Motion and Manipulation

1. Motion and Manipulation Kinematics: Orientations

2. Axis-Angle Rotation • Any rotation Φ of a rigid body with fixed point O about an axis û can be decomposed into three rotations about three (non-planar) axes • The final orientation of a rigid body after a finite number of rotations is equivalent to that obtained after a unique rotation about a unique axis

3. Axis-Angle Rotation Body/moving frame M rotates by an angle Φabout a line with direction given by the global unit vector û=(u1,u2,u3)T, so

4. Angle-Axis Rotation Rotation matrix R that maps coordinates in M to coordinates in F is given by where I is the identity matrix and and

5. Angle-Axis Rotation Note that the matrix ũ is skew-symmetric: it satisfies Observe also that

6. Angle-Axis Rotation After substitutions, we obtain the rotation matrix

7. Angle-Axis Rotation If the transformation matrix R for the rotation is given then we can also obtain the rotation angle Φ and axis û using where

8. Exercise The unit cube is rotated by π/4 about the line through its corners A and G. What are the coordinates of the cube’s corners after the rotation?

9. Exercise A body frame M undergoes Euler rotations φ=π/6 (about the global/local z-axis), then by θ=π/4 (about the local x-axis), and then by ψ=π/3 (about the local z-axis). Determine angle and axis of rotation. Recall:

10. Euler Parameters Given rotation angle Φand axis represented by global unit vector û=(u1,u2,u3)T, the Euler parameters (e0,e1,e2,e3) are given by • e0 = cos Φ/2 • e1 = u1 sin Φ/2 • e2 = u2 sin Φ/2 • e3 = u3 sin Φ/2

11. Euler Parameters to Matrix The transformation matrix R corresponding to the Euler parameters (e0,e1,e2,e3)is

12. Matrix to Euler Parameters If the transformation matrix R for the rotation is given then we can also obtain the Euler parameters using where also

13. Exercise Determine the Euler parameters for a rotation angle Φ=π/3 and axis given by Use the Euler parameters to find the transformation matrix R and then use R to get the Euler parameters back again

14. Matrix to Euler Parameters We can identify the Euler parameters (e0,e1,e2,e3) from transformation matrix R = (rij)1≤i,j≤3 using:

15. Matrix to Euler Parameters or, alternatively, one of the following three sets of equalities:

16. Quaternions • Very similar to Euler parameters: rotation axis is encoded using a system based on a generalization of complex numbers • Elementary arithmetic operations become easy using the specific conventions of the complex number system