SE(3): 3D Transformations

The group of all transformations in the 3D Cartesian space is SE(3) (SE: special Euclidean group). Transformations consist of a rotation and a translation. Those can be represented in different ways just like rotations can be expressed in different ways.

For most representations of orientations we can find an analogous representation of transformations:

  • A transformation matrix \boldsymbol T is similar to a rotation matrix \boldsymbol R.

  • A screw axis \mathcal S is similar to a rotation axis \hat{\boldsymbol{\omega}}.

  • A screw matrix \left[\mathcal{S}\right] is similar to a cross-product matrix of a unit rotation axis \left[\hat{\boldsymbol{\omega}}\right].

  • The logarithm of a transformation \left[\mathcal{S}\right] \theta is similar to a cross-product matrix of the angle-axis representation \left[\hat{\boldsymbol{\omega}}\right] \theta.

  • The exponential coordinates \mathcal{S} \theta for rigid body motions are similar to exponential coordinates \hat{\boldsymbol{\omega}} \theta for rotations (axis-angle representation).

  • A twist \mathcal V = \mathcal{S} \dot{\theta} is similar to angular velocity \hat{\boldsymbol{\omega}} \dot{\theta}.

  • A (unit) dual quaternion p_w + p_x i + p_y j + p_z k + \epsilon (q_w + q_x i + q_y j + q_z k) is similar to a (unit) quaternion w + x i + y j + z k.

Here is an overview of the representations and the conversions between them that are available in pytransform3d.

Transformations

Transformation Matrix

One of the most convenient ways to represent transformations are transformation matrices. A transformation matrix is a 4x4 matrix of the form

\boldsymbol T =
\left( \begin{array}{cc}
    \boldsymbol R & \boldsymbol t\\
    \boldsymbol 0 & 1\\
\end{array} \right)
=
\left(
\begin{matrix}
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{matrix}
\right)

It is a partitioned matrix with a 3x3 rotation matrix \boldsymbol R and a column vector \boldsymbol t that represents the translation. It is also sometimes called the homogeneous representation of a transformation. All transformation matrices form the special Euclidean group SE(3).

pytransform3d uses a numpy array of shape (4, 4) to represent transformation matrices and typically we use the variable name A2B for a transformation matrix, where A corrsponds to the frame from which it transforms and B to the frame to which it transforms.

It is possible to transform position vectors or direction vectors with it. Position vectors are represented as a column vector \left( x,y,z,1 \right)^T. This will activate the translation part of the transformation in a matrix multiplication. When we transform a direction vector, we want to deactivate the translation by setting the last component to zero: \left( x,y,z,0 \right)^T.

We can use a transformation matrix \boldsymbol T_{AB} to transform a point {_B}\boldsymbol{p} from frame B to frame A. For example, transforming a position vector p will give the following result:

\boldsymbol{T}_{AB}  {_B}\boldsymbol{p} =
\left( \begin{array}{c}
    \boldsymbol{R} {_B}\boldsymbol{p} + \boldsymbol t\\
    1\\
\end{array} \right)

Position and Quaternion

An alternative to transformation matrices is the representation in a 7-dimensional vector that consists of the translation and a rotation quaternion:

\left( \begin{array}{c}
    x\\y\\z\\q_w\\q_x\\q_y\\q_z
\end{array} \right)

This representation is more compact than a transformation matrix and is particularly useful if you want to represent a sequence of poses in a 2D array.

pytransform3d uses a numpy array of shape (7,) to represent position and quaternion and typically we use the variable name pq.

Exponential Coordinates

(Source code, png, hires.png, pdf)

_images/plot_screw.png

Just like any rotation can be expressed as a rotation by an angle about a 3D unit vector, any transformation (rotation and translation) can be expressed by a motion along a screw axis. The screw parameters that describe a screw axis include a point vector \boldsymbol{q} through which the screw axis passes, a (unit) direction vector \hat{\boldsymbol{s}} that indicates the direction of the axis, and the pitch h. The pitch represents the ratio of translation and rotation. A screw motion translates along the screw axis and rotates about it.

pytransform3d uses two vectors q and s_axis of shape (3,) and a scalar h to represent the parameters of a screw.

Screw axis

A screw axis is typically represented by \mathcal{S} = \left[\begin{array}{c}\boldsymbol{\omega}\\\boldsymbol{v}\end{array}\right] \in \mathbb{R}^6, where either

  1. ||\boldsymbol{\omega}|| = 1 or

  2. ||\boldsymbol{\omega}|| = 0 and ||\boldsymbol{v}|| = 1 (only translation).

pytransform3d uses a numpy array of shape (6,) to represent a screw axis and typically we use the variable name S or screw_axis.

In case 1, we can compute the screw axis from screw parameters (\boldsymbol{q}, \hat{\boldsymbol{s}}, h) as

\mathcal{S} = \left[ \begin{array}{c}\hat{\boldsymbol{s}} \\ \boldsymbol{q} \times \hat{\boldsymbol{s}} + h \hat{\boldsymbol{s}}\end{array} \right]

In case 2, h is infinite and we directly translate along \hat{\boldsymbol{s}}.

By multiplication with an additional parameter \theta we can then define a complete transformation through its exponential coordinates \mathcal{S} \theta = \left[\begin{array}{c}\boldsymbol{\omega}\theta\\\boldsymbol{v}\theta\end{array}\right] \in \mathbb{R}^6. This is a minimal representation as it only needs 6 values.

pytransform3d uses a numpy array of shape (6,) to represent a exponential coordinates of transformation and typically we use the variable name Stheta.

Logarithm of Transformation

Alternatively, we can represent a screw axis \mathcal S in a matrix

\left[\mathcal S\right]
=
\left( \begin{array}{cc}
    \left[\boldsymbol{\omega}\right] & \boldsymbol v\\
    \boldsymbol 0 & 0\\
\end{array} \right)
=
\left(
\begin{matrix}
0 & -\omega_3 & \omega_2 & v_1\\
\omega_3 & 0 & -\omega_1 & v_2\\
-\omega_2 & \omega_1 & 0 & v_3\\
0 & 0 & 0 & 0\\
\end{matrix}
\right)
\in \mathbb{R}^{4 \times 4}

that contains the cross-product matrix of its orientation part and its translation part. This is the matrix representation of a screw axis and we will also refer to it as screw matrix in the API.

pytransform3d uses a numpy array of shape (4, 4) to represent a screw matrix and typically we use the variable name screw_matrix.

By multiplication with \theta we can again generate a full description of a transformation \left[\mathcal{S}\right] \theta \in se(3), which is the matrix logarithm of a transformation matrix and se(3) is the Lie algebra of Lie group SE(3).

pytransform3d uses a numpy array of shape (4, 4) to represent the logarithm of a transformation and typically we use the variable name transform_log.

Twist

We call spatial velocity (translation and rotation) twist. Similarly to the matrix logarithm, a twist \mathcal{V} = \mathcal{S} \dot{\theta} is described by a screw axis S and a scalar \dot{\theta} and \left[\mathcal{V}\right] = \left[\mathcal{S}\right] \theta \in se(3) is the matrix representation of a twist.

Dual Quaternions

Similarly to unit quaternions for rotations, unit dual quaternions are an alternative to represent transformations. They support similar operations as transformation matrices.

A dual quaternion consists of a real quaternion and a dual quaternion:

\boldsymbol{p} + \epsilon \boldsymbol{q} = p_w + p_x i + p_y j + p_z k + \epsilon (q_w + q_x i + q_y j + q_z k),

where \epsilon^2 = 0. We use unit dual quaternions to represent transformations. In this case, the real quaternion is a unit quaternion and the dual quaternion is orthogonal to the real quaternion. The real quaternion is used to represent the rotation and the dual quaternion contains information about the rotation and translation.

Dual quaternions support similar operations as transformation matrices, they can be renormalized efficiently, and interpolation between two dual quaternions is possible.

Warning

The unit dual quaternions \boldsymbol{p} + \epsilon \boldsymbol{q} and -\boldsymbol{p} - \epsilon \boldsymbol{q} represent exactly the same transformation.

The reason for this ambiguity is that the real quaternion \boldsymbol{p} represents the orientation component, the dual quaternion encodes the translation component as \boldsymbol{q} = 0.5 \boldsymbol{t} \boldsymbol{p}, where \boldsymbol{t} is a quaternion with the translation in the vector component and the scalar 0, and rotation quaternions have the same ambiguity.

References

Lynch, Park: Modern Robotics (Section 3.3); available at http://hades.mech.northwestern.edu/index.php/Modern_Robotics

Wikipedia: Dual Quaternion; available at https://en.wikipedia.org/wiki/Dual_quaternion

Yan-Bin Jia: Dual Quaternions; available at http://web.cs.iastate.edu/~cs577/handouts/dual-quaternion.pdf

Ben Kenwright: A Beginners Guide to Dual-Quaternions; available at http://wscg.zcu.cz/WSCG2012/!_WSCG2012-Communications-1.pdf