MECH572 Introduction To Robotics

1. MECH572Introduction To Robotics Lecture 11

2. Review Recursive Inverse Dynamics Inverse Dynamics � Known joint angles compute joint torques 1) Outward Recursion � Kinematic Computation Known Compute From 0 to n, recursively based on geometrical and differential relationship associated with each link. 2) Inward Recursion � Dynamics Computation Compute wrench wi based on wi+1 and kinematic quantities obtained from 1) From n+1 to 0, recursively using Newton-Euler equation

3. Review The Natural Orthogonal Compliment Each link � 6-DOF; Within the system � 1-DOF 5-DOF constrained Kinematic Constraint equation T : Natural Orthogonal Complement (Twist Shape Function)

4. Review Natural Orthogonal Complement (cont'd) Use T in the Newton-Euler Equation, the system equation of motion becomes: where Consistent with the result obtained from Euler-Lagrange equation

5. Natural Orthogonal Complement Constraint Equations & Twist-Shape Matrix 1) Angular velocity Constraint Ei : Cross-product matrix of ei 2) Linear Velocity Constraints ci = ci-1+ ?i-1 + ?i Differentiate:

6. Natural Orthogonal Complement Constraint Equations & Twist Shape Matrix � R Joint Equations (6.63) and (6.64) pertaining to the first link:

7. Natural Orthogonal Complement Constraint Equations & Twist Shape Matrix � R Joint 6n? 6n matrix

8. Natural Orthogonal Complement Constraint Equations & Twist Shape Matrix � R Joint Define partial Jacobian 6? n matrix with its element defined as Mapping the first i joint rates to ti of the ith link

9. Natural Orthogonal Complement Constraint Equations & Twist Shape Matrix � R Joint

10. Natural Orthogonal Complement Constraint Equation and Twist Shape Matrix � R Joint Easy to verify Recall

11. Natural Orthogonal Complement Constraint equation and Twist Shape Matrix � P Joint

12. Natural Orthogonal Complement Constraint equation and Twist Shape Matrix � P Joint Regroup (6.74a) and (6.77):

13. Natural Orthogonal Complement Constraint equation and Twist Shape Matrix � P Joint If the first joint is prismatic, then where Define partial Jacobian

14. Natural Orthogonal Complement Constraint equation and Twist Shape Matrix Compute If kth joint is prismatic, then

15. Natural Orthogonal Complement Noninertial Base Link Include it in the joint rate vector - 6(n+1) The generalized velocity:

16. Forward Dynamics Overview Purpose of forward dynamics � Simulation, Model-based control Method � Solving Ordinary Differential equation (System E.O.M):

17. Forward Dynamics Problem Description Known: at To find: at Solution: Integration to compute at Need to compute I, ?, and

18. Forward Dynamics Computation Procedure (1) Compute I Using T, the Natural Orthogonal Complement Recall M � Positive Semi-Definite Factoring:

19. Forward Dynamics Computation Procedure

20. Forward Dynamics Computation Procedure (2) Compute Rewrite system equation as the problem can be solved as an inverse dynamics problem using the recursive algorithm. Know current compute

21. Forward Dynamics Computation Procedure (3) Solving Equations Cholesky decomposition of the generalized inertia matrix Solving two linear systems of equations Alternative solution

22. Planar Manipulator Fundamentals Basic definitions in 2-D Newton-Euler Equation in 2-D Matrix forms: Element level: System level:

23. Planar Manipulators Fundamentals Constraint equations/Natural Orthogonal Compliment K � 3n?3n matrix T � 3n?n matrix Equation of Motion

24. Planar Manipulators Example

25. Planar Manipulators Example Solution: Angular velocities: Twist-Shape matrix

26. Planar Manipulators Example

27. Planar Manipulators Example The inertial matrix Elements Generalized Inertial Matrix

28. Planar Manipulators Example Twist Shape Matrix Rate Let represent (i,j) entry of

29. Planar Manipulators Example Now define

30. Planar Manipulators Example Gravity wrench

31. Planar Manipulators Example Final form

32. Dynamic Model Review Summary Dynamic Model of a system

33. Gravity Term in E.O.M Model Gravitational Force Incorporate gravity into recursive inverse dynamics algorithm Using the natural orthogonal complement T No change in the algorithm.

34. Dissipative Term in E.O.M Model Friction Forces Viscous Friction � Solid vs viscous fluids Coulomb Friction � Solid vs Solid (Dry friction) (1) Viscous Friction Velocity field v = v(r, t) v vanishes at the interface surface

35. Dissipative Term in E.O.M Model Friction Forces Only the symmetric part of the gradient is responsible for power dissipation ? - Viscosity coefficient of the fluids For revolute joint pair velocity field can be modeled as pure tangential The dissipative function At each joint System level

36. Dissipative Term in E.O.M Model Friction Forces The dissipative force (2) Coulomb Friction Simplified model Constant determined experimentally (R � Force; P � Torque) Dissipative function: At joint i Overall

37. Dissipative Term in E.O.M Model Friction Forces Property: Lower relative speed -> Coulomb friction is high High relative speed -> Coulomb friction is low Enhanced model:

38. Course Review Overview of Robotics

39. Course Review Robotics Topics

40. Course Review Analysis & Modelling Tools

41. Office Hour Next Week Mon/Tues (Dec 6, 7) 17:00-18:00 MD 457 Assignment #4 Due on Dec 6. Submit your assignment during the office hour and get the solution. Final Exam (Open Book): 14:00 � 17:00 Dec 8, 2004