# Nonlinear Trajectory Tracking Controller for Underwater Vehicles with Shifted Center of Mass Model
## Appendix B
The velocity errors are determined by  
\[\quad+(q-q_{d}-u_{d}z_{e}c\psi_{e})s\theta_{e}+(r/c\theta-r_{d}/ c\theta_{d}+u_{d}y_{e}c\theta_{d})s\psi_{e}. \tag{106}\]3 (A32)  
Taking into account (8), the IQVs are as follows:  
\[\quad+(q-q_{d}-u_{d}z_{e}c\psi_{e})s\theta_{e}+(r/c\theta-r_{d}/ c\theta_{d}+u_{d}y_{e}c\theta_{d})s\psi_{e}. \tag{106}\]4 \[\quad+(q-q_{d}-u_{d}z_{e}c\psi_{e})s\theta_{e}+(r/c\theta-r_{d}/ c\theta_{d}+u_{d}y_{e}c\theta_{d})s\psi_{e}. \tag{106}\]5 (A33)  
For the signal \(\tau_{u}\), the first integral sliding surface is taken:  
\[\quad+(q-q_{d}-u_{d}z_{e}c\psi_{e})s\theta_{e}+(r/c\theta-r_{d}/ c\theta_{d}+u_{d}y_{e}c\theta_{d})s\psi_{e}. \tag{106}\]6 (A34)  
where \(\lambda_{1}>0\) is a positive control gain. Making use of (9), one obtains the time derivative of \(S_{D1}\), i.e., \(\dot{S}_{D1}=\dot{\zeta}_{1e}+\lambda_{1}\dot{\zeta}_{1e}\), where \(\dot{\zeta}_{1e}=\dot{\zeta}_{1}-\dot{\zeta}_{1d}=\dot{u}-\dot{u}_{d}-\hat{ \Upsilon}_{14}(\dot{q}-\dot{q}_{d})\):  
\[\quad+(q-q_{d}-u_{d}z_{e}c\psi_{e})s\theta_{e}+(r/c\theta-r_{d}/ c\theta_{d}+u_{d}y_{e}c\theta_{d})s\psi_{e}. \tag{106}\]7 (A35)  
_Control input_\(\tau_{u}\). To guarantee that the error converges to zero along the sliding surface \(S_{D1}\), the control input is designed as follows:  
\[\quad+(q-q_{d}-u_{d}z_{e}c\psi_{e})s\theta_{e}+(r/c\theta-r_{d}/ c\theta_{d}+u_{d}y_{e}c\theta_{d})s\psi_{e}. \tag{106}\]8 (A36)  
The final term, according to Assumption 4, refers to the perturbation reduction in parameters, and \(\hat{f}_{u}\) denotes the evaluated value of the external disturbance \(f_{u}\). Next, the following LFC is considered:  
\[\quad+(q-q_{d}-u_{d}z_{e}c\psi_{e})s\theta_{e}+(r/c\theta-r_{d}/ c\theta_{d}+u_{d}y_{e}c\theta_{d})s\psi_{e}. \tag{106}\]9 (A37)  
where \(\tilde{f}_{u}=f_{u}-\hat{f}_{u}\) means the estimated disturbance force error and \(\beta_{1}>0\) is a fixed control parameter. Calculating the time derivative of \(V_{D1}\) and applying (A35)-(A36) yields the following:  
\[\mu_{dt} =u_{d}(c\psi_{e}c\theta c\theta_{d}+s\theta s\theta_{d})-k_{x}x_{e}, \tag{107}\]0 \[\mu_{dt} =u_{d}(c\psi_{e}c\theta c\theta_{d}+s\theta s\theta_{d})-k_{x}x_{e}, \tag{107}\]1 (A38)  
where  
\[\mu_{dt} =u_{d}(c\psi_{e}c\theta c\theta_{d}+s\theta s\theta_{d})-k_{x}x_{e}, \tag{107}\]2 \[\mu_{dt} =u_{d}(c\psi_{e}c\theta c\theta_{d}+s\theta s\theta_{d})-k_{x}x_{e}, \tag{107}\]3 (A39)  
Referring to Assumption 2, the gain term \(\Pi_{1}\) of the form  
\[\mu_{dt} =u_{d}(c\psi_{e}c\theta c\theta_{d}+s\theta s\theta_{d})-k_{x}x_{e}, \tag{107}\]4 \[\mu_{dt} =u_{d}(c\psi_{e}c\theta c\theta_{d}+s\theta s\theta_{d})-k_{x}x_{e}, \tag{107}\]5 (A40)  
is applied. Using Assumption 3 (if uncertainties \(\dot{f}_{u}\) are close to zero or \(\dot{f}_{u}\
eq 0\) are arbitrarily large and rapidly changing in time), the time derivative of (A38) can be written as  
\[\mu_{dt} =u_{d}(c\psi_{e}c\theta c\theta_{d}+s\theta s\theta_{d})-k_{x}x_{e}, \tag{107}\]6 \[\mu_{dt} =u_{d}(c\psi_{e}c\theta c\theta_{d}+s\theta s\theta_{d})-k_{x}x_{e}, \tag{107}\]7 (A41)By choosing the adaptation component of \(\dot{\hat{f}}_{u}=\beta_{1}^{-1}S_{D1}\), one has  
\[\mu_{dt} =u_{d}(c\psi_{e}c\theta c\theta_{d}+s\theta s\theta_{d})-k_{x}x_{e}, \tag{107}\]8  
Thus, one obtains \(\check{V}_{D1}=\epsilon_{1}\) provided that \(\zeta_{1e}\) tends to zero along the sliding surface \(S_{D1}\). However, \(\zeta_{1e}=u_{e}-\check{\Upsilon}_{14}q_{e}\), so \(u_{e}=\check{\Upsilon}_{14}q_{e}\). Therefore, to ensure that \(u_{e}\) converges to zero, it is also necessary to ensure that \(q_{e}\) goes to zero. This will be demonstrated later for \(S_{D2}\). For the bounded signal \(\hat{f}_{u}\) (Assumption 3) and a well-designed controller, there exists a positive scalar \(\epsilon_{1}\geq|\beta_{1}\tilde{f}_{u}\dot{f}_{u}|\). The condition \(|\beta_{1}\tilde{f}_{u}\dot{f}_{u}|<\rho_{1}|S_{D1}|\) must be fulfilled to guarantee uniformly ultimate boundedness. This problem was considered in [37; 45].