1. Find the equivalent x-y-z roll-pitch-yaw angles associated with the rotation matrix
.
2. A rigid body is rotated about the fixed z-axis by 60°followed by a translation of 5 units in x-direction and 3 units in the -y-direction. Find the transformation matrix .
3. Figure 1 shows a spatial manipulator where six identical limbs connect the moving platform to the fixed base by spherical joints. Each limb consists of an upper arm and a lower arm connected by a prismatic joint. Find the number of degrees of freedom of the manipulator.

Figure 1 A spatial manipulator
4.Sketch as many feasible spatial 4-dof parallel manipulators with a connectivity listing of (6, 6, 5, 5) as possible.
5. Figure 2 shows a schematic diagram of Stanford arm. Find the joint angles needed to bring the end effector to a given position and orientation, q and u, v, w.

Figure 2 D-H coordinate systems associated with the Stanford manipulator

6. Figure 3 shows a spatial 3-dof, 3R manipulator where the second joint axis is perpendicular to the first while the third joint axis parallel to the second. Find the joint angles needed to bring the end effector of the manipulator to a given position q using the method of successive screw displacements.

Figure 3 Spatial 3-dof, 3R manipulator
7.(20 score) Figure 4 shows a planar 3-dof, 3R manipulator. We wish to move the end effector, Q, along the x-axis at 2.0 m/s and, at the same time, keep the direction of approach, the x3-axis, in the –y0-dirction. Calculate the joint rates required to accomplish this task.

Figure 4 Planar 3-dof manipulator