1. 3D Kinematics Eric Whitman 1/24/2010

2. Rigid Body State: 2D p

3. Rigid Body State: 3D p

4. Add a Reference Frame p

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

6. Unit Vectors p

7. Using the Rotation Matrix p A a

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

9. Simple Rotation Matrices 2D3D

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

11. 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, 3-2-1 body is common – Yaw, pitch, roll For spacecraft, 3-1-3 body is common

12. Construct a Rotation Matrix 3-1-3 Body Convention – Common for spacecraft

13. Recover Euler Angles

14. Gimbal Lock Physically: two gimbal axes line up, making movement in one direction impossible Mathematically describes a singularity in Euler angle systems For the 3-1-3 body convention, this occurs when angle 2 equals 0 or pi For the 3-1-2 body convention, this occurs when angle 2 +/- pi/2 Switching helps

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

16. 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

17. 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

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

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

20. Constraint Easy to enforce

21. Conversion with RM

22. Composition

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

24. 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

25. Homogeneous Transformations Define:

26. Composition