Planar rigid-body motions

A planar rigid body motion is defined as:

Where:

• $T \in SE(2)$: A $3 \times 3$ real matrix representig a rigid body motion in $\mathbb{R}^2$
• $R \in SO(2)$: Rotational part of the motion (Rotation matrix)
• $p \in \mathbb{R}^2$: Linear displacement of the motion (and roation axis)
• $\theta \in [0,\pi)$: Rotation angle

The set of all planar homogeneous transforamtion matrices is called special Euclidean group $SE(2)$

Spacial rigid-body motions

A homogeneous transformation matrix (or rigid body motion) $T$ in $\mathbb{R}^3$ is defined as:

Where:

• $T \in SE(3)$: A $4 \times 4$ real matrix representig a rigid body motion in $\mathbb{R}^3$
• $R \in SO(3)$: Rotational part of the motion (Rotation matrix)
• $p \in \mathbb{R}^3$: Linear displacement of the motion (and rotation axis)

The set of all spacial homogeneous transformation matices is called the special Euclidian group $SE(3)$

Multiplication with a Vector

Sometimes it’s useful to calculate $Rx + p$ (where $x \in \mathbb{R}^3$ and $(R,p) = T$).

We then need to append a $1$ to $x$ to make it a $4 \times 1$ vector.

Where

• $[x^T 1]^T$: homogeneous coordinates representation of $x$
• $T \in SE(3)$: homogenous transformation

Properties of Transformation Matrices

These properties hold for $SE(2)$ and $SE(3)$.

• Inverse:

  • $T^{-1} = \begin{bmatrix} R & p \\ 0 & 1 \\ \end{bmatrix}^{-1} = \begin{bmatrix} R^T & -R^Tp \\ 0 & 1 \\ \end{bmatrix} \in SE(n)$

• Closure: $T_1 T_2 \in SE(n)$

• Associative $(T_1 T_2) T_3 = T_1 (T_2 T_3)$

• Not commutative: $T_1 T_2 \neq T_2 T_1$

• Composition Rule for Rotations (Combining by matrix multiplication): $T_{ac} = T_{ab} T_{bc}$ (subscript cancellation)

• $T = (R, p) \in SE(3)$ transforms a point $x \in \mathbb{R}^3$ to $Tx$

  • $T$ preserves distances: $\left\| T_x - T_y \right\| = \left\| x - y \right\|$
    • $\left\| a \right\| = \sqrt{a^Ta}$: standard Euclidian norm in $\mathbb{R}^3$
  • $T$ preserves angles: $\left< T_x - T_z, T_y - T_z\right> = \left< x - z, y - z\right>$ for all $z \in \mathbb{R}^3$
    • $\left< a, b\right> = a^Tb$: standard Euclidian inner product in $\mathbb{R}^3$

Uses of Transformation Matrices

There are three uses for a transformation matrix:

• To represent a configuration (position and orientation) of a rigid body
• To change the reference frame in which a vector or a frame is represented (operator for passive rotation)
• To displace a vector or a frame (operator for active rotation)

Representing a Configuration

Any frame can be expressed relative to any other frame.

$T_{sa} = (R_{sa}, p_{sa})$ represents the configuration of frame $a$ relative to frame $s$.

Changing the reference Frame of a Vector or Frame

Changing the reference frame of a vector or a frame is analogous to rotation matirces using the subscript cancellation rule:

Displace a Frame or Vector

A displacement (rotation and translaation) of a frame or a vector can be seen as a translation along a vector $p$ ($Trans(p)$) and a rotation around the axis $\hat{\omega}$ with angle $\theta$ ($Rot(\hat{\omega}, \theta)$).

With:

Space-Frame Transformation

When a frame $T_{sb}$ is premultiplied by a transformation matrix $T$ the vectors $p$ and $\hat{\omega}$ are interpreted in the coordinate system of the $\{s\}$ frame (first subscript of $T_{sb}$):

The order of the operations is:

1. rotation (this will cause the origin of $\{b\}$ to move if it is not coincident with the origin of $\{s\}$)
2. translation

Body-Frame Transformation

When the frame $T_{sb}$ is postmultiplied by a transformation matrix $T$ the vectors $p$ and $\hat{\omega}$ are interpreted in the coordinate system of the $\{b\}$ frame (second subscript of $T_{sb}$):

The order of the operations is:

1. translation
2. rotation (this does not move the origin of the frame)

Literature

Notes taken from:

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