# rod2angle

Convert Euler-Rodrigues vector to rotation angles

## Description

example

[R1 R2 R3]=rod2angle(rod) function calculates the set of rotation angles, R1, R2, and R3, for a given Euler-Rodrigues (also known as Rodrigues) vector, rod. The rotation used in this function is a passive transformation between two coordinate systems.

[R1 R2 R3]=rod2angle(rod,S) function calculates the set of rotation angles for a given Rodrigues vector and a specified rotation sequence, S.

## Examples

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Determine rotation angles from vector, [.1 .2 -.1].

r = [.1 .2 -.1];
[yaw, pitch, roll] = rod2angle(r)
yaw =

-0.1651

pitch =

0.4074

roll =

0.1651

## Input Arguments

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M-by-3 matrix containing M Rodrigues vector.

Data Types: double

Rotation angles, in radians, from which to determine Rodrigues vector. For the default rotation sequence, ZYX, the rotation angle order is:

• R1 — z-axis rotation

• R2 — y-axis rotation

• R3 — x-axis rotation

Data Types: char | string

## Output Arguments

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M-by-1 array of first rotation angles, in radians.

M-by-1 array of second rotation angles, in radians.

M-by-1 array of third rotation angles, in radians.

## 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.

## References

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

## Version History

Introduced in R2017a