3.5 Embeddings of Manifolds in Rn

You can’t always embed a manifold neatly into a visual space.

3.5.1 Matrix Representations of Rigid-Body Configuration

Representing the position and orientation of a rigid body using an $mXm$ matrix. The manifold this matrix represents is a submanifold of ${\mathbb{R}}^{m^2}$. A convenient feature of this representation is the ability to multiply matrices to get new matrices within the manifold.

The orientation of a rigid body in $n$-dimensional space (n=2 or 3) by the matrix groups $SO(n)$, and the position by the matrix groups $SE(n)$.

Here’s a concrete example.

Orientation: SO(2) and SO(3)

$R=\left[\begin{array}{ccc}{R}_{11}& R_{12}& R_{13}\\ R_{21}& R_{22}& R_{23}\\ R_{31}& R_{32}& R_{33}\end{array}\right]\in SO(3)$ The matrix $R$ is often called a Rotation Matrix.

The matrix is used to represent three angular degrees of freedom, so there are six independent constraints on the matrix entries: that each column and row is a unit vector, (3 constraints) and that the columns and rows are orthogonal to each other (3 more constraints).

Matrices satisfying these conditions belong to the special orthogonal group, because (in a right handed frame) the determinant is +1 (meaning that orientation and “volume” are preserved)

Orientations in $n$-dimensional space can be written

$SO(n)=\{R\in {\mathbb{R}}^{nxn}|R{R}^T=\iota ,det(R)=1\}$ where $\iota$ is the identity matrix.

Position and Orientation: SE(2) and SE(3)

Let $p=[p_1,p_2,p_3{]}^T\in {\mathbb{R}}^3$ be the position of the rigid body relative to the stationary frame, and let $R$ be the rotation matrix as described above. Then we represent position and orientation as the 4x4 transform matrix

$T=\left[\begin{array}{cc}R& p\\ 0& 1\end{array}\right]\in SE(3)$ Where the bottom row consists of three zeros and a one.

Uses of Matrix Representations

• Represent rigid-body configurations
• change the reference frame for the representation of a configuration or a point
  • called a frame
• displace a configuration or a point.
  • called a transform