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Consensus-Based Distributed Filtering for Angles-Only Tracking in Cislunar Space

Giovanni Romagnoli, Fabrizio Pilone, Giordana Bucchioni

arXiv:2610.30864v1eess.SYcs.MAcs.SY

TL;DR

Estimating a target’s inertial state from LOS-only measurements is difficult because those measurements provide no direct range information. This paper compares three consensus-based distributed filters in a cislunar tracking scenario, finding that ICF remains closest to the centralized reference across finite consensus and prior-misalignment tests, while all three improve on independent local filters.

  • Problem

    LOS-only sensing provides no direct range information, limiting single-observer state reconstruction, while cislunar nonlinear dynamics further complicate navigation.

    Source: §I · p. 1

  • Method

    The study implements and compares DEKF, ICF, and KCF for cislunar angles-only tracking, using a centralized EKF benchmark and independent local filters.

    Source: §I · p. 1

  • Results

    ICF remained closest to the centralized reference under finite consensus iterations and increasing prior misalignment, while all distributed filters improved on independent local filters.

    Source: §V · p. 5, 1) Nominal Scenario · p. 5, 2) Sensitivity to the Number of… · p. 5

  • Takeaways & Limitations

    The ICF provided the most reliable distributed approximation of the centralized reference across the tested communication and prior-misalignment conditions.

    Source: §V · p. 5

    The KCF is theoretically sub-optimal because it neglects cross-covariances between local estimates arising from overlapping neighborhood information.

    Source: 3) Kalman Consensus Filter (KCF) · p. 4

Abstract

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This paper investigates consensus-based distributed filtering strategies for multi-observer angles-only navigation in the cislunar space. The objective is to estimate the inertial state of a target spacecraft using only line-of-sight measurements collected by a network of observer spacecraft. This problem is particularly relevant for autonomous cislunar navigation, where ground support may be limited, and centralized fusion architectures may be problematic due to communication and computational constraints, and single-point failure risks. Three distributed filtering approaches are considered: the Distributed Extended Kalman Filter, the Information-Weighted Consensus Filter, and the Kalman Consensus Filter. All methods share the same local prediction and measurement-processing steps, but differ in the way information is exchanged and fused across the network. Their performance is evaluated and compared to a centralized Extended Kalman Filter, adopted as benchmark, and independent local filters in a high-fidelity cislunar scenario. Numerical simulations show that the distributed approaches significantly improve estimation accuracy with respect to independent filters. Monte Carlo sensitivity analyses are then conducted to characterize the robustness of the considered methods to finite consensus iterations and prior misalignment among the agents.

I. INTRODUCTION

Cislunar navigation motivates cooperative angles-only tracking because LOS sensors provide no direct range, while single-observer reconstruction is limited by observability and nonlinear dynamics. The study compares three consensus filters in a cislunar scenario and uses a high-fidelity reference propagation model.

I. INTRODUCTION

  • Ground communication limits make autonomous onboard navigation important, and consensus filters exchange information locally without a central fusion node.DEKF, ICF, and KCF are among the approaches considered; finite communication and prior mismatch can affect their performance.Source: §I · p. 1
  • Distributed filtering is motivated by the single-observer observability limitation of LOS-only sensing and the difficulty of cislunar nonlinear dynamics.The work compares DEKF, ICF, and KCF against a centralized EKF and independent local filters, with Monte Carlo analyses of finite consensus and prior misalignment.Source: §I · pp. 1–2
  • The scenario uses a formation of NRHO observers to estimate a target state from noisy LOS measurements, benchmarking against a centralized EKF and independent local filters.The study also examines sensitivity to finite consensus iterations and prior misalignment.Source: §I · pp. 1–2

A. High-Fidelity Ephemeris Model

  • Reference trajectories are propagated in ECI using SPICE ephemerides and Earth-Moon gravity, with deterministic solar third-body acceleration included only in reference propagation.Earth, Moon, and spacecraft positions are obtained from the ephemeris catalog at each integration step.Source: A. High-Fidelity Ephemeris Model · p. 2

B. Simplified Filter Model

The onboard filter uses a simplified dynamical model that retains dominant gravitational effects while analytically approximating the Moon’s trajectory. Its passive optical measurements are noisy LOS unit vectors, and their covariance weights the filtering update.

B. Simplified Filter Model

  • The onboard model retains dominant gravitational effects but analytically approximates the Moon’s trajectory to reduce reliance on expensive full ephemeris propagation.Under the planar CR3BP assumptions, Earth and Moon follow circular coplanar orbits; the lunar orbit plane and angular velocity are fixed at the initial epoch.Source: B. Simplified Filter Model · p. 2

C. Measurement Model

  • Each observer’s passive optical sensor provides a noisy LOS unit vector, modeled as a small random rotation of the true direction.The rotation vector is zero-mean isotropic Gaussian, with a shared noise scale and independent realizations across sensors and epochs.Source: C. Measurement Model · p. 2
  • The LOS covariance matrix is used as the measurement weighting matrix in the filtering update.The supplied passage does not specify further covariance properties.Source: C. Measurement Model · p. 2

III. NAVIGATION ALGORITHMS

The navigation network consists of observers that maintain local target-state estimates and covariances, with communication represented by an undirected graph.

III. NAVIGATION ALGORITHMS

  • Each agent maintains a local target-state estimate and an associated 6×6 covariance matrix.Source: §III · p. 2
  • The communication network is an undirected graph whose vertices are observers and whose edges represent pairs able to communicate directly.Source: §III · p. 2
  • The neighbor set Ni denotes the agents directly connected to agent i, and the total agent count is N = |V|.Source: §III · p. 2

A. Shared Prediction Step

Each observer propagates its estimate locally, then computes information quantities from its LOS measurement and predicted state. The measurement covariance is singular along the LOS direction and receives a small regularization for numerical stability.

A. Shared Prediction Step

  • At each navigation step, each observer locally propagates its current estimate using the simplified dynamical model.The prediction formulation includes process noise, a process Jacobian, and a gravity-gradient contribution evaluated at the current state estimate.Source: A. Shared Prediction Step · pp. 2–3

B. Local Information Terms Computation

  • Each agent acquires a LOS measurement, linearizes the measurement function around its predicted state, and computes local information quantities.The information-weighted LOS quantities and Ui characterize the measurement’s information about the target state.Source: B. Local Information Terms Computation · p. 3
  • The ideal LOS covariance is singular because its nonzero variance lies in the plane orthogonal to the measured LOS, so a small LOS-direction regularization supports numerical stability.The regularization has negligible effect on angular information content.Source: B. Local Information Terms Computation · p. 3
  • The innovation information vector is defined from the locally computed information quantities.Source: B. Local Information Terms Computation · p. 3

C. Average Consensus Protocol

The protocol drives locally computed quantities toward their network-wide average through repeated consensus updates. With finite iterations, each navigation step uses an approximation to that average.

C. Average Consensus Protocol

  • Average consensus moves locally computed quantities toward their average across the communication graph.Each agent starts from its local value and updates it through consensus iterations.Source: C. Average Consensus Protocol · p. 3
  • Metropolis weights preserve the network average and converge to it as the iteration count approaches infinity on an undirected connected graph.The weights depend on the degrees of directly connected agents.Source: C. Average Consensus Protocol · p. 3
  • Each navigation step uses a finite number of consensus iterations, so the resulting value approximates rather than necessarily equals the network average.The number of iterations per step is denoted Nc.Source: C. Average Consensus Protocol · p. 3

D. Distributed Filtering Strategies

The distributed filters share prediction and local measurement-information computation, but differ in how they fuse information. In the DEKF, average consensus is applied to local measurement information before the posterior update.

1) Distributed Extended Kalman Filter (DEKF):

D. Distributed Filtering Strategies

1) Distributed Extended Kalman Filter (DEKF):

  • Under aligned prior estimates and infinitely many consensus iterations, the DEKF converges to centralized performance, with process noise scaled to preserve covariance scaling.The DEKF interprets its local covariance as approximately N times the centralized covariance under ideal consensus conditions.Source: 1) Distributed Extended Kalman Filter… · p. 3

2) Information-Weighted Consensus Filter (ICF):

The ICF includes prior information alongside local measurement contributions in the quantities exchanged through consensus. It converges to the centralized information update under ideal consensus conditions and is generally more robust to finite iterations and small prior misalignments.

2) Information-Weighted Consensus Filter (ICF):

  • The ICF applies consensus to quantities that include both prior information and each agent’s local measurement contribution.Consensus is performed on a pair of quantities, whose outputs are used to reconstruct the posterior information matrix and vector.Source: 2) Information-Weighted Consensus… · pp. 3–4
  • Under infinitely many consensus iterations and consistent prior information, the ICF converges to the centralized information update.Source: 2) Information-Weighted Consensus… · p. 4
  • Including prior information in the consensus variables makes the ICF generally more robust to finite iterations and small prior misalignments.The comparison is with the DEKF, which does not include local prior information directly in its consensus variables.Source: 2) Information-Weighted Consensus… · p. 4

3) Kalman Consensus Filter (KCF):

The KCF uses measurement information from each agent’s one-hop neighborhood and adds an explicit correction to its predicted state estimate. It omits cross-covariances arising from overlapping neighborhood information, making it theoretically sub-optimal.

3) Kalman Consensus Filter (KCF):

  • The KCF combines one-hop neighborhood measurement information with an explicit state-consensus correction on predicted estimates.Unlike the DEKF and ICF, it does not perform iterative average-consensus reconstruction.Source: 3) Kalman Consensus Filter (KCF) · p. 4
  • The KCF’s state-consensus correction is scaled by a gain that depends on the predicted covariance norm and a tuning parameter.The gain is defined using the Frobenius norm and the scalar parameter ε_kcf.Source: 3) Kalman Consensus Filter (KCF) · p. 4
  • Because it neglects cross-covariances from overlapping neighborhood information, the KCF is theoretically sub-optimal.Under the assumptions considered in, stability and asymptotic agreement have been demonstrated for sufficiently small γ.Source: 3) Kalman Consensus Filter (KCF) · p. 4

E. Centralized Extended Kalman Filter (CEKF)

The CEKF fuses all observers’ measurements at one node, while the reference scenario uses three observers in a Southern NRHO and a connected but not fully connected communication topology.

E. Centralized Extended Kalman Filter (CEKF)

  • The CEKF processes all agents’ measurements at a single fusion node, directly summing their information contributions before computing posterior estimates.Its prediction step follows the approach described in Section III-A.Source: E. Centralized Extended Kalman Filter… · p. 4

A. Reference Scenario

  • Three observers share a Southern NRHO around the Earth-Moon L2 point, with different initial phases creating a bounded in-track formation.The target trajectory is generated by perturbing position and velocity relative to the observer-formation centroid.Source: A. Reference Scenario · p. 4
  • The undirected line-graph topology is not fully connected, limiting inter-agent communication while preserving connectivity for average consensus.Source: A. Reference Scenario · pp. 4–5

B. Simulation Setup

In the nominal scenario, ICF and DEKF approach centralized performance, while KCF improves on independent local filters but remains less aligned with the CEKF. Sensitivity tests show differing dependence on consensus iterations.

B. Simulation Setup

  • Observers collect LOS measurements every 0.1 min within 500 km visibility, with 10^-4 rad angular noise and identical filter initial estimates.The nominal ICF and DEKF use Nc = 20 consensus iterations; KCF tuning uses ϵkcf = 0.1 and ϵreg = 10^-6.Source: B. Simulation Setup · p. 5

1) Nominal Scenario:

  • ICF and DEKF closely approach CEKF performance; KCF improves over independent local filters but has higher error and less alignment with CEKF.Independent filters have significantly larger errors over the visibility window.Source: 1) Nominal Scenario · p. 5

2) Sensitivity to the Number of Consensus Iterations:

  • At Nc ≃15, DEKF approaches the CEKF, while ICF remains close with few iterations and KCF is insensitive to Nc.Across 100 simulations with aligned initial estimates, DEKF degrades at small Nc; ICF outperforms KCF even there. KCF’s remaining gap may reflect one-hop measurement availability and neglected cross-covariances, which this comparison does not isolate.Source: 2) Sensitivity to the Number of… · p. 5

3) Sensitivity to Misaligned Initial Estimates:

With consensus iterations fixed at 20, the study varies prior mismatch among agents. ICF remains closest to the centralized reference across the tested mismatch range, while DEKF is highly sensitive.

3) Sensitivity to Misaligned Initial Estimates:

  • With Nc fixed at 20, each local prior is independently perturbed from aligned initial estimates, and 100 random simulations assess each mismatch level.The scalar α controls mismatch magnitude, and mean position error is computed over the visibility window.Source: 3) Sensitivity to Misaligned Initial… · p. 5
  • As prior mismatch increases, DEKF error grows rapidly, whereas ICF and KCF depend less on mismatch; ICF remains closest to CEKF.KCF remains bounded and more robust than DEKF, but retains a larger residual gap from CEKF.Source: 3) Sensitivity to Misaligned Initial… · p. 5

V. CONCLUSIONS

  • Across the considered cases, ICF is the most reliable distributed approximation of CEKF; DEKF needs aligned priors and sufficient consensus, while KCF improves over independent filters.KCF is described as communication-efficient, though it retains a larger residual error than the centralized reference.Source: §V · p. 5
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