Definition

A configuration space, $\mathcal{C}$, is a complete specification of the position of every point in the robotic system.

Examples

Let $\mathbb{S}:=\{ z\in\mathbb{C}\mid \lVert z \rVert=1 \}$ be the circle group - Mobile robot translating in the plane: $\mathbb{R}^{2}$ - Mobile robot translating and rotating in the plane: $\mathbb{R}^{2}\times \mathbb{S}$ - Rigid body translating in three-space: $\mathbb{R}^{3}$ - A spacecraft: $SE(3)=\mathbb{R}^{3}\times SO(3)$ - An $n$-joint revolute arm: $\mathbb{T}^{n}=\underbrace{ \mathbb{S}\times\dots \times \mathbb{S} }_{ n\text{ times}}$ - A drone with an attached $n$-joint arm: $SE(3)\times \mathbb{T}^{n}$