Spatial Description & Transformation

Descriptions

Descriptions of a position, an orientation

In a coordinate system \(\{A\}\) (three orthogonal unit vectors),

(i) A point \(^AP\) is expressed by a vector

\[ ^AP=\left[\begin{array}{c} p_x\\p_y\\p_z \end{array}\right] \]

(ii) Attach another coordinate system \(\{B\}\), with its unit vectors \(\hat{X}_B\), \(\hat{Y}_B\), and \(\hat{Z}_B\). If they represented by system \(\{A\}\), as \(^A\hat{X}_B\), \(^A\hat{Y}_B\), and \(^A\hat{Z}_B\). Stack these three vectors together, and we have a rotation matrix from \(A\) to \(B\)

\[ ^A_BR=\left[\begin{array}{ccc}^A\hat{X}_B& ^A\hat{Y}_B& ^A\hat{Z}_B\end{array}\right]. \]

If we consider from linear algebra, denoting \(\hat{e}_1,\hat{e}_2,\hat{e}_3\) as basis of \(\{B\}\), \(e_1,e_2,e_3\) as basis of \(\{A\}\), so

\[ \hat{e}_{i}=\sum_{j=1}^3 r_{ij} e_j, \]

from inner product, we have

\[ \langle\hat{e}_i,e_j\rangle=r_{ij}. \]

So a rotation matrix from \(\{A\}\) to \(B\), we have

\[ ^A_BR=\left[\begin{array}{ccc} r_{11}&r_{12}&r_{13}\\ r_{21}&r_{22}&r_{23}\\ r_{31}&r_{32}&r_{33} \end{array}\right] \]

we can express the relationship between the two basis

\[ \left[\begin{array}{ccc}\hat{e}_1&\hat{e}_2&\hat{e}_3\end{array}\right]=\left[\begin{array}{ccc}e_1&e_2&e_3\end{array}\right]^A_BR \]

or

\[ \left[\begin{array}{c}\hat{e}_1\\\hat{e}_2\\\hat{e}_3\end{array}\right]=^A_BR^T\left[\begin{array}{c}e_1\\e_2\\e_3\end{array}\right] \]

So if we have a point \(\vec{v}=\left[\begin{array}{ccc}v_1&v_2&v_3\end{array}\right]\left[\begin{array}{ccc}e_1&e_2&e_3\end{array}\right]^T\) in frame \(A\), assume \(^A_BR\) is inversible, then we have its expression in frame \(B\)

\[ \vec{v}=\left[\begin{array}{ccc}v_1&v_2&v_3\end{array}\right]\left(^A_BR^T\right)^{-1}\left[\begin{array}{c}\hat{e}_1\\\hat{e}_2\\\hat{e}_3\end{array}\right] \]

Homogenious Transformation Matrix

Does every matrix could be expressed by \(Z-Y-X\) angle?

When \(\beta=\frac{\pi}{2}\), then we lose a degree of freedom, solution is not unique.

gamble lock.