## Midterm

### 24-354 General Robotics Spring-02. Prof. Howie Choset

#### You have 1 hour and 15 minutes to complete the exam. Please write all answers either on the exam or ina blue book You must attempt all four problems Good Luck

Problem #1 (30 points)

(a.) For the following configuration space, draw the visibility graph and the Voronoi diagram (using the L2 metric).
(b.) For each method, highlight the path the a robot would take assuming it had full knowledge of the world.
(c.) Comment on the significant features of each path and explain why they are different.
(d.) Given that the configuration space was two dimensional, how many degrees of freedom does the robot have? Visibility graph Voronoi Diagram

Problem #2 (30 points)

(a.) Given the following workspace (with robot shown at start configuration) and configuration space, use the Wavefront planner to find a path in configuration space from the start configuration to the goal configuration. You may use any metric you want, but be sure to state which you are using. Draw the path on the figure. Note: Theta 1 is the first joint angle and Theta 2 is the second joint angle. The black dot is the end effector, and the joints have no joint limits.

(b.) Draw in the workspace five intermediate configurations between the start and goal configurations, and show their location in the configuration space. (We have provided extra copies of the workspace and configuration space)   Problem #3 (10 points)

In the motion planning lab, some groups used eight-point connectivity when growing their wave front, i.e., their wave front expanded according to the generator as opposed to four-point connectivity, i.e., a wavefront expanded according to the generator .

(a.) Which wavefront generator gives rise to the shortest path in the L1 metric and why?

(b.) Why does the other not give rise to the shortest path in the L1 metric?

(c.) Does the other generator give rise to the shortest path with the L2 metric? Why?

Problem #4 (15 points)

(a.) Given the following input signal, match each mask with its corresponding convolved output. Fill in the correct mask number...

[-1 -1 -1 -1 1 1 1 1 ]:___________     [-1 -1 1 1]:___________    [-1 1]:____________

(b.) Match each mask to its description (draw lines from the mask to description)
Description
1   1   1
0   0   0
-1 -1 -1

1  0   -1
1  0   -1
1  0   -1

0   1  0
1  -4  1
0   1  0
horizontal edge detector

vertical edge detector

Laplacian detector

Problem #5 (15 points)

In class most of the mobile robot discussion was limited to robots that can only translate in the plane, i.e. were parameterized by configuration Q = (x, y).
Consider a non-circularly symmetric robot which can translate and rotate in the plane.

(a.) What is the dimension of the configuration space of this robot?

(b.) What does the L1 metric look like for the robot? (write it out algebraically) Note: In class, the L1 metric was defined as D(A, B) = |Ax - Bx| - |Ay - By|.

(c.) If we were to use the wavefront planner to generate a path in the configuration space for this robot, how could we adapt the L1 metric to favor translation over rotational motion?