1. INTRODUCTION TO
DYNAMICS
ANALYSIS OF
ROBOTS
(Part 4)

2. Introduction to Dynamics Analysis of Robots (4)
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 acceleration analysis
Solve problems of robot instantaneous motion using joint variable interpolation
Calculate the Jacobian of a given robot

3. Summary of previous lecture
Acceleration tensor and angular acceleration vector

4. Summary of previous lecture
Moving FORs

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

6. Example: Acceleration and moving FORs
We did the following:
To get

7. Example: Acceleration and moving FORs
We should get the same answer if we use transformation matrix method. We know that
For

8. Example: Acceleration and moving FORs
The answer is the same as that obtained earlier:

9. Relative angular acceleration
We can differentiate the relative angular velocity to get the relative angular acceleration:
where

10. Example: Relative Angular Acceleration
B=2
C=1
Y0, Y1
X0, X1
Z0, Z1 Z2 X2 Y2 Z3 X3 Y3
Example:
The 3 DOF RRR Robot: P
What is after 1 second if all the joints are rotating at

11. Example: Relative angular acceleration
Solution: We re-used the following data obtained from the previous lecture

12. Example: Relative angular acceleration

13. Example: Relative angular acceleration
You should get the same answer from the overall rotational matrix and its derivative, i.e.

14. Example: Relative angular acceleration
Using the data from the previous example:

15. Example: Relative angular acceleration
The answer is the same as that obtained earlier:

16. Instantaneous motion of robots
So far, we have gone through the following exercises:
Given the robot parameters, the joint angles and their rates of rotation, we can find the following:
The linear (translation) velocities w.r.t. base frame of a point located at the end of the robot arm
The angular velocities w.r.t. base frame of a point located at the end of the robot arm
The linear (translation) acceleration w.r.t. base frame of a point located at the end of the robot arm
The angular acceleration w.r.t. base frame of a point located at the end of the robot arm
We will now use another approach to solve the linear velocities and linear acceleration problem.

17. Jacobian for Translational Velocities
In general, the position and orientation of a point at the end of the arm can be specified using
Note that the position of the point w.r.t. {0} is
The velocities of the point w.r.t. frame {0} is

18. Jacobian for Translational Velocities

19. Example: Jacobian for Translational Velocities
Y0, Y1
X0, X1
Z0, Z1
What is the Jacobian for translational velocities of point “P”? Z2 X2 Y2 Z3 X3 Y3 P
Given:

20. Example: Jacobian for Translational Velocities
The transformation matrix of point “P” w.r.t. frame {3} is

21. Example: Jacobian for Translational Velocities

22. Example: Jacobian for Translational Velocities
What is the velocity of point “P” after 1 second if all the joints are rotating at

23. Example: Jacobian for Translational Velocities
Given a=3, B=2, C=1.
At t=1,
The answer is similar to that obtained previously using another approach! (refer to velocity and moving FORs)

24. Getting the Translational Acceleration
If the angular acceleration for 1, 2, …, n are 0s then

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

26. Example: Getting the Translational Acceleration

27. Example: Getting the Translational Acceleration
At t=1,
Given a=3, B=2, C=1.

28. Example: Getting the Translational Acceleration
All the angular acceleration for 1, 2, …, n are 0s:
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:
Principles of relative motion between bodies in terms of acceleration analysis
Robot instantaneous motion using joint variable interpolation
Jacobian of a robot