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Name
1. Given two frames {B} and {C} that are initially coincident with each other. First, we rotate {C} about ZˆC by θ1 degrees. Then, we rotate the resulting frame {C} about the new YˆC by θ2.
(a) (3 points) Determine the 3 × 3 rotation matrix, BCR, that will change the description of a vector P in frame {C}, CP, to frame {B}, BP.
(b) (2 Points) What is the value of BCR, if θ1 = 45◦, θ2 = 60◦?
(c) (3 Points) We then define a new frame A which translates from the frame B along the vector of Bq = [q1,q2,q3]T. Write down the homogeneous transformation ACT from frame C to frame A.
2. Consider the following manipulator with two revolute joint and one prismatic joint.

(a) (3 Points) Draw the frames of this manipulator. Define l1 to the length connecting points g and a, and l2 to be the length connecting points a and b. Note that frame 3 has been done for you, and your solution needs to be consistent with the given frame 3. Hint: Frame 0 is not located at point g
(b) (4 Points) Find the Denavit-Hartenberg parameters for this manipulator and fill in the entries of the following table
i ai−1 αi−1 di θi
1
2
3
i ai−1 αi−1 di θi
1
2
3
4
3. You are presented with the RRR manipulator below. L1, L2, and L3 are strictly positive.

(a) (3 Points) Find the Denavit-Hartenberg parameters for this manipulator. Assign the frames such that all your ai are positive.
(b) (3 Points) The position of the end-effector is:
,
where c12 = cos(θ1 + θ2).
Derive the linear Jacobian 0Jv.
(c) (6 Points) Find the singular configurations of this manipulator. For each singularity, draw the robot configuration and clearly state how the movement is restricted (in terms of frame axes).
Hint: The linear Jacobian in frame {2} is given to you here:

4. (10 Points) Let us consider the manipulator RPRP shown below, find the linear jacobian 0Jv and the angular jacobian 0Jω for the end effector point (origin of frame {4}), expressed in frame {0}.