## 2 Towed Cable Kinematic Model
### Coordinate System  
As shown in Figure 1, the lumped mass method is employed to discretize the towed cable into a series of elastic, frictionless springs interconnected by nodes. To mathematically establish the motion model for the towed cable, three reference coordinate systems are defined: the ground inertial coordinate system \((E-XYZ)\), the cable local coordinate system \((o-x_{1}y_{1}z_{1})\)\((m=1,\cdot\cdot\cdot,N)\), and the vehicle motion coordinate system \((S-xyz)\). The spring is defined between Nm and Nm\({}^{-1}\). Damp between adjacent nodes is considered as a Rayleigh damp model. Figure 1: Composition of underwater towed system. From Figure 1, we define the position vector of the vehicle in \(S-xyz\) as \(\mathbf{P}_{S}^{E}=\left[X_{S}\quad Y_{S}\quad Z_{S}\right]^{\mathsf{T}}\). Additionally, to determine the posture of the vehicle in \(S-xyz\), we refine the Euler angle into pitch angle \(\theta\), yaw angle \(\psi\), and roll angle \(\varphi\). These Euler angles represent the relationship between \(S-xyz\) and \(E-XYZ\):  
\[ \begin{bmatrix}x\quad y\quad z\end{bmatrix} = \begin{bmatrix}X\quad Y\quad Z \end{bmatrix} \mathbf{R}_{E}^{S}(\theta,\psi,\varphi) \tag{1}\]  
where  
\[\mathbf{R}_{E}^{S}(\theta,\psi,\varphi)= \begin{bmatrix}\cos\psi\cos\theta&\sin \psi\cos\varphi+\cos\psi\sin\theta\sin\varphi&-\sin\psi\sin\varphi+\cos\psi\sin \theta\cos\varphi\\ \sin\psi\cos\theta&\cos\psi\cos\varphi+\sin\psi\sin\theta\sin\varphi&-\cos\psi \sin\varphi+\sin\psi\sin\theta\cos\varphi\\ -\sin\theta&\cos\theta\sin\varphi&\cos\theta\cos\varphi\end{bmatrix} \]  
The towed cable is connected to the vehicle; therefore, the position of the link point between vehicle and the cable is defined as \(\mathbf{P}_{L}^{S}= \begin{bmatrix}x_{L}\quad y_{L}\quad z_{L}\end{bmatrix} ^{ \mathsf{T}}\). The position of the link point in the ground inertial coordinate system \((E-XYZ)\) is represented as \(\mathbf{P}_{L}^{E}=\left(X_{L}\quad Y_{L}\quad Z_{L}\right)^{\mathsf{T}}\). The relationship between \(\mathbf{P}_{L}^{S}\) and \(\mathbf{P}_{L}^{E}\) can be determined by Equation (1) and \(\mathbf{P}_{S}^{E}\), which can be expressed as  
\[\mathbf{P}_{L}^{E}=\mathbf{P}_{S}^{E}+\mathbf{R}_{S}^{E}(\theta,\psi,\varphi)\mathbf{P}_{L}^{S} \tag{2}\]  
The lumped mass method simulates the attitude and position of the towed cable approximately. The spatial location of cable profile is determined by a discrete system specification with consideration of buoyancy, gravity, hydrodynamics, and inertia force. The resolution of force is introduced in Section 2.3. The position of the node m in the ground inertial coordinate system can be expressed as \(\mathbf{P}_{m}^{E}=\left(X_{m}\quad Y_{m}\quad Z_{m}\right)^{\mathsf{T}}\). The attitude and position of the elastic cable segment can be determined using Euler angles \((\xi_{m},\eta_{m})\). \[\xi_{m}=\arcsin(Z_{m}-Z_{m-1}/l_{m}) \tag{3}\]  
\[\eta_{m}= \begin{cases}\arcsin\left(\frac{Y_{m}-Y_{m-1}}{l_{m}\cos\xi_{m}} \right)\quad X_{m}-X_{m-1}\leq 0\\ \pi-\arcsin\left(\frac{Y_{m}-Y_{m-1}}{l_{m}\cos\xi_{m}}\right)\quad X_{m}-X_{ m-1}>0\end{cases}  \tag{4}\]  
where \(l_{m}\) is the length from node m to node m\({}^{-1}\), expressed as  
\[l_{m}=\sqrt{\left(X_{m}-X_{m-1}\right)^{2}+\left(Y_{m}-Y_{m-1}\right)^{2}+ \left(Z_{m}-Z_{m-1}\right)^{2}} \tag{5}\]  
The relationship between \(o-x_{1}y_{1}zlational velocity and angular velocity vectors between coordinate systems \(E-XYZ\) and \(S-xyz\) can be written as follows  
\[ \begin{cases}\mathbf{V}_{S}^{E}=\mathbf{R}_{E}^{S}\mathbf{V}_{S}\\ \mathbf{W}_{S}^{E}=\mathbf{T}_{E}^{E}\mathbf{W}_{S}\end{cases}  \tag{7}\]  
where \(\mathbf{T}_{E}^{S}= \begin{pmatrix}0&\sin\varphi&\cos\varphi\\ \cos\varphi/\tan\theta&0&-\sin\varphi/\tan\theta\\ 1&-\tan\theta\cos\varphi&\tan\theta\sin\varphi\end{pmatrix} \). Owing to the neglect of the bending and torsional stiffness of the towed cable, we posit that the node possesses only instantaneous translational velocity.