Project 2

Robert Argus, Aneesh Malhotra, Kathleen McLane

This project looked at how GPS works, and how the process works. GPS is a common commodity in society today helping people get from point a to point b. The actual workings of GPS provide an interesting problem as to how to find the location of the satellites, making sure the timing is correct, and finally making sure the error is minimal.

Each satellite that is circling earth has to transmit signals from positions in space to a GPS receiver on Earth. The user of the GPS receiver then uses this information to determine the coordinates of the receiver. The receiver then takes the transmission time, and using the speed of light is able to compute the distance of the satellite from the receiver. If there are three satellites, then there are three spheres, but an issue occurs with the timing accuracy. The image below shows the interestion of three spheres.

To solve this a fourth satellite is adding, and an additional variable of d is added. This is to account for the time. The location of the satellites is denoted \(A_i,B_i,C_i\), and the true intersection point is denoted (x,y,z). The four equations can be seen below: \begin{align*} (x-A_1)^2+(y-B_1)^2+(z-C_1)^2 = [x(t_1-d)]^2 \\ (x-A_2)^2+(y-B_2)^2+(z-C_2)^2 = [x(t_2-d)]^2\\ (x-A_3)^2+(y-B_3)^2+(z-C_3)^2 = [x(t_3-d)]^2\\ (x-A_4)^2+(y-B_4)^2+(z-C_4)^2 = [x(t_4-d)]^2\\ \end{align*} Solving for (x,y,z,d) shows the location of the satellite, and the correct time. While four spheres may not have an intersection point, when the radii are increased enough, an intersection point will be found. By squaring the above equations, it can be seen that two solutions will be found.

The individual parts, with additional background information, can be seen in the links below.