1. Joint Velocity and the Jacobian
How did I
do that?
Look! I can
move!

2. Chapter Objectives By the end of the Chapter, you should be able to:
Characterize frame velocity
Compute linear and rotational velocity
Compute Jacobian and robot singularities
Bibliography:
Craig’s book
Handout

3. Velocity of a Point
The position of a point in frame B in terms of frame A is
Point velocity in A = derivative with respect to time:
When differentiating, two frames come into play:
The frame with respect to which we differentiate
The frame in which result is expressed, e.g.:

4. Rotational Velocity Suppose now that B is rotating w.r.t. A:
Differentiating:
A trick to get an economic representation:
It can be shown that:

5. Rotational Velocity (cont.)
We write:
Wedge operator
Where:
Rotational Velocity
A “cool” expression of velocity due to time varying rotation:

6. Linear + Rotational Velocity
If we have simultaneous time varying rotation & translat.:
Using Homogeneous Coordinates, we can show that:

7. Link Differential Transformation
Recall the general link transformation:
Differentiating:

8. Differential Transformation (cont.)
Using the formula for the inverse:
Therefore:

9. Differential Transformation (cont.)
The last expression can be written as:
The angular and linear velocities are:

10. Velocity Propagation from link 2 link

11. Velocity Propagation (cont.)
Rotational velocities may be added as vectors:
Where:
Also:
With respect to the linear velocity:

12. Velocity Propagation: Prismatic Joint
For the case i+1 is a prismatic joint:

13. An Example: V3 L2 L1

14. The Jacobian Jacobian = Multidimensional Derivative Example:
6 functions fi , i=1,…,6
6 variables xi , i=1,…,6
Write: y1 = f1(x1, x2, …, x6)
y2 = f2(x1, x2, …, x6)
 = 
y6 = f6(x1, x2, …, x6)
In vector form:
Y = F(X)

15. The Jacobian (cont.) Taking derivatives: Jacobian
Dividing by t on both sides:
The Jacobian is a time varying transformation mapping velocities to velocities

16. Jacobian for a Manipulator
Robot kinematics give:
frame of EE = F(joint variables)
Using the Jacobian:
velocity of EE = Jjoint variable derivatives
If all joints rotational, and calling:
then we write:

17. Singularities of a Robot
If J is invertible, we can compute joint velocities given Cartesian velocities:
Important relationship: shows how to design joint velocities to achieve Cartesian ones
Most robots have joint values for which J is non-invertible
Such points are called singularities of the robot.

18. Singularities (cont.) Two classes of singularities:
Workspace boundary singularities
Workspace interior singularities
Robot in singular configuration: it has lost one or more degrees of freedom in Cartesian space