Electronic Navigation Systems - MATLAB- Module 4 - Errors and Geometry

Instructions

1. (10 points) In module 1, the last homework exercise was an example of "theta-theta" positioning.  That is, angle measurements from two stations were used to determine position (this was a classic case of so-called "triangulation").  Although a closed-form solution exists (and was provided to you with the excerpt from the Groves book), an alternative solution may be found by linearizing the measurement equation and proceeding in a manner similar to that outlined in the screencast for the case of range measurements.  The first step is to linearize the measurement equation.  For this exercise, linearize the non-linear measurement equation:

where X1 and Y1 are the known coordinates of reference station number 1,  is measured, and X and  Y are the user coordinates to be solved for.  Obviously this can be generalized to the "i-th" reference station.  If we were going to do the full user position solution, we would need measurements from at least two different reference stations.  For this exercise, though, you can stop after linearizing the above.  NOTE: You must do this derivation 'by hand' and thus you must show every step in determining the partial derivatives.  You may use a symbolic math software package to double-check your work if you wish but do not 'skip' steps in what you turn in by relying solely on the software.

2. (6 points) Download the following MAT-files:  case1.mat, case2.mat 

Each file contains the position errors (x and y components) for a particular case of rho-rho positioning.

a) Create a so-called 'scatter plot' of the position errors for each case.  For case 1, for example: 

plot(xu_err_1,yu_err_1,'.')

axis equal

You will see that in Case 1, the x and y error components appear to be uncorrelated whereas there is clearly a high degree of correlation for Case 2.  NOTE: The "axis equal" command is important to see this clearly.

b) Determine the sample means of the x and y errors for each case.  For each of these sample means, determine the 95.5% uncertainty bounds (i.e., plus and minus two sigma).

c) Do the same as (b) but for the sample standard deviations and their bounds.

d) Determine the covariance matrix of the x and y errors for each case.  Feel free to use the MATLAB "cov" command.

e) Determine the correlation coefficient between the x and y errors for each case.  Feel free to use the MATLAB "corrcoef" command.

f) Do the covariance matrices and correlation coefficients confirm the initial suspicion regarding amount of correlation in each of the two cases?  Why or why not?

3. (4 points) You can simulate a discrete-time Random Walk process in MATLAB as follows:

e = randn(1000,1);

rw = cumsum(e);

And you can, of course, look at the result simply by plotting it out.

Experimentally verify the fact that the variance of a random walk process increases linearly with time.  You will need to simulate a large number of random walk processes (formally: ‘realizations’) and then calculate the variance of the ensemble of all the realizations as a function of time.  In other words, you will NOT calculate the variance of each random walk process (see note below).  Instead, for EACH time step (i.e., 1:1000), you will calculate the variance OVER ALL the random walk processes that you have simulated.  The final result should be a plot of estimated variance versus 'time' (we're equating index steps with time steps). NOTE: Since the random walk is an example of a non-stationary random process, time averages DO NOT equal ensemble averages.

Submit a scan, PDF or Word doc containing your work for #1; submit the results obtained for #2 and #3, and submit any m-files (or list of commands) which you used.