Sensor Placement for Flapping Wing Model Using Stochastic Observability Gramians

TL;DR

The paper addresses sensor placement for nonlinear, stochastic systems by developing a stochastic empirical Gramian framework to quantify observability under process noise. It proposes a Monte Carlo PSO-based workflow to optimize sensor locations with objectives based on observability metrics such as the $n$-th root of the determinant $([\det(W_o)]^{1/n})$ and the unobservability index, using two case studies: a low-dimensional UAV wind-tracking model and a high-dimensional bioinspired flapping-wing FE model with neural-encoding measurements. Key contributions include introducing the stochastic empirical Gramian for nonlinear systems, analyzing how noise can reveal observability, and delivering a practical sensor-placement pipeline that balances output energy and estimation conditioning. The work advances sensor deployment strategies under uncertainty and informs filter design in complex, high-dimensional systems, with potential extensions to more realistic wing models and neural decoding schemes.

Abstract

Systems in nature are stochastic as well as nonlinear. In traditional applications, engineered filters aim to minimize the stochastic effects caused by process and measurement noise. Conversely, a previous study showed that the process noise can reveal the observability of a system that was initially categorized as unobservable when deterministic tools were used. In this paper, we develop a stochastic framework to explore observability analysis and sensor placement. This framework allows for direct studies of the effects of stochasticity on optimal sensor placement and selection to improve filter error covariance. Numerical results are presented for sensor selection that optimizes stochastic empirical observability in a bioinspired setting.

Figures (5)

• Figure 1: (a) Details of structural model and boundary conditions for flapping wing model in COMSOL, (b) meshing of the wing plate and normal strain distribution on the top surface of wing.
• Figure 2: The probabilistic firing model of a neuron where the input is the strain information, $\epsilon$, and the output is the probability of firing, $P_\text{fire}$Boyacioglu2021.
• Figure 3: The change of two unobservability metrics as the noise level increases. Here, horizontal lines with color indicate the median value, and the plus sign denotes the mean. $I_2$ is the $2 \times 2$ identity matrix.
• Figure 4: The average distribution of two unobservability measures ($K=40$). Yellow regions are more observable than the dark blue ones.
• Figure 5: (top) The change of the observability costs by the number of sensors with fourth-degree polynomial fitting, (bottom) The optimal neural-inspired sensor placement for $r=12$. The figures use the same legend.