Post Mid-Term Sample Questions

Note that for some of these questions the answer is actually in the notes.  However, you should try and arrive at the answer with minimal reference to the notes.  Also, you should expect questions pertaining to the assignments.
  1. Data points from a laser range finder have a range and bearing denoted (ρi, θi).  Derive the perpendicular distance di for an observed point (ρi, θi) from a line given by the parameters r and α.

  2. Derive Bayes rule from the definition of conditional probability.

  3. On slide 21 in the notes "Localization: Part 1" there is a "probability pizza" with three ingredients:

    (a) From the definition of independence show why the events "Mushrooms" and "Anchovies" are not independent.
    (b) Given that the event "Mushrooms" has ocurred, we can say that "Pepperoni" and "Anchovies" are independent.  That is, "Pepperoni" and "Anchovies" are conditionally independent.  If we add anchovies to slice 3, are these events still conditionally independent?

  4. Derive equation 1 of Bayes filter (try not to look at your notes!).  Clearly state all assumptions and justifications.

  5. Question 6 from the 2006 final exam: PDF document

  6. Question 7 from the 2006 final exam: PDF document

  7. What are the advantages of Monte Carlo localization over grid localization?

  8. Your favourite restaurant in the University Centre has decided to build a new pizza-turning robot.  This new version will represent its belief that the heat lamp is over a particular slice by a continous variable.  This variable is an angle, θ.  Angle θ will be tracked by a Kalman filter.  As the robot has no sensors installed yet, the Kalman filter will apply only the prediction step.  The state vector xt is defined as [θ, dθ/dt]T.  At every timestep the control input ut gives a constant acceleration which is added to dθ/dt.  Meanwhile, the speed of rotation decays due to friction by a constant factor of 0.5 at every time step.  The motion of the rotating pizza is defined by the following equations:

    θ = θ + Δt dθ/dt
    dθ/dt = 0.5 dθ/dt + Δt ut

    (Note that these equations are not particularly realistic.  See section 16.3 of "Engineering Mechanics: Statics and Dynamics" by R.C. Hibbeler, 1992, for a more realistic model.)

    Give the matrices At and Bt for the system update equation: xt = At xt-1 + Bt ut + εt.

  9. Fill in slide 18 in the notes "Swarm Robotics: Part 2" which gives blanks for the actions needed in each sensory configuration to support clustering. Slide 19 provides the answers.

  10. Provide an algorithm (written in pseudocode) to synchronize the flashing of a robot in a swarm with its neighbours.