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{{Short description|Equations of motion for marine craft}}
{{Multiple issues|
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Fossen's marine craft model provides a mathematical framework for describing the nonlinear motion of marine craft in a compact matrix-vector form. [[Thor I. Fossen]] first introduced these [[equations of motion]] in his 1991 doctoral thesis,<ref name="fossen1991">{{cite thesis
 | last = Fossen
 | first = Thor I.
 | year = 1991
 | title = Nonlinear Modelling and Control of Underwater Vehicles
 | publisher = Norges Tekniske Høgskole (NTH) (Now part of the Norwegian University of Science and Technology, NTNU)
 | location = Trondheim, Norway 
 | type = Dr. Ing. thesis
}}</ref> 
marking a significant advancement over classical hydrodynamic models, traditionally expressed in component form. These earlier models often involved hundreds of elements, making it challenging to exploit inherent system properties such as symmetry and energy conservation. Fossen's work unified these dynamics into a streamlined representation, enabling more effective analysis and design. This approach has since become a cornerstone in the field of marine craft hydrodynamics and motion control, widely adopted in both academic research and industry applications.<ref name="fossen1994">{{cite book
 | last = Fossen
 | first = Thor I.
 | year = 1994
 | title = Guidance and Control of Ocean Vehicles
 | publisher = Wiley
 | isbn = 978-0471941132
}}</ref><ref name="fossen2011">{{cite book
 | last = Fossen
 | first = Thor I.
 | year = 2011
 | title = Handbook of Marine Craft Hydrodynamics and Motion Control
 | edition = 1st
 | publisher = Wiley
 | isbn = 978-1119991496
}}</ref><ref name="fossen2021">{{cite book
 | last = Fossen
 | first = Thor I.
 | year = 2021
 | title = Handbook of Marine Craft Hydrodynamics and Motion Control
 | edition = 2nd
 | publisher = Wiley
 | isbn = 978-1119575054
}}</ref>

This matrix-vector representation is particularly valuable for designing guidance, navigation, and control (GNC) systems. It is widely applied to marine craft, including ships, floating offshore structures, submarines, autonomous underwater vehicles (AUVs), and uncrewed surface vehicles (USVs). By leveraging system properties such as symmetry, skew symmetry, and the positive definiteness of matrices, Fossen's marine craft model simplifies nonlinear stability analysis and provides an intuitive and robust framework for control system design.

== Mathematical formulation ==

The [[Six degrees of freedom|six-degrees-of-freedom]] (DOFs) marine craft equations of motion are expressed in matrix-vector form using [[generalized coordinates]] <math>\boldsymbol{\eta} = [x, y, z, \phi, \theta, \psi]^\top</math> to represent the position and orientation, and the [[generalized velocity]] <math>\boldsymbol{\nu} = [u, v, w, p, q, r]^\top</math> to describe the linear and angular velocities.  The [[generalized forces]] acting on the craft, arising from propulsion, wind, waves, and ocean currents, are denoted   <math>\boldsymbol{\tau} = [X, Y, Z, K, M, N]^\top</math>. 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).<ref name="sname1950">{{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
}}</ref>

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;"
| 
<math>
\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}
</math>
|}

where

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

=== Kinematics ===

The kinematic equation can be represented using [[Euler angles]] or [[quaternion|unit quaternions]]<ref name="fossen2021"/>  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):

<math>
\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]
</math>

<math>
\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}
</math>

<math>
\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}
</math>

where <math>\mathrm{s}\;\cdot=\sin(\cdot)</math>, <math>\mathrm{c}\;\cdot=\cos(\cdot)</math> and
<math>\mathrm{t}\;\cdot=\tan(\cdot)</math>. The matrix <math>\mathbf{R}_{zyx}(\boldsymbol{\eta})</math> is the rotation matrix for translational velocities and <math>\mathbf{T}_{zyx}(\boldsymbol{\eta})</math> is the transformation matrix for rotational velocities.

=== Kinetics ===

The system inertia matrix <math>\boldsymbol M = \boldsymbol M_{RB} + \boldsymbol M_A</math> and Coriolis and centripetal matrix <math>\boldsymbol C(\boldsymbol{\nu}) = \boldsymbol C_{RB}(\boldsymbol{\nu}) + \boldsymbol C_A(\boldsymbol{\nu})</math> consist of contributions from both the [[rigid body dynamics|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 <math>\boldsymbol D(\boldsymbol{\nu})</math> while <math>\boldsymbol{g}(\boldsymbol{\eta})</math> is the vector of gravitational and buoyancy forces. Let <math>\boldsymbol r_G^b = [x_G, y_G, z_G]^\top</math> and  <math>\boldsymbol r_B^b = [x_B, y_B, z_B]^\top</math> 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 <math>\mathbf{a} \times \mathbf{b} = \boldsymbol S(\mathbf{a}) \mathbf{b}</math>, where <math> \boldsymbol S(\mathbf{a})</math> is a [[skew-symmetric matrix]]:

<math>
\boldsymbol{S}(\mathbf{a}) = \begin{bmatrix}
      \,\,0 &  \!-a_3 & \,\,\,a_2 \\
  \,\,\,a_3 &       0 &    \!-a_1 \\
     \!-a_2 & \,\,a_1 &     \,\,0
\end{bmatrix}
</math>

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 equations|Kirchhoff's equations]]. This approach was formalized in Sagatun and Fossen's 1991 theorem on the Lagrangian formulation of vehicle dynamics.<ref name="sagatun1991">{{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
}}</ref> The specific matrices that govern the dynamics are detailed below, each representing a fundamental aspect of the system's behavior:

<math>
\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}
</math>

<math>
\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}
</math>

Here <math>m</math> is the rigid-body mass, <math>\boldsymbol I_b^b</math> is the [[inertia tensor]] about the CO, which is related to the inertia tensor about the CG, <math>\boldsymbol{I}_g^b</math>,  by [[Parallel axis theorem|Huygens–Steiner's parallel-axis theorem]] according to <math>\boldsymbol I_b^b = \boldsymbol I_g^b - m \boldsymbol{S}^2(\boldsymbol{r}_G)</math>. The linear and angular velocity vectors are denoted by 
<math>\boldsymbol \nu_1 = [u, v, w]^\top</math> and <math>\boldsymbol \nu_2 = [p,  q,  r]^\top</math>, respectively. As discussed in,<ref name="fossen2011"/><ref name="fossen2021"/> there exists several matrix parametrizations of <math>\boldsymbol{C}_{RB}(\boldsymbol \nu)</math> and as shown later it is advantageous to choose a parametrization, which is independent of linear velocity <math>\boldsymbol{\nu}_1</math>, when including irrotational ocean currents using the relative velocity vector. The linear velocity-independent parametrization was derived by Fossen and Fjellstad in 1995:<ref name="fossenfjellstad1995">{{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
 }}</ref>

<math>
\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]
</math>

The corresponding added mass matrices can be expressed as functions of the hydrodynamic derivatives, derived using a Lagrangian formulation based on [[Kirchhoff equations|Kirchhoff's equations]]:<ref name="sagatun1991"/>

<math>
\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]  
</math>

where <math>\boldsymbol{M}_A^{(12)} = \boldsymbol{M}_A^{(21)}</math>. The Coriolis and centripetal matrix, due to hydrodynamic added mass, is:<ref name="sagatun1991"/>

<math>
\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}
</math>

where

<math>
\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}
</math>

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

<math>
\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})
</math>

where <math>\boldsymbol{D}_n(\boldsymbol \nu)</math> captures the nonlinear velocity-dependent damping effects. If <math> x_G = x_B</math> and <math> y_G = y_B</math> (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, <math>\boldsymbol g(\boldsymbol \eta) = \boldsymbol{G} \boldsymbol \eta</math>, for a surface craft can be expressed using the following restoring matrix:

<math>
\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]
</math>

where <math>\rho</math> is the density of water, <math>g</math> is the acceleration of gravity, <math>A_{wp}</math> is the waterplane area, <math>\mathrm{LCF}</math> is the x-distance from the CO to the centroid of the waterplane, and <math>\nabla</math> is the displaced volume. At the same time, <math>GM_T</math> and <math>GM_L</math> are the transverse and lateral [[metacentric height]]s, respectively. For underwater vehicles, the waterplane area diminishes, and the restoring forces and moments take the following form:

<math>
\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] 
</math>

where <math>W = m g</math> is the weight and <math>B = \rho g \nabla</math> is the buoyancy force.

=== Properties ===

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

<div style="border: 1px solid #ccc; background-color: #f9f9f9; padding: 10px;">
<ul>
    <li>Positive definite mass matrix:
    <math>\boldsymbol{M} = \boldsymbol{M}^\top > 0</math> and 
<math> \dot{\boldsymbol{M}} = \boldsymbol{0}</math>
    </li>
    <li>Skew-symmetric Coriolis and centripetal matrix:
    <math>\boldsymbol{\nu}^\top \boldsymbol{C}(\boldsymbol{\nu}) \boldsymbol{\nu} \equiv \boldsymbol{0}</math>
    </li>
    <li>Strictly positive hydrodynamic damping matrix:
    <math>\boldsymbol{\nu}^\top \boldsymbol{D}(\boldsymbol{\nu}) \boldsymbol{\nu} > 0, \quad \text{for all } \boldsymbol{\nu} \neq \boldsymbol{0}</math>
    </li>
</ul>
</div>

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

<math>
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} 
</math>

Exploiting the properties above, it can be shown that the time derivative satisfies the passivity condition<ref name="brogliato2006">{{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
}}</ref>

<math>
\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}
</math>

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

== Equations of relative motion ==

Environmental forces and moments can be included using relative velocity for ocean currents. At the same time, wind and wave loads <math>\boldsymbol{\tau}_{\textrm{wind}}</math> and <math>\boldsymbol{\tau}_{\textrm{wave}}</math> can be added by linear superposition. The relative velocity, <math>\boldsymbol{\nu}_r = \boldsymbol{\nu} - \boldsymbol{\nu}_c</math>, accounts for the influence of an irrotational ocean current with velocity <math>\boldsymbol{\nu}_c = [u_c, v_c, w_c, 0, 0, 0]^\top</math>. 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;"
| 
<math>
\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}}
</math>
|}

where

** Rigid-body forces: <math>\boldsymbol{M}_{RB}\dot{\boldsymbol{\nu}} + \boldsymbol{C}_{RB}(\boldsymbol{\nu})\boldsymbol{\nu}</math>  
** Hydrodynamic forces: <math>\boldsymbol{M}_{A}\dot{\boldsymbol{\nu}}_{r} + \boldsymbol{C}_{A}(\boldsymbol{\nu}_{r})\boldsymbol{\nu}_{r} + \boldsymbol{D}(\boldsymbol{\nu}_{r})\boldsymbol{\nu}_{r}</math>  
** Hydrostatic forces: <math>\boldsymbol{g}(\boldsymbol{\eta})</math>  
** Propulsion forces: <math>\boldsymbol{\tau}</math> 
** Wind forces: <math>\boldsymbol{\tau}_{\textrm{wind}}</math>  
** Wave-induced forces: <math>\boldsymbol{\tau}_{\textrm{wave}}</math>

=== Simplified relative equations of motion ===

The relative equations of motion can be simplified by adopting the rigid-body Coriolis and centripetal matri<math>\boldsymbol{C}^{\,\boldsymbol{\nu}_2}_{RB}(\boldsymbol \nu)</math>, which is independent of the linear velocity component <math>\boldsymbol\nu_1 = [u, v, w]^\top</math>. This key property was exploited by Hegrenæs in 2010,<ref name="fossen2021"/><ref name="hegrenaes2010">{{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
}}</ref> who showed that:

<div style="border: 1px solid #ccc; background-color: #f9f9f9; padding: 10px;">
<math>
\boldsymbol{M}_{RB} \dot{\boldsymbol{\nu}}
+\boldsymbol{C}^{\,\boldsymbol{\nu}_2}_{RB} (\boldsymbol{\nu})\boldsymbol{\nu}\equiv\boldsymbol{M}_{RB}\dot{\boldsymbol{\nu}}_{r}
+\boldsymbol{C}^{\,\boldsymbol{\nu}_2}_{RB}(\boldsymbol{\nu}_{r}) \boldsymbol{\nu}_{r}
</math>
</div>

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

{| style="border: 2px solid green; padding: 10px; background-color: #f9fff9;"
| 
<math>
\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}} 
</math> 
|}

where <math>\boldsymbol M = \boldsymbol M_{RB} + \boldsymbol M_A</math> and <math>\boldsymbol C(\boldsymbol \nu_r) = \boldsymbol C_{RB}(\boldsymbol \nu_r) + \boldsymbol C_A(\boldsymbol \nu_r)</math>.

=== Numerical integration of the relative equations of motion ===

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, <math>\boldsymbol{\nu}_c = [u_c, v_c, w_c, 0, 0, 0]^\top</math>, remains spatially uniform and constant (or nearly constant) in the NED frame, with no rotational components <math>(p_c = q_c = r_c = 0)</math>. Hence, the application of <math>\dot{\boldsymbol{R}}_{zyx} = \boldsymbol{R}_{zyx}\boldsymbol{S}(\boldsymbol{\nu}_2)</math> implies that the ocean current velocity vector satisfies:

<math>
\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}
</math>

where <math>\boldsymbol{v}_{c}^{b} = [u_c, v_c, w_c]^\top</math> is the ocean current linear velocity vector expressed in the BODY frame. The numerical solution proceeds by integrating the differential equation for absolute velocity:

<math>
\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}
</math>

== Applications ==

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,<ref>{{cite web
 | title = Marine Systems Simulator (MSS)
 | url = https://github.com/cybergalactic/MSS
 | website = GitHub
 | publisher = Cybergalactic
}}</ref> 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:<ref name="fossen2021"/>

<math>
\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}}
</math>

where <math>\boldsymbol{\nu}_r = [u_r, v_r, r]^\top</math> and <math>\boldsymbol{\eta} = [x, y, \psi]^\top</math>. The model matrices for 3-DOF surface vessels take the following form:

<math>
\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}
</math>

== References ==

<references>
</references>



{{Draft categories|
[[Category:Equations of fluid dynamics]]
}}

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