Last updated on Feb 20, 2026.

netsse.tools.kinematics#

Functions to manipulate the body kinematics of, e.g., marine vessels.

Copyright (C) 2023-2026 Technical University of Denmark, R.E.G. Mounet

This code is part of the NetSSE software.

NetSSE is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

NetSSE is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see https://www.gnu.org/licenses/.

To credit the author, users are encouraged to use below reference:

Mounet, R. E. G., & Nielsen, U. D. NetSSE: An open-source Python package
for network-based sea state estimation from ships, buoys, and other
observation platforms (version 2.2). Technical University of Denmark,
GitLab. February 2026. https://doi.org/10.11583/DTU.26379811.

Last updated on 20-02-2026 by R.E.G. Mounet

Functions#

Rzyx(phi, theta, psi)

Computes the Euler angle rotation matrix \(R\) using the zyx convention,

Tzyx(phi, theta)

Computes the Euler angle transformation matrix \(T\) for attitude using the zyx convention, as per Fossen (2011).

accel_rigid_body(a_O, r_P, omega, omega_dot[, axis])

Calculates the acceleration \(a_P\) of a point P in a rigid body.

eulerang(phi, theta, psi[, unit])

Computes the Euler angle transformation matrix \(J\) in full using the zyx

Module Contents#

netsse.tools.kinematics.Rzyx(phi, theta, psi)[source]#

Computes the Euler angle rotation matrix \(R\) using the zyx convention, as per Fossen (2011).

Note

This function can be used to transform velocity vectors from a body-fixed frame to the North-East-Down (NED) reference frame, using specified Euler angles.

Parameters:

  • phi (Scalar, or numpy array of size (N,) or (N,M)) – Roll angle [rad].

  • theta (Scalar, or numpy array of size (N,) or (N,M)) – Pitch angle [rad].

  • psi (Scalar, or numpy array of size (N,) or (N,M)) – Yaw angle [rad].

Returns:

R – Euler angle rotation matrix \(R\).

Return type:

Numpy array of size (shape(phi),3,3)

See also

Tzyx

Euler angle transformation matrix for attitude.

eulerang

Euler angle transformation matrix in full.

References

Fossen, T. I. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley and Sons.

Example

>>> R = Rzyx(phi, theta, psi)
netsse.tools.kinematics.Tzyx(phi, theta)[source]#

Computes the Euler angle transformation matrix \(T\) for attitude using the zyx convention, as per Fossen (2011).

Note

This function can be used to transform velocity vectors from a body-fixed frame to the North-East-Down (NED) reference frame, using specified Euler angles.

Parameters:

  • phi (Scalar, or numpy array of size (N,) or (N,M)) – Roll angle [rad].

  • theta (Scalar, or numpy array of size (N,) or (N,M)) – Pitch angle [rad].

Raises:

ValueError – The \(T\) matrix has a singularity for \(\theta = \pm 90^\circ\).

Returns:

T – Euler angle transformation matrix \(T\) for attitude.

Return type:

Numpy array of size (shape(phi),3,3)

See also

Rzyx

Euler angle rotation matrix.

eulerang

Euler angle transformation matrix.

References

Fossen, T. I. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley and Sons.

Example

>>> T = Tzyx(phi, theta)
netsse.tools.kinematics.accel_rigid_body(a_O, r_P, omega, omega_dot, axis=0)[source]#

Calculates the acceleration \(a_P\) of a point P in a rigid body.

The acceleration, \(a_O\) of a point O in the rigid body, as well as the body’s angular velocity, \(\omega\), and acceleration, \(\dot{\omega}\), are assumed to be known.

Parameters:

  • a_O (Numpy array of vector of size (_,3,_)) – Linear acceleration vector of the reference point O [m/s²].

  • r_P (Numpy vector of size (3,)) – Position vector of point P relative to the reference point O [m].

  • omega (Numpy array of vector of size (_,3,_)) – Angular velocity vector [rad/s].

  • omega_dot (Numpy array of vector of size (_,3,_)) – Angular acceleration vector [rad/s²].

  • axis (int, optional) – Axis along which to compute the cross products. Default is 0.

Returns:

a_P – Linear acceleration vector of the target point P [m/s²].

Return type:

Numpy array of size (_,3,_)

Example

>>> a_P = accel_rigid_body(a_O, r_P, omega, omega_dot)
netsse.tools.kinematics.eulerang(phi, theta, psi, unit='rad')[source]#

Computes the Euler angle transformation matrix \(J\) in full using the zyx convention, as per Fossen (2011):

\[J = \left[ \begin{array}{cc} J_1 & 0 \newline 0 & J_2 \end{array} \right],\]

where \(J_1 = R_{zyx}\) and \(J_2 = T_{zyx}\).

Note

This function can be used to transform velocity vectors from a body-fixed frame to the North-East-Down (NED) reference frame, using specified Euler angles.

Parameters:

  • phi (Scalar, or numpy array of size (N,) or (N,M)) – Roll angle.

  • theta (Scalar, or numpy array of size (N,) or (N,M)) – Pitch angle.

  • psi (Scalar, or numpy array of size (N,) or (N,M)) – Yaw angle.

  • unit ({'rad','deg'}, optional) – Unit of the input angles phi, theta, and psi.

Returns:

  • J (Numpy array of size (shape(phi),6,6)) – Euler angle transformation matrix.

  • J1 (Numpy array of size (shape(phi),3,3)) – \(J_1 = R_{zyx}\), the Euler angle rotation matrix.

  • J2 (Numpy array of size (shape(phi),3,3)) – \(J_2 = T_{zyx}\), the Euler angle transformation matrix for attitude.

See also

Rzyx

Euler angle rotation matrix.

Tzyx

Euler angle transformation matrix for attitude.

References

Fossen, T. I. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley and Sons.

Example

>>> [J, Rzyx, Tzyx] = eulerang(phi, theta, psi)