dcm2rod

Convert direction cosine matrix to Euler-Rodrigues vector

Description

example

R = dcm2rod(dcm) function calculates the Euler-Rodrigues vector (R) from the direction cosine matrix. This function applies only to direction cosine matrices that are orthogonal with determinant +1.

R = dcm2rod(dcm,action) performs action if the direction cosine matrix is invalid (not orthogonal).

R = dcm2rod(dcm,action,tolerance) uses a tolerance level to evaluate if the direction cosine matrix, n, is valid (orthogonal).

Examples

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Determine the Rodrigues vector from the direction cosine matrix.

DCM = [0.433 0.75 0.5;-0.25 -0.433 0.866;0.866 -0.5 0.0];
r = dcm2rod(DCM)
r =

1.3660    0.3660    1.0000

Determine the Rodrigues vector from the direction cosine matrix validated within tolerance.

DCM = [0.433 0.75 0.5;-0.25 -0.433 0.866;0.866 -0.5 0.0];
r = dcm2rod(DCM,'Warning',0.1)
r =
1.3660    0.3660    1.0000

Input Arguments

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3-by-3-by-M containing M direction cosine matrices.

Data Types: double

Function behavior when direction cosine matrix is invalid (not orthogonal).

• Warning — Displays warning and indicates that the direction cosine matrix is invalid.

• Error — Displays error and indicates that the direction cosine matrix is invalid.

• None — Does not display warning or error (default).

Data Types: char | string

Tolerance of direction cosine matrix validity, specified as a scalar. The function considers the direction cosine matrix valid if these conditions are true:

• The transpose of the direction cosine matrix times itself equals 1 within the specified tolerance (transpose(n)*n == 1±tolerance)

• The determinant of the direction cosine matrix equals 1 within the specified tolerance (det(n) == 1±tolerance).

Data Types: double

Output Arguments

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M-by-3 matrix containing M Rodrigues vectors.

Data Types: double

Algorithms

An Euler-Rodrigues vector $\stackrel{⇀}{b}$ represents a rotation by integrating a direction cosine of a rotation axis with the tangent of half the rotation angle as follows:

$\stackrel{\to }{b}=\left[\begin{array}{ccc}{b}_{x}& {b}_{y}& {b}_{z}\end{array}\right]$

where:

$\begin{array}{l}{b}_{x}=\mathrm{tan}\left(\frac{1}{2}\theta \right){s}_{x},\\ {b}_{y}=\mathrm{tan}\left(\frac{1}{2}\theta \right){s}_{y},\\ {b}_{z}=\mathrm{tan}\left(\frac{1}{2}\theta \right){s}_{z}\end{array}$

are the Rodrigues parameters. Vector $\stackrel{⇀}{s}$ represents a unit vector around which the rotation is performed. Due to the tangent, the rotation vector is indeterminate when the rotation angle equals ±pi radians or ±180 deg. Values can be negative or positive.

 Dai, J.S. "Euler-Rodrigues formula variations, quaternion conjugation and intrinsic connections." Mechanism and Machine Theory, 92, 144-152. Elsevier, 2015.

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