Configuration Space Topology and Representation
What is a Space?
A space is a set of all possible “states” or “locations” something can be in.
A point in space represents one specific configuration. A configuration space (C-space) is the set of all possible configurations of a system.
Understanding the C-Space of a System
To understand the configuration space of a system, consider its degrees of freedom (DOF) and the topology of the types of motion allowed.
- Topology describes spaces based on their fundamental shape, not size.
- Two shapes are topologically equivalent if one can be stretched or bent into the other without tearing or gluing.
- This helps us represent DOFs independently of physical geometry.
Euclidean Space (\(\mathbb{R}^n\))
Euclidean space, denoted \(\mathbb{R}^n\), is an \(n\)-dimensional space representing infinite degrees of freedom along each of its coordinate axes. Each point in \(\mathbb{R}^n\) is described by an \(n\)-tuple of real numbers.
Spherical Space (\(S^n\))
Spherical space, denoted \(S^n\), is defined as the set of all points in \(\mathbb{R}^{n+1}\) that are exactly a distance of 1 from the origin. In other words,
\[S^n = \{ x \in \mathbb{R}^{n+1} : \|x\| = 1 \}\]- \(S^0\) is two discrete points (the “sphere” in 1D).
- \(S^1\) is a circle (the set of points at unit distance from the origin in 2D).
- \(S^2\) is the surface of a sphere in 3D.
Common Topologies in C-Space
| Topology Type | Symbol | Description | Dimensionality |
|---|---|---|---|
| Euclidean | \(\mathbb{R}^1\) | Infinite line (1 DOF along an axis) | 1D |
| Euclidean | \(\mathbb{R}^2\) | Infinite plane (2 DOF) | 2D |
| Euclidean | \(\mathbb{R}^3\) | Infinite 3D space (3 DOF) | 3D |
| Spherical | \(S^0\) | Two discrete points | 0D |
| Spherical | \(S^1\) | Circle, looping 1 DOF (e.g., revolute joint) | 1D |
| Abstract | \(\mathbb{R}^n\) | \(n\) DOF in Euclidean space | \(n\)D |
| Abstract | \(S^n\) | \(n\)-sphere: surface of a sphere in \((n+1)\)D | \(n\)D |
- \(S^1\) represents all angles from \(0\) to \(2\pi\) in a circular loop.
- \(S^0\) has no continuity—just two isolated points.
- \(S^1\) has 1 DOF along the surface.
Joint Type and Configuration Space
Robotic joints contribute to C-space based on the type of motion they allow:
| Joint Type | DOF | Configuration Space | Description |
|---|---|---|---|
| Revolute | 1 | \(S^1\) | Circular rotation |
| Prismatic | 1 | \(\mathbb{R}^1\) | Linear translation |
C-Space Examples
-
1 Revolute + 1 Prismatic:
Represents a cylinder: a circle extended infinitely in one direction. -
2 Revolute joints:
Represents a torus (like a donut): one circle wrapped around another. -
2 Revolute + 1 Prismatic:
Represents a cylindrical torus: torus extended infinitely in one direction.
Summary
- Configuration space captures all possible states of a system.
- Topology helps us express configuration spaces abstractly, using symbols like \(S^1\) and \(\mathbb{R}^1\).
- The Cartesian product of joint spaces defines the full system’s C-space.
- These spaces can grow beyond visualization, but math notation keeps them tractable.