A Mathematical Review A Mathematical Review A.2 Rigid transforms

A.1 Rotation transforms

Rotation matrices are used to describe the orientation of 3D coordinate frames in space, and to transform vectors between these coordinate frames.

Figure A.1: Two coordinate frames A and B rotated with respect to each other.

Consider two 3D coordinate frames A and B that are rotated with respect to each other (Figure A.1). The orientation of B with respect to A can be described by a $3\times 3$ rotation matrix ${\bf R}_{BA}$ , whose columns are the unit vectors giving the directions of the rotated axes ${\bf x}^{\prime}$ , ${\bf y}^{\prime}$ , and ${\bf z}^{\prime}$ of B with respect to A.

${\bf R}_{BA}$ is an orthogonal matrix, meaning that its columns are both perpendicular and mutually orthogonal, so that

${\bf R}_{BA}^{T}\,{\bf R}_{BA}={\bf I}$

(A.1)

where ${\bf I}$ is the $3\times 3$ identity matrix. The inverse of ${\bf R}_{BA}$ is hence equal to its transpose:

${\bf R}_{BA}^{-1}={\bf R}_{BA}^{T}.$

(A.2)

Because ${\bf R}_{BA}$ is orthogonal, $|\det{\bf R}_{BA}|=1$ , and because it is a rotation, $\det{\bf R}_{BA}=1$ (the other case, where $\det{\bf R}_{BA}=-1$ , is not a rotation but a reflection). The 6 orthogonality constraints associated with a rotation matrix mean that in spite of having 9 numbers, the matrix only has 3 degrees of freedom.

Now, assume we have a 3D vector ${\bf v}$ , and consider its coordinates with respect to both frames A and B. Where necessary, we use a preceding superscript to indicate the coordinate frame with respect to which a quantity is described, so that ${}^{A}{\bf v}$ and ${}^{B}{\bf v}$ and denote ${\bf v}$ with respect to frames A and B, respectively. Given the definition of ${\bf R}_{AB}$ given above, it is fairly straightforward to show that

${}^{A}{\bf v}={\bf R}_{BA}\,{}^{B}{\bf v}$

(A.3)

and, given (A.2), that

${}^{B}{\bf v}={\bf R}_{BA}^{T}\,{}^{A}{\bf v}.$

(A.4)

Hence in addition to describing the orientation of B with respect to A, ${\bf R}_{BA}$ is also a transformation matrix that maps vectors in B to vectors in A.

It is straightforward to show that

${\bf R}_{BA}^{-1}={\bf R}_{BA}^{T}={\bf R}_{AB}.$

(A.5)

Figure A.2: Schematic illustration of three coordinate frames A, B, and C and the rotational transforms relating them.

A simple rotation by an angle $\theta$ about one of the basic coordinate axes is known as a basic rotation. The three basic rotations about x, y, and z are:

${\bf R}_{x}(\theta)=\left(\begin{matrix}1&0&0\\ 0&\cos(\theta)&-\sin(\theta)\\ 0&\sin(\theta)&\cos(\theta)\end{matrix}\right),$

${\bf R}_{y}(\theta)=\left(\begin{matrix}\cos(\theta)&0&\sin(\theta)\\ 0&1&0\\ -\sin(\theta)&0&\cos(\theta)\end{matrix}\right),$

${\bf R}_{z}(\theta)=\left(\begin{matrix}\cos(\theta)&-\sin(\theta)&0\\ \sin(\theta)&\cos(\theta)&0\\ 0&0&1\end{matrix}\right).$

Next, we consider transform composition. Suppose we have three coordinate frames, A, B, and C, whose orientation are related to each other by ${\bf R}_{BA}$ , ${\bf R}_{CB}$ , and ${\bf R}_{CA}$ (Figure A.6). If we know ${\bf R}_{BA}$ and ${\bf R}_{CA}$ , then we can determine ${\bf R}_{CB}$ from

${\bf R}_{CB}={\bf R}_{BA}^{-1}\,{\bf R}_{CA}.$

(A.6)

This can be understood in terms of vector transforms. ${\bf R}_{CB}$ transforms a vector from C to B, which is equivalent to first transforming from C to A,

${}^{A}{\bf v}={\bf R}_{CA}\,{}^{C}{\bf v},$

(A.7)

and then transforming from A to B:

${}^{B}{\bf v}={\bf R}_{BA}^{-1}\,{}^{A}{\bf v}={\bf R}_{BA}^{-1}\;{\bf R}_{CA}% {}^{C}{\bf v}={\bf R}_{CB}\,{}^{C}{\bf v}.$

(A.8)

Note also from (A.5) that ${\bf R}_{CB}$ can be expressed as

${\bf R}_{CB}={\bf R}_{AB}\,{\bf R}_{CA}.$

(A.9)

In addition to specifying rotation matrix components explicitly, there are numerous other ways to describe a rotation. Three of the most common are:

Roll-pitch-yaw angles: There are 6 variations of roll-pitch-yaw angles. The one used in ArtiSynth corresponds to older robotics texts (e.g., Paul, Spong) and consists of a roll rotation about the z axis, followed by a pitch rotation about the new y axis, followed by a yaw rotation about the new x axis. The net rotation can be expressed by the following product of basic rotations: ${\bf R}_{z}(r)\,{\bf R}_{y}(p)\,{\bf R}_{x}(y)$ .
Axis-angle: An axis angle rotation parameterizes a rotation as a rotation by an angle $\theta$ about a specific axis ${\bf u}$ . Any rotation can be represented in such a way as a consequence of Euler’s rotation theorem.
Euler angles: There are 6 variations of Euler angles. The one used in ArtiSynth consists of a rotation $\phi$ about the z axis, followed by a rotation $\theta$ about the new y axis, followed by a rotation $\psi$ about the new z axis. The net rotation can be expressed by the following product of basic rotations: ${\bf R}_{z}(\phi)\,{\bf R}_{y}(\theta)\,{\bf R}_{z}(\psi)$ .

A Mathematical Review Bibliography A.2 Rigid transforms

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