1. 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:

2. Example

3. Gimbal Lock Gimbal Lock Animation

4. Euler Summary Video

5. Quaternions Quaternions are a number system that extends the complex numbers They were first described by Irish mathematician William Rowan Hamilton in 1843 The quaternions H are equal to, a four-dimensional vector space over the real numbers

6. Quaternions A quaternion has 4 components Of the 4 components one is ‘real’ scalar number, and the other 3 form a vector in imaginary ijk space

7. Quaternions Sometimes, they are written as the combination of a scalar value s and a vector v Where

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

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

10. 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.

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

12. 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) !

13. 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 )

14. Rotation Example If we want to do a rotation by x, y,z : This is equal to:

15. Denavit-Hartenberg Specialized description of articulated figures Each joint has only one degree of freedom rotate around its z-axis translate along its z-axis What’s so interesting about 6 DOF ?

16. Denavit-Hartenberg 1. Compute the link vector a i and the link length 2. Attach coordinate frames to the joint axes 3. Compute the link twist α i 4. Compute the link offset d i 5. Compute the joint angle φ i 6. Compute the transformation (i-1) T i which transforms entities from link i to link i-1

17. Denavit-Hartenberg This transformation is done in several steps : Rotate the link twist angle α i around the axis x i Translate the link length a i along the axis x i Translate the link offset d i along the axis z i Rotate the joint angle φ i around the axis z i 17

18. Denavit-Hartenberg 18

19. Denavit-Hartenberg Multiplying the matrices : In DH only φ and d are allowed to change. 19

20. Denavit-Hartenberg Video

21. Example 1 D-H Link Parameter Table : rotation angle from X i-1 to X i about Z i-1 : distance from origin of (i-1) coordinate to intersection of Z i-1 & X i along Z i-1 : distance from intersection of Z i-1 & X i to origin of i coordinate along X i : rotation angle from Z i-1 to Z i about X i a0a0 a1a1 Z0Z0 X0X0 Y0Y0 Z3Z3 X2X2 Y1Y1 X1X1 Y2Y2 d2d2 Z1Z1 X3X3 Z2Z2 Joint 1 Joint 2 Joint 3 http://opencourses.emu.edu.tr/file.php/32/lecture%20notes/Denavit-Hartenberg%20Convention.ppt

22. Example 1 : rotation angle from X i-1 to X i about Z i-1 : distance from origin of (i-1) coordinate to intersection of Z i-1 & X i along Z i-1 : distance from intersection of Z i-1 & X i to origin of i coordinate along X i : rotation angle from Z i-1 to Z i about X i