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 (Om tentan går i flera salar ska du bifoga ett försättsblad till varje sal och ringa in vilken sal som avses) Tid 08:00 12:00 Kurskod TSRT14 Provkod DAT1 Kursnamn/benämning Sensor Fusion Institution ISY Antal uppgifter som ingår 4 i tentamen Jour/kursansvarig Fredrik Gustafsson (Ange vem som besöker salen) Telefon under skrivtiden Besöker salen cirka kl. 08:00, 09:00, 10:00, 11:00 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. 4. Lecture notes and slides. Digital versions are available at /site/edu/rt/tsrt14/ 5. The matlab toolbox manual. A digital version is available at /site/edu/rt/tsrt14/

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3 EXAMINATION IN TSRT14 SENSOR FUSION ROOM: ISY:s datorsalar TIME: at 08:00 12:00 COURSE: TSRT14 Sensor Fusion PROVKOD: DAT1 DEPARTMENT: ISY NUMBER OF EXERCISES: 4 RESPONSIBLE TEACHER: Fredrik Gustafsson, tel VISITS: cirka 08:00, 09:00, 10:00, 11:00 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. 4. Lecture notes and slides. Digital versions are available at /site/edu/rt/tsrt14/ 5. The matlab toolbox manual. A digital version is available at /site/edu/rt/tsrt14/ 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 the examiners room, B-house, entrance 25, A corridor directly 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 Figur 1: Sensor network with three sensors located at p S,1 = ( 1, 0), p S,2 = (0, 0) and p S,3 = (1, 0), respectively. An object at p T = (X, Y ) T and velocity v = (v X, v Y ) T is approaching. 1. Consider the sensor network scenario in Figure 1. Assume that each sensor computes the distance from the target to the sensor with an unknown bias according to the sensor model y k = r k + b + e k,, k = 1, 2, 3, E(e k ) = 0, Var(e k ) = R (a) Define a sensor network in Matlab according to Figure 1, and simulate measurements for sensor k = 1, 2, 3 for one time instant. Use target position p T,1 = (0.5, 5) T and Gaussian measurement noise with variance R = Estimate the target position using WLS using both pair-wise differences y i y j and a model with the bias b as a parameter. Present a plot of the network with a confidence ellipsoid for the target position. (7p) (b) Compute the CRLB for this problem and present it in a plot as an ellipsoid in the sensor network. (3p) 2. In a famous experiment, the examiner is throwing his smartphone four meters up in the air. Before that, there was a calibration phase in which the smartphone was moved from the ground to a table. Can the height of the table be computed from the accelerometer signal alone by deadreckoning? The logged sensor signals can be seen in Figure 2, and are available as June2014.mat. Only the accelerometer signal will be used in this exercise, and only the two parts in the intervals (for bias estimation) and (for dead-reckoning), respectively. 2

5 Figur 2: Smartphone data of pressure in millibar and acceleration in m/s 2 (a) Neglect horizontal acceleration and assume that the smartphone is exactly horizontal all the time. Suggest a simple motion model, and use it to compute vertical speed and position, respectively as a function of time. (8p) (b) Again, neglect horizontal acceleration, but now account for that the smartphone is not exactly horizontal all the time. The vertical acceleration is with this assumption the magnitude of the measured acceleration. Modify the motion model in (a), and re-compute the vertical speed and position, respectively as a function of time. (2p) 3. Consider the object motion in Figure 3. Assume that we know that the object is moving in a circle with constant speed, but we do not know the actual trajectory (including the initial angle ϕ 0, speed ϕ k = ω and the parameters L, R). (a) Assume that we have a directional sensor providing a noisy measurement of the angle to the object (DOA sensor). Derive the sensor model on the form y k = h k (θ) + e k for this problem, where θ = (ϕ 0, ω, L, R) T. Assume that the noise is normally distributed, e k N (0, σ 2 ). (4p) 3

6 Figur 3: An object is moving with constant speed in a circle with radius R, with origin located at distance L from the observer (depicted by the cross). (b) Assume that θ = (0, 2π/10, 5, 1) T. Compute an analytic expression for the FIM and use this to numerically compute the CRLB for the parameters θ given the measurements y 0:9. Assume the sampling interval T = 1 and σ 2 = Is there anything particular about the lower bound? Try computing CRLB for the case when R is known (which means evaluating CRLB for θ new = (ϕ 0, ω, L) T instead). Compare the two CRLBs and comment on your observations. (6p) Hint: feel free to use the symbolic toolbox for Matlab. Functions like syms, eval, subs might be handy. 4. Consider the following motion and sensor model: x k+1 = x k + v k, { 1 P vk (v) = c (9 v2 ) if v < 3 0 otherwise, y k = sign (x k + e k ), e k N (0, 1), where c is unknown normalization constant. Since both the motion and sensor models have non-gaussian distributions (saturated distribution 4

7 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) 5