System of ordinary differential equations of translational and angular ship motion.

• Uses Earth-fixed coordinate system.
• Uses quaternions instead of Euler angles to alleviate Gimbal lock problems.
• Uses centre of floatation as a point around which the ship rotates.
• Every term in the equation is divided by ship mass.
• Uses the formulae adapted from [1]. $$\frac{d}{dt}\left[ \begin{array}{c} \vec{x}\\ q\\ \vec{v}\\ \vec\omega \end{array} \right] = \left[ \begin{array}{c} \vec{v} + \vec{r}\times\vec\omega\\ \frac{1}{2}\vec\omega q\\ \vec{F}/m - \left(\vec{r}\times\vec\omega\right)\times\vec\omega - \vec{r}\times\vec\alpha\\ \mathcal{I}^{-1}\left(\vec\tau - \mathcal{I}_r'\vec\omega - \mathcal{I}_r\vec\alpha\right) \end{array} \right].$$ Here $\vec{x}$ is ship position, $q$ is ship orientation in quaternion form, $\vec{v}$ is linear velocity, $\vec\omega$ is angular velocity, $\vec{F}$ is a vector of external forces, $\vec\tau$ is a vector of external force moments, $\vec\omega$ is angular velocity, $\vec\alpha$ is angular acceleration, $\vec{r}$ is the vector from centre of gravity to centre of floatation, $\mathcal{I}_\text{body}$ is ship hull inertia matrix computed with respect to centre of gravity in body-fixed coordinate system, $\mathcal{I}$ is the same matrix in Earth-fixed coordinate system, $\mathcal{I}_r$ is translation matrix computed for centre of floatation, $R$ is rotation matrix for quaternion $q$. Formulae for auxiliary physical quantities: $$\mathcal{I} = R \, \mathcal{I}_{\text{body}} \, R^{T} \qquad \mathcal{I}_r' = \left[\begin{array}{ccc} 2 M_y + 2 M_z & -M_x r_y - r_x M_y & -M_x r_z - r_x M_z \\ -M_x r_y - r_x M_y & 2 M_x + 2 M_z & -M_y r_z - r_y M_z \\ -M_x r_z - r_x M_z & -M_y r_z - r_y M_z & 2 M_x + 2 M_y \\ \end{array}\right] \qquad \vec{M} = \vec{r}\times\vec\omega.$$
• Froude—Krylov, gravity and wind force moments are computed with respect to centre of floatation (see [11]).

Types

• using matrix_type = vtb::geometry::Matrix< T, 3 >
• using quaternion_type = typename Ship< T >::quaternion_type
• using rotation_matrix_type = typename Ship< T >::rotation_matrix_type
• using variables_type = typename Ship< T >::variables_type
• using state_vector_type = typename Ship< T >::state_vector_type
• using scalar_type = T

Methods

• clear() -> void
• calc_forces_and_torques(const Ship< T > & ship, Vector< T, 3 > buoyancy_force, Vector< T, 3 > wind_force) -> void
• delta() const -> const Vector< T, 3 > &
• delta(const Vector< T, 3 > & rhs) -> void
• total_torque() const -> const Vector< T, 3 > &
• total_force() const -> const Vector< T, 3 > &
• angular_damping() const -> const Vector< T, 3 > &
• angular_damping(const Vector< T, 3 > & rhs) -> void
• linear_damping() const -> const Vector< T, 3 > &
• linear_damping(const Vector< T, 3 > & rhs) -> void
• total_torque(const Vector< T, 3 > & rhs) -> void
• total_force(const Vector< T, 3 > & rhs) -> void
• gravity_force(const Vector< T, 3 > & rhs) -> void
• translation_matrix(const matrix_type & rhs) -> void
• translation_matrix() const -> const matrix_type &
• inertia_matrix(const rotation_matrix_type & rhs) -> void
• inertia_matrix() const -> const rotation_matrix_type &
• operator()(T t, const state_vector_type & vars) -> state_vector_type
• operator()(T t, const variables_type & vars) -> variables_type
• solve(T t, const state_vector_type & v, state_vector_type & dv) -> void
• Ship_motion_equation(const rotation_matrix_type & inertia_matrix)explicit
• Ship_motion_equation()