## 3 Controller Design  
### 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:  
1. The input layer consists of an input vector \(X_{i}\), \(X_{i}=[x_{i1}\,\ x_{i2}\dots x_{im}]^{T}\in R^{m\times 1}\), \((i=1,\dots,n)\), \((z=1,\ \dots,m)\), where \(i\) indicates the number of neural networks and \(z\) indicates the number of inputs for each neural network.
2. 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  
Figure 4: Relationship between \(U\) and \(V(U)\).  
Figure 3: Schematic illustration of backlash associated with the links between input torque and output torque.  
the radial basis function, which is generally a Gaussian function (Figure 5b) and can be expressed as follows:  
\[\mu_{ij}=\exp\left(\frac{-\big{(}x_{iz}-b_{ij}\big{)}^{2}}{\sigma_{ij}^{2}} \right),\ (j=1,\ \ldots,\ k) \tag{17}\]  
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:  
\[\hat{\mathcal{N}}(X_{i})=[f_{1},\ldots,f_{n}]^{T}=W^{T} \tag{18}\]  
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:  
\[W=\begin{bmatrix}w_{11}&\ldots&w_{1k}&\ldots&0&0&0\\ \vdots&\vdots&\vdots&\ldots&\vdots&\vdots&\vdots\\ 0&0&0&\ldots&w_{n1}&w_{n2}&w_{nk}\end{bmatrix}^{T}\in R^{nk\times n} \tag{19}\]  
\[\mu=\begin{bmatrix}\mu_{11}&\cdots&\mu_{1k}&\cdots&\mu_{n1}&\cdots&\mu_{nk} \end{bmatrix}^{T}\in R^{nk\times 1} \tag{20}\]  
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:  
\[\begin{cases}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=z\end{cases} \tag{21}\]  
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:  
\[S(x_{1},x_{2})=0 \tag{22}\]  
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  
Figure 5: (**a**) Structure of RBNN. (**b**) Corresponding Gaussian function.  
the sliding surface (i.e., \(S\
eq 0\)). At this point, the following conditions must be met to ensure that the system reaches the sliding surface within a set period of time:  
\[S\dot{S}<-K|S|,\ \ K>0 \tag{23}\]  
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:  
\[\underset{s\to 0}{\lim}S\dot{S}<0\begin{cases}(1)\ S>0,\ \dot{S}<0\\ (2)\ S<0,\ \dot{S}>0\end{cases} \tag{24}\]