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Rotation

The configuration of a point is fully described by a position, bodies additionally require a rotation to define their pose.

A rotation $R$ in 3D is specified by a rotation angle $\theta$ and a unit vector u (the rotation axis).

Frames

• Space frame: $\{s\}$
• Body frame: $\{b\}$

Expressing orientation of $\{b\}$ relative to $\{s\}$:

• writing the unit coordinate axis of $\{b\}$ as column vectors in the coordinates of $\{s\}$
• Form a matrix from the column vectors

Example:

Where:

• $\hat{x}_b$, $\hat{y}_b$, $\hat{z}_b$: Unit vectors of $\{b\}$ represented in coordinates of $\{s\}$

Combined unit vectors:

Where:

• $R_{sb}$:
  • Subscript $s$: Reference frame
  • Subscript $b$: Frame whose orientation is being represented

Constraints

There are only $3$ dimensions for orientation of a rigid body in space. But the $3 \times 3$ rotation matrix has $9$ numbers. So $6$ constraints are required:

• All $3$ column vectors are unit vectors
• Dot product of any $2$ column vectors is zero (they are all mutually orthogonal to each other)

These 6 constraints can also be written as:

Where:

• $I$: Identity matrix

Furthermore for right-handed frames the following must hold:

This means that all rotation matrices are elements of the Special Orthogonal Group $SO(n)$.

Special Orthogonal Group SO(2) and SO(3)

The Special Orthogonal Group $SO(2)$ and $SO(3)$ is the set of all possible $2 \times 2$ or $3 \times 3$ real matrices $R$ that satisfy:

We refer to $SO(3)$ as the rotation group of $\mathbb{R}^3$

The rotation group $SO(3)$ is referred to as the configuration space of the system and a trajectory of the system is a curve $R(t) \in SO(3)$ for $t \in [0,T]$.

Because a rotation matrix $R$ lives in $SO(3)$, there is no numerical equivalent to a position such as angular position.

Properties of Rotation Matrices

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

• Inverse:
  • $R^{-1} = R^T \in SO(n)$
  • $R R^{-1} = R^{-1} R = I$
• Closure: $R_1 R_2 \in SO(n)$
• Associative $(R_1 R_2) R_3 = R_1 (R_2 R_3)$
• Not commutative: $R_1 R_2 \neq R_2 R_1$
• Identity element: $R I = I R = R$
• Composition Rule for Rotations (Combining by matrix multiplication): $R_{ac} = R_{ab} R_{bc}$ (subscript cancellation)
• Rotating a vector doesn’t change its length: $x \in \mathbb{R}^3, \left \| Rx \right \| = \left \| x \right \|$
• A rotation matrix (in 2D) has 3 eigenvectors:
  • One real eigenvector corresponding to to eigenvalue $1$
  • Two complex eigenvectors with eigenvalues: $\lambda = \cos \theta \pm j \sin \theta$ where $\theta$ is the rotation angle

Composition of Rotations

The coordinates of a vector $\mathbf{u}$ can be mapped from frame $B$ to frame $A$ by writing:

Where:

• ${}_A\mathbf{u}$: Vector $\mathbf{u}$ expressed in frame $\{A\}$ coordinates
• ${}_B\mathbf{u}$: Vector $\mathbf{u}$ expressed in frame $\{B\}$ coordinates
• $R_{AB}$: Rotation matrix that converts the representation of the vector from $\{B\}$ frames coordiante representation to $\{A\}$ coordinates representation

The vector $\mathbf{u}$ can also be mapped from frame $C$ to frame $B$ by writing:

Combining these equations:

The resulting rotation matrix $R_{AC} = R_{AB} \cdot R_{BC}$ (s.a. subscript cancellation) can be interpreted as the rotation obtained by rotating frame $A$ until it coincides with frame $B$, and then rotating frame $B$ until it coincides with frame $C$.

Uses of Roatation Matrices

There are three uses for a rotation matrix:

• To represent an orientation
• To change the reference frame in which a vector or a frame is represented (operator for passive rotation)
• To rotate a vector or a frame (operator for active rotation)

In the first use, $R$ is thought of as representing a frame; in the second and third uses, $R$ is thought of as an operator that acts on a vector or frame.

Representing Orientation

$R_c$ means implicit the orientation of frame ${c}$ relative to the fixed reference frame ${s}$. Explicit notation would be $R_{sc}$.

Properties:

• $R_{ac} R_{ca} = I$
• $R_{ac} = R_{ca}^{-1} = R_{ca}^T$

Changing the reference Frame (Passive Rotation)

Meaning: $R_{ac} = change\_reference\_frame\_from\_b\_to\_a(R_{bc})$

$R_{bc}$ can be viewed as a representation of the orientation of ${c}$. $R_{ab}$ can be viewed as a mathematical operator that changes the reference frame from ${b}$ to ${a}$.

Subscript cancellation rule: $R_{ab} R_{bc} = R_{a\not{b}} R_{\not{b}c} = R_{ac}$

The reference frame of a vector can also be changed by a rotation matrix: $R_{ab} p_b = R_{a\not{b}} p_{\not{b}} = p_a$

Rotating a vector or a frame (Active Rotation)

$R$ as a rotation operator can be written:

Elementary Rotations

General form of Rotation

with: $\hat{\omega} = (\hat{\omega}_1, \hat{\omega}_2, \hat{\omega}_3)$

Properties

• Any $R \in SO(3)$ can be obtained by rotating from the identity matrix $I$ by some $\theta$ about some $\hat{\omega}$
• $Rot(\hat{\omega},\theta) = Rot(-\hat{\omega},-\theta)$

Reference Frame for Rotation

When rotating a frame ${b}$ by $Rot(\hat{\omega}, \theta)$ and $R_{sb}$ represents the orientation of ${b}$ relative to ${s}$, it’s important to express if the rotation axis $\hat{\omega}$ is defined in ${s}$ or in ${b}$ coordinates.

Given $R = Rot(\hat{\omega}, \theta)$:

• Rotate by $R$ in ${s}$ frame:
  • $R_{s{b}'} = RR_{sb}$
  • Premultiplying by $R$: $\hat{\omega}$ considered in the fixed frame
• Rotate by $R$ in ${b}$ frame:
  • $R_{s{b}''} = R_{sb}R$
  • Postmulitplying by $R$: $\hat{\omega}$ considered in the body frame.

To rotate a vector $v$, there is only one frame involved. $\hat{\omega}$ has to be interpreted as being in the frame in which $v$ is represented:

Literature

Notes taken from:

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

A Mathematical Introduction to Robotic Manipulation by Richard M. Murray, Zexiang Li and S. Shankar Sastry, CRC Press, 1994