1. INTRODUCTION TO
DYNAMICS
ANALYSIS OF
ROBOTS
(Part 5)

2. Introduction to Dynamics Analysis of Robots (5)
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:
Solve problems of robot instantaneous motion using joint variable interpolation
Calculate the Jacobian of a given robot
Investigate robot singularity and its relation to Jacobian

3. Summary of previous lecture
Jacobian for translational velocities

4. 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 angular velocities problem.

5. Jacobian for Angular Velocities
In general, the position and orientation of a point at the end of the arm can be specified using

6. Jacobian for Angular Velocities

7. Jacobian for Angular Velocities
Similarly:

8. Jacobian for Angular Velocities
Similarly:
Jacobian for angular velocities

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

10. Example: Jacobian for Angular Velocities

11. Example: Jacobian for Angular Velocities

12. Example: Jacobian for Angular Velocities

13. Example: Jacobian for Angular Velocities

14. Example: Jacobian for Angular Velocities

15. Example: Jacobian for Angular Velocities
What is after 1 second if all the joints are rotating at
The answer is similar to that obtained previously using another approach! (refer to the example on relative angular velocity)

16. Clarification
Why r1 r2
Note: every point on the link will rotate at the same angular velocity! However, the linear velocities at different points on the link are not the same!

17. Getting the Angular Acceleration
If the joint angular acceleration for 1, 2, …, n are 0s then

18. Example: Getting the 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

19. Getting the Angular Acceleration
All the joints angular acceleration for 1, 2, …, n are 0s:
The answer is similar to that obtained previously using another approach! (refer to the example on relative angular acceleration)

20. Transformation between Joint variables and the general motion of the last link
We can combine the Jacobians for the linear and angular velocities to get:

21. What is the Jacobian for the 3 DOF RRR robot?
Example: Transformation between Joint variables and the general motion of the last link
A=3
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 Jacobian for the 3 DOF RRR robot?

22. Example: Transformation between Joint variables and the general motion of the last link

23. Jacobian and Singularities
We know that
The above is true only if the Jacobian is invertible. From algebra, we now that a matrix cannot be inverted if its determinant is zero (i.e. the matrix is singular)

24. Example: Jacobian and Singularities
Y0, Y1
X0, X1
Z0, Z1 Z2 X2 Y2 Z3 X3 Y3
Example:
The 3 DOF RRR Robot: P
Investigate the singularities of the 3 DOF RRR robot

25. Example: Jacobian and Singularities

26. Example: Jacobian and Singularities
Under these two conditions, we cannot determine the joint angular velocities using the Jacobian

27. 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:
Robot instantaneous motion using joint variable interpolation
The Jacobian of a given robot
Robot singularity and its relation to Jacobian