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Homotopic information gain for sparse active target tracking

Jennifer Wakulicz, Ki Myung Brian Lee, Teresa Vidal-Calleja, Robert Fitch

TL;DR

The paper addresses active target tracking by planning in a discrete homotopic belief space rather than the full low level trajectory, leveraging homotopic information gain as a sparse, high level information objective. It formalizes a homotopic GMM to model multi modal trajectories, derives a tractable KL based gain over partial h signatures, and proves a bound to metric information gain. An online planning framework using heatcube based rewards and Monte Carlo tree search solves the orienteering problem with time windows and updates beliefs online. Empirical evaluation on simulated and real pedestrian datasets demonstrates that the approach achieves competitive trajectory accuracy with significantly fewer measurements and reduced computation relative to metric information planning, validating the practical efficiency and effectiveness of planning over homotopic beliefs.

Abstract

The problem of planning sensing trajectories for a mobile robot to collect observations of a target and predict its future trajectory is known as active target tracking. Enabled by probabilistic motion models, one may solve this problem by exploring the belief space of all trajectory predictions given future sensing actions to maximise information gain. However, for multi-modal motion models the notion of information gain is often ill-defined. This paper proposes a planning approach designed around maximising information regarding the target's homotopy class, or high-level motion. We introduce homotopic information gain, a measure of the expected high-level trajectory information given by a measurement. We show that homotopic information gain is a lower bound for metric or low-level information gain, and is as sparsely distributed in the environment as obstacles are. Planning sensing trajectories to maximise homotopic information results in highly accurate trajectory estimates with fewer measurements than a metric information approach, as supported by our empirical evaluation on real and simulated pedestrian data.

Homotopic information gain for sparse active target tracking

TL;DR

The paper addresses active target tracking by planning in a discrete homotopic belief space rather than the full low level trajectory, leveraging homotopic information gain as a sparse, high level information objective. It formalizes a homotopic GMM to model multi modal trajectories, derives a tractable KL based gain over partial h signatures, and proves a bound to metric information gain. An online planning framework using heatcube based rewards and Monte Carlo tree search solves the orienteering problem with time windows and updates beliefs online. Empirical evaluation on simulated and real pedestrian datasets demonstrates that the approach achieves competitive trajectory accuracy with significantly fewer measurements and reduced computation relative to metric information planning, validating the practical efficiency and effectiveness of planning over homotopic beliefs.

Abstract

The problem of planning sensing trajectories for a mobile robot to collect observations of a target and predict its future trajectory is known as active target tracking. Enabled by probabilistic motion models, one may solve this problem by exploring the belief space of all trajectory predictions given future sensing actions to maximise information gain. However, for multi-modal motion models the notion of information gain is often ill-defined. This paper proposes a planning approach designed around maximising information regarding the target's homotopy class, or high-level motion. We introduce homotopic information gain, a measure of the expected high-level trajectory information given by a measurement. We show that homotopic information gain is a lower bound for metric or low-level information gain, and is as sparsely distributed in the environment as obstacles are. Planning sensing trajectories to maximise homotopic information results in highly accurate trajectory estimates with fewer measurements than a metric information approach, as supported by our empirical evaluation on real and simulated pedestrian data.
Paper Structure (27 sections, 1 theorem, 21 equations, 12 figures, 2 tables, 1 algorithm)

This paper contains 27 sections, 1 theorem, 21 equations, 12 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Homotopy classes in robotics
  4. Informative path planning for active target tracking
  5. Background and problem formulation
  6. The $h$-signature as a homotopy invariant
  7. The homotopic belief space
  8. Problem formulation
  9. Homotopic information gain for active target tracking
  10. Gaussian mixture models for trajectory belief modelling
  11. Homotopic information gain
  12. Complexity analysis
  13. Theoretical analysis
  14. Planning framework
  15. Active target tracking as an orienteering problem
  16. ...and 12 more sections

Key Result

Theorem 1

Homotopic information gain is a lower bound for low-level, metric information gain. That is, where $\mathcal{I}(\mathbf{Y};\rho_t,\mathbf{z}_{1:t})$ is the expected low-level information gained from measurements $\mathbf{z}_{1:t}$ and partial $h$-signature $\rho_t$ regarding the full trajectory $Y$.

Figures (12)

  • Figure 1: Heatmaps showing visitation frequency at sensing locations for the active target tracking problem on the ATC dataset. A conventional tracking approach (left) is compared to the sparse homotopic approach proposed in this paper (right). With our proposed approach, the sensing robot is only concerned with how a target manoeuvres around obstacles, reducing planning and sensing resources while still producing accurate estimates of the target trajectory.
  • Figure 2: Illustrations of homotopy classes and $h$-signature calculation. wakulicz_topological_2023
  • Figure 3: Heatcubes of homotopic information gain at different sensing locations and times ($z$-axis). Red line shows the test trajectory travelling upwards in time.
  • Figure 4: Left: one of the environments used to simulate pedestrian trajectories, here with three obstacles. Middle: The real pedestrian ATC mall dataset with five obstacles present. Right: the THÖR dataset with both real and augmented pedestrian trajectories with three obstacles present.
  • Figure 5: Results for active target tracking on the simulated pedestrian dataset with three obstacles.
  • ...and 7 more figures

Theorems & Definitions (1)

  • Theorem 1