Space Topology

• Euclidian space: $\mathbb{E}^n$ or $\mathbb{R}^n$
• Sphere: $S^n$
• Torus: $T^n$ ($T^2 = S^1 \times S^1$)

$\mathbb{R}^n$ is the $n$-dimensional Euclidian space, $S^n$ the $n$-dimensional surface of a sphere and $T^n$ is the $n$-dimensional surface of a torus in a $(n + 1) dimensional space.

$S^1 \times S^1 \times S^1 \times \cdots \times S^1 = T^n$ for $n$ copies of $S^1$

Some spaces (c-spaces) can be represented as Cartesian product of two or more lower dimensional spaces.

Space Representation

A numerical representation is not fundamental as the topology of a space. It involves always a choice.

Explicit parametrisation

• Min. Number of coordinates needed
• For example
  • Cartesian coordinates $(x, y, z)$
  • latitude/longitude

Implicit representation

• Surface embedded in a higher dimensional space with constraints
• For example
  • One constraint on three coordinates results in two degrees of freedom (2-D c-space)
  • $(x, y, z)$ such that $x^2 + y^2 + z^2 = 1$

Implicit representations don’t have singularities but have more numbers than the number of degrees of freedom.

• Unit quaternion
• Rotation matrix

Literature

Notes taken from:

Modern Robotics: Mechanics, Planning, and Control by Kevin M. Lynch and Frank C. Park, Cambridge University Press, 2017