EXAMINATION IN TSRT14 SENSOR FUSION

Transcription

1 EXAMINATION IN TSRT14 SENSOR FUSION ROOM: ISY:s computer rooms 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, 18:00 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 slides; a digital version will be available at /site/edu/rt/tsrt14/ 5. Signal and Systems toolbox manual; a digital version will be 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 at in Gustaf Hendeby s office, room 2A:536, B-house, entrance 25, A corridor to the right. PRELIMINARY GRADE LIMITS: grade 3 15 points grade 4 23 points grade 5 30 points NB! Solutions should include code and plots and clear cross references between these. Mark all print-outs with your name. Good luck!

2 UTSKRIFTSTIPS (LINUX): Utskrifter av vanliga filer kan skickas till en viss skrivare genom att man skriver kommandon som till exempel lp -d printername file.pdf i ett terminalfönster. (Byt ut printername mot den aktuella skrivarens namn.) Om man väljer File/Print i ett simulinkschema kan man ange en viss skrivare genom att lägga till -Pprintername i rutan vid Device option. NAMN PÅ UTSKRIFTER: Man kan lägga in text i matlabplottar med kommandona title och gtext och i scopeplottar i Simulink genom att högerklicka i dem och välja Axes properties. I simulinkscheman kan man dubbelklicka på något blankt ställe och sedan skriva in text. 2

3 1. Consider the problem of estimating the state x based on the prior information x 0 N (ˆx 0, P 0 ), and the two independent measurements y i = x + e i and e i = N (0, R i ), for i = 1, 2. Assume: ˆx 0 = P = (1a) y 1 = R 1 = (1b) y 2 = 0.1 ( ) R 2 = (1c) Both analytic and numeric answers are accepted as long as they are properly motivated. (a) Compute the optimal estimates x 1 and x 2 (mean ˆx i and covariance P i ) of x resulting from combining x 0 with y 1 and y 2, respectively! (2p) (b) Use the two individual estimates (x 1 and x 2 ) and derive a combined estimates x 3 of x (mean and covariance)! Assume the only thing that you know about the two estimates is their mean and covariance, e.g., they come from unknown sources in a sensor network. (c) Compute an optimal estimate x 4 of x (mean and covariance)! This time you are allowed to use all information given in the exercise. (d) Plot estimates x 0, x 1, x 2, x 3, and x 4 (mean and covariance)! Compare x 3 and x 4. Explain why they are the same or why they differ depending on your findings! Hint: plot2 can be used to plot mean and covariance of an ndist object. (2p) 3

4 1 S4 S3 0.5 x2 [m] S1 T1 S x1 [m] Figure 1: TOA network and target trajectory as described in Exercise Consider the sensor network of four TOA sensors arranged according to Figure 1. The task is to track an object moving around between the sensors using some of the different filters discussed during the course. The target is assumed to move on an almost circular path, starting in ( 0.8, 1), initially heading straight up with the speed 0.1 m/s and turning with angular rate 0.08 rad/s. (a) Start by creating a sensor model in MATLAB for the sensor network depicted in Figure 1, let R = for all the sensors! Plot the resulting sensor network! (Make sure it is similar to Figure 1.) (2p) (b) Create a motion model for the object in MATLAB. Simulate measurements for 20 s from the model without initial uncertainty and process noise using the just designed sensor model! Add the resulting trajectory to the plot! (c) Use an EKF to estimate the trajectory! Add the estimated trajectory and covariance ellipses to the plot! (Hint: Make sure to update the initial uncertainty and process noise.) (2p) (d) Use a PF to estimate the trajectory! Add the estimated trajectory and covariance ellipses to the plot! 4

5 3. Consider the problem of obtaining a scalar parameter, x, from N independent but noisy measurements, y k = x + e k, (2a) where cov(e k ) = R. The least squares (LS) solution is then given by ˆx ls = arg min x N (y k x) 2. (2b) k=1 (a) Derive the maximum likelihood (ML) estimate, ˆx ml-n, of x assuming that e k N (0, R) where R is assumed known! Use this to show that in this specific case ˆx ls = ˆx ml-n! (b) Derive the ML estimate ˆx ml-l of x assuming e k L(λ), where λ is a function of R (hence assumed known)! Show that the result can be written as N ˆx ml-l = arg min x f(y k x), (2c) where f(e) is a cost function and give f(e)! (In the Gaussian case above, f(e) = e 2.) Here L(λ) represents the Laplace distribution given by the probability density function (pdf) p(e; λ) = 1 2λ exp( e ) λ. (2d) k=1 (5p) (c) Plot the cost function f(e) used in the two cases above and give a reason why one would sometimes like to use the Laplacian noise assumption. Hint: What is the impact of a single large outlier to the end result? (2p) 5

6 IMU Z Y X gravity Figure 2: Experimental setup with IMU mounted close to the user s foot. Figure 3: Schematic problem formulation. 4. Capturing the motion of humans can be of interest in many situations, for instance to create animations in movies. One way to accomplish this is to place several inertial measurement units (IMUs) on a person s body and use the measurements to estimate his/her motion. In this exercise a simplified version of the problem is considered, where the lower leg is tracked using one IMU placed close to the foot and assuming the subject s knee is not moving. (See Figure 2.) The problem is further simplified by assuming the IMU is only rotating directly around its own z-axis, which is aligned with the horizontal plane, as depicted in Figure 3. The task is to estimate the angle denoted x in the figure using measurements from the IMU. The file data mat contains the measurements at 100 Hz (acc contains the accelerometer measurements in the x, y, and z direction, respectively, in the columns, and gyr contains the gyroscope measurements in a similar way.) (a) Suggest a measurement model to estimate x using the accelerometer! Here it can be assumed the accelerometer only measures the gravity. Use this to estimate the orientation and plot it! Hint: (1) The repeated usage of estimate for each measurement can be a bit slow, start with just a few measurements. (2) This is not a standard exsensor model. (4p) (b) Suggest a motion model that uses the gyroscope measurements as input! Use this to estimate x assuming the IMU starts with the x-axis pointing down. Plot the new estimate in the same figure as above! (c) Combine the measurement model and the motion model, and apply a filter to estimate x! Plot the result in the same figure as the two previous estimates. Hint: (1) Make sure to specify the frequency as the 4 th argument when creating the nl object to make the model time discrete. (2) To create a sig object with both measurements and input, use the syntax sig(y, fs, u). 6