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PhD - I year (part 1)

The f‌irst problem of the bugged version of LOMEM is about the sampling time: if this parameter is set suf‌f‌iciently large then the tracker returns tracks that do not make any sense.
I've start to investigate this issue in November 2021, discovering in the middle of December that the problem was in the discretization process used to def‌ine the prediction step. The incriminated discretization method, actually implemented in my master thesis work and called there Heuristic Discretization, is one original (but, unfortunately, pretty bad) idea developed by me.
An easy solution is to replace heuristic discretization with the standard (forward) [Euler's method], which provides numerical stability even for large sampling times. However, provided that the sampling time is not too large, Euler tends to perform worse than heuristic.
Still unsatisf‌ied, I've then searched for a third and independent solution, ending up in the [Tustin method], which is numerically stable likewise Euler but, at the same time, also more performant than heuristic.

heuristic discretizationaccuratenumerically unstable
Euler's methodnot accuratenumerically stable
Tustin's methodaccuratenumerically stable

Here "numerically stable" means that the discretization method is able to mantain constant the distance between the discretized sample points whenever the reference track is drawn at constant speed; "accurate" means that the overall dif‌f‌erence between the reference sampled points and the discretized sample points is small.

Sampling time problem

Simple test about the dif‌f‌erent discretization methods: in white is represented the reference track, an example of [Euler's spiral]; in yellow are represented the dif‌f‌erent discretizations of the reference track. The white (yellow) dots are the reference (discretized) sample points. The starting points are located in the north-west whirls of the spirals, the ending point are located in the south-est ones.
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