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Trajectory Optimization for Unknown Maneuvering Target Tracking with Bearing-only Measurements

Yingbo Fu, Ziwen Yang, Liang Xu, Yi Guo, Shanying Zhu, Xinnping Guan

TL;DR

The paper addresses tracking an unknown maneuvering target using bearing-only measurements by marrying online Gaussian-process learning with trajectory optimization for an AUV. It introduces the GP-based Bearing-only Tracking (GBT) framework, deriving a probabilistic tracking error bound that depends on the bearing data and enabling analytic design of optimal bearing conditions. A differential-flatness-based parameterization and Unscented Transform-based planning are employed to transform the continuous-time trajectory problem into a tractable optimization, with LMC extensions enabling cross-axis motion modeling. Numerical experiments in 2D and 3D demonstrate rapid error reduction, robustness to disturbances, and superiority over several bearing-only trackers and IMM-based methods. The approach reduces reliance on predefined target dynamics while providing formal performance guarantees applicable to real-world underwater surveillance and tracking tasks.

Abstract

This paper studies trajectory optimization of an autonomous underwater vehicle (AUV) to track an unknown maneuvering target both in the 2D and 3D space. Due to the restrictions on sensing capabilities in the underwater scenario, the AUV is limited to collecting only bearing measurements to the target. A framework called {\it GP-based Bearing-only Tracking (GBT)} is proposed with integration of online learning and planning. First, a Gaussian process learning method is proposed for the AUV to handle unknown target motion, wherein pseudo linear transformation of bearing measurements is introduced to address nonlinearity of bearings. A probabilistic bearing-data-dependent bound on tracking error is then rigorously established. Based on it, optimal desired bearings that can reduce tracking uncertainty are obtained analytically. Finally, the trajectory optimization problem is formulated and transformed into an easily solved one with parametric transformation. Numerical examples and comparison with existing methods verify the feasibility and superior performance of our proposed framework.

Trajectory Optimization for Unknown Maneuvering Target Tracking with Bearing-only Measurements

TL;DR

The paper addresses tracking an unknown maneuvering target using bearing-only measurements by marrying online Gaussian-process learning with trajectory optimization for an AUV. It introduces the GP-based Bearing-only Tracking (GBT) framework, deriving a probabilistic tracking error bound that depends on the bearing data and enabling analytic design of optimal bearing conditions. A differential-flatness-based parameterization and Unscented Transform-based planning are employed to transform the continuous-time trajectory problem into a tractable optimization, with LMC extensions enabling cross-axis motion modeling. Numerical experiments in 2D and 3D demonstrate rapid error reduction, robustness to disturbances, and superiority over several bearing-only trackers and IMM-based methods. The approach reduces reliance on predefined target dynamics while providing formal performance guarantees applicable to real-world underwater surveillance and tracking tasks.

Abstract

This paper studies trajectory optimization of an autonomous underwater vehicle (AUV) to track an unknown maneuvering target both in the 2D and 3D space. Due to the restrictions on sensing capabilities in the underwater scenario, the AUV is limited to collecting only bearing measurements to the target. A framework called {\it GP-based Bearing-only Tracking (GBT)} is proposed with integration of online learning and planning. First, a Gaussian process learning method is proposed for the AUV to handle unknown target motion, wherein pseudo linear transformation of bearing measurements is introduced to address nonlinearity of bearings. A probabilistic bearing-data-dependent bound on tracking error is then rigorously established. Based on it, optimal desired bearings that can reduce tracking uncertainty are obtained analytically. Finally, the trajectory optimization problem is formulated and transformed into an easily solved one with parametric transformation. Numerical examples and comparison with existing methods verify the feasibility and superior performance of our proposed framework.

Paper Structure

This paper contains 23 sections, 3 theorems, 45 equations, 9 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. System Models
  4. Bearing Measurement Model
  5. Objectives
  6. Learning-based Bearing-only Target Tracking
  7. GP Learning for Unknown Target Tracking
  8. Probabilistic Bearing-data-dependent Tracking Error Bound
  9. Extension to Linear Model of Coregionalization
  10. Trajectory Optimization for Unknown Target Tracking
  11. Trajectory Optimization Problem
  12. Differential-flatness-based Parameterization
  13. Numerical Validation
  14. Simulation Setup
  15. Effectiveness of the Proposed GBT
  16. ...and 8 more sections

Key Result

Theorem 1

Consider the unknown target position $P_T(t)$ in the $d$-dimensional space ($d=2$ or $d=3$) and a prior Gaussian process $\mathcal{G} \mathcal{P}(\boldsymbol{0}_d, \boldsymbol{k}(t, t'))$ satisfying Assumption ass2. For a finite set $\mathcal{T}_{k}$ to achieve target tracking, the posterior mean fu holds for $\delta \in (0, 1]$, where $\beta(\delta) = \sqrt{\chi^2_{d, 1-\frac{\delta}{|\mathcal{T}

Figures (9)

  • Figure 1: Schematic framework of GBT.
  • Figure 2: Illustrations of the problem statement.
  • Figure 3: Optimal desired bearings.
  • Figure 4: Simulation results.
  • Figure 5: Comparison of results with different AUV motion modes.
  • ...and 4 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Theorem 2
  • Remark 4
  • Remark 5
  • Lemma 1