se3

SE(3) homogeneous transformation

Since R2022b

Description

The se3 object represents an SE(3) transformation as a 3-D homogeneous transformation matrix consisting of a translation and rotation for a right-handed Cartesian coordinate system.

For more information, see the 3-D Homogeneous Transformation Matrix section.

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

Creation

Syntax

transformation = se3

transformation = se3(rotation)

transformation = se3(rotation,translation)

transformation = se3(transformation)

transformation = se3(euler,"eul")

transformation = se3(euler,"eul",sequence)

transformation = se3(quat,"quat")

transformation = se3(quaternion)

transformation = se3(axang,"axang")

transformation = se3(angle,axis)

transformation = se3(___,translation)

transformation = se3(translation,"trvec")

transformation = se3(pose,"xyzquat")

Description

Rotation Matrices, Translation Vectors, and Transformation Matrices

transformation = se3 creates an SE(3) transformation representing an identity rotation with no translation.

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

transformation = se3(rotation) creates an SE(3) 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.

$$rotation=\left[\begin{array}{ccc}{r}_{11}& r_{12}& r_{13}\\ r_{11}& r_{22}& r_{23}\\ r_{31}& r_{32}& r_{33}\end{array}\right]$$

$$transformation=\left[\begin{array}{cccc}{r}_{11}& r_{12}& r_{13}& 0\\ r_{21}& r_{22}& r_{23}& 0\\ r_{31}& r_{32}& r_{33}& 0\\ 0& 0& 0& 1\end{array}\right]$$

transformation = se3(rotation,translation) creates an SE(3) transformation representing a rotation defined by the orthonormal rotation rotation and the translation translation. The function applies the rotation matrix first, then translation vector to create the transformation.

$$rotation=\left[\begin{array}{ccc}{r}_{11}& r_{12}& r_{13}\\ r_{11}& r_{22}& r_{23}\\ r_{31}& r_{32}& r_{33}\end{array}\right]$$, $$translation=\left[\begin{array}{c}{t}_1\\ t_2\\ t_3\end{array}\right]$$

$$transformation=\left[\begin{array}{cccc}{r}_{11}& r_{12}& r_{13}& t_1\\ r_{21}& r_{22}& r_{23}& t_2\\ r_{31}& r_{32}& r_{33}& t_3\\ 0& 0& 0& 1\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& t_1\\ 0& 1& 0& t_2\\ 0& 0& 1& t_3\\ 0& 0& 0& 1\end{array}\right]·\left[\begin{array}{cccc}{r}_{11}& r_{12}& r_{13}& 0\\ r_{21}& r_{22}& r_{23}& 0\\ r_{31}& r_{32}& r_{33}& 0\\ 0& 0& 0& 1\end{array}\right]$$

transformation = se3(transformation) creates an SE(3) transformation representing a translation and rotation as defined by the homogeneous transformation transformation.

Other 3-D Rotation Representations

transformation = se3(euler,"eul") creates an SE(3) transformation from the rotations defined by the Euler angles euler.

transformation = se3(euler,"eul",sequence) specifies the sequence of the Euler angle rotations sequence. For example, the sequence "ZYX" rotates the z-axis, then the y-axis and x-axis.

transformation = se3(quat,"quat") creates an SE(3) transformation from the rotations defined by the numeric quaternions quat.

transformation = se3(quaternion) creates an SE(3) transformation from the rotations defined by the quaternion quaternion.

transformation = se3(axang,"axang") creates an SE(3) transformation from the rotations defined by the axis-angle rotation axang.

transformation = se3(angle,axis) creates an SE(3) transformation from the rotations angle about the rotation axis axis.

transformation = se3(___,translation) creates an SE(3) transformation from the translation vector translation along with any other type of rotation input arguments.

Other Translations and Transformation Representations

transformation = se3(translation,"trvec") creates an SE(3) transformation from the translation vector translation.

transformation = se3(pose,"xyzquat") creates an SE(3) transformation from the 3-D compact pose pose.

Note

If any inputs contain more than one rotation, translation, or transformation, then the output transformation is an N-element array of se3 objects corresponding to each of the N input rotations, translations, or transformations.

Input Arguments

rotation — Orthonormal rotation

3-by-3 matrix | 3-by-3-by-N matrix | so3 object | N-element array of so3 objects

Orthonormal rotation, specified as a 3-by-3 matrix, a 3-by-3-by-N array, a scalar so3 object, or an N-element array of so3 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.

Example: eye(3)

Data Types: single | double

translation — Translation

three-element row vector | N-by-3 matrix

Translation, specified as a three-element row vector or an N-by-3 array. N is the total number of translations and each translation is of the form [x y z].

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.

Example: [1 4 3]

Data Types: single | double

transformation — Homogeneous transformation

4-by-4 matrix | 4-by-4-by-N array | se3 object | N-element array of se3 objects

Homogeneous transformation, specified as a 4-by-4 matrix, a 4-by-4-by-N array, a scalar se3 object, or an N-element array of se3 objects. N is the total number of transformations specified.

If transformation is an array, the resulting number of created se3 objects is equal to N.

Example: eye(4)

Data Types: single | double

quaternion — Quaternion

quaternion object | N-element array of quaternion objects

Quaternion, specified as a scalar quaternion object or as an N-element array of quaternion objects. N is the total number of specified quaternions.

If quaternion is an N-element array, the resulting number of created se3 objects is equal to N.

Example: quaternion(1,0.2,0.4,0.2)

euler — Euler angles

N-by-3 matrix

Euler angles, specified as an N-by-3 matrix, in radians. Each row represents one set of Euler angles with the axis-rotation sequence defined by the sequence argument. The default axis-rotation sequence is ZYX.

If euler is an N-by-3 matrix, the resulting number of created se3 objects is equal to N.

Example: [pi/2 pi pi/4]

Data Types: single | double

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 se3 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: se3([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 se3(pi/2,"rotz") * se3(pi/3,"roty") * se3(pi/4,"rotz")

Data Types: string | char

quat — Numeric quaternion

N-by-4 matrix

Numeric quaternion, specified as an N-by-4 matrix. N is the number of specified quaternions. Each row represents one quaternion of the form [qw qx qy qz], where qw is a scalar number.

If quat is an N-by-4 matrix, the resulting number of created se3 objects is equal to N.

Note

The se3 object normalizes the input quaternions before converting the quaternions to a rotation matrix.

Example: [0.7071 0.7071 0 0]

Data Types: single | double

axang — Axis-angle rotation

N-by-4 matrix

Axis-angle rotation, specified as an N-by-4 matrix in the form [x y z theta]. N is the total number of axis-angle rotations. x, y, and z are vector components from the x-, y-, and z-axis, respectively. The vector defines the axis to rotate by the angle theta, in radians.

If axang is an N-by-4 matrix, the resulting number of created se3 objects is equal to N.

Example: [.2 .15 .25 pi/4] rotates the axis, defined as 0.2 in the x-axis, 0.15 along the y-axis, and 0.25 along the z-axis, by pi/4 radians.

Data Types: single | double

angle — Single-axis-angle rotation

N-by-M matrix

Single-axis-angle rotation, specified as an N-by-M matrix. Each element of the matrix is an angle, in radians, about the axis specified using the axis argument, and the se3 object creates an se3 object for each angle.

If angle is an N-by-M matrix, the resulting number of created se3 objects is equal to N.

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

Data Types: single | double

axis — Axis to rotate

"rotx" | "roty" | "rotz"

Axis to rotate, specified as one of these options:

• "rotx" — Rotate about the x-axis:

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

• "roty" — Rotate about the y-axis:

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

• "rotz" — Rotate about the z-axis:

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

Use the angle argument to specify how much to rotate about the specified axis.

Example: Rx = se3(phi,"rotx");

Example: Ry = se3(psi,"roty");

Example: Rz = se3(theta,"rotz");

Data Types: string | char

pose — 3-D compact pose

N-by-7 matrix

3-D compact pose, specified as an N-by-7 matrix, where N is the total number of compact poses. Each row is a pose, comprised of a xyz position and quaternion, in the form [x y z qw qx qy qz]. x, y, and z are the positions in the x-, y-, and z-axes, respectively. qw, qx, qy, and qz together are the quaternion rotation.

If pose is an N-by-7 matrix, the resulting number of created se3 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

axang	Convert transformation or rotation into axis-angle rotations
eul	Convert transformation or rotation into Euler angles
rotm	Extract rotation matrix
quat	Convert transformation or rotation to numeric quaternion
trvec	Extract translation vector
tform	Extract homogeneous transformation
xyzquat	Convert transformation or rotation to compact 3-D pose representation

Object Conversions

so3	SO(3) rotation
quaternion	Create quaternion array

Examples

Create SE(3) Transformation Using Euler Angles and Translation

Define an Euler-angle rotation of [pi/2 0 pi/7] with a "XYZ" rotation sequence, and a xyz translation of [6 4 1].

angles = [pi/2 0 pi/7];
trvec = [6 4 1];

Create an SE(3) transformation using the Euler angles and the translation.

TF = se3(angles,"eul","XYZ",trvec)

TF = se3
    0.9010   -0.4339         0    6.0000
    0.0000    0.0000   -1.0000    4.0000
    0.4339    0.9010    0.0000    1.0000
         0         0         0    1.0000

Algorithms

3-D Homogeneous Transformation Matrix

3-D homogeneous transformation matrices consist of both an SO(3) rotation and an xyz-translation.

To read more about SO(3) rotations, see the 3-D Orthonormal Rotation Matrix section of the so3 object.

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

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

The SO(3) 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 3-by-3 submatrix, and the translation vector is present as a three-element vector in the last column.

$$T=\left[\begin{array}{cc}R& t\\ 0_{1x3}& 1\end{array}\right]=\left[\begin{array}{cc}{I}_3& t\\ 0_{1x3}& 1\end{array}\right]·\left[\begin{array}{cc}R& 0\\ 0_{1x3}& 1\end{array}\right]$$

Extended Capabilities

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

Version History

Introduced in R2022b

R2023a: New methods and syntaxes

se3 supports new methods and syntaxes for converting to and from other transformations and rotations.

The new methods are:

• axang

• eul

• so3

• quat

• quaternion

• xyzquat