Example

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: $SE(3)\times \mathbb{T}^{n}$