Rotations and Translations

1. Rotations and Translations Euler Theorem + Quaternions

2. Representing a Point 3D • A three-dimensional point A is a reference coordinate system here

3. Rotation along the Z axis • In general:

4. Using Rotation Matrices

5. Combining Rotation and Translation

6. Extension to 4x4 • We can define a 4x4 matrix operator and use a 4x1 position vector

7. Formula

8. Notes • Homogeneous transforms are useful in writing compact equations; a computer program would not use them because of the time wasted multiplying ones and zeros. This representation is mainly for our convenience. • For the details turn to chapter 2.

9. Euler’s Theorem • Any two independent orthonormal coordinate frames can be related by a sequence of rotations (not more than three) about coordinate axes, where no two successive rotations may be about the same axis.

10. Euler Angles • This means, that we can represent an orientation with 3 numbers • Assuming we limit ourselves to 3 rotations without successive rotations about the same axis:

11. Example

12. Another Example • Suppose we want to use ZXZ rotation, • Rotation along Z axis, • Rotation along X axis, • Rotation along Z axis,

13. Example - Cont • Let’s see what happens if,

14. Example – Cont 2 • Changing 's and 's values in the above matrix has the same effects: the rotation's angle changes, but the rotation's axis remains in the direction

15. Gimbal Lock • Gimbal Lock Animation

16. Euler Angle - Matlab • If we want to rotate Roll,Pitchand Yaw • Roll 0.1 degrees • Pitch 0.2 degrees • Yaw 0.3 degrees >> rotx(0.1)*roty(0.2)*rotz(0.3) ans = 0.9363 -0.2896 0.1987 0.3130 0.9447 -0.0978 -0.1593 0.1538 0.9752

17. Euler Angle – Matlab cont. >> rpy2r(0.1,0.2,0.3) ans = 0.9363 -0.2896 0.1987 0.3130 0.9447 -0.0978 -0.1593 0.1538 0.9752

18. Euler Theorem • In three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point.

19. Euler Theorem - Matlab R = 0.9363 -0.2896 0.1987 0.3130 0.9447 -0.0978 -0.1593 0.1538 0.9752 [theta, v] = tr2angvec(R) theta = 0.3816 v = 0.3379 0.4807 0.8092

20. Euler Theorem – Matlab cont. >> angvec2r(0.3816, [0.3379,0.4807,0.8092]) ans = 0.9363 -0.2897 0.1987 0.3130 0.9447 -0.0979 -0.1593 0.1538 0.9752

21. 3D Rotations - Matlab R = rotx(pi/2); trplot(R) tranimate(R)

22. HW

23. Quaternions • The quaternion group has 8 members: • Their product is defined by the equation:

24. Example • Calculate

25. Quaternions - Algebra • Using the same methods, we can get to the following:

26. Quaternions Algebra We will call the following linear combination a quaternion. It can be written also as: All the combinations of Q where a,b,c,s are real numbers is called the quaternion algebra.

27. Quaternion Algebra By Euler’s theorem every rotation can be represented as a rotation around some axis with angle . In quaternion terms: Composition of rotations is equivalent to quaternion multiplication.

28. Example We want to represent a rotation around x-axis by 90 , and then around z-axis by 90 :

29. Rotating with quaternions We can describe a rotation of a given vector v around a unit vector u by angle : this action is called conjugation. * Pay attention to the inverse of q (like in complex numbers) !

30. Rotating with quaternions • The rotation matrix corresponding to a rotation by the unit quaternion z = a + bi + cj + dk (with |z| = 1) is given by: Its also possible to calculate the quaternion from rotation matrix: Look at Craig (chapter 2 p.50 )