Numerical Simulation of Adaptive Radial Basis NN-Based Non-Singular Fast Terminal Sliding Mode Control with Time Delay Estimator for Precise Control of Dual-Axis Manipulator

Featured Application

The proposed method is suitable for application in robotic arm systems.

Abstract

Robotic manipulators can reduce the cost of production and improve productivity; however, controlling a manipulator to follow a desired trajectory is a thorny problem. In this study, we introduced various forms of interference to facilitate the modeling of a dual-axis manipulator. The interference associated with the payload is handled by an adaptive radial basis neural network (ARBNN) controller, while other interference is estimated by a time delay estimator (TDE). The control signal is output by a non-singular fast terminal sliding mode controller (NFTSMC) to minimize further interference. Since the proposed controller can deal with the payload, system uncertainties, external disturbances, friction, and backlash, compared with conventional control methods, it has better tracking accuracy and stability.

1. Introduction

The application of industrial manipulators can largely eliminate human error, boost product quality, and increase productivity [1,2]. Numerous researchers have sought to develop control methods capable of ensuring that the manipulator follows a desired trajectory. Su et al. [3] proposed a saturated PID controller to deal with motor failure in mechanical arms resulting from excessive input torque. Martinez et al. [4] proposed a static neural network controller that uses a novel Gaussian basis function to facilitate adaptive tracking control. Their controller switches between a neural network controller and a robust proportional derivative (PD) controller. Wai et al. [5] developed an SMC-based position tracking method based on a high-order dynamics model followed by a fuzzy neural network to simulate the original sliding mode control for stability analysis. Zhu et al. [6] developed a novel composite control scheme that introduces uncertainties into the dynamic modeling of a manipulator and then uses a novel dead zone compensation method to attenuate uncertainties and the corresponding nonlinearities. He et al. [7] proposed an adaptive neural network controller to ensure control over the tracking of uncertain n-links manipulators with full-state constraints. Van et al. [8] achieved trajectory tracking control in a system that combines the advantages of a sliding mode controller with those of a backstepping controller. Liu et al. [9] proposed an adaptive bias radial basis neural network controller with local and global bias to improve approximation accuracy in dynamic models with significant bias. When the input deviates from the approximation domain due to excessive payload, a neural network controller degenerates into a PID controller to return the system to its original state. Note, however, that most previous approaches to manipulator control systems considered a limited number of interference sources and dealt with them as a whole.

Our objective in the current paper was to deal with the above issues from two perspectives. First, we established an enhanced second-order dynamic model of the manipulator, which considers payload-related issues as well as those pertaining to model uncertainties, friction, external disturbances, and backlash. We then developed a composite controller based on NFTSMC, which deals with the two classes of interference (payload and other), respectively, using ARBNN and TDE. The proposed control strategy enhances accuracy while reducing the influence of variations in interferences. It also reduces dependence on parameter adjustment and does not require an interference upper bound.

The remainder of this paper is organized as follows: Section 2 introduces the process of designing our enhanced dynamic model of the dual-axis manipulator. Section 3 outlines our advanced control strategy and stability analysis. Section 4 outlines the simulations used to assess the performance of the proposed algorithm. Conclusions are presented in Section 5.

2. Enhanced Dynamic Model

In this section, we examine a dynamic model of a dual-axis manipulator, which can be expressed as follows:

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

(1)

where $M\left(\theta \right)$ refers to the inertia matrix, $C\left(\theta ,\dot{\theta }\right)$ is the centripetal and Coriolis force vector, $G\left(\theta \right)$ is the gravity vector, $U$ is control torque, and $q, \dot{q},\ddot{q}$ respectively indicate the angle, angular velocity, and angular acceleration of the joint. We first consider friction [10], which can be divided into static friction, viscous friction, and Coulomb friction.

In Figure 1, $\mathcal{B}$, ${\mathcal{Q}}_C$, and ${\mathcal{Q}}_s$, respectively, describe the slope of viscous friction, static friction, and Coulomb friction. Note that when the velocity of the manipulator is increased, static friction decreases rapidly, such that viscous friction dominates. Thus, we consider only viscous friction and Coulomb friction in the current study, the formulas of which can be expressed as follow

$$f\left(\dot{q}\right)=\mathcal{B}\dot{q}+{\mathcal{Q}}_C$$

(2)

and

$${\mathcal{Q}}_C=\left\{\begin{array}{c} 0 \dot{q}=0 \\ {\mathcal{Q}}_C^{+} \dot{q}>0\\ {\mathcal{Q}}_C^{-} \dot{q}<0\end{array}\right.$$

(3)

Thus, the dynamic model of the manipulator can be rewritten as:

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

(4)

Next, we consider the external disturbance $d\left(t\right)$ to which the manipulator is most susceptible in practical applications. There are numerous forms of external disturbance, and they can be very difficult to model; therefore, we used white noise with an upper bound to represent external disturbances acting on the two joints of the manipulator, which can be expressed as follows:

$$\left|d\left(t\right)\right|\le \overline{d}\left(t\right)$$

(5)

where $\overline{d}\left(t\right)$ is an unknown positive constant vector. Thus, the dynamic model of the manipulator can be rewritten as

$$M\left(q\right)\ddot{q}+C\left(q,\dot{q}\right)\dot{q}+G\left(q\right)+f\left(\dot{q}\right)+d\left(t\right)=U$$

(6)

We also considered the effects of payload properties on the manipulator (a common issue in practical applications). Consider the example in Figure 2, illustrating the process of grasping the payload, moving it to the destination, and putting it down. During this process, the payload is converted into forces and torques acting on the joints and links, such that the dynamic model of the manipulator can be rewritten as follows:

$$M\left(q\right)\ddot{q}+C\left(q,\dot{q}\right)\dot{q}+G\left(q\right)+f\left(\dot{q}\right)+d\left(t\right)+U_p=U$$

(7)

Next, we considered the uncertainties in the manipulator model associated with system parameters. The basic dynamic model outlined above includes three main parameters: inertia matrix $M\left(q\right)$, the centripetal force/Coriolis force matrix $C\left(q,\dot{q}\right)$ and the gravity vector $G\left(q\right)$. In the process of establishing a dynamic model, these parameters are generally derived empirically. Thus, we decompose the system parameters into two classes (known items and model uncertainties), which can be expressed as follows:

$$M\left(q\right)=M_0\left(q\right)+\Delta M\left(q\right)$$

(8)

$$C\left(q,\dot{q}\right)=C_0\left(q,\dot{q}\right)+\Delta C\left(q,\dot{q}\right)$$

(9)

$$G\left(q\right)=G_0\left(q\right)+\Delta G\left(q\right)$$

(10)

Thus, the dynamic model of the manipulator can be expressed as follows:

$$\begin{gathered} M_0\left(q\right)\ddot{q}+C_0\left(q,\dot{q}\right)\dot{q}+G_0\left(q\right)+\Delta M\left(q\right)\ddot{q}+\Delta C\left(q,\dot{q}\right)\dot{q} \\ +\Delta G\left(q\right)+f\left(\dot{q}\right)+d\left(t\right)+U_p=U \end{gathered}$$

(11)

Finally, we consider backlash [11,12], which refers to the combined time delay effects of occlusions among gears, chains, and belts used in the transmission of power from the motor to other components (see Figure 3). This phenomenon can be described using a class of nonlinear equations as follows [13]:

$$V\left(U\right)=\left\{\begin{array}{l}\mathrm{{\rm Y}}\left(U-\frac{b}{2}\right) , if \dot{U}>0 and U\ge U\left(t-1\right)+b/2\\ \mathrm{{\rm Y}}\left(U+\frac{b}{2}\right) , if \dot{U}<0 and U\le U\left(t-1\right)-b/2 \\ V{\left(U\right)}_{\left(t^{-}\right)} , otherwise\end{array}\right.$$

(12)

where ${\rm Y}$ refers to the slope, $b$ indicates the size of the dead zone, $U$ indicates the control torque, and $V\left(U\right)$ indicates the control torque after backlash. This relationship is illustrated in Figure 4.

We can rewrite Equation (12) as follows:

$$V\left(U\right)={\rm Y}U+\Gamma \left(U\right)$$

(13)

where $\mathrm{\Gamma }\left(U\right)$ is expressed as

$$\mathrm{\Gamma }\left(U\right)=\left\{\begin{array}{l}-\mathrm{{\rm Y}}\frac{b}{2}, if \dot{U}>0 and U\ge U\left(t-1\right)+b/2\\ \mathrm{{\rm Y}}\frac{b}{2}, if \dot{U}<0 and U\le U\left(t-1\right)-b/2\\ V{\left(U\right)}_{\left(t^{-}\right)}-\mathrm{{\rm Y}}U , otherwise\end{array}\right.$$

(14)

Our enhanced dynamic model of a dual-axis manipulator under the effects of multiple forms of interference can be expressed as follows:

$$M_0\ddot{q}+C_0\dot{q}+G_0+\Xi \left(q,\dot{q},\ddot{q},t\right)+U_P=V\left(U\right)={\rm Y}U+\Gamma \left(U\right)$$

(15)

where, for convenience, we specify that

$$\Xi \left(q,\dot{q},\ddot{q},t\right)=\Delta M\ddot{q}+\Delta C\dot{q}+\Delta G+f\left(\dot{q}\right)+d\left(t\right)$$

(16)

3. Controller Design

3.1. Preliminaries

As shown in Figure 5a, the radial basis neural network (RBNN) provides a simple control structure and the ability to approximate any nonlinear function with arbitrary accuracy without the need for lengthy calculations [14,15]. RBNN employs an input layer, a hidden layer, and an output layer:

• The input layer consists of an input vector $X_i$, $X_i={\left[x_{i1},x_{i2}\dots x_{im}\right]}^T\in R^{m\times 1},\left(i=1,\dots ,n\right),\left(z=1,\dots ,m\right)$, where $i$ indicates the number of neural networks and $z$ indicates the number of inputs for each neural network.
• Note that there is no weight connection between the hidden and input layers, and the hidden layer is composed of neurons. The activation function in a neuron comprises the radial basis function, which is generally a Gaussian function (Figure 5b) and can be expressed as follows:

  $${\mu }_{ij}=\exp \left(\frac{-{\left(x_{iz}-b_{ij}\right)}^2}{{\sigma }_{ij}^2}\right), \left(j=1, \dots , k\right)$$

  (17)

  where 𝑗 indicates the number of neurons in the hidden layer of each neural network, while $b_{ij}\in R^{n\times k}$ and ${\sigma }_{ij}\in R^{n\times k}$ refer to the mean vector and the width of the Gaussian function of the $j^{th}$ node in the hidden layer in the $i^{th}$ neural network.
• The output layer describes the control signal of RBNN as follows:

  $$\widehat{\mathcal{N}}\left(X_i\right)={\left[f_1,\dots ,f_n\right]}^T=W^T$$

  (18)

  where $W$ refers to the weight between the hidden layer and the output layer and $\mu$ is the output of the hidden layer, as follows:

  $$W={\left[\begin{array}{ccccccc}{w}_{11}& \dots & w_{1k}& \dots & 0& 0& 0\\ ⋮& ⋮& ⋮& \dots & ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & w_{n1}& w_{n2}& w_{nk}\end{array}\right]}^T\in R^{nk\times n}$$

  (19)

  $$\mu ={\left[\begin{array}{ccccccc}{\mu }_{11}& \cdots & {\mu }_{1k}& \cdots & {\mu }_{n1}& \cdots & {\mu }_{nk}\end{array}\right]}^T\in R^{nk\times 1}$$

  (20)

The simple structure of the RBF neural network described above allows it to be used online to estimate expected values over time, the accuracy of which is proportional to the number of neurons in the hidden layer.

We also employed a sliding mode control (SMC) [16,17,18] nonlinear control method. As shown in Figure 6, SMC uses a discontinuous control signal to slide the nonlinear system between two states, forcing the system to finally enter a desired control result through the sliding surface. The system to be controlled is first written out in the form of a state-space equation, an example of which is presented as follows:

$$\left\{\begin{array}{c}{\dot{x}}_1=x_2 \\ {\dot{x}}_2=z \end{array}\right.$$

(21)

where $x_i$, $i=1,2$ refers to system states and $z$ is the control input. The next step involves designing sliding surface $S$, which is a function related to the system state satisfying the following:

$$S\left(x_1,x_2\right)=0$$

(22)

The sliding mode control method can be divided into two states: approaching and sliding. The approaching state indicates a situation in which the system has not yet reached the sliding surface (i.e., $S\ne 0$). At this point, the following conditions must be met to ensure that the system reaches the sliding surface within a set period of time:

$$S\dot{S}<-K\left|S\right|, K>0$$

(23)

The sliding state indicates a situation in which the system has reached the sliding surface ($S=0$). At this point, the system rapidly switches between the upper and lower sides of the sliding surface until converging to the point of equilibrium, at which point the system achieves the required control performance. The conditions required for this process are as follows:

$$\underset{s\to 0}{\lim }S\dot{S}<0 \left\{\begin{array}{c}\left(1\right) S>0, \dot{S}<0\\ \left(2\right) S<0, \dot{S}>0\end{array}\right.$$

(24)

3.2. Proposed Controller Design

Figure 7 is the proposed controller, which simultaneously combines the adaptive radial basis neural network (ARBNN) [19], time delay estimator (TDE) [20], and non-singular fast terminal sliding mode control (NFTSMC) [21]. The ARBNN function is to estimate the payload part, using adaptive control to adjust the weights of neurons in the hidden layer. The TDE is used to deal with system uncertainties, external disturbances, friction, and backlash. Finally, the NFTSMC is used to ensure that the entire system remains stable. The proposed controller is to control the output torque of the motor of each joint based on the predefined desired trajectory in the joint space. The predefined trajectory is based on the dynamic model of the manipulator in Section 2. The force and position relationship could be found in [22].

3.2.1. Adaptive Radial Basis NN (ARBNN)

The dynamic model of the dual-axis manipulator is established as follows:

$$M_0\left(q\right)\ddot{q}+C_0\left(q,\dot{q}\right)\dot{q}+G_0\left(q\right)+\Xi \left(q,\dot{q},\ddot{q},t\right)+U_p=V\left(U_{DM}\right)$$

(25)

where ${\rm Y}$ is set to a constant value of 1.0 for the sake of convenience and $U_{DM}$ refers to the control torque of dual-axis manipulator. We can rewrite Equation (25) in state-space form divided into four parts, as follows:

$$\ddot{q}=M_0^{-1}{U}_{DM}+M_0^{-1}\left(-C_{0}q-G_0\right)+M_0^{-1}\left(-U_p\right)+M_0^{-1}\left(-\Xi +\Gamma \left(U_{DM}\right)\right)$$

(26)

where $M_0^{-1}{U}_{DM}$ refers to the control signal, $M_0^{-1}\left(-C_{0}q-G_0\right)$ is the known term, $M_0^{-1}\left(-U_p\right)$ is the unknown payload term, and $M_0^{-1}\left(-\Xi +\Gamma \left(U_{DM}\right)\right)$ is the unknown interferences and uncertainties beyond the payload term. Equation (26) can then be replaced by the following:

$$\ddot{q}=M_0^{-1}{U}_{DM}+F+\mathcal{N}+\Delta \left(q,\dot{q},U,t\right)$$

(27)

where we define the following for convenience: $F=M_0^{-1}\left(-C_{0}q-G_0\right)$, $\mathcal{N}=M_0^{-1}\left(-U_p\right)$, and $\Delta \left(q,\dot{q},U,t\right)=M_0^{-1}\left(-\Xi +\Gamma \left(U_{DM}\right)\right)$.

The trajectory tracking error is defined as $e\triangleq q-q_d$ and the actual and desired trajectories are indicated by $q$ and $q_d$, respectively.

We select an NFTSMC surface that combines the advantages of the conventional sliding film and the terminal sliding mode, allowing the error to converge quickly to zero in a finite time [23]. The NFTSMC surface is as follows:

$$s=e+c_1{e}^{\alpha }+c_2{\dot{e}}^{\left(\frac{p}{l}\right)}={\left[s_1 s_2\right]}^T$$

(28)

where $c_1$, $c_2\in R^{n\times n}$ are diagonal positive definite matrixes, and 𝑝 and 𝑙 are positive odd numbers satisfying the relation $1<p/l < 2 \mathrm{and} \alpha > p/l$, such that Equation (28) can be derived as

$$\dot{s}=\dot{e}+c_1\alpha {\left|e\right|}^{\alpha -1}\otimes \dot{e}+c_2\left(\frac{p}{l}\right){\left|\dot{e}\right|}^{\left(\frac{p}{l}-1\right)}\otimes \ddot{e}={\left[{\dot{s}}_1 {\dot{s}}_2\right]}^T$$

(29)

where $\left(·\right)\otimes \left(·\right)$ means that $\mathcal{A}\otimes \mathcal{B}={\left[a_1{b}_1 a_2{b}_2\dots a_n{b}_n\right]}^T$ if $\mathcal{A},\mathcal{B}\in R^{n\times 1}$. For the sake of simplicity, we stipulate that $O_1\triangleq c_1\alpha {\left|e\right|}^{\alpha -1}\in R^{n\times 1}$, $O_2\triangleq c_2\left(\frac{p}{l}\right){\left|\dot{e}\right|}^{\left(\frac{p}{l}-1\right)}\in R^{n\times 1}$, which are both positive. In accordance with the design procedures for sliding mode control, let $\dot{s}=0$ and substitute Equation (27) into Equation (29) to obtain the following:

$$\dot{s}=\dot{e}+O_1\otimes \dot{e}+O_2\otimes \left(\begin{array}{c}{M}_0^{-1}{U}_{DM}+F+\mathcal{N}+\\ \Delta \left(q,\dot{q},U,t\right)-{\ddot{q}}_d\end{array}\right)=0$$

(30)

Thus, if the control strategy based on NFTSMC satisfies $\left|\Delta +\mathcal{N}\right|\le \eta \in R^{n\times 1}$ (where $\eta$ is a small positive vector), then the equivalent control signal $U_{eq}$ of this system can be derived as follows:

$$U_{eq}=O_2^{-1}\otimes \left(-\dot{e}-O_1\otimes \dot{e}\right)+M_0{\ddot{q}}_d-F={\left[u_{eq1} u_{eq2}\right]}^T$$

(31)

At the same time, the uncertainties and disturbances are dealt with using switching control signal $U_r$, which ensures that the system continuously moves along the sliding surface:

$$U_r=-\left(\eta +\delta \right)\otimes \mathrm{sign}\left(s\right)={\left[u_{r1} u_{r2}\right]}^T$$

(32)

where $\delta \in R^{n\times 1}$ is a small vector of positive constants. Thus, the overall control input can be expressed as follows:

$$U_{DM1}=M_0{U}_{eq}+M_0{U}_r$$

(33)

Based on the design of the control input signal, we selected a candidate Lyapunov function as $V_{DM1}=\frac{1}{2}{s}^{T}s$, which, when differentiated, gives us ${\dot{V}}_{DM1}=s^T\dot{s}$. When we combine Equations (30)–(33), the resulting function satisfies the inequality ${\dot{V}}_{DM1}\le -O_2^T\delta \otimes \left|s\right|$. Furthermore, both $O_2$ and $\delta$ are positive, while $\left|s\right|\ge 0$, such that $-O_2^T\delta \otimes \left|s\right|\le 0$, thereby indicating that the entire system is Lyapunov stable.

The NFTSMC-based controller is able to provide the basic control required for nonlinear systems; however, it cannot ensure stability and robustness in the face of multiple complex disturbances and uncertainties. On the one hand, switching control input $U_r$ requires a known maximum upper bound on overall interferences; however, this is seldom satisfied in practical scenarios. Note also that this control strategy depends on the design of the sliding surface parameters $c_1$ and $c_2$, the debugging of which would be a tedious process.

We sought to overcome the shortcomings of this controller by employing an additional controller to deal with other (multiple) forms of disturbance and uncertainty while reducing dependence on the NFTSM control strategy. As mentioned previously, the dual-axis manipulator addressed in this paper includes payload $U_p$, external disturbances $d\left(t\right)$, system uncertainties $\Delta M\left(q\right)\ddot{q}, \Delta C\left(q,\dot{q}\right)\dot{q}, \Delta G\left(q\right)$, friction force $f\left(\dot{q}\right)$, and backlash $V\left(U\right)$. We assigned the payload to an estimator and the other factors to the controller.

In dealing with the payload, ARBNN was used for estimation and compensation tasks. Manipulators commonly encounter problems related to the payload when grasping heavy objects or using end-effectors. We previously defined $\mathcal{N}=M_0^{-1}\left(-U_p\right)$ as the portion of the payload to be estimated by the ARBNN, which can be expressed as follows:

$$\mathcal{N}={\mathcal{N}}^{*}+\Delta \mathcal{N}=W^{*}{}^T\mu +\Delta \mathcal{N}$$

(34)

where ${\mathcal{N}}^{*}$ is the optimal estimate achievable by the neural network, $\Delta \mathcal{N}$ is the neural network reconstruction error, and $W^{*}$ is the optimal weight. We can express the control input $U_{NN}$ estimate $\widehat{\mathcal{N}}$ of the neural network as follows:

$$U_{NN}=\widehat{\mathcal{N}}={\widehat{W}}^T\mu ={\left[u_{NN1} u_{NN2}\right]}^T$$

(35)

where $\widehat{W}$ is the estimated weight, such that estimation error $\tilde{\mathcal{N}}$ can be derived in a straightforward manner, as follows:

$$\tilde{\mathcal{N}}=\mathcal{N}-\widehat{\mathcal{N}}={\mathcal{N}}^{*}+\Delta \mathcal{N}-\widehat{\mathcal{N}}=-{\tilde{W}}^T\mu +\Delta \mathcal{N}$$

(36)

$$\tilde{W}=\widehat{W}-W^{*}$$

(37)

3.2.2. Time Delay Estimator (TDE)

The time delay estimator (TDE) is used to estimate the magnitudes of the other (i.e., non-payload-related) disturbances and uncertainties, which is a type of disturbance estimator. TDE is a simple, highly capable estimation method for the estimation of uncertainty terms using dynamic model equations in accordance with known state variables and their derivatives. In other words, TDE modifies the control signal directly using previous observations of system responses and control inputs rather than by adjusting the controller gain. It follows, therefore, that estimation accuracy is inversely proportional to time delay $\Delta t$.

For the dual-axis manipulator system dealt with in this paper, Equation (27) stipulates that the target term $\Delta \left(q,\dot{q},U,t\right)$ can be expressed as follows:

$$\Delta \left(q,\dot{q},U,t\right)=\ddot{q}-M_0^{-1}U+F+\mathcal{N}$$

(38)

In estimating this term, we assume that its value at the current time (i.e., $t_0$) is very close to the value at $t_0-\Delta t$, if time delay $\Delta t$ is sufficiently small, such that

$$\Delta \left(q,\dot{q},U,t\right){|}_{t=t_0}\cong \Delta \left(q,\dot{q},U,t\right){|}_{t=t_0-\Delta t}$$

(39)

Thus, we define the output value of the TDE as

$$\widehat{\Delta }\left(q,\dot{q},U,t\right){|}_{t=t_0}\triangleq \Delta \left(q,\dot{q},U,t\right){|}_{t=t_0-\Delta t}=\left(\ddot{q}-\left(M_0^{-1}U\right)-F-\mathcal{N}\right){|}_{t=t_0-\Delta t}$$

(40)

The estimation process generates estimation error $\epsilon$ satisfying the following equation:

$$\Delta \left(q,\dot{q},U,t\right){|}_{t=t_0}=\widehat{\Delta }\left(q,\dot{q},U,t\right){|}_{t=t_0}+\epsilon$$

(41)

The output value of the time delay estimator is

$$U_{TDE}=\widehat{\Delta }\left(q,\dot{q},U,t\right){|}_{t=t_0}={\left[u_{TDE1} u_{TDE2}\right]}^T$$

(42)

Accordingly, we can use ARBNN ($U_{NN}$) to estimate the payload portion without a known upper bound in conjunction with TDE ($U_{TDE}$) to estimate factors other than payload. The equivalent control ($U_{eq}$) and switching control ($U_r$) of NFTSMC can then be combined to output the final control signal as long as the estimation error of ARBNN ($\Delta N$) and estimation error of TDE ($\epsilon$) satisfy the following inequalities:

$$\left|\Delta \mathcal{N}\right|\le \phi \in R^{ n\times 1}$$

(43)

$$\left|\epsilon \right|\le \varrho \in R^{n\times 1}$$

(44)

where $\phi$ and $\varrho$ are both positive vectors. Accordingly, the control input of the proposed controller can be redesigned as

$$U_{DM2}=M_0{U}_{eq}-M_0{U}_{NN}-M_0{U}_{TDE}-M_0{U}_r$$

(45)

where $U_{eq}=O_2^{-1}\otimes \left(-\dot{e}-O_1\otimes \dot{e}\right)-F-\varpi s+{\ddot{q}}_d$, $\varpi \in R^{n\times n}$ is a small positive definite diagonal matrix, and $U_r=\left(\delta +\phi +\varrho \right)\otimes \mathrm{sign}\left(s\right)$.

3.3. Stability Analysis

We define a candidate Lyapunov function $V_{DM2}$ (i.e., a positive definite function) as follows:

$$V_{DM2}=\frac{1}{2}{s}^{T}s+\frac{1}{2}tr\left({\tilde{W}}^T{\Lambda }^{-1}\tilde{W}\right)$$

(46)

where $\Lambda \in R^{nk\times nk}$ is a positive definite matrix. Differentiating Equation (46) with respect to time, we obtain

$${\dot{V}}_{DM2}=s^T\dot{s}+tr\left({\tilde{W}}^T{\Lambda }^{-1}\dot{\tilde{W}}\right)$$

(47)

According to Equation (30), the first term in Equation (47) can be rewritten as follows:

$$\begin{array}{c}{s}^T\dot{s}=s^T\left(\dot{e}+O_1\otimes \dot{e}+O_2\otimes \ddot{e}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=s^T\left(\dot{e}+O_1\otimes \dot{e}+O_2\otimes \left(\begin{array}{c}{M}_0^{-1}{U}_{DM2}+F+\mathcal{N}\\ +\Delta \left(q,\dot{q},U,t\right)-{\ddot{q}}_d\end{array}\right)\right)\end{array}$$

(48)

Substituting Equation (45) into Equation (48), we obtain

$$s^T\dot{s}=s^T{O}_2\otimes \left(\tilde{\mathcal{N}}-\varpi s+\epsilon -\left(\left(\delta +\phi +\varrho \right)\otimes \mathrm{sign}\left(s\right)\right)\right)$$

(49)

Then, substituting Equation (49) into Equation (47), we obtain

$$\begin{array}{c}{\dot{V}}_{DM2}=s^T{O}_2\otimes \left(\begin{array}{c}\tilde{\mathcal{N}}-\varpi s+\epsilon \\ -\left(\left(\delta +\phi +\varrho \right)\otimes \mathrm{sign}\left(s\right)\right)\end{array}\right)+tr\left({\tilde{W}}^T{\Lambda }^{-1}\dot{\tilde{W}}\right)\\ \hspace{1em}\hspace{1em}=s^T{O}_2\otimes \left(\begin{array}{c}-{\tilde{W}}^T\mu +\Delta N-\varpi s+\epsilon \\ -\left(\left(\delta +\phi +\varrho \right)\otimes \mathrm{sign}\left(s\right)\right)\end{array}\right)+tr\left({\tilde{W}}^T{\Lambda }^{-1}\dot{\tilde{W}}\right)\end{array}$$

(50)

Based on Equation (37), we know that $\dot{\tilde{W}}=\dot{\widehat{W}}-{\dot{W}}^{*}=\dot{\widehat{W}}$, such that we obtain the following:

$${\dot{V}}_{DM2}=s^T{O}_2\left(\begin{array}{c}-{\tilde{W}}^T\mu +\Delta N-\varpi s+\epsilon \\ -\left(\left(\delta +\phi +\varrho \right)\otimes \mathrm{sign}\left(s\right)\right)\end{array}\right)+tr\left({\tilde{W}}^T{\Lambda }^{-1}\dot{\widehat{W}}\right)$$

(51)

The adaptive law used to adjust the RBNN weights is as follows:

$$\dot{\widehat{W}}=\Lambda \left(\mu {(O_2\otimes s)}^T-\varphi \widehat{W}\right)$$

(52)

where $\varphi \in R^{nk\times nk}$ is a positive definite matrix. Substituting Equation (52) into Equation (51) yields

$${\dot{V}}_{DM2}=-s^T\left(O_2\otimes \varpi s\right)-\Psi -tr\left({\tilde{W}}^T\varphi \widehat{W}\right)$$

(53)

where $\Psi =s^T\left(O_2\otimes \left(\left(\delta +\phi +\epsilon \right)\otimes \mathrm{sign}\left(s\right)-\Delta \mathcal{N}-\epsilon \right)\right)$.

Thus, we obtain the following:

$$-\left({\tilde{W}}^T\varphi \widehat{W}\right)\le \frac{1}{2}\left(W^{*}{}^T\varphi W^{*}-W^T\varphi \tilde{W}\right)$$

(54)

Accordingly, we can conclude that

$${\dot{V}}_{DM2}\le -s^T{O}_2\varpi s-\Psi +\frac{1}{2}tr\left(W^{*}{}^T\varphi W^{*}-{\tilde{W}}^T\varphi \tilde{W}\right)\le -\rho V_{DM2}+\frac{1}{2}tr\left(W^{*}{}^T\varphi W^{*}\right)$$

(55)

where $\rho =\min \left({\lambda }_{min}\left(O_2\varpi \right), {\lambda }_{min}\left(\varphi \Lambda \right)\right)$. Based on Equation (54), we can conclude that the error will eventually be bounded.

4. Simulations

In this section, we outline simulations of the dual-axis manipulator system (implemented in MATLAB/Simulink) aimed at assessing the feasibility of the proposed controller. The simulation results were then compared with those of existing controllers.

4.1. Simulation Settings

Simulations were performed using the dynamic parameters of a dual-axis manipulator [24] (see

Table 1. Parameter values of basic manipulator model.

Parameters	Descriptions	Values
$g$	Gravity	9.81 $\mathrm{m}/{\mathrm{s}}^2$
$m_1$	Mass of link 1	2.0 kg
$m_2$	Mass of link 2	0.85 kg
$l_1$	Length of link 1	0.35 m
$l_2$	Length of link 2	0.31 m
$l_{c1}$	Half Length of link 1	0.175 m
$l_{c2}$	Half Length of link 2	0.155 m
$I_1$	Inertia of link 1	$61.25\times {10}^{-3} {\mathrm{kgm}}^2$
$I_2$	Inertia of link 2	$20.42\times {10}^{-3}{\mathrm{kgm}}^2$

) under the following uncertainties and disturbances:

$$\Delta M=0.2M, \Delta C=0.2C, \Delta G=0.2G, \mathcal{B}=0.3, d_1=d_2=0.01, b=0.1$$

$${\mathcal{Q}}_C=\left\{\begin{array}{l}0 \dot{q}=0\\ 0.285 \dot{q}>0\\ -0.285 \dot{q}<0\end{array}\right.,U_p=\left\{\begin{array}{c}0.5\sin \left(t\right)\\ 0.3\sin \left(t\right)\end{array}\right.$$

The proposed controller was compared against four existing controllers, including a Proportional–Integration–Derivative (PID) controller, NFTSMC, and NFTSMC in conjunction with TDE. All controller parameters were obtained by trial and error, as follows: PID ($P=100$,$I=50$, $D=100$); NFTSMC $(p=9$, $l=7$, $\alpha =1.5$, $c_1=diag\left[1120320\right]$, $c_2=diag\left[2515\right]$); NFTSMC in conjunction with TDE $(\Delta t$ = ${10}^{-4}$s); proposed controller ($\Lambda =I$, $\varphi =I$, $\varpi =diag\left[1 1\right]$).

As shown as Figure 8, the performance of the controllers in the simulations was evaluated while following periodic trajectories ($q_d={\left[0.5\sin \left(t\right),0.5\cos \left(t\right)\right]}^T$) and aperiodic trajectories

$$q{d}_1=\left\{\begin{array}{c}0.5, t=0\\ \frac{0.5\sin \left(\pi t\right)}{\pi t}, t\ne 0\end{array}\right. \mathrm{and} q{d}_2=\left\{\begin{array}{c}0.3, t=0\\ \frac{0.3\sin \left(\pi t\right)}{\pi t}, t\ne 0\end{array}\right.$$

where $t\in \left[0, t_f\right]$ and $t_f=10 \mathrm{s}$.

4.2. Simulation Results

4.2.1. Tracking Periodic Trajectories

In the simulations, we assumed that the robotic manipulator would be performing repetitive tasks; i.e., tracking periodic trajectories. Figure 9a presents the position tracking performance for joint 1. Note that in Figure 9a, the PID controller clearly underperformed the other controllers. The reason is that the PID cannot handle the nonlinear model well. Figure 9b presents a partial enlargement of Figure 9a near $t=6.59 \mathrm{s}$. We can see that NFTSMC and NFTSMC+TDE poorly dealt with disturbances that occurred at the points of position tracking, where the corresponding disturbances were mainly caused by backlash and friction. In contrast, the proposed controller handled these two disturbances very well; thus, the result clearly outperformed the other controllers. Figure 10a presents the position tracking performance for joint 2. Note that we obtain similar results in the use of different controllers. Figure 10b presents a partial enlargement of Figure 10a near $t=4.92 \mathrm{s}$, from which we can see that the proposed controller is still relatively the best among the four controllers.

Since this simulation uses the differential method to obtain the velocity value of the system, we consider that this method will amplify the noise. Therefore, a simple first-order low-pass filter is used with a time constant of 0.2 for all controllers’ velocity signals in order to obtain more realistic velocity tracking results. Figure 11 and Figure 12 present the velocity tracking performance for joints 1 and 2. Overall, the proposed system was far less effective in controlling velocity than in controlling position. The reason for this phenomenon is that the velocity signal is delayed after the low-pass filter, but it can filter most of the noise caused by the differential methods so that we can compare the tracking results of each controller more clearly. Nonetheless, we can see in Figure 11 and Figure 12 that the proposed controller outperformed the other controllers in controlling the velocity of joint 1.

In order to show the tracking situation of the four controllers more intuitively, the root mean square error (RMSE) of the four controllers in tracking periodic trajectories is calculated, and the results are shown in

Table 2. RMSE of tracking results for periodic trajectories.

Controller	Joint 1		Joint 2
	Position $1\times {10}^{-4}\left(\mathbf{rad}\right)$	Velocity $1\times {10}^{-4}(\mathbf{rad}/\mathbf{s})$	Position $1\times {10}^{-4}\left(\mathbf{rad}\right)$	Velocity $1\times {10}^{-4}(\mathbf{rad}/\mathbf{s})$
PID	282.000	1143.000	59.000	730.000
NFTSMC	15.000	694.000	65.000	717.000
NFTSMC+TDE	2.716	676.000	3.892	693.000
Proposed	0.080	674.000	0.214	692.000

. We can see that the RMSE results of the proposed controller are the best. The values of position tracking RMSE for the proposed controller joints 1 and 2 are 0.080 $\times {10}^{-4}$ rad and 0.214 $\times {10}^{-4}$ rad, respectively. In terms of velocity tracking, although affected by noise, the proposed controller still has pretty good performances, proving the proposed controller’s effectiveness and superiority in tracking periodic trajectories.

4.2.2. Tracking Aperiodic Trajectories

We also examined the performance in controlling aperiodic trajectories. Figure 13a,b present the position tracking results for joint 1 and Figure 14a,b present the position tracking results for joint 2. In the process of tracking aperiodic trajectories, the position tracking results of joint 1 show that the PID controller performs relatively worst, and the proposed controller performs the best. In addition, the position tracking results of joint 2 show that the proposed controller is also the best. The results validate that the proposed controller can also track aperiodic position trajectories well.

Figure 15a,b present the velocity tracking results for joint 1 and Figure 16a,b present the velocity tracking results for joint 2. The results show that the NFTSMC controller performs best in tracking the velocity of aperiodic trajectories, which is better than the proposed controller and NFTSMC+TDE. The reason for this situation is that the TDE is based on a dynamic model to estimate the disturbance term. This process requires acceleration $\ddot{q}$; however, the differentiation used will further amplify the noise and affect the estimation accuracy of the TDE. Therefore, the accuracy of the velocity tracking is reduced. However, we can see the results of Figure 15b and Figure 16b that under the cases of both using TDE, the velocity tracking accuracy of the proposed controller is better than that of NFTSMC+TDE.

Table 3. RMSE of tracking results for aperiodic trajectories.

Controller	Joint 1		Joint 2
	Position $1\times {10}^{-3}\left(\mathbf{rad}\right)$	Velocity $1\times {10}^{-3}(\mathbf{rad}/\mathbf{s})$	Position $1\times {10}^{-3}\left(\mathbf{rad}\right)$	Velocity $1\times {10}^{-3}(\mathbf{rad}/\mathbf{s})$
PID	26.600	110.200	6.400	52.700
NFTSMC	1.900	81.100	7.800	54.300
NFTSMC+TDE	2.200	107.900	15.500	56.800
Proposed	1.200	82.600	3.800	53.800

shows the RMSE of the proposed controller and other controllers when tracking aperiodic trajectories. Compared with tracking periodic trajectories, we can find that the tracking aperiodic trajectories vary more within a short period of time, and the noise changes are more complicated, affecting the tracking accuracy of each controller. However, the proposed controller’s position tracking is still relatively the best, with 1.2 $\times {10}^{-3}$ rad and 3.8 $\times {10}^{-3}$ rad, respectively. In terms of velocity tracking, because of the influence of noise in the tracking process of TDE, the RMSE results of the proposed controller and NFTSMC+TDE are not as good as NFTSMC. However, comparing the proposed controller with NFTSMC+TDE, we can find that the proposed controller performs better, thus proving that the proposed controller has better tracking accuracy and stability.

We further discussed the weight variances for the periodic and aperiodic trajectories. Figure 17 presents the weight change; the results can be seen in Figure 17a. Since the periodic trajectory is easier to track, thus, the weight changes are minor. The magnitude of the weight changes is about around $1\times {10}^{-5}$, which is presented in Figure 17b. Conversely, the aperiodic trajectory has a more significant vibration in the weight changes of Figure 17a,b because it is not easy to track. The results of weight changes can help us understand why the tracking performances of the two trajectories are different.

5. Conclusions

This study established an enhanced second-order dynamic model of manipulators, which considers payload-related issues as well as model uncertainties, friction, external disturbances, and backlash. We then developed a composite controller based on NFTSMC, which deals with the two classes of interference (payload and other), respectively, using ARBNN and TDE. In simulations, the proposed system outperformed state-of-the-art control systems when applied to periodic as well as aperiodic trajectories.