1. An ultrasound sensor measures distance x = c∆t/2. Here, c is the speed of sound and ∆t is the difference in time between emitting and receiving a signal. Let the variance of your time measurement ∆t be σ 2 t . What can you say about x, when c is assumed to be constant? Hint: how does a change in ∆t affect x? 2. Consider a unicycle that turns with angular velocity φ˙ and has radius r. Its speed is thus a function of φ˙ and r and is given by v = f(φ, r ˙ ) = rφ˙ Assume that your measurement of φ˙ is noisy and has a standard deviation σφ. Use the error propagation law to calculate the resulting variance of your speed estimate σ 2 v . 3. Assume that the ceiling is equipped with infra-red markers that the robot can identify with some certainty. Your task is to develop a probabilistic localization scheme, and you would like to calculate the probability p(marker|reading) to be close to a certain marker given a certain sensing reading and information about how the robot has moved. (a) Derive an expression for p(marker|reading) assuming that you have an estimate of the probability to correctly identify a marker p(reading|marker) and the probability p(marker) of being underneath a specific marker. (b) Now assume that the likelihood that you are reading a marker correctly is 90%, that you get a wrong reading is 10%, and that you do not see a marker when passing right underneath it is 20%. Consider a narrow corridor that is equipped with 4 markers. You know with certainty that you started from the entry closests to marker 1 and move right in a straight line. The first reading 1 you get is “ Marker 3”. Calculate the probability to be indeed underneath marker 3. (c) Could the robot also possibly be underneath marker 4? 2
