Rigid Body Motions

Table of Contents

Rotation Matrices and SO(3)

The Special Orthogonal Group SO(3)

Rotation matrices belong to the Special Orthogonal Group SO(3), which has the properties:

• $R^T R = I$ (6 orthogonality constraints)
• $\det(R) = 1$ (determinant constraint)
• Uses 9 numbers with 6 constraints - an implicit representation

Group Properties:

• Associativity: $(R_1 R_2) R_3 = R_1 (R_2 R_3)$
• Non-commutativity: Generally $R_1 R_2 \neq R_2 R_1$
• Identity element: $I$ exists
• Inverse: Every rotation has an inverse $R^{-1} = R^T$

Orientation Representation Comparison

Quaternions

• Type: Implicit representation (4 numbers for 3D space)
• Constraint: Unit quaternion ($||q|| = 1$)
• Pros: No singularities, fewest numbers without singularities, easy interpolation
• Cons: Hard to understand, dual representation (antipodal quaternions represent same orientation)

Roll-Pitch-Yaw (Euler Angles)

• Type: Explicit representation (3 numbers - minimal)
• Pros: Minimum number of parameters, intuitive
• Cons: Singularities exist (gimbal lock)

Rotation Matrices

• Type: Implicit representation (9 numbers with 6 constraints)
• Pros: No singularities in representation
• Cons: Many parameters, “snapping back” to satisfy constraints is non-trivial

Angular Velocities and so(3)

The Lie Algebra so(3)

The space so(3) is the tangent space to SO(3) - it represents the space of angular velocities. Since velocities are local quantities, they do not have singularities.

Any rotational velocity can be represented as an angular velocity $\omega \in \mathbb{R}^3$, which is the product of:

• A unit axis of rotation
• A scalar speed

The relationship is: $\dot{x} = \omega \times x$

Skew-Symmetric Matrix Representation

The cross product can be converted to matrix multiplication using the skew-symmetric matrix $[\omega]$: $\omega \times x = [\omega]x$

where $[\omega]$ is the skew-symmetric matrix representation of $\omega$. The space so(3) is called the Lie algebra of the Lie group SO(3).

Rate of Change of Rotation Matrices

The rate of change of a rotation matrix is found by pre-multiplying with the so(3) representation. Similar to the cancellation of subscripts in rotation matrices, we can change frames of velocity vectors.

Exponential Coordinates and Axis-Angle Representation

Exponential Representation

The axis-angle representation describes orientation as:

• Axis: Direction to rotate about
• Angle θ: How far to rotate

This represents the orientation of one frame relative to another.

Differential Equations and Matrix Exponentials

Scalar case: $\dot{x} = ax(t) \rightarrow x(t) = e^{at}x_0$

where $e^{at} = 1 + at + \frac{(at)^2}{2!} + \frac{(at)^3}{3!} + \cdots$

Vector case: $\dot{x} = Ax \rightarrow x(t) = e^{At}x_0$

Integrating angular velocity: $\dot{p} = \omega \times p = [\omega]p \rightarrow p(t) = e^{[\omega]t}p(0)$

This integrates the angular velocity to find position over both time and angular displacement.

Rodrigues’ Formula

There exists a closed-form solution for the matrix exponential known as Rodrigues’ Formula.

Matrix Exponential and Logarithm

Exponential map: $[\hat{\omega}]\theta \in \text{so}(3) \rightarrow R \in \text{SO}(3)$

Logarithm map: $R \in \text{SO}(3) \rightarrow [\hat{\omega}]\theta \in \text{so}(3)$

These mappings allow us to integrate angular velocities to find orientation in space over time, transitioning between different orientations rather than decaying or growing.

Homogenous Transformation Matrices

Special Euclidean group (SE(3)), constists of a rotation matrix and a 3 vector of positions. T = [R p; 0 1], T^-1 = [RT -RTp; 0 1].

3 uses of HT matrices: Tab tells us where b is relative to a. 2) change a refernce frame of a vector or frame 3) Displace a vector or frame: T = (R,p) = Trans(p)Rot($\hat{\omega}, \theta$). Any rigid body transformation can be represented on a screw axis (linear speed + angular speed) -> pitch = linear / angular speed, and speed along the screw. If the pitch is infinite (then we dont neeed any angualar motion to advance linearly). Generalization of angular velcocity…

Twists

• What is the instantaneous linwae morion of the whole frame. Linear velocty of origin ofa frame that will be refeerebcedd bu the frame.

if theta.dot = 1

Can use transformation to change framew of refernce of a framee.

Va - [Ad_td] as an adjoiner.

Twists are 6 vectors. se(3) is lie algebra of SE(3), the lset of all possible Tdot when T = I

Important Concepts

• for a screw axis {q, shat, h}. The unit twist axis is so that he angular momemntum is hard to change due to precesssions’

Thw dimension of space of screws: 5.

Application: screw axis of Sb in kinova robot arm:

SB = [0 1 0 0 0 0]’

Ss = [0 1 0 9.8 0 0]’

J2: Sb = [0 0 1 0 0 -717.3]

Ss = [0 0 1 275.5 0 0]

Wrenches and Transformations

Power is independent

F_b = [Adj_T_ab]^T * F_a

Example 1:

R = [ 0 0 1 1 0 0 0 1 0 ]

p = (0 4 -2)’

T = [R p 0 1]

Sa = (0 0 1 0 0 5)’ Sb = (0 1 0 0 5 -4)’

Tab’ = e^{[S_a]\theta} _ Tab = Tab _ e^{[S_b]\theta}

Tab’ = [0 0 -1 0 -1 0 0 4 0 -1 0 0.5 0 0 0 1 ]