1. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 1 Lecture 2. Rigid Body Motion Main Concepts: Configuration Space Rotational motion, SO(3), Exponential Coordinates Rigid motion in exponential Coordinates Velocity of Rigid Body Coordinate Transformation Wrenches and Twists Notations: z y x p Matrix:

2. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 2 Motion of a point mass: Initial position Def: A trajectory is a curve Position at Rigid motion z y x p q p q p q p q z y x P(t)

3. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 3 Rotational Motions of Coordinates of x b in A unit vector Rotation matrix Let be a rotation matrix Linear algebra:(Verify!) z y x A z y x z y x B z y x

4. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 4 Fact: SO(3) is a group under matrix multiplication. Def:is a group if Examples:

5. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 5 For SO(3): 1. If then 2. 3. Configuration Space: 1.Set a reference frame A. 2.Attach a body frame B. 3.Write down Def: A trajectory of a rigid body rotation is a curve Conversely, given write down a config. ConfigurationElements of SO(3)

6. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 6 Rigid transformation: y z y x q z x A B Note:

7. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 7 Linear algebra Def: (Rigid transformation) is a rigid transformation if Lemma:

8. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 8 Exponential Coordinates of SO(3) 6 constraints 3 independent parameters ! How to parameterize SO(3) ? i.e. q(0) q(t)

9. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 9 Letwrite Lemma :Let then Rodrigue’s formula

10. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 10 In general, Fact : Let Continuity of determinant function

11. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 11 Given and

12. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 12 Exponential Coordinates of Exp: I exp SO(3) Thm: (Euler) Any orientation is equivalent to a rotation about a fixed axis through an angle A B

13. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 13 Other representations: Euler Angles A A’ B x y z x’x’ y’ z’

14. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 14 y x Quaternions

15. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 15 Rigid Motions in A B P ab Position Orientation Config. Space Homogeneous Representation Points: vectors:

16. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 16 1. 2. 3. O.K. Meaningless. Homogeneous transformation. Composition Rule: A B C

17. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 17 Fact:

18. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 18 Fact: is a rigid transform, E.q.: A B l

19. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 19 Exponential Coordinates of SE(3) A P(t) q For translational motion : v p p(t) A

20. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 20 Define: twist coordinates of Prop: Proof:

21. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 21 If there is offset, Subtle concept! A q A’

22. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 22 Prop: is onto. Proof: Let Case 1: Let Case 2: Solve for from previous section.

23. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 23 Let Claim: Example:

24. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 24 3.3 Screws Screw motion: Pitch: Axis: Magnitude: Def: A screw S consists of an axis l, pitch h,, and mag. M. A screw motion is a rotation by about l, followed by translation by parallel to l. p Pure translation d A C q p v

25. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 25 If then, translation about v by for a screw motion: If we let

26. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 26 For pure rotation: h=0. Pure translation: Screw associated with a twist:

27. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 27 Screw:Twist: Case 1: Screw Motion: Translation along v by M. Case 2: Screw motion: Rigid motion by Special cases: Pure rotation (Revolution joints) Pure translation (prismatic joints) Thm:(Chasles) Every rigid body motion can be realized By a rotation about an axis combined with a translation Parallel to that axis. q p

28. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 28 4. Velocity of a Rigid Body Motion of a point mass: Rotational Motion: Spatial Angular velocity z y x A z y x q

29. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 29 Body Angular velocity E.q.

30. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 30 Rigid Body Motion. q B A

31. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 31 Note: Relation:

32. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 32 Note: Also, Lemma: Summary: Spatial velocity:

33. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 33 Body velocity: E.q. A B

34. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 34 Velocity of Screw Motion.

35. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 35 Note: If Coordinate Transformation. AC B Similarly:

36. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 36 E.q.

37. EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 37 5. Wrenches A B C f Let be force or moment applied at the origin of C. Generalized power: work: