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Problem 1:(20 points) A. lb. weight hangs from linear spring as shown in Figure XA below At equilibrium, thespring stretched Winch by the weight. a. Calculate the stiffness of the spring. b. What the frequency at which the weight will vibrate up and down ifth spring stretched and then released? Give formula for the oscillation frequency in which neither the mass nor the spring constant appears. M Figure Mass suspended Ninear spring fromo fixed block Problem 2: (20points) Given two link manipulator described bythe DH diagram in Figure 2 Write Az matrices describing this manipulator b Write the 6X2 Jacobian matrix for this manipulator. Z0 701* X1 Z1 d2* a1 Figure 2:A twolin manipulator with arotary joint followed by prismatic joint Problem (20 points) Given general homogeneous transformation matrix H : if where N' an 3 > rotation matrix and isa3 vector. a Derive an expression for II in terms of R and < b Given the transformation matrices TH Tf. n). where TY for i=1,2 and j=0,2,3 Derive an expression for 13 in terms of the given po and Problem 4:(20 points) Given the transformation matrix $1 210 +ages -e 823 II II for manipulator that has three rotary joints The two a's are link lengths for links and 2. a. Draw the DH diagram for this manipulator. Be suretolabel the axes, the link lengths that are not zero, and the joint riables b. Write the and 02) matrices for this manipulator Problem 5: (20 points)Given the DH diagram for t wo joint manipulator in Figure .Note that s fixed (not variable) but 3his variable. a. Specify the set of points reachable by the endpoint of the manipulator (point Ain the figure) when both rotary joints can rotate unconstrained and the prismatic joint must satisfy 0 d' b. Specify the number of ways that each reac hable point can be reached. Z0 d1 Z1 62* Z2 d2* X0,X1 X2 Figure Threejoint manipulator with 2 rotary joints followed by aprismatic joint

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