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Ultra Low-Power, Lightweight, Probabilistic RSS-Based Path Reconstruction: A System for Landscape-Scale Bee Tracking

Christopher J. Noroozi, Joseph L. Woodgate, Michael Mangan, Michael T. Smith

arXiv:2608.27152v1cs.LG

TL;DR

Very small, power-constrained devices need landscape-scale localization without the mass and energy demands of GNSS or measurement-heavy RSS methods. The paper combines sparse RSS bursts from rotating high-gain transmitters with probabilistic AoA inference and Gaussian-process path reconstruction, achieving approximately 15m accuracy at <180µW for a 38mg receiver and approximately 10m at <600µW with more measurements.

  • Problem

    Landscape-scale localization methods for very small devices are limited by GNSS power and mass demands, while RSS-based alternatives face range and RSS-measurement constraints.

  • Method

    The system probabilistically infers AoA from sparse RSS bursts emitted by rotating high-gain transmitters, then reconstructs movement paths with a Gaussian process and doubly stochastic variational inference.

  • Results

    Approximately 15m path reconstruction accuracy is achieved at <180µW with as few as 3 RSS measurements per AoA observation, improving to approximately 10m at <600µW with more measurements.

  • Takeaways & Limitations

    The system tracks a 38mg receiver across complex landscapes and demonstrates a use case in Bombus terrestris return-flight tracking.

  • Takeaways & Limitations

    The receiver is archival, so re-observation of the tracked subject is required to read the received data.

Abstract

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Applications in fields such as movement ecology, Internet of Things or robotics share the need for systems that localize devices that are too small and power constrained to implement GNSS (Global Navigation Satellite Systems). Alternative low-power localization methods often rely on only measurements of RSS (Received Signal Strength) to infer the AoA (Angle of Arrival) of a transmitted radio frequency signal, but are limited by range and the power demand of the large number of RSS measurements required to infer an accurate AoA. In this paper we address these issues with a novel RSS-based method for tracking ultra lightweight and low-power moving receivers across a complex landscape, achieved by using a minimal number of RSS measurements from simple rotating high-gain transmitters with a range of 300m, and applying probabilistic modelling to infer their AoA. The receiver's movement path is then modelled using a Gaussian process and reconstructed using doubly stochastic variational inference, resulting in approximately 15m accuracy tracking of receivers weighing 38mg (including power source) over a scalable landscape range while consuming less than 180uW, increased to approximately 10m accuracy at less than 600uW by taking more RSS measurements. We anticipate that this method will support fields such as the behavioural study of flying insect species, which we demonstrate by applying the system to track Bombus terrestris nest return flights.

I. INTRODUCTION

Ultra-lightweight landscape tracking requires localization methods that avoid GNSS-scale power and mass while overcoming the range and measurement costs of existing RSS approaches. The paper introduces probabilistic RSS-based AoA inference and Gaussian-process path reconstruction using sparse measurements from rotating high-gain transmitters.

  • GNSS units require substantial hardware and battery capacity, with the cited U-Blox UBX-M10150-CC requiring 8mW to function.
  • Landscape-scale insect tracking must use tags weighing < 70mg and consume minimal power because heavier tags impose energetic costs.
  • Existing RSS-based methods are constrained by either too many measurements for ultra-low-power receivers or insufficient range and cumbersome antennas at landscape scale.
  • A 38mg receiver achieves approximately 15m mean absolute error while consuming <180µW, improving to approximately 10m at <600µW with more RSS measurements.
  • The system was validated in ground-truthed localization tasks and demonstrated by tracking Bombus terrestris return flights.
  • The proposed system infers AoA from bursts of as few as 3 RSS measurements received from rotating high-gain transmitters with a 300m range.
  • The receiver’s path is modelled with a Gaussian process and reconstructed through doubly stochastic variational inference from uncertain, non-concurrent AoA predictions.

III. HARDWARE

The system combines a 38mg archival receiver with rotating high-gain transmitters to collect sparse RSS measurements for landscape-scale AoA-based path reconstruction.

  • Receiver: The 38mg archival receiver logs packets and RSS from rotating transmitters at known stationary landscape positions.The stored data are later used for AoA inference.
  • Receiver: The receiver uses a DA14531 BLE system-on-chip, crystal oscillator, capacitors, monopole antenna, and a 24mg supercapacitor on a flexible polyimide PCB.The supercapacitor provides the receiver’s power source.
  • Receiver: A 7ms receiving period captures packets from transmitters in range before the receiver enters hibernation mode.At 5mA receiving power, the supercapacitor supports over 500 receiving duty cycles.
  • Receiver: The receiver varies burst duration and receiving intervals to trade power consumption and device lifespan against AoA and path-reconstruction accuracy.These parameters determine k, the number of RSS measurements per burst.
  • Transmitters: The rotating transmitters use 14dBi Yagi-Uda antennas on tripods and broadcast directed BLE packets for receiver localization.Packets encode transmitter rotation angle, transmission time, and transmitter identity; reception is possible up to 300m.
  • AoA inference: AoA inference seeks the receiver’s unknown angle θ from k RSS measurements y collected at known transmitter orientations γ.The method presents three approaches for inferring AoA from limited RSS measurements.

A. Full Probability Distribution Method

The full probability distribution method compares observed RSS changes with a calibrated antenna radiation pattern to infer a distribution over the receiver’s AoA. It retains prediction uncertainty for subsequent path reconstruction.

  • AoA model: The antenna’s anisotropic radiation pattern creates distinctive RSS changes during rotation, allowing θ to be inferred from observations y and known orientations γ.The informative pattern occurs around the antenna’s approximately 40° H-plane HPBW.
  • AoA model: The model assumes burst measurements share constant attenuation a, which can arise from distance, obstructions, or multipath interference.The constant-attenuation assumption is supported by the short time required for the narrow main lobe to pass the receiver.
  • AoA model: The method empirically profiles the transmitter’s antenna radiation pattern before using it for AoA inference.The profile supplies the expected signal-strength pattern against which RSS observations are compared.
  • Inference: For each candidate θ on a uniform 0°–360° grid, the method predicts signal strengths from θ−γ_i, compares mean-centred predictions with observations, and normalizes the resulting probabilities.This produces p(θ|y, γ), the predicted distribution of the receiver’s angle relative to North.
  • Path reconstruction: The full AoA probability distribution is used as an observation for path reconstruction, preserving uncertainty that may still inform the inferred movement path.The approach therefore passes more than a single point estimate to the path model.

B. Rejection Sampling Method

Rejection sampling estimates AoA distributions by matching mean-centred RSS patterns to training data, yielding confident angles near the antenna’s peak but uncertainty elsewhere. The section also situates this method within probabilistic path reconstruction using Gaussian processes and an integrated EQ kernel.

  • Rejection Sampling Method: Rejection sampling approximates p(θ|y, γ) by selecting training rows whose mean-centred RSS patterns match the observed pattern.The associated training angles form empirical samples of the AoA distribution.
  • Rejection Sampling Method: Five RSS packets spaced from −1s to +1s can estimate p(θ|y, γ) at each observation time.The sampling window spans enough time for the transmitter to rotate one radian on either side of the prediction time.
  • Rejection Sampling Method: RSS measurements around the antenna’s HPBW produce samples densely centred on the true θ, whereas measurements elsewhere yield many likely angles.The figure demonstrates confident inference near the RSS peak and high uncertainty away from it.
  • Rejection Sampling Method: Single-angle reconstruction can retain only low-standard-deviation predictions and combine them using a circular mean.This provides a way to avoid using the full AoA distribution in later path inference.
  • Rejection Sampling Method: Peak-based AoA estimation is noise-sensitive because short bursts or too few measurements may miss RSS values around the antenna’s HPBW.The method smooths RSS across rotation angles before selecting the peak.
  • Choice of Model: The receiver path is inferred probabilistically with Gaussian-process priors, using an integrated EQ kernel whose uncertainty grows over time without observations.This kernel is preferred for subjects that do not reliably return to a nominal position when unobserved.

B. Variational Inference

The variational-inference procedure approximates the unknown posterior over paths with a computationally efficient surrogate distribution. It optimizes an inducing-variable ELBO using sampled paths, likelihood estimates, and a Gaussian KL-divergence term.

  • Variational Inference: Variational inference approximates the unknown posterior p(F|Y) with a computationally inexpensive surrogate posterior q(F).The method is presented as cheaper for inference than Markov Chain Monte Carlo or Expectation Propagation.
  • Variational Inference: Maximizing the ELBO is equivalent to minimizing KL divergence because the marginal likelihood p(Y) is constant with respect to q(F).The ELBO consists of a data-fitting term and a prior term.
  • Variational Inference: Inducing variables u selected at evenly spaced time inputs Z are used to represent the path F and augment the ELBO.The approach assumes u sufficiently determines F so that p(F|u,Y)=p(F|u).
  • Variational Inference: Doubly stochastic variational inference samples paths from q(F), averages their observation log likelihoods, and optimizes the variational parameters.The variational parameters are the mean m and lower diagonal matrix R defining covariance RR⊤.
  • Variational Inference: The Gaussian prior and surrogate posterior make the KL divergence term analytically tractable.This closed-form term contributes to the ELBO optimization.
  • Variational Inference: Automatic differentiation updates m and R until the surrogate distribution approximates the true posterior and yields location distributions over time.JAX computes the gradients used for these updates.

C. Likelihood

The likelihood formulation converts AoA observations into path-consistency terms for variational inference. It supports both full AoA distributions and single-angle observations, with interpolation enabling differentiable evaluation.

  • Likelihood: The likelihood implementation differs for full AoA distributions and observations represented by a single angle.Full distributions use p(θ|y, γ), while single-angle methods require an alternative likelihood.
  • Likelihood: For a sampled path Fi, the full-distribution likelihood evaluates the probability of the angle ϕ from the transmitter to the sampled receiver location.The transmitter location is subtracted from the sampled path point before computing ϕ relative to North.
  • Likelihood: Linear interpolation between precomputed log p(θ|y, γ) values provides normalized, fast, autodifferentiable likelihood evaluations for arbitrary ϕ.For example, an angle between grid points is weighted by its distances to the neighboring values.
  • Likelihood: Single-angle observations are converted into two-dimensional bearing lines so likelihood depends on a predicted location’s perpendicular distance from the line.The line uses cos θ and sin θ to form a unit vector extending from the transmitter.
  • Likelihood: A Gaussian likelihood over distance from the bearing line evaluates how compatible each sampled path location is with the single-angle observation.Locations farther from the line receive lower assumed likelihood.
  • Likelihood: The likelihood normalization is approximately valid for the problem rather than exactly valid for the stated likelihood function.The derivation is reported as being provided in supplementary material.

VI. EXPERIMENTAL DESIGN

The experiments evaluate AoA accuracy under different RSS burst sizes and assess path reconstruction in locations distinct from the training site. The design also includes ground-truthed tests and a movement-ecology deployment.

  • VI. EXPERIMENTAL DESIGN: Training data comprises 10 minutes of continuous packet and RSS scanning from a single rotating transmitter with the receiver 30m due North and in clear line of sight.The receiver used an external power supply during training-data collection.
  • VI. EXPERIMENTAL DESIGN: All experiments use locations different from the training site to test generalization to unknown landscapes.This separates training-data collection from evaluation environments.
  • VI. EXPERIMENTAL DESIGN: The experiments assess AoA accuracy for three inference approaches while varying k, the number of RSS packets per burst.The goal is to examine how far k can be reduced to save receiver power.
  • VI. EXPERIMENTAL DESIGN: The AoA test places the receiver 100m from the transmitter at a known angle and collects 100 bursts containing 200 packets each.Packets are captured in 2000ms bursts at the highest sampling rate available on the DA14531.
  • VI. EXPERIMENTAL DESIGN: AoA error is measured as the angular difference from the known true angle, repeated 1000 times to calculate MAE and its standard deviation.The three inference methods are compared using the mode for the full probability distribution method.
  • VI. EXPERIMENTAL DESIGN: The RSS noise standard deviation is fixed at σ=6dB based on empirical observations.This assumption is used in the experimental tests.

C. Path Reconstruction Experiment

The experiments evaluate how RSS sampling and AoA inference affect path reconstruction under realistic landscape conditions and constrained power budgets. Ground-truthed and synthetic paths test accuracy, sampling trade-offs, and a bee-tracking deployment.

  • Path Reconstruction Experiment: Six ground-truthed paths in a complex 200m × 200m park compare reconstruction accuracy across AoA methods and RSS samples per burst.The paths included five complex trajectories and one straight-line trajectory, with four transmitters placed approximately 125m apart.
  • Path Reconstruction Experiment: The receiver was externally powered during dense data collection, then uniformly subsampled across bursts to compare low- and high-k performance on identical measurements.This preserves the same movement paths and RSS attenuation while varying the number of measurements used per burst.
  • Path Reconstruction Experiment: A Gaussian-process path reconstructed by variational inference is evaluated point-for-point against GNSS, using MAE and its standard deviation across paths.The algorithm uses 60 inducing points per axis and a kernel choice reflecting smooth movement and walking-consistent velocity.
  • Duty Cycle Parameter Investigation: A synthetic sine-wave path tests whether many uncertain AoA observations or fewer accurate observations better use a fixed RSS budget.The experiment simulates 100 RSS measurements distributed across bursts with different k values and compares AoA and path accuracy.
  • Bee Tracking Field Trial: A field trial applies the system to Bombus terrestris return flights using four transmitters around a nest and k = 10 RSS measurements every 10 seconds.The 38mg receiver was attached to a bee, which was released approximately 100m from the nest before its reconstructed return path was retrieved.

VII. RESULTS

The results show that probabilistic AoA inference supports path reconstruction with low-power sampling, while very small RSS bursts substantially reduce angular accuracy. The full probability distribution method is especially robust when measurements are sparse, reaching approximately 15m path accuracy at low power.

  • A. AoA Inference Accuracy: At high k, full-distribution and peak-RSS methods achieve MAE below 2° with approximately 1° standard deviation, whereas rejection sampling reaches 5° MAE with approximately 40° standard deviation.Rejection sampling becomes inconsistent because few training points match all observed RSS values, while more training data would increase computational complexity exponentially.
  • A. AoA Inference Accuracy: Below 15 RSS measurements, all AoA methods begin to lose accuracy; at k = 5, full-distribution and rejection sampling reach approximately 50° MAE.At these low values, all methods produce highly uncertain AoA predictions with standard deviation above approximately 50°.
  • B. Path Reconstruction Accuracy: The GNSS ground-truth uncertainty and transmitter-position uncertainty establish an estimated lower bound on achievable path MAE in the evaluation.Figure 11 represents variability around MAE with one-standard-deviation shading and marks this estimated lower bound in gray.
  • B. Path Reconstruction Accuracy: The full probability distribution method reconstructs paths with approximately 15m MAE using as few as 3 RSS samples per burst, while high-k operation reaches approximately 10m MAE.Peak-finding path error increases sharply below 10 RSS measurements per burst because bursts may miss the transmitters’ antenna half-power beamwidth.
  • B. Path Reconstruction Accuracy: Both assessed methods place 80% of path predictions within approximately 10m error at high k, but full-distribution inference remains within approximately 16m for 80% of predictions at k = 3.Errors above 30m are uncommon and occur mainly near path beginnings and ends, where the posterior reverts toward the prior.
  • B. Path Reconstruction Accuracy: The inferred path uncertainty grows at the start and end of reconstructed trajectories because observations are absent before or after the tracked period.The cumulative error distributions otherwise show only a small proportion of predictions exceeding 30m.

C. Trade-offs in Duty Cycle Parameters

At fixed energy, path accuracy depends on balancing the number of AoA observations against their precision. Frequent uncertain observations generally outperform too few highly precise observations, while burst timing also affects reconstruction.

  • Measurement allocation: ∼10m path reconstruction accuracy is achieved with both numerous uncertain AoA observations and fewer more accurate observations at fixed energy.Accuracy degrades mainly at the extremes of the measurement allocation.
  • Measurement allocation: ∼15m MAE remains relatively accurate when 100 RSS measurements are distributed across many low-k bursts rather than fewer high-k bursts.Error rises sharply as burst count falls and k increases.
  • Measurement allocation: Frequent uncertain AoA observations generally outperform too few accurate observations because very few inferred AoAs cannot represent the movement path.The comparison keeps total energy use constant.
  • Burst length: 2000ms bursts give the best path and angle reconstruction at low k because transmitters rotate every two seconds.Shorter bursts may miss samples from the highly informative HPBW portion of the antenna pattern.
  • Field demonstration: The system demonstrates landscape-scale bee tracking with a 38mg receiver and reconstructed return flight consistent with field observations.The receiver used simple rotating transmitters and achieved ∼15m accuracy below 180µW, improving to ∼10m below 600µW with more RSS measurements.

ETHICS STATEMENT

The study used Bombus terrestris, an invertebrate outside the protected-animal definition under the cited UK legislation, while applying three-R principles and careful handling.

  • ETHICS STATEMENT: Bombus terrestris falls outside the cited UK protected-animal definition because the legislation covers living vertebrates and cephalopods.The researchers nevertheless applied Replacement, Reduction and Refinement principles.
  • ETHICS STATEMENT: Tags were removed after recapture, and tagged bees were observed to resume flying and navigation after release.

STRAIGHT INFINITE LINE

The derivation marginalizes a random position along an infinite line, reducing the likelihood to depend on the perpendicular distance from the observation to that line.

  • STRAIGHT INFINITE LINE: x is constrained to an infinite line through the origin by writing x = β ˆx, with β assigned an improper uniform prior.
  • STRAIGHT INFINITE LINE: d is the shortest distance between y and the line defined by β ˆx, illustrated geometrically in the supplementary figures.
  • STRAIGHT INFINITE LINE: y is decomposed as αˆx + v, where αˆx is the closest point on the line and v is perpendicular to the line.
  • STRAIGHT INFINITE LINE: The derivation integrates out β from the Gaussian likelihood while treating its prior as improper and constant.

A. Applying this result to integrating out attenuation

The attenuation derivation represents all attenuated signal-strength predictions as a line and evaluates observations by their perpendicular distance to that line.

  • A. Applying this result to integrating out attenuation: d measures how far observed signal strengths y lie from the closest attenuated version of predicted strengths t for a given AoA θ.
  • A. Applying this result to integrating out attenuation: The method removes the component of t − y along the all-ones direction to compute the perpendicular distance d.
  • A. Applying this result to integrating out attenuation: Marginalizing attenuation a converts p(y|t + a1_k, σ^2) into a likelihood based on d under an improper uniform prior for a.
  • A. Applying this result to integrating out attenuation: The resulting distance expression is d = ||(t −¯t) −(y −¯y)||2.
  • APPROXIMATELY VALID: The likelihood approximation assumes the error-distance standard deviation is much smaller than the predicted transmitter-receiver distance.
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