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Distributed Estimation for a 3-D Moving Target in Quaternion Space with Unknown Correlation

Yizhi Zhou, Xufan Liu, Xuan Wang

TL;DR

The paper tackles consistent distributed estimation for a 3-D moving target represented by augmented quaternion states when inter-sensor correlations are unknown. It extends Inverse Covariance Intersection (ICI) within an error-state EKF to fuse quaternion-based states across a time-varying sensor network, yielding a fully distributed framework. Key contributions include formulating quaternion-aware ICI fusion, deriving a nine-dimensional error-state fusion mechanism, and validating performance via Monte Carlo simulations that show improved consistency and near-centralized accuracy under higher communication rates. The work enables robust 3-D target tracking in non-Euclidean state spaces for distributed camera networks and similar sensor arrays.

Abstract

For distributed estimations in a sensor network, the consistency and accuracy of an estimator are greatly affected by the unknown correlations between individual estimates. An inconsistent or too conservative estimate may degrade the estimation performance and even cause divergence of the estimator. Cooperative estimation methods based on Inverse Covariance Intersection (ICI) can utilize a network of sensors to provide a consistent and tight estimate of a target. In this paper, unlike most existing ICI-based estimators that only consider two-dimensional (2-D) target state estimation in the vector space, we address this problem in a 3-D environment by extending the ICI algorithm to the augmented quaternion space. In addition, the proposed algorithm is fully distributed, as each agent only uses the local information from itself and its communication neighbors, which is also robust to a time-varying communication topology. To evaluate the performance, we test the proposed algorithm in a camera network to track the pose of a target. Extensive Monte Carlo simulations have been performed to show the effectiveness of our approach.

Distributed Estimation for a 3-D Moving Target in Quaternion Space with Unknown Correlation

TL;DR

The paper tackles consistent distributed estimation for a 3-D moving target represented by augmented quaternion states when inter-sensor correlations are unknown. It extends Inverse Covariance Intersection (ICI) within an error-state EKF to fuse quaternion-based states across a time-varying sensor network, yielding a fully distributed framework. Key contributions include formulating quaternion-aware ICI fusion, deriving a nine-dimensional error-state fusion mechanism, and validating performance via Monte Carlo simulations that show improved consistency and near-centralized accuracy under higher communication rates. The work enables robust 3-D target tracking in non-Euclidean state spaces for distributed camera networks and similar sensor arrays.

Abstract

For distributed estimations in a sensor network, the consistency and accuracy of an estimator are greatly affected by the unknown correlations between individual estimates. An inconsistent or too conservative estimate may degrade the estimation performance and even cause divergence of the estimator. Cooperative estimation methods based on Inverse Covariance Intersection (ICI) can utilize a network of sensors to provide a consistent and tight estimate of a target. In this paper, unlike most existing ICI-based estimators that only consider two-dimensional (2-D) target state estimation in the vector space, we address this problem in a 3-D environment by extending the ICI algorithm to the augmented quaternion space. In addition, the proposed algorithm is fully distributed, as each agent only uses the local information from itself and its communication neighbors, which is also robust to a time-varying communication topology. To evaluate the performance, we test the proposed algorithm in a camera network to track the pose of a target. Extensive Monte Carlo simulations have been performed to show the effectiveness of our approach.
Paper Structure (10 sections, 1 theorem, 39 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 10 sections, 1 theorem, 39 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. preliminaries and Problem Formulation
  3. Quaternion
  4. Track-to-Track Fusion for Vector Space Variables
  5. Problem Formulation: Cooperative 3-D Target Tracking with Quaternion States using Distributed Fusion
  6. Proposed Algorithm
  7. Simulations
  8. Conclusion
  9. IMU propagation model
  10. Camera measurement model

Key Result

Lemma 1

NOACK2017 Let $\mathbf{\hat{P}}_{CI}^k$ and $\mathbf{\hat{P}}_{ICI}^k$ be, respectively, the fused covariance matrices with minimal traces using CI and ICI algorithms at timestep $k$. We have $\mathbf{\hat{P}}_{CI}^k \geq \mathbf{\hat{P}}_{ICI}^k$.

Figures (5)

  • Figure 1: Cooperative tracking of a drone in 3-D environments over camera networks. The red cameras represent the blind cameras which are not able to sense the drone, while the blue ones indicate that the cameras can directly detect the drone.
  • Figure 2: Position RMSE of each camera using the proposed algorithm with different communication rates.
  • Figure 3: Orientation RMSE of each camera using the proposed algorithm with different communication rates.
  • Figure 4: Comparison between the ICI-based and CI-based approach of averaged NEES with $30\%$ communication.
  • Figure 5: Quadrotor's grountruth path and the tracking performance over 50 Monte-Carlo simulation with $30\%$ communication rate. We plot the estimated trajectories of the first 20 trails, which shows that the estimated trajectories are close to the groundtruth.

Theorems & Definitions (3)

  • Definition 1
  • Lemma 1
  • Remark 1: Implementation and Effectiveness of Algorithm \ref{['alg']} under Time-varying Graphs with Blind Agents