[Paper Review] Mobile Sensor Path Planning for Kalman Filter Spatiotemporal Estimation

Jiazhong Mei, Steven L. Brunton+1 more|arXiv (Cornell University)|Dec 16, 2022

Target Tracking and Data Fusion in Sensor Networks4 citations

TL;DR

This paper proposes a greedy path planning algorithm for mobile sensors that optimizes Kalman filter performance by minimizing the condition number of the discrete observability matrix, leveraging dynamic trajectories to achieve estimation accuracy equivalent to many stationary sensors. The method improves convergence speed and limiting error, especially for systems with localized spatial structures.

ABSTRACT

The estimation of spatiotemporal data from limited sensor measurements is a required task across many scientific disciplines. The sensor selection problem, which aims to optimize the placement of sensors, leverages innovations in greedy algorithms and low-rank subspace projection to provide model-free, data-driven estimates. Alternatively, Kalman filter estimation balances model-based information and sparsely observed measurements to collectively make an estimation, with many related optimization algorithms developed for selecting optimal sensors. The majority of methods have been developed for stationary sensors, with relatively limited work estimating spatiotemporal data using mobile sensors that leverage both Kalman filtering and low-rank features. We show that mobile sensing along dynamic trajectories can achieve the equivalent performance of a larger number of stationary sensors, with performance gains related to three distinct timescales: (i) the timescale of the spatio-temporal dynamics, (ii) the velocity of the sensors, and (iii) the rate of sampling. Taken together, these timescales strongly influence how well-conditioned the estimation task is. Mobile sensing is particularly effective for spatio-temporal data that contain spatially localized structures, whose features are captured along dynamic trajectories. We draw connections between the Kalman filter performance and the observability of the state space model, and propose a greedy path planning algorithm based on minimizing the condition number of the observability matrix. Through a series of examples of increasing complexity, we show that mobile sensing improves Kalman filter performance in terms of better limiting estimation and faster convergence.

Motivation & Objective

• To address the limitation of stationary sensor placement in spatiotemporal estimation by enabling mobile sensing with dynamic trajectories.
• To improve Kalman filter estimation performance—specifically limiting error and convergence rate—using mobile sensors instead of fixed sensor networks.
• To formulate sensor path planning as a submatrix selection problem based on the condition number of the discrete observability matrix.
• To develop a time-forwarding greedy algorithm that selects optimal sensor locations along a periodic trajectory using QR with column pivoting and GappyPOD principles.
• To investigate the influence of three key timescales—spatiotemporal dynamics, sensor velocity, and sampling rate—on estimation performance.

Proposed method

• The method formulates sensor path planning as minimizing the condition number of the discrete observability matrix to enhance Kalman filter observability and estimation stability.
• A time-forwarding greedy algorithm is proposed to iteratively select sensor positions along a periodic trajectory, using the same selection rules as QR with column pivoting (QRcp) and GappyPOD+E.
• The algorithm operates on a candidate subset of spatial locations, ensuring computational efficiency while maintaining near-optimal performance.
• The approach integrates Kalman filtering with low-rank subspace projections to estimate spatiotemporal dynamics from sparse, mobile sensor measurements.
• The method accounts for three critical timescales: the dynamics' Nyquist rate, sampling rate, and sensor velocity, which jointly determine estimation conditioning.
• The path planning is constrained to periodic trajectories to enable predictable, repeatable sensor motion and stable estimation over time.

Experimental results

Research questions

• RQ1 Can mobile sensor trajectories achieve Kalman filter estimation performance equivalent to a larger number of stationary sensors?
• RQ2 How do the three timescales—spatiotemporal dynamics, sensor velocity, and sampling rate—influence the conditioning and performance of Kalman filter estimation?
• RQ3 To what extent does minimizing the condition number of the observability matrix improve estimation accuracy and convergence speed?
• RQ4 Can a greedy path planning algorithm based on observability metrics outperform static sensor placement in complex, non-linear spatiotemporal systems?
• RQ5 How does mobile sensing enhance the capture of localized spatial features in a linearized dynamical model?

Key findings

• Mobile sensing along optimized dynamic trajectories achieves estimation performance equivalent to a larger number of stationary sensors, particularly for systems with localized spatial structures.
• The estimation error converges faster with mobile sensors compared to stationary configurations, enabling quicker adaptation to changing dynamics.
• The limiting squared error of the Kalman filter is significantly reduced when sensor paths are planned to minimize the condition number of the observability matrix.
• The method demonstrates improved performance in both synthetic and real-world experiments, including a 14-week periodic trajectory for sea surface temperature estimation.
• The convergence rate of the estimation error is notably faster with mobile sensors, with error remaining low even after system changes, as shown in the first two years of SST data.
• The proposed greedy algorithm effectively captures local features in the spatiotemporal field, outperforming static placement in terms of both accuracy and adaptability.