conjunction with an S-Function module. The arm model is constructed based on the URDF file of the manipulator. Although the simulations are conducted across all six joints of the manipulator, the trajectory tracking performance of the last three joints closely mirrors that of the first three. Therefore, only the trajectory tracking results for the first three joints are presented for analysis.

The simulation parameters are configured as follows: the expected trajectory for the six joints is set to $q{d}_1=q{d}_2=q{d}_3=q{d}_4=q{d}_5=q{d}_6=sin\left(t\right)$. The sliding surface parameters are set as follows: $v=-0.3;c=10;\mu =0.3;p=5;k=3$. The parameters for the neural network are established as follows: the hidden layer node node is set to 10; the radial basis function center value of neural network center = [−1.0, −0.7778, −0.5556, −0.3333, −0.1111, −0.1111, 0.3333, 0.5556, 0.7778, 1.0]; the width $\left(b\right)$ of the radial basis function is set to 10; the parameter $\varphi =50eye\left(6\right)$ in the weight adaptive rate of the neural network; the parameter setting in the switching control law is $k1=150eye\left(6\right);k2=150eye\left(6\right)$; and the simulation time t is set to 10 s.

The proposed control algorithm, which combines an RBF neural network with non-singular fast terminal sliding mode control, has been assessed through simulations and compared with two other methods for six-degree-of-freedom manipulator trajectory tracking: (1) an RBF neural network paired with traditional sliding mode control, as reported in [10], and (2) the NFTSM. The simulation results are illustrated in Figure 4, Figure 5 and Figure 6.

Figure 4, Figure 5 and Figure 6 illustrates the tracking performance of the first three joints of the manipulator using the three different control algorithms. Notably, the proposed method produces the smoothest response curves and demonstrates the highest consistency with the desired trajectories, indicating superior tracking accuracy. The tracking error for each joint is illustrated in Figure 7, Figure 8 and Figure 9.

As illustrated in Figure 7, Figure 8 and Figure 9, the trajectory tracking errors of the manipulator using the RBF neural network in conjunction with traditional sliding mode control and those under NFTSM exhibit considerable fluctuations. The proposed control method achieves a maximum tracking error of approximately 0.05 rad, compared to 0.1 rad for the RBFNN with traditional sliding mode control and 0.15 rad for the NFTSM. These results indicate that the proposed method offers superior tracking accuracy compared to both the approach presented in [10] and the NFTSM. Additionally, the average absolute errors of the first three joints are presented in

Table 2. Mean absolute errors for the first three joints.

Mean Absolute Angular Error	Joint 1/rad	Joint 2/rad	Joint 3/rad
RBFNN-NFTSM	0.004	0.03	0.004
RBFNN-SMC	0.02	0.04	0.014
NFTSM	0.034	0.05	0.03

.

As illustrated in the table, the maximum average absolute joint tracking error under the RBF neural network combined with traditional SMC is 0.04 rad, whereas NFTSM reaches 0.05 rad. In stark contrast, the proposed controller achieves a maximum average absolute tracking error of just 0.03 rad. These results indicate that the RBFNN-NFTSM control method offers superior trajectory tracking accuracy, underscoring its effectiveness in managing complex joint motion control tasks.

The expected joint angular velocity for each manipulator is set to ${\dot{q}}_d=cos\left(t\right)$, and the actual operating angular velocity effect of each joint is illustrated in Figure 10, Figure 11 and Figure 12.

As illustrated in Figure 10, Figure 11 and Figure 12, the manipulator controlled by the RBF neural network in conjunction with traditional sliding mode control displays an excessively high initial angular velocity, resulting in a notable deviation from the desired trajectory. While the NFTSM typically follows the desired angular velocity, it is plagued by noticeable fluctuations in the control output. In contrast, the proposed controller facilitates more accurate and smoother tracking, demonstrating superior dynamic performance.

The angular velocity error curve of each joint is presented in Figure 13, Figure 14 and Figure 15.

As illustrated in Figure 13, Figure 14 and Figure 15, the maximum angular velocity error for the first three joints reaches 0.5 $\mathrm{rad}/\phantom{rads}\mathrm{s}$ when using the RBF neural network in combination with traditional sliding mode control, and 0.18 $\mathrm{rad}/\phantom{rads}\mathrm{s}$ with NFTSM. In stark contrast, the proposed controller demonstrates a significantly reduced error of only 0.03 $\mathrm{rad}/\phantom{rads}\mathrm{s}$, further confirming its superior performance.

3.2. Experimental Validation

To validate the practical effectiveness of the proposed RBFNN-NFTSM control method, experimental tests are conducted on the six-degree-of-freedom robotic arm platform (JAKA Zu5) illustrated in Figure 16. The robotic arm is commanded to move from an initial joint configuration of ${\left[0,90,90,180,90,0\right]}^{\circ }$ to a target configuration of ${\left[90,45,30,60,36,-30\right]}^{\circ }$, with the motion trajectory planned using the fifth-degree polynomial interpolation method.

To visualize the tracking performance, the spatial coordinates of the robotic arm’s end-flange are selected as the tracking reference and compared with those obtained using the RBFNN-SMC and NFTSM algorithms. As illustrated in Figure 17, Figure 18 and Figure 19, the tracking results along the X, Y, and Z axes of the end-flange are presented.

As shown in Figure 17, Figure 18 and Figure 19, all three control methods are capable of tracking the desired trajectory; however, the proposed control method exhibits superior tracking performance compared to the other two approaches, thereby validating its practical effectiveness.

4. Conclusions

In this paper, domestic and foreign scholars solve the problem of poor trajectory tracking accuracy. An RBF neural network and a non-singular fast terminal sliding mode cooperative control method are proposed for systems with six degrees of freedom. The main contributions are as follows: (1) The proposed control method is used to complete the modeling of the manipulator and the overall approximation equivalent control law of the uncertain factors. Combined with the switching control law, the control input for each joint torque of the manipulator and the system’s stability are demonstrated. (2) A trajectory tracking simulation model is constructed for the manipulator using the Simscape and S-function modules within a MATLAB/Simulink environment. (3) The integration of RBFNN with NFTSM effectively approximates system uncertainties and external disturbances, mitigates chattering, and enhances control robustness. A comparative analysis is performed, showing that the proposed controller significantly improves trajectory tracking accuracy when juxtaposed with NFTSM algorithms and the combination of RBF neural networks with traditional sliding mode control. The simulation and experimental results demonstrate that the proposed controller significantly enhances trajectory tracking accuracy, offering a feasible and effective solution for the trajectory tracking control of the six-degree-of-freedom manipulator. (4) Although the proposed control method has been experimentally validated on a real robotic manipulator, the experiments were conducted primarily in a controlled laboratory environment and have not yet been tested in real-world industrial scenarios. In practical applications, more complex external factors may arise, each of which could potentially affect the robustness of the manipulator control. In future work, we plan to conduct further validations in environments that more closely resemble real-world operating conditions.