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Horizon-Dependent Tube MPC for Elliptical-Orbit Rendezvous Under Mass Uncertainty

Omer Burak Iskender, Keck-Voon Ling

arXiv:2608.27659v1eess.SYastro-ph.IMphysics.space-ph

TL;DR

The paper addresses robustness gaps in tube MPC for eccentric-orbit rendezvous with linear time-varying prediction dynamics. It derives a sampling- and orbit-dependent validity bound, couples tightening with horizon selection, and evaluates re-posed phases against a benchmark, finding propellant savings and near-universal capture compliance while leaving key guarantees unclaimed.

  • Problem

    Standard tube MPC formulations were not built to handle eccentric-orbit rendezvous with jointly varying prediction dynamics and robustness challenges.

  • Method

    The paper compares one-step linearisation residuals with the certified disturbance bound, uses the resulting range threshold as a design rule, and couples constraint tightening to variable-horizon selection.

  • Results

    The controller reached the 0.20 m capture requirement on 499 and 500 of 500 elliptical-scenario draws and used 24.1 and 31.4 kg less propellant on 472 and 497 matched seeds.

  • Takeaways & Limitations

    The validity rule disqualifies long-range homing at the tested sampling period, while element-space re-posing restores the certificate and a more accurate closing-phase frame changes propellant by 0.09%.

  • Takeaways & Limitations

    The reported campaign sets mass_err_max = 0, so it does not exercise the mass-uncertainty lifting; recursive feasibility is also not delivered by the stated certificate.

Abstract

from arXiv · show

A spacecraft closing on a target from three hundred kilometres to contact flies one guidance law across five orders of magnitude of range, and a controller that is provably safe at close range can lose that guarantee completely at long range while continuing to fly as though nothing were wrong. This paper derives the range at which the guarantee lapses and uses it as a design rule. The bound compares the prediction model's own linearisation error against the disturbance set the controller was built to reject, and needs only the sampling period, the orbit and that disturbance bound, so it can be evaluated before any simulation. On a Mars Sample Return approach it disqualifies the homing phase, where most of the propellant is spent, and clears the other two. Re-posing the disqualified phase in relative orbital elements restores the guarantee; re-posing a phase the rule already clears, in a frame two orders of magnitude more accurate, changes propellant by under a tenth of one per cent, and it is that second prediction that makes the rule falsifiable rather than descriptive. The constraint tightening also ties the prediction horizon to feasibility, so the horizon search limit becomes a mission parameter rather than a solver setting. Against a reimplementation of a published benchmark that reproduces its propellant to within one per cent, over five hundred dispersed Monte Carlo transfers per case on matched seeds, the controller saves 29% of the propellant on a circular target orbit and 40% on an eccentric one, docking inside the 0.20 m capture requirement on essentially every draw at a median miss near 5 cm. Two results run the other way: the saving is bought with time of flight and computation, and it comes from what the guarantee demanded of the terminal condition rather than from better disturbance rejection. Recursive feasibility and asymptotic stability are not claimed.

1 Introduction

The paper addresses eccentric-orbit rendezvous with time-varying YA dynamics and mass-dependent input authority, proposing horizon-dependent tube MPC that certifies predictor validity and couples tightening with horizon selection.

  • Problem: Eccentric rendezvous combines time-varying orbital dynamics with mass-dependent input uncertainty, breaking constant-matrix assumptions behind rigid-tube MPC.YA replaces inadmissible CW dynamics, while propellant depletion and payload exchange change the input matrix.
  • Contribution: The method replaces per-phase nominal MPC with horizon-dependent tube MPC while retaining the published phased architecture.Tightening grows with prediction step, and the benchmark’s phase schedule and other settings are preserved for comparison.
  • Certificate: The predictor-validity bound makes the disturbance-based certificate conditional on separation, sampling period, orbit, and disturbance budget.For YA dynamics, the bound is used to identify where linearisation residuals exceed the disturbance set.
  • Horizon selection: The horizon search jointly selects reach and robustness margin because terminal tolerance shrinks as horizon length increases.At a 50-step cap, all 21 stress cells flew, whereas a 25-step cap left one draw stalled 908 km out.
  • Evaluation: The controller reduces propellant relative to benchmark replications while improving docking compliance, with outcomes split by target-orbit eccentricity.Across 500 dispersed transfers per scenario, the proposed form met the 0.20 m requirement on 499 circular and 500 elliptical draws.
  • Interpretation: The reported savings trade against time of flight and computation and are attributed to terminal-condition demands rather than improved disturbance rejection.The paper separately reports that recursive feasibility and asymptotic stability are not claimed.

2 Relative Dynamics on an Eccentric Orbit

The relative-dynamics model uses true anomaly and YA state transitions for eccentric-orbit motion, while representing mass uncertainty as multiplicative input-matrix uncertainty within a tube-MPC decomposition.

  • Frames and orbital parameterisation: The chaser state is expressed in the target LVLH frame, and true anomaly is the independent variable for the Keplerian elliptical orbit.The model distinguishes radial, along-track, and cross-track geometry carried by the target frame.
  • YA prediction model: YA provides a time-varying discrete prediction model whose input matrix depends on spacecraft mass and whose state matrices vary with true anomaly.The impulsive Δv is applied at the interval start, and disturbances are bounded element-wise.
  • Model validity: At close range the orbit is sufficiently straight for a linearised relative model, while long-range linearisation failure motivates the validity analysis.The target-path sag in the close-range depiction is 0.72 m, or 0.03% of the panel height.
  • Model comparison: YA and CW have about 7 m error against nonlinear truth on a circular orbit, whereas their errors separate only when eccentricity is nonzero.This motivates YA for eccentric-orbit rendezvous rather than selecting it merely as a modeling preference.
  • Mass uncertainty: Mass uncertainty is modeled as a persistent, unknown multiplicative scalar on the input matrix, with δmmax ranging from 0.02 to 0.10.The reported 500-draw campaign set mass_err_max to zero and therefore exercised only the additive disturbance bound.
  • Tube decomposition: Tube MPC decomposes the state into nominal and error trajectories, with ancillary feedback determining closed-loop dynamics and constraint tightening.The nominal state resets to the measurement each step, so the applied control equals the nominal control at that step.

3 Phased Rendezvous Mission Design

The mission uses three guidance phases with phase-specific sampling and admissibility, and hand-over requires simultaneous range and orbit-match conditions rather than a simple range partition.

  • Mission architecture: The campaign begins 300 km behind the target after phasing, then uses three guidance phases through contact.The schedule follows the published Hartley scenario to enable matched-baseline comparison.
  • Phase schedule: The two long-range phases enumerate horizons from 8 to 50, while the terminal phase uses a fixed horizon.Each phase inherits admissibility from its own sampling period.
  • Phase admissibility: The sampling period is part of the certificate: an orbit at e=0.5 passes at a fast rate but fails at 600 s, where the threshold is 0.362.The same vehicle and orbit can therefore receive different certification outcomes across phases.
  • Hand-over: Homing-to-closing hand-over requires 10–30 km in-track separation and a five-row orbit-match residual below Rh=2000 m.The conjunction preserves synchronisation established by homing because the closing YA model cannot recover inclination error.
  • Constraint scope: The flown Mars approach does not include a rotating keep-out constraint, and the campaign’s ±109 m state box is inert.The tube instead acts through terminal element tolerance and related constraints described in the paper or companion treatment.

4 Horizon-Dependent Tube MPC and the Eccentricity Certificate

The controller propagates horizon-dependent tube bounds under a sufficient Perron–Frobenius condition, yielding an eccentricity certificate that depends strongly on sampling period. The tightening guarantees robust constraint satisfaction, but the formulation does not establish recursive feasibility or asymptotic stability.

  • Tube construction: The tube decomposes uncertainty into additive disturbances and mass effects, propagated step by step through the closed-loop dynamics.Mass uncertainty scales with the sampling period twice and dominates additive disturbance at the homing period, but not at the FINAL period; reported campaigns switch the mass term off.
  • Robust guarantee: Feasible tightened constraints ensure the true state and input satisfy the original box constraints by the triangle inequality.The guarantee is independent of the MPC objective.
  • Scope of guarantee: Recursive feasibility is not delivered by the printed program because the theorem requires a terminal robust positively invariant set.Adding the ellipsoidal terminal constraint changes the problem from a QP to a quadratically constrained program.
  • Tube behavior: The reported recursion approaches the mRPI width within one step because the closed-loop spectral radius is small on this orbit.From j=1 onward, the horizon-dependent profile is close to the constant-width limit.
  • Certificate condition: ρ(Ā)<1 is sufficient but stronger than pointwise closed-loop stability, so it must be checked independently for the time-varying orbit dynamics.For the MSRE design, the two reported spectral-radius values are 0.129 and 0.128.
  • Eccentricity certificate: e★=0.6455 at Ts=200 s, while the threshold falls from 0.8514 at 50 s to 0.2492 at 900 s.The threshold decreases by a factor of 3.4 across the tested sampling periods and must be reported with its period.
  • Orbit dependence: The MSRE orbit clears the condition at every tested period, but e=0.5 loses certification above 368 s.At Mars eccentricity 0.204, ρ(Ā) ranges from 0.029 at 50 s to 0.791 at 900 s.
  • Design sensitivity: The threshold varies only from 0.635 to 0.664 across six gain-design anomalies, while changing sampling period shifts it by a factor of 3.4.Changing the state and input weights leaves the threshold unchanged to four decimals at every tested period.

5 Where the Tube Is Valid, and What Follows

The validity rule compares predictor linearisation residuals with the disturbance bound and identifies where the tube certificate can be trusted. It disqualifies the long-range homing phase, while reformulation helps there but has negligible propellant effect in an already-valid closing phase.

  • Validity rule: The predictor residual grows quadratically with separation, so the tube is valid only within a range determined by sampling period and disturbance bound.The resulting limit is 26.4 km at Ts=300 s and 13.2 km at Ts=600 s.
  • Homing phase: The homing phase begins 23 times beyond its 13.2 km validity limit and is therefore disqualified.Its Cartesian predictor residuals are 2486 m and 4260 m against a 5 m disturbance bound, exceeding that bound by factors of 497 and 852.
  • Homing reformulation: Relative orbital elements restore the homing certificate, with residual-to-bound ratios of 0.856 and 0.716 and 15–28% margin.The mechanism is the terminal condition—five element equalities rather than a position box—not merely the coordinate change.
  • Closing phase: The closing phase already satisfies the validity bound, so a cylindrical relative frame 11–128 times more accurate changes propellant by only 0.09%.This tests the rule in the opposite direction: removing an error that was already inside the disturbance bound has negligible fuel effect.
  • Cost of invalidity: Ignoring the validity bound raises propellant from 108 to 1185 kg in the circular scenario and from 92 to 264 kg in the elliptical scenario.The Cartesian formulation also fails to reach the target approach point on one draw in five in the reported comparison.
  • Certificate comparison: Figure 3 compares e★(Ts) under the Perron–Frobenius condition with the weaker pointwise condition reported in earlier work.Compare the two threshold curves across sampling periods, especially where the stronger certificate falls below the operating eccentricity.

6 Horizon Selection and the Tightened Terminal Set

The benchmark’s variable-horizon search is retained, but horizon-dependent tightening makes feasibility an interval: horizons can be too short to reach or too long to remain feasible. Stress tests show that increasing the cap changes which dispersed cases can fly and can bind selection in practice.

  • Horizon selection: The benchmark minimizes J = N + w_u||u||_1 over candidate horizons, while the tube changes the constraint set evaluated at each N.The variable horizon and objective are inherited; horizon-dependent robustness tightening is the new design feature.
  • Tightened terminal set: At N = 8, tube tightening closes 2.5 km of a 10 km half-window, increasing to 20.3 km at N = 25.Feasibility therefore occupies an interval rather than a half-line: short horizons cannot reach, while long horizons can be tightened out of existence.
  • Stress behavior: One draw in twenty-five at n_max = 25 found every candidate horizon infeasible, abandoned 908 km out, and spent no propellant.The untightened fallback was also infeasible, so the stalled transfer reflects the reachable feasible set rather than a solver-only failure.
  • Stress behavior: 34 km elliptical and 26 km circular are the maximum one-sample drift distances for a 20 km window under the stated bound.Above the corresponding semi-major-axis threshold, one step can overshoot the whole window and the failure boundary depends jointly on semi-major-axis error and target true anomaly.
  • Stress behavior: At n_max = 25, 11 of 21 full-fidelity stress cells flew; at 50, all 21 flew, with no cell made worse.Across 71,398 guidance calls, the selected horizon was within one step of the cap on 22% of calls, so the cap remained practically binding.

7 Simulation Setup

The evaluation reconstructs the published benchmark before comparing controllers, keeping plant, frames, phase schedule, propulsion assumptions, and matched disturbances aligned. It uses a full-fidelity nonlinear simulator with dispersed initial conditions, navigation errors, execution errors, and coupled attitude effects.

  • Benchmark reconstruction: The reimplementation returns 102.27 kg versus 102.96 kg on the circular scenario and 93.98 kg versus 89.29 kg on the elliptical scenario.These correspond to ratios of 0.993 and 1.053 against the published reference, establishing the comparison baseline.
  • Benchmark reconstruction: The comparison preserves the plant, frames, phase ladder, thruster model, and specific impulse on both sides of every controller comparison.The reimplementation is used as the ruler because it reproduces the published value rather than serving as a deliberately weak baseline.
  • Controller implementation: The reference program uses linear programs for the two long-range phases and a quadratic program for the final hundred metres, switching once at the target approach point.Fuel-weight sweeps saturate above w_u ≈ 300, indicating that the reported shortfall is attributed to the constraint set rather than objective weighting.
  • Simulation fidelity: All 5000 campaign records use the Table 4 simulator with environment terms enabled, vehicle parameters redrawn per seed, and the Table 6 error model.Truth propagation remains distinct from the optimizer’s YA or element-space prediction model.
  • Dispersions and disturbances: Initial chaser semi-major-axis errors span [-50, 50] km, while target semi-major-axis errors span [-10, 10] km and target true anomaly spans [0, 320]°.The tube is designed against per-step disturbance bounds, including transfer-phase translational bounds of 5 m × 3 and 5 × 10^-3 m/s × 3.
  • Matched evaluation: Paired seeds give each controller the same dispersion, navigation stream, and actuator execution error, enabling per-trial difference comparisons.Attitude work is charged through the same propellant budget as translation, and the final-phase program reserves it conservatively with a 1-norm upper bound.
  • Translation–attitude coupling: The linear predictor uses small-angle attitude dynamics, while the plant integrates the full nonlinear rigid body with gravity-gradient and radiation-pressure torques.The uncoupled comparison executes a saturated quaternion-feedback PD after translation has committed the interval.

8 Results

Across 500 paired dispersed transfers per scenario, the proposed tube controller uses less propellant and achieves stronger elliptical-orbit capture compliance than the reimplemented benchmark. The gains trade against longer flight times and computation, while coupling and hand-over analyses identify where tail performance and savings arise.

  • Evaluation protocol: 500 paired dispersed transfers per scenario evaluate free-drift plane crossing separately from compliance with the 0.20 m capture circle.On the elliptical scenario, up to 48 draws crossed the target plane but failed the requirement, so the metrics are reported separately.
  • Capture compliance: 500 of 500 elliptical transfers met the requirement with the proposed form, versus 468, 463, and 452 for the three replication variants.On the circular scenario, every form was at or within one draw of 500 of 500.
  • Propellant: 29.2% circular and 39.6% elliptical are the proposed form’s propellant savings against the best replication, equal to 24.05 kg and 31.39 kg per matched campaign.The proposed means are 58.18 and 47.97 kg against 82.23 and 79.37 kg, respectively, with the ranking unchanged across replication variants.
  • Trade-offs: 15.41 and 9.03 orbits are the proposed form’s flight times versus 5.55 and 4.47 for replication, factors of 2.8 and 2.0.A mission unable to spend the additional time cannot realize the propellant saving.
  • Miss distribution: 10.39 cm elliptical 95th-percentile miss contrasts with 32.55 cm for the best replication and 43.97 cm for the faithful reimplementation.Circular 95th-percentile misses reverse the ordering with only a small margin, and medians differ by at most 0.2 cm across cells.
  • Miss distribution: One circular proposed-form draw reaches 179.17 cm, the only proposed arrival outside the requirement in 1000 draws.Its hand-over lateral rate is 13 times the same campaign’s 99.8th percentile and correlates with eventual miss at r = 0.963.
  • Attitude coupling: Translation–attitude coupling changes the 95th-percentile miss from 10.87 to 8.70 cm circular and from 12.18 to 7.94 cm elliptical, at a 0.02 kg cost.This 25-draw paired experiment supports a tail improvement without a propellant change, but is not directly comparable to the 500-draw cells.
  • Phase breakdown: Homing consumes 86% circular and 85% elliptical of charged propellant, while closing consumes 11% in both scenarios.The dominant homing share identifies where controller changes can have the greatest mission-level effect.

9 Discussion

The YA-based tube is ultimately bounded by eccentric-orbit kinematics and sampling period, while the threshold provides a practical boundary for selecting richer formulations. Its benefits remain conditional on modeling assumptions and mission time available.

  • Certificate boundary: The YA asymmetry that necessitates the model also bounds the tube as eccentricity increases.Keplerian rate variation makes the worst-case step dominate the error bound.
  • Certificate boundary: At T_s=200 s, the constant-gain LTV tube is admissible up to e≈0.65, but at 600 s and 900 s only to 0.362 and 0.249.An e=0.5 orbit therefore passes at the fast rate and fails at the slow one.
  • Design implications: GTO at e=0.73 lies outside the tested envelope at every period and requires disturbance-budget review before flight.The paper identifies periodic gains or variable-rate sampling as possible richer formulations near e≈0.7.
  • Limitations: The threshold is sufficient rather than tight because it assumes constant K, uniform T_s, and an additive box disturbance bound.A ν-scheduled gain or ellipsoidal tube could extend the envelope by a body-dependent constant.
  • Benchmark context: The held variant uses 101.8 versus 77.6 kg in the circular case and 69.0 versus 57.7 kg in the elliptical case.Its extra propellant buys a detour that recovers the published geometry the released law does not.
  • Limitations: The propellant saving requires two to three times the time of flight, limiting it to missions that can spend orbits rather than fixed arrival windows.The companion journal treatment contains the corridor, cross-body, and geostationary studies.

10 Conclusion

The conclusion turns the tube bound into a phase-specific reformulation rule and couples tightening with horizon selection. The resulting controller improves propellant and capture outcomes, but with longer flight, higher computational cost, and limited formal guarantees.

  • Validity rule: 26.4 km at 300 s and 13.2 km at 600 s are the predictor validity limits, disqualifying homing but not the closing phase.Homing begins at 300 km and exceeds its limit by 23 times, with residual overruns of 497 and 852; element space restores a 15–28% margin.
  • Validity rule: A cylindrical frame 128 times more accurate changes closing-phase propellant by 0.09%, validating the bound’s predictive discrimination.Element space restores the disqualified homing phase’s certificate with a 15–28% margin.
  • Horizon selection: Raising the horizon cap from 25 to 50 steps increases flown stress-grid approaches from 11 to 21, costs 2.5 times wall clock, and saves 15.8 m/s on shared cells.The cap remains binding: the selected horizon is within one step of 50 during 22% of 71,398 guidance calls.
  • Dispersed evaluation: Over 500 dispersed transfers per scenario, the controller saves 24.1 and 31.4 kg on 472 and 497 of 500 matched seeds.It arrives inside the 0.20 m requirement on 499 and 500 of 500 draws, versus 452–468 of 500 for elliptical replication variants.
  • Trade-offs: The saving costs 2.8 and 2.0 times the time of flight, while the circular case uses 81.4 m/s versus the 38.9 m/s idealized two-impulse floor.The proposed form remains below the replication’s 116.1 m/s.
  • Interpretation and limits: The eccentricity certificate falls by a factor of 3.4 across scheduled sampling periods, and 85–86% of propellant is spent in homing.The paper attributes the scenario’s tube benefit to the horizon rule rather than to corridor margin.
  • Interpretation and limits: Robust constraint satisfaction is proved, but recursive feasibility and asymptotic stability are not claimed.The terminal ellipsoid that would deliver recursive feasibility makes the program a QCQP the solver cannot accept.
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