Adaptive Robust RBF-NN Nonsingular Terminal Sliding Mode Control Scheme for Application to Snake Robot’s Head for Image Stabilization

Abstract

Image stabilization is important for snake robots to be used as mobile robots. In this paper, we propose an adaptive robust RBF neural network nonsingular terminal sliding mode control to reduce swinging in the snake robot’s head while it is being driven. To avoid complex dynamic problems and reduce interference during driving, we propose a 2-DOF snake robot’s head system. We designed the control system based on the nonsingular terminal sliding mode control, which ensures a fast response and finite time convergence. To reduce chattering, we incorporated an RBF neural network that can compensate for disturbances. Additionally, we included an adaptive robust term to address the disadvantages of neural network-based control. The adaptive robust term generates control inputs based on the error and is used in conjunction with the reverse saturation function to eliminate chattering. The update law of the neural network and the adaptive robust term is designed based on Lyapunov’s theory. We proved the stability of the proposed controller by investigating finite time convergence before and after the reverse saturation function operation section. Finally, we verified the performance of the proposed controller through computer simulation. The simulation evaluates the controllers using a sinusoidal reference signal similar to snake robot movement and a mixed reference signal considering the controller’s waste case. The proposed controller has excellent tracking performance and improved chattering compared with the previous controller.

1. Introduction

The snake robot is inspired by the locomotion of biological snakes and can overcome various environmental conditions such as narrow spaces and rough ground. Due to these advantages, snake robots are currently attracting attention as robots to be used in environments where it is difficult for humans to enter [1,2,3,4,5,6,7,8,9,10]. The snake robot moves with a combination of sinusoidal waves with different phases [11,12,13,14,15,16,17,18]. As a result, the robot’s head swings a lot when in motion. Figure 1 shows that the snake robot’s head swings, making it difficult for the robot operator to understand the camera image. Shaky camera images seriously affect the robot’s maneuverability and make it difficult to operate remotely. This issue becomes a major obstacle for using snake robots as mobile robots [19,20,21,22,23,24,25,26,27,28]. This issue is a very important problem in the utilization of the snake robot and must be solved.

To address this issue, we reviewed previous research, which can be divided into two parts: Firstly, there is a method that uses image processing techniques; however, this approach requires significant computing power, and it is challenging to compensate for the large swing of the snake robot [20,21]. Secondly, many researchers use kinematics-based locomotion control [22,23,24,25,26,27,28]. Their research reduces the swing of the snake robot’s head by altering its locomotion through the snake robot’s drive module. However, changing the locomotion of the snake robot may result in decreased driving speed or drift during driving, and other factors such as ground friction and other locomotion are not considered. Moreover, their findings lack a rigorous analysis of driving performance. As far as the author is aware, there has been no detailed study on the dynamic control method yet. To consider the dynamics, the snake robot system has many degrees of freedom (DOF) and is subject to complex nonlinearity. Additionally, ground friction also significantly affects the robot’s motion, making control challenging. Nevertheless, for effective control considering various conditions, it is necessary to develop a robust nonlinear controller considering disturbance.

Sliding mode control is a well-known nonlinear controller used to handle system nonlinearity and unmeasurable disturbance [29,30,31,32]. The fundamental aspect of widely used sliding mode control is the generating of the domain of attraction around a defined switching surface, also known as a sliding manifold, by using a discontinuous control input. The desired control performance is achieved by setting the switching term in the ideal sliding mode. Sliding mode control is based on the principles of asymptotic stability and Lyapunov’s theory, with the linear sliding surface specifying the expected control performances. The Lipschitz condition for ordinary differential equations underpins asymptotic stability. However, a characteristic of asymptotic stability is that the closer the system dynamics are to the equilibrium point, the slower the state convergence and it does not guarantee that the system state can reach the equilibrium point in a finite time. In actual application, this may not pose a problem for sliding mode control, but it may not be feasible when high precision is required. To address this issue, a control method called terminal sliding mode control was developed [33,34,35]. Terminal sliding mode control includes an exponential term in the sliding surface, enabling the system to reach within a finite time. However, there was a problem of singularity, which led researchers to propose nonsingular terminal sliding mode control [36,37]. The nonsingular terminal sliding mode exhibits good control performance, but chattering exists. If the control gain of the switching term is increased in order to be robust against external disturbances, chattering occurs significantly. Conversely, if the control gain is lowered, it is affected by disturbances. Therefore, a method to measure the disturbance is needed to reduce the chattering by lowering the control gain.

Nonlinearity, disturbances, and mathematical model uncertainties are major challenges in robotic system controller design. To solve this issue, researchers are using nonlinear control theory, disturbance observer, robust control, and adaptive control. Among them, intelligent control is being studied in many systems due to the development of computer technology. Among them, the artificial neural network (in this paper, called a neural network) can evade complex mathematical analyses such as high-order systems and can resolve nonlinear problems using multilayer neural networks and nonlinear activation functions [38,39,40,41,42]. In this study, the disturbance is approximated using the Radial Basis Function (RBF) neural network. The RBF neural network can approximate nonlinear functions with relatively simple mathematical analysis and simple structure compared with Multi-Layer Perceptron (MLP) [43,44,45,46]. Due to these characteristics, it is helpful for online control in combination with a robot system. Nonetheless, the artificial neural network has problems such as non-continuous signals, large system output due to an error at the beginning of the system operation, delay in convergence speed, or system instability. Therefore, research and applications have been applied and studied in combination with control engineering. Among them, there is a study case that demonstrated excellent performance in combination with a controller [46,47,48,49,50,51].

On the other hand, as mentioned above, neural networks have the problem of convergence delay or instability due to large signal changes or parameter changes or discontinuous signals in the initial weight update stage. This problem can also appear when combined with a controller. Therefore, in previous studies, to compensate for this problem, an additional input called the ‘robust term’ has been used [39,42,43,44,45,46,47,48]. The robust term is similar to the switching term and consists of an error sign function and constant control gain. The robust term additionally applies a constant control input according to the sign of the error function. The controller with the robust term was effective in improving control performance but did not respond rapidly to large signal changes, and chattering occurred when the control gain was increased to respond rapidly. Consequently, to solve all these problems, the control gain of the nonsingular terminal sliding mode control switching term must be increased. Therefore, the problem of large chattering exists. It is important to solve this issue for network-based control.

Other recent research are [47,48,52,53,54,55,56]. Studies by [47,48,52,53,54] have employed a backstepping-based control scheme, which is a useful method for nonlinear systems. Their results proved effective in enhancing control performance, but backstepping control requires a complicated design process and faces the problem of ‘explosion of term’ problems. The studies by [55,56] proposed prescribed performance control with a fuzzy-neural network for nonlinear systems with uncertainty and constraint conditions. Their research results demonstrated effectiveness through simulation; however, the implementation of the system requires strict performance function settings and a complex design process. For research the other way, there are [57,58]. Ref. [57] compensated tremors using a learning network-based filter integrating a deep learning network and modified incremental learning algorithms. Their study presents interesting results, but the swing to be solved in this paper is quite large, and the convolutional neural network they use is not easy to apply to the controller. Ref. [58] suggested control of model uncertainty using a depth-independent composite Jacobian matrix to make visual parameters and robotic physical parameters appear linearly in a parametrized uniform form so that it is an adaptive algorithm. This study also presents interesting results, nevertheless, it is difficult to apply visual servoing control to our general issue.

In this paper, we propose a head system for a snake robot and an error-based RNF neural network with nonsingular terminal sliding mode control for image stabilization of snake robot’s head. We propose the snake robot’s head system to simplify the control objective by avoiding the complex dynamics of the entire robot. Next, we use RBF neural network to estimate disturbance. The neural network consists of an error-based input and a minimum node for online learning in the combined snake robot’s head system. We propose an adaptive robust term to solve the problem that occurs during the initial weights update process of the neural network. The adaptive robust term is composed of the adaptive coefficient and error function. The adaptive coefficient responds to signal changes by changing the control gain, and the reverse saturation function is used to prevent chattering in the steady state. The proposed control is designed based on Lyapunov’s theory and proves stability. We prove finite time convergence by investigating system boundness. Finally, we verify the control performance by comparing the proposed controller with the previous controller through computer simulation.

This paper is structured as follows: Section 2 introduces the snake robot’s head system and dynamics. Section 3 shows the design of the nonsingular terminal sliding mode control with RBF neural network and the design the adaptive robust term, it also proves the finite time convergence. Section 4 compares and verifies the performance of the proposed controller through computer simulation. Section 5 discusses conclusions and future work.

2. Snake Robot’s Head System

In this paper, the snake robot has two axes (yaw, pitch). The Snake Robot’s whole system dynamics is nonlinear and very complex. To avoid the complexity of the snake robot’s whole system dynamics, we propose a two degree of freedom (2-DOF) snake robot’s head system. The roll rotation of the image can be easily processed by image processing, but the yaw and pitch rotation are difficult to correct. Therefore, the snake robot’s head system has an actuator capable of yaw and pitch rotation. In addition, it is composed of a smaller size than the driving module to reduce the impact on driving. The proposed snake robot’s head system is shown in Figure 2.

$$M(\theta )\ddot{\theta }+C(\theta ,\dot{\theta })+G(\theta )=\tau +{\tau }_{Disturbance}$$

(1)

The snake’s robot head system is similar to the 2-DOF robot arm system. We designed the snake robot’s head system. $M(\theta )$ is 2 × 2 inertia matrix, $C(\theta ,\dot{\theta })$ is 2 × 1 Coriolis matrix, $G(\theta )$ is 2 × 1 gravity matrix, $\tau$ is 2 × 1 system input matrix, and ${\tau }_{Disturbance}$ is unmeasurable disturbance. The snake robot’s head angles, snake robot’s head angular velocity, and snake robot’s head angular acceleration of module are $\theta ={\left[\begin{array}{cc}{\theta }_1& {\theta }_2\end{array}\right]}^T$, $\dot{\theta }={\left[\begin{array}{cc}{\dot{\theta }}_1& {\dot{\theta }}_2\end{array}\right]}^T$, and $\ddot{\theta }={\left[\begin{array}{cc}{\ddot{\theta }}_1& {\ddot{\theta }}_2\end{array}\right]}^T$, respectively.

The system expressed in Equation (1) is expressed as strict feedback form as follows in Equations (2)–(5) [52].

$$x_1=\theta$$

(2)

$${\dot{x}}_1=x_2=\dot{\theta }$$

(3)

$${\ddot{x}}_1={\dot{x}}_2=\ddot{\theta }$$

(4)

$${\dot{x}}_2=-M^{-1}(C+G)+M^{-1}u+D$$

(5)

The control input is $u={\left[{\tau }_1 {\tau }_2\right]}^T$. $D$ is an external disturbance. The snake robot’s head system does not contain snake robot dynamics. Therefore, the swing from the snake robot is included in $D$, and contains the snake robot’s head system.

3. Controller Design

We used nonsingular terminal sliding mode control as the main controller of the snake robot’s head system. The concept of terminal sliding mode control was rooted in the notion of terminal attractors. Terminal sliding mode control is able to carry out finite time convergence; however, it had a problem that led to singularity under certain conditions. To overcome the singularity in terminal sliding mode control, some papers have proposed nonsingular terminal sliding mode control. This control method can be achieved by changing the sliding surface. In this paper, we propose a controller based on a nonsingular terminal sliding mode that can converge within a finite time and improve control performance without singularity problems.

3.1. Nonsingular Terminal Sliding Mode Control

The nonsingular terminal sliding surface used in this paper is shown in Equation (6).

$$s=\alpha e+{\dot{e}}^{\lambda }$$

(6)

$$\dot{s}=\alpha \dot{e}+\lambda \ddot{e}\cdot {\dot{e}}^{(\lambda -1)}=\alpha \dot{e}+({\ddot{x}}_d-\ddot{x})\cdot \lambda {\dot{e}}^{(\lambda -1)}$$

(7)

where $e$ and $\dot{e}$ are the system error and a derivative of the error, $\alpha$ > 0, 1 < $\lambda$ < 2. Equation (7) is the time derivative Equation (6). ${\ddot{x}}_d$ is the second derivative of the reference signal, and $\ddot{x}$ is the second derivative of the output. The sliding surface Equation (6) uses only the exponential term for the error derivative term. The conventional sliding surfaces use exponential terms and control gains. However, we use it as shown in Equation (6) to maintain the characteristics of NTSMC while resolving the disadvantage because it is difficult to tune the control gains for use in the additional controller described later.

Next, to ensure system stability and design the controller, we consider a Lyapunov function candidate such as Equation (8).

$$V_1=s^2/2$$

(8)

Its time derivative is

$${\dot{V}}_1=s\dot{s}$$

(9)

Substitute Equation (7) into Equation (9)

$${\dot{V}}_1=s\dot{s}=s\cdot \{\alpha \dot{e}+({\ddot{x}}_d-\ddot{x})\cdot \lambda {\dot{e}}^{(\lambda -1)}\}$$

(10)

Substitute Equation (5) into Equation (10)

$${\dot{V}}_1=s\dot{s}=s\cdot \{\alpha \dot{e}+({\ddot{x}}_d+M^{-1}(C+G)-M^{-1}u-D)\cdot \lambda {\dot{e}}^{(\lambda -1)}\}$$

(11)

To realize asymptotic stability ${\dot{V}}_1<0$, we obtain the control input as

$$u=M\cdot ({\ddot{x}}_d+\alpha \dot{e}/\lambda {\dot{e}}^{\lambda -1}-D+k\cdot \mathrm{sgn}(s))+(C+G)$$

(12)

Equation (12) is the nonsingular terminal sliding mode control input. $k\cdot \mathrm{sgn}(s)$ is switching term, and $k$ is switching term positive control gain. If $D$ can be measured perfectly, the control gain $k$ can be lowered, and we can design a controller that produces good control performance and less chattering. However, it is difficult to measure $D$. Therefore, in this paper, we propose to compensate by approximating $D$ using the RBF neural network.

3.2. RBF Neural Network

Equation (12) does not calculate and measure the disturbance. Disturbances generated during snake robot driving are irregular and difficult to measure; therefore, when a disturbance occurs, the control gain must be increased to compensate. In this case, large chattering may occur and adversely affect the system; therefore, in this paper, the disturbance is estimated and compensated by using the RBF network.

A neural network consists of an input layer, a hidden layer, and an output layer. The hidden layer of a neural network consists of many nodes. In general, the larger the number of nodes in the hidden layer, the better the approximation, but it has the disadvantage of increasing the computational speed. The RBF neural network has one layer each, it is a simple structure. So, the RBF neural network has the advantage of approximating nonlinearity with a small number of nodes by using a Gaussian function-based activation function. In this paper, we use the RBF network structure to apply a neural network to a snake robot’s head system, and implement the neural network with the minimum number of nodes for online learning. The structure of the RBF neural network is shown in Figure 3a.

$$Y(Z)=W^{T}H(Z)$$

(13)

$$h_{ij}=\exp (-{‖x_i-c_j‖}^2/2b_j{}^2)$$

(14)

Equation (13) is RBF neural network output. $Z={\left[e_1 e_2\right]}^T$ is the input vector, $W={\left[w_1 \dots w_5\right]}^T$ is the weight vector, and $H={\left[h_1 \dots h_5\right]}^T$ is the activation function vector. The activation function is expressed as a Gaussian function as follows in Equation (14). $c_j$ represents the coordinate value of center point of Gaussian function, and $b_j$ represents the width of Gaussian function. In this paper, according to the scope of the input value, we use $c=0.5\times [-1,-0.5, 0, 0.5, 1]$, $b=15$. The activation function according to the change in $c_j$ is shown in Figure 3b.

Next, to compensate for disturbances, we assume $D$ as in Equation (15).

$$D\cong d(•)\cdot \Lambda$$

(15)

The disturbance is approximated by the product of the nonlinear term $d(•)$, and the error function $\Lambda =\varpi e+\dot{e}$. The nonlinear term $d(•)$ is approximated RBF neural network as in Equation (16).

$$d^{*}(•)=W^{*}{}^T\cdot H(Z)$$

(16)

where $d^{*}(•)$ is an optimal approximated nonlinear term, and $W^{*}$ is an optimal weight vector.

$$\widehat{d}(•)={\hat{W}}^T\cdot H(Z)$$

(17)

where $\widehat{d}(•)$ is the neural network approximated nonlinear term, and $\widehat{W}$ is approximated weight vector. Substituting Equations (15) and (17) into the system input, we can obtain Equation (18).

$$u=M\cdot ({\ddot{x}}_d+\alpha \dot{e}/\lambda {\dot{e}}^{\lambda -1}-\widehat{d}(•)\Lambda +k\cdot \mathrm{sgn}(s))+(C+G)$$

(18)

Next, to ensure system stability and design the RBF neural network update law, we consider the Lyapunov function candidate such as Equation (19).

$$V_2=s^2/2+{\tilde{W}}^T\tilde{W}/2\gamma$$

(19)

Its time derivative is

$${\dot{V}}_2=\dot{s}s+{\tilde{W}}^T\dot{\widehat{W}}/\gamma$$

(20)

where $\tilde{W}=\widehat{W}-W^{*}$, $\gamma$ is the positive network update control gain. Equations (7) and (18) are substituted into Equation (20). We can obtain Equations (21) and (22).

$${\dot{V}}_2=s(-\lambda {\dot{e}}^{\lambda -1}\cdot k\cdot \mathrm{sgn}(s)+\lambda {\dot{e}}^{\lambda -1}\cdot \Lambda {\tilde{W}}^{T}H)+{\tilde{W}}^T\dot{\widehat{W}}/\gamma$$

(21)

$${\dot{V}}_2=-\lambda {\dot{e}}^{\lambda -1}\cdot k\cdot s\cdot \mathrm{sgn}(s)+{\tilde{W}}^T(\lambda {\dot{e}}^{\lambda -1}\cdot s\cdot \Lambda H+\dot{\widehat{W}}/\gamma )$$

(22)

where $\lambda {\dot{e}}^{\lambda -1}\cdot k\cdot s\cdot \mathrm{sgn}(s)$ is always positive. To realize asymptotic stability ${\dot{V}}_2<0$, we can obtain RBF neural network update law Equation (23).

$$\dot{\widehat{W}}=-\gamma (\lambda {\dot{e}}^{\lambda -1}\cdot s\cdot \Lambda H+\sigma \tilde{W})$$

(23)

where $\sigma$ is positive momentum control gain.

3.3. Adaptive Robust Term

The neural network causes neural network estimation error during the learning process. The neural network estimation error decreases to a small value when training is complete; however, during learning, the neural network estimation error has valid values and can affect transient states. Network estimation errors usually occur during initial weight learning or when the large signal changes, which causes system errors. Compensating for the neural network estimation error is important in robotic system control using neural networks.

Previous research has used an additional control input similar to the switching term called the robust term $\psi \mathrm{sgn}(e)$. $\psi$ is a positive constant, $\mathrm{sgn}()$ is a sign function, and $e$ is an error function; therefore, it is a controller that generates a control gain according to the sign of the error function. If tuning well, the robust term is effective and simple. Nevertheless, this method is not effective in some situations. For example, using a small control gain to reduce chattering has little improvement over large instantaneous signal changes. This can be improved by using a large control gain, but it causes chattering on the steady state output or control input; therefore, we need to increase the robustness of the system by adaptively compensating for the neural network estimation error. In other words, the adaptive robust term should be able to respond to large signal changes by generating control force according to the error. Finally, we need an output that does not chatter in a steady state [52].

In this section, we design an adaptive robust term using the Lyapunov function candidate to consider the stability of the controller. To remove the chattering, we use the reverse saturation function to eliminate chattering or residual control inputs in the steady state.

$$D^{*}=d^{*}(•)\cdot \Lambda +{\epsilon }^{*}$$

(24)

$$u=M\cdot ({\ddot{x}}_d+\alpha \dot{e}/\lambda {\dot{e}}^{\lambda -1}-\widehat{d}(•)\Lambda +k\cdot \mathrm{sgn}(s)-\widehat{\epsilon })+(C+G)$$

(25)

Due to the neural network estimation error, we redesign optimal disturbance approximation $D^{*}$, such as in Equation (24). ${\epsilon }^{*}$ is the optimal neural network estimation error, and $\widehat{\epsilon }$ is the adaptive robust term output. Equation (25) is the proposed control input with the adaptive robust term. To ensure system stability and design the adaptive law, we consider Lyapunov function candidate such as Equation (27)

$$V_3=s^2/2+{\tilde{W}}^T\tilde{W}/2\gamma +{\tilde{\epsilon }}^T\tilde{\epsilon }/2\beta$$

(26)

Its time derivative is

$${\dot{V}}_3=\dot{s}s+{\tilde{W}}^T\dot{\widehat{W}}/\gamma +{\tilde{\epsilon }}^T\dot{\widehat{\epsilon }}/\beta$$

(27)

where $\tilde{\epsilon }=\widehat{\epsilon }-{\epsilon }^{*}$, $\beta$ is the positive adaptive control gain. Equations (7) and (25) are substituted into Equation (27). We can obtain Equations (28) and (29).

$${\dot{V}}_3=s\{-\lambda {\dot{e}}^{\lambda -1}\cdot k\cdot \mathrm{sgn}(s)+\lambda {\dot{e}}^{\lambda -1}\cdot \Lambda ({\tilde{W}}^{T}H)+{\tilde{\epsilon }}^T)\}+{\tilde{W}}^T\dot{\widehat{W}}/\gamma +{\tilde{\epsilon }}^T\dot{\widehat{\epsilon }}/\beta$$

(28)

$${\dot{V}}_3=-\lambda {\dot{e}}^{\lambda -1}\cdot k\cdot s\cdot \mathrm{sgn}(s)+{\tilde{W}}^T(\lambda {\dot{e}}^{\lambda -1}\cdot s\cdot \Lambda H+\dot{\widehat{W}}/\gamma )+{\tilde{\epsilon }}^T(\lambda {\dot{e}}^{\lambda -1}\cdot s+\dot{\widehat{\epsilon }}/\beta )$$

(29)

To realize asymptotic stability ${\dot{V}}_3<0$, we can obtain the adaptive robust term update law Equation (30).

$$\dot{\widehat{\epsilon }}=-\beta (\lambda {\dot{e}}^{\lambda -1}\cdot s+\xi \tilde{\epsilon })$$

(30)

The adaptive robust term can improve control performance by compensating neural network approximation errors, although Equation (30) satisfied the Lyapunov theorem, the adaptive robust term contains many error-related terms, which can suffer convergence and convergence speed. Tuning is not easy due to the characteristic of the term of ${\dot{e}}^{\lambda -1}$. Therefore, we substitute a constant term that satisfies Equation (31) by using the fact that it is always a positive.

$${\dot{e}}^{\lambda -1}<C \mathrm{and} \left|{\dot{e}}^{\lambda -1}\cdot k\cdot s\cdot \mathrm{sgn}(s)+\sigma {\tilde{W}}^T\tilde{W}+\xi {\tilde{\epsilon }}^T\tilde{\epsilon }\right|>\left|{\tilde{\epsilon }}^T\lambda \cdot s({\dot{e}}^{\lambda -1}-C)\right|$$

(31)

Equation (32) is the adaptive robust term update law replaced by a constant term. Next, we combine the reverse saturation function as Equation (33) with the adaptive robust term. The reverse saturation function generates outputs larger than the threshold value and removes outputs less than the threshold value. Therefore, phenomena such as chattering can be prevented by removing the additional control input in steady state through the reverse saturation function.

$$\dot{\widehat{\epsilon }}=-\beta (\lambda \cdot C\cdot s+\xi \tilde{\epsilon })$$

(32)

$$SA{T}_{rev}(u)=\left\{\begin{array}{c}-\left|u\right|\\ 0\\ +\left|u\right|\end{array}\right.\begin{array}{c} \mathrm{if}:\\ \mathrm{if}:\\ \mathrm{if}:\end{array}\begin{array}{c}u<-N\\ -N\le u\le N\\ u>N\end{array}$$

(33)

Finally, we can obtain the proposed control input as Equation (34). The control flow is in Figure 4. Figure 4 is the controller block diagram using Equation (34). The controller creates an error signal using the reference signal and the output, and generates the output of the network and the adaptive robust term output. Finally, the generated outputs are integrated with the NTSMC controller and applied to the snake robot’s head.

$$u=(C+G)+M\cdot \{{\ddot{x}}_d+\alpha \dot{e}/\lambda {\dot{e}}^{\lambda -1}+\underset{Switching term}{\underbrace{k\cdot \mathrm{sgn}(s)}}-\underset{\mathrm{RBF} \mathrm{NN} \mathrm{output}}{\underbrace{\widehat{d}(•)\Lambda }}-\underset{\mathrm{Adaptive} \mathrm{robust} \mathrm{term}}{\underbrace{SA{T}_{rev}(\widehat{\epsilon })\}}}$$

(34)

3.4. Stability Analysis and Finite Time Convergence

This section analyzes the stability of the proposed controller. The proposed controller uses the adaptive robust tern and the reverse saturation function in a steady state. When the adaptive robust term is larger than the reverse saturation function threshold value, stability is guaranteed according to the Equation (27), and used in the adaptive robust term design. When the adaptive robust is below the threshold value, it does not generate control force at a steady state. Therefore, stability can be guaranteed according to Equation (19) used in the previous RBF neural network controller. In this section, finite time convergence is investigated to strictly prove the stability of the system including the reverse saturation function.

First, we consider when the adaptive robust term is larger than the threshold value. To investigate stability, and finite time convergence, we ponder the Lyapunov function candidate as in Equation (35).

$$V_4=s^2/2+{\tilde{W}}^T\tilde{W}/2\gamma +{\tilde{\epsilon }}^T\tilde{\epsilon }/2\beta$$

(35)

Its time derivative is

$${\dot{V}}_4=\dot{s}s+{\tilde{W}}^T\dot{\widehat{W}}/\gamma +{\tilde{\epsilon }}^T\dot{\widehat{\epsilon }}/\beta$$

(36)

By substituting the proposed control input and using $\tilde{W}=\widehat{W}-W^{*}$, and $\tilde{\epsilon }=\widehat{\epsilon }-{\epsilon }^{*}$, we can obtain Equation (37)

$${\dot{V}}_4=-\lambda {\dot{e}}^{\lambda -1}\cdot k\cdot s\cdot \mathrm{sgn}(s)-\sigma ({\tilde{W}}^T\tilde{W}+{\widehat{W}}^T\widehat{W}-W^{*T}{W}^{*})/2-\xi ({\tilde{\epsilon }}^T\tilde{\epsilon }+{\widehat{\epsilon }}^T\widehat{\epsilon }-{\epsilon }^{*T}{\epsilon }^{*})/2$$

(37)

$${\dot{V}}_4\le -\lambda {\dot{e}}^{\lambda -1}\cdot k\cdot s^2/\left|s\right|-\sigma ({\tilde{W}}^T\tilde{W})/2-\xi ({\tilde{\epsilon }}^T\tilde{\epsilon })/2+\sigma (W^{*T}{W}^{*})/2+\xi ({\epsilon }^{*T}{\epsilon }^{*})/2$$

(38)

By replacing $\kappa =\lambda {\dot{e}}^{\lambda -1}\cdot k/\left|s\right|$, $\kappa >0$ and ${\rho }_1=\sigma (W^{*T}{W}^{*})/2+\xi ({\epsilon }^{*T}{\epsilon }^{*})/2$, we can obtain Equation (39)

$${\dot{V}}_4\le -\kappa s^2-\sigma ({\tilde{W}}^T\tilde{W})/2-\xi ({\tilde{\epsilon }}^T\tilde{\epsilon })/2+{\rho }_1$$

(39)

$\sigma \ge \kappa /\gamma$, and $\beta \ge \kappa /\xi$ can obtain Equations (40) and (41)

$${\dot{V}}_4\le -\kappa s^2-\kappa \cdot {\tilde{W}}^T\tilde{W}/\gamma -\kappa \cdot {\tilde{\epsilon }}^T\tilde{\epsilon }/\beta +{\rho }_1$$

(40)

$${\dot{V}}_4\le -2\kappa (s^2/2+{\tilde{W}}^T\tilde{W}/2\gamma +{\tilde{\epsilon }}^T\tilde{\epsilon }/2\beta )+{\rho }_1$$

(41)

By substituting Equation (35), $\psi \ge 2\kappa$ we can obtain Equation (42)

$${\dot{V}}_4\le -\psi V_4+{\rho }_1$$

(42)

From Equation (42) we can obtain Equation (43)

$$V_4(t)\le (V_4(t_0)-{\rho }_1/\psi )e^{-\psi (t-t_0)}+{\rho }_1/\psi , \forall t\ge t_0$$

(43)

$V_4(t_0)$ is the Lyapunov function candidate’s initial value. $V_4$ is bounded as ${\rho }_1/\psi$, and all system variables related to $V_4$. So, the closed loop system is bounded; therefore, the system is stable.

Next, to define the finite time, we use Equation (42), and we can obtain Equations (44)–(46)

$$d{V}_4/dt=-\psi V_4+{\rho }_1$$

(44)

$$d{V}_4/(\psi V_4-{\rho }_1)=-dt$$

(45)

$$\left[{\displaystyle {\int }_{V_4(t_0)}^{V_4(t_N)}\psi /(\psi V_4-{\rho }_1)\cdot dV}\right]/\psi =-{\displaystyle {\int }_0^{t_N}dt}$$

(46)

where $t_N$ is the finite time to reach the threshold value. At last, we can obtain finite time as Equation (47)

$$t_N=\{\ln (\left|\psi V_4(t_0)-{\rho }_1\right|)-\ln (\left|\psi V_4(t_N)-{\rho }_1\right|)\}/{\psi }_1$$

(47)

Second, when the adaptive robust term control is below the reverse saturation function threshold value, we can prove stability and finite time convergence using the same method and Equation (48).

We can obtain Equations (49) and (50)

$$V_5=s^2/2+{\tilde{W}}^T\tilde{W}/2\gamma$$

(48)

$$V_5(t)\le (V_5(t_0)-{\rho }_2/\psi )e^{-\psi (t-t_0)}+{\rho }_2/\psi , \forall t\ge t_0$$

(49)

$$t_f=\left\{\begin{array}{c}\ln (\psi V_5(t_0)/{\rho }_2-1)/\psi , \psi V_5(t_0)\ge {\rho }_2\\ \ln (1-\psi V_5(t_0)/{\rho }_2)/\psi , \psi V_5(t_0)<{\rho }_2\end{array}\right.$$

(50)

where ${\rho }_2=\sigma (W^{*T}{W}^{*})/2$, $V_5(t_0)$ is the Lyapunov function candidate’s initial value, $V_5(t_0)$ is the same value as $V_4(t_N)$, and $t_f$ is finite time of proposed controller. We used the reverse saturation function to improve the control performance. Because of this, a finite time is different when the adaptive robust term is larger than the threshold value or lower than the threshold value. However, in both cases, convergence in finite time is satisfied. Therefore, our closed loop system is stable and satisfies finite time convergence [33,36,37,49,59,60,61,62].

4. Simulation

4.1. Simulation Setup

We conducted a computer simulation to compare and verify the proposed controller with the previous controller. The locomotion of the snake robot is implemented as a combination of sinusoidal waves of different phases. Consequently, the snake robot’s head also swings in a sinusoidal wave while it is being driven. Therefore, it needs to evaluate the sinusoidal signal tracking performance of the controller. Next, our proposed controller used a neural network. As mentioned above, neural networks are vulnerable to sudden signal changes. Therefore, we configured disturbance and control inputs with large signal changes to test whether the controller has satisfactory control performance [52].

Figure 5 is the continuous control input. This control input is similar to snake robot’s head movement. Figure 6 is a control input with rapid signal change. Although this control input is not similar to the movement of the snake robot’s head, it is used to test the control performance by implementing a situation that is a disadvantage of the proposed controller using a neural network.

Figure 7 is position disturbance. The disturbance resembles the snake robot’s head movement. Moreover, we combine sinusoidal signal and high-frequency noise that for unknown reasons such as topographical conditions.

The snake robot’s head system parameter is $m_1=0.1 \mathrm{kg}$, $m_2=0.2 \mathrm{kg}$, $l_{1,2}=0.07 \mathrm{m}$, initial angle ${\theta }_1=1.5 \mathrm{rad}$, ${\mathsf{\theta }}_2=1.8 \mathrm{rad}$, ${\dot{\theta }}_{1,2}=0 \mathrm{rad}/\mathrm{s}$, and ${\ddot{\theta }}_{1,2}=0 \mathrm{rad}/{\mathrm{s}}^2$. The control gain parameter is $\alpha =10$, $\lambda =1.67$, $k=8$, $c=0.5\times [-1,-0.5, 0, 0.5, 1]$, $b=15$, $\varpi =3.5$, $\gamma =80$, $\sigma =0.15$, $\beta =0.2$, $C=21$, $\xi =0.001$, and $N=0.1$. Assuming no offline training is performed, all network initial weights are set to 0.1. All control gain parameters can be obtained through simulation iterations. The simulation sampling time is $1 \mathrm{ms}$. For comparison of the control performance, the control gain of the previous robust term is used as 20 to achieve similar performance to the proposed controller in all situations. While in simulation, nonsingular terminal sliding mode control (NTSMC), and nonsingular terminal sliding mode control with RBF neural network (NTSMC-NN), and previous robust term (RT-NTSMC-NN) are used.

4.2. Continouous Input Simulation Result

The simulation proceeds using Figure 5. For verification of the control performance, we consider output performance, input performance, and error performance.

Figure 8 is the previous nonsingular terminal sliding mode control output result [36,37]. As can be seen in Figure 8, compared with other controllers, it generates based on the maximum reference input, the overshoot is 19.5% and 27%, respectively, and the convergence speed is slow. Large steady-state errors are also generated during the process. As shown in Figure 9, using a neural network can effectively improve overshoot, convergence speed, and steady state errors for continuous input. Figure 10 and Figure 11 use additional control input as the previous robust term, and adaptive robust term. Compared with Figure 9, there is more improved convergence speed. In addition, although the control performance of the previous robust term and the adaptive robust term is similar, chattering occurs in the previous robust term.

Figure 12 and Figure 13 are compared for control input performance. As shown in Figure 12 and Figure 13b–d, the controller is improved, it generates a greater control force than the NTSMC. However, as shown in Figure 12 and Figure 13b, chattering increased more than NTSMC, and as shown in Figure 13c, the control performance was more improved, but the previous robust term caused larger chattering. As shown in Figure 12 and Figure 13d, the proposed controller can generate a control input suitable for the situation, and the control input is removed when it is close to the steady state due to the reverse saturation function, and chattering is improved.

Figure 14 and Figure 15 compare errors for continuous input. It can be seen that all controllers generate a small steady-state error for continuous input. As shown in Figure 8, Figure 9, Figure 10 and Figure 11, it can also be seen in Figure 14 and Figure 15 that the performance of proposed controllers and NTSMC-NN-RT are improved compared with NTSMC-NN. NTSMC-NN-RT converges about 0.2 s faster on average than the proposed controller, though it causes the greatest chattering. The proposed controller has the least chattering among the other controllers. Chattering can be a major threat in robotic systems. Therefore, the proposed controller shows the best performance.

4.3. Mixed Control Input Simulation Result

The simulation proceeds using Figure 6. The mixed control input is not an actual control input, assuming the worst case of a controller using neural network. Therefore, we verify through simulation whether the proposed controller shows satisfactory control performance even in the worst case.

Figure 16, Figure 17, Figure 18 and Figure 19 are the result of controller performance for mixed input. As shown in Figure 16, when only NTSMC is used, the sinusoidal wave reference signal part shows the same characteristics as Figure 8. Nevertheless, in a constant reference signal with a large bias, it follows the steady state well. However, it is slow to reach a steady state. In Figure 17 using RBF NN, the sinusoidal wave reference signal part is improved as shown Figure 9. The effect of sudden signal change, which is a weak part of the neural network, seems to be suppressed by NTSMC, so overshoot does not appear. Whereas, as with the previous results, there is a delay. Figure 18 shows the previous robust term, and Figure 19 shows the adaptive robust term. Both controllers improved control performance for suddenly changing signals. Based on 15 s when the signal changes suddenly, the previous robust term is 0.73 s faster than NTSMC-NN, and the adaptive robust term is 0.93 s faster than NTSMC-NN. As a result, the proposed controller was able to increase the convergence speed compared with the previous robust term.

Next, we proceed to compare the control inputs of the controllers to the mixed inputs and the error comparisons.

Figure 20 and Figure 21 are compared for control input performance result. Results from the mixed input are similar to those from continuous input. As shown is Figure 20 and Figure 21b, the controller using the neural network generates a large control force, and the chattering of the control input is also increased by the switching term because the effect of the neural network is included in the system output. Figure 20 and Figure 21c are results for the controller using neural network and robust term. As shown in Figure 18, the control performance is much better than NTSMC-NN; however, input chattering is larger than that of NTSMC-NN. The proposed controller, as shown in Figure 20 and Figure 21d, significantly reduced chattering compared with the other controller. Due to the reverse saturation function and the adaptive coefficient, the control input decreases as it approaches the steady state, and it can be confirmed that chattering occurs less by removing the robust term, which can cause chattering in the vicinity of the steady state. Additionally, as shown in 15 s and 22.5 s, the proposed controller can generate control input according to errors; therefore, the proposed controller is faster than the previous controller.

Figure 22 and Figure 23 compare error performance for mixed input. A large error change occurs at 7.5 s, 15 s, and 22.5 s, when the signal changes suddenly. On average, the proposed controller showed a faster convergence speed of about 0.2 s than the previous robust term. Moreover, it can be seen that the steady state error converges to 0 for a constant signal, unlike the sinusoidal signal.

5. Conclusions

In this paper, we proposed an error-based adaptive robust RBF neural network nonsingular terminal sliding mode control scheme for application to 2-DOF snake robot’s head system for image stabilization.

We designed the nonsingular terminal sliding mode control-based controller to cope with the nonlinearity and disturbance of the snake robot’s head dynamics. The sliding surface uses only the exponential term for the error derivative term to avoid tuning complexity for additional control gain. To reduce chattering the disadvantage of the sliding mode-based controller, we minimized the coefficients of the switching term to minimize chattering. Afterward, we used RBF neural network to compensate for the disturbance. The disturbance is composed of the product of the nonlinear term and the error function, and the nonlinear term is approximated through a neural network estimation error. Then, an error-based adaptive robust term is designed to compensate for the neural network estimation error. The adaptive robust terms generate control inputs according to the error function, and near the steady state, it is removed by the reverse saturation function to prevent chattering. We designed neural network adaptive laws and adaptive robust term update laws based on the Lyapunov Theory. The proposed controller proves stability by investigating boundness and proving finite time convergence.

Next, to verify the performance of the proposed controller, we proceeded with computer simulation considering two inputs of snake robot head movement control input and the worst case of the controller. The proposed controller obtained satisfactory performance by comparing the previous robust terms with the controller used in the design step, and minimized the chattering of control input and control output.

In the future, we plan to produce a real snake robot and a snake robot’s head and prove the control performance in real situations.