Analysis and exploitation of dispersive waves for source localization on solids - Arun, Kattukandy Rajan (2018)
This thesis mainly addresses different classes of algorithms for impact source localization in solids with particular focus on developing a human-computer interface (HCI) through touch impact. The challenge of this research lies in the fact that transverse propagating waves on a thin elastic plate are dispersive in nature. As opposed to wave propagation in a non-dispersive medium such as air, a dispersive medium results in a wave propagation speed that is frequency dependent. This results in the spreading of impact waveform as it propagates. Although this research is conducted for touch/tap localization in human-interaction technology, results arising from this work can potentially be applied to the areas of identification of impact damage, loose part monitoring in industrial applications and crack analysis in non-destructive evaluation. Depending on material properties and geometry of the body, several modes of wave propagation in solids are possible and are characterized by their specific particle oscillatory patterns. Based on research derived from this thesis, amplitudes of these mode responses are shown to vary with source location. In view of this, algorithms derived in this thesis are based on time-frequency analysis of the signal. This first class of algorithms is based on the location template matching (LTM) method which estimates the index of input data by comparing features of the received signal with features obtained from a pre-recorded set of signals. LTMcan be extended to localization on an interactive surface by creating a library of signal features generated by exciting each point of interest with an impact. During operation, an impact is localized by determining the index of the template signal that best matches the pattern of the signal received under test. In this inter-disciplinary research, mechanical vibration theories that model wave propagation of the flexural modes of vibration generated by an impact on the elastic plate surface are studied and analyzed. By analyzing the mathematical model of flexural vibration derived, a Zak transform based time-frequency classifier is proposed for the localization of a finger-tap based on LTM. The research is then extended to localization using time-difference-of-arrival (TDOA) between two spatially separated sensors. The TDOA-based technique is one of the most widely used methods in acoustic source localization and its main advantage compared to LTM-based methods is that it does not require training of known points. This implies that such algorithm can achieve a large number of distinguishable tap locations with minimal setup. The proposed triangulation based method employs phase transit TDOA between the received signals received by the vibration sensors. A mathematical model of the wave propagation of flexural vibration is used to explain the frequency spectrum diversity of vibration signals and background noise. An algorithm is subsequently developed using the Kullback-Leibler discrimination information to determine the phase transit time of the vibration signal. The velocity for each frequency mode needs to be estimated for phase transit based algorithm. The next stage of this research focuses on deriving a source localization algorithm which does not require pre-calibration. The proposed method exploits the similarity of warped dispersive signal in the time-frequency domain. The wave propagation of flexural vibration due to an impact on a plate surface is analyzed and the change in dispersion between two signals is shown to be dependent on the range difference between the propagation path of the signals. Utilizing this, a source localization algorithm is developed for impact localization on solid surfaces. As the proposed source localization algorithm jointly estimates the model parameters and source location, this method does not require pre-calibration of the velocity parameter. Although pre-calibration of velocity parameter is avoided in warping-based algorithm, the mechanical parameter of a solid has to be estimated before localization. The final part of this research uses the cross-spectrum that is independent of any pre-calibration steps or material parameter for source localization. The wave propagation model is used to show how the non-linear wavenumber function represents the dispersive nature of wave propagation. The non-linear wavenumber function is then approximated using the Taylor series to derive an expression for the phase derivative of the cross-spectrum. Moreover, it is noted that the phase derivative contains coupled terms pertaining to the range difference between sensor signals and material-dependent constant. A function is then proposed to map the estimated phase derivatives to a domain that facilitates in the estimation of the source location. This is achieved by employing the Radon transform and subsequently particle swarm optimization in the new domain. The validity of all the proposed algorithms is evaluated via experiments conducted on aluminum and glass plates, for both stylus and finger taps.