eul

R2026b

Convert transformation or rotation into Euler angles

Since R2026b

Syntax

angles = eul(transformation)

angles = eul(rotation)

angles = eul(___,sequence)

Description

angles = eul(transformation) converts the rotation of the transformation transformation to the Euler angles angles.

angles = eul(rotation) converts the rotation rotation to the Euler angles angles.

angles = eul(___,sequence) specifies the sequence of the Euler-angle rotations sequence using any of the input arguments in previous syntaxes. For example, a sequence of "ZYX" first rotates the z-axis, followed by the y-axis and x-axis.

Examples

Convert SE(3) Transformation to Euler Angles

Create SE(3) transformation with no translation but with a rotation defined by a Euler angles.

eul1 = [pi/4 pi/3 pi/8]

eul1 = 1×3

    0.7854    1.0472    0.3927

T = se3(eul1,"eul")

T = se3
    0.3536   -0.4189    0.8364         0
    0.3536    0.8876    0.2952         0
   -0.8660    0.1913    0.4619         0
         0         0         0    1.0000

Get the Euler angles from the transformation.

eul2 = eul(T)

eul2 = 1×3

    0.7854    1.0472    0.3927

Convert SO(3) Rotation to Euler Angles

Create SO(3) rotation defined by a Euler angles.

eul1 = [pi/4 pi/3 pi/8]

eul1 = 1×3

    0.7854    1.0472    0.3927

R = so3(eul1,"eul")

R = so3
    0.3536   -0.4189    0.8364
    0.3536    0.8876    0.2952
   -0.8660    0.1913    0.4619

Get the Euler angles from the transformation.

eul2 = eul(R)

eul2 = 1×3

    0.7854    1.0472    0.3927

Input Arguments

transformation — Transformation

se3 object | N-element array of se3 objects

Transformation, specified as an se3 object or as an N-element array of se3 objects. N is the total number of transformations.

rotation — Rotation

so3 object | N-element array of so3 objects

Rotation, specified as an so3 object or as an N-element array of so3 objects. N is the total number of rotations.

sequence — Axis-rotation sequence

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

Axis-rotation sequence for the Euler angles, specified as one of these string scalars:

• "ZYX" (default)

• "ZYZ"

• "ZXY"

• "ZXZ"

• "YXY"

• "YZX"

• "YXZ"

• "YZY"

• "XYX"

• "XYZ"

• "XZX"

• "XZY"

These are orthonormal rotation matrices for rotations of ϕ, ψ, and θ about the x-, y-, and z-axis, respectively:

$$R_x(\varphi )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \varphi & \sin \varphi \\ 0& -\sin \varphi & \cos \varphi \end{array}\right]$$, $$R_y(\psi )=\left[\begin{array}{ccc}\cos \psi & 0& \sin \psi \\ 0& 1& 0\\ -\sin \psi & 0& \cos \psi \end{array}\right]$$, $$R_z(\theta )=\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]$$

When constructing the rotation matrix from this sequence, each character indicates the corresponding axis. For example, if the sequence is "XYZ", then the eul object constructs the rotation matrix R by multiplying the rotation about x-axis with the rotation about the y-axis, and then multiplying that product with the rotation about the z-axis:

$$R=R_x(\varphi )·R_y(\psi )·R_z(\theta )$$

Example: eul([pi/2 pi/3 pi/4],"eul","ZYZ") rotates a point by pi/4 radians about the z-axis, then rotates the point by pi/3 radians about the y-axis, and then rotates the point by pi/2 radians about the z-axis. This is equivalent to eul(pi/2,"rotz") * eul(pi/3,"roty") * eul(pi/4,"rotz")

Data Types: string | char

Output Arguments

angles — Euler angles

M-by-3 matrix

Euler angles, returned as an M-by-3 matrix of Euler rotation angles. Each row represents one Euler angle set.

Version History

Introduced in R2026b