Figure 1 depicts a three-link rotational, planar manipulator. Let the link lengths be l1 = 1,
(a) Work out the 2D forward kinematics for the manipulator. This should be a function from the three joint angles to
planar coordinates (the output is only the point coordinates in E(2) with orientation ignored). The full symbolic
answer should be worked out.
(b) Given the joint angles a = ( (--), what is the end-effector position (Xe,Ye)?
(c) Work out the (manipulator) Jacobian for the function from part (a).
(d) Using the joint angles from part (b), and the joint velocities & = (1,-1,1)T what is the end-effector velocity?
There should be a function called planarR3_display that can be used to visualize the manipulator. It is available
through the class wiki page. You can even plot the vector in the figure using the quiver function in Matlab. Given
a set of base points and an associated set of vector coordinates, the quiver function will plot the vectors at their
associated base points. In this example case, the forward kinematics position would be the base point, and the end-
effector velocity would be the vector.
Figure 1: Planar 3R Manipulator.
Every planar transformation going from one frame to another is equivalent to a pure rotation
about a unique point in the plane called the pole, see Figure 2. Basically, the pole is the point in the plane whose
coordinates do not change when the point is rigidly transformed by g. Let qp denote the location of the pole.
(a) First off, take the above statements and write down what they mean as an equation. In particular, what is the
mathematical equation associated to the english of the second sentence? I neglected to include the super-
and sub-scripts for the frames. Fill those in properly in your equation. (There are actually two equivalent
interpretations, I just want one of them)
(b) If the planar transformation is given symbolically by g = (d,R), find the location of the pole symbolically as a
function of d and R. In what frame did you compute the location of the pole?
(c) Suppose that the initial configuration of the object was go = (7.0,2.0,-3x/4) and the final configuration was
gig = (0, 8.0,/2). Where is the pole located? In what frame did you find the location of the pole?
Figure 2: Pole of a planar transformation.
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