## 2 Underwater Glider Modeling
### Underwater Glider Kinematic Modeling  
The 'Petrel-L' underwater glider is elliptical, and it navigates through the Earth's oceans by pitching, rolling, and yawing. Therefore, it is necessary to establish body coordinate \(B_{0}(b_{1},b_{2},b_{3})\), velocity coordinate \(V_{0}(v_{1},v_{2},v_{3})\), and inertial coordinate systems \(E_{0}(e_{1},e_{2},e_{3})\), as shown in Figure 3.  
The vector \(\mathbf{b}=[X,Y,Z]^{\mathrm{T}}\) and \(\mathbf{\Phi}=[\phi,\theta,\psi]^{T}\) represent the position and attitude angles of the underwater glider's buoyancy center in the inertial coordinate system, respectively. Among them, \(X\) and \(Y\) are the horizontal plane coordinates, \(Z\) is the depth; \(\phi\) is the roll angle, which is the angle between \(B_{0}b_{1}b_{2}\) and plane \(E_{0}e_{1}e_{2}\). A positive value indicates a right roll when viewed from the tail to the nose of the vehicle; \(\theta\) is the pitch angle, which is  
Figure 2: Operational phases of an underwater glider. the angle between the longitudinal axis \(B_{0}b_{1}\) of the underwater glider and the plane \(E_{0}e_{1}e_{2}\). A positive value indicates a pitch-down angle; a negative value indicates a pitch-up angle; \(\psi\) is the yaw angle, which is the angle between the projection of the longitudinal axis \(B_{0}b_{1}\) of the underwater glider on the plane \(E_{0}e_{1}e_{2}\) and \(E_{0}e_{2}\). A positive value indicates a right yaw when viewed from the tail to the nose of the vehicle. The vector \(\mathbf{V}=[u,v,w]^{T}\) and \(\mathbf{\Omega}=[p,q,r]^{T}\) represent the velocity and angular velocity of the underwater glider's buoyancy center in the body coordinate system, respectively. Among them, \(u\), \(v\), and \(w\) are the linear velocities of the glider's buoyancy center along the \(b_{1}\), \(b_{2}\), and \(b_{3}\) directions, and \(p\), \(q\), and \(r\) are the angular velocities of the glider's buoyancy center rotating around the \(b_{1}\), \(b_{2}\), and \(b_{3}\) axes, respectively. The angle of attack \(\alpha\) and sideslip angle \(\beta\) are defined in the velocity coordinate system. The projection of the glider's buoyancy center velocity vector on the \(B_{0}b_{1}b_{3}\) plane, the angle between the projection and the \(b_{1}\) axis is the angle of attack \(\alpha\). A positive value indicates that the projection is below the \(b_{1}\) axis, and a negative value indicates that it is above the \(b_{1}\) axis; the angle between the glider's buoyancy center velocity vector and the projection is the sideslip angle \(\beta\). When viewed from the tail to the nose of the vehicle, a positive value indicates that the velocity vector is to the right of the \(b_{1}\) axis, and a negative value indicates that it is to the left of the \(b_{1}\) axis. To design control strategies for UGs, it is necessary to comprehensively consider the glider's position, angle, velocity, and angular velocity information. Therefore, these variables must be transformed into the same coordinate system. First, rotate the inertial coordinate system around the \(e_{3}\) axis by an angle \(\psi\), with the rotation matrix \(T_{Z}(\psi)\). Then, rotate around the \(e_{2}\) axis by an angle \(\theta\), with the rotation matrix \(T_{Y}(\theta)\). Finally, rotate around the \(e_{1}\) axis by an angle \(\phi\), with the rotation matrix \(T_{X}(\phi)\), to obtain the body coordinate system. Let the rotation matrix from the inertial frame to the body frame be denoted as \(T_{B}^{E}\): then, \(T_{B}^{E}\) is specifically expressed as follows:  
\[\mathbf{T}_{B}^{E}=T_{Z}(\psi)T_{Y}(\theta)T_{X}(\phi) \tag{1}\]  
\[= \begin{bmatrix}\cos\psi&-\sin\psi&0\\ \sin\psi&\cos\psi&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}\cos\theta&0&\sin\theta\\ 0&1&0\\ -\sin\theta&0&\cos\theta\end{bmatrix}\begin{bmatrix}1&0&0\\ 0&\cos\phi&-\sin\phi\\ 0&-\sin\phi&\cos\phi\end{bmatrix} \]  
\[=