*3.2. Model Decoupling*

Note that the degrees of freedom of the AUV are coupled in nonlinear model (18). In order to simplify the design of the controller, the 5 DOF nonlinear dynamic model (18) of the AUV is decoupled for surge speed control, heading control, and depth control. Considering that the AUV always maintains a constant surge speed for path tracking, the nominal surge speed *us* in the heading nominal control model and depth nominal control model above is set as a constant. Then, these decoupled models can all be described as a Lipschitz nonlinear system. The hydrodynamic coefficients in these models are given in our previous research [24].

1. Surge speed nominal control model:

$$
\dot{\overline{\mathfrak{X}}}\_{\mathfrak{u}} = A\_{\mathfrak{u}} \overline{\mathfrak{x}}\_{\mathfrak{u}} + B\_{\mathfrak{u}} \overline{\mathfrak{U}}\_{\mathfrak{u}} + \underline{\mathfrak{g}}\_{\mathfrak{x}}(\overline{\mathfrak{x}}\_{\mathfrak{u}}) \tag{19}
$$

where the state is denoted by *xu* = *us*. The control input is denoted by the nominal stern thruster force *Uu* = *Fx*. *Au* = *xu*/(*m* − *X* . *<sup>u</sup>*), *Bu* = 1/(*m* − *X* . *<sup>u</sup>*), and *gx*(*xu*) = *Xuuus*|*us*|. *m* is the mass of the AUV. *X* . *<sup>u</sup>* is the added mass. *Xuu* is hydrodynamic damping coefficient.

2. Heading nominal control model:

$$
\dot{\overline{\overline{x}}}\_y = A\_y \overline{\overline{x}}\_y + B\_y \overline{\overline{U}}\_y + g\_y(\overline{\overline{x}}\_y) \tag{20}
$$

where the state is denoted by *xy* = (*v*,*r*) *<sup>T</sup>*. The control input is denoted by the nominal vertical plane deflection: *Uy* = *δr*.

$$\begin{aligned} A\_{\mathcal{Y}} &= \begin{bmatrix} m - Y\_{\dot{\upsilon}} & -Y\_{\dot{r}} \\ -N\_{\dot{\upsilon}} & I\_{zz} - N\_{\dot{r}} \end{bmatrix}^{-1} \begin{bmatrix} Y\_{\text{uv}}\overline{u}\_{s} & (m + Y\_{\text{ur}})\overline{u}\_{s} \\ N\_{\text{uv}}\overline{u}\_{s} & N\_{\text{uv}}\overline{u}\_{s} \end{bmatrix}, \\ B\_{\mathcal{Y}} &= \begin{bmatrix} m - Y\_{\dot{\upsilon}} & -Y\_{\dot{r}} \\ -N\_{\dot{\upsilon}} & I\_{zz} - N\_{\dot{r}} \end{bmatrix}^{-1} \begin{bmatrix} Y\_{\text{un}\delta\_{s}} \\ N\_{\text{un}\delta\_{s}} \end{bmatrix}, \\ \mathcal{G}\_{\mathcal{Y}}(\overline{\mathbf{x}}\_{\mathcal{Y}}) &= \begin{bmatrix} m - Y\_{\dot{\upsilon}} & -Y\_{\dot{r}} \\ -N\_{\dot{\upsilon}} & I\_{zz} - N\_{\dot{r}} \end{bmatrix}^{-1} \begin{bmatrix} Y\_{\upsilon\upsilon}\overline{\nu}|\overline{\upsilon}| + Y\_{\upsilon\dot{r}}\overline{\tau}|\overline{\tau}| \\ N\_{\upsilon\upsilon}\overline{\nu}|\overline{\upsilon}| + N\_{\upsilon r}\overline{\tau}|\overline{\tau}| \end{bmatrix} \end{aligned} \tag{21}$$

where *Izz* is the rotational inertia. *Y*. *<sup>v</sup>*, *N*. *<sup>r</sup>*, *Y*. *<sup>r</sup>* and *N*. *<sup>v</sup>* are the added mass. *Yuv*, *Yur*, *Nuv*, *Nur*, *Yuu<sup>δ</sup><sup>s</sup>* and *Nuu<sup>δ</sup><sup>s</sup>* are hydrodynamic coefficients. *Yvv*, *Yrr*, *Nvv* and *Nrr* are hydrodynamic damping coefficients.

3. Depth nominal control model:

$$
\dot{\overline{\overline{x}}}\_z = A\_z \overline{\overline{x}}\_z + B\_{\overline{z}} \overline{U}\_z + g\_z(\overline{\overline{x}}\_z) \tag{22}
$$

where the state is denoted by *xz* = (*w*, *q*) *<sup>T</sup>*. The control input is denoted by the nominal translational plane deflection: *Uz* = *δs*.

$$\begin{aligned} A\_z &= \begin{bmatrix} m - Z\_{\dot{w}} & -Z\_{\dot{q}} \\ -M\_{\dot{w}} & I\_{yy} - M\_{\dot{q}} \end{bmatrix}^{-1} \begin{bmatrix} Z\_{\text{uw}} \overline{u}\_s & \left(-m + Z\_{\text{uq}}\right) \overline{u}\_s \\ M\_{\text{uw}} \overline{u}\_s & M\_{\text{uq}} \overline{u}\_s \end{bmatrix}, \\ B\_z &= \begin{bmatrix} m - Z\_{\dot{w}} & -Z\_{\dot{q}} \\ -M\_{\dot{w}} & I\_{yy} - M\_{\dot{q}} \end{bmatrix}^{-1} \begin{bmatrix} Z\_{\text{uw}\delta\_s} \\ M\_{\text{uw}\delta\_s} \end{bmatrix} \end{aligned}$$

$$\begin{aligned} g\_z(\overline{x}\_z) = \begin{bmatrix} m - Z\_{\dot{w}} & -Z\_{\dot{q}} \\ -M\_{\dot{w}} & I\_{\mathcal{Y}\mathcal{Y}} - M\_{\dot{q}} \end{bmatrix}^{-1} \begin{bmatrix} Z\_{ww}\overline{w}|\overline{w}| + Z\_{qq}\overline{q}|\overline{q}| \\ M\_{ww}\overline{q}|\overline{q}| + M\_{qq}\overline{q}|\overline{q}| \end{bmatrix} \tag{23}$$

where *Iyy* is the rotational inertia. *Z* . *<sup>w</sup>*, *M* . *<sup>w</sup>*, *Z*. *<sup>q</sup>* and *M*. *<sup>q</sup>* are the added mass. *Zuw*, *Zuq*, *Muw*, *Muq*, *Zuu<sup>δ</sup><sup>s</sup>* and *Muu<sup>δ</sup><sup>s</sup>* are hydrodynamic coefficients. *Zww*, *Zqq*, *Mww* and *Mqq* are hydrodynamic damping coefficients.

## *3.3. Problem Statement*

**Problem 1.** *Given a vector <sup>η</sup><sup>r</sup>* <sup>∈</sup> *<sup>R</sup>*5×<sup>1</sup> *that stands for the reference position and attitude angle and a vector <sup>ν</sup><sup>r</sup>* <sup>∈</sup> *<sup>R</sup>*5×<sup>1</sup> *that stands for the reference speed, the nominal path-tracking deviation vector is denoted by e<sup>η</sup>* := *η* − *η<sup>r</sup> and the surge speed deviation vector is denoted by eus* := *us* − *usr*. *The speed control law ν<sup>d</sup>* := (*usd*, *vd*, *wd*, *qd*,*rd*) *<sup>T</sup>* = *κη eη*,*eν needs to be obtained to converge the nominal path-tracking deviation*:lim*t*→∞*eη*(*t*) <sup>=</sup> 0.

**Problem 2.** *The speed control law deviation vector is denoted by eν<sup>d</sup>* = *ν* − *ν<sup>d</sup> and the actual path-tracking deviation is denoted by e<sup>η</sup>* := *η* − *η<sup>r</sup>* = *ex*,*ey*,*ez*,*eθ*,*e<sup>ψ</sup> T* . *Based on decoupling models (19), (20), and (22), AUV's control vector τ* = (*Fx*, *δr*, *δs*) *<sup>T</sup> needs to be obtained to respond to the speed control law*: lim*t*→∞*eν<sup>d</sup>* (*t*) <sup>=</sup> 0. *Finally, the actual path-tracking deviation <sup>e</sup><sup>η</sup> can be converged:* lim*t*→∞*eη*(*t*) <sup>=</sup> 0.
