Fault-tolerant Model Predictive Control for Spacecraft

Dimos V. Dimarogonas; Maria Charitidou; Pedro Roque; Raphael St\"ockner

REVIEW 3 major objections 4 minor 24 references

Fault-tolerant Model Predictive Control for Spacecraft

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A spacecraft with multiple failed thrusters can still track a reference trajectory by steering a virtual orbit center, provided one torque-free bias input remains available.

desk verdict Clever orbit-based FTC idea and solid experiments, but the printed error dynamics have a feedforward inconsistency that hits the stability proof; needs fixing before I'd trust Theorem 1. read the letter →

arxiv 2509.02527 v1 pith:XBWXJPLN submitted 2025-09-02 cs.RO

Raphael Stöckner , Pedro Roque , Maria Charitidou , Dimos V. Dimarogonas This is my paper · ORCID

classification cs.RO

keywords fault-tolerantcontrolmodelpredictivespacecraftunderactuatedactuatorfailuresterminalsetexplicitMPCtrajectorytracking

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that a spacecraft with several known thruster failures can still track a reference trajectory with a guaranteed stability certificate, as long as one torque-free bias input lies strictly inside the set of achievable thruster inputs. The central trick is to stop steering the spacecraft's position directly: the bias input $u_v$ that Assumption 1 guarantees is reinterpreted as the centripetal force of a small orbit, so the spacecraft continuously orbits a virtual center, and the controller stabilizes the orbit center instead. The authors prove that the resulting MPC problem is recursively feasible and that the orbit error dynamics are asymptotically stable, while the physical spacecraft position converges to a tube of radius $r_s \ge r + \beta$ around the reference. If correct, this gives a single control law that can keep a damaged spacecraft on a safe path for servicing or deorbiting without full actuation.

What carries the argument

The load-bearing object is the orbit transformation (4), $c = p_r + R^T r_\diamond$, together with the centripetal balance condition (5), $f = m \omega_d^2 r_\diamond$. These two equations rewrite the faulty spacecraft as a fully controllable orbit system (7) in which the bias input $u_v$ no longer acts as a disturbance: because $u_v \in \mathrm{Int}(U_r)$, the shifted input set satisfies $0 \in \mathrm{Int}(U)$, so the orbit error dynamics (10) are controllable. The second mechanism is the terminal pair: the feedback-linearizing terminal controller (14) cancels the nonlinear term $f(e_\omega)$ and reduces the position error to the double integrator (15a), stabilized by an offline-computed explicit MPC law (an explicit MPC is a parametric quadratic program solved offline into a piecewise-affine feedback law), with the terminal set sized through the convex program (22).

What would settle it

Give a spacecraft whose failure pattern satisfies all of Assumptions 1–3, then solve the terminal-set sizing problem (22) numerically; if the constraints are infeasible for every $\hat{u}_{\max}$ and $e_{\max}$, or if a closed-loop simulation of the proposed MPC loses feasibility or fails to converge after a finite number of steps, the claimed stability certificate is disproved.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that underactuation from actuator failures does not have to be treated as a tracking obstruction: it can be absorbed into the geometry of a periodic orbit. Given Assumption 1, the authors fix a body-frame vector $r_\diamond = [0, r, 0]^T$ and choose an orbit angular velocity $\omega_d$ so that the bias input satisfies $f = m \omega_d^2 r_\diamond$ (5); the orbit center $c = p_r + R^T r_\diamond$ (4) then has error dynamics (10) that are controllable because $0 \in \mathrm{Int}(U)$. They build a terminal set $T = \hat{X}_f \times E$ and terminal cost $l_T$ using a feedback-linearizing terminal controller (14) whose linear position part is an explicit MPC law, and prove in Theorem 1 that the closed-loop orbit error $e_{k+1} = \phi(e_k, \mu_k^{MPC})$ is asymptotically stable while (13) remains recursively feasible. Theorem 2 converts this into the spacecraft-level guarantee $\|p_r(t) - \chi_{[1:3]}(t)\| \le r_s$ with $r_s \ge r + \beta$ as $t \to \infty$.

Load-bearing premise

Assumption 1, that after the known failures there exists a torque-free bias input aligned with a principal axis and strictly inside the achievable input set, is the load-bearing premise; if the only available bias lies on the boundary or carries residual torque, the orbit construction and the stability proof no longer go through.

Editorial extensions

If this is right

If the assumptions hold, the MPC needs no runtime re-verification: recursive feasibility from $k=0$ is enough to guarantee feasibility at every later step.
The orbit-center tracking error goes to zero asymptotically, so the damaged spacecraft settles into a bounded tube of radius $r_s \ge r + \beta$ around the reference instead of drifting away.
The method handles trajectory tracking, not just setpoint stabilization, so the same controller can steer a faulty spacecraft along a collision-free path toward servicing or deorbiting.
The eMPC-based terminal set is larger than the standard LQR-based terminal set, which enlarges the region of attraction and lets the designer shape it through (22).
The controller remains valid with known (detected) failures and does not require the spacecraft to remain fully actuated.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same orbit-compensation idea should transfer to any vehicle with a persistent unidirectional thrust bias, such as a rotorcraft with a stuck propeller, as long as the reduced error dynamics remain controllable; the paper itself only demonstrates spacecraft cases.
Assumption 1 is a polytope-containment condition, so a designer could precompute all failure sets that admit a torque-free interior force and switch the controller on only for those, making the fault-tolerant property testable offline.
If the bias force magnitude drifts during operation (for example through mass loss), the balance condition (5) is violated; an adaptive variant that re-computes $r$ and $\omega_d$ online would be a natural extension.
Because the position bound is $r_s \ge r + \beta$, choosing the orbit radius $r$ is a direct trade-off between input feasibility and tracking accuracy: larger orbits ease feasibility but widen the guaranteed tube.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a genuinely clever way to handle stuck thrusters—put the spacecraft on a small orbit so the fault-induced force becomes part of the orbit dynamics, then track with MPC. The eMPC terminal set with feedback linearization is a real improvement over the zero-terminal-set approach in [7], and the ATMOS experiments plus open-source code give it credibility. I'd send it to a serious referee.

The soft spots matter. The biggest is in the equations. In (10a) the error dynamics subtract M u_ref, but the control law (12) and the constraint (13b) feed in \tilde R u_ref as part of the input. Plugging one into the other leaves an extra (\tilde R^T M \tilde R - M) u_ref term. That term is zero only if u_ref has no torque components; in general e=0 is not an equilibrium, and the cancellation that Lemma 1 and Theorem 1 rely on is not exact. The fix is straightforward in principle—the feedforward should appear as \tilde R^T M \tilde R u_ref, and you probably need to restrict u_ref's angular part or keep it in the reference dynamics—but as printed, the central theorem doesn't follow from the written equations. This is an internal algebraic issue, not a missing assumption.

Second, Assumption 1 (existence of a torque-free interior input aligned with a principal axis) does a lot of work, and the paper gives no general characterization of which failure patterns satisfy it. The 2D claim for up to three failures is argued, but the 3D example is hand-picked. That limits the fault-tolerance claim.

Third, the proof of Theorem 2 uses a strict monotonicity of the position error that asymptotic stability doesn't guarantee. That particular step is not load-bearing—the result follows from convergence plus the triangle inequality—but it should be cleaned up.

The comparative claim about larger terminal sets is cited to a thesis; I didn't recompute it, but it looks plausible given the eMPC saturations.

Bottom line: the idea is good, the experiments help, and with the feedforward term corrected and Assumption 1's scope made precise, this would be a solid paper. I'd send it to peer review, with a strong request to fix (10a) before final. If you work on FTC or MPC, worth reading alongside the code.

Referee Report

3 major / 4 minor

Summary. This paper proposes a model predictive controller for a spacecraft with known thruster failures. The key idea is to exploit Assumption 1, the existence of a torque-free force vector inside the input set, to place the spacecraft on a small circular orbit, and then to control the orbit center rather than the spacecraft position directly. The controller solves MPC problem (13) for the orbit error dynamics (10) using a terminal set built from an explicit MPC of a double integrator and a linear angular-rate feedback. The authors claim recursive feasibility and asymptotic stability of the orbit error (Theorem 1) and a bounded positional tracking error with radius r_s >= r + beta (Theorem 2). Numerical 2D experiments on the ATMOS platform and a 3D simulation illustrate the approach.

Significance. If the main theorem were correct, this would be a meaningful advance over prior work: it handles coupled translational and rotational dynamics under multiple thruster failures with input constraints, avoids the small zero terminal set of earlier underactuated MPC designs, and provides larger terminal sets via explicit MPC. The open-source implementation and ATMOS experiments are concrete strengths. However, the central algebraic derivation connecting the error dynamics to the MPC and terminal controller is currently inconsistent, and the proof of Theorem 2 contains an unjustified monotonicity claim. The contribution is therefore promising but not yet established as written.

major comments (3)

[Section III-B, Eq. (10a) and (13b)] The error dynamics and the MPC/terminal control law are algebraically inconsistent. Equation (10a) defines the second block of phi(e,u) as \tilde{R}^T (f(e_omega) + M u) - M u_ref, while constraint (13b) evaluates this map at u = u_o,s + \tilde{R}_s u_s^ref, and the terminal controller (14) contains the same + \tilde{R} u_ref term. Substitution yields the extra term (\tilde{R}^T M \tilde{R} - M) u_ref in the velocity-error update. Since M in (8) does not commute with \tilde{R} = diag(R, I_3) for a generic attitude, this term is nonzero: at e=0 and u_o=0, the one-step prediction is nonzero whenever u_ref is nonzero, so the origin is not an equilibrium of the closed-loop error dynamics. The cancellations leading to the linear dynamics (15), the terminal invariance (19), and the Lyapunov decrease in Proposition 1 and Theorem 1 all rely on the erroneous cancellation. This is the central load-bearing step and must be corrected; a local notational fix is not sufficient.
[Section IV-D, proof of Theorem 2] The proof asserts that \|c_{k+1} - \chi_{k+1,[1:3]}\| < \|c_k - \chi_{k,[1:3]}\| on the strength of Theorem 1. Theorem 1 proves asymptotic stability of the full error vector e_k = (e_p,k, e_omega,k); it does not imply that the position component e_p,k,[1:3] decreases monotonically. The eventual bound \|p_r(t) - \chi_{[1:3]}(t)\| <= r_s can be obtained by a limsup argument from e_p,k -> 0 and Assumption 4, so the step should be replaced rather than assumed. As written, the proof is logically invalid.
[Section III-A, Eq. (7)] The orbit dynamics derivation uses the identity d/dt R^T = R^T omega^times, which is not consistent with the standard quaternion kinematics in (1). With R the rotation from body to global frame, \dot{R} = R omega^times and hence \dot{R}^T = -omega^times R^T, not R^T omega^times. This changes the sign of the R^T omega^times r_diamonds term in the definition of v and of the cross-coupling terms in (7). Unless a nonstandard convention is explicitly introduced and consistently used in the quaternion kinematics, the claimed cancellation in (5) and the controllability statement following (7) need to be re-derived. Please state the convention and redo the derivation.

minor comments (4)

[Assumption 2] The phrase 'u_k^ref in N' should read u_k^ref in U_chi subset R^6; the surrounding notation 'u^k_ref,k in N' is also garbled.
[Figure 3 and Section V] The text says the blue terminal set corresponds to the proposed controller (14) and the red one replaces the eMPC part with an LQR, while the figure caption says 'Red: Linear controller. Cyan: eMPC controller.' Please reconcile the color descriptions.
[Section V, 2D implementation] The statement 'Qe = I5x5' is unclear because the full error state is nine-dimensional; since this is a simplified planar model, the dimensions of all matrices in the 2D implementation should be stated explicitly.
[Section IV-B, Eq. (22)] The optimization problem (22) contains malformed expressions, in particular the term with subscript 2 and the terms 'm u_max' and 'sqrt(f_max)' that are not consistent with (18)-(21). Please provide a readable and correct statement of the constraints.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the MPC stability proof is self-contained from Assumptions 1–4; self-citations are illustrative or provide an assumed bound, and the 'by construction' terminal ingredients are standard MPC design.

full rationale

The claimed stability and tracking results are derived through a standard MPC terminal-ingredient argument, not by re-using the target result. Assumption 1 guarantees 0 ∈ Int(U), making the orbit dynamics (7) controllable; the orbit transformation (4)–(5) is an explicit coordinate change rather than a renamed version of the conclusion. The terminal controller (14) is deliberately chosen to cancel f(eω) and reduce (10a) to the linear double-integrator dynamics (15), and the terminal set T = X̂f × E and terminal cost (24) are constructed from the feasibility set and parametric cost of the eMPC problem (16), following external references [13,16]. Recursive feasibility and asymptotic stability are then obtained from the standard theorem [24, Thm. 5.13], with Lemma 1 relying on the standard recursive-feasibility property of eMPC; none of these steps presupposes Theorem 1 or Theorem 2. The self-citations are not load-bearing: [15] is one way to compute an inter-sample bound that Assumption 4 already assumes; [19] documents the experimental platform; [22] supplies the comparison figure in Fig. 3. No fitted parameter is renamed as a prediction, and no uniqueness claim is imported from the authors' prior work. The skeptical algebraic concern about (10a) versus (13b)—that the reference feedforward should enter as R̃ᵀM R̃ u_ref rather than M u_ref unless the torque component of u_ref vanishes—is a correctness/consistency issue for the model, not circularity: it does not make the stability conclusion equivalent to its own inputs by construction. The paper is also tested against open-source code, Gazebo simulation, and ATMOS experiments, providing external demonstration beyond the analytical claims.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The scheme rests on five explicit domain assumptions (Assumptions 1-5) about the fault pattern, reference trajectory, discretization, and weight structure, plus the standard MPC terminal-conditions theorem. The orbit radius r and terminal-set sizing parameters emax, uhat_max are designer-chosen and affect, but do not compromise, the stability conclusion. No new physical entities are postulated.

free parameters (4)

orbit radius r (or orbit angular velocity omega_d) = r = 0.47 m (2D), 0.58 m (3D); omega_d = 0.5/0.6 rad/s
Chosen by the designer via f = m omega_d^2 r (Eq. 5). It sets the orbit size and directly enters the tracking bound rs = r + beta in Theorem 2.
terminal set scaling parameters emax, uhat_max = computed from optimization (22); numerical values not reported
These size the terminal set T = Xhat_f x E and are chosen by a volume-maximizing optimization (22); they must satisfy the inclusion (23), which underpins recursive feasibility and input constraint satisfaction in Lemma 1.
gain K = diag(kappa1, kappa2, kappa3) = K = 0.1 (2D), K = I (3D)
Design gain in the terminal controller (14); chosen with 0 < kappa_i < 1/delta to ensure e_omega dynamics (15b) decay. It affects the size of the terminal set E but not the existence of the stability proof.
MPC weights Qe, Qu, Qhat_e, Qhat_u and horizons N, Nhat = listed in Section V
Tuning weights and horizons; positive definite by construction. Theorem 1 requires Qe block diagonal (Assumption 5) and Qu > 0, but not specific values.

assumptions (7)

domain assumption Assumption 1: there exists uv = [f^T 0_{1x3}]^T in Int(Ur) with f aligned with a principal axis
Ensures a torque-free constant acceleration direction is available inside the post-failure input set; the orbit construction and the property 0 in Int(U) in (7) rely on it.
domain assumption Assumption 2: reference trajectory is piecewise constant over the sampling period, generated by G, with u_ref in U_chi subset Ur setminus uv
Defines the class of trackable references and guarantees the feedforward input respects constraints after subtracting uv.
domain assumption Assumption 3: G(y1,y2) = [Gbar[1:10](y1, tilde R(y1,[7:10]) y2); omega_des^0]
Matching condition between reference dynamics (3) and the Euler-discretized orbit dynamics (9); needed to derive the error dynamics (10) and to ensure the reference is trackable by the orbit system.
domain assumption Assumption 4: there exists beta > 0 with ||c(t)-c_k|| <= beta for t in [t_k, t_{k+1})
Bounding the inter-sample error of the Euler discretization; required for the continuous-time tracking bound rs = r + beta in Theorem 2. The paper cites [14, Thm 3.4] for existence but does not compute beta.
domain assumption Assumption 5: Qe = diag(Qhat_e, Qe_omega)
Block-diagonal structure of the state weight needed in Proposition 1 to split the terminal cost into position and angular-velocity parts.
standard math MPC stability criterion [24, Thm 5.13, Rem. 5.17] using standard terminal cost and terminal set conditions
Invoked in the proof of Theorem 1 to conclude asymptotic stability of the closed loop from the terminal cost inequality (Proposition 1) and forward invariance (Lemma 1).
standard math Lyapunov equation solution P exists for the linear dynamics (15b) with cost Qe_omega + 2 ||Qbar_u||_2 K^T K
Used to define Theta_3(e_s) in the terminal cost (24); existence is standard for a Schur-stable A = I - delta K.

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Cite this review

Pith. "Pith review of Fault-tolerant Model Predictive Control for Spacecraft." pith.science (2026). https://pith.science/paper/XBWXJPLN

@misc{pith2026250902527,
  author       = {Pith},
  title        = {Pith review of: Fault-tolerant Model Predictive Control for Spacecraft},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XBWXJPLN}},
  note         = {Machine review of arXiv:2509.02527}
}

read the original abstract

Given the cost and critical functions of satellite constellations, ensuring mission longevity and safe decommissioning is essential for space sustainability. This article presents a Model Predictive Control for spacecraft trajectory and setpoint stabilization under multiple actuation failures. The proposed solution allows us to efficiently control the faulty spacecraft enabling safe navigation towards servicing or collision-free trajectories. The proposed scheme ensures closed-loop asymptotic stability and is shown to be recursively feasible. We demonstrate its efficacy through open-source numerical results and realistic experiments using the ATMOS platform.

Figures

Figures reproduced from arXiv: 2509.02527 by the authors.

**Figure 2.** The ATMOS free-flyer available at the KTH Space [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Spacecraft path and orbit center. Plotted is the position [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: The system under failures is quickly stabilized to the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [7]

Autonomous thruster failure recovery on underactuated spacecraft using model predictive control

C. M. Pong, A. Saenz Otero, and D. W. Miller. “Autonomous thruster failure recovery on underactuated spacecraft using model predictive control”. In: Guidance and Control 2011: Proceedings of the 34th Annual AAS Rocky Mountain Section Guidance and Control Conference 141 (Feb. 2011), pp. 107–126. ISSN : 0065-3438

work page 2011

[2] [1]

A Survey on Active Fault-Tolerant Control Systems

A. Abbaspour et al. “A Survey on Active Fault-Tolerant Control Systems”. In: Electronics 9.9 (2020). ISSN : 2079-9292

work page 2020

[3] [2]

Review and comparison of active space debris capturing and removal methods

M. Shan, J. Guo, and E. Gill. “Review and comparison of active space debris capturing and removal methods”. In: Progress in Aerospace Sciences 80 (2016), pp. 18–32. ISSN : 0376-0421

work page 2016

[4] [3]

Spacecraft attitude control and stabilization: Applica- tions of geometric control theory to rigid body models

P. Crouch. “Spacecraft attitude control and stabilization: Applica- tions of geometric control theory to rigid body models”. In: IEEE Transactions on Automatic Control 29.4 (1984), pp. 321–331

work page 1984

[5] [4]

Detumbling and reorienting underactuated rigid spacecraft

V . Coverstone-Carroll. “Detumbling and reorienting underactuated rigid spacecraft”. In: Journal of Guidance, Control, and Dynamics 19.3 (1996), pp. 708–710

work page 1996

[6] [5]

Global asymptotic stabilization of the attitude and the angular rates of an underactu- ated non-symmetric rigid body

D. Casagrande, A. Astolfi, and T. Parisini. “Global asymptotic stabilization of the attitude and the angular rates of an underactu- ated non-symmetric rigid body”. In: Automatica 44.7 (July 2008), pp. 1781–1789. ISSN : 0005-1098

work page 2008

[7] [6]

Quaternion feedback regulator for large angle maneuvers of underactuated spacecraft

J. S. Hall, M. Romano, and R. Cristi. “Quaternion feedback regulator for large angle maneuvers of underactuated spacecraft”. In: Proceed- ings of the 2010 American Control Conference. 2010, pp. 2867–2872

work page 2010

[8] [8]

Finite-time fault-tolerant attitude stabilization for spacecraft with actuator saturation

Q. Shen et al. “Finite-time fault-tolerant attitude stabilization for spacecraft with actuator saturation”. In: IEEE Transactions on Aerospace and Electronic Systems 51.3 (2015), pp. 2390–2405

work page 2015