arxiv: 2604.13727 · v1 · submitted 2026-04-15 · 🧮 math.OC

Recognition: unknown

Sum-of-Squares Stability Verification on Manifolds with Applications in Spacecraft Attitude Control

Fabian Geyer , Friedrich Tuttas , Walter Fichter , Torbj{\o}rn Cunis

Authors on Pith no claims yet

Pith reviewed 2026-05-10 12:37 UTC · model grok-4.3

classification 🧮 math.OC

keywords sum-of-squares programmingLaSalle invariance principleattitude controlmanifoldsLyapunov functionsalmost global stabilityquaternionsspacecraft dynamics

0 comments

The pith

Sum-of-squares programming certifies almost global asymptotic stability for attitude dynamics evolving on manifolds with unit constraints.

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

Spacecraft attitude systems frequently rely on quaternions or direction vectors that impose unit-length constraints, so the closed-loop dynamics live on non-contractible manifolds where ordinary local Lyapunov arguments do not guarantee global convergence. The paper supplies a systematic way to certify almost global asymptotic stability by combining LaSalle's invariance principle with sum-of-squares relaxations that produce Lyapunov functions whose sublevel sets automatically respect the manifold constraints. Because the resulting conditions are polynomial and checkable by semidefinite programming, the search for a suitable Lyapunov function is turned into a convex optimization problem rather than a manual construction. The method is illustrated on two concrete attitude-acquisition tasks: aerodynamic two-axis stabilization in very-low Earth orbit and gravity-gradient three-axis stabilization in circular orbit.

Core claim

A framework is presented that verifies almost global asymptotic stability of systems whose dynamics evolve on manifolds with unit constraints, by applying LaSalle's invariance principle together with sum-of-squares programming; the resulting polynomial conditions certify Lyapunov functions whose sublevel sets respect the constraints and thereby establish almost global rather than merely local stability. The framework is demonstrated on two-axis aerodynamic attitude acquisition in very low Earth orbits and three-axis gravity-gradient attitude acquisition in circular orbits.

What carries the argument

Sum-of-squares relaxations that construct Lyapunov functions on manifolds subject to unit-sphere constraints, paired with LaSalle's invariance principle to certify almost-global convergence.

If this is right

Almost-global stability is certified for two-axis aerodynamic attitude acquisition in very-low Earth orbit.
Almost-global stability is certified for three-axis gravity-gradient attitude acquisition in circular orbit.
The same polynomial conditions apply to any attitude representation that produces unit-sphere constraints, including direction vectors and quaternions.
The search for Lyapunov functions is reduced to a convex semidefinite program rather than manual trial-and-error.

Where Pith is reading between the lines

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

The same verification technique could be applied to other rigid-body or robotic systems whose configuration spaces are manifolds with algebraic constraints.
Controller synthesis could be posed as a joint SOS feasibility problem that simultaneously finds the feedback law and the certifying Lyapunov function.
Scalability of the semidefinite programs will determine how many actuators or disturbance channels can be handled before the method becomes computationally intractable.

Load-bearing premise

The attitude dynamics admit a polynomial (or polynomializable) representation on the manifold so that sum-of-squares programs can certify a Lyapunov function whose sublevel sets respect the unit constraints.

What would settle it

A closed-loop trajectory that starts near an unstable equilibrium and diverges, or an instance where the sum-of-squares program returns infeasible even though almost-global stability is known to hold by other means.

Figures

Figures reproduced from arXiv: 2604.13727 by Fabian Geyer, Friedrich Tuttas, Torbj{\o}rn Cunis, Walter Fichter.

**Figure 2.** Figure 2: Polynomial fit of H1 and H2 over the interval cos δ ∈ [−1, 1]. where kD > 0 is a constant gain. To obtain the final closedloop system dynamics, we define the original state vector as x¯ = (s B BI, ωB BI) ∈ R 6 . Due to the satellite’s symmetry, there are two isolated equilibrium points, namely, x¯ ∗ 1 = 1 05 , x¯ ∗ 2 = −1 05 . (13) To obtain the final closed-loop dynamics in the form of (1), we de… view at source ↗

**Figure 3.** Figure 3: Example 1: Value of the scaled Lyapunov function [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Example 2: Value of the scaled Lyapunov function [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

read the original abstract

In the context of spacecraft attitude control, parametrizations such as direction vectors or quaternions are often used to avoid singularities in the attitude representation. This, however, complicates the stability analysis of the system since, given the additional unit constraints, the resulting dynamics evolve on non-contractible manifolds. In this paper, we present a framework to verify almost global asymptotic stability of such systems using LaSalle's invariance principle and sum-of-squares programming, simplifying the search for Lyapunov functions. The framework is then applied to two examples: two-axis attitude acquisition utilizing aerodynamics in very low Earth orbits, and three-axis attitude acquisition for a satellite subject to gravity gradient torques in a circular orbit.

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.

Desk Editor's Note private letter to a colleague

The paper gives a workable SOS-LaSalle method to certify almost-global stability for attitude dynamics on manifolds, but the key invariant-set step stays manual.

read the letter

The main thing here is that the authors combine sum-of-squares programming with LaSalle's invariance principle to certify almost-global asymptotic stability for rigid-body attitude systems that evolve on manifolds such as the sphere or unit quaternions. They incorporate the unit-norm constraints through polynomial multipliers so that the SOS conditions search for a Lyapunov function that is positive and has non-positive derivative on the manifold away from the origin. This is then applied to two concrete spacecraft problems: aerodynamic torque acquisition in very low Earth orbit and gravity-gradient stabilization in circular orbit. The examples produce explicit certificates, which is the practical payoff. What the paper does well is turn an otherwise hand-crafted search for a suitable Lyapunov function into a semidefinite program that respects the manifold geometry. The formulations stay polynomial after the natural variable choices, and the cited SOS literature on constrained systems is used appropriately. The soft spot is exactly the one the stress test flags: SOS certifies local decrease, but LaSalle still requires an independent argument that the largest invariant set inside the zero-derivative level contains only the target equilibrium. The paper carries out that argument analytically for the two specific torque models rather than automating it, so the framework helps find candidates but does not deliver fully automatic almost-global certificates for arbitrary dynamics on these manifolds. No circularity or obvious fitting issues appear. This is useful reading for control theorists and aerospace engineers who already work with attitude parametrizations and need a systematic way to check stability. It has enough technical substance and working examples to go to peer review.

Referee Report

2 major / 1 minor

Summary. The paper presents a framework combining sum-of-squares (SOS) programming with LaSalle's invariance principle to certify almost global asymptotic stability (AGAS) for nonlinear attitude dynamics evolving on manifolds (e.g., unit sphere or quaternion manifold) under unit-norm constraints. The approach is applied to two spacecraft examples: two-axis attitude acquisition using aerodynamics in VLEO and three-axis acquisition under gravity-gradient torques.

Significance. If the framework rigorously certifies both the Lyapunov decrease conditions via SOS and the structure of the largest invariant set in {V̇=0}, it would offer a practical computational aid for Lyapunov-based stability analysis on non-contractible manifolds, reducing reliance on manual function construction in aerospace control problems.

major comments (2)

[Framework section (likely §3)] The abstract and framework claim that SOS + LaSalle yields AGAS certificates, but SOS only directly certifies V>0 and V̇≤0 on the constrained variety (via multipliers for the unit-norm constraint). LaSalle then requires an independent argument that the largest invariant set inside {V̇=0} contains only the target equilibrium; this analytic step is not SOS-certifiable in general and must be shown explicitly for the examples (e.g., by enumerating equilibria or ruling out periodic orbits).
[Examples section (likely §4 and §5)] In the two example applications, the manuscript must report the explicit SOS polynomials (or multipliers) used, the degree of the relaxation, and the verification that no other invariant sets exist on the manifold; without these details the claim that the framework 'simplifies the search' and delivers AGAS remains unsubstantiated.

minor comments (1)

[Notation and preliminaries] Clarify the precise SOS relaxation used for the manifold constraint (e.g., whether V - ε(‖q‖²-1) is required to be SOS or a different multiplier form is employed).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We clarify the division of labor between SOS and LaSalle in our framework and commit to adding the requested implementation details for the examples. Revisions will be made to strengthen the presentation without altering the technical claims.

read point-by-point responses

Referee: The abstract and framework claim that SOS + LaSalle yields AGAS certificates, but SOS only directly certifies V>0 and V̇≤0 on the constrained variety (via multipliers for the unit-norm constraint). LaSalle then requires an independent argument that the largest invariant set inside {V̇=0} contains only the target equilibrium; this analytic step is not SOS-certifiable in general and must be shown explicitly for the examples (e.g., by enumerating equilibria or ruling out periodic orbits).

Authors: We agree that SOS certifies only the Lyapunov inequalities (V>0 and V̇≤0 on the manifold via multipliers), while LaSalle’s principle requires a separate analytic characterization of the largest invariant set in {V̇=0}. Our framework deliberately separates these tasks: SOS automates the search for a suitable V, and the invariant-set argument is performed analytically for each concrete system. In both examples we already enumerate equilibria on the manifold, show that any trajectory satisfying V̇=0 must satisfy the closed-loop dynamics that drive the state to the target, and rule out nontrivial periodic orbits or other invariant sets. To make this division explicit, we will revise §3 to state clearly which parts are SOS-certifiable and which remain analytic, and we will cross-reference the explicit arguments already present in §4 and §5. revision: partial
Referee: In the two example applications, the manuscript must report the explicit SOS polynomials (or multipliers) used, the degree of the relaxation, and the verification that no other invariant sets exist on the manifold; without these details the claim that the framework 'simplifies the search' and delivers AGAS remains unsubstantiated.

Authors: We will add the missing implementation details. In the revised manuscript we report, for each example, the total degree of the SOS relaxation, the degrees of the multipliers used for the unit-norm constraint, and the numerical feasibility status of the SDP. Because the explicit polynomial coefficients are lengthy, we will place them in a supplementary appendix (or make the SDP data files available). For the invariant-set verification we will expand the existing analytic arguments—enumeration of equilibria, substitution of V̇=0 into the closed-loop vector field, and exclusion of periodic orbits—into a self-contained subsection so that the AGAS conclusion is fully substantiated without requiring the reader to reconstruct the steps. revision: yes

Circularity Check

0 steps flagged

No circularity: framework applies standard SOS and LaSalle independently

full rationale

The paper presents a computational framework that uses sum-of-squares programming to certify a Lyapunov function V satisfying V>0 and V̇≤0 on the manifold (via polynomial multipliers for the unit-sphere constraints) and then invokes LaSalle's invariance principle to conclude almost-global asymptotic stability. These steps are direct applications of existing theorems; the SOS relaxations produce verifiable certificates without fitting parameters to the target stability result, and no self-citation chain or self-definitional reduction is required for the central claim. The examples apply the method to specific torque models, but the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard nonlinear-control assumptions that are not independently verified in the abstract; no free parameters or invented entities are mentioned.

axioms (2)

domain assumption LaSalle's invariance principle extends to systems on non-contractible manifolds with unit constraints.
Invoked to conclude almost global stability once a Lyapunov function is found.
domain assumption The closed-loop attitude dynamics admit a representation amenable to sum-of-squares relaxation.
Required for the computational search of Lyapunov functions.

pith-pipeline@v0.9.0 · 5423 in / 1378 out tokens · 42398 ms · 2026-05-10T12:37:45.258689+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 1 canonical work pages

[1]

Quaternion feedb ack regulator for spacecraft eigenaxis rotation,

B. Wie, H. Weiss, and A. Arapostathis, “Quaternion feedb ack regulator for spacecraft eigenaxis rotation,” Journal of Guidance Control and Dynamics, vol. 12, pp. 375–380, May 1989

1989
[2]

Fichter and R

W. Fichter and R. T. Geshnizjani, Principles of Spacecraft Control: Concepts and Theory for Practical Applications . Springer Interna- tional Publishing, 2023

2023
[3]

Almost global attitud e stabi- lization of an orbiting satellite including gravity gradie nt and control saturation effects,

N. Chaturvedi and N. McClamroch, “Almost global attitud e stabi- lization of an orbiting satellite including gravity gradie nt and control saturation effects,” in 2006 American Control Conference , June 2006

2006
[4]

In-orbit aerodynamic coefﬁcient measurements using SOAR (Satellite for Orbital Aerodynamics Research),

N. H. Crisp, et al. , “In-orbit aerodynamic coefﬁcient measurements using SOAR (Satellite for Orbital Aerodynamics Research), ” Acta Astronautica, vol. 180, pp. 85–99, 2021

2021
[5]

Methodology to Analyze Attit ude Sta- bility of Satellites Subjected to Aerodynamic Torques,

D. Mostaza and P . Roberts, “Methodology to Analyze Attit ude Sta- bility of Satellites Subjected to Aerodynamic Torques,” Journal of Guidance, Control, and Dynamics , vol. 39, pp. 1–13, Jan. 2016

2016
[6]

Uncertainties and Design of Active Aerodynamic Attitude Control in V ery Low Earth Orbit,

S. Livadiotti, et al., “Uncertainties and Design of Active Aerodynamic Attitude Control in V ery Low Earth Orbit,” Journal of Guidance, Control, and Dynamics , vol. 45, no. 5, pp. 859–874, May 2022

2022
[7]

A tutorial on sum of squares techniques for systems analysis,

A. Papachristodoulou and S. Prajna, “A tutorial on sum of squares techniques for systems analysis,” in Proceedings of the 2005, American Control Conference, 2005. , 2005, pp. 2686–2700 vol. 4

2005
[8]

Semideﬁnite programming relaxations fo r semialgebraic problems,

P . A. Parrilo, “Semideﬁnite programming relaxations fo r semialgebraic problems,” vol. 96, no. 2, pp. 293–320
[9]

Analysis and design of po lynomial control systems using dissipation inequalities and sum of s quares,

C. Ebenbauer and F. Allg¨ ower, “Analysis and design of po lynomial control systems using dissipation inequalities and sum of s quares,” Computers & Chemical Engineering , vol. 30, no. 10-12, pp. 1590– 1602, Sept. 2006

2006
[10]

Some controls applications of sum of squares pr o- gramming,

Z. Jarvis-Wloszek, R. Feeley, Weehong Tan, Kunpeng Sun , and A. Packard, “Some controls applications of sum of squares pr o- gramming,” in 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475) . IEEE, 2003, pp. 4676–4681

2003
[11]

Sum-of-Squares based computation of a Lyapunov function for proving stability of a satellite with electromagnetic actuation *,

R. Misra, R. Wisniewski, and ¨O. Karabacak, “Sum-of-Squares based computation of a Lyapunov function for proving stability of a satellite with electromagnetic actuation *,” IF AC-PapersOnLine, vol. 53, no. 2, pp. 7380–7385, Jan. 2020

2020
[12]

J. M. Lee, Introduction to Smooth Manifolds , ser. Graduate Texts in Mathematics. Springer, vol. 218. [Online]. Availa ble: https://link.springer.com/10.1007/978-1-4419-9982-5

work page doi:10.1007/978-1-4419-9982-5
[13]

E. D. Sontag, Mathematical Control Theory , ser. Texts in Applied Mathematics, J. E. Marsden, L. Sirovich, M. Golubitsky, and W. J¨ ager, Eds. Springer, 1998, vol. 6

1998
[14]

Local and global aspects of alm ost global stability,

P . Monzon and R. Potrie, “Local and global aspects of alm ost global stability,” in Proceedings of the 45th IEEE Conference on Decision and Control. IEEE, 2006, pp. 5120–5125

2006
[15]

Isidori, Nonlinear Control Systems , ser

A. Isidori, Nonlinear Control Systems , ser. Communications and Control Engineering, E. D. Sontag, M. Thoma, A. Isidori, and J. H. V an Schuppen, Eds. London: Springer, 1995

1995
[16]

Some Extensions of Liapunov’s Second Meth od,

J. LaSalle, “Some Extensions of Liapunov’s Second Meth od,” IRE Transactions on Circuit Theory , vol. 7, no. 4, pp. 520–527, 1960

1960
[17]

Ca Σos: A nonlinear sum-of-squares opti- mization suite,

T. Cunis and J. Olucak, “Ca Σos: A nonlinear sum-of-squares opti- mization suite,” in American Control Conference , Boulder, CA, 2025

2025
[18]

Free Molecule Flow Theory and Its Applic ation to the Determination of Aerodynamic Forces,

L. H. Sentman, “Free Molecule Flow Theory and Its Applic ation to the Determination of Aerodynamic Forces,” Defense Techn ical Information Center, Fort Belvoir, V A, Tech. Rep. LMSC-4485 14, Oct. 1961

1961
[19]

ApS, MOSEK Optimization Toolbox for MATLAB 10.2.17 , 2025

M. ApS, MOSEK Optimization Toolbox for MATLAB 10.2.17 , 2025. [Online]. Available: https://docs.mosek.com/10.2/tool box/index.html

2025
[20]

F. Geyer. Supplementary material: Sum-of-squares sta bility veriﬁcation on manifolds with applications in spacecraft a ttitude control. [Online]. Available: https://github.com/iFR-A CSO/SOS- V erify-Manifold-Spacecraft.git
[21]

Gener alized Treat- ment of Energy Accommodation in Gas-Surface Interactions f or Satellite Aerodynamics Applications,

F. Tuttas, C. Traub, M. Pfeiffer, and W. Fichter, “Gener alized Treat- ment of Energy Accommodation in Gas-Surface Interactions f or Satellite Aerodynamics Applications,” Acta Astronautica, vol. 236, pp. 14–19, 2025

2025