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Mission-Centric Requirements Analysis of Model Predictive Control in Spacecraft Rendezvous and Proximity Operations

Nils Maier, Torbjørn Cunis, Tam W. Nguyen

arXiv:2610.02782v1eess.SYcs.SY

TL;DR

Mission requirements are typically translated into phase-specific MPC design choices ad hoc. This report applies a common EMPC design framework to three sequential ADRIOS / ClearSpace-1 proximity operations; for far-range closing, its economic variants use less fuel than Standard MPC, while a pure-fuel cost can leave the controller inactive.

  • Problem

    The translation from mission-level requirements to MPC design parameters is typically performed ad hoc for each mission and phase.

    Source: §1 · p. 5

  • Method

    The report instantiates a shared condensed-QP EMPC formulation across three sequential operations, identifies each pure-fuel dissipativity gap, and selects a mission-motivated terminal or stage-cost ingredient.

    Source: §8 · p. 17

  • Results

    31.72 m/s for Standard MPC versus 3.29 m/s (Run A1) to 18.33 m/s (Run B) for the economic variants in the closing scenario.

    Source: §6.1.2 · p. 13

  • Takeaways & Limitations

    The report presents a first step toward traceably relating mission requirements to MPC design choices across an entire mission sequence.

    Source: §8 · p. 17

    The analysis assumes noise-free full-state feedback and idealised continuously throttleable low-thrust actuation, and excludes proximity manoeuvring and its six-degree-of-freedom pose tracking.

    Source: §8 · p. 18

Abstract

from arXiv · show

Rendezvous and proximity operations (RPO) missions are commonly designed as a sequence of distinct phases each governed by its own sample time, constraint set, and cost structure, with the translation from mission-level requirements to model predictive control (MPC) design parameters typically performed ad hoc. This report takes a first step toward a systematic, traceable mapping between mission requirements and MPC design choices, using the ADRIOS / ClearSpace-1 active debris removal mission as a case study across three sequential operations. Its central hypothesis is that Economic MPC (EMPC), in which the stage cost reflects propellant consumption directly, is structurally better suited to proximity operations than fixed-point regulation or reference tracking, because it can exploit the natural, fuel-free relative-orbital-motion trajectories admitted by the Clohessy-Wiltshire-Hill (CWH) dynamics instead of fighting them. We show that a pure-fuel stage cost is fundamentally degenerate for this class of problem as the closed-loop optimum is not strictly dissipative with respect to any single target state or orbit, and analyse, operation by operation, which terminal ingredient (a terminal region, a terminal cost, a terminal equality, or a stage-cost regularisation) is required to repair this degeneracy, at what fuel and time cost, and under what formal guarantee. Across all three operations the economic controllers cut fuel consumption by roughly 40-99% relative to standard quadratic-tracking MPC, at the cost of longer manoeuvre times, and we synthesise the results into a cross-scenario table of mission requirements, MPC ingredients, and their residual limitations.

Nomenclature

The nomenclature defines the relative state and control input, then lists the MPC design variables and mission-specific quantities used in the report.

State and control

  • The Hill-frame state x contains relative position [x, y, z] and velocity [ẋ, ẏ, ż], while u is specific thrust.The state and input are measured in metres, metres per second, and metres per second squared, respectively.Source: Nomenclature · p. 4

MPC design quantities

  • The MPC model uses sampling time Δt, horizon length N, stage-cost weights Q and R, terminal-cost matrix P, and stabilising gain K.The passage distinguishes the stage-cost weights from the DARE design weights Q_dare and R_dare.Source: Nomenclature · p. 4

Mission-specific quantities

  • Mission quantities include total characteristic velocity Δv for fuel usage, the keep-out-sphere radius r_KOS, and the docking and handover target states x_dock and x_switch.The nomenclature also identifies the NMC phase angle ϕ₀ and along-track drift centre y_c.Source: Nomenclature · p. 4

1 Introduction

The report addresses the ad hoc translation of phase-specific mission requirements into MPC design choices. It uses ADRIOS / ClearSpace-1 to take a first step toward a traceable mapping, examining EMPC across sequential proximity operations.

1 Introduction

  • MPC is presented as suitable for phased ADR missions because it handles state and input constraints over a receding horizon and can be re-parameterised between operations.The mission sequence includes far-range closing, inspection or fly-around, and final docking or capture.Source: §1 · p. 5
  • Mission-level requirements such as approach geometry, fuel budgets, and keep-out zones are typically translated into MPC parameters ad hoc for each mission and phase.The design choices include horizon length, cost structure, terminal ingredients, and constraint formulation.Source: §1 · p. 5
  • The report’s eventual goal is a traceable, reusable requirements framework spanning a mission sequence rather than one isolated manoeuvre.This report presents a first step toward that goal.Source: §1 · p. 5
  • The report’s working hypothesis is that EMPC, with a stage cost reflecting the resource of interest, can exploit natural relative-orbital motion instead of fighting it.The hypothesis contrasts EMPC with fixed-point regulation and reference-tracking MPC.Source: §1 · p. 5
  • ADRIOS / ClearSpace-1 is used to examine problems in operation-specific MPC design and identify where EMPC is useful and which mission requirements motivate it.The report treats closing, NMC fly-around, and final docking as sequential operations.Source: §1 · p. 5

2 Mission Overview & Scenarios

The report applies MPC to a simplified, sequential version of ADRIOS / ClearSpace-1’s proximity operations. Each operation imposes different requirements that drive its design instance.

2.1 ADRIOS ClearSpace-1

  • ADRIOS / ClearSpace-1 is an ESA mission to rendezvous with, capture, and remove the non-cooperative PROBA-1 satellite from low Earth orbit.The report identifies PROBA-1 as an ESA microsatellite in a Sun-synchronous orbit at roughly 400 km altitude.Source: §2.1 · p. 5

2.2 Concept of Operations

  • The ConOps proceeds through closing from 30 km, fly-around in 50–100 m NMC orbits, proximity manoeuvring toward the FAP, and capture followed by deorbit.Closing uses passively safe impulsive manoeuvres; fly-around observes and characterises the client, while proximity manoeuvring tracks its rotation.Source: §2.2 · pp. 5–6

2.3 Scope of This Report

  • The report studies a simplified version of the first three phases, excluding FKP-to-FAP pose tracking, and treats its operations as one continuous mission sequence.Sampling time, horizon, and terminal ingredient are selected for each phase; Table 3 is described as linking each operation to its driving mission requirement.Source: §2.3 · p. 6, Table 3 · p. 6

3 Rendezvous Dynamics

The report models relative rendezvous motion with the Clohessy–Wiltshire equations and identifies both their natural drift and a special bounded-motion regime. Under the drift-free condition, the chaser can follow a fuel-free periodic NMC orbit.

3.1 Rendezvous Mechanics and the CW Equations

  • For small separations and near-circular reference orbits, the nonlinear relative two-body dynamics are linearised as the Clohessy–Wiltshire equations.The resulting model has six states and three inputs and assumes a circular, unperturbed target orbit with a first-order gravity-field linearisation.Source: §3.1 · p. 7
  • In the CW dynamics, x and y are coupled through Coriolis terms, z is a decoupled harmonic oscillator, and unforced in-plane motion can drift along-track.The system is not asymptotically stable; the drift-free initial-condition constraint removes the secular term and permits a closed periodic orbit.Source: §3.1 · p. 7

3.2 Natural Motion Circumnavigation

  • Natural Motion Circumnavigations are closed, periodic relative orbits that require no propellant to maintain, making them a natural target regime for fly-around inspection.The along-track secular term must vanish for bounded motion; violating the drift-free condition allows drift to accumulate because the y-dynamics have no restoring force.Source: §3.2 · p. 7
  • With the drift-free condition and zero radial velocity, the in-plane NMC is an ellipse with radial semi-axis |x₀| and along-track semi-axis 2|x₀|.The report connects this geometry to the orbit acquired in Operation 2.Source: §3.2 · p. 7

4 Motivation: Why (Economic) Model Predictive Control?

MPC suits proximity operations because it handles mission constraints and can be re-parameterised across sequential phases. Economic MPC can exploit fuel-free natural-motion trajectories, but a pure-fuel objective does not select a desired state or orbit and needs additional structure for convergence.

4 Motivation: Why (Economic) Model Predictive Control?

  • MPC explicitly handles state and input constraints, which are central to proximity operations.RPO requirements include keep-out zones, passive-safety envelopes, thrust and total-Δv budgets, and line-of-sight cones.Source: §4 · pp. 7–8
  • The same underlying optimisation can be re-parameterised across mission phases by changing its horizon, constraints, and cost structure.The relative-motion model, condensed QP structure, and solver architecture remain fixed across phases.Source: §4 · p. 8

4.1 Regulation, Reference Tracking, and Economic Cost

  • Fuel-free periodic natural-motion trajectories motivate economic costs, while fixed-point regulation fights their free motion and tracking requires a known, supplied reference.The quadratic regulation and tracking designs both retain a state penalty; tracking replaces a fixed setpoint with a time-varying reference.Source: §4.1 · pp. 8–9

4.2 Economic MPC

  • The report uses a quadratic control-effort surrogate for fuel, which preserves a well-conditioned QP but does not minimise Δv exactly.The cost omits state tracking to preserve freedom to exploit fuel-free motion; convergence therefore requires added structure such as a terminal ingredient or stage-cost regularisation.Source: §4.2 · p. 9
  • A pure propellant cost is degenerate for Clohessy–Wiltshire dynamics because a continuum of zero-cost equilibria and periodic orbits prevents selection of one target.Without a state penalty, the controller can drift toward whichever equilibrium is cheapest to reach.Source: §4.2 · p. 9

5 MPC Formulation & Concepts

The formulation uses a shared condensed-QP structure while adapting costs and constraints to each operation. Economic controllers use fuel-based costs with terminal ingredients tailored to the operation, including phase-synchronised fly-around targets.

5.1 System Model

  • The continuous-time CWH model uses relative position and velocity with specific thrust as acceleration; scenario-specific zero-order-hold discretisation supplies the MPC dynamics.The DARE solution provides the terminal-cost matrix P.Source: §5.1 · p. 10

5.2 Condensed QP Formulation

  • Condensing the predictions eliminates states, leaving a strictly convex finite-horizon QP over the input sequence with quadratic stage and terminal costs and an optional reference.State and input constraints are likewise collected as linear inequalities in the input sequence.Source: §5.2 · p. 10
  • Approach constraints keep the chaser on the correct side of the docking port and within a sensor line-of-sight cone, approximated by inward flat faces that conservatively contain no points outside the true cone.Any trajectory satisfying the flat-face approximation also satisfies the actual cone constraint.Source: §5.2 · pp. 10–11
  • The final predicted state can be fixed exactly at the docking state or constrained to a box inside the true target region.The box is used instead of a curved target region to retain a QP-compatible constraint.Source: §5.2 · p. 11
  • Every controller applies only the first optimal input and then resolves the QP after the horizon recedes.The same condensed construction is shared across controllers, which differ in costs, terminal ingredients, references, and active constraints.Source: §5.2 · p. 11

5.3 Economic MPC Formulation

  • Economic regulation sets the state-stage weight Q to zero so the stage cost reflects fuel use, while retaining a DARE terminal cost whose descent condition holds across the reported sampling intervals.The report states that condition (16) holds with margin at every sampling interval used.Source: §5.3 · p. 11
  • For fly-around, phase-synchronised terminal equality pins all six states to the NMC reference, while ε-regularisation instead restores strict P-periodic dissipativity without a terminal constraint.The reference advances one sample per controller step and wraps every period.Source: §5.3 · p. 12

5.4 Software Availability

  • The implementation, simulation scripts, and supporting materials for reproducing the numerical studies are available in an open-source repository under the BSD 3-Clause License.Source: §5.4 · p. 12

6 Mission Operations

Across three sequential proximity operations, the report examines how mission requirements determine MPC terminal ingredients and cost structure. Economic controllers address pure-fuel degeneracy and generally reduce fuel use at the cost of longer manoeuvres.

6.1 Operation 1: Far-Range Closing

  • Operation 1 closes from a 10 km V-bar hold to a 300 m handover box, using a 120 s sample time, 20-step horizon, and 10^-2 m/s2 thrust bound.The simulation stops within a 50 m ball around the switch point; five orbital periods are a maximum duration, not the realised closing time.Source: §6.1 · p. 12

6.1.1 Dissipativity Gap and the Design Axis

  • Pure fuel cost leaves the 10 km zero-input equilibrium optimal, so the chaser never closes without a terminal constraint, terminal cost, or state regularisation.The value function is not positive definite with respect to the handover target because it reaches zero at every V-bar equilibrium.Source: §6.1.1 · p. 12

6.1.2 Results

  • 3.29–18.33 m/s of fuel for the economic closing variants versus 31.72 m/s for Standard MPC; Run A1 takes 61 steps (2.0 h) versus 21 steps (0.67 h).The terminal ball is the least costly economic option; the fuel saving comes with slower closing. Table 5 compares the runs.Source: §6.1.2 · p. 13, Table 5 · p. 13

6.2 Operation 2: NMC Fly-Around

  • For fly-around acquisition, pure fuel cost coasts without acquiring an orbit, while a hard phase-consistent periodic terminal equality selects the target orbit; adding state cost and a range band can produce an expensive powered limit cycle.The equality reaches the target orbit at 0.071 m/s total ∆v; phase-synchronised regularisation reaches the same orbit at 7.519 m/s while freeing the horizon from a full-period constraint.Source: §6.2 · p. 14, Table 6 · p. 14, Table 7 · p. 14

6.3 Operation 3: Final Docking

  • Docking is confined to an along-track corridor by a 30° line-of-sight cone about +V-bar and the constraint y ≥ 0.The trajectory figure shows the Hill-frame approach from the 150 m hold to the port and a terminal zoom within 10 m.Source: §6.3 · p. 15, Figure 4 · p. 16
  • Docking EMPC uses 1.31 m/s versus Standard MPC’s 2.45 m/s, a 46.4% saving, but takes 43 versus 24 steps; the terminal equality tightens distance from 0.99 m to 0.69 m.The 150 m start and 24-step horizon are mission-imposed; 24 steps was the minimum feasible horizon found empirically for the equality-constrained run.Source: §6.3 · p. 15, Table 8 · p. 15

7 Cross Table

The cross-scenario synthesis combines the three operations’ controller choices and requirements analysis. It distinguishes end-to-end mission costs from each operation’s guarantees, observed results, and residual limitations.

7 Cross Table

  • The three operations are analysed separately, each with its own dissipativity gap, terminal ingredient, and certificate.The tables bring those operation-specific analyses together across the mission sequence.Source: §7 · p. 16
  • Table 9 compares end-to-end fuel and duration across controller combinations for closing, fly-around, and docking.Total ∆v excludes the fixed departure burn; duration covers the complete sequence from each operation’s initial conditions.Source: Table 9 · p. 16, §7 · p. 16
  • Table 10 aligns each operation’s mission requirement with its MPC ingredient, claimed theoretical property, observed simulation result, and residual limitation.The claimed properties come from cited EMPC literature and are not re-derived in the report.Source: §7 · p. 16, Table 10 · p. 17

8 Conclusion

The report applies a mission-centric EMPC framework to three sequential ADRIOS / ClearSpace-1 proximity operations, matching operation-specific dissipativity gaps to terminal ingredients and mission requirements. Its conclusions are bounded by simplifying assumptions and by guarantees taken from prior literature rather than re-derived for the specific systems.

8 Conclusion

  • The report applies one condensed-QP EMPC formulation across three operations, identifying each pure-fuel cost’s dissipativity gap and selecting a terminal ingredient for its mission requirement.It synthesises the cases in a cross-scenario table of requirements, ingredients, expected properties, and residual limitations.Source: §8 · p. 17
  • The CWH model assumes near-circular reference orbits and small separations, omits perturbations and several spacecraft dynamics, and excludes proximity manoeuvring.The report also assumes noise-free state feedback and uses quadratic control effort as a convex surrogate for the ∆v objective.Source: §8 · p. 18
  • The terminal-ingredient guarantees are drawn from EMPC literature, not re-derived or verified against the report’s specific dynamics and constraints.A system-specific check of theorem hypotheses is left as future work.Source: §8 · p. 18
  • The Operation 3 minimum feasible horizon, N = 24, was found by simulation sweep; a closed-form bound is left for future work.All controllers were validated only in simulation with quadprog.Source: §8 · p. 18
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