INTRODUCTION TO DYNAMICS ANALYSIS OF ROBOTS (Part 2)

1. INTRODUCTION TO DYNAMICS ANALYSIS OF ROBOTS (Part 2)

2. Introduction to Dynamics Analysis of Robots (2) • This lecture continues the discussion on the analysis of the instantaneous motion of a rigid body, i.e. the velocities and accelerations associated with a rigid body as it moves from one configuration to another. • After this lecture, the student should be able to: • Derive the principles of relative motion between bodies in terms of relative velocity

3. ISA parallel to The sliding velocity Axis of rotation passes through the point If a point on the rigid body is fixed, i.e. The rate of rotation The ISA has to pass through this fixed point. Summary of previous lecture

4. e3-axis Q Z-axis e2-axis e1-axis Frame {b} Y-axis Frame {a} X-axis Moving FORs Consider a point “Q” on a body and two FORs as follow: If the two frames are only translated and “inline” as shown , then

5. Z-axis Q e3-axis e2-axis e1-axis Y-axis Frame {b} Frame {a} X-axis Moving FORs If the two frames have undergone a rotation We have to rotate frame {b} back to be “in-line” with frame {a} before adding, i.e.

6. Y2 Y3 X2 X3 Z0, Z1 Z2 Z3 Y0, Y1 X0, X1 Example: Moving FORs A=3 B=2 C=1 Example: The 3 DOF RRR Robot: P What is the position of point “P” after 1 second if all the joints are rotating at

7. Example: Moving FORs From the previous lecture, we know that for this robot: At t=1,

8. Example: Moving FORs Similarly: At t=1,

9. Example: Moving FORs Similarly: At t=1,

10. Example: Moving FORs Given We need to find

11. Example: Moving FORs We should get the same answer if we use transformation matrix method.

12. Example: Moving FORs

13. Example: Moving FORs

14. Example: Moving FORs The answer is the same as that obtained earlier:

15. Velocity and moving FORs Consider the general case where = rotation of frame {b} w.r.t. frame {a} = position of point “Q” w.r.t. frame {b} = position of point “Q” relative to frame {b} w.r.t. frame {a} = origin of frame {b} w.r.t. frame {a} = Absolute position of point “Q” w.r.t. frame {a}

16. Velocity and moving FORs To get the instantaneous linear velocity of point “Q” w.r.t. frame {a}, we have to differentiate its absolute position where

17. Y2 Y3 X2 X3 Z0, Z1 Z2 Z3 Y0, Y1 X0, X1 Example: Velocity and moving FORs A=3 B=2 C=1 Example: The 3 DOF RRR Robot: P What is the velocity of point “P” after 1 second if all the joints are rotating at

18. Example: Velocity and moving FORs At t=1,

19. Example: Velocity and moving FORs At t=1,

20. Example: Velocity and moving FORs At t=1,

21. Example: Velocity and moving FORs Given Find

22. Example: Velocity and moving FORs

23. Example: Velocity and moving FORs There is no translation velocity between frames {3} and {2} and no translation velocity of point “P” in frame {3}

24. Example: Velocity and moving FORs There is no translation velocity between frames {2} and {1}

25. Example: Velocity and moving FORs There is no translation velocity between frames {1} and {0}

26. Example: Velocity and moving FORs We should get the same answer if we use transformation matrix method. We know that But

27. Example: Velocity and moving FORs where

28. Example: Velocity and moving FORs The answer is the same as that obtained earlier:

29. Summary • This lecture continues the discussion on the analysis of the instantaneous motion of a rigid body, i.e. the velocities and accelerations associated with a rigid body as it moves from one configuration to another. • The following were covered: • The principles of relative motion between bodies in terms of relative velocity