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## Section: Application Domains

### Self calibration problem & Gear fault diagnosis $-$ collaboration with Safran Tech

#### Self calibration problem

Due to numerous applications (e.g. sensor network, mobile robots), sources and sensors localization has intensively been studied in the literature of signal processing. The anchor position self calibration problem, a well-known problem in signal processing, consists in estimating the positions of both the moving sources and a set of fixed sensors (anchors) when only the distance information between the points from the two different sets is available. The position self-calibration problem is a particular case of the Multidimensional Unfolding (MDU) problem for the Euclidean space of dimension 3.

Based on computer algebra methods for polynomial systems, we have recently proposed a new approach for the MDU problem which yields closed-form solutions and an efficient algorithm for the estimation of the positions [56] only based on linear algebra techniques. This first result, obtained in collaboration with Dagher (Research Engineer, Inria Chile) and Zheng (Defrost , Inria Lille - Nord Europe), yields a recent patent [55]. Real tests are now carried out. Our first results will be further developed, improved, tested, and demonstrated.

The MDU problem is just one instance of localization problems: more problems can be addressed for which a computer algebra expertise can brought new interesting results, especially in finding closed-form solutions, yielding new estimation techniques which avoid the use of optimization algorithms as commonly done in the signal processing literature. The main differences between these localization problems can essentially be read on a certain matrix of distance called the Euclidean distance matrix [56].

#### Gear fault diagnosis

We have a collaboration with Barau (Safran Tech) and Hubert (Safran Tech), and Dagher (Research Engineer, Inria Chile) on the symbolic-numeric study of the new multi-carrier demodulation method developed in [71]. Gear fault diagnosis is an important issue in aeronautics industry since a damage in a gearbox, which is not detected in time, can have dramatic effects on the safety of a plane.

Since the vibrations of a spur gear can be modeled as a product of two periodic functions related to the gearbox kinematic, [71] has proposed to recover each function from the global signal by means of an optimal reconstruction problem which, by means of Fourier analysis, can be rewritten as

${\mathrm{argmin}}_{u\in {ℂ}^{n},{v}_{1},{v}_{2}\in {ℂ}^{m}}\parallel M-u\phantom{\rule{0.166667em}{0ex}}{v}_{1}^{☆}-D\phantom{\rule{0.166667em}{0ex}}u\phantom{\rule{0.166667em}{0ex}}{v}_{2}^{☆}{\parallel }_{F},$

where $M\in {ℂ}^{n×m}$ (resp. $D\in {ℂ}^{n×n}$) is a given (resp. diagonal) matrix with a special shape, $\parallel ·{\parallel }_{F}$ denotes the Frobenius norm, and ${v}^{☆}$ the Hermitian transpose of $v$. Based on closed-form solutions of the exact problem $-$ which are defined by a system of polynomial equations in the unknowns $-$ we have recently proposed efficient numerical algorithms to numerically solve the problem. The first results are interesting and they will be further developed and tested on different data sets. Finally, we shall continue to study the extremal solutions of the corresponding polynomial problem by means of symbolic and numeric methods, etc.