## 3 Position Tracking Control Design  
To simplify the expression, the established AUV actual dynamic model with a nonlinear random disturbance term is re-expressed as:  
\[\left\{ \begin{array}{l}\dot{u}=M_{1}(-X_{u}u+a_{11}vr+f_{1}(u,t)+F_{u})\\ \dot{v}=M_{2}(-Y_{v}v+a_{22}ur)\\ \dot{r}=M_{3}(-N_{r}r+a_{33}uv+f_{2}(r,t)+F_{r})\end{array} \right. \tag{9}\]  
where  
\[ \begin{array}{l}M_{1}=\frac{1}{m-X_{\dot{v}}},\ M_{2}=\frac{1}{m-Y_{\dot{v} }},\ M_{3}=\frac{1}{I_{z}-N_{r}}\\ a_{11}=m-Y_{\dot{v}},\ a_{22}=X_{\dot{\mu}}-m,\ a_{33}=Y_{\dot{v}}-X_{\dot{\mu}} \end{array} \]  
The reference model needed in the MRAC system is the desired model, which is a reference model with the desired dynamic performance. This model is selected according to the desired system dynamic performance index. The purpose of adaptive control is to make the output of the controlled system asymptotically follow the output of the specification (reference) model so that the system can meet the dynamic performance required by the system and finally make the output error (also called adaptive control error) asymptotically consistent and guaranteed to converges to zero. The control method proposed in this paper is shown in Figure 2.  
The reference model of the MRAC system is obtained by discarding the nonlinear term in the existing AUV nonlinear dynamic model (10). The output of this model is the reference output for RBF-supervised training. \[\left\{ \begin{array}{rl}\dot{u}_{m}=M_{1}(-X_{u}u_{m}+a_{11}v_{m}r_{m}+F_{u} )\\ \dot{v}_{m}=M_{2}(-Y_{v}v_{m}+a_{22}u_{m}r_{m})\\ \dot{r}_{m}=M_{3}(-N_{r}r_{m}+a_{33}4m_{v}v_{m}+F_{r})\end{array} \right. \tag{10}\]  
The trajectory tracking error is defined as:  
\[\left\{ \begin{array}{rl}x_{e}=x-x_{m}\\ y_{e}=y-y_{m}\end{array} \right. \tag{11}\]  
where \(x_{m},y_{m}\) are the expected time-varying position coordinates. Through the (5) AUV kinematics equation, the dynamic equation of the position error can be obtained, and the velocity tracking error is given. \[\left[ \begin{array}{rl}\dot{x}_{e}\\ \dot{y}_{e}\end{array} \right]=\left[ \begin{array}{rl}\cos\psi&-\sin\psi\\ \sin\psi&\cos\psi\end{array} \right]\left[ \begin{array}{rl}u\\ v\end{array} \right]-\left[ \begin{array}{rl}\dot{x}_{m}\\ \dot{y}_{m}\end{array} \right] \tag{12}\]  
\[\left\{ \begin{array}{rl}e_{u}=u-u_{m}\\ e_{v}=v-v_{m}\end{array} \right. \tag{13}\]  
Because biological neurons have local responses, some researchers introduced RBF into the neural network design, resulting in an RBF neural network. And, it is found that an RBF neural network has good approximation performance to nonlinear continuous functions. Since the RBF network has only one hidden layer, the control effect is excellent, and it is easier to deploy in controllers and computing modules with low computing power. \(R_{j}(x)\) is the radial basis function, \(j=1,2,3,\ldots,n\). The input of the RBF network is \(x=[x_{1}(k),x_{2}(k),\cdots x_{n}(k)]^{T}\), so the number of nodes in the input layer is \(n\). The AUV system input has the surge force \(F_{u}\) and yaw moment \(F_{r}\). The two propellers installed at the tail can simultaneously provide horizontal force and moment, so the horizontal and surge forces are controlled separately. Since the calculation  
Figure 2: Schematic layout of MRAC. process of the two networks is relatively similar, only the controller design process related to the horizontal force is given in this article. The output of RBF is \(y_{m}(k)\), and the activation function of the hidden layer is the Gaussian kernel function, also known as the radial basis (RBF) function, which is a scalar function symmetrical along the radial direction, which is used to map finite-dimensional data to high-dimensional space, in the form:  
\[R_{j}(x)=exp\left(-\frac{\left\|x-c_{j}\right\|^{2}}{2b_{j}^{2}}\right),j=1,2, \cdots,n \tag{14}\]  
where \(c_{j}\) is the center point of the j basis function, and \(c_{j}=\left[c_{j1},c_{j2},\cdots,c_{jn}\right]^{T}\).