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arxiv: 2604.20485 · v1 · submitted 2026-04-22 · 🧮 math.DS

Co-State Based Data Fusion and Risk Aware Filtering for Spacecraft Navigation and Hazard Prediction

Surya Ratna Prakash D , Soumyendu Raha This is my paper

Pith reviewed 2026-05-09 23:18 UTC · model grok-4.3

classification 🧮 math.DS
keywords spacecraft navigationdata fusionco-stateconsistency monitoringrisk forecastingMarkov generatorlunar descenthazard prediction
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The pith

A differential co-state multiplier detects spacecraft navigation inconsistencies earlier than an Extended Kalman Filter by learning regime transitions from trajectory data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a fusion framework that treats a differential algebraic co-state as an instantaneous Lagrange multiplier to enforce compatibility between measurements and dynamics at each instant, yielding a direct signal of geometric mismatch. Over longer intervals, sequences of these co-states together with filter innovations are used to identify a continuous-time Markov generator that governs switches between coarse behavioral modes, from which mode probabilities and mean first-passage times supply intrinsic risk forecasts. The resulting pipeline combines geometric projection, stochastic filtering, and probabilistic hazard assessment in one online process and requires neither predefined fault models nor labeled failure data. When applied to real lunar powered-descent telemetry, the co-state signals and derived risk measures rise coherently before physical divergence or statistical inconsistency appears in a standard Extended Kalman Filter, supplying earlier, interpretable warnings for autonomous landing.

Core claim

The central claim is that a differential algebraic co-state, acting as the instantaneous Lagrange multiplier that enforces measurement-dynamics compatibility, supplies both a real-time geometric consistency diagnostic and, when its trajectory is combined with the innovation process, the data needed to learn a continuous-time Markov generator over coarse behavioral regimes; the resulting mode probabilities and mean first-passage times then furnish probabilistic risk forecasts that precede Extended Kalman Filter divergence on actual lunar descent telemetry.

What carries the argument

The differential algebraic co-state, defined as the instantaneous Lagrange multiplier enforcing measurement dynamics compatibility at the differential level, serves as the primary signal that carries both short-term geometric inconsistency detection and the input trajectories for learning the continuous-time Markov generator over behavioral regimes.

If this is right

  • Geometric inconsistency, stochastic drift, and probabilistic risk signals increase together well before physical failure.
  • The single pipeline operates online without requiring predefined fault models, labeled failure data, or heuristic thresholds.
  • Mean first-passage times computed from the learned Markov generator supply operationally interpretable early-warning horizons for autonomous landing.
  • The same architecture unifies geometric projection, stochastic inference, and risk assessment for any navigation system that produces an innovation sequence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same co-state-plus-Markov construction could be tested on other autonomous platforms where model mismatch arises gradually, such as aerial or underwater vehicles.
  • If the learned generator proves stable across missions, it might reduce dependence on hand-tuned safety thresholds in certification processes.
  • Direct comparison of mean first-passage time predictions against actual time-to-failure on more datasets would quantify how much earlier the risk measure typically activates.

Load-bearing premise

Co-state and innovation trajectories contain enough information to learn a reliable continuous-time Markov generator for behavioral regime transitions without any external fault models or labeled data.

What would settle it

On additional independent lunar or planetary descent telemetry sets, the co-state-derived risk measures and Markov mode probabilities fail to rise ahead of physical divergence or Extended Kalman Filter statistical inconsistency.

Figures

Figures reproduced from arXiv: 2604.20485 by Soumyendu Raha, Surya Ratna Prakash D.

Figure 1
Figure 1. Figure 1: Risk-aware closed-loop architecture integrating projected dynamics, [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Learned hazard probability, Lyapunov geometric stress, and early [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of EKF innovation and co-state-based consistency signals [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

This paper develops a co-state based fusion frame work for spacecraft navigation, consistency monitoring, and hazard forecasting. A differential algebraic co-state is introduced as an instantaneous Lagrange multiplier that enforces measurement dynamics compatibility at the differential level and provides a physically interpretable signal of geometric inconsistency. On a longer time scale, co-state and innovation trajectories are used to learn a continuous time Markov generator governing transitions between coarse behavioural regimes, enabling intrinsic probabilistic risk forecasting through mode probabilities and mean first-passage time (MFPT). The resulting architecture unifies geometric projection, stochastic inference, and probabilistic risk assessment in a single online pipeline without requiring predefined fault models, labelled failure data, or heuristic thresholds. The framework is demonstrated on real lunar powered-descent telemetry, where it detects structural internal model inconsistency significantly earlier than physical divergence or statistical inconsistency in an Extended Kalman Filter (EKF). The results show that geometric inconsistency, stochastic drift, and probabilistic risk rise coherently prior to failure, yielding interpretable and operationally meaningful early-warning capability for autonomous landing systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to develop a co-state based data fusion framework for spacecraft navigation, consistency monitoring, and hazard forecasting. A differential algebraic co-state is introduced as an instantaneous Lagrange multiplier enforcing measurement dynamics compatibility at the differential level, providing a physically interpretable signal of geometric inconsistency. On longer timescales, co-state and innovation trajectories are used to learn a continuous-time Markov generator governing transitions between coarse behavioural regimes, enabling intrinsic probabilistic risk forecasting via mode probabilities and mean first-passage times (MFPT). The architecture unifies geometric projection, stochastic inference, and probabilistic risk assessment in a single online pipeline without predefined fault models, labelled failure data, or heuristic thresholds. It is demonstrated on real lunar powered-descent telemetry, where it detects structural internal model inconsistency significantly earlier than physical divergence or statistical inconsistency in an Extended Kalman Filter (EKF), with geometric inconsistency, stochastic drift, and probabilistic risk rising coherently prior to failure.

Significance. If the central claims hold, this work could advance dynamical systems methods in aerospace engineering by providing an integrated, online framework for navigation, consistency monitoring, and risk-aware filtering that operates without labelled data or predefined faults. The demonstration on actual lunar telemetry data and the emphasis on interpretable early-warning signals are strengths that enhance potential operational relevance for autonomous landing systems. The unification of geometric, stochastic, and probabilistic components addresses practical challenges in real-time hazard prediction.

major comments (2)
  1. The estimation of the continuous-time Markov generator from a single unlabelled trajectory of co-state and innovation data (as described in the stochastic inference component) lacks any reported quantitative validation such as bootstrap variability of the rate matrix, out-of-sample regime prediction accuracy, or sensitivity analysis to coarse-graining choices. This is load-bearing for the MFPT-based risk forecasts, since the generator is conditioned on the observed sample path and under-sampling of rare transitions would propagate directly into unreliable mean first-passage times.
  2. In the demonstration on real lunar powered-descent telemetry, the claim of detecting structural internal model inconsistency 'significantly earlier' than the EKF is stated qualitatively without supporting quantitative metrics, such as specific detection time differences, false-alarm rates, or statistical comparison of the co-state/innovation signals against EKF divergence thresholds. This undermines assessment of the headline performance advantage.
minor comments (2)
  1. The abstract introduces the 'differential algebraic co-state' without providing its defining equations or derivation steps, which reduces immediate clarity even if these appear in the main text.
  2. Notation for the Markov generator and MFPT calculations could be standardized with explicit references to standard continuous-time Markov chain literature to aid readers from the dynamical systems community.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and insightful comments on our manuscript. We address each major comment below and will revise the paper to incorporate additional quantitative analyses that strengthen the validation of the Markov generator and the performance claims.

read point-by-point responses

  1. Referee: The estimation of the continuous-time Markov generator from a single unlabelled trajectory of co-state and innovation data (as described in the stochastic inference component) lacks any reported quantitative validation such as bootstrap variability of the rate matrix, out-of-sample regime prediction accuracy, or sensitivity analysis to coarse-graining choices. This is load-bearing for the MFPT-based risk forecasts, since the generator is conditioned on the observed sample path and under-sampling of rare transitions would propagate directly into unreliable mean first-passage times.

    Authors: We agree that explicit quantitative validation of the generator estimation is important for assessing the reliability of the MFPT risk forecasts. With only a single real telemetry trajectory available, traditional out-of-sample regime prediction is constrained; however, we can and will add bootstrap resampling of trajectory segments to quantify variability in the estimated rate matrix, together with a sensitivity study over the coarse-graining parameter. These results will be reported in the revised manuscript to provide a clearer picture of uncertainty in the probabilistic forecasts. revision: yes

  2. Referee: In the demonstration on real lunar powered-descent telemetry, the claim of detecting structural internal model inconsistency 'significantly earlier' than the EKF is stated qualitatively without supporting quantitative metrics, such as specific detection time differences, false-alarm rates, or statistical comparison of the co-state/innovation signals against EKF divergence thresholds. This undermines assessment of the headline performance advantage.

    Authors: The earlier detection is currently illustrated via direct comparison of the time series in the lunar descent data. We will revise the results section to include explicit quantitative metrics: the lead time (in seconds) between the first statistically significant rise in the co-state-based inconsistency signal and the corresponding EKF divergence threshold crossing, together with a simple statistical comparison of signal-to-threshold ratios. While a full false-alarm rate analysis would require additional non-failure trajectories not present in the current dataset, we will discuss this limitation and provide the lead-time quantification to support the performance claim. revision: yes

Circularity Check

1 steps flagged

CTMC generator learning from trajectories renders MFPT risk metrics fitted by construction

specific steps
  1. fitted input called prediction [Abstract]
    "On a longer time scale, co-state and innovation trajectories are used to learn a continuous time Markov generator governing transitions between coarse behavioural regimes, enabling intrinsic probabilistic risk forecasting through mode probabilities and mean first-passage time (MFPT)."

    The generator is estimated (fitted) from the co-state/innovation trajectories of the real telemetry. The risk forecasting is then obtained by deriving mode probabilities and MFPT directly from this estimated generator. On a single trajectory without reported cross-validation or sensitivity checks, the MFPT risk values are necessarily computed from a model conditioned on the observed data path, making the 'forecast' a derived quantity of the input fit rather than an independent forward prediction.

full rationale

The paper's central pipeline introduces a co-state construct independently but then learns the continuous-time Markov generator directly from the same co-state and innovation trajectories extracted from the single demonstrated lunar descent telemetry segment. The probabilistic risk (mode probabilities and MFPT) is subsequently computed from this learned generator. Because the generator estimation is conditioned on the observed sample path and no out-of-sample, bootstrap, or held-out validation is described in the abstract, the risk-forecasting step reduces to a post-hoc calculation on the fitted model rather than an independent prediction. This matches the fitted-input-called-prediction pattern at a moderate level; the geometric co-state and EKF comparison remain independent, preventing a higher circularity score.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

Ledger is provisional and based solely on abstract; full paper would likely reveal additional parameters and assumptions in the derivations.

free parameters (1)
  • Continuous-time Markov generator parameters
    Learned from co-state and innovation trajectories to model regime transitions; specific fitting process and values not detailed in abstract.
axioms (1)
  • domain assumption Differential algebraic co-state enforces measurement dynamics compatibility at the differential level as an instantaneous Lagrange multiplier
    Central introduction for geometric consistency in the fusion framework.
invented entities (1)
  • Differential algebraic co-state no independent evidence
    purpose: Enforces differential-level measurement compatibility and provides signal of geometric inconsistency
    Newly introduced construct in the paper.

pith-pipeline@v0.9.0 · 5480 in / 1397 out tokens · 64303 ms · 2026-05-09T23:18:36.212720+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    A new approach to linear filtering and prediction problems,

    R. E. Kalman, “A new approach to linear filtering and prediction problems,”ASME Trans., J. Basic Eng., vol. 82, no. 1, pp. 35–45, 1960

    work page 1960

  2. [2]

    New results in linear filtering and prediction theory,

    R. E. Kalman and R. S. Bucy, “New results in linear filtering and prediction theory,”ASME Trans., J. Basic Eng., vol. 83, no. 1, pp. 95– 108, 1961

    work page 1961

  3. [3]

    Doucet, N

    A. Doucet, N. de Freitas, and N. Gordon, Eds.,Sequential Monte Carlo Methods in Practice. Springer, 2001

    work page 2001

  4. [4]

    Feature density based terrain hazard detection for planetary landing,

    A. Gao and S. Zhou, “Feature density based terrain hazard detection for planetary landing,”IEEE Transactions on Aerospace and Electronic Systems, vol. 54, no. 5, pp. 2411–2420, 2018

    work page 2018

  5. [5]

    Robust trajectory optimization for guided powered descent and landing,

    G. E. Calkins, D. C. Woffinden, and Z. R. Putnam, “Robust trajectory optimization for guided powered descent and landing,” inAAS/AIAA Astrodynamics Specialists Conference, 2022, paper AAS 22-660, NASA NTRS 20220010431

    work page 2022

  6. [6]

    Fault and failure tolerant model predictive control of quadrotor uav,

    W. Jung and H. Bang, “Fault and failure tolerant model predictive control of quadrotor uav,”International Journal of Aeronautical and Space Sciences, vol. 22, no. 3, pp. 663–675, 2021

    work page 2021

  7. [7]

    A survey on fault diagnosis and fault-tolerant control methods for unmanned aerial vehicles,

    G. K. Fourlas, “A survey on fault diagnosis and fault-tolerant control methods for unmanned aerial vehicles,”Aerospace, vol. 9, no. 9, p. 197, 2021

    work page 2021

  8. [8]

    Real-time fault detection for uav based on model acceleration engine,

    B. Wang, X. Peng, M. Jiang, and D. Liu, “Real-time fault detection for uav based on model acceleration engine,”IEEE Transactions on Instrumentation and Measurement, vol. 69, pp. 9505–9516, 2020

    work page 2020

  9. [9]

    Estimation of stochastic systems: Arbitrary system process with additive noise observation errors,

    G. Kallianpur and C. Striebel, “Estimation of stochastic systems: Arbitrary system process with additive noise observation errors,”Ann. Math. Stat., vol. 39, no. 3, pp. 785–801, 1968

    work page 1968

  10. [10]

    Xiong,An introduction to stochastic filtering theory

    J. Xiong,An introduction to stochastic filtering theory. OUP Oxford, 2008, vol. 18

    work page 2008

  11. [11]

    U. M. Ascher and L. R. Petzold,Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, 1998

    work page 1998

  12. [12]

    Kunkel and V

    P. Kunkel and V . Mehrmann,Differential-Algebraic Equations: Analysis and Numerical Solution. EMS Press, 2006

    work page 2006