3D Motion

Rigid Body Motion

• an object $$O$$ is a set of points $$O\subset\mathbb{R}^3$$
• Movement (or displacement) is a map $$g:\mathbb{R}^3\to\mathbb{R}^3$$ acting on all points in $$O$$.
• Motion is a displacement $$g(t)$$ for every point $$t$$ in time
• $$g(t): \mathbb{R}^3\to\mathbb{R}^3; \mathbf{X}\mapsto g(t)(\mathbf{X})$$

Euclidean Transformation

• A rigid body does not change shape when it moves.
• In particular, the distance between two points does not change

$|| \mathbf{X} - \mathbf{Y} || = ||g(\mathbf{X})-g(\mathbf{Y})||$

If this identity is satisfied for all points $$\mathbf{X},\mathbf{Y}$$, we have a Euclidean Transformation and write $$g\in E(3)$$

The map $$g$$ induces a map $$g_*$$ on free vectors,

$g_*(v) = g(\mathbf{X}) - g(\mathbf{Y})\quad\text{where}\quad v = \mathbf{X}) -\mathbf{Y})$

Special Euclidean Transformation

• Consider mirroring: $$[X_1,X_2,X_3]\mapsto[X_1,X_2,-X_3]$$
• Euclidean transformation, but not a rigid body displacement
• Rigid body motion preserves orientation, i.e.
• For three points $$\mathbf{X},\mathbf{Y},\mathbf{Z}$$
• define vectors $$u=\mathbf{Y}-\mathbf{X}$$ and $$v=\mathbf{Z}-\mathbf{X}$$
• To preserve orientation we preserve the cross product

$g_*(u)\times g_*(v) = g_*(u\times v),\quad\forall u,v\in\mathbb{R}^3$

Summary

A Special Euclidean Transformation $$g\in SE(3)$$ satisfies

$\langle u,v\rangle = \langle g_*(u),g_*(v)\rangle$

$g_*(u)\times g_*(v) = g_*(u\times v),\quad\forall u,v\in\mathbb{R}^3$

Translation

• Vectors act on points

$v: x \mapsto x+v$

$[x_1,x_2,x_3]^\mathrm{T} + [v_1,v_2,v_3]^\mathrm{T} = [x_1+v_1,x_2+v_2,x_3+v_3]^\mathrm{T}$

• The vector defines a translation
• Translate object by translating every point in the object

Rotation

Rotational Matrix

• The Rotational Matrix is orthogonal: $$R(t)\cdot R^\mathrm{T}(t)=I$$
• Differentiation gives

$\dot R(t)R\mathrm{T}(t) + R(t)\dot R^\mathrm{T}=0 \Longrightarrow \dot R(t)R\mathrm{T}(t) = -(\dot R(t) R^\mathrm{T})^\mathrm{T}$

• I.e. skew-symmetric $$\dot R(t) = \hat\omega(t)R(t)$$ for some $$\omega$$
• This gives a first-order approximation $$R(t_0+dt)\approx I + \hat\omega(t_0)dt$$

Matrix Exponential

ODE: $$\dot x(t) = \hat\omega x(t)$$

$x(t) = e^{\hat\omega t}x(0)$

$e^{\hat\omega t} = I + \hat\omega t + \frac{(\hat\omega t)^2}{2!} + \cdots \frac{(\hat\omega t)^n}{n!} + \cdots$

• Assume $$||\omega||=1$$. $$R(t)=e^{\hat\omega t}$$ rotates by $$t$$ radians around the axis $$\omega$$

$$\mathrm{ad}_g: \mathrm{se}(3)\to\mathrm{se}(3);\quad \hat\xi\mapsto g\hat\xi g^{-1}$$