nwtc_num::dcm_logmap Interface Reference

For any direction cosine matrix (DCM), \(\Lambda\), this routine calculates the logarithmic map, \(\lambda\), which a skew-symmetric matrix: More...

Public Member Functions
subroutine	dcm_logmapr (DCM, logMap, ErrStat, ErrMsg, thetaOut)
	For any direction cosine matrix (DCM), \(\Lambda\), this routine calculates the logarithmic map, \(\lambda\), which a skew-symmetric matrix: More...
subroutine	dcm_logmapd (DCM, logMap, ErrStat, ErrMsg, thetaOut)
	For any direction cosine matrix (DCM), \(\Lambda\), this routine calculates the logarithmic map, \(\lambda\), which a skew-symmetric matrix: More...

Detailed Description

For any direction cosine matrix (DCM), \(\Lambda\), this routine calculates the logarithmic map, \(\lambda\), which a skew-symmetric matrix:

\begin{equation} \lambda = \log( \Lambda ) = \begin{bmatrix} 0 & \lambda_3 & -\lambda_2 \\ -\lambda_3 & 0 & \lambda_1 \\ \lambda_2 & -\lambda_1 & 0 \end{bmatrix} \end{equation}

The angle of rotation for \(\Lambda\) is

\begin{equation} \theta= \begin{matrix} \cos^{-1}\left(\frac{1}{2}\left(\mathrm{trace}(\Lambda)-1\right)\right) & \theta \in \left[0,\pi\right]\end{matrix} \end{equation}

And the logarithmic map is

\begin{equation} \lambda = \left\{ \begin{matrix} 0 & \theta = 0 \\ \frac{\theta}{2\sin\theta} \left( \Lambda - \Lambda^T\right) & \theta \in \left(0,\pi\right) \\ \pm\pi v & \theta = \pi \end{matrix} \right. \end{equation}

where \(v\) is the skew-symmetric matrix associated with the unit-length eigenvector of \(\Lambda\) associated with the eigenvalue 1. However, this equation has numerical issues near \(\theta = \pi\), so for \(\theta > 3.1\) we instead implement a separate equation to find lambda * sign(lambda(indx_max)) and use \(\Lambda - \Lambda^T\) to choose the appropriate signs.

This routine is the inverse of DCM_exp (nwtc_num::dcm_exp).
Use DCM_logMap (nwtc_num::dcm_logmap) instead of directly calling a specific routine in the generic interface.

Parameters

[in]	dcm	the direction cosine matrix, \(\Lambda\)
[out]	logmap	vector containing \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\), the unique components of skew-symmetric matrix \(\lambda\)
[out]	thetaout	the angle of rotation, \(\theta\); output only for debugging
[out]	errstat	Error status of the operation
[out]	errmsg	Error message if ErrStat /= ErrID_None

Member Function/Subroutine Documentation

dcm_logmapd()

subroutine nwtc_num::dcm_logmap::dcm_logmapd	(	real(dbki), dimension(3,3), intent(in)	DCM,
		real(dbki), dimension(3), intent(out)	logMap,
		integer(intki), intent(out)	ErrStat,
		character(*), intent(out)	ErrMsg,
		real(dbki), intent(out), optional	thetaOut
	)

For any direction cosine matrix (DCM), \(\Lambda\), this routine calculates the logarithmic map, \(\lambda\), which a skew-symmetric matrix:

\begin{equation} \lambda = \log( \Lambda ) = \begin{bmatrix} 0 & \lambda_3 & -\lambda_2 \\ -\lambda_3 & 0 & \lambda_1 \\ \lambda_2 & -\lambda_1 & 0 \end{bmatrix} \end{equation}

The angle of rotation for \(\Lambda\) is

\begin{equation} \theta= \begin{matrix} \cos^{-1}\left(\frac{1}{2}\left(\mathrm{trace}(\Lambda)-1\right)\right) & \theta \in \left[0,\pi\right]\end{matrix} \end{equation}

And the logarithmic map is

\begin{equation} \lambda = \left\{ \begin{matrix} 0 & \theta = 0 \\ \frac{\theta}{2\sin\theta} \left( \Lambda - \Lambda^T\right) & \theta \in \left(0,\pi\right) \\ \pm\pi v & \theta = \pi \end{matrix} \right. \end{equation}

where \(v\) is the skew-symmetric matrix associated with the unit-length eigenvector of \(\Lambda\) associated with the eigenvalue 1. However, this equation has numerical issues near \(\theta = \pi\), so for \(\theta > 3.1\) we instead implement a separate equation to find lambda * sign(lambda(indx_max)) and use \(\Lambda - \Lambda^T\) to choose the appropriate signs.

This routine is the inverse of DCM_exp (nwtc_num::dcm_exp).
Use DCM_logMap (nwtc_num::dcm_logmap) instead of directly calling a specific routine in the generic interface.

Parameters

[in]	dcm	the direction cosine matrix, \(\Lambda\)
[out]	logmap	vector containing \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\), the unique components of skew-symmetric matrix \(\lambda\)
[out]	thetaout	the angle of rotation, \(\theta\); output only for debugging
[out]	errstat	Error status of the operation
[out]	errmsg	Error message if ErrStat /= ErrID_None

dcm_logmapr()

subroutine nwtc_num::dcm_logmap::dcm_logmapr	(	real(reki), dimension(3,3), intent(in)	DCM,
		real(reki), dimension(3), intent(out)	logMap,
		integer(intki), intent(out)	ErrStat,
		character(*), intent(out)	ErrMsg,
		real(reki), intent(out), optional	thetaOut
	)

For any direction cosine matrix (DCM), \(\Lambda\), this routine calculates the logarithmic map, \(\lambda\), which a skew-symmetric matrix:

\begin{equation} \lambda = \log( \Lambda ) = \begin{bmatrix} 0 & \lambda_3 & -\lambda_2 \\ -\lambda_3 & 0 & \lambda_1 \\ \lambda_2 & -\lambda_1 & 0 \end{bmatrix} \end{equation}

The angle of rotation for \(\Lambda\) is

\begin{equation} \theta= \begin{matrix} \cos^{-1}\left(\frac{1}{2}\left(\mathrm{trace}(\Lambda)-1\right)\right) & \theta \in \left[0,\pi\right]\end{matrix} \end{equation}

And the logarithmic map is

\begin{equation} \lambda = \left\{ \begin{matrix} 0 & \theta = 0 \\ \frac{\theta}{2\sin\theta} \left( \Lambda - \Lambda^T\right) & \theta \in \left(0,\pi\right) \\ \pm\pi v & \theta = \pi \end{matrix} \right. \end{equation}

where \(v\) is the skew-symmetric matrix associated with the unit-length eigenvector of \(\Lambda\) associated with the eigenvalue 1. However, this equation has numerical issues near \(\theta = \pi\), so for \(\theta > 3.1\) we instead implement a separate equation to find lambda * sign(lambda(indx_max)) and use \(\Lambda - \Lambda^T\) to choose the appropriate signs.

This routine is the inverse of DCM_exp (nwtc_num::dcm_exp).
Use DCM_logMap (nwtc_num::dcm_logmap) instead of directly calling a specific routine in the generic interface.

Parameters

[in]	dcm	the direction cosine matrix, \(\Lambda\)
[out]	logmap	vector containing \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\), the unique components of skew-symmetric matrix \(\lambda\)
[out]	thetaout	the angle of rotation, \(\theta\); output only for debugging
[out]	errstat	Error status of the operation
[out]	errmsg	Error message if ErrStat /= ErrID_None

---

The documentation for this interface was generated from the following file:

• NWTC_Num.f90