Draft:Fossen's marine craft model

The six-degrees-of-freedom (DOFs) marine craft equations of motion are expressed in matrix-vector form using generalized coordinates $\backslash boldsymbol\{\backslash eta\}\; =\; [x,\; y,\; z,\; \backslash phi,\; \backslash theta,\; \backslash psi]^\backslash top$ to represent the position and orientation, and the generalized velocity $\backslash boldsymbol\{\backslash nu\}\; =\; [u,\; v,\; w,\; p,\; q,\; r]^\backslash top$ to describe the linear and angular velocities. The generalized forces acting on the craft, arising from propulsion, wind, waves, and ocean currents, are denoted $\backslash boldsymbol\{\backslash tau\}\; =\; [X,\; Y,\; Z,\; K,\; M,\; N]^\backslash top$. These variables represent the craft's translational and rotational dynamics, with the vector elements following the standard terminology established by the Society of Naval Architects and Marine Engineers (SNAME).{{cite book

| last = SNAME

| year = 1950

| title = Nomenclature for Treating the Motion of a Submerged Body Through a Fluid

| series = Technical and Research Bulletin

| volume = 1–5

}}

The kinematics and kinetics of Fossen's marine craft model are described by the following equations:

style="border: 2px solid green; padding: 10px; background-color: #f9fff9;" | $$ \begin{aligned} \dot{\boldsymbol{\eta}} & = \boldsymbol{J}(\boldsymbol{\eta}) \boldsymbol{\nu} \\ \boldsymbol{M} \dot{\boldsymbol{\nu}} + \boldsymbol{C}(\boldsymbol{\nu}) \boldsymbol{\nu} + \boldsymbol{D}(\boldsymbol{\nu}) \boldsymbol{\nu} + \boldsymbol{g}(\boldsymbol{\eta}) & = \boldsymbol{\tau} \end{aligned}

where

• $\backslash boldsymbol\{M\}$: Inertia matrix, combining rigid-body and added mass effects.
• $\backslash boldsymbol\{C\}(\backslash boldsymbol\{\backslash nu\})$: Coriolis and centripetal matrix, combining rigid-body and added mass effects.
• $\backslash boldsymbol\{D\}(\backslash boldsymbol\{\backslash nu\})$: Hydrodynamic damping matrix.
• $\backslash boldsymbol\{g\}(\backslash boldsymbol\{\backslash eta)\}$: Hydrostatic forces and moments.
• $\backslash boldsymbol\{J\}(\backslash boldsymbol\{\backslash eta\})$: Transformation matrix relating velocities in the BODY and North-East-Down (NED) frames.

The kinematic equation can be represented using Euler angles or unit quaternions to describe the transformation from the BODY frame to the NED frame. The Tait–Bryan angles is the Euler angle representation typically used for marine craft. This involves sequential rotations in the order of yaw (z-axis), pitch (y-axis), and roll (x-axis):

$$

\boldsymbol{J}(\boldsymbol{\eta}) = \left[

\begin{array}[c]{cc}

\mathbf{R}_{zyx}(\boldsymbol{\eta}) & \mathbf{0}_{3\times3}\\

\mathbf{0}_{3\times3} & \mathbf{T}_{zyx}(\boldsymbol{\eta})

\end{array}

\right]

$$

\begin{aligned}

\mathbf{R}_{zyx}(\boldsymbol{\eta}) & = \boldsymbol{R}_z(\psi) \boldsymbol{R}_y(\theta) \boldsymbol{R}_x(\phi) = \left[

\begin{array}[c]{ccc}

\mathrm{c}\psi\mathrm{c}\theta & -\mathrm{s}\psi\mathrm{c}\phi+\mathrm{c}

\psi\mathrm{s}\theta\mathrm{s}\phi & \mathrm{s}\psi\mathrm{s}\phi+\mathrm{c}\psi\mathrm{c}\phi\mathrm{s}\theta\\

\mathrm{s}\psi\mathrm{c}\theta & \mathrm{c}\psi\mathrm{c}\phi+\mathrm{s}

\phi\mathrm{s}\theta\mathrm{s}\psi & -\mathrm{c}\psi\mathrm{s}\phi+\mathrm{s}\theta\mathrm{s}\psi\mathrm{c}\phi\\

-\mathrm{s}\theta & \mathrm{c}\theta\mathrm{s}\phi & \mathrm{c}\theta\mathrm{c}\phi

\end{array}

\right]

\end{aligned}

$$

\mathbf{T}_{zyx}(\boldsymbol{\eta}) =\left[

\begin{array}[c]{ccc}

1 & \mathrm{s}\phi\mathrm{t}\theta & \mathrm{c}\phi\mathrm{t}\theta\\

0 & \mathrm{c}\phi & -\mathrm{s}\phi\\

0 & \mathrm{s}\phi/\mathrm{c}\theta & \mathrm{c}\phi/\mathrm{c}\theta

\end{array}

\right],\quad \theta \neq \pm (\pi/2 + k\pi), \quad \text{for all } k \in \mathbb{Z}

where $\backslash mathrm\{s\}\backslash ;\backslash cdot=\backslash sin(\backslash cdot)$, $\backslash mathrm\{c\}\backslash ;\backslash cdot=\backslash cos(\backslash cdot)$ and

$\backslash mathrm\{t\}\backslash ;\backslash cdot=\backslash tan(\backslash cdot)$. The matrix $\backslash mathbf\{R\}\_\{zyx\}(\backslash boldsymbol\{\backslash eta\})$ is the rotation matrix for translational velocities and $\backslash mathbf\{T\}\_\{zyx\}(\backslash boldsymbol\{\backslash eta\})$ is the transformation matrix for rotational velocities.

The system inertia matrix $\backslash boldsymbol\; M\; =\; \backslash boldsymbol\; M\_\{RB\}\; +\; \backslash boldsymbol\; M\_A$ and Coriolis and centripetal matrix $\backslash boldsymbol\; C(\backslash boldsymbol\{\backslash nu\})\; =\; \backslash boldsymbol\; C\_\{RB\}(\backslash boldsymbol\{\backslash nu\})\; +\; \backslash boldsymbol\; C\_A(\backslash boldsymbol\{\backslash nu\})$ consist of contributions from both the rigid-body dynamics of the vehicle and the hydrodynamic effects due to interaction with the surrounding fluid, also known as the added mass effect. The hydrodynamic damping matrix is denoted by $\backslash boldsymbol\; D(\backslash boldsymbol\{\backslash nu\})$ while $\backslash boldsymbol\{g\}(\backslash boldsymbol\{\backslash eta\})$ is the vector of gravitational and buoyancy forces. Let $\backslash boldsymbol\; r\_G^b\; =\; [x\_G,\; y\_G,\; z\_G]^\backslash top$ and $\backslash boldsymbol\; r\_B^b\; =\; [x\_B,\; y\_B,\; z\_B]^\backslash top$ denote the vectors from the body-fixed coordinate origin (CO) to the center of gravity (CG) and the center of buoyancy (CB), respectively. Let the cross product of two vectors be expressed as a matrix multiplication $\backslash mathbf\{a\}\; \backslash times\; \backslash mathbf\{b\}\; =\; \backslash boldsymbol\; S(\backslash mathbf\{a\})\; \backslash mathbf\{b\}$, where $\backslash boldsymbol\; S(\backslash mathbf\{a\})$ is a skew-symmetric matrix:

$$

\boldsymbol{S}(\mathbf{a}) = \begin{bmatrix}

\,\,0 & \!-a_3 & \,\,\,a_2 \\

\,\,\,a_3 & 0 & \!-a_1 \\

\!-a_2 & \,\,a_1 & \,\,0

\end{bmatrix}

This matrix encodes the antisymmetric nature of the cross-product operation. The mass matrix, as well as the Coriolis and centripetal matrix, can be derived from Kirchhoff's equations. This approach was formalized in Sagatun and Fossen's 1991 theorem on the Lagrangian formulation of vehicle dynamics.{{cite conference

| last1 = Sagatun

| first1 = S. I.

| last2 = Fossen

| first2 = Thor I.

| year = 1991

| title = Lagrangian Formulation of Underwater Vehicles' Dynamics

| book-title = Proceedings of the IEEE International Conference on Systems, Man and Cybernetics

| location = Charlottesville, VA

| pages = 1029–1034

}} The specific matrices that govern the dynamics are detailed below, each representing a fundamental aspect of the system's behavior:

$$

\begin{aligned}

\boldsymbol{M}_{RB} & =\left[

\begin{array}[c]{cc}

m\boldsymbol{I}_{3} & -m\boldsymbol{S}(\boldsymbol{r}_{G}^{b})\\

m\boldsymbol{S}(\boldsymbol{r}_{G}^{b}) & \boldsymbol{I}_b^b

\end{array}

\right] \\

& = \left[

\begin{array}[c]{cccccc}

m & 0 & 0 & 0 &

mz_G & -my_G\\

0 & m & 0 & -mz_G &

0 & mx_G\\

0 & 0 & m & my_G &

-mx_G & { 0}\\

0 & -mz_G & my_G &

I_x & -I_{xy} & -I_{xz}\\

mz_G & 0 & -mx_G &

-I_{yx} & I_y & -I_{yz}\\

-my_{G} & mx_G & 0 &

-I_{zx} & -I_{zy} & I_z

\end{array}

\right]

\end{aligned}

$$

\begin{aligned}

\boldsymbol{C}_{RB}(\boldsymbol{\nu}) & =

\left[

\begin{array}{cc}

\boldsymbol{0}_{3\times3} & -m\boldsymbol{S}(\boldsymbol{\nu}_{1}) - m\boldsymbol{S}(\boldsymbol{S}(\boldsymbol{\nu}_{2})\boldsymbol{r}_{G}^{b}) \\

-m\boldsymbol{S}(\boldsymbol{\nu}_{1}) - m\boldsymbol{S}(\boldsymbol{S}(\boldsymbol{\nu}_{2})\boldsymbol{r}_{G}^{b}) &

m\boldsymbol{S}(\boldsymbol{S}(\boldsymbol{\nu}_{1})\boldsymbol{r}_{G}^{b}) - \boldsymbol{S}(\boldsymbol{I}_{b}^b\boldsymbol{\nu}_{2})

\end{array}

\right] \\

& = \left[

\begin{array}{ccc}

0 & 0 & 0 \\

0 & 0 & 0 \\

0 & 0 & 0 \\

-m(y_G q + z_G r) & m(y_G p + w) & m(z_G p - v) \\

m(x_G q - w) & -m(z_G r + x_G p) & m(z_G q + u) \\

m(x_G r + v) & m(y_G r - u) & -m(x_G p + y_G q)

\end{array}

\right. \\

& \qquad \qquad

\left.

\begin{array}{ccc}

m(y_G q + z_G r) & -m(x_G q - w) & -m(x_G r + v) \\

-m(y_G p + w) & m(z_G r + x_G p) & -m(y_G r - u) \\

-m(z_G p - v) & -m(z_G q + u) & m(x_G p + y_G q) \\

0 & -I_{yz}q - I_{xz}p + I_{z}r & I_{yz}r + I_{xy}p - I_{y}q \\

I_{yz}q + I_{xz}p - I_{z}r & 0 & -I_{xz}r - I_{xy}q + I_{x}p \\

-I_{yz}r - I_{xy}p + I_{y}q & I_{xz}r + I_{xy}q - I_{x}p & 0

\end{array}

\right]

\end{aligned}

Here $m$ is the rigid-body mass, $\backslash boldsymbol\; I\_b^b$ is the inertia tensor about the CO, which is related to the inertia tensor about the CG, $\backslash boldsymbol\{I\}\_g^b$, by Huygens–Steiner's parallel-axis theorem according to $\backslash boldsymbol\; I\_b^b\; =\; \backslash boldsymbol\; I\_g^b\; -\; m\; \backslash boldsymbol\{S\}^2(\backslash boldsymbol\{r\}\_G)$. The linear and angular velocity vectors are denoted by

$\backslash boldsymbol\; \backslash nu\_1\; =\; [u,\; v,\; w]^\backslash top$ and $\backslash boldsymbol\; \backslash nu\_2\; =\; [p,\; q,\; r]^\backslash top$, respectively. As discussed in, there exists several matrix parametrizations of $\backslash boldsymbol\{C\}\_\{RB\}(\backslash boldsymbol\; \backslash nu)$ and as shown later it is advantageous to choose a parametrization, which is independent of linear velocity $\backslash boldsymbol\{\backslash nu\}\_1$, when including irrotational ocean currents using the relative velocity vector. The linear velocity-independent parametrization was derived by Fossen and Fjellstad in 1995:{{cite journal

| last1 = Fossen

| first1 = Thor I.

| last2 = Fjellstad

| first2 = O. E.

| year = 1995

| title = Nonlinear Modelling of Marine Vehicles in 6 Degrees of Freedom

| journal = International Journal of Mathematical Modelling of Systems

| volume = 1

| issue = 1

| pages = 17–28

| doi = 10.1080/13873959508837004

}}

$$

\boldsymbol{C}^{\,\boldsymbol{\nu}_2}_{RB}(\boldsymbol{\nu}) =\left[

\begin{array}[c]{cc}

m\boldsymbol{S}(\boldsymbol \nu_2) & -m\boldsymbol{S}(\boldsymbol \nu_2)\boldsymbol{S}(\boldsymbol{r}_{G}^{b})\\

m\boldsymbol{S}(\boldsymbol{r}_{G}^{b})\boldsymbol{S}(\boldsymbol \nu_2) & -\boldsymbol{S}( \boldsymbol{I}_{b}^b \boldsymbol \nu_2)

\end{array} \right]

The corresponding added mass matrices can be expressed as functions of the hydrodynamic derivatives, derived using a Lagrangian formulation based on Kirchhoff's equations:

$$

\boldsymbol{M}_{A} =

\left[

\begin{array}[c]{ccc}

\boldsymbol{M}_A^{(11)} & \boldsymbol{M}_A^{(12)}\\

\boldsymbol{M}_A^{(21)} & \boldsymbol{M}_A^{(22)}

\end{array} \right] = -\left[

\begin{array}[c]{cccccc}

X_{\dot{u}} & X_{\dot{v}} & X_{\dot{w}} & X_{\dot{p}} & X_{\dot{q}} &

X_{\dot{r}}\\

Y_{\dot{u}} & Y_{\dot{v}} & Y_{\dot{w}} & Y_{\dot{p}} & Y_{\dot{q}} &

Y_{\dot{r}}\\

Z_{\dot{u}} & Z_{\dot{v}} & Z_{\dot{w}} & Z_{\dot{p}} & Z_{\dot{q}} &

Z_{\dot{r}}\\

K_{\dot{u}} & K_{\dot{v}} & K_{\dot{w}} & K_{\dot{p}} & K_{\dot{q}} &

K_{\dot{r}}\\

M_{\dot{u}} & M_{\dot{v}} & M_{\dot{w}} & M_{\dot{p}} & M_{\dot{q}} &

M_{\dot{r}}\\

N_{\dot{u}} & N_{\dot{v}} & N_{\dot{w}} & N_{\dot{p}} & N_{\dot{q}} &

N_{\dot{r}}

\end{array}

\right]

where $\backslash boldsymbol\{M\}\_A^\{(12)\}\; =\; \backslash boldsymbol\{M\}\_A^\{(21)\}$. The Coriolis and centripetal matrix, due to hydrodynamic added mass, is:

$$

\begin{aligned}

\boldsymbol{C}_{A}(\boldsymbol{\nu}) & =\left[

\begin{array}[c]{cc}

\boldsymbol{0}_{3\times3} & -\boldsymbol{S}(\boldsymbol M_A^{(11)}\boldsymbol{\nu}_{1}

+\boldsymbol{M}_A^{(12)}\boldsymbol{\nu}_{2})\\

-\boldsymbol{S}(\boldsymbol M_A^{(11)}\boldsymbol{\nu}_{1}+\boldsymbol M_A^{(12)}\boldsymbol{\nu}_{2}) &

-\boldsymbol{S}(\boldsymbol M_A^{(21)}\boldsymbol{\nu}_{1}+\boldsymbol M_A^{(22)}\boldsymbol{\nu}_{2})

\end{array}

\right] \\

& = \left[

\begin{array}[c]{cccccc}

{ 0} & { 0} & { 0} & { 0} & { -a}_{3} &

{ a}_{2}\\

{ 0} & { 0} & { 0} & { a}_{3} & { 0} &

{ -a}_{1}\\

{ 0} & { 0} & { 0} & { -a}_{2} & { a}_{1} &

{ 0}\\

{ 0} & { -a}_{3} & { a}_{2} & { 0} & { -b}_{3} &

{ b}_{2}\\

{ a}_{3} & { 0} & { -a}_{1} & { b}_{3} & { 0} &

{ -b}_{1}\\

{ -a}_{2} & { a}_{1} & { 0} & { -b}_{2} &

{ b}_{1} & { 0}

\end{array}

\right]

\end{aligned}

where

$$

\begin{aligned}

a_{1} & = X_{\dot{u}}u+X_{\dot{v}}v+X_{\dot{w}}w+X_{\dot{p}}p+X_{\dot{q}}q+X_{\dot{r}}r\\

a_{2} & = Y_{\dot{u}}u+Y_{\dot{v}}v+Y_{\dot{w}}w+Y_{\dot{p}}p+Y_{\dot{q}}q+Y_{\dot{r}}r\\

a_{3} & = Z_{\dot{u}}u+Z_{\dot{v}}v+Z_{\dot{w}}w+Z_{\dot{p}}p+Z_{\dot{q}}q+Z_{\dot{r}}r\\

b_{1} & = K_{\dot{u}}u+K_{\dot{v}}v+K_{\dot{w}}w+K_{\dot{p}}p+K_{\dot{q}}q+K_{\dot{r}}r\\

b_{2} & = M_{\dot{u}}u+M_{\dot{v}}v+M_{\dot{w}}w+M_{\dot{p}}p+M_{\dot{q}}q+M_{\dot{r}}r\\

b_{3} & = N_{\dot{u}}u+N_{\dot{v}}v+N_{\dot{w}}w+N_{\dot{p}}p+N_{\dot{q}}q+N_{\dot{r}}r

\end{aligned}

The hydrodynamic damping matrix depends on linear and quadratic damping and even higher-order terms. This can be expressed by

$$

\boldsymbol{D} = -\left[

\begin{array}{cccccc}

X_{u} & X_{v} & X_{w} & X_{p} & X_{q} & X_{r} \\

Y_{u} & Y_{v} & Y_{w} & Y_{p} & Y_{q} & Y_{r} \\

Z_{u} & Z_{v} & Z_{w} & Z_{p} & Z_{q} & Z_{r} \\

K_{u} & K_{v} & K_{w} & K_{p} & K_{q} & K_{r} \\

M_{u} & M_{v} & M_{w} & M_{p} & M_{q} & M_{r} \\

N_{u} & N_{v} & N_{w} & N_{p} & N_{q} & N_{r}

\end{array}

\right] + \boldsymbol{D}_n(\boldsymbol{\nu})

where $\backslash boldsymbol\{D\}\_n(\backslash boldsymbol\; \backslash nu)$ captures the nonlinear velocity-dependent damping effects. If $x\_G\; =\; x\_B$ and $y\_G\; =\; y\_B$ (the CG is aligned with the CB in both longitudinal and lateral directions), and the craft has starboard-port symmetry, the restoring forces and moments, $\backslash boldsymbol\; g(\backslash boldsymbol\; \backslash eta)\; =\; \backslash boldsymbol\{G\}\; \backslash boldsymbol\; \backslash eta$, for a surface craft can be expressed using the following restoring matrix:

$$

\boldsymbol{G} = \left[

\begin{array}[c]{cccccc}

0 & 0 & 0 & 0 & 0 & 0 \\

0 & 0 & 0 & 0 & 0 & 0 \\

0 & 0 & \rho g A_{wp} & 0 & -\rho g A_{wp} \mathrm{LCF} & 0 \\

0 & 0 & 0 & \rho g \nabla \mathrm{GM}_{T} & 0 & 0 \\

0 & 0 & -\rho g A_{wp} \mathrm{LCF} & 0 & \rho g \left( A_{wp} \mathrm{LCF}^2 + \nabla \mathrm{GM}_{L} \right) & 0 \\

0 & 0 & 0 & 0 & 0 & 0

\end{array}

\right]

where $\backslash rho$ is the density of water, $g$ is the acceleration of gravity, $A\_\{wp\}$ is the waterplane area, $\backslash mathrm\{LCF\}$ is the x-distance from the CO to the centroid of the waterplane, and $\backslash nabla$ is the displaced volume. At the same time, $GM\_T$ and $GM\_L$ are the transverse and lateral metacentric heights, respectively. For underwater vehicles, the waterplane area diminishes, and the restoring forces and moments take the following form:

$$

\boldsymbol{g}(\boldsymbol{\eta})=\left[

\begin{array}

[c]{clcl}

& (W-B)\sin(\theta) & & \\

- & (W-B)\cos(\theta)\sin(\phi) & & \\

- & (W-B)\cos(\theta)\cos(\phi) & & \\

- & (y_G W-y_B B)\cos(\theta)\cos(\phi) & + & (z_G W-z_B B)\cos(\theta)\sin(\phi)\\

& (z_G W-z_B B)\sin(\theta) & + & (x_G W-x_B B)\cos(\theta)\cos({\phi)}\\

- & (x_G W-x_B B) \cos(\theta)\sin(\phi) & - & (y_G W-y_B B)\sin(\theta)

\end{array}

\right]

where $W\; =\; m\; g$ is the weight and $B\; =\; \backslash rho\; g\; \backslash nabla$ is the buoyancy force.

The matrices in Fossen's marine craft model satisfy the following properties:

The dissipative nature of the marine craft model is verified by the time differentiation of the Lyapunov function:

$$

V = \frac{1}{2} \boldsymbol \nu^T \boldsymbol M \boldsymbol \nu + \int_{\boldsymbol{0}}^{\boldsymbol{\eta}}

(\boldsymbol{J}(\boldsymbol{\xi})^\top \boldsymbol{g}(\boldsymbol{\xi}))^\top d \boldsymbol{\xi}

Exploiting the properties above, it can be shown that the time derivative satisfies the passivity condition{{cite book

| last1 = Brogliato

| first1 = Bernard

| last2 = Lozano

| first2 = Rogelio

| last3 = Maschke

| first3 = Bernard

| last4 = Egeland

| first4 = Olav

| year = 2006

| title = Dissipative Systems Analysis and Control: Theory and Applications

| edition = 2nd

| publisher = Springer

| isbn = 978-3030194192

}}

$$

\begin{aligned}

\dot{V} & = \boldsymbol{\nu}^\top \left( \boldsymbol{M} \dot{\boldsymbol{\nu}} + \boldsymbol{g}(\boldsymbol{\eta}) \right) \\

& = \boldsymbol{\nu}^\top \boldsymbol{\tau} - \boldsymbol{\nu}^\top \boldsymbol{D}(\boldsymbol{\nu}) \boldsymbol{\nu}

\end{aligned}

This equation demonstrates the passivity property of the system, as the rate of change of the Lyapunov function, $\backslash dot\{V\}$, depends on the input power $\backslash boldsymbol\{\backslash nu\}^\backslash top\; \backslash boldsymbol\{\backslash tau\}$ and the dissipation term $-\backslash boldsymbol\{\backslash nu\}^\backslash top\; \backslash boldsymbol\{D\}(\backslash boldsymbol\{\backslash nu\})\; \backslash boldsymbol\{\backslash nu\}$. Since $\backslash boldsymbol\{D\}(\backslash boldsymbol\{\backslash nu\})$ is strictly positive, the dissipation term is strictly negative, ensuring energy dissipation and contributing to the asymptotic stability of the system.

Environmental forces and moments can be included using relative velocity for ocean currents. At the same time, wind and wave loads $\backslash boldsymbol\{\backslash tau\}\_\{\backslash textrm\{wind\}\}$ and $\backslash boldsymbol\{\backslash tau\}\_\{\backslash textrm\{wave\}\}$ can be added by linear superposition. The relative velocity, $\backslash boldsymbol\{\backslash nu\}\_r\; =\; \backslash boldsymbol\{\backslash nu\}\; -\; \backslash boldsymbol\{\backslash nu\}\_c$, accounts for the influence of an irrotational ocean current with velocity $\backslash boldsymbol\{\backslash nu\}\_c\; =\; [u\_c,\; v\_c,\; w\_c,\; 0,\; 0,\; 0]^\backslash top$. This relative velocity modifies the hydrodynamic forces and moments, as the interaction of the vehicle or vessel with the surrounding fluid depends on the velocity relative to the water. The resulting model is:

style="border: 2px solid green; padding: 10px; background-color: #f9fff9;" | $$ \boldsymbol{M}_{RB}\dot{\boldsymbol{\nu}} + \boldsymbol{C}_{RB}(\boldsymbol{\nu})\boldsymbol{\nu} + \boldsymbol{M}_{A}\dot{\boldsymbol{\nu}}_{r} + \boldsymbol{C}_{A}(\boldsymbol{\nu}_{r})\boldsymbol{\nu}_{r} + \boldsymbol{D}(\boldsymbol{\nu}_{r})\boldsymbol{\nu}_{r} + \boldsymbol{g}(\boldsymbol{\eta}) = \boldsymbol{\tau} + \boldsymbol{\tau}_{\textrm{wind}} + \boldsymbol{\tau}_{\textrm{wave}}

where

• Rigid-body forces: $\backslash boldsymbol\{M\}\_\{RB\}\backslash dot\{\backslash boldsymbol\{\backslash nu\}\}\; +\; \backslash boldsymbol\{C\}\_\{RB\}(\backslash boldsymbol\{\backslash nu\})\backslash boldsymbol\{\backslash nu\}$
• Hydrodynamic forces: $\backslash boldsymbol\{M\}\_\{A\}\backslash dot\{\backslash boldsymbol\{\backslash nu\}\}\_\{r\}\; +\; \backslash boldsymbol\{C\}\_\{A\}(\backslash boldsymbol\{\backslash nu\}\_\{r\})\backslash boldsymbol\{\backslash nu\}\_\{r\}\; +\; \backslash boldsymbol\{D\}(\backslash boldsymbol\{\backslash nu\}\_\{r\})\backslash boldsymbol\{\backslash nu\}\_\{r\}$
• Hydrostatic forces: $\backslash boldsymbol\{g\}(\backslash boldsymbol\{\backslash eta\})$
• Propulsion forces: $\backslash boldsymbol\{\backslash tau\}$
• Wind forces: $\backslash boldsymbol\{\backslash tau\}\_\{\backslash textrm\{wind\}\}$
• Wave-induced forces: $\backslash boldsymbol\{\backslash tau\}\_\{\backslash textrm\{wave\}\}$

The relative equations of motion can be simplified by adopting the rigid-body Coriolis and centripetal matri$\backslash boldsymbol\{C\}^\{\backslash ,\backslash boldsymbol\{\backslash nu\}\_2\}\_\{RB\}(\backslash boldsymbol\; \backslash nu)$, which is independent of the linear velocity component $\backslash boldsymbol\backslash nu\_1\; =\; [u,\; v,\; w]^\backslash top$. This key property was exploited by Hegrenæs in 2010,{{cite thesis

| last = Hegrenæs

| first = Øyvind

| year = 2010

| title = Autonomous Navigation for Underwater Vehicles

| publisher = Department of Engineering Cybernetics, Norwegian University of Science and Technology

| location = Trondheim, Norway

| type = PhD thesis

}} who showed that:

Using this result, the relative equations of motion are simplified to:

style="border: 2px solid green; padding: 10px; background-color: #f9fff9;" | $$ \boldsymbol{M} \dot{\boldsymbol{\nu}}_r + \boldsymbol{C}(\boldsymbol{\nu}_r)\boldsymbol{\nu}_r +\boldsymbol{D}(\boldsymbol{\nu}_r)\boldsymbol{\nu}_r+\boldsymbol{g}(\boldsymbol{\eta)} = \boldsymbol{\tau}+\boldsymbol{\tau}_{\textrm{wind}} +\boldsymbol{\tau}_{\textrm{wave}}

where $\backslash boldsymbol\; M\; =\; \backslash boldsymbol\; M\_\{RB\}\; +\; \backslash boldsymbol\; M\_A$ and $\backslash boldsymbol\; C(\backslash boldsymbol\; \backslash nu\_r)\; =\; \backslash boldsymbol\; C\_\{RB\}(\backslash boldsymbol\; \backslash nu\_r)\; +\; \backslash boldsymbol\; C\_A(\backslash boldsymbol\; \backslash nu\_r)$.

An irrotational ocean current implies its velocity field has no curl, leading to a potential flow description. In practical terms, this means the ocean current velocity, $\backslash boldsymbol\{\backslash nu\}\_c\; =\; [u\_c,\; v\_c,\; w\_c,\; 0,\; 0,\; 0]^\backslash top$, remains spatially uniform and constant (or nearly constant) in the NED frame, with no rotational components $(p\_c\; =\; q\_c\; =\; r\_c\; =\; 0)$. Hence, the application of $\backslash dot\{\backslash boldsymbol\{R\}\}\_\{zyx\}\; =\; \backslash boldsymbol\{R\}\_\{zyx\}\backslash boldsymbol\{S\}(\backslash boldsymbol\{\backslash nu\}\_2)$ implies that the ocean current velocity vector satisfies:

$$

\dot{\boldsymbol{v}}_{c}^{n} = \dot{\boldsymbol{R}}_{zyx} \boldsymbol{v}_{c}^{b} + \boldsymbol{R}_{zyx} \dot{\boldsymbol{v}}_{c}^{b} \equiv \boldsymbol{0} \quad \implies \quad \dot{\boldsymbol{v}}_{c}^{b} = -\boldsymbol{S}(\boldsymbol{\nu}_2) \boldsymbol{v}_{c}^{b}

where $\backslash boldsymbol\{v\}\_\{c\}^\{b\}\; =\; [u\_c,\; v\_c,\; w\_c]^\backslash top$ is the ocean current linear velocity vector expressed in the BODY frame. The numerical solution proceeds by integrating the differential equation for absolute velocity:

$$

\begin{aligned}

\dot{\boldsymbol{\nu}} & =

\left[

\begin{array}{c}

-\boldsymbol{S}(\boldsymbol{\nu}_2)\boldsymbol{v}_{c}^{b} \\

\boldsymbol{0}_{3\times1}

\end{array}

\right] \\

& + \boldsymbol{M}^{-1} \left( \boldsymbol{\tau} +

\boldsymbol{\tau}_{\textrm{wind}}

+\boldsymbol{\tau}_{\textrm{wave}}

- \boldsymbol{C}(\boldsymbol{\nu}_{r})\boldsymbol{\nu}_{r}-\boldsymbol{D}(\boldsymbol{\nu}_{r})\boldsymbol{\nu}_{r}

-\boldsymbol{g}(\boldsymbol{\eta})\right)

\end{aligned}

Since its introduction in 1991, Fossen's marine craft model has been cited in thousands of research papers and technical references. It has become a cornerstone in studying and developing dynamic models for various types of marine craft, including ships, semisubmersibles, USVs, AUVs, submarines, and offshore structures. The model and its associated tools are available for implementation and further exploration through the "Marine Systems Simulator" (MSS) GitHub repository,{{cite web

| title = Marine Systems Simulator (MSS)

| url = https://github.com/cybergalactic/MSS

| website = GitHub

| publisher = Cybergalactic

}} providing a valuable resource for researchers and practitioners.

One of the most common applications of the model is in describing the surge-–sway-–yaw motions of a starboard-port symmetrical ship. For such vessels, the equations of relative motion can be expressed by:

$$

\boldsymbol{M} \dot{\boldsymbol{\nu}}_r + \boldsymbol{C}(\boldsymbol{\nu}_r)\boldsymbol{\nu}_r +\boldsymbol{D}(\boldsymbol{\nu}_r)\boldsymbol{\nu}_r =\boldsymbol{\tau}+\boldsymbol{\tau}_{\textrm{wind}}

+\boldsymbol{\tau}_{\textrm{wave}}

where $\backslash boldsymbol\{\backslash nu\}\_r\; =\; [u\_r,\; v\_r,\; r]^\backslash top$ and $\backslash boldsymbol\{\backslash eta\}\; =\; [x,\; y,\; \backslash psi]^\backslash top$. The model matrices for 3-DOF surface vessels take the following form:

$$

\begin{aligned}

\boldsymbol{M}_{RB} & = \left[

\begin{array}[c]{ccc}

m & 0 & 0\\

0 & m & mx_G \\

0 & mx_G & I_{z}

\end{array} \right] \\

\boldsymbol{M}_A & = \left[

\begin{array}[c]{ccc}

- X_{\dot{u}}& 0 & 0\\

0 & -Y_{\dot{v}} & -Y_{\dot{r}} \\

0 & -N_{\dot{v}} & -N_{\dot{r}}

\end{array} \right] \\

\boldsymbol{C}(\boldsymbol{\nu}_r) & = \left[

\begin{array}[c]{ccc}

0 & -mr & -mx_G r\\

mr & 0 & 0\\

mx_G r & 0 & 0

\end{array}

\right] \\

\boldsymbol{C}_{A}(\boldsymbol{\nu}_{r}) & = \left[

\begin{array}[c]{ccc}

0 & 0 & Y_{\dot{v}}v_{r}+Y_{\dot{r}}r\\

0 & 0 & -X_{\dot{u}}u_{r}\\

-Y_{\dot{v}}v_{r}-Y_{\dot{r}}r & X_{\dot{u}}u_{r} & 0

\end{array} \right] \\

\boldsymbol{D} & = -\left[

\begin{array}

[c]{ccc}

X_{{u}} & 0 & 0\\

0 & Y_{{v}} & Y_{{r}} \\

0 & N_{{v}} & N_{{r}}

\end{array}

\right] \\

\boldsymbol{D}_n(\boldsymbol{\nu}_r) & = - \left[

\begin{array}[c]{ccc}

X_{|u|u} |u_r| & 0 & 0 \\

0 & { Y}_{|v|v}\left\vert v_{r}\right\vert +{ Y}_{|r|v}\left\vert

r\right\vert & { Y}_{|v|r}\left\vert v_{r}\right\vert +{ Y}_{|r|r}\left\vert r\right\vert \\

0 & { N}_{|v|v}\left\vert v_{r}\right\vert +{ N}_{|r|v}\left\vert

r\right\vert & { N}_{|v|r}\left\vert v_{r}\right\vert + N

_{|r|r}\left\vert r\right\vert

\end{array} \right]

\end{aligned}

{{Draft categories|

Category:Equations of fluid dynamics

}}

{{Drafts moved from mainspace|date=December 2024}}