Parameter Estimation for Model-Based Sensing of Magneto-Mechanical Resonators

Abstract

Magneto-mechanical resonators (MMRs) represent a recently proposed type of passive sensor that enables the estimation of its pose as well as sensing other parameters in its environment. The working principle of MMRs entails an excitation of the sensors by oscillating magnetic fields, followed by a readout process facilitated by inductive receiver coils. The sensing technology relies on real-time parameter estimation. This encompasses the solution of a nonlinear inverse problem, with the induced signals and a suitable forward model as inputs. The aim of this paper is twofold: first, to introduce a reference model and simplified models for the MMR dynamics and inductive readout, and second, to provide robust and real-time capable methods to estimate the model parameters. The effectiveness of the presented methods is evaluated in terms of their real-time potential, precision, and accuracy. All presented methods demonstrate the capacity to estimate the measured signal, with the simplified methods reducing the corresponding parameter estimation time by up to two orders of magnitude at the expense of less than 4 % deviation for large maximum deflection angles.

Figures (15)

• Figure 1: General principle of the MMR sensing system. Left: The MMR is composed of two permanent magnets, within a local $uvw$-coordinate system. The lower magnet (stator) is attached to the housing, while the upper one (rotor) is attached to a thin filament shown in black. In the upper left the magnets are in equilibrium alignment, while in the top view of the rotor, the rotor's magnetic moment is displaced by the deflection angle $\varphi$ from its equilibrium position. Center: The schematic representation shows the cyclical flow of measurements and a model experimental configuration for a sensing application. A control loop facilitates continuous operation by establishing control parameters (CP) for the subsequent excitation. The estimation of the sensing parameters (SP) is conducted independently from the control loop on the receive signal. The experimental configuration is equipped with an MMR sensor, a sensing system comprising an inductive coil array, and a real-time signal processing platform. Right: An exemplary signal measured by a single coil and the temporal division of the measurement are illustrated. Each frame is divided into an excitation (TX) and receive (RX) window, during which the respective operations are performed.
• Figure 2: A typical voltage signal detected at the MMR sensing platform. In the time domain, the oscillating nature of the signal is evident in the shorter time frame on the left-hand side. As demonstrated in the time-frequency representation (i.e., the spectrogram $\lvert S_w(t, \tilde{\omega})\rvert^2$ in the center), the signal's multi-component nature is apparent. In analyzing the lowest frequency component, the instantaneous amplitude $\alpha_1(t)$ and frequency $\omega_1(t)$, shown on the right-hand side, are obtained. The former characterizes the signal's decay over time, whereas the latter describes a subtle frequency shift undergone by the signal.
• Figure 3: Basic principle of signal generation. Top: Within the receive window, the magnetic moment of the rotor oscillates freely. Its deflection angle $\varphi$ from its equilibrium position $\varphi = 0$ and corresponding orientation are shown. Center: The movement causes two fundamental oscillation frequencies of the magnetic moment in the $uvw$-coordinate system: the $v$-component, which oscillates at the same frequency around zero, and the $u$-component, which oscillates with twice the frequency around a nonzero offset. Bottom: If we align a unit sensitivity sensing system to pick up the time derivative of each of the individual components, we would observe $\vartheta_u$ and $\vartheta_v$, respectively. A typical signal will be a superposition of these, such as $s$, shown in the same plot.
• Figure 4: Overview of the models and methods. The abbreviations T (time) and F (time-frequency) indicate the corresponding domain for the parameter estimation method.
• Figure 5: Measurement setup, MMR, and exemplary signals for the two experiment types (un-/controlled). Left: 3D coil arrangement in Helmholtz configuration with the global coordinate system (top) and an image of MMR L and MMR S (bottom). Right: Exemplary filtered signals of MMR S measured with the $x$-coil. Orange dots indicate the instantaneous frequency acquired from the signal demodulation, the orange line during TX the excitation frequency of the current frame. The vertical dashed lines correspond to the cut-off times ranging from $T_\text{30}$ to $T_\text{100}$ for Experiment 1. The index denotes the proportion of the dataset considered, ranging from 30100, which corresponds to the time interval over which the signal amplitude decreased by 70%. Note that in Experiment 1 the receive window was not divided into RX1 and RX2 during the measurement, however, in the analysis afterwards. For Experiment 2, a vertical dashed line indicates the RX1 and RX2 division for the control loop during measurement.