3D Kinematics Eric Whitman 1/24/2010

Rigid Body State: 2D p

Rigid Body State: 3D p

Add a Reference Frame p

Rotation Matrix Linear Algebra definition – Orthogonal matrix: R -1 = R T square – det(R) = 1 2D: 4 numbers 3D: 9 numbers

Unit Vectors p

Using the Rotation Matrix p A a

Pros and Cons Rotates Vectors Directly Easy composition 9 numbers Difficult to enforce constraints

Simple Rotation Matrices 2D3D

Degrees of Freedom 2D – 2x2 matrix has 4 numbers – Only one DoF 3D – 3x3 matrix has 9 numbers – 6 constraints – 3 DoF

Euler Angle Combinations Can use body or world coordinates 2 consecutive angles must be different – Can alternate (3-1-3) or be all different (3-1-2) 24 possibilities (12 pairs of equivalent) For aircraft, body is common – Yaw, pitch, roll For spacecraft, body is common

Construct a Rotation Matrix Body Convention – Common for spacecraft

Recover Euler Angles

Gimbal Lock Physically: two gimbal axes line up, making movement in one direction impossible Mathematically describes a singularity in Euler angle systems For the body convention, this occurs when angle 2 equals 0 or pi For the body convention, this occurs when angle 2 +/- pi/2 Switching helps

Pros and Cons Minimal Representation Human readable Gimbal Lock Must convert to RM to rotate a vector No easy composition

Axis Angle (4 numbers) A special case of Euler’s Rotation Theorem: any combination of rotations can be represented as a single rotation 3 numbers to represent the axis of rotation 1 number to represent the angle of rotation Has singularity for small rotations

Rotation Vector (3 numbers) The axis can be a unit vector (only 2 DoF) Multiply axis by angle of rotation Can easily extract axis angle – Axis = rotation vector Normalize if desired – Angle = ||rotation vector|| Same singularity – small rotations

Pros and Cons Minimal Representation Human readable (sort of) Singularity for small rotations Must convert to RM to rotate a vector No easy composition

(Unit) Quaternions All schemes with 3 numbers will have a singularity – So says math (topology)

Constraint Easy to enforce

Conversion with RM

Composition

Pros and Cons No Singularity Almost minimal representation Easy to enforce constraint Easy composition Interpolation possible Not quite minimal Somewhat confusing

Summary of Rotation Representations Need rotation matrix to rotate vectors Often more convenient to use something else and convert to rotation matrix Euler angles good for small angular deviations Quaternions good for free rotation

Homogeneous Transformations Define:

Composition