Midterm: Introduction to Robotics 16-311

Fall 2005

1hr  15 mins, use one 8.5 x 11 cheat sheet

Name:

Team:

Problem #1 [25 pts]

(a)  Draw the generalized Voronoi diagram for the configuration space for the square shaped robot in the orientation shown. [20 pts]

(b)  Draw the shortest path with respect to the L2 metric between the start and goal locations shown. [5 pts]

Extra

Problem #2 [15 pts]

The following describes two convolution scenarios.

• Convolve a signal with the mask [-1 1] and then convolve the result again with the same mask [-1 1].

• Convolve a signal with the mask [ -1 2 -1].

(a)  Is the result the same or different? [5 pts]

(b)  Why? [5 pts]

(c)  What algebraic operation does convolving with the mask [-1 2 -1] do? [5 pts]

Problem #3 [9 pts]

(a)  How many DOF does a circular robot which can translate and rotate on some point other than its center have? [3 pts]

(b)  How many DOF does a robot with a fixed base and 3 revolute joints have? [3 pts]

(c)  What is the dimension of the configuration space for a robot which can only translate and has a boom on it that can rotate?  [3 pts]

Problem #4 [30 pts]

A 1 DOF prismatic joint is a linear DOF, which as its name suggests, provides motion along a line.  Think of it as a telescoping arm.  The robot below has a revolute joint with angle θ at the base, which rotates a prismatic joint with length s whose range of motion is 0 to 100 cm.  The base joint has no limits.  The robot is shown in its initial position.

There are 3 obstacles: two point obstacles at (0,50) and (0,-50), and a curved quarter circular wall with radius 75 cm.

Pick a metric and draw the shortest path with respect to that metric.  Draw the end position in the workspace along with 2 intermediate points on the shortest path.

Workspace:

Configuration Space

Problem #5 [21 pts]

Draw a line to match the resulting signal with the corresponding mask that was used to convolve the original signal.

Original signal:

 [1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9] [-1 0 1] [-1 -1 -1 -1 0 1 1 1 1]