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Reference-Filter-Driven Transition Probabilities for IMM-Based Satellite Maneuver Detection

Euiseok Han, Sangheon Choi, Seung-Hyun Kong

arXiv:2610.11242v1eess.SYastro-ph.IMcs.SYeess.SP

TL;DR

Fixed-TPM IMM filters require a transition-probability choice despite unknown, irregular maneuver rates, and adaptive methods can feed their own TPM back through IMM mixing. RD-IMM instead drives the TPM continuously from an unmixed reference coast filter’s NIS. In the reported scenario, it detects small burns missed by the fixed-TPM IMM, avoids closed-loop adaptation failures, and cuts post-burn position error by about half versus single filters.

  • Problem

    A fixed IMM transition matrix must be tuned despite unknown maneuver rates, while internal-statistic adaptation creates a TPM feedback loop that can cause stuck-high probabilities or desensitization under dynamics mismatch.

    Source: §1 · p. 1, §3.1 · p. 5

  • Method

    RD-IMM maps the NIS from a reference coast filter that is not mixed with the IMM continuously to the coast-to-maneuver transition probability.

    Source: §1 · p. 2, §5 · p. 13

  • Results

    RD-IMM detects small burns missed by the fixed-TPM IMM, avoids closed-loop adaptation failures, and reduces post-burn position error by about half versus single filters.

    Source: §1 · p. 2

  • Takeaways & Limitations

    In the evaluated space-based optical scenario, reference-driven adaptation avoids both reported closed-loop failure modes while improving small-burn detection and post-burn position accuracy.

    Source: §1 · p. 2, §4.7 · p. 13

    The simulated observation scenario assumes two to four sensors observe the target every minute for several hours; with longer intervals, RD-IMM’s latency advantage over threshold-based detection is expected to diminish.

    Source: §4.7 · p. 13

Abstract

from arXiv · show

Maneuver detection of non-cooperative satellites is an essential task of space situational awareness. The interacting multiple model (IMM) filter has been applied to detect unannounced maneuvers, but the standard IMM assumes a fixed Markov transition probability matrix (TPM) that must be tuned without knowledge of the maneuver rate of the target. To avoid this tuning, adaptive-TPM approaches update the TPM online from statistics computed inside the IMM. However, these statistics already depend on the TPM through the mixing step, and this dependence forms a feedback loop. Under the dynamics mismatch of orbit tracking, the loop can keep the maneuver probability high in the absence of a maneuver or suppress its rise when a maneuver occurs. In this paper, the TPM is driven by the normalized innovation squared (NIS) of a reference coast filter that is not mixed with the IMM, and the NIS is mapped continuously to the coast-to-maneuver transition probability. The reference-driven IMM (RD-IMM) is evaluated with four space-based optical sensors tracking a geosynchronous target. RD-IMM detects small burns that the fixed-TPM IMM fails to detect, with few false declarations in maneuver-free runs, and avoids both failures of the closed-loop adaptation. Moreover, RD-IMM reduces the position error after a small in-track burn by more than 50% compared with the single-filter approaches.

1. Introduction

Fixed transition probabilities trade coast accuracy against burn response, while adaptive approaches driven by internal IMM statistics can form a feedback loop under dynamics mismatch. The paper proposes RD-IMM, which drives transition probabilities with an unmixed reference filter’s NIS, and reports improved detection and post-burn position accuracy.

1. Introduction

  • Timely detection of unannounced maneuvers matters because corrupted orbit estimates can undermine catalogue correlation and conjunction assessments, while false declarations also degrade estimates.The paper identifies collision avoidance as a reason that detection should be timely and false declarations rare.Source: §1 · p. 1
  • A fixed coast-to-maneuver probability trades coast accuracy against response time, and can fail to balance them when the target’s maneuver rate is unknown.A small probability preserves coast accuracy but delays switching after a burn; a large one shortens delay but raises maneuver probability during coast.Source: §1 · pp. 1–2
  • Adaptive-TPM methods update transition probabilities from IMM-internal statistics, but those statistics depend on the TPM and can cause stuck-high probabilities or desensitization under dynamics mismatch.Prior evaluations did not measure false declarations without maneuvers.Source: §1 · p. 2
  • RD-IMM detects small burns missed by the fixed-TPM IMM, avoids closed-loop adaptation failures, and reduces post-burn position error by about half versus single filters.The evaluation uses a space-based optical tracking scenario with dynamics mismatch.Source: §1 · p. 2
  • RD-IMM maps the NIS of a reference coast filter, not mixed with the IMM, continuously to the coast-to-maneuver transition probability.The mapping uses χ2 surprisal, a fading maximum, and a logistic function.Source: §1 · p. 2

2. Problem formulation

The formulation models a geostationary satellite with coast and maneuver dynamics, angle-only optical observations, and an IMM that combines mode-specific filters and estimates.

2.1. Orbital dynamics and maneuver model

  • The satellite uses a 6-state coast model and a 9-state maneuver model that augments the coast state with constant thrust acceleration.The filters propagate only the J2 gravity term; omitted higher zonals, third-body effects, and radiation pressure are treated as unmodeled accelerations through process noise.Source: §2.1 · pp. 2–3
  • Coasting sets thrust acceleration to zero, while a maneuver is an impulse at an unknown time approximated by constant acceleration whose process-noise block models a random walk.The impulse is a velocity change Δv; the coast process noise compensates for neglected dynamics terms.Source: §2.1 · p. 3

2.2. Measurement model

  • Space-based optical sensors track the geostationary target using angle-only measurements, and the observation function depends on each model’s position components.Sensors are available when Earth does not block line of sight and the bistatic angle is below 90°; available measurements are concatenated.Source: §2.2 · p. 3
  • The observation setup acquires measurements every 60 s and integrates filter dynamics in 10 s steps between measurement times.The angle-noise scaling factor varies with bistatic solar angle, from 0.8 below 20° to 1.2 between 75° and 90°.Source: §2.2 · p. 3

2.3. Interacting multiple model filter

  • The coast and maneuver modes follow a Markov chain with coast-to-maneuver and maneuver-to-coast transition probabilities, and each IMM cycle includes interaction, filtering, probability updates, and combination.Mode-matched estimates are mapped between state spaces before mixing because the models have different dimensions.Source: §2.3 · p. 4
  • Each mode-matched filter is a cubature Kalman filter; cubature points are propagated through the dynamics, with discrete process noise derived from continuous-time noise using the Van Loan method.Prediction is repeated between measurement times using the state-transition matrix.Source: §2.3 · p. 4
  • Measurement updates use mode likelihoods and NIS, then update mode probabilities and combine the shared position–velocity estimate; detection occurs when the maneuver mode dominates.Latency is measured from maneuver initiation to the first measurement time meeting the declaration condition.Source: §2.3 · p. 5

3. Reference-driven IMM

The section diagnoses failures from adapting the IMM transition matrix using statistics that depend on that matrix, then presents RD-IMM, which drives adaptation with an unmixed reference coast filter. It maps the reference filter’s NIS continuously to the coast-to-maneuver probability and sets the mapping constant to balance latency and false declarations.

3.1. TPM adaptation driven by internal statistics

  • Internal-statistic adaptation forms a feedback loop through IMM mixing; under dynamics mismatch, it can keep maneuver probability above 0.5 during coast or suppress the drive after a burn.The first failure occurs with mode-probability drive; the second occurs with coast-filter NIS drive.Source: §3.1 · p. 5

3.2. Decoupled drive from a reference coast filter

  • RD-IMM uses a parallel reference coast filter that receives the IMM’s measurements but is not mixed with it, so its innovation statistic does not depend on the TPM or mode probabilities.This avoids the internal-statistic feedback loop.Source: §3.2 · p. 6
  • When the reference filter’s NIS exceeds Ψmax, RD-IMM resets its covariance; the filter then reconverges within a few steps and its statistic returns to coast level.The reset threshold is used only for resetting in RD-IMM, not for declaring a maneuver.Source: §3.2 · p. 6

3.3. Continuous mapping to the transition probability

  • The reference NIS is converted to a sensor-count-independent χ²-tail surprisal, then a fading maximum retains elevated drive briefly before a logistic map sets pCM between bounds.The fading factor ρ reduces αk after a large surprisal; the logistic function is centered at αk = 1.Source: §3.3 · pp. 6–7
  • A moderately large innovation raises pCM continuously without a threshold, reducing the detection delay associated with detect-then-reset approaches.The IMM’s model likelihoods select the maneuver mode.Source: §3.3 · p. 7

3.4. Mapping constant

  • With c = 4, the design rule gives a coast false-alarm bound for αk reaching 1, but does not bound false declarations caused by IMM model likelihoods.The chosen c rounds the rule’s value of 3.9; c trades detection latency against false declarations.Source: §3.4 · p. 7

4. Simulation results and analysis

Across the simulated GEO tracking cases, RD-IMM detects small maneuvers missed by the fixed-TPM IMM, avoids the two failures of internal-statistics adaptation, and balances detection performance against coast accuracy. Its results are evaluated with four optical sensors and compared with single-filter, fixed-TPM, adaptive-TPM, and NIS-test baselines.

4.1. Simulation setup

  • Four space-based optical sensors track a 42,164 km geostationary target with angle measurements every 60 s, over a 400 min simulation.Cases use 0.5, 1, 3, or 10 m/s impulses in-track or out-of-plane, plus maneuver-free runs, with 50 Monte Carlo replications.Source: §4.1 · pp. 7–8, Figure 2 · p. 8

4.2. Compared filters and metrics

  • Comparisons include EKF, fixed-TPM IMM-CKF, adaptive-TPM AIMM, and the single-coast-filter NIS test; metrics cover detection, latency, false declarations, and position RMSE.The binary-NIS-gate and internal-NIS-drive variants isolate the continuous-map and NIS-source choices.Source: §4.2 · pp. 8–9

4.3. Failure modes of internal-statistics adaptation

  • AIMM declares maneuvers in every maneuver-free run, while the internal-NIS drive detects only 4% of 1 m/s burns, matching its maneuver-free declaration rate.Mixing holds AIMM’s maneuver probability above 0.5 during coast; it also suppresses the coast sub-filter innovation after a burn.Source: §4.3 · p. 9

4.4. Effect of the reference filter and the continuous map

  • After an in-track 1 m/s burn, reference-filter surprisal rises to 1.18 at 44 min while the IMM coast sub-filter stays near its coast level; the continuous map detects in 88 min versus 108 min for the binary gate.For out-of-plane burns, the two maps have the same 45 min latency.Source: §4.4 · p. 9, Figure 4 · p. 10, Figure 3 · p. 10

4.5. Detection and false declaration

  • At 1 m/s in-track, RD-IMM detects maneuvers in 0.74 of runs after a median 88 min, compared with 0.58 after 148 min for the NIS test.Source: §4.5 · p. 10
  • RD-IMM’s false-declaration rate is 0.04; setting c = 3 cuts 1 m/s in-track latency from 88 to 42 min but raises that rate to 0.26.With c = 5, 1 m/s in-track detection probability falls from 0.74 to 0.26; process-noise scaling also changes detection probabilities.Source: §4.5 · p. 11

4.6. Tracking accuracy

  • After a 1 m/s in-track burn, post-burn RMSE is 1.04 km for RD-IMM versus 2.23 km for the NIS test, while the NIS test is more accurate after reconverging.In maneuver-free runs, the coast-only NIS test and EKF have 0.44 km RMSE versus 0.96 km for RD-IMM; mixing the noisier maneuver estimate degrades IMM accuracy in coast.Source: §4.6 · pp. 12–13, Figure 6 · p. 12

4.7. Discussion

  • RD-IMM avoids feedback-loop failures by using an unmixed reference-filter NIS and lowers detection latency through a continuous TPM map, but its latency advantage is expected to diminish with longer observation intervals.The simulated setting has two to four sensors observing every minute for several hours.Source: §4.7 · p. 13

5. Conclusion

RD-IMM breaks the feedback loop in adaptive-TPM IMM filters by driving the transition probability from an unmixed reference coast filter. In the evaluated geosynchronous-target scenario, it detects small burns missed by a fixed-TPM IMM, reduces post-burn position error after a small in-track burn, and is tested under dedicated minute-interval tracking.

5. Conclusion

  • RD-IMM detected small burns that a fixed-TPM IMM missed, with fewer false declarations than fixed-TPM and adaptive-TPM IMM filters.In the maneuver-free runs, single filters remained more accurate; after a small in-track burn, RD-IMM reduced position error compared with them.Source: §5 · p. 13
  • RD-IMM drives the coast-to-maneuver transition probability from an unmixed reference coast filter, eliminating the feedback loop of internal-statistics adaptation.The loop can keep maneuver probability high during coast when driven by mode probability, or suppress its rise after a burn when driven by the coast sub-filter NIS.Source: §5 · p. 13
  • The results concern geosynchronous-target tracking with measurements every minute, corresponding to dedicated tracking of a high-interest target.Sparse observations with long gaps, and multiple or finite burns, remain for future work.Source: §5 · p. 13
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