## 2 Problem Formulation and Preliminaries
### Kinematic and Dynamic Modeling
 \tag{23}\]  
\[\mathbf{D}_{a}(\mathbf{v})=diag\Big{(}\Big{[}X_{u|u|}\Big{|}u\Big{|}+X_{u},M_{q|q|} \Big{|}\Big{|}q\Big{|}+M_{q},N_{i\
u|}\Big{|}r\Big{|}+N_{r}\Big{]}\Big{)} \tag{24}\]  
\[\mathbf{G}_{a}= \begin{bmatrix}0,Z_{B}G\sin\theta,0\end{bmatrix} ^{T} \tag{25}\]  
\[\mathbf{\tau}_{a}= \begin{bmatrix}\tau_{u},\tau_{q},\tau_{r}\end{bmatrix} ^{T} \tag{26}\]  
\[\mathbf{f}_{a}= \begin{bmatrix}f_{1},f_{2},f_{3}\end{bmatrix} ^{T} \tag{27}\]Due to the limited maximum moment that the actuator can handle, the input moment \(\mathbf{\tau}_{a}\) is considered to be subject to the following asymmetric constraints. \[\tau_{Lu}\leq\tau_{u}\leq\tau_{Hu},\tau_{Lq}\leq\tau_{q}\leq\tau_{Hq},\tau_{Lr} \leq\tau_{r}\leq\tau_{Hr} \tag{28}\]  
Before presenting the research objectives of this paper, the following assumptions were made:  
**Assumption 1**. : _The transformation matrix \(\mathbf{J}_{1}(\mathbf{\eta})\) is invertible, and there is a known constant \(\mathbf{\hat{j}}\), such that \(\sup_{\eta}\|\mathbf{J}_{1}(\mathbf{\eta})\|\leq\mathbf{\hat{j}}\)._  
**Assumption 2**. : _The derivative of the transformation matrix \(\mathbf{J}_{1}(\mathbf{\eta})\) with respect to time has the following form:_  
\[\mathbf{\hat{j}}_{1}=\mathbf{J}_{1}\mathbf{\omega} \tag{29}\]  
_where \(\mathbf{\omega}= \begin{bmatrix}0&-r&q\\ r&0&0\\ -q&0&0\end{bmatrix} \) is an antisymmetric matrix for vector \(\mathbf{I}\). The following theorem follow._  
\[\mathbf{I}^{T}\mathbf{\omega}\mathbf{I}=0 \tag{30}\]  
**Assumption 3**. : _For the parameter \(\mathbf{M}_{a},\mathbf{C}_{a},\mathbf{D}_{a},\mathbf{G}_{a},\mathbf{f}_{a}\), the following constraints apply:_  
\[m_{1}\leq\|\mathbf{M}_{a}\|\leq m_{2},\|\mathbf{C}_{a}\|\leq C\parallel\mathbf{v} \parallel,\|\mathbf{D}_{a}\|\leq d_{1}+d_{2}\parallel\mathbf{v}\parallel,\|\mathbf{G}_{a }\|\leq g,\|\mathbf{f}_{a}\|\leq f \tag{31}\]  
_where \(m_{1},m_{2},c,d_{1},d_{2},g,f\), is an unknown constant._  
**Remark 1**. : _Some of the system parameters or external influences were not adequately considered due to environmental uncertainties and technical measurement limitations. In addition, the complex dynamic behavior of real systems is difficult to capture entirely through mathematical models. Therefore, the model parameters \(\mathbf{M}_{a},\mathbf{C}_{a},\mathbf{D}_{a},\mathbf{G}_{a}\) are not modeled accurately enough. In this paper, we explore the effect of the unmodeled part on the system and derive controllers that do not depend on the model parameters through the described decoupling algorithm._  
This study aimed to investigate the problem of three-dimensional trajectory tracking for underactuated autonomous underwater vehicles (AUVs) in the presence of unmodeled dynamics and external disturbances. The primary objective was to ensure that the tracking error converges to a small bounded region within a predefined time frame. Under the assumption that the desired trajectory is smooth and continuously differentiable, the tracking error can be mathematically expressed as  
\[\eta_{e}=\mathbf{\eta}_{1}-\mathbf{\eta}_{1d}, \tag{32}\]  
A predefined-time adaptive controller without model parameters is derived such that the trajectory error satisfies the following conditions:  
\[\lim_{t\to T}\lvert\eta_{ei}\rvert\leq\zeta,(i=1,2,3) \tag{33}\]  
\(T\) is the settling time constant, and the competition is \(\zeta\) positive parameter.