Due February 25.
1) Here is the example object from the grasping lecture:
a) Using a ruler, measure the locations of the fingers, and write the numerical Jacobian matrix relating the motion of the part to the motion of the points on the object nearest the fingers.
b) Use your ruler to estimate the normals, and write the normal matrix N.
c) Write the constraint equation that describes how the fingers constrain 
. Your answer should contain exactly one numerical matrix.
d) Give an example of a motion of the part (a numerical vector for 
) that does not cause collision with the fingers, and an example of a motion that does cause collision. Prove that your examples are correct using the constraint equation from c).
2) In middle school math, we divided 
into four quadrants: I, II, III, IV. Write a matrix inequality that expresses the statement "the point (x,y) lies in the second quadrant".
3) Use Lagrangian dynamics to write the equations of motion for a "double pendulum" (2R planar arm) like the one from the movie about the "brachiating" robot we saw in class. The configuration of the arm is 
. Assume the lengths of the arms are 
and 
, and that the masses of the arm are 
and 
. Assume that the mass of each link is "concentrated" at the end of each link. (That is, the elbow and and hand are heavy, but the rest of the arm weighs nothing.)
You do not have to invert the mass matrix; an answer in the form
(1)
is sufficient. You do have to take all necessary derivatives, and give the expressions for each of the matrices. (Hint: if you are uncomfortable with taking partials of matrices, etc, then just write out a separate equation of motion for 
and 
without any matrices (similarly to what we did for the 1R arm. When you're done, collect terms and write in matrix form.)
Extra credit. Write a simulator in the programming language of your choice for the 2R planar arm.
