*3.1. AUV Motion Model*

The global coordinate and the local coordinate frame are defined, and the coordinate transformation relationship is shown in Figure 1. Here *E* − *ξηζ* denotes the global coordinate system, and *O* − *xyz* denotes the local coordinate system [24].

**Figure 1.** AUV coordinate system [24].

Note that the roll motion is self-stable, and the roll motion attitude is also small, meaning that the roll angle *φ* and the roll speed *p* can all be regarded as 0. Therefore, the roll motion is not considered in this paper. The speed vector is denoted by *ν* = (*us*, *v*, *w*, *q*,*r*) *T* in the motion coordinate *O* − *xyz*, where *us*, *v*, *w*, *q*, and *r* are respectively surge speed, sway speed, heave speed, pitch speed, and yaw speed. The position and attitude angle (pitch and yaw angles) vector is denoted by *η* = (*x*, *y*, *z*, *θ*, *ψ*) *<sup>T</sup>* in global coordinate system *E* − *ξηζ*. The kinematics model is given as:

$$
\eta^T = f\_{\nu \eta} \nu^T \tag{16}
$$

where *Jνη* is a transformation matrix from *O* − *xyz* to *E* − *ξηζ*:


The kinetic model is given as:

$$
\mathcal{M}\,\dot{\nu} = (\mathbb{C}(\nu) + D(\nu) + \Delta F\_{\text{CD}})\nu + F + G(\pi) + \mathfrak{r}\_d \tag{18}
$$

where M ∈ *<sup>R</sup>*5×<sup>5</sup> is the inertial matrix, *<sup>C</sup>*(*ν*) <sup>∈</sup> *<sup>R</sup>*5×<sup>5</sup> is the Coriolis and centripetal matrix, *<sup>D</sup>*(*ν*) <sup>∈</sup> *<sup>R</sup>*5×<sup>5</sup> is the hydrodynamic damping matrix, and *<sup>F</sup>* <sup>∈</sup> *<sup>R</sup>*5×<sup>1</sup> is the hydrostatic force and moment. *τ* = (*Fx*, *δr*, *δs*) *<sup>T</sup>* is the AUV's control vector, where *Fx* is the stern thruster force, *δ<sup>r</sup>* is the vertical plane deflection, and *δ<sup>s</sup>* is the translational plane deflection. *<sup>G</sup>*(*τ*) : *<sup>R</sup>*3×<sup>1</sup> <sup>→</sup> *<sup>R</sup>*5×<sup>1</sup> is the active control force in the AUV's motion coordinate system *<sup>O</sup>* <sup>−</sup> *xyz*. *<sup>τ</sup><sup>d</sup>* <sup>∈</sup> *<sup>R</sup>*5×<sup>1</sup> is the external disturbance. <sup>Δ</sup>*FCD* <sup>∈</sup> *<sup>R</sup>*5×<sup>5</sup> represents the disturbance brought by parametric uncertainties [10].
