Matlab API
Warning
MATLAB® scripts are a work in progress and will generally lag behind the available Python scripts.
Main module
The functions in the main module can be called with MR.function-name.
- R_p_to_SE3(R, p)
Constructs a homogeneous transformation matrix from a rotation matrix and displacement vector.
- Parameters
R – 3 by 3 rotation matrix \(\RotationMatrix\in\SOthree\)
p – 3 by 1 displacement vector
- Returns
4 by 4 homogeneous transformation matrix \(\HomogeneousTransformationMatrix \in \SEthree\)
- SE3_to_vec(T)
‘Differentiation’ of a transformation matrix by using the matrix log of its rotation matrix to determine the screw velocity vector (twist) that reaches this transformation in unit time.
- Parameters
T – Homogeneous transformation matrix
- Returns
Velocity twist vector \(\Twist\) with length \(\theta\)
See also
- SO3_to_vec(R)
‘Differentiation’ of a rotation matrix by using the matrix log to determine the angular velocity that reaches that orientation in unit time.
- Parameters
R – 3 by 3 rotation matrix \(\RotationMatrix \in \SOthree\).
- Returns
Angular velocity vector \(\omega\) with length \(\theta\).
See also
so3_to_vec()vec_to_SO3()
- big_adjoint(T)
Constructs the ‘big adjoint’ form of the transformation matrix T.
\[\begin{split}\HomogeneousTransformationMatrix = \begin{bmatrix} \RotationMatrix & p \\ 0 & 1 \end{bmatrix} \quad\Rightarrow\quad \Adjoint{\HomogeneousTransformationMatrix} = \begin{bmatrix} \RotationMatrix & 0 \\ \TildeSkew{p}\RotationMatrix & \RotationMatrix \end{bmatrix}\end{split}\]- Parameters
T – 4 by 4 homogeneous transformation matrix \(\HomogeneousTransformationMatrix \in \SEthree\)
- Returns
6 by 6 ‘big adjoint’ form of the transformation matrix T
Example
>> T = [eye(3), [1; 2; 3]; 0, 0, 0, 1]; >> MR.big_adjoint(T) ans = 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 -3 2 1 0 0 3 0 -1 0 1 0 -2 1 0 0 0 1
- inv_SE3(T)
Inverts the transformation matrix T.
\[\begin{split}\HomogeneousTransformationMatrix = \begin{bmatrix} \RotationMatrix & p \\ 0 & 1 \end{bmatrix}\in\SEthree \quad\Rightarrow\quad \HomogeneousTransformationMatrix^{-1} = \begin{bmatrix} \RotationMatrix\Transposed & -\RotationMatrix\Transposed p \\ 0 & 1 \end{bmatrix}\in\SEthree\end{split}\]- Parameters
T – 4 by 4 homogeneous transformation matrix \(\HomogeneousTransformationMatrix \in \SEthree\)
- Returns
4 by 4 inverse of the homogeneous transformation matrix \(\HomogeneousTransformationMatrix^{-1} \in \SEthree\)
- little_adjoint(V)
Constructs the ‘little adjoint’ form of the velocity twist V.
\[\begin{split}\Twist = \begin{bmatrix} \omega \\ v \end{bmatrix} \quad\Rightarrow\quad \LittleAdjoint{\Twist} = \begin{bmatrix} \TildeSkew{\omega} & 0 \\ \TildeSkew{v} & \TildeSkew{\omega} \end{bmatrix}\end{split}\]- Parameters
V – 6 by 1 velocity twist
- Returns
6 by 6 ‘little adjoint’ form of the twist V
Example
>> V = [1; 2; 3; 4; 5; 6]; >> MR.little_adjoint(V) ans = 0 -3 2 0 0 0 3 0 -1 0 0 0 -2 1 0 0 0 0 0 -6 5 0 -3 2 6 0 -4 3 0 -1 -5 4 0 -2 1 0
- little_so3_to_vec(v_tilde)
Reconstruct a 3 vector from its skew-symmetric matrix form. No check is performed to see if v_tilde is really skew-symmetric.
- Parameters
v_tilde – 3 by 3 skew-symmetric matrix form of v, \(\TildeSkew{v}\).
- Returns
3 vector.
See also
vec_to_so3()
- vec_to_SE3(S, theta)
‘Integration’ of a velocity twist to determine the configuration (expressed as transformation matrix) that this velocity twist reaches in unit time.
- Parameters
S – 6 vector velocity twist
theta – Distance travelled along the velocity twist screw axis
- Returns
4 by 4 homogeneous transformation matrix \(\HomogeneousTransformationMatrix \in \SEthree\)
See also
- vec_to_SO3(omega, theta)
‘Integration’ of an angular velocity to determine the orientation (expressed as rotation matrix) that this angular velocity reaches in unit time.
Uses Rodrigues’ formula to solve the matrix exponential of the vector’s skew-symmetric matrix form.
\[\RotationMatrix = \exp(\TildeSkew{\UnitLength{\omega}}\theta) = I + \TildeSkew{\UnitLength{\omega}}\sin\theta + \TildeSkew{\UnitLength{\omega}}^2(1-\cos\theta) \in \SOthree\]- Parameters
omega – 3 vector angular velocity. Can be unit length, in that case theta should also be provided.
theta – If omitted, the Euclidean norm of omega is assumed to be theta.
- Returns
3 by 3 rotation matrix \(\RotationMatrix \in \SOthree\).
See also
vec_to_so3()SO3_to_vec()
- vec_to_little_so3(v)
Build a skew-symmetric matrix from a 3 vector.
- Parameters
v – 3 vector.
- Returns
3 by 3 skew-symmetric matrix form of v, \(\TildeSkew{v}\).
See also
so3_to_vec()
gen submodule
The +MR.+gen submodule contains functions to easiliy construct homogeneous transformation matrices with base transformations. The functions in this submodule can be called with MR.gen.function-name.
- TRx(theta)
Creates a transformation matrix with a single rotation around the \(x\)-axis.
- Parameters
theta – Angle to rotate with around \(x\), in radians.
- Returns
Homogeneous transformation matrix with \(x\)-rotation and zero translation.
Example
>> MR.gen.TRx(pi/6) ans = 1.0000 0 0 0 0 0.8660 -0.5000 0 0 0.5000 0.8660 0 0 0 0 1.0000
- TRy(theta)
Creates a transformation matrix with a single rotation around the \(y\)-axis.
- Parameters
theta – Angle to rotate with around \(y\), in radians.
- Returns
Homogeneous transformation matrix with \(y\)-rotation and zero translation.
Example
>> MR.gen.TRy(pi/6) ans = 0.8660 0 0.5000 0 0 1.0000 0 0 -0.5000 0 0.8660 0 0 0 0 1.0000
- TRz(theta)
Creates a transformation matrix with a single rotation around the \(z\)-axis.
- Parameters
theta – Angle to rotate with around \(z\), in radians.
- Returns
Homogeneous transformation matrix with \(z\)-rotation and zero translation.
Example
>> MR.gen.TRx(pi/6) ans = 0.8660 -0.5000 0 0 0.5000 0.8660 0 0 0 0 1.0000 0 0 0 0 1.0000
- Tt(varargin)
Creates a transformation matrix with only a translational part.
- Parameters
t – Translation three vector. Should be a column vector. Optional, if omitted use one of the following to create
t.x – \(x\)-coordinate of the translation. Optional and only used if
tis not provided.y – \(y\)-coordinate of the translation. Optional and only used if
tis not provided.z – \(z\)-coordinate of the translation. Optional and only used if
tis not provided.
- Returns
Homogeneous transformation matrix without rotation, only translation.
Examples
>> MR.gen.Tt([1;2;3]) ans = 1 0 0 1 0 1 0 2 0 0 1 3 0 0 0 1
>> MR.gen.Tt('y', 2, 'z', 6) ans = 1 0 0 0 0 1 0 2 0 0 1 6 0 0 0 1