Homotopic information gain for sparse active target tracking
Authors: Jennifer Wakulicz, Ki Myung Brian Lee, Teresa Vidal-Calleja, Robert Fitch
Organizations: Australian Centre for Robotics, School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, Camperdown, 2006, NSW, Australia · Department of Electrical and Computer Engineering, UC San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA · School of Mechanical and Mechatronics Engineering, University of Technology Sydney, Ultimo, 2007, NSW, Australia
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.
Figures & tables
Fig. 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.
Fig. 2 : Illustrations of homotopy classes and h -signature calculation. [ 9 ]
Fig. 3 : Heatcubes of homotopic information gain at different sensing locations and times ( z -axis). Red line shows the test trajectory travelling upwards in time.
Fig. 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.
Dataset
Simulated
THÖR
ATC
O
3
4
5
6
7
3
5
H
4
6
7
11
12
6
13
TABLE I : The number of obstacles O and homotopy classes H present in each dataset used for evaluation.
Fig. 5 : Results for active target tracking on the simulated pedestrian dataset with three obstacles.
Fig. 6 : Results for active target tracking on the simulated pedestrian datasets increasing number of obstacles.
Fig. 7 : Results for active target tracking on the ATC mall pedestrian dataset with five obstacles.
Fig. 8 : Results for active target tracking on the augmented THÖR pedestrian dataset with three obstacles.
Fig. 9 : Total tracking experiment length for the real datasets.
Fig. 10 : Snapshots of predictions output by a homotopic GMM during an active target tracking experiment using homotopic information gain 10(a) - 10(e) and metric information gain 10(f) - 10(j) . Black solid path is the elapsed ground truth test trajectory at the given time. Dotted path indicates future ground truth trajectory remaining. Coloured crosses denote all measurements taken by the sensing robot up to time t , where colours serve to differentiate between approaches (orange for the metric approach, blue for the homotopic approach). Shaded regions show the variance of the mean shown in solid colour, where colours differentiate GMM components. Transparency is proportional to component weight.
Fig. 11 : Distributions of displacement errors averaged over the full trajectory length for each dataset with (labelled with ∅ ) and without the empty h -signature.
Fig. 12 : Distributions of the number of measurements taken per tracking experiment for each dataset with (labelled with ∅ ) and without the empty h -signature.
Datasets without h=()
Sim-3
Sim-4
Sim-5
Sim-6
Sim-7
ATC
THÖR
Ours
100
100
100
99
91
81
100
Metric
100
100
100
99
98
95
100
Datasets with h=()
Sim-3 ∅
Sim-4 ∅
Sim-5 ∅
Sim-6 ∅
Sim-7 ∅
ATC ∅
THÖR ∅
Ours
96.67
100
98.9
90
91
85
81
TABLE II : Success rates (%) for our homotopic approach versus the metric approach before and after the empty h -signature is introduced.
Informative path planning chooses robot routes and sensing actions that gather useful information while respecting travel, energy, time, and safety constraints.
We study trajectory optimization in mobile sensing systems that must identify which member of a finite candidate set is the true target, while maintaining reachability to all potential candidate targets, under resource constraints. Deferred-Decision Trajectory Optimization (DDTO) addresses this setting by computing trajectories that reach individual targets but remain coincident for as long as possible before separating toward different targets. We propose Active-Sensing DDTO (AS-DDTO), which extends DDTO by adding a trajectory-dependent information-acquisition term to the planning objective. The resulting planner maintains reachability to candidate targets while biasing the coincident portion of the trajectories toward regions that enable earlier target identification. The framework supports Bayesian updates and conformal candidate-set updates for distance-dependent sensing. We derive a mixed-integer conic reformulation and provide guarantees on recursive feasibility, belief concentration, and fixed-time coverage for the raw conformal candidate set. Numerical simulations show improved target identification compared with standard DDTO under distance-dependent sensing uncertainty and limited sensing budget.
Farbod Siahkali, Mengxue Hou, Vijay Gupta
Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47906, USA · Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
Existing active reconstruction systems with Gaussian-splatting maps select observations greedily, optimizing a single next-best-view (NBV) at each step and connecting the chosen views by short-horizon path planning. This greedy decoupling disregards the global structure of scene information, producing inefficient trajectories that waste sensing capacity in transit between selected views. In this work, we study active reconstruction as an ergodic coverage problem: the time-averaged spatial statistics of the sensor trajectory should match a target information distribution induced by the current map. Our approach derives this target distribution online from uncertainty and visibility, and calculates ergodic trajectories via a kernel-ergodic horizon planner with gradient flow and footprint depletion, closing the loop between mapping and trajectory optimization. We thoroughly evaluate TRACE on the Replica dataset against the Next-Best-View (NBV) baselines, improving PSNR by 1.5 dB. Code: https://github.com/spikelab-jhu/trace-active-reconstruction.
Ziyue Zheng, Linli Shi, Bingkun He +2
1Johns Hopkins University, USA · University of Pennsylvania, USA
Offline reference trajectories for active target tracking are needed both for building multi-modal tracking datasets and for benchmarking online tracking planners under repeatable conditions. We present Track A star (TA star), an offline search-based trajectory planner that targets the visibility-aware target tracking objective on a discretized four-dimensional spatio-temporal grid (x, y, z, t). TA star combines a layered Directed Acyclic Graph (DAG) search with three engineering optimizations: cross-time obstacle distance caching against a Bounding Volume Hierarchy (BVH), per-layer beam pruning, and a configurable multi-ray visibility evaluator. TA star employs a beam-pruned heuristic search on this discrete graph to efficiently find high-quality tracking trajectories. While it trades strict theoretical optimality for practical scalability, our empirical results demonstrate robust, near-baseline visibility performance at a fraction of the computational cost. On a 1000-scenario stress test across eight CARLA Optimized maps, TA star converges on all scenarios and completes in 45 s using 32 workers; on a 248-scenario controlled comparison against an unoptimized priority-queue A star baseline (BinaryHeap implementation) under identical scenario inputs and a 5 x 10^6 expansion cap, TA star reduces mean planning time by 23.0x and worst-case planning time by 11.8x, while raising convergence from 56.9% to 100%. On the n=141 baseline-converged subset, TA star changes average visibility by only -0.15 percentage points (pp), with no scenario exceeding a 5 pp drop. We position TA star as a practical offline reference planner under these specific conditions, with limitations and failure cases discussed for environments such as Town07 dense vegetation.