Dirac Framework for Robotics Tuesday, July 8 th , (4 hours) Stefano Stramigioli

1. Dirac FrameworkforRoboticsTuesday, July 8th, (4 hours)Stefano Stramigioli

2. 1D Mechanics: as introduction 3D Mechanics Points, vectors, line vectors screws Rotations and Homogeneous matrices Screw Ports Rigid Body Kinematics and Dynamics Springs Interconnection and Mechanisms Dynamics Contents

3. 1D Mechanics

4. In 1D Mechanics there is no geometry for the ports: efforts/Forces and flows/velocities are scalar Starting point to introduce the basic elements for 3D 1D Mechanics

5. Mass Energy Co-Energy where is the momenta the applied force and its velocity.

6. The dynamics Equations The second Law of dynamics is: Diff. form Integral Form

7. The Kernel PCH representation Interconnection port

8. Spring Energy Co-Energy where is the displacement the applied force to the spring and its relative velocity.

9. The dynamics Equations The elastic force on the spring is: Diff. form Integral Form

10. The Kernel PCH representation Interconnection port

11. Spring Mass-Spring System • Mass

12. Together….

13. Interconnection of the two subsystems (1 junc.) Or in image representation

14. Combining… There exists a left orthogonal

15. Finally

16. All possible 1D networks of elements can be expressed in this form Dissipation can be easily included terminating a port on a dissipating element Interconnection of elements still give the same form Summary and Conclusions

17. 3D Mechanics

18. Set of points in Euclidean Space Free Vectors in Euclidean Space Right handed coordinate frame I Coordinate mapping associated to Notation

19. Rotations

20. Rotations It can be seen that if and are purely rotated where

21. If is a differentiable function of time are skew-symmetric and belong to : Theorem

22. Tilde operator

23. The linear combination of skew-symmetric matrices is still skew-symmetric To each matrix we can associate a vector such that … It is a vector space It is a Lie Algebra !! is a Lie algebra

24. SO(3) is a Group • It is a Group because • Associativity • Identity • Inverse

25. where where Lie Algebra Commutator It is a Lie Group (group AND manifold)

26. Lie Groups Common Space thanks to Lie group structure

27. For any finite dimensional vector space we can define the space of linear operators from that space to Dual Space co-vector The space of linear operators from to (dual space of ) is indicated with

28. In our case we have Configuration Independent Port !

29. General Motion

30. General Motions It can be seen that in general, for right handed frames where ,

31. Due to the group structure of it is easy to compose changes of coordinates in rotations Can we do the same for general motions ? Homogeneous Matrices

32. SE(3)

33. If is a differentiable function of time belong to where Theorem

34. Tilde operator

35. Elements of se(3): Twists The following are vector and matrix coordinate notations for twists: The following are often called twists too, but they are no geometrical entities ! 9 change of coordinates !

36. SE(3) is a Group • It is a Group because • Associativity • Identity • Inverse

37. where where Lie Algebra Commutator SE(3) is a Lie Group (group AND manifold)

38. Lie Groups Common Space thanks to Lie group structure

39. Intuition of Twists Consider a point fixed in : and consider a second reference where and

40. Possible Choices For the twist of with respect to we consider and we have 2 possibilities

41. Left and Right Translations

42. Possible Choices and

43. Notation used for Twists For the motion of body with respect to body expressed in the reference frame we use or • The twist is an across variable ! • Point mass geometric free-vector • Rigid body geometric screw + Magnitude

44. Chasle's Theorem and intuition of a Twist Any twist can be written as:

45. Examples of Twists

46. Examples of Twists

47. It can be proven that Changes of Coordinates for Twists

48. Twists belong geometrically to Wrenches are DUAL of twist: Wrenches are co-vectors and NOT vectors: linear operators from Twists to Power Using coordinates: Wrenches

49. Poinsot's Theorem and intuition of a Wrench Any wrench can be written as:

50. Chasles vs. Poinsot Charles Theorem Poinsot Theorem The inversion of the upper and lower part corresponds to the use of the Klijn form