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.

$$transformation=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

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.

$$rotation=\left[\begin{array}{cc}{r}_{11}& r_{12}\\ r_{21}& r_{22}\end{array}\right]$$

$$transformation=\left[\begin{array}{ccc}{r}_{11}& r_{12}& 0\\ r_{21}& r_{22}& 0\\ 0& 0& 1\end{array}\right]$$

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.

$$rotation=\left[\begin{array}{cc}{r}_{11}& r_{12}\\ r_{21}& r_{22}\end{array}\right]$$, $$translation=\left[\begin{array}{c}{t}_1\\ t_2\end{array}\right]$$

$$transformation=\left[\begin{array}{ccc}{r}_{11}& r_{12}& t_1\\ r_{21}& r_{22}& t_2\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& t_1\\ 0& 1& t_2\\ 0& 0& 1\end{array}\right]·\left[\begin{array}{ccc}{r}_{11}& r_{12}& 0\\ r_{21}& r_{22}& 0\\ 0& 0& 1\end{array}\right]$$

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.

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

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)

Data Types: single | double

translation — Translation

two-element row vector | N-by-2 matrix

Translation, specified as an N-by-2 matrix. N is the total number of translations and each translation is of the form [x y].

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

If you specify more than one translation but only one rotation, the resulting transformations contain the same rotation specified in rotation and the corresponding translation in translation. The resulting number of created se2 objects is equal to the value of the translation.

Example: [1 4]

Data Types: single | double

transformation — Homogeneous transformation

3-by-3 matrix | 3-by-3-by-N array | se2 object | N-element array of se2 objects

Homogeneous transformation, specified as a 3-by-3 matrix, a 3-by-3-by-N array, a scalar se3 object, or an N-element array of se2 objects. N is the total number of transformations specified.

If transformation is an array, the resulting number of created se2 objects is equal to N.

Example: eye(3)

Data Types: single | double

angle — z-axis rotation angle

N-by-M matrix

z-axis rotation angle, specified as an N-by-M matrix. Each element of the matrix is an angle, in radians, about the z-axis. The se2 object creates an se2 object for each angle.

If angle is an N-by-M matrix, the resulting number of created se2 objects is equal to N.

The rotation angle is counterclockwise positive when you look along the axis toward the origin.

Data Types: single | double

pose — 2-D compact pose

N-by-3 matrix

3-D compact pose, specified as an N-by-3 matrix, where N is the total number of compact poses. Each row is a pose, comprised of an xy position and a rotation about the z-axis, in the form [x y theta]. x, y are the xy-positions and theta is the rotation about the z-axis.

If pose is an N-by-3 matrix, the resulting number of created se2 objects is equal to N.

Data Types: single | double

Object Functions

Mathematical Operations

mtimes, *	Transformation or rotation multiplication
mrdivide, /	Transformation or rotation right division
rdivide, ./	Element-wise transformation or rotation right division
times, .*	Element-wise transformation or rotation multiplication

Utilities

interp	Interpolate between transformations
dist	Calculate distance between transformations
normalize	Normalize transformation or rotation matrix
transform	Apply rigid body transformation to points

Numerical Conversions

rotm	Extract rotation matrix
trvec	Extract translation vector
tform	Extract homogeneous transformation
theta	Convert transformation or rotation to 2-D rotation angle
xytheta	Convert transformation or rotation to compact 2-D pose representation

Object Conversions

so2	SO(2) rotation

Examples

Create SE(2) Transformation Using Angle and Translation

Define an angle rotation of pi/4 and a xyz translation of [6 4].

angle = pi/6;
trvec = [2 1];

Create an SE(2) transformation using the angle and translation.

TF = se2(angle,"theta",trvec)

TF = se2
    0.8660   -0.5000    2.0000
    0.5000    0.8660    1.0000
         0         0    1.0000

Algorithms

2-D Homogeneous Transformation Matrix

2-D homogeneous transformation matrices consist of both an SO(2) rotation and an xy-translation.

To read more about SO(2) rotations, see the 2-D Orthonormal Rotation Matrix section of the so2 object.

The translation is along the x-, and y-axes as a two-element column vector:

$$t=\left[\begin{array}{c}x\\ y\end{array}\right]$$

The SO(2) rotation matrix R is applied to the translation vector t to create the homogeneous translation matrix T. The rotation matrix is present in the upper-left of the transformation matrix as 2-by-2 submatrix, and the translation vector is present as a two-element vector in the last column.

$$T=\left[\begin{array}{cc}R& t\\ 0_{1\times 2}& 1\end{array}\right]=\left[\begin{array}{cc}{I}_2& t\\ 0_{1\times 2}& 1\end{array}\right]·\left[\begin{array}{cc}R& 0\\ 0_{1\times 2}& 1\end{array}\right]$$

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2022b

R2023a: New methods and syntaxes

se2 supports new methods and syntaxes for converting to and from other transformations and rotations.

The new methods are:

• se3

• so2

• theta

• xytheta