ECE Senior Capstone Project 2019 Tech NotesDoppler Shift Navigation using LEO Satellites

Obtaining a Position Fix UsingDoppler Shift MeasurementsBy Tyler Klein, ECE ‘19

AbstractThe original Transit Doppler navigation system useda complex communications system[5] to augmentand significantly enhance the position fix algorithm.The system, which was the precursor to GPS, usedtwo carrier frequencies to remove ionosphericeffects, with both transmissions broadcasting orbitaldata to provide accurate fixes. This paper describesa simple algorithm to obtain a position fix givenDoppler shift measurements from arbitrary satelliteconstellations. Simulations are described todemonstrate the performance of the algorithm.

Problem StatementObtaining a position fix is simple exercise inoptimization. The algorithm determines the positionon earth that minimizes the difference between themeasured Doppler shift and the expected Dopplershift at that location. This is a simple concept, but inmost cases an algebraic solution does not exist.Therefore numerical optimization must be used.Additionally, algebraic expressions for the expectedDoppler shifts aren’t obtainable, which results in anexpensive-to-evaluate cost function, or function wewould like to minimize to find our position.

Optimization AlgorithmsOptimization algorithms are divided into twocategories – stochastic methods and iterativemethods. Stochastic methods are not guaranteed toproduce an exact solution, but frequently have largesearch spaces.[6] Iterative methods are locallyconvergent, but frequently find only local extrema.Figure 1 shows the “view” from an iterative method

- it is very clear which direction to go in, but it is notclear whether or not the minimum will be global.Figure 2 shows the result from a stochastic

Figure 1: Rastrigin Function - Contour Plot

algorithm - it is clear which region contains theglobal minima, but the precise location is unclear.

Newton’s MethodNewton’s method is a well-studied root findingalgorithm with quadratic convergence.[2] It requiresa continuous first derivative, and requires an initialguess that is in the neighborhood of a root. Whilethe first requirement isn’t an issue in most cases, thesecond requirement is a major shortcoming ofNewton’s method. Consider the functionf (x) = arctan(x). Figure 3 shows two lines, |xk| and|xk+1|, for f (x) = arctan(x), where xk is the current

Department of Electrical and Computer EngineeringSenior Project Handbook: http://sites.tufts.edu/eeseniordesignhandbook/

Figure 2: Rastrigin Function

approximation of the root, and xk+1 is the newapproximation after one step of Newton’s method. If|xk| ≥ 1.39, then |xk+1| > |xk| and the algorithmmoves further away from the solution. This meansthat the initial guess must satisfy |x0| < 1.39. If wewere given a black-box with f (x) = arctan(x), wewould need to start extremely close to the solution inorder to find the solution. This is circular logic.

The need to provide an initial guess near thesolution is far-reaching, and has serious implicationsfor optimization. Figure 4 shows a simple fourthdegree polynomial f (x) = −x4− x3 + 7x2 + 3x− 16and its derivative. The colored lines denote thebasins of attraction for each root of the derivative.To find the local minimum, the initial guess must bein the blue region, otherwise the algorithm willconverge to a local maximum. In my initial Doppleranalysis simulations, I experienced extremely poorconvergence to the global minimum using Newton’smethod. As a result, I began a search foroptimization techniques less prone to prematureconvergence to local extrema.

Differential EvolutionWhile researching algorithms to avoid localextrema, I stumbled upon the Differential Evolution(DE) algorithm. DE is a simple stochastic algorithmdesigned to provide “an efficient framework to solvecomplicated optimization tasks.”[3] It works by

Figure 3: Region of Convergence - f (x) = arctan(x)

Figure 4: Region of Convergence - f (x) = −x4 − x3 +7x2 +3x−16

iteratively improving a set of candidate solutions bycomparing the candidates to random linearcombinations of other candidates. That is, a newcandidate solution is generated, and if the newcandidate is an improvement, it replaces the originalpopulation member. This happens for every memberof the population for each iteration. DE convergesmuch slower than Newton’s method, but isextremely simple to implement, doesn’t requireknowledge of the gradient of the cost function, andcan search large solution spaces. In the examplefrom Figure 4, my implementation of DifferentialEvolution found the global minimum in every trial.

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The main disadvantage of DE algorithms is the needto evaluate the cost function many times, while theadvantage is the ability to search large spaces forextrema.

Adaptations

DE is also prone to converging to local extrema.According to an experiment in [3], the default DEalgorithm converged to the global minimum in just2% of trials. As a result, the algorithm was adaptedto maintain population diversity. Populationdiversity prevents premature convergence(contraction of the solution space) to a localextrema.[4] The proposed modification was toreplace the worst candidates with new randomcandidates from the original solution space. In thesame experiment, this modification improvedconvergence from 2% to 83%[3]. Replacing theworst candidates with random samples ensures thatthe solution space will never shrink prematurely, ifat all, and significantly improves the ability to jumpout of a basin of attraction. Additionally, I furthermodified the algorithm to replace any candidatesoutside a desired solution space.

My AlgorithmThe task was to optimize an expensive-to-evaluatecost function, with no algebraic representation, for areasonably large number of data points. DifferentialEvolution provides the ability to search a largesolution space, many kilometers wide. To reduce thecomputational burden of DE, I use only the mostsignificant data points in the initial search. In theory,one can pinpoint the location of a receiver in theorbital plane using just the point of inflection in theDoppler curve.[1] The angle to the orbital plane canbe determined by the width of the Doppler curve.Only the first two measurements, the last twomeasurements, and two midpoints of the Dopplercurve are used in the initial DE computations. Thisprovides a crude estimate of the receiver position.This initial guess is used as the starting point forNewton’s method with all of the data points,resulting in quick and precise convergence ofNewton’s method to the global extremaapproximated by the DE algorithm.

Initial ResultsTo simulate the system, I generated Doppler data fora single satellite pass for a fixed receiver position,then added 1Hz of Additive White Gaussian Noise,or AWGN, to the measurements. I then ran theoptimization algorithm on the simulated data.Results should be taken with a grain of salt, as thesimulation assumed that the TLE data waserror-free. TLE error directly translates to positionfix error. To mitigate this issue, I plan to usemultiple satellites to average out the TLE error andobtain a more accurate fix. With an initial guessprovided within 2◦ of longitude and latitude, thealgorithm converged 74/75 times using thesimulated data.

SummaryThe problem was to globally optimize anexpensive-to-evaluate cost function. While countlessoptimization techniques exist, many are prone topremature convergence, or convergence to localextrema. Optimization algorithms can generally bedivided into two categories – fast, local methods,and slow, global methods. I used the exploratorycapabilities of the modified DifferentialEvolution[3] algorithm to avoid local extrema; andtook advantage of the extremely fast, localconvergence of Newton’s method to pinpoint theglobal extrema from the “best guess” of the DEalgorithm. This produced a reliable algorithm thatcan be used to generate an accurate position fix fromDoppler shift measurements.

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References[1] G. Boudouris. “A method for interpreting the doppler curves of artificial satellites”. In: Journal of the British

Institution of Radio Engineers 20.12 (Dec. 1960), pp. 933–935. DOI: 10.1049/jbire.1960.0108.

[2] Anne Greenbaum and Timothy P. Chartier. Numerical Methods: Design, Analysis, and ComputerImplementation of Algorithms. Princeton University Press, 2012. ISBN: 0691151229. URL:http://a.co/d/1ooFyT9.

[3] Roman Knobloch, Jaroslav Mlynek, and Radek Srb. “Convergence rate of the modified differential evolutionalgorithm”. In: 2017. DOI: 10.1063/1.5013964. URL: http://dx.doi.org/10.1063/1.5013964.

[4] Population Diversity. Nov. 2018. URL: https : / / www . mathworks . com / help / gads / population -diversity.html.

[5] Thomas A. Stansell. “Transit, the Navy Navigation Satellite System”. In: Navigation 18.1 (), pp. 93–109.DOI: 10.1002/j.2161-4296.1971.tb00077.x. eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/j.2161-4296.1971.tb00077.x. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/j.2161-4296.1971.tb00077.x.

[6] Z. Caner Taskın. Optimization vs. heuristics: Which is the right approach for your business? Nov. 2018.URL: https://icrontech.com/blog%5C_item/optimization-vs-heuristics-which-is-the-right-approach-for-your-business/.

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