Figure 1. Carrier phase measurement geometry.
Problem 1. Integer Dilution of Precision Calculation.
Consider the estimation of the integer ambiguity (N) and baseline length (d) from carrier phase measurements (D0 and D1) for the idealized relative positioning case depicted in Figure 1. Derive the expression given in class for the integer-ambiguity dilution of precision (IDOP):
IDOP = (cos2q0 + cos2q1)0.5 / | cosq0 - cosq1|
by modeling the carrier phase measurement errors as uncorrelated and distributed dentically with zero mean and variance s2(D).
Figure 1. Carrier phase measurement geometry.
Problem 2. Attitude Determination Using GPS Carrier Phase Measurements.
Consider an idealized attitude estimation problem as shown in Figure 2. Two antennas lie on a plane at the ends of a baseline, which is at an unknown angle q from the x-axis. The distance between the antenna phase centers is exactly 16 cm. There is an extra complication that the cables connecting the two antennas to a common receiver may be of unequal lengths. Three satellites are in view with the following line of sight vectors:
s1 = 1(1) = (0.707, 0.707)T
s2 = 1(2) = (0, 1)T
s3 = 1(3) = (-0.707, 0.707)T
The single difference carrier phase measurements at L1 from the three satellites are given in centimeters as lf = (-9.1404, -3.001, 8.5475)T. There is some residual measurement noise for each measurement. Determine the angle q, the integer ambiguities, and the line bias between the cables.
Figure 2. Two dimensional attitude determination problem.
Problem 3. Effect of Reference Station Errors on a DGPS User.
Consider a local DGPS implementation in which there is a significant error (say 1 km) in the estimated position of the reference antenna. The differential corrections computed at the reference station would, therefore, be in error, and a user applying these corrections would get her absolute position wrong. Show that the error in the user's position would be approximately the same as that in the reference antenna position, and the estimated relative position vector between the reference and the user would be essentially unaffected by this error. In other words, the user position is still accurate relative to the position of the reference station, and any error in the reference position would be harmless if the user is only interested in the relative solution.
If you intend to perform a research project in lieu of the final exam and remaining homework assignments, you may substitute this alternate problem for any one of the above problems (your choice). This is an individual assignment to write a 2-page proposal for your research project that you will perform in the second half of the course.
Your proposal should describe a problem that you wish to analyze and state specifically what you plan to do by the end of the project. Think of your proposal as a statement of work that commits you to perform a certain action by the end of the semester. Identify what research products you expect to have in your report at the end of your project (i.e. a table of measurements, a demonstration of an algorithm, a sensitivity study, etc.).
In your proposal, provide a complete list of all the hardware (be specific!) that you think you will need to perform your experiment, including receivers, antennas, cables, power supplies, etc. Indicate where the resources are that you intend to use, and any logistical issues such as taking the equipment to an experiment site, required access to a site or the GPS simulator, coordinating measurements in different locations, collecting and interpreting data, etc. You are welcome to use any of the facilities in the GPS lab for your research, or any other equipment provided you have a source to provide the equipment. You may use any data products or tools available anywhere on the Internet in your research. Try to resolve all anticipated problems or needs associated with conducting your experiment and performing your research.
The level of difficulty that you propose should be scaled to a 5 week research project. Pick a topic that is challenging but that you can also complete within the allotted time. In addition to completing your research, you will be expected to write a web-based report documenting your project on the Internet, and to give a 20 minute verbal presentation on your project to the class.
Your research proposal will not be formally graded, but it will be checked
for the following content:
- is the topic and final research product well defined?
- are all the required resources identified?
- is the experimentation plan well constructed?
- is the goal achievable, but also challenging (i.e., no low-ball topics)
- is the topic interesting and relevant?
- is the research creative and original?
Feel free to contact me as you are preparing your topic. If I do not feel that your topic is adequate for some reason, I will suggest modifications to make it acceptable. Once we have agreed to your proposal it will become a binding statement of work for your research project.
By selecting this problem, you are committing yourself to a course of action that involves completing the project instead of taking the final exam. Once selected there is no turning back, so make sure you are willing to complete the project before submitting your proposal.