Försättsblad till skriftlig tentamen vid Linköpings universitet

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1 Försättsblad till skriftlig tentamen vid Linköpings universitet Datum för tentamen Sal (1) ISY:s datorsalar (Egypten) (Om tentan går i flera salar ska du bifoga ett försättsblad till varje sal och ringa in vilken sal som avses) Tid 14:00 18:00 Kurskod TSRT14 Provkod DAT1 Kursnamn/benämning Sensor Fusion Institution ISY Antal uppgifter som ingår 4 i tentamen Jour/kursansvarig Gustaf Hendeby (Ange vem som besöker salen) Telefon under skrivtiden Besöker salen cirka kl. 15:00, 16:00, 17:00 Kursadministratör/ kontaktperson Ninna Stensgård, , ninna.stensgard@liu.se (Namn, telefonnummer, mejladress) Tillåtna hjälpmedel Övrigt Vilken typ av papper Rutigt ska användas, rutigt eller linjerat Antal exemplar i påsen 1. F. Gustafsson: Statistical Sensor Fusion 2. C. Lundqvist, Z. Sjanic, F. Gustafsson: Statistical Sensor Fusion Exercises 3. Other course books and tables.

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3 EXAMINATION IN TSRT14 SENSOR FUSION ROOM: ISY:s datorsalar (Egypten) TIME: at 14:00 18:00 COURSE: TSRT14 Sensor Fusion PROVKOD: DAT1 DEPARTMENT: ISY NUMBER OF EXERCISES: 4 RESPONSIBLE TEACHER: Gustaf Hendeby, tel VISITS: cirka 15:00, 16:00, 17:00 COURSE ADMINISTRATOR: Ninna Stensgård, , ninna.stensgard@liu.se APPROVED TOOLS: 1. F. Gustafsson: Statistical Sensor Fusion 2. C. Lundqvist, Z. Sjanic, F. Gustafsson: Statistical Sensor Fusion Exercises 3. Other course books and tables. MATLAB FILES: The files that are needed for the exam are available at /site/edu/rt/sigsys. Execute initcourse( tsrt14 ) to set the search path. SOLUTIONS: Available at the course homepage after the exam. The exam can be inspected and checked out kl in Fredrik Gustafsson s office, room 2A:574, B-house, entrance 25, A corridor to the right. PRELIMINARY GRADE LIMITS: grade 3 15 points grade 4 23 points grade 5 30 points OBS! Solutions should include code and plots and clear cross references between these. Mark all print-outs with your name. Good luck!

4 1. Consider three sensors located in (0, 1), (0, 0) and (1, 0), respectively, and the three TOA measurements r 1 = x (x 2 1) 2, r 2 = x x2 2, r 3 = (x 1 1) 2 + x 2 2. We are going to a SLS approach similar to (4.40) (4.42), but in a more direct way. (a) Express the squared distances ri 2 with i = 1, 2, 3 as a system of linear equations in the parameters x 1, x 2, R 2 = x x2 2, and derive an explicit solution for the target position (x 1, x 2 ). (4p) (b) Suppose that the ranges are observed as y i = r i + e i with Gaussian noise e i N (0, σ 2 ). What is the mean and variance of yi 2 for each sensor i? What is then the mean and variance of (x 1, x 2 ) in (a), if each r i is simply replaced with the measurement y i? (6p) Hint: Note that for a scalar, normally distributed variable e N (0, σ 2 ), its moments are given by E (e) = 0, E ( e 2) = σ 2, E ( e 3) = 0, E ( e 4) = 3σ Suppose we observe the Cartesian position of a target that is known to be located on the unit circle. The sensor model for this problem can be written ( ) cos(x) y = + e, (1a) sin(x) e N (0, σ 2 I 2 ). (1b) (a) Show that the ML estimate (MLE) for the polar position (angle) is given by ˆx = arctan(y 2 /y 1 ). (2p) (b) Show that Gauss approximation formula (the TT1 approximation) gives Var (ˆx) = σ 2. (3p) (c) Simulate N = data for x = 0, σ 2 = 0.1 and apply the MLE. What is the MSE? Compare to the approximation in Exercise 1(b). (5p) 2

5 Figur 1: Walkabout in Ngulia, showing GPS positions, where the color is proportional to barometric pressure. 3. Consider the GPS trajectory and barometer data illustrated in Figure 1. The matlab file nguliawalkabout.mat contains longitude, latitude and altitude from GPS as well as barometric pressure. The GPS altitude z k is quite inaccurate, while the barometric pressure has an unknown linear relation p = θ 1 + θ 2 h to altitude h. Assume that each measurement is observed with independent noise σ 2 p and σ 2 z σ 2 p. (a) Design an extended Kalman filter for estimating the altitude, and plot the resulting estimate with confidence bounds. Tune σp 2 and σz, 2 and motivate your choices. (6p) (b) How can the estimates from (a) be improved in case we know that the same location has been passed twice? Outline the strategy (implementation is not needed). (4p) 3

6 4. Consider the following motion and sensor model: x k+1 = x k + v k, { 1 P vk (v) = c (2 v ) if v < 1 0 otherwise, 1 if x k + e k < 1, y k = 2 if 1 x k + e k < 1, 3 if x k + e k 1, e k N (0, 1), where c is unknown normalization constant. Since both the motion and sensor models have non-gaussian distributions (saturated distribution and quantized Gaussian distribution, respectively), we would like to implement the standard SIR particle filter. For this, we need to implement somewhat non-standard time and measurement updates. (a) Time update: Suggest an algorithm for generating samples vk i using accept-reject algorithm. Implement the algorithm, generate many (e.g ) samples and present a histogram. (5p) Hint: Note that there is a typo in (B.1) in the course book. The normalization constant is c 1 but is written as c. (b) Measurement update: Derive how the weights w (i) k k 1 should be updated after a measurement y k becomes available. (5p) 4