Chapter 3: 3D Rigid Body Motion

3.1 Rotation Matrix

3.1.1 Points, Vectors and Coordinate Systems

Suppose in linear space, we find a set of base $(\pmb{e}_1,\pmb{e}_2,\pmb{e}_3)$, then, an arbitrary vector $\pmb{a}$ has a coordinate under this base: $$\pmb{a} = \left[ { {\pmb{e}_1}, {\pmb{e}_2}, {\pmb{e}_3} } \right] \left[ \begin{array}{l} {a_1}\\ {a_2}\\ {a_3} \end{array} \right] = {a_1}{\pmb{e}_1} + {a_2}{\pmb{e}_2} + {a_3}{\pmb{e}_3}.$$ Inner Product $$\pmb{a} \cdot \pmb{b} = { \pmb{a}^T }\pmb{b} = \sum\limits_{i = 1}^3 { {a_i} {b_i} } = \left| \pmb{a} \right|\left| \pmb{b} \right|\cos \left\langle {\pmb{a},\pmb{b}} \right\rangle$$ The inner product can also describe the projection relationship between vectors.

Outer Product $$\pmb{a} \times \pmb{b} = \left\| {\begin{array}{*{20}{c}} \pmb{e}_1 & \pmb{e}_2 & \pmb{e}_3 \\ { {a_1}}&{ {a_2}}&{ {a_3}}\\ { {b_1}}&{ {b_2}}&{ {b_3}} \end{array}} \right\| = \left[ \begin{array}{l} {a_2}{b_3} - {a_3}{b_2}\\ {a_3}{b_1} - {a_1}{b_3}\\ {a_1}{b_2} - {a_2}{b_1} \end{array} \right] = \left[ {\begin{array}{*{20}{c}} 0&{ - {a_3}}&{ {a_2}}\\ { {a_3}}&0&{-{a_1}}\\ {-{a_2}}&{ {a_1}}&0 \end{array}} \right] \pmb{b} \buildrel \Delta \over = { \pmb{a}^ \wedge } \pmb{b}.$$ The result of the outer product is a vector whose direction is perpendicular to the two vectors, and the length is $\left | \pmb{a} \right | \left | \pmb{b} \right | \left \langle { \pmb {a}, \pmb {b}} \right \rangle$, which is also the area of the quadrilateral of the two vectors.

From the outer product operations, we introduce the $^ \wedge$ operator here, which means writing $a$ as a skew-symmetric matrix. $$\pmb{a}^\wedge = \left[ {\begin{array}{*{20}{c}} 0&{-{a_3}}&{ {a_2}}\\ { {a_3}}&0&{ - {a_1}}\\ { - {a_2}}&{ {a_1}}&0 \end{array}} \right].$$

3.1.2 Euclidean Transforms Between Coordinate Systems

The Euclidean transform consists of rotation and translation.

Rotation Matrix

$$\left[ \begin{array}{l} {a_1}\\ {a_2}\\ {a_3} \end{array}\right]=\underbrace{\left[{\begin{array}{*{20}{c}} {\pmb{e}_1^T\pmb{e}_1'} & {\pmb{e}_1^T\pmb{e}_2'} & {\pmb{e}_1^T\pmb{e}_3'}\\ {\pmb{e}_2^T\pmb{e}_1'} & {\pmb{e}_2^T\pmb{e}_2'} & {\pmb{e}_2^T\pmb{e}_3'}\\ {\pmb{e}_3^T\pmb{e}_1'} & {\pmb{e}_3^T\pmb{e}_2'} & {\pmb{e}_3^T\pmb{e}_3'} \end{array}} \right]}_{\text{rotation matrix}}\left[ \begin{array}{l} a_1'\\ a_2'\\ a_3' \end{array} \right] \buildrel \Delta \over = \pmb{R} \pmb{a}'$$

Special property: rotation matrix is an orthogonal matrix with a determinant of 1. Conversely, an orthogonal matrix with a determinant of 1 is also a rotation matrix. So we can define a set of n dimensional rotation matrices as follows: $$\mathrm{SO}(n) = \{ \pmb{R} \in \mathbb{R}^{n \times n} | \pmb{R R}^T = \pmb{I}, \mathrm{det} (\pmb{R})=1 \}$$ $\mathrm{SO}(n)$ refers to the special orthogonal group

Since the rotation matrix is orthogonal, its inverse (i.e., transpose) describes an opposite rotation. $$\pmb{a} '= \pmb{R}^{-1} \pmb{a} = \pmb{R}^{T} \pmb{a}.$$

Translation Matrix

$$\pmb{a} '= \pmb{R} \pmb{a} + \pmb{t},$$

3.1.3 Transform Matrix and Homogeneous Coordinates

$$\left[\begin{array}{l} \pmb {a} ' \\ 1 \end{array} \right] = \left[ {\begin{array}{*{20}{c}} \pmb{R}&\pmb{t}\\ { {\pmb{0}^T}}&1 \end{array}} \right] \left[ \begin{array}{l} \pmb {a} \\ 1 \end{array} \right] \buildrel \Delta \over = \pmb{T} \left[ \begin{array}{l} \pmb {a} \\ 1 \end{array} \right]$$

Suppose we have two transformations: $\pmb {R}_ 1, \pmb {t}_ 1$ and $\pmb {R}_ 2, \pmb {t}_ 2$ $$\tilde{\pmb{b}} = \pmb{T}_1 \tilde{\pmb{a}}, \ \tilde{\pmb{c}} = \pmb{T}_2 \tilde{\pmb{b}} \quad \Rightarrow \tilde{\pmb{c}} = \pmb{T}_2 \pmb{T_1} \tilde{\pmb{a}}$$ Note that if homogeneous coordinate transformation is not performed, the matrix multiplication here does not make sense.

The transformation matrix $\pmb{T}$ has a special structure: the upper left corner is the rotation matrix, the right side is the translation vector, the lower-left corner is $\pmb{0}$ vector, and the lower right corner is 1. This set of transform matrix is also known as the special Euclidean group: $$\mathrm{SE}(3) = \left\{ \pmb{T} = \left[ {\begin{array}{*{20}{c}} \pmb{R} & \pmb{t} \\ { {\pmb{0}^T}} & 1 \end{array}} \right] \in \mathbb{R}^{4 \times 4} | \pmb{R} \in \mathrm{SO}(3), \pmb{t} \in \mathbb{R}^3\right\}$$ Like $SO(3)$, the inverse of the transform matrix represents an inverse transformation: $${ \pmb{T}^{ - 1}} = \left[ {\begin{array}{*{20}{c}} { {\pmb{R}^T}}&{ - {\pmb{R}^T}\pmb{t}}\\ { {\pmb{0}^T}}&1 \end{array}} \right]$$