Final Exam Question

Figure 1, robot parameter

Figure 2, robot parameter

The transformation from Cartesian coordinate to  Cylindrical coordinate is

 According to the equation we derived from Newton's Law of Motion, 

  

and the flight time equation,

The range of the ball will be at

Now , suppose we have a addition actuator for ping pong ball robot to control the initial velocity Vo. Under the Cylindrical coordinate (r, theta) , what will be the new corresponding Jacobian Matrix for the falling spot of the ball? Where the form of Jacobian will be like

 

Solution

Since 

the new Jacobian Matrix will be

So, the overall Jacobian matrix will look like