se2

SE(2) homogeneous transformation

Since R2022b

Description

The se2 object represents an SE(2) transformation as a 2-D homogeneous transformation matrix consisting of a translation and rotation.

For more information, see the 2-D Homogeneous Transformation Matrix section.

This object acts like a numerical matrix, enabling you to compose poses using multiplication and division.

Creation

Syntax

transformation = se2

transformation = se2(rotation)

transformation = se2(rotation,translation)

transformation = se2(transformation)

transformation = se2(angle,"theta")

transformation = se2(angle,"theta",translation)

transformation = se2(translation,"trvec")

transformation = se2(pose,"xytheta")

Description

Rotation Matrices, Translation Vectors, and Transformation Matrices

transformation = se2 creates an SE(2) transformation representing an identity rotation with no translation.

$t r a n s f o r m a t i o n = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

transformation = se2(rotation) creates an SE(2) transformation representing a pure rotation defined by the orthonormal rotation rotation with no translation. The rotation matrix is represented by the elements in the top left of the transformation matrix.

$r o t a t i o n = [\begin{matrix} r_{11} & r_{12} \\ r_{21} & r_{22} \end{matrix}]$

$t r a n s f o r m a t i o n = [\begin{matrix} r_{11} & r_{12} & 0 \\ r_{21} & r_{22} & 0 \\ 0 & 0 & 1 \end{matrix}]$

transformation = se2(rotation,translation) creates an SE(2) transformation representing a rotation defined by the orthonormal rotation rotation and the translation translation. The function applies the rotation matrix first, then the translation vector, to create the transformation.

$r o t a t i o n = [\begin{matrix} r_{11} & r_{12} \\ r_{21} & r_{22} \end{matrix}]$ , $t r a n s l a t i o n = [\begin{matrix} t_{1} \\ t_{2} \end{matrix}]$

$t r a n s f o r m a t i o n = [\begin{matrix} r_{11} & r_{12} & t_{1} \\ r_{21} & r_{22} & t_{2} \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 & t_{1} \\ 0 & 1 & t_{2} \\ 0 & 0 & 1 \end{matrix}] \cdot [\begin{matrix} r_{11} & r_{12} & 0 \\ r_{21} & r_{22} & 0 \\ 0 & 0 & 1 \end{matrix}]$

transformation = se2(transformation) creates an SE(2) transformation representing a translation and rotation as defined by the homogeneous transformation transformation.

Other 2-D Rotations and Transformation Representations

transformation = se2(angle,"theta") creates SE(2) transformations transformation from rotations around the z-axis, in radians. The transformation contains zero translation.

transformation = se2(angle,"theta",translation) creates SE(2) transformations from rotations around the z-axis, in radians, with translations translation.

example

transformation = se2(translation,"trvec") creates an SE(2) transformation from the translation vector translation.

transformation = se2(pose,"xytheta") creates an SE(2) transformation from the 2-D compact pose pose.

Input Arguments

expand all

`rotation` — Orthonormal rotation
2-by-2 matrix | 2-by-2-by-N matrix | `so2` object | N-element array of `so2` objects

Orthonormal rotation, specified as a 2-by-2 matrix, a 2-by-2-by-N array, a scalar so2 object, or an N-element array of so2 objects. N is the total number of rotations.

If rotation contains more than one rotation and you also specify translation at construction, the number of translations in translation must be one or equal to the number of rotations in rotation. The resulting number of transformation objects is equal to the value of the translation or rotation argument, whichever is larger.

If rotation contains one rotation and you also specify translation as an N-by-2 matrix, then the resulting transformations contain the same rotation specified by rotation and the corresponding translation vector in translation. The resulting number of transformation objects is equal to the number of translations in translation.

Example: eye(2)