Adaptive neural network iterative learning control of long-stroke hybrid robots with initial errors and full state constraints

Abstract

Research on initial errors and constraint restrictions is one of the main research directions in the field of control of uncertain robotic systems. An adaptive iterative learning control (AILC) method based on radial basis function (RBF) neural network is proposed to address the trajectory tracking problem of the long-stroke hybrid robot system with random initial errors and full state constraints. The RBF neural network is used to approximate the unknown nonlinear terms, and the network weights are updated using an iterative learning law that incorporates a projection mechanism. Additionally, a robust learning strategy is used to compensate for both the approximation error of the neural network and the external disturbances that vary with each iteration. To relax the requirement of traditional iterative learning control (ILC) for identical initial condition, an equivalent error function is constructed based on the time-varying boundary layer. The tangent-type barrier Lyapunov function (BLF) is designed to ensure that the joint position and speed of the robot system are bounded within a predetermined range. Through stability analysis based on barrier composite energy function (BCEF), it can be proved that the boundedness of all signals in the closed-loop system and the tracking error of the robot system will converge to an adjustable residual set asymptotically. Finally, through simulation experiments conducted on the MATLAB platform, the results demonstrate that the method overcomes the random initial errors of the system effectively, ensures that the system satisfies the full-state constraints, and realizes high-precision trajectory tracking.

Introduction

With the development of robotics technology and the arrival of the intelligent manufacturing era, the application demand for robots in modern industry has become increasingly complex. Robots with variable structures have emerged.^1,2 A long-stroke hybrid robot is a special structural robot that combines translational and rotational joints, mainly used in the field of high-precision automatic machining and assembly. In actual production processes, robot manipulators are often used on assembly lines to perform tasks such as handling, painting, and injection molding. The action trajectories executed by the robot manipulators are all singular and repetitive.^3,4 With increasing of the system operation time and the impact from the external environment, the tracking error will increase gradually, and it will become more difficult to improve the control effect of the system.

Nowadays, the actual working environment of robot systems is becoming more complex, and there are many uncertain factors such as vibration, ⁵ friction, ⁶ load variation, ⁷ random disturbances, and unmodeled dynamics. ⁸ Those uncertain factors make it impossible to obtain a complete and accurate robot system model. In order to overcome adverse effects from the uncertainty and unknown interference, and also to improve their anti-interference ability, various control strategies have been developed and studied for robot systems, such as adaptive control, ⁹ fuzzy control,^10,11 neural network control,^12–14 and sliding mode control.^15–17 Adaptive control is suitable for systems with parameter uncertainty and can compensate for parameter uncertainty effectively. Sliding mode control is widely used in uncertain control systems due to its simple design and good robustness, but it often suffers from chattering. At the same time, various approximation tools are often used to address unknown nonlinear problems in systems, such as fuzzy logic system (FLS) and neural network (NN). Therefore, many scholars have combined these approximation tools with adaptive control to design controllers. Jiang et al. ¹⁸ employed a radial basis function neural network (RBFNN) to cope with inevitable dynamic uncertainties and utilized a new composite learning law to update the weights of the neural network. Sun et al. ¹⁹ used an adaptive NN method to approximate the unknown nonlinearity, while using a tangent-type BLF to satisfy the constraint. Ruan et al. ²⁰ used the error transformation method to convert constrained system variables into unconstrained variables, used the FLS to approximate the unknown nonlinear terms, and designed a fault-tolerant control scheme to ensure that the system satisfies the constraints. Overall, trajectory tracking control of uncertain robot systems remains a meaningful fundamental control problem.

Generally, when a robot system performs repetitive tasks, the reference trajectory is consistent within a fixed operating time. However, with the time increasing of system operation, the tracking error increases gradually. Iterative learning control (ILC) is an intelligent control method that can reduce the system error gradually during each operation, and can also improve the system tracking performance with the increasing of operation number. Therefore, the ILC strategy is a suitable choice for application in such robot systems. It is the core design idea of the ILC method that update and optimize the controller constantly during each iteration. The previous control input information and trajectory tracking error are used in the current controller to adjust and modify the controller, reduce the tracking error gradually, and achieve the expected trajectory tracking of the controlled object. The primary objectives of utilizing contraction mappings in traditional ILC theory are to enhance the stability, convergence, and robustness, and reduce oscillations of the control system. As the requirements for the control accuracy and speed of robot systems increase gradually, the traditional ILC method can no longer meet the application requirements. In recent years, some scholars have applied Lyapunov-like function theory to the design of ILC and proposed adaptive iterative learning control (AILC). AILC not only possesses the characteristics of improving control performance gradually found in traditional ILC but also incorporates the advantages of adaptive control. Liu et al. ²¹ used an adaptive controller to compensate for the uncertain parameters in the system and used a feedback ILC method to estimate the external disturbances. Bensidhoum and Bouakrif ²² designed an AILC scheme to address the problem of reduced stability in a robot system caused by varying disturbances and input dead zones. They used an adaptive iterative control law to estimate the optimal weights of the RBFNN. The AILC algorithms proposed above assume that the control object satisfies perfect reset condition. However, in engineering application, the actual mechanical structure will limit the accuracy of the physical resetting. When there is an initial bias, it is extremely challenging for a robot system to ensure that the system state remains strictly on the desired trajectory every time it executes a repetitive task. The existence of non-zero initial errors prevents the system from satisfying the requirement of having identical initial conditions in ILC. This requirement of identical initial condition has restricted the further development of ILC. Overcoming the impact of initial errors on the system has become a research focus for scholars both domestically and internationally. The existing methods for overcoming the effect of initial errors include initial rectification, ²³ time-varying boundary layers,^24,25 and error tracking strategies.^26,27 Zhang et al. ²³ constructed an initial rectification auxiliary reference signal to overcome obstacles caused by non-zero initial errors during ILC design. Chien et al. ²⁴ first proposed the method of time-varying boundary layers, and Guan et al. ²⁵ overcame the difficulty of non-zero initial errors of a pneumatic artificial muscle drive mechanism. Yan et al. ²⁶ constructed an expected error trajectory and designed a controller to drive the actual system error to track the constructed expected error trajectory gradually, thereby breaking through the requirement of the traditional ILC algorithm that the initial error must be zero. Hong et al. ²⁷ dealt with the random initial position of a tank gun system by using the error tracking strategy, and approximated the unknown nonparametric uncertainties of the tank gun system by using a FLS. Hence, further research is required to explore more effective and efficient methods for addressing the initial state problem in uncertain robot systems that perform repetitive tasks.

In practical application of robot system, due to the influence of the working environment and hardware performance, robots are limited by various conditions during operation, such as input constraints, ²⁸ output constraints,^29,30 and state constraints.^31,32 Failure to comply with these constraints may result in a decline in control performance, potential hardware damage, and even endanger the personal safety of operators. Hence, it is essential to take these constraints into account when designing controllers for robot systems. In recent years, an analysis method known as the barrier Lyapunov function (BLF) has been developed, and the barrier function is often employed to ensure that the objective function remains within the predetermined region consistently, thereby achieving the goal of constraining the objective variable. Yu et al. ³³ introduced a composite energy function ³⁴ for theoretical proof, demonstrating that the proposed AILC algorithm can confine system states and inputs within predefined limits. Min et al. ³⁵ designed a backstepping control method based on RBFNN and BLF, and Cruz-Ortiz et al. ³⁶ proposed a BLF for stability analysis of the designed robust finite time controller. Both their methods ensure that the system meets the full state constraints. During industrial production, the actual state variables change with time, and the time-varying restriction can be more complex and general than the previously mentioned restriction problems. To address the time-varying restriction in the robot system, Lu et al. ³⁷ used a time-varying BLF to ensure that the system state remains within a predetermined time-varying range. From the above analysis, research on achieving improved and faster control performance is a central focus of robotic control systems under constraints that ensure system safety.

So far, many scholars have conducted research on uncertain robotic systems with initial errors or constraint limitations. However, in the field of robotic control, there still exists a gap in research addressing uncertainty, random initial errors, and constraint limitations simultaneously. On the one hand, the practical working environment of long-stroke hybrid robots is complex, and the changing heavy loads may cause unpredictable deformations in the robotic manipulator. At the same time, there are uncertain factors such as vibration, friction, and unmodeled dynamics, which can also affect the accuracy of robotic manipulator reset, leading to a decrease in the stability of the long-stroke hybrid robot system. On the other hand, considering the safety and motion range restrictions during the operation of long-stroke hybrid robots, the positions and speeds of each joint of the robotic manipulator must be limited within specified boundaries. To address the issue of poor robustness in systems with both system uncertainty and random initial errors, and where the system needs to satisfy full-state constraints, this paper proposes a robust iterative control algorithm based on adaptive learning neural networks for long-stroke hybrid robot under full state constraints. The main contributions of this paper are as follows:

• (1) In response to system uncertainty and random external disturbances, an RBF neural network is used to approximate unknown nonlinear terms in uncertain robot systems. A robust term is designed to compensate for the approximation error of the neural network and random external disturbances. The network weights and the unknown upper bound of both the approximation error and random external disturbances are continuously updated according to the projection iterative learning law.

• (2) To address the non-zero and random initial errors of uncertain robot systems in each iteration, a time-varying boundary layer is constructed to counteract the random expansion error at the initial time. An equivalent error function is designed with an initial value of zero, and this equivalent error function is utilized as the main control variable for the iterative controller to relax identical initial conditions.

• (3) To address the simultaneous existence of system uncertainty and random initial errors, while also ensuring the system satisfies full-state constraints, a tangent-type BLF is introduced in the design of the robust iterative control algorithm. This BLF constrains the system states within predefined bounds, allowing for high-precision and robust trajectory tracking while ensuring the safe operation of the robotic system.

The rest of this paper is organized as follows: In Section “System description and preliminaries”, a dynamic model of the long-stroke hybrid robot system is established, the system is transformed, and basic assumptions are provided. In Section “Controller design and stability analysis”, an AILC scheme based on the RBF neural network is designed, a barrier composite energy function is constructed, and stability analysis is conducted on the robot system. In Section “Simulation results”, simulation experiments based on the MATLAB platform are presented. This paper is summarized in Section “Conclusion”.

System description and preliminaries

Robot system dynamics

A long-stroke hybrid robot is composed of a combination of translational and rotational joints, and Figure 1 shows its structural schematic diagram. We define the base coordinate system $\left\{O\right\}$ , the translational joint rigid robot manipulator coordinate system $\left\{P_r\right\}(r=1,2,3)$ , and the rotational joint rigid robotic manipulator coordinate system $\left\{R_j\right\}$ . The number of axes j of the rotational joint can be selected as two or three based on the actual application situation.

Figure 1.

Structure diagram of the long-stroke hybrid robot.

As shown in Table 1, after determining these coordinate systems, we can obtain the D-H parameter table of the long-stroke hybrid robot.

Table 1.

D-H parameter table of the long-stroke hybrid robot.

$r$	$a_{(r-1)}/m$	${\alpha }_{(r-1)}/\left(\mathrm{rad}\right)$	$d_r/m$	${\theta }_r/\left(\mathrm{rad}\right)$
1	0	$-\pi /2$	$d_1$	$\pi /2$
2	0	$\pi /2$	$d_2$	$-\pi /2$
3	0	$-\pi /2$	$d_3$	0
4	0	0	$d_4$	${\theta }_4(\pi /2)$
5	0	$\pi /2$	$d_5$	${\theta }_5\left(\pi \right)$
6	0	$\pi /2$	$d_6$	${\theta }_6\left(\pi \right)$

To facilitate the analysis and experimental comparison, we analyze and study a long-stroke hybrid robot composed of three-axis translational joints and three-axis rotational joints. The moving base, consisting of translational joints, and the multi-axis mechanical manipulator with rotational joints are treated as two subsystems. Considering the interaction forces between the rigid manipulators, their dynamic equations are expressed as:

$$\mathit{D}\left({\mathit{q}}_P\right){\stackrel{··}{\mathit{q}}}_P+\mathit{C}({\mathit{q}}_P,{\stackrel{·}{\mathit{q}}}_P){\stackrel{·}{\mathit{q}}}_P+\mathit{G}\left({\mathit{q}}_P\right)={\mathit{u}}_P+{\mathit{J}}_P^T{\mathit{f}}_R-{\mathit{o}}_P$$

(1)

$$\mathit{D}\left({\mathit{q}}_R\right){\stackrel{··}{\mathit{q}}}_R+\mathit{C}({\mathit{q}}_R,{\stackrel{·}{\mathit{q}}}_R){\stackrel{·}{\mathit{q}}}_R+\mathit{G}\left({\mathit{q}}_R\right)={\mathit{u}}_R+{\mathit{J}}_R^T{\mathit{f}}_P-{\mathit{o}}_R$$

(2)

Equation (1) describes a moving base subsystem composed of translational joints, and equation (2) describes a robot manipulator subsystem composed of rotational joints. ${\mathit{f}}_R\in {\mathit{R}}^m$ is the reaction force exerted by the rotating-joint robot manipulator on the translational-joint moving base. ${\mathit{f}}_P\in {\mathit{R}}^{3K\times 1}$ is the inertial force generated when the moving base moves with acceleration $a_x={\stackrel{··}{d}}_1,a_y={\stackrel{··}{d}}_2,a_z={\stackrel{··}{d}}_3$ . We assume that the mass of the rotating-joint robot manipulator is concentrated in a finite discrete particle, and according to the d′ Alembert principle, ${\mathit{f}}_P$ can be represented as follows:

$${\mathit{f}}_P=\left[\begin{array}{c}{\mathit{f}}_x\hfill \\ \multicolumn{1}{c}{{\mathit{f}}_u}\\ \multicolumn{1}{c}{{\mathit{f}}_z}\end{array}\right]=\left[\begin{array}{ccc}\overline{\mathit{m}}\hfill & \mathit{0}\hfill & \mathit{0}\hfill \\ \multicolumn{1}{c}{\mathit{0}}& \overline{\mathit{m}}\hfill & \mathit{0}\hfill \\ \multicolumn{1}{c}{\mathit{0}}& \mathit{0}\hfill & \overline{\mathit{m}}\hfill \end{array}\right]\left[\begin{array}{c}{a}_x\hfill \\ \multicolumn{1}{c}{a_y}\\ \multicolumn{1}{c}{a_z}\end{array}\right]$$

(3)

where $\overline{\mathit{m}}={[m_1,\cdots ,m_K]}^T$ , $0={[0,\cdots ,0]}^T$ is the K-dimension zero vector, $K$ is the number of finite discrete points, and $m_i$ is the discrete point in the i-th mass set. ${\mathit{f}}_R$ consists of two parts: one is the reaction force of the inertial force applied to the rotating-joint robot manipulator, and the other is the force required for the accelerating motion of the rotating joint robot manipulator, which can be expressed as:

$$\begin{array}{c}{\mathit{f}}_r=\left[\begin{array}{c}{\mathit{f}}_x\hfill \\ \multicolumn{1}{c}{{\mathit{f}}_y}\\ \multicolumn{1}{c}{{\mathit{f}}_z}\end{array}\right]=-\left[\begin{array}{c}\sum _{i=1}^K{m}_i{a}_{\mathrm{xi}}\hfill \\ \multicolumn{1}{c}{\sum _{i=1}^K{m}_i{a}_{\mathrm{yi}}}\\ \multicolumn{1}{c}{\sum _{i=1}^K{m}_i{a}_{\mathrm{zi}}}\end{array}\right]\hfill \\ +{\left[\begin{array}{ccc}\stackrel{K}{\overbrace{-\cos \theta ,\cdots ,-\cos \theta }}\hfill & \stackrel{K}{\overbrace{-\sin \theta ,\cdots ,-\sin \theta }}\hfill & \stackrel{K}{\overbrace{0,\cdots ,0}}\hfill \\ \multicolumn{1}{c}{-\sin \theta ,\cdots ,-\sin \theta }& -\cos \theta ,\cdots ,-\cos \theta \hfill & 0,\cdots ,0\hfill \\ \multicolumn{1}{c}{0,\cdots ,0}& 0,\cdots ,0\hfill & -1,\cdots ,-1\hfill \end{array}\right]}_{3\times 3K}{\mathit{f}}_p\hfill \end{array}$$

(4)

${\mathit{o}}_P$ and ${\mathit{o}}_R$ represent the external environmental disturbances of the two subsystems.

This paper primarily focuses on the subsystem of the rotating-joint robot manipulator of the long-stroke hybrid robot. To ensure the universality of the conclusions in this paper, a rotating-joint robot manipulator with n-DOF is considered to operate repeatedly in $t\in [0,T]$ , and its dynamic equation is expressed as:

$$\mathit{D}\left({\mathit{q}}_l\right){\stackrel{··}{\mathit{q}}}_l+\mathit{C}({\mathit{q}}_l,{\stackrel{·}{\mathit{q}}}_l){\stackrel{·}{\mathit{q}}}_l+\mathit{G}\left({\mathit{q}}_l\right)={\mathit{u}}_l+{\mathit{d}}_l$$

(5)

where $l$ represents the number of iterations, ${\mathit{q}}_l={[q_{1,l},q_{2,l},\cdots ,q_{n,l}]}^T\in {\mathit{R}}^n$ is the joint position vector at the l-th iteration, ${\stackrel{.}{\mathit{q}}}_l\in {\mathit{R}}^n$ is the joint velocity vector, ${\stackrel{··}{\mathit{q}}}_l\in {\mathit{R}}^n$ is the joint acceleration vector, $\mathit{D}\left({\mathit{q}}_l\right)\in {\mathit{R}}^{n\times n}$ is a known symmetric positive definite inertia matrix, $\mathit{C}\left({\mathit{q}}_l,{\stackrel{.}{\mathit{q}}}_l\right)\in {\mathit{R}}^{n\times n}$ is the unknown centripetal force and Coriolis force matrix, $\mathit{G}\left({\mathit{q}}_l\right)\in {\mathit{R}}^n$ is the gravity matrix, ${\mathit{u}}_l\in {\mathit{R}}^n$ is the input torque, and ${\mathit{d}}_l={\mathit{J}}_R^T{\mathit{f}}_P-{\mathit{o}}_R={\left[d_{1,l},d_{2,l},\cdots ,d_{n,l}\right]}^T\in {\mathit{R}}^n$ represents the unmodeled dynamic part of the system and the unknown external disturbances that vary with iteration.

• Property 1. The matrix $\mathit{D}\left({\mathit{q}}_l\right)$ and its inverse matrix ${\mathit{D}}^{-1}({\mathit{q}}_l)$ are symmetric and positive definite.

For simplicity, the time-independent variable $t$ of the function is omitted from the text.

System transformation and basic assumptions

We define ${\mathit{x}}_{1,l}={\mathit{q}}_l$ , ${\mathit{x}}_{2,l}={\stackrel{·}{\mathit{q}}}_l$ , and obtain the following equation through variable transformation:

$$\{\begin{array}{c}{\stackrel{.}{\mathit{x}}}_{1,l}={\mathit{x}}_{2,l}\\ {\stackrel{.}{\mathit{x}}}_{2,l}={\mathit{D}}^{-1}\left({\mathit{x}}_{1,l}\right)[{\mathit{u}}_l-\mathit{C}\left({\mathit{x}}_{1,l},{\mathit{x}}_{2,l}\right){\mathit{x}}_{1,l}-\mathit{G}\left({\mathit{x}}_{1,l}\right)+{\mathit{d}}_l]\end{array}$$

(6)

Equation (6) can be expanded to:

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{\stackrel{·}{\mathit{x}}}_{1,l}={\mathit{x}}_{2,l}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathit{x}}}_{2,l}={\mathit{D}}^{-1}\left({\mathit{x}}_{1,l}\right){\mathit{u}}_l+{\mathit{D}}^{-1}\left({\mathit{x}}_{1,l}\right)[-\mathit{C}({\mathit{x}}_{1,l},{\mathit{x}}_{2,l}){\mathit{x}}_{1,l}-\mathit{G}\left({\mathit{x}}_{1,l}\right)]}\\ \multicolumn{1}{c}{+{\mathit{D}}^{-1}\left({\mathit{x}}_{1,l}\right){\mathit{d}}_l}\end{array}}\end{array}$$

(7)

In this equation, $f({\mathit{x}}_{1,l},{\mathit{x}}_{2,l})={\mathit{D}}^{-1}\left({\mathit{x}}_{1,l}\right)[-\mathit{C}({\mathit{x}}_{1,l},{\mathit{x}}_{2,l}){\mathit{x}}_{1,l}-\mathit{G}\left({\mathit{x}}_{1,l}\right)]$ is defined as an unknown nonlinear term and is approximated using an RBF neural network.

The RBF neural network possesses excellent generalization ability and a simple network structure, consisting of three layers: input layer, hidden layer, and output layer. The neurons in the hidden layer of RBF neural network use activation functions, which have nonlinear mapping ability. This mapping enables the RBF neural network to approximate any nonlinear function within a compact set and with any accuracy. Consequently, in control system design, RBF neural networks are often used as function approximators for unknown functions. When integrating the RBF neural network into the controller, relying on the self-learning ability of the neural network, the neural network controller can improve the control performance of the uncertain systems and achieve the desired tracking accuracy.^38,39 The adaptive and learning capabilities of neural networks make them suitable for handling complex and uncertain control tasks.

The output $\hat{f}({\mathit{x}}_l,{\hat{\ell }}_l)$ of the neural network is an estimation of $f({\mathit{x}}_{1,l},{\mathit{x}}_{2,l})$ , which can be expressed as follows:

$$\hat{f}({\mathit{x}}_l,{\hat{\ell }}_l)=\sum _{j=1}^m{\hat{\ell }}_{j,l}{\phi }_j\left({\mathit{x}}_l\right)={\hat{\ell }}_l^T\mathit{\phi }\left({\mathit{x}}_l\right)$$

(8)

where ${\mathit{x}}_l={[{\mathit{x}}_{1,l},{\mathit{x}}_{2,l}]}^T$ is the input of the neural network, m is the number of hidden layer neurons, ${\hat{\ell }}_l={[{\hat{\ell }}_{1,l},{\hat{\ell }}_{2,l},\cdots ,{\hat{\ell }}_{m,l}]}^T$ is an estimation of the optimal weight of the neural network, and $\mathit{\phi }\left({\mathit{x}}_l\right)={[{\mathit{\phi }}_1\left({\mathit{x}}_l\right),{\mathit{\phi }}_2\left({\mathit{x}}_l\right),\cdots ,{\mathit{\phi }}_m\left({\mathit{x}}_l\right)]}^T$ is a Gaussian basis function vector.

There exists an optimal set of neural network weights $\ell$ such that the nonlinear function $f({\mathit{x}}_{1,l},{\mathit{x}}_{2,l})$ is represented as follows:

$$f({\mathit{x}}_{1,l},{\mathit{x}}_{2,l})={\ell }^T\mathit{\phi }\left({\mathit{x}}_l\right)+{\mathit{\epsilon }}_l$$

(9)

The approximation error ${\mathit{\epsilon }}_l$ satisfies $\left|{\mathit{\epsilon }}_l\right|\le {\mathit{\epsilon }}^{*}$ , where ${\mathit{\epsilon }}^{*}$ is the normal number. Then, the robot system (7) can be expressed as:

$$\{\begin{array}{c}{\stackrel{·}{\mathit{x}}}_{1,l}={\mathit{x}}_{2,l}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\mathit{x}}}_{2,l}={\mathit{D}}_l^{-1}{\mathit{u}}_l+{\ell }^T\mathit{\phi }\left({\mathit{x}}_l\right)+{\mathit{D}}_l^{-1}{\mathit{d}}_l+{\mathit{\epsilon }}_l}\end{array}$$

(10)

where ${\mathit{x}}_{1,l}={[x_{11,l},x_{12,l},\cdots ,x_{1n,l}]}^T$ , ${\mathit{x}}_{2,l}={[x_{21,l},x_{22,l},\cdots ,x_{2n,l}]}^T$ , and n is the number of joints.

• Assumption 1. The expected state vectors ${\mathit{x}}_{d1}={[x_{d11},x_{d12},\cdots ,x_{d1n}]}^T$ and ${\mathit{x}}_{d2}={[x_{d21},x_{d22},\cdots ,x_{d2n}]}^T$ are bounded; that is, $\left|x_{d1r}\right|\le b_{d1}$ and $\left|x_{d2r}\right|\le b_{d2},r=1,2,\cdots ,n$ , where $b_{d1}$ and $b_{d2}$ are the known normal numbers.

• Assumption 2. There are random bounded initial state errors ${\mathit{e}}_{1,l}\left(0\right)$ and ${\mathit{e}}_{2,l}\left(0\right)$ , at any iteration, $\left|e_{1r,l}\left(0\right)\right|=|x_{d1r}\left(0\right)-x_{1r,l}\left(0\right)|\le b_{1r}$ , and $\left|e_{2r,l}\left(0\right)\right|=|x_{d2r}\left(0\right)-x_{2r,l}\left(0\right)|\le b_{2r},r=1,2,\cdots ,n$ , where $b_{1r}$ and $b_{2r}$ are the known normal numbers.

• Assumption 3. The unmodeled dynamic part of the system and the unknown external interference $d_l$ that varies with iteration are also bounded.

For the rotating-joint manipulator subsystem described in (5), this paper designs an adaptive iterative learning controller ${\mathit{u}}_l$ . As the system states are constrained within $\left|x_{1r,l}\right|\le x_{1r,max}$ and $\left|x_{2r,l}\right|\le x_{2r,max}$ , where $x_{1r,max}$ and $x_{2r,max}$ represent the upper bounds of states $x_{1r,l}$ and $x_{2r,l}$ , with increasing iterations, the control algorithm guarantees that the tracking error of the system converges to an adjustable boundary, while ensuring that all variables of the closed-loop system remain uniformly bounded.

Controller design and stability analysis

Design of an iterative learning controller based on the RBF neural network

First, we define the state tracking error ${\mathit{e}}_{1,l}={\mathit{x}}_{d1}-{\mathit{x}}_{1,l}$ , ${\mathit{e}}_{2,l}={\mathit{x}}_{d2}-{\mathit{x}}_{2,l}$ .

Then, we can define the extension error ${\mathit{s}}_l$ :

$${\mathit{s}}_l=\lambda {\mathit{e}}_{1,l}+{\mathit{e}}_{2,l}={[s_{1,l},s_{2,l},\cdots ,s_{n,l}]}^T$$

(11)

where $\lambda$ is a normal number.

If ${\mathit{s}}_l$ can approach zero in runtime $[0,T]$ , then ${\mathit{e}}_{1,l}$ and ${\mathit{e}}_{2,l}$ will converge to zero. However, due to the random and non-zero initial state tracking errors, $s_{r,l}\left(0\right)\ne 0,r=1,2,\cdots ,n$ . To address the impact of initial errors, an equivalent error function ${\mathit{s}}_l^{\psi }$ is defined:

$${\mathit{s}}_l^{\psi }={\mathit{s}}_l-{\mathit{\psi }}_l\mathrm{sat}\left(\frac{{\mathit{s}}_l}{{\psi }_l}\right)={[s_{1,l}^{\psi },s_{2,l}^{\psi },\cdots ,s_{n,l}^{\psi }]}^T$$

(12)

The following equation holds for the design of time-varying boundary layer ${\mathit{\psi }}_l$ :

$${\psi }_{r,l}=\left|s_{r,l}\left(0\right)\right|e^{-\mu t}$$

(13)

$${\stackrel{·}{\psi }}_{r,l}+\mu {\psi }_{r,l}=0$$

(14)

$$\left|s_{r,l}\left(0\right)\right|=\lambda e_{1r,l}\left(0\right)+e_{2r,l}\left(0\right)\le \lambda b_{1r}+b_{2r}=b_{\delta r}$$

(15)

where $\mu$ is a known normal number.

From equation (13), it can be inferred that the value of time-varying function ${\mathit{\psi }}_l$ will gradually decrease to overcome the random error generated at the initial time of each iteration. According to the definition of the equivalent error function, $s_{r,l}^{\psi }\left(0\right)=0$ satisfies the identical initial condition required for ILC. Therefore, the equivalent error function ${\mathit{s}}_l^{\psi }$ serves as the primary control variable of the controller in this paper.

In equation (12), $\mathrm{sat}(·)$ is the saturation function, defined as follows:

$$\mathrm{sat}\left(\frac{s_{r,l}}{{\psi }_{r,l}}\right)=\{\begin{array}{c}1s_{r,l}>{\psi }_{r,l}\hfill \\ \frac{s_{r,l}}{{\psi }_{r,l}}\left|s_{r,l}\right|\le {\psi }_{r,l}\hfill \\ -1s_{r,l}<-{\psi }_{r,l}\hfill \end{array}$$

(16)

To facilitate stability and convergence analysis, the following properties of ${\mathit{s}}_l^{\psi }$ are useful:

$$s_{r,l}^{\psi }\left(0\right)=0$$

(17)

$$S_{r,l}^{\psi }\mathrm{sat}\left(\frac{s_{r,l}}{{\psi }_{r,l}}\right)=\left|s_{r,l}^{\psi }\right|=\{\begin{array}{c}{s}_{r,l}-{\psi }_{r,l}{s}_{r,l}>{\psi }_{r,l}\hfill \\ 0\left|s_{r,l}\right|\le {\psi }_{r,l}\hfill \\ -(s_{r,l}+{\psi }_{r,l})s_{r,l}<-{\psi }_{r,l}\hfill \end{array}$$

(18)

According to the variables defined above, the controller ${\mathit{u}}_l$ is designed to make the equivalent error function ${\mathit{s}}_l^{\psi }$ converge to zero on $[0,T]$ .

$${\mathit{u}}_l=\mathit{D}\left({\mathit{x}}_{1,l}\right)[\mu {\mathit{s}}_l+\lambda \left|{\mathit{e}}_{2,l}\right|\mathrm{sat}\left(\frac{{\mathit{s}}_l}{{\psi }_l}\right)-{\hat{\ell }}_l^T\mathit{\phi }\left({\mathit{x}}_l\right)+{\hat{\mathbf{\zeta }}}_l\mathbf{sat}\left(\frac{{\mathit{s}}_l}{{\mathit{\psi }}_l}\right)]$$

(19)

where ${\hat{\ell }}_l^T\mathit{\phi }({\mathit{x}}_l)$ is used to estimate the nonlinear function $f({\mathit{x}}_{1,l},{\mathit{x}}_{2,l})$ in the system. ${\mathbf{\zeta }}_l$ is the unknown upper bound of both the random external interference and the RBF neural network approximation error, which satisfies the inequality $|{\stackrel{·}{\mathit{x}}}_{d2,l}-{\mathit{D}}^{-1}({\mathit{x}}_{1,l}){\mathit{d}}_l-{\mathit{\epsilon }}_l|\le {\mathbf{\zeta }}_l$ , and ${\hat{\zeta }}_l$ is the estimation of ${\mathbf{\zeta }}_l$ .

For unknown neural network weights ${\ell }_{\mathrm{rj},l},r=1,2,\cdots ,n,j=1,2,\cdots ,m$ , and unknown upper bound ${\zeta }_{\mathrm{rl}},r=1,2,\cdots ,n$ , the following parameter learning laws are designed:

$${\hat{\ell }}_{r,l}^{*}={\hat{\ell }}_{r,l-1}-\mathbf{\Gamma }\mathit{\phi }\left({\mathit{x}}_l\right){\mathit{\vartheta }}_{r,l}$$

(20)

$${\hat{\ell }}_{r,l}=\mathrm{pro}{\mathrm{j}}_{\ell }\left({\hat{\ell }}_{r,l}^{*}\right),{\hat{\ell }}_{r,-1}=0$$

(21)

$${\hat{\zeta }}_{r,l}^{*}={\hat{\zeta }}_{r,l-1}+\upsilon \left|{\vartheta }_{r,l}\right|$$

(22)

$${\hat{\zeta }}_{r,l}=\mathrm{pro}\mathrm{j}\zeta \left({\hat{\zeta }}_{r,l}^{*}\right),{\hat{\zeta }}_{r,-1}=0$$

(23)

where ${\vartheta }_{r,l}=\frac{s_{r,l}^{\psi }}{{\cos }^2\left(\frac{\pi {s_{r,l}^{\psi }}^2}{2k_{r,v}^2}\right)},r=1,2,\cdots ,n$ , $k_{r,v}>0$ is a constraint on the amplitude of $s_{r,l}^{\psi }$ , $\mathbf{\Gamma }\in {\mathit{R}}^{n\times n}$ is a symmetric positive definite parameter gain matrix, and $\upsilon >0$ is the learning gain.

To ensure that the estimated values of the parameters are always within the appropriate range, a projection mechanism is introduced as follows:

$$\begin{array}{l}{\mathrm{proj}}_{\ell }\left({\widehat{\ell }}_{r,l}^{\ast }\right)=\left\{{\mathrm{proj}}_{\ell }\left({\widehat{\ell }}_{rj,l}^{\ast }\right)\right\}\left(j=1,2,\cdots ,m\right)\\ {\mathrm{proj}}_{\ell }\left({\widehat{\ell }}_{rj,l}^{\ast }\right)=\{\begin{array}{cc}{\ell }_{rj,\min }& {\widehat{\ell }}_{rj,l}^{\ast }<{\ell }_{rj,\min }\\ {\widehat{\ell }}_{rj,l}^{\ast }& {\ell }_{rj,\min }\le {\widehat{\ell }}_{rj,l}^{\ast }\le {\ell }_{rj,\max }\\ {\ell }_{rj,\max }& {\widehat{\ell }}_{rj,l}^{\ast }>{\ell }_{rj,\max }\end{array}\end{array}$$

(24)

$$\mathrm{pro}{\mathrm{j}}_{\zeta }\left({\hat{\zeta }}_{r,l}^{*}\right)=\{\begin{array}{cc}{\zeta }_{r,min}\hfill & {\hat{\zeta }}_{r,l}^{*}<{\zeta }_{r,min}\hfill \\ \multicolumn{1}{c}{{\hat{\zeta }}_{r,l}^{*}}& {\zeta }_{r,min}\le {\hat{\zeta }}_{r,l}^{*}\le {\zeta }_{r,max}\hfill \\ \multicolumn{1}{c}{{\zeta }_{r,max}}& {\hat{\zeta }}_{r,l}^{*}>{\zeta }_{r,max}\hfill \end{array}$$

(25)

where ${\ell }_{\mathrm{rj},min}$ and ${\ell }_{\mathrm{rj},max}$ are the lower and upper bounds of parameter estimation ${\hat{\ell }}_{\mathrm{rj},l}$ , and ${\zeta }_{r,min}$ and ${\zeta }_{r,max}$ are the lower and upper bounds of ${\hat{\zeta }}_{r,l}$ .

The robot system diagram using the proposed method is shown in Figure 2. In the diagram, the feedback term, neural network approximation term, and robust compensation term together constitute the proposed AILC (18). The parameter learning law is implemented by (20)–(23).

• Remark 1. We need to choose appropriate control parameters to achieve excellent control performance. $\lambda$ is a positive constant. If it is too small or too large, it can result in a slower convergence speed and lower accuracy of the system. $\mu$ is a positive constant. If it is too small, it can affect the convergence speed, and if it is too large, it can lead to a decrease in the control performance of joint speed. $\mathbf{\Gamma }\in {\mathit{R}}^{n\times n}$ is a symmetric positive definite parameter gain matrix. If it is too small, it can lead to a decrease in the tracking accuracy of the system; if it is too large, it can lead to a decrease in the control performance of joint speed, along with the occurrence of error spikes. $\upsilon$ is the learning gain, and being too small or too large can both lead to a decrease in the control performance of joint speed.

Figure 2.

Block diagram of robot system using the proposed method.

Stability analysis

For the convenience of analysis and proof, the lemma that will be used in this paper is given below.

• Lemma 1. For $\forall \ell \in {[{\ell }_{min},{\ell }_{max}]}^T$ and $\zeta \in \{{\zeta }_{min},{\zeta }_{max}\}$ , we have:

$${[\ell -\mathrm{pro}{\mathrm{j}}_{\ell }\left({\hat{\ell }}_{r,l}^{*}\right)]}^T{\mathbf{\Gamma }}^{-1}[{\hat{\ell }}_{r,l}^{*}-\mathrm{pro}{\mathrm{j}}_{\ell }\left({\hat{\ell }}_{r,l}^{*}\right)]\le 0$$

(26)

$$(\zeta -\mathrm{pro}{\mathrm{j}}_{\zeta }\left({\hat{\zeta }}_l^{*}\right))({\hat{\zeta }}_l^{*}-\mathrm{pro}{\mathrm{j}}_{\zeta }\left({\hat{\zeta }}_l^{*}\right))\le 0$$

(27)

The theoretical summary of the system is shown in Theorem 1.

• Theorem 1. For the rotational joint subsystem (5) of a long-stroke hybrid robot that satisfies the assumption, under the action of control law (19) and parameter learning laws (20)–(23), we have the following:

• (T1) The equivalent error function ${\mathit{s}}_l^{\psi }$ is bounded and gradually converges to zero along the iteration axis, and $\underset{l\to \infty }{lim}{s}_{r,l}^{\psi }\left(t\right)=0$ , $\forall t\in [0,T]$ .

• (T2) The extension error ${\mathit{s}}_l$ is bounded and $\underset{l\to \infty }{lim}\left|s_{r,l}\left(t\right)\right|\le b_{\delta r}$ , $\forall t\in [0,T]$ .

• (T3) The error vector ${\mathit{e}}_l$ is bounded, and $\underset{l\to \infty }{lim}\left|e_{1r,l}\left(t\right)\right|\le \frac{b_{\delta r}}{\lambda }+b_{1r}$ , $\underset{l\to \infty }{lim}\left|e_{2r,l}\left(t\right)\right|\le 2b_{\delta r}+\lambda b_{1r}$ , $\forall t\in [0,T]$ .

• (T4) The state vector ${\mathit{x}}_l$ is bounded and satisfies the constraints $\left|x_{1r,l}\right|\le x_{1r,max}$ and $\left|x_{2r,l}\right|\le x_{2r,max}$ , $\forall t\in [0,T]$ .

• Proof. First, we construct a positive definite BCEF for the l-th iteration:

$$\begin{array}{c}\multicolumn{1}{c}{E_l=\sum _{r=1}^n\frac{k_{r,v}^2}{\pi }\tan \left(\frac{\pi {\left(s_{r,l}^{\psi }\right)}^2}{2k_{r,v}^2}\right)+\frac{1}{2}\sum _{r=1}^n{\int }_0^t{\tilde{\ell }}_{r,l}^T{\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,l}d\tau +\frac{1}{2\upsilon }\sum _{r=1}^n{\int }_0^t{\tilde{\zeta }}_{r,l}^{2}d\tau }\\ \stackrel{\mathrm{\Delta }}{=}{T}_l+W_l+L_l\hfill \end{array}$$

(28)

where $T_l=\sum _{r=1}^n\frac{k_{r,v}^2}{\pi }\tan \left(\frac{\pi {\left(s_{r,l}^{\psi }\right)}^2}{2k_{r,v}^2}\right)$ is the tangent-type barrier Lyapunov function used to analyze error convergence that satisfies the state constraints. In Remark 1, we will explain how the BLF constrains the system states. As long as the equivalent initial errors $\left|s_{r,l}^{\psi }\left(0\right)\right|<k_{r,v}$ is present, $T_l$ is non-negative. $\tan (·)$ is the tangent function, ${\tilde{\ell }}_{r,l}=\ell -{\hat{\ell }}_{r,l}$ , ${\tilde{\zeta }}_{r,l}=\zeta -{\hat{\zeta }}_{r,l}$ , $W_l=\frac{1}{2}\sum _{r=1}^n{\int }_0^t{\tilde{\ell }}_{r,l}^T{\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,l}\mathrm{d}\tau$ , and $L_l=\frac{1}{2\upsilon }\sum _{r=1}^n{\int }_0^t{\tilde{\zeta }}_{r,l}^2\mathrm{d}\tau$ .

The proof of Theorem (T1) is as follows:

We will prove Theorem (T1) in three steps. The first step proves the negative definition of the BCEF, the second step proves the boundedness of the BCEF, and the third step proves the boundedness and convergence of ${\mathit{s}}_l^{\psi }$ .

• Step 1. The negative definition of the BCEF.

Considering a difference in $E_l$ yields:

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }{E}_l=E_l-E_{l-1}}\\ =T_l-T_{l-1}+W_l-W_{l-1}+L_l-L_{l-1}\hfill \end{array}$$

(29)

According to equations (11) and (12), $T_l$ can be expressed as:

$$\begin{array}{c}\multicolumn{1}{c}{T_l=\sum _{r=1}^n{\int }_0^t\frac{s_{r,l}^{\psi }}{{\cos }^2\left(\frac{\pi {\left(s_{r,l}^{\psi }\right)}^2}{2k_{r,v}^2}\right)}{\stackrel{·}{s}}_{r,l}^{\psi }\mathrm{d}\tau =\sum _{r=1}^n{\int }_0^t{\vartheta }_{r,l}({\stackrel{·}{s}}_{r,l}-\mathrm{sgn}\left(s_{r,l}^{\psi }\right){\stackrel{·}{\psi }}_{r,l})\mathrm{d}\tau }\\ =\sum _{r=1}^n{\int }_0^t({\vartheta }_{r,l}{\stackrel{·}{s}}_{r,l}-\left|{\vartheta }_{r,l}\right|{\stackrel{·}{\psi }}_{r,l})\mathrm{d}\tau =\sum _{r=1}^n{\int }_0^t[{\vartheta }_{r,l}(\lambda e_{2r,l}+{\stackrel{·}{e}}_{2r,l})-\left|{\vartheta }_{r,l}\right|{\stackrel{·}{\psi }}_{r,l}]\mathrm{d}\tau \hfill \end{array}$$

(30)

Using the system model (10), equation (30) can be simplified as:

$$\begin{array}{c}\multicolumn{1}{c}{T_l=\sum _{r=1}^n{\int }_0^t[{\vartheta }_{r,l}(\lambda e_{2r,l}+{\stackrel{·}{x}}_{d2r}-{\stackrel{·}{x}}_{2r,l})-\left|{\vartheta }_{r,l}\right|{\stackrel{·}{\psi }}_{r,l}]\mathrm{d}\tau }\\ =\sum _{r=1}^n{\int }_0^t[{\vartheta }_{r,l}(\lambda e_{2r,l}+{\stackrel{·}{x}}_{d2r}-{\ell }_{r,l}^T\mathit{\phi }\left({\mathit{x}}_l\right)-{\mathit{D}}^{-1}\left({\mathit{x}}_{1r,l}\right){\mathit{u}}_l-{\mathit{D}}^{-1}\left({\mathit{x}}_{1r,l}\right){\mathit{d}}_l-{\mathit{\epsilon }}_{r,l})-\left|{\vartheta }_{r,l}\right|{\stackrel{·}{\psi }}_{r,l}]\mathrm{d}\tau \hfill \end{array}$$

(31)

Substituting the control law (19) into equation (31), we obtain:

$$\begin{array}{c}\multicolumn{1}{c}{T_l=\sum _{r=1}^n{\int }_0^t[{\vartheta }_{r,l}(\lambda e_{2r,l}+{\stackrel{·}{x}}_{d2r}-{\ell }_{r,l}^T\mathit{\phi }\left({\mathit{x}}_l\right)-\mathbf{\mu }{\mathbf{s}}_{r,l}-\mathbf{\lambda }\left|{\mathbf{e}}_{2r,l}\right|\mathrm{sat}\left(\frac{{\mathbf{s}}_{r,l}}{{\mathit{\psi }}_{r,l}}\right)+{\hat{\mathbf{\ell }}}_{r,l}^T\mathit{\phi }\left({\mathit{x}}_l\right)}\\ -{\hat{\zeta }}_{r,l}\mathrm{sat}\left(\frac{s_{r,l}}{{\psi }_{r,l}}\right)-{\mathit{D}}^{-1}\left(x_{1r,l}\right){\mathit{d}}_l-{\epsilon }_{r,l})-\left|{\vartheta }_{r,l}\right|{\stackrel{·}{\psi }}_{r,l}]\mathrm{d}\tau \hfill \end{array}$$

(32)

Since ${\vartheta }_{r,l}\mathrm{sat}\left(\frac{s_{r,l}}{{\psi }_{r,l}}\right)=\left|{\vartheta }_{r,l}\right|$ , $\lambda {\vartheta }_{r,l}{e}_{2r,l}\le \lambda \left|{\vartheta }_{r,l}\right|\left|e_{2r,l}\right|$ , and using equation (12), equation (32) can be simplified as:

$$\begin{array}{l}{T}_l=\sum _{r=1}^n{\int }_0^t\lambda {\vartheta }_{r,l}{e}_{2r,l}-\lambda [{\vartheta }_{r,l}\left|e_{2r,l}\right|\mathrm{sat}\left(\frac{s_{r,l}}{{\psi }_{r,l}}\right)-{\vartheta }_{r,l}{\tilde{\ell }}_{r,l}^T\mathit{\phi }\left({\mathit{x}}_l\right)+{\vartheta }_{r,l}{\tilde{\zeta }}_{r,l}-\mu s_{r,l}{\vartheta }_{r,l}-\left|{\vartheta }_{r,l}\right|{\stackrel{.}{\psi }}_{r,l}]\mathrm{d}\tau \\ \le \sum _{r=1}^n{\int }_0^t\left[-{\vartheta }_{r,l}{\tilde{\ell }}_{r,l}^T\mathit{\phi }\left({\mathit{x}}_l\right)+{\vartheta }_{r,l}{\tilde{\zeta }}_{r,l}-\mu s_{r,l}{\vartheta }_{r,l}-\left|{\vartheta }_{r,l}\right|{\stackrel{.}{\psi }}_{r,l}\right]\mathrm{d}\tau \end{array}$$

(33)

Since $-\mu {\vartheta }_{r,l}{s}_{r,l}^{\psi }\le 0$ , using equations (12) and (14), equation (33) can be simplified as:

$$\begin{array}{c}\multicolumn{1}{c}{T_l\le \sum _{r=1}^n{\int }_0^t[-{\vartheta }_{r,l}{\tilde{\ell }}_{r,l}^T\mathit{\phi }\left({\mathit{x}}_l\right)+{\mathit{\vartheta }}_{r,l}{\tilde{\mathbf{\zeta }}}_{r,l}-\mathbf{\mu }{\mathit{\vartheta }}_{r,l}{\mathbf{s}}_{r,l}^{\mathit{\psi }}-\mathbf{\mu }{\mathit{\vartheta }}_{r,l}{\mathit{\psi }}_{r,l}\mathrm{sat}\left(\frac{{\mathbf{s}}_{r,l}}{{\mathit{\psi }}_{r,l}}\right)-\left|{\mathit{\vartheta }}_{r,l}\right|{\stackrel{·}{\mathit{\psi }}}_{r,l}]\mathrm{d}\tau }\\ \le \sum _{r=1}^n{\int }_0^t[-{\vartheta }_{r,l}{\tilde{\ell }}_{r,l}^T\mathit{\phi }\left({\mathit{x}}_l\right)+\left|{\mathit{\vartheta }}_{r,l}\right|{\tilde{\mathbf{\zeta }}}_{r,l}]\mathrm{d}\tau \hfill \end{array}$$

(34)

Considering a difference in $W_l$ yields:

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }{W}_l=W_l-W_{l-1}}\\ =\frac{1}{2}\sum _{r=1}^n{\int }_0^t[{\tilde{\ell }}_{r,l}^T{\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,l}-{\tilde{\ell }}_{r,l-1}^T{\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,l-1}]\mathrm{d}\tau \hfill \\ =-\frac{1}{2}\sum _{r=1}^n{\int }_0^t{(2\ell -{\hat{\ell }}_{r,l}-{\hat{\ell }}_{r,l-1})}^T{\mathbf{\Gamma }}^{-1}({\hat{\ell }}_{r,l}-{\hat{\ell }}_{r,l-1})\mathrm{d}\tau \hfill \\ =-\sum _{r=1}^n{\int }_0^t{(\ell -{\hat{\ell }}_{r,l})}^T{\mathbf{\Gamma }}^{-1}({\hat{\ell }}_{r,l}-{\hat{\ell }}_{r,l-1})\mathrm{d}\tau -\frac{1}{2}\sum _{r=1}^n{\int }_0^t{({\hat{\ell }}_{r,l}-{\hat{\ell }}_{r,l-1})}^T{\mathbf{\Gamma }}^{-1}({\hat{\ell }}_{r,l}-{\hat{\ell }}_{r,l-1})\mathrm{d}\tau \hfill \end{array}$$

(35)

According to the parameter learning laws (20) and (21), equation (35) can be simplified as:

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }{W}_l=-\sum _{r=1}^n{\int }_0^t{(\ell -\mathrm{pro}{\mathrm{j}}_{\ell }\left({\hat{\ell }}_{r,l}^{*}\right))}^T{\mathbf{\Gamma }}^{-1}(\mathrm{pro}{\mathrm{j}}_{\ell }\left({\hat{\ell }}_{r,l}^{*}\right)-{\hat{\ell }}_{r,l}^{*}-\mathbf{\Gamma }\mathit{\phi }\left({\mathit{x}}_l\right){\mathit{\vartheta }}_{r,l})\mathrm{d}\tau }\\ -\frac{1}{2}\sum _{r=1}^n{\int }_0^t{({\hat{\ell }}_{r,l}-{\hat{\ell }}_{r,l-1})}^T{\mathbf{\Gamma }}^{-1}({\hat{\ell }}_{r,l}-{\hat{\ell }}_{r,l-1})\mathrm{d}\tau \hfill \\ \le \sum _{r=1}^n{\int }_0^t{(\ell -\mathrm{pro}{\mathrm{j}}_{\ell }\left({\hat{\ell }}_{r,l}^{*}\right))}^T{\mathbf{\Gamma }}^{-1}({\hat{\ell }}_{r,l}^{*}-\mathrm{pro}{\mathrm{j}}_{\ell }\left({\hat{\ell }}_{r,l}^{*}\right))\mathrm{d}\tau +\sum _{r=1}^n{\int }_0^t{(\ell -{\hat{\ell }}_{r,l})}^T\mathit{\phi }\left({\mathit{x}}_l\right){\mathit{\vartheta }}_{r,l}\mathrm{d}\mathbf{\tau }\hfill \\ -\frac{1}{2}\sum _{r=1}^n{\int }_0^t{({\hat{\ell }}_{r,l}-{\hat{\ell }}_{r,l-1})}^T{\mathbf{\Gamma }}^{-1}({\hat{\ell }}_{r,l}-{\hat{\ell }}_{r,l-1})\mathrm{d}\tau \hfill \end{array}$$

(36)

According to equation (26) in Lemma 1, equation (36) can be simplified as:

$$\mathrm{\Delta }{W}_l\le \sum _{r=1}^n{\int }_0^t{\tilde{\ell }}_{r,l}^T\mathit{\phi }({\mathit{x}}_l){\mathit{\vartheta }}_{r,l}\mathrm{d}\mathbf{\tau }-\frac{1}{2}\sum _{r=1}^n{\int }_0^t{({\hat{\ell }}_{r,l}-{\hat{\ell }}_{r,l-1})}^T{\mathbf{\Gamma }}^{-1}({\hat{\ell }}_{r,l}-{\hat{\ell }}_{r,l-1})\mathrm{d}\tau$$

(37)

Considering a difference in $L_l$ yields:

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }{L}_l=L_l-L_{l-1}=\frac{1}{2\upsilon }\sum _{r=1}^n{\int }_0^t({\tilde{\zeta }}_{r,l}^2-{\tilde{\zeta }}_{r,l-1}^2)\mathrm{d}\tau }\\ =-\frac{1}{\upsilon }\sum _{r=1}^n{\int }_0^t(\zeta -{\hat{\zeta }}_{r,l})({\hat{\zeta }}_{r,l}-{\hat{\zeta }}_{r,l-1})\mathrm{d}\tau -\frac{1}{2\upsilon }\sum _{r=1}^n{\int }_0^t{({\hat{\zeta }}_{r,l}-{\hat{\zeta }}_{r,l-1})}^2\mathrm{d}\tau \hfill \end{array}$$

(38)

According to the parameter learning laws (22) and (23), equation (38) can be simplified as:

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }{L}_l=-\frac{1}{\upsilon }\sum _{r=1}^n{\int }_0^t(\zeta -\mathrm{pro}{\mathrm{j}}_{\zeta }\left({\hat{\zeta }}_{r,l}^{*}\right))(\mathrm{pro}{\mathrm{j}}_{\zeta }\left({\hat{\zeta }}_{r,l}^{*}\right)-{\hat{\zeta }}_{r,l}^{*}+\upsilon \left|{\vartheta }_{r,l}\right|)\mathrm{d}\tau -\frac{1}{2\upsilon }\sum _{r=1}^n{\int }_0^t{({\hat{\zeta }}_{r,l}-{\hat{\zeta }}_{r,l-1})}^2\mathrm{d}\tau }\\ =\frac{1}{\upsilon }\sum _{r=1}^n{\int }_0^t(\zeta -\mathrm{pro}{\mathrm{j}}_{\zeta }\left({\hat{\zeta }}_{r,l}^{*}\right))({\hat{\zeta }}_{r,l}^{*}-\mathrm{pro}{\mathrm{j}}_{\zeta }\left({\hat{\zeta }}_{r,l}^{*}\right))\mathrm{d}\tau -\sum _{r=1}^n{\int }_0^t{\tilde{\zeta }}_{r,l}\left|{\vartheta }_{r,l}\right|\mathrm{d}\tau -\frac{1}{2\upsilon }\sum _{r=1}^n{\int }_0^t{({\hat{\zeta }}_{r,l}-{\hat{\zeta }}_{r,l-1})}^2\mathrm{d}\tau \hfill \end{array}$$

(39)

According to equation (27) in Lemma 1, equation (39) can be simplified as:

$$\mathrm{\Delta }{L}_l\le -\sum _{r=1}^n{\int }_0^t{\tilde{\zeta }}_{r,l}\left|{\vartheta }_{r,l}\right|\mathrm{d}\tau -\frac{1}{2\upsilon }\sum _{r=1}^n{\int }_0^t{({\hat{\zeta }}_{r,l}-{\hat{\zeta }}_{r,l-1})}^2\mathrm{d}\tau$$

(40)

Substituting equations (34), (37), and (40) into equation (29) yields:

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }{E}_l\le -T_{l-1}-\frac{1}{2}\sum _{r=1}^n{\int }_0^t{({\hat{\ell }}_l-{\hat{\ell }}_{l-1})}^T{\mathbf{\Gamma }}^{-1}({\hat{\ell }}_l-{\hat{\ell }}_{l-1})\mathrm{d}\tau -\frac{1}{2\upsilon }\sum _{r=1}^n{\int }_0^t{({\hat{\zeta }}_l-{\hat{\zeta }}_{l-1})}^2\mathrm{d}\tau }\\ \le -T_{l-1}\le 0\hfill \end{array}$$

(41)

Adding equation (41) from 1 to $l$ yields:

$$E_l\le E_0-\sum _{j=1}^l{T}_{j-1}=E_0-\sum _{r=1}^n\sum _{j=1}^l\frac{k_{r,v}^2}{\pi }\tan \left(\frac{\pi {\left(s_{r,j-1}^{\psi }\right)}^2}{2k_{r,v}^2}\right)$$

(42)

When the number of iterations is infinite, namely, $l\to \infty$ , equation (42) can be simplified as:

$$\underset{l\to \infty }{lim}{E}_l\le E_0-\underset{l\to \infty }{lim}\sum _{r=1}^n\sum _{j=1}^l\frac{k_{r,v}^2}{\pi }\tan \left(\frac{\pi {\left(s_{r,j-1}^{\psi }\right)}^2}{2k_{r,v}^2}\right)$$

(43)

According to equation (41), $E_l$ is a non-increasing function. Equation (43) shows that if $E_0$ is bounded, then it can be inferred that $E_l$ is bounded for any number of iterations, and $\underset{l\to \infty }{lim}{s}_{r,l}^{\psi }\left(t\right)=0$ holds for $\forall t\in [0,T]$ .

• Step 2. The boundedness of the BCEF.

When $l=0$ , equation (29) can be expressed as:

$$\begin{array}{c}\multicolumn{1}{c}{E_0=\sum _{r=1}^n\frac{k_{r,v}^2}{\pi }\tan \left(\frac{\pi {\left(s_{r,0}^{\psi }\right)}^2}{2k_{r,v}^2}\right)+\frac{1}{2}\sum _{r=1}^n{\int }_0^t{\tilde{\ell }}_{r,0}^T{\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,0}\mathrm{d}\tau }\\ +\frac{1}{2\upsilon }\sum _{r=1}^n{\int }_0^t{\tilde{\zeta }}_{r,0}^2\mathrm{d}\tau \hfill \end{array}$$

(44)

According to equation (44), the derivative of $E_0$ can be obtained as:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{E}}_0=\sum _{r=1}^n\frac{s_{r,0}^{\psi }}{{\cos }^2\left(\frac{\pi {\left(s_{r,0}^{\psi }\right)}^2}{2k_{r,v}^2}\right)}{\stackrel{·}{s}}_{r,0}^{\psi }+\frac{1}{2}\sum _{r=1}^n{\tilde{\ell }}_{r,0}^T{\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,0}+\frac{1}{2\upsilon }\sum _{r=1}^n{\tilde{\zeta }}_{r,0}^2}\\ \le -\sum _{r=1}^n{\vartheta }_{r,0}{\tilde{\ell }}_{r,0}^T\mathit{\phi }\left({\mathit{x}}_0\right)+\sum _{r=1}^n\left|{\vartheta }_{r,0}\right|{\tilde{\zeta }}_{r,0}+\frac{1}{2}\sum _{r=1}^n{\tilde{\ell }}_{r,0}^T{\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,0}\hfill \\ +\frac{1}{2\upsilon }\sum _{r=1}^n{\tilde{\zeta }}_{r,0}^2\hfill \end{array}$$

(45)

Based on equations (20), (21), (22), and (23), equation (45) can be simplified as:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{E}}_0\le \sum _{r=1}^n{({\hat{\ell }}_{r,0}^{*}-{\hat{\ell }}_{r,-1})}^T{\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,0}+\frac{1}{2}\sum _{r=1}^n{\tilde{\ell }}_{r,0}^T{\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,0}}\\ +\frac{1}{\upsilon }\sum _{r=1}^n({\hat{\zeta }}_{r,0}^{*}-{\hat{\zeta }}_{r,-1}){\tilde{\rho }}_{r,0}+\frac{1}{2\upsilon }\sum _{r=1}^n{\tilde{\zeta }}_{r,0}^2\hfill \\ \le \sum _{r=1}^n{{\hat{\ell }}_{r,0}^{*}}^T{\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,0}+\sum _{r=1}^n(\ell -{\hat{\ell }}_{r,0}){\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,0}\hfill \\ +\frac{1}{\upsilon }\sum _{r=1}^n{\hat{\zeta }}_{r,0}^{*}{\tilde{\zeta }}_{r,0}+\frac{1}{\upsilon }\sum _{r=1}^n(\zeta -{\hat{\zeta }}_{r,0}){\tilde{\zeta }}_{r,0}\hfill \end{array}$$

(46)

By using equations (26) and (27) in Lemma 1, equation (46) can be simplified as:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{E}}_0\le \sum _{r=1}^n{({\hat{\ell }}_{r,0}^{*}-\mathrm{pro}{\mathrm{j}}_{\ell }\left({\hat{\ell }}_{r,0}^{*}\right))}^T{\mathbf{\Gamma }}^{-1}(\ell -\mathrm{pro}{\mathrm{j}}_{\ell }\left({\hat{\ell }}_{r,0}^{*}\right))+\sum _{r=1}^n{\ell }^T{\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,0}}\\ +\frac{1}{\upsilon }\sum _{r=1}^n({\hat{\zeta }}_{r,0}^{*}-\mathrm{pro}{\mathrm{j}}_{\zeta }\left({\hat{\zeta }}_{r,0}^{*}\right))(\zeta -\mathrm{pro}{\mathrm{j}}_{\zeta }\left({\hat{\zeta }}_{r,0}^{*}\right))+\frac{1}{\upsilon }\sum _{r=1}^n\zeta {\tilde{\zeta }}_{r,0}\hfill \\ \le \sum _{r=1}^n{\ell }^T{\mathbf{\Gamma }}^{-1}{\tilde{\ell }}_{r,0}+\frac{1}{\upsilon }\sum _{r=1}^n\zeta {\tilde{\zeta }}_{r,0}\hfill \end{array}$$

(47)

Using Young’s inequality for products, equation (47) can be simplified as:

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{E}}_0\le {\lambda }_{max}\left({\mathbf{\Gamma }}^{-1}\right)\sum _{r=1}^n(c_1{\Vert {\tilde{\ell }}_{r,0}\Vert }^2+\frac{1}{4c_1}{\Vert \ell \Vert }^2)}\\ +\frac{1}{\upsilon }\sum _{r=1}^n(c_2{\left|{\tilde{\zeta }}_{r,0}\right|}^2+\frac{1}{4c_2}{\left|\zeta \right|}^2)\hfill \end{array}$$

(48)

where $c_1$ and $c_2$ are normal numbers.

According to Lemma 1, both $\ell$ and $\zeta$ are bounded in $t\in [0,T]$ . According to the projection mechanism, both ${\tilde{\ell }}_{r,0}$ and ${\tilde{\zeta }}_{r,0}$ are bounded. Therefore, for $t\in [0,T]$ , ${\stackrel{·}{E}}_0$ is bounded, $E_0$ is continuous, and it is easy to see that $E_0$ is bounded. Ultimately, it can be concluded that $E_l$ is bounded.

• Step 3. The boundedness and convergence of ${\mathit{s}}_l^{\psi }$ .

According to equation (43), since $E_0$ and $E_l$ are bounded, it can be inferred that $\underset{l\to \infty }{lim}\sum _{r=1}^n\sum _{j=1}^l\frac{k_{r,v}^2}{\pi }\tan \left(\frac{\pi {\left(s_{r,j-1}^{\psi }\right)}^2}{2k_{r,v}^2}\right)$ is bounded and convergent. Therefore, for $t\in [0,T]$ , $s_{r,l}^{\psi }\left(t\right)$ will approach zero along the iteration axis, $\underset{l\to \infty }{lim}{s}_{r,l}^{\psi }\left(t\right)=0$ .

The proof of Theorem (T2) is as follows:

Based on equations (13) and (15), $0<b_{\delta r}{e}^{-\mu T}\le {\psi }_{r,l}(t)\le b_{\delta r}$ . According to equation (11) and Theorem (T1), it can be obtained that for $\forall t\in [0,T]$ , we have:

$$\begin{array}{c}\multicolumn{1}{c}{\underset{l\to \infty }{lim}|s_{r,l}(t)|=\underset{l\to \infty }{lim}|s_{r,l}^{\psi }\left(t\right)+{\psi }_{r,l}\left(t\right)\mathrm{sat}\left(\frac{s_{r,l}\left(t\right)}{{\psi }_{r,l}\left(t\right)}\right)|}\\ \le \underset{l\to \infty }{lim}\left|s_{r,l}^{\psi }\left(t\right)\right|+{\psi }_{r,l}\left(t\right)\hfill \\ ={\psi }_{r,l}\left(t\right)\le b_{\delta r}\hfill \end{array}$$

(49)

The proof of Theorem (T3) is as follows:

According to equation (11), we can obtain $e_{2r,l}\left(t\right)+\lambda e_{1r,l}\left(t\right)=s_{r,l}\left(t\right)$ , and solving the differential equation yields:

$$e_{1r,l}\left(t\right)={\int }_0^t{s}_{r,l}\left(\tau \right)e^{-\lambda (t-\tau )}\mathrm{d}\tau +e_{1r,l}\left(0\right)e^{-\lambda t}$$

(50)

Combining Theorem (T2) and Assumption 2, the following can be concluded:

$$\underset{l\to \infty }{lim}\left|e_{1r,l}\left(t\right)\right|\le b_{\delta r}{\int }_0^t{e}^{-\lambda (t-\tau )}\mathrm{d}\tau +b_{1r}{e}^{-\lambda t}\le \frac{b_{\delta r}}{\lambda }+b_{1r}$$

(51)

$$\underset{l\to \infty }{lim}\left|e_{2r,l}\left(t\right)\right|\le \underset{l\to \infty }{lim}\left|s_{r,l}\left(t\right)\right|+\underset{l\to \infty }{lim}\lambda \left|e_{1r,l}\left(t\right)\right|\le 2b_{\delta r}+\lambda b_{1r}$$

(52)

The proof of Theorem (T4) is as follows:

When $\left|s_{r,l}^{\psi }\right|\to k_{r,v}$ , $T_l$ will approach infinity, and it has been proven in Theorem (T1) that $T_l$ is bounded and $s_{r,l}^{\psi }\left(0\right)=0$ , thus obtaining $\left|s_{r,l}^{\psi }\right|<k_{r,v}$ . According to equation (12), $\left|s_{r,l}\right|<k_{r,v}+b_{\delta r}$ . Therefore, $s_{r,l}$ is bounded.

Similar to the derivation process of Theorem (T3), it can be obtained that $\left|e_{1r,l}\right|<\frac{k_{r,v}+b_{\delta r}}{\lambda }+b_{1r}$ and $\left|e_{2r,l}\right|<2(k_{r,v}+b_{\delta r})+\lambda b_{1r}$ . According to Assumption 1, if we know the upper bounds of the expected state to be $\left|x_{d1r}\right|\le b_{d1}$ and $\left|x_{d2r}\right|\le b_{d2}$ , then the system states satisfy $\left|x_{1r,l}\right|<b_{d1}+\frac{k_{r,v}+b_{\delta r}}{\lambda }+b_{1r}$ and $\left|x_{2r,l}\right|<b_{d2}+2(k_{r,v}+b_{\delta r})+\lambda b_{1r}$ . To ensure that the system states satisfy constraints $\left|x_{1r,l}\right|\le x_{1r,max}$ and $\left|x_{2r,l}\right|\le x_{2r,max}$ , it is necessary to select appropriate parameters that satisfy the following two equations:

$$b_{d1}+\frac{k_{r,v}+b_{\delta r}}{\lambda }+b_{1r}\le x_{1r,max}$$

(53)

$$b_{d2}+2(k_{r,v}+b_{\delta r})+\lambda b_{1r}\le x_{2r,max}$$

(54)

The above is the entire proof process.

• Remark 1: For the barrier Lyapunov function $T_l=\sum _{r=1}^n\frac{k_{r,v}^2}{\pi }\tan \left(\frac{\pi {\left(s_{r,l}^{\psi }\right)}^2}{2k_{r,v}^2}\right)$ , if $\left|s_{r,l}^{\psi }\right|\to k_{r,v}$ , then $T_l$ will approach infinity. If we can demonstrate that $T_l$ is bounded, then $\left|s_{r,l}^{\psi }\right|<k_{r,v}$ can be obtained based on $s_{r,l}^{\psi }\left(0\right)=0$ , thereby establishing the constraint range for the system state and input.

Simulation results

In this paper, an AILC method based on RBF neural network is proposed for the rotating joint subsystem of the long-stroke hybrid robot. To verify the effectiveness of the proposed algorithm, a 2-DOF rotating joint robotic manipulator is used as the validation model for the algorithm. The structural parameters of the model are $m_1=8.5\mathrm{kg}$ , $m_2=2.4\mathrm{kg}$ , $L_1=1.5m$ , $L_2=0.75m$ , $L_{q1}=0.75m$ , $L_{q2}=0.35m$ , $I_1=0.68\mathrm{kg}·m^2$ , and $I_2=0.12\mathrm{kg}·m^2$ . The model can be described as follows:

$$\begin{array}{c}{\mathit{D}}_l=\left[\begin{array}{cc}{D}_{11}\hfill & D_{12}\hfill \\ \multicolumn{1}{c}{D_{21}}& D_{22}\hfill \end{array}\right],{\mathit{C}}_l=\left[\begin{array}{cc}{C}_{11}\hfill & C_{12}\hfill \\ \multicolumn{1}{c}{C_{21}}& C_{22}\hfill \end{array}\right],{\mathit{G}}_l=\left[\begin{array}{c}G1\hfill \\ \multicolumn{1}{c}{G2}\end{array}\right],\hfill \\ {\mathit{u}}_l=\left[\begin{array}{c}{u}_{1,l}\hfill \\ \multicolumn{1}{c}{u_{2,l}}\end{array}\right],{\mathit{d}}_l=\left[\begin{array}{c}{d}_{1,l}\hfill \\ \multicolumn{1}{c}{d_{2,l}}\end{array}\right]\hfill \end{array}$$

(55)

where $D_{11}=m_1{L}_{q1}^2+m_2(L_1^2+L_{q2}^2+2L_1{L}_{q2}\cos \left(q_{2,l}\right))+I_1+I_2$ , $D_{12}=m_2(L_{q2}^2+L_1{L}_{q2}\cos \left(q_{2,l}\right))+I_2$ , $D_{21}=m_2(L_{q2}^2+L_1{L}_{q2}\cos \left(q_{2,l}\right))+I_2$ , $D_{21}=m_2{L}_{q2}^2+I_2$ , $C_{11}=-m_2{L}_1{L}_{q2}\sin \left(q_{2,l}\right){\stackrel{·}{q}}_{2,l}$ , $C_{12}=-m_2{L}_1{L}_{q2}\sin \left(q_{2,l}\right)({\stackrel{·}{q}}_{1,l}+{\stackrel{·}{q}}_{2,l})$ , $C_{21}=m_2{L}_1{L}_{q2}\sin \left(q_{2,l}\right){\stackrel{·}{q}}_{1,l}$ , $C_{22}=0$ , $G_1=(m_1{L}_{q1}+m_2{L}_1)g\cos \left(q_{1,l}\right)+m_2{L}_{q2}g\cos (q_{1,l}+q_{2,l})$ , and $G_2=m_2{L}_{q2}g\cos (q_{1,l}+q_{2,l})$ . The total unknown itemset of the system is $d_{1,l}=0.5\mathrm{rand}\left(l\right)\cos \left(2t\right)$ and $d_{2,l}=0.5\mathrm{rand}\left(l\right)\sin \left(3t\right)$ .

The mathematical model of the robot system mentioned above is used for simulation verification in MATLAB, and the effectiveness of the controller is analyzed. This simulation is conducted on a personal computer running the Windows 10 operating system with an Intel i5 processor, using MATLAB R2020a software. We use fourth-order Runge-Kutta equation solving in the simulation, with a sampling time of 1 ms. The simulation time is 8 s. The controller and parameter learning laws proposed in this paper are shown in equations (19)–(23). The parameter settings are $\mu =2$ , $\lambda =4$ , $\mathbf{\Gamma }=5{\mathit{I}}_{5\times 5}$ , and $\upsilon =0.01$ . The number of neurons in the RBFNN is $m=5$ , the center of the Gauss function is evenly distributed in the interval $[-2,2]\times [-2,2]\times [-2,2]\times [-2,2]\times [-2,2]\times [-2,2]\times [-2,2]\times [-2,2]\times [-2,2]\times [-2,2]$ , and the width of the Gaussian function is 3.

The expected position trajectory is $q_{1,d}=1+0.2\sin (0.5\pi t)$ , $q_{2,d}=1-0.2\cos (0.5\pi t)$ , and the expected velocity trajectory is ${\stackrel{·}{q}}_{1,d}=0.1\pi \cos (0.5\pi t)$ , and ${\stackrel{·}{q}}_{2,d}=0.1\sin (0.5\pi t)$ . The initial state of the robot system is as follows:

$$\{\begin{array}{c}{q}_{1,l}\left(0\right)=0.87+0.1\mathrm{rand}\left(l\right)\hfill \\ \mathrm{d}{q}_{1,l}\left(0\right)=0.15+0.08\mathrm{rand}\left(l\right)\hfill \\ q_{2,l}\left(0\right)=0.85+0.1\mathrm{rand}\left(l\right)\hfill \\ \mathrm{d}{q}_{2,l}\left(0\right)=0.05\mathrm{rand}\left(l\right)\hfill \end{array}$$

(56)

where $\mathrm{rand}\left(l\right)$ is a random function with values in the range $[-1,1]$ .

Figures 3 to 5 show the trajectory tracking results of system after 20 iterations. The Figure 3(a) to (d) respectively represent the position, speed tracking curves, and tracking error curves of the two joints. The result of Figure 3 indicates that even with non-zero initial errors, the robot system states track their expected trajectory successfully. From the results of Figure 3(a) and (b), we can see that both the position and speed variables remain within the predetermined range, satisfying the constraint restrictions. Figure 4 shows the learning convergence curve of the equivalent error function of the system, which gradually converges to zero with increasing iterations. Figure 5 shows the system control inputs, and overall, the control inputs are relatively smooth.

Figure 3.

System tracking curve: (a) joint position tracking, (b) joint speed tracking, (c) joint position tracking error, and (d) ioint speed tracking error.

Figure 4.

Convergence curve of $s_l^{\psi }$ .

Figure 5.

Control input.

We use two comparative experiments to verify the progressiveness of the proposed algorithm.

Comparative experiment 1

The experiment utilizes the AILC scheme proposed by Tayebi. ⁴⁰ Notably, the impact of non-zero initial errors on the robot system has not been effectively overcame by the controller in this scheme.

The experimental simulation equipment and platform are consistent with the experimental requirements of the proposed method. The expected trajectory and interference term of the system are also the same as in the previous experiment.

Figure 6 shows the position trajectory tracking under zero initial errors condition, while Figure 7 shows the position trajectory tracking under non-zero initial errors condition, respectively. Comparing Figures 6 and 7, non-zero initial errors have a significant impact on the control effectiveness of the system. When there is non-zero initial errors in the robot system, even after adjusting the controller parameters, the system state consistently fails to track the desired trajectory. Consequently, this AILC method is not effective in addressing the trajectory tracking problem in robot systems with non-zero initial errors.

Figure 6.

Position trajectory tracking under zero initial errors: (a) joint 1 position tracking and (b) joint 2 position tracking.

Figure 7.

Position trajectory tracking under non-zero initial errors: (a) joint 1 position tracking and (b) joint 2 position tracking.

Comparative Experiment 2

The simulation employs the AILC scheme with initial rectification.^41,42 The difference from the method proposed is the use of an initial correction strategy to relax the identical initial condition of robotic system. Despite this modification, the algorithm still employs the RBF neural network to approximate unknown nonlinear terms in the system.

The experimental simulation equipment and platform are consistent with the experimental requirements of the method proposed, and the parameter settings of the RBF neural network and parameter learning laws remain consistent with the aforementioned experiment.

Figures 8 to 10 show the trajectory tracking results of system after 20 iterations. The Figure 8(a) to (d) represent the position, speed tracking curves, and tracking error curves of the two joints, respectively. Figure 8 demonstrates that the initial rectification strategy addresses the problem of initial errors effectively, and the system states can track their expected trajectory ultimately. Figure 9 shows the convergence curve of the corresponding equivalent error function, which eventually converges to zero as the number of iterations increases. Figure 10 shows the control input of system, which undergoes significant changes during the initial period and may cause damage to the robot system.

Figure 8.

System tracking curve (Comparative Experiment 2): (a) joint position tracking, (b) joint speed tracking, (c) joint position tracking error, and (d) joint speed tracking error.

Figure 9.

Convergence curve of $s_{\omega ,l}$ .

Figure 10.

Control input (Comparative Experiment 2).

Figure 11 shows the tracking performance of the system using the proposed method and the initial correction method used in comparative experiment 2. As shown in Figure 11, after 20 iterations, both control methods successfully drive the system state to track the required trajectory within the allowed error range. It can be concluded that compared to the initial correction method, the proposed method in this paper can track the desired trajectory in a shorter time and with higher tracking accuracy. For joint 1 position and speed tracking, the method proposed in this paper can converge the tracking error in 2.24 s and 1.57 s, shorten the convergence time by 4 s and 3 s, respectively, and achieve tracking accuracy of 10⁻⁴ and 4 × 10⁻⁴. For joint 2 position and speed tracking, the method proposed in this paper can converge the tracking error in 2.66s and 1.96s, shorten the convergence time by 2.3s and 3s respectively, and achieve a tracking accuracy of 3 × 10⁻⁵ and 2 × 10⁻⁴. More importantly, the method proposed in this paper ensures that the system state remains within the constraint range. However, as shown in Figure 11(b), under the control of the initial correction method, the joint speed exceeded the constraint range. Figure 12 shows the system control input using the method proposed in this paper and the initial correction method. It can be seen that the control input of the method proposed in this article is smoother and has less jitter in the initial stage.

Figure 11.

System tracking performance under two comparative methods: (a) joint position tracking and (b) joint speed tracking.

Figure 12.

System control input under two comparative methods.

Although both the initial rectification strategy and the time-varying boundary layer can relax the identical initial condition, the design of the time-varying boundary layer is simpler. Therefore, we introduce the time-varying boundary layer to solve the problem of random initial errors.

The results of the above simulation experiments verify the effectiveness of the method proposed.

Conclusion

For the trajectory tracking problem of the long-stroke hybrid robot with initial errors and full state constraints, an AILC method based on RBF neural network is proposed. The adaptive learning neural network is designed to approximate the unknown nonlinear terms in the robot system, and the robust learning method is adopted to compensate for both the approximation error and external interference. An equivalent error function is designed by introducing the time-varying boundary layer to address non-zero random initial errors. The tangent-type BLF is constructed to prevent the system from violating the full state constraints. Through stability analysis based on the BCEF, the proposed control strategy ensures that the system tracking error remains within a very small range, and all signals of the robot system are bounded. Finally, the feasibility of the proposed method is verified by simulation examples on the MATLAB platform. The results of these experiments highlight the significance of overcoming the initial errors and satisfying the constraints of the robotic system. At present, the algorithm has only been simulated numerically using MATLAB. In the future, we plan to establish a real long-stroke hybrid robot experimental platform to verify the effectiveness of the algorithm and make further improvements.