Project 2

GPS, Conditioning, and Nonlinear Least Square

Group Member: Alex Kayser and Kim Nguyen

Part 1:

Solve the system (4.37) by using Multivariate Newtons Method. Find the receiver position (x, y, z) near earth and time correction d for known, simultaneous satellite positions (15600, 7540, 20140), (18760, 2750, 18610), (17610, 14630, 13480), (19170, 610, 18390) in km, measured time intervals 0.07074, 0.07220, 0.07690, 0.07242 in seconds, repectively. Set the initial vector to be (x0, y0, z0, d0) = (0, 0, 6370,0). As a check, the answers are approximately (x, y, z) = (-41.77271, -16.78919, 6370.0596), and d = -3.201566*10^-3 seconds.

  • Code when initial vector (x0, y0, z0, d0) = (0, 0, 6370,0)
  • Command Window
  • Output

    Part 2:

    Write a MATLAB program to carry out the solution via the quadratic formula.

    Step 1: Calculate r by using the equation (4.37) on textbook.

    Step 2: Assign b = [r15, r25, r35]

    Step 3: Assign A = [r11 r12 r13 r14; r21 r22 r23 r24; r31 r32 r33 r34];

    Step 4: Then, set r = [A b]. Then using the rref command applied to the augmented matrix [A | b ] returns r.

    Step 5: Finally, calculate constant m, d x, y, and z.

  • Code and output
  • Part 3: Skip

    Part 4:

    Set up a test of conditioning of the GPS problem. Define satellite positions (Ai, Bi, Ci) from spherical coordinates (ρ φi, θi) where ρ = 26570 km is fixed, 0 ≤ φi ≤ π/2 and 0 ≤ θi ≤ 2π. Compute the difference in position ||(Δx, Δy, Δz)||∞ and the error magnification factor. Try different variations of the Δti's. What is the maximum position error found, in meter? Estimate the condition number of the problem, on the basic of the error magnification factors we have computed.

  • Code
  • Helper Functions:
    1. Calculate Range
    2. Generate Ones
    3. Replace Zeroes
  • Output
  • Part 5:

    Repeat step 4 with a more tightly grouped set of satellites. Choose all φi within 5% of one another and all θi within 5 percent of one another. Solve with and without the same input error as Step 4. Find the maximum position error and error magnification factor. Compare the conditioning of the GPS problem when the satellies are tightly loosely bunched.

    In this question, we used the same code in part 4 but different input.

  • Output
  • Comment: Our results show that when the satellites are close together, our condition number gets very large.
  • Part 6:

    Design a Gauss-Newton iteration to solve the least squares system of eight equations in four variables (x, y, z, d). Note that add any number BUT 4 satellites, or investigate adding as many as you want.

  • Code
  • Output and Graph:

    Output

    Graph

  • Comment: The more satellites we added, the more error reduced. The Gauss Newton method can handle huge amount of satellies as appropriately spaced out satellites are available. As we added more sattelites, the condition number is also reduced. There is no loss of precision occurs. The Gauss Newton method makes sure that clustering issues can be avoided and error can be reduced significantly by using the maximum available satellites in the GPS system.