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3D Kinematics

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 d et (R) = 1 2D: 4 numbers 3D: 9 numbers. Unit Vectors. p. Using the Rotation Matrix.

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3D Kinematics

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  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 = RT • square • det(R) = 1 • 2D: 4 numbers • 3D: 9 numbers

  6. Unit Vectors p

  7. Using the Rotation Matrix a A p

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

  9. Simple Rotation Matrices 2D 3D

  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

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