angle2rod

Convert rotation angles to Euler-Rodrigues vector

Syntax

rod=angle2rod(R1,R2,R3)

rod=angle2rod(R1,R2,R3,S)

Description

rod=angle2rod(R1,R2,R3) function converts the rotation described by the three rotation angles, R1, R2, and R3, into an M-by-3 Euler-Rodrigues (Rodrigues) matrix, rod. The rotation angles represent a passive transformation from frame A to frame B. The resulting angles represent a series of right-hand intrinsic passive rotations from frame A to frame B.

rod=angle2rod(R1,R2,R3,S) function converts the rotation described by the three rotation angles and a rotation sequence, S, into an M-by-3 Euler-Rodrigues array, rod, that contains the M Rodrigues vector.

Examples

Determine the Rodrigues Vector from One Rotation Angle

Determine the Rodrigues vector from rotation angles.

yaw = 0.7854;
pitch = 0.1;
roll = 0;
r = angle2rod(yaw,pitch,roll)

r =

   -0.0207    0.0500    0.4142

Determine Rodrigues Vectors from Multiple Rotation Angles

Determine the Rodrigues vectors from multiple rotation angles.

yaw = [0.7854 0.5];
pitch = [0.1 0.3];
roll = [0 0.1];
r = angle2rod(pitch,roll,yaw,'YXZ')

r =

    0.0207    0.0500    0.4142
    0.0885    0.1381    0.2473

Input Arguments

R1 — First rotation angle

M-by-1 array

First rotation angle, in radians, from which to determine Euler-Rodrigues vector. Values must be real.

Data Types: double | single

R2 — Second rotation angle

M-by-1 array

Second rotation angle, in radians, from which to determine Euler-Rodrigues vector. Values must be real.

Data Types: double | single

R3 — Third rotation angle

M-by-1 array

Third rotation angle, in radians, from which to determine Euler-Rodrigues vector. Values must be real.

Data Types: double | single

S — Rotation sequence

ZYX (default) | ZYZ | ZXY | ZXZ | YXZ | YXY | YZX | YZY | XYZ | XYX | XZY | XZX

Rotation sequence. 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

rod — Euler-Rodrigues vector

3-element vector

Euler-Rodrigues vector determined from rotation angles.

Algorithms

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

$$\overrightarrow{b}=\left[\begin{array}{ccc}{b}_x& b_y& b_z\end{array}\right]$$

where:

$$\begin{array}{l}{b}_x=\tan \left(\frac{1}{2}\theta \right)s_x,\\ b_y=\tan \left(\frac{1}{2}\theta \right)s_y,\\ b_z=\tan \left(\frac{1}{2}\theta \right)s_z\end{array}$$

are the Rodrigues parameters. Vector $$\stackrel{\rightharpoonup }{s}$$ 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