(10 points) In the motion planning lab, some groups used eightpoint connectivity when
growing their wave front, i.e., their wave front expanded according to the
generator in Fig. 1 as opposed to fourpoint connectivity,
i.e., a wavefront expanded according to the generator in Fig. 2.
1
1
1
1
1
1
1
1
1
Figure 1

1

1
1
1

1

Figure 2
Which wavefront generator gives rise to the shortest path in the L1 metric and why?
Why does the other not give rise to the shortest path in the L2 metric?
(10 points) For the homogenous tranformation S shown in Fig. 3, demonstrate that S is either a translation followed by a rotation or a rotation
following by a translation.
(i.e., show that it is or is not a rotation followed by a translation, and show that it is or is not a translation followed by a rotation)
α_{x}
β_{x}
γ_{x}
P_{x}
α_{y}
β_{y}
γ_{y}
P_{y}
α_{z}
β_{z}
γ_{z}
P_{z}
0
0
0
1
Figure 3
(10 points) Perform the following transformations on a wedge. Draw each transformation (before and after pictures of wedge) on a separate set of
axes. Make sure to label your axes, distances and angles on each drawing.
Trans(Z,10inches)
Rot(Z, 45 degrees)
Trans(X, 10 inches)
Rot(X, 45 degrees)
(10 points) Write the transformation matrix for each of the four transformations in the previous problem.
(15 points) During the USAR lab you experienced the joys of remote teleoperation. Describe three problems with remote teleoperation, and propose design changes which would help overcome each problem.
Note: the design change doesn't have to be to the robot itself, you should consider the entire system. You cannot, however, change the scenario (i.e., it is a remote teleoperation operation in an unknown and possibly chaotic environment.)
(5 points)How many degrees of freedom does an arbitrarily shaped planar object have? (No explanation required)
(5 points)How many degrees of freedom are necessary for a serial linkage, i.e., a sequence of links connected by revolute joints, to
arbitrarily place an arbitrarily shaped object in the plane? (No explanation required)
(15 points) A mobile robot has a planar twolink manipulator. The length of the first link is 3 units and the second link is 2 units in length.
Determine an (x,y) coordinate pair for the mobile base (call it (α,β)) such that once the base is placed at (α,β), the twolink manipulator can
reach the locations (0,0), (6,0), and (6,6).
Note: the mobile base cannot move once it is placed.
With the mobile base fixed at your (α,β), perform the inverse kinematics for each of the three target locations for the twolink manipulator.
(20 points)Consider the conveyor belt system shown in Fig. 4. Arbitarily shaped objects are placed on the conveyor
and they move with the conveyor. Downstream along the conveyor, there is a fence which can only rotate about a pivot point.
In class, when dealing with nonholonomic constraint problems, we first derived the constraints w_{i}(q) and then
from the constraints we derived the initial set of allowable motions g_{i}(q). We then performed Lie bracket
operations on the original set of motions to obtain the full set of motions achievable by the robot.
Sometimes, it is easier to directly start with the initial set of allowable motions: