The document describes two algorithms, gradient descent and Gauss-Newton, for determining location from GPS satellite signals. It outlines the process of linearizing the pseudorange equations and deriving the iterative algorithms to minimize error between measured and calculated ranges. Simulation results on synthetic noiseless data show Gauss-Newton converges much faster than gradient descent, in 4 iterations versus over 50,000 iterations. Gauss-Newton is thus determined to be a more optimal algorithm for GPS positioning.

1. Project 2: Algorithms
for Global Positioning
Kevin Le (A11727120)
Professor: Ken Kreutz-Delgado
ECE 174
12/1/2016

2. Introduction
The Global Positioning System (GPS) is a network of dozens of satellites that hover out in space
with the purpose of allowing people to identify their location on earth. Signals from the GPS
satellites are transmitted to a GPS receiver on earth’s surface to pinpoint the satellite’s location
in space. With knowledge of the satellite’s orbit and utilizing time information, a GPS receiver is
able to determine its own location under the condition that four satellites are within range.
However, due to the inaccuracy of the receiver’s clock when utilizing commercial GPS units for
low cost, the distances calculated, called pseudo-ranges, are not accurate. Ideally, these four
pseudo-ranges should intersect at a single point for a true receiver-satellite distance, but the
unsynchronized clocks prevent this. To accurately determine a location, a few algebraic
computations are necessary to make the adjustment for the imperfect information. These
algebraic computations consist of deriving, implementing, and testing two algorithms, the
Gradient Descent and Gauss Newton algorithms. Throughout this project, we will be exploring
how these algorithms contribute to resolving the clock error when determining the true pseudo-
range under noiseless conditions.
Procedure
Step 1 Linerazation:
Both algorithms begin with minimizing the nonlinear weighted least squares loss function: given
by
where 𝑦 is the pseudo-range vector to the satellites, consisting of a collection of vector-
measurement equations, and ℎ 𝑥 is the vector of true ranges calculated from the position vector
𝑥. The pseudo-range measurement for satellite 𝑙 is denoted by 𝑦 and modeled as
𝑦 = 𝑅 𝑆 + 𝑏 + 𝑣
where 𝑅 𝑆 is a nonlinear function of the receiver location, 𝑏 is the (constant) systematic clock
bias error, and 𝑦 is the random noise. To proceed with determining the pseudo-range, we use the
multivariate Taylor Series Expansion to estimate 𝑅 𝑆
𝑅 𝑆 ≈ 𝑅 𝑆- +
𝜕
𝜕𝑆
𝑅 𝑆- Δ𝑆
where Δ𝑆 = 𝑆 − 𝑆-, and the higher order terms have been discarded. With this expansion, we are
able to linearize 𝑅 𝑆 , turning into the following form through vector derivative identities.
∇2 𝑅 𝑆 =
𝜕
𝜕𝑆
𝑅 𝑆 =
∆𝑆4
𝑅(𝑆)
The pseudorange equation can now be expressed as

3. 𝑦 = 𝑅 𝑆- +
∆𝑆4
𝑅(𝑆)
Δ𝑆 + 𝑏 + 𝑣
It can be seen that as the true range increases, the linearized approximation becomes more
accurate, because the difference term is driven to zero.
Step 2 Algorithm Development:
Now to derive the gradient descent algorithm, we begin with the loss function.
If we take the gradient of l(x), then we also end up with the derivative of l(x) with respect to x
𝑙7
𝑥 = ∇𝑙 𝑥 =
𝑑
𝑑𝑥
𝑙 𝑥 = −𝐻4
(𝑥)(𝑦 − ℎ 𝑥 )
where ℎ7
𝑥 = −𝐻4
. If we linearize about the current estimate 𝑥:, we can claim that
allowing us to replace the difference term of x with the gradient of l(x) and completing our
algorithm.
αk is the step size parameter and is critical for good performance, because of the potential
overshoot and instability. For this reason, αk was chosen as 0.25. H(X) is the Jacobian matrix
and can be determined from Rl (S).
Q(X) is chosen as I, because the correction is given along the direction of the steepest descent.
As mentioned before, both algorithms are derived from the loss function, however, for deriving
the Gauss Newton algorithm, we need to expand h(x) about the current estimate of xk, which
yields
and therefore the loss function would be
Minimizing the loss function with respect to xk would yield the unique correction of
which provides a solution to our linearized inverse problem where

4. is the pseudoinverse H(ˆxk). The iterative algorithm can now be derived as
Similar to before, αk is used as the step-size parameter, but for the Gauss-Newton algorithm, it is
less sensitive. Therefore, it was recommended to have a choice of αk = 1.
While this is the usual case, our problem states that H(X) is a square matrix, implying that it is
invertible, which changes the pseudoinverse in our algorithm to and overall
yielding the Gauss Newton to be
Step 3 Simulation:
Applying these algorithms to accurately determining the location, we first begin by setting the
actual receiver position as S = S0 and satellite positions as Sl with l = 1-4.
A total of four satellites will be used. Our clock bias error,b, is taken as b = 2.35478806 x 10^-3
ER. As for initial estimates, we use
as the vehicle location and b(0) = 0. Since we are dealing with a zero noise case, where vl = 0
and sigma = 0, our pseudorange vector equation becomes
𝑦 = 𝑅 𝑆 + 𝑏 + 𝑣, l = 1,2,3,4
Given the previous steps on linearizing the pseudorange equation and deriving the algorithms, to
begin testing the algorithms, we need to determine a termination condition. The termination
condition is designed to terminate once it reaches past the maximum number of steps. For the
gradient descent, the max iteration was 50,000 while the Gauss Newton was set to 4.
Results
Testing the algorithms with the synthetic data, we can now present our findings through plots of
the loss function l(x), receiver position estimate error in units of meters, and clock bias estimate
error in units of meters as a function of the iteration step. From our results, we will be observing
for the observed convergence rates of the gradient descent verses the Gauss-Newton method.

5. Final Receiver Position Values
Gradient Descent Gauss Newton
1.00000000000031 0.999999999999999
5.71887423485670e-13 -1.57293703511202e-15
5.74338363736638e-13 -1.42373846869791e-15
Clock Bias Error
0.00234547880742619 0.00234547880679832
Given the vast difference in iterations between the two algorithms, it can be seen that the
Gradient Descent and Gauss Newton both return similar answers. These answers are quite
satisfactory in the accuracy of determining the location, presumably, because there is no noise in
generation of the synthetic data. The noise is meant to represent the modelling errors that occur
in the real world. If we were to consider noise when determining the location, we would have to

6. approximate the noise with the mean of m, variance s2
, and the normally distributed numbers to
each pseudorange. We would expect that with the noise, we are unable to calculate the accurate
location, because the algorithms are focused on minimizing the loss function and does not
correspond with the location. Luckily, we are not dealing with the noise case and are able to
produce optimistic results.
Focusing on the accuracy between the two algorithms, it could be seen that the Gaussian Newton
is much more superior than the Gradient Descent, as the final position values converge much
faster based on the number of iterations taken to converge. This can be supported through an
analysis of the loss function, position error, and clock bias error plots between the two
algorithms. To begin with, we see that if we take the log of the loss function, the Gauss Newton
converges right at the maximum number of steps, 4, while the Gradient has still not converged
with iterations over 50,000. Minimizing the loss function is important in determining the
location, because it is also minimizing the difference of the pseudorange equation and the vector
of the true range. The minimization brings us to our convergence and location. The position error
also follows a similar minimization structure that will give us an error of how far our estimated
position is from the true position. Within 2 iterations, the Gauss Newton is already approaching a
convergence while the Gradient Descent takes approximately 10,000 iterations, demonstrating
the Gauss Newton’s superiority. Finally, we observe the clock bias error between the two
algorithms to come to our final justification. Already anticipating the clock bias error to also
converge, we can confirm this by observing that the Gauss Newton converges at 3 iterations
while the Gradient converges at 10,000. From all these plot observations, we can justify the
Gaussian Newton algorithm as a more optimal choice than the Gradient Descent algorithm.
Conclusion
From our results, it was determined that the Gaussian Newton algorithm was far superior than
the Gradient Descent. While the Gaussian Newton did involve fairly complicated calculations,
the number of iterations in comparison to the Gradient Descent proves its efficiency by hundreds
or thousands of times better. Having more measurements from each satellite generally reduces
the error in our calculation and brought us to our final results.