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Explicit Globally Asymptotically Stable Attitude Control with Reaction Wheel Angular Momentum and Torque Constraints

Javier Urrios, Rafael Vazquez, Guillermo Pacheco-Ramos

arXiv:2610.34922v1math.OCcs.SYeess.SY

TL;DR

Spacecraft attitude control must achieve stability while enforcing hard limits on reaction-wheel momentum and torque, but existing approaches do not combine explicit feedback, GAS, and both constraints. This paper presents an explicit Lyapunov controller with barrier-inspired momentum feedback and torque scaling. It proves GAS and constraint satisfaction for regulation, and reports a roughly 200x per-step speedup over an OD-CLF-CBF-QP baseline in simulations.

Abstract

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We present an explicit Lyapunov attitude controller for rigid spacecraft actuated by three orthogonally mounted reaction wheels, parameterized using Modified Rodrigues Parameters (MRPs), which enforces hard bounds on wheel angular momentum and applied torque at every instant without solving any online optimization. A barrier-inspired momentum feedback prevents wheel saturation; a continuous positive scalar uniformly scales the feedback whenever a torque limit would otherwise be breached, preserving Lyapunov convergence. For the regulation problem, global asymptotic stability (GAS) is established via LaSalle's invariance principle, and constraint satisfaction is proved analytically for arbitrary symmetric positive-definite inertia matrices. Replacing the barrier by an error-state counterpart extends the design to tracking, with asymptotic (though no longer global) stability and the same safety guarantees under explicit feasibility conditions on the reference. Simulations confirm convergence and constraint satisfaction, and compare performance against other controllers.

I. INTRODUCTION

The paper addresses the gap between explicit, globally stabilizing attitude control and rigorous enforcement of both reaction-wheel momentum and torque limits. It states the spacecraft model and assumptions underlying its regulation and tracking designs.

I. INTRODUCTION

  • Existing approaches do not simultaneously provide explicit feedback, global asymptotic stability, and hard wheel-momentum and torque constraints.Torque-saturation methods ignore wheel momentum, CLF-CBF-QP methods rely on online optimization and relax the CLF constraint, and BLF or ERG approaches have stability or scope limits.Source: §I · p. 1
  • The proposed explicit law combines barrier-inspired momentum feedback with scalar torque scaling, proving constraint satisfaction and GAS for regulation; tracking is asymptotically stable under explicit feasibility conditions.The introduction also reports O(1) per-cycle computation and a 200x speedup over CLF-CBF-QP baselines.Source: §I · p. 1

A. Problem Statement

  • The problem assumes three orthogonal reaction wheels, bounded momentum and torque, full-state and inertia feedback, and no disturbances or model uncertainties.The wheels need not align with the principal inertia axes.Source: A. Problem Statement · p. 2

B. Attitude Representation

  • Attitude is represented by Modified Rodrigues Parameters, with a shadow-set switch at unit norm to avoid the kinematic singularity.The switch maps σ to −σ/∥σ∥² when ∥σ∥₂=1.Source: B. Attitude Representation · p. 2

C. Dynamics and Conservation Law

  • Inertial total angular momentum is conserved, and Assumption (A1) requires its initial magnitude H0 to be below the wheel limit hmax.This leaves every attitude holdable within the momentum bound, while conservation slaves wheel momentum to attitude and angular velocity, so stability claims concern a six-dimensional state.Source: C. Dynamics and Conservation Law · p. 2

III. FEEDBACK CONTROL LAW

The regulation law uses a Lyapunov design with momentum-dependent damping and a scalar that limits torque. The scalar preserves nonincreasing Lyapunov value while potentially changing convergence rate.

A. Lyapunov Function and Regulation Control Law

  • The gain bound reserves torque authority for damping to dominate near momentum saturation despite inertia cross-couplings.The Lyapunov derivative uses the MRP identity and the fact that ω is orthogonal to ω ×·.Source: A. Lyapunov Function and Regulation… · p. 2
  • The positive-diagonal matrix D encodes each wheel’s proximity to its momentum limit, and the momentum-feedback vector z supplies the damping term.The supplied passages identify these roles but do not provide their full expressions.Source: A. Lyapunov Function and Regulation… · p. 2
  • The scale s is the minimum of 1 and the per-axis values s_i, and the control law uses it to scale the damping feedback.The scale is positive and at most 1; it preserves ˙V ≤ 0 while affecting convergence rate.Source: A. Lyapunov Function and Regulation… · p. 2

B. Stability

Under Assumption (A1) and the stated gain conditions, the regulation equilibrium is globally asymptotically stable. The proof uses a nonincreasing Lyapunov function and LaSalle’s invariance principle in MRP coordinates.

B. Stability

  • Under Assumption (A1), the equilibrium (σ, ω) = (0, 0) is globally asymptotically stable for the stated control and gain conditions.The Lyapunov derivative is nonpositive and vanishes exactly when ω=0; the largest invariant set there is the equilibrium, so LaSalle’s principle gives convergence.Source: B. Stability · p. 2
  • At the shadow-set switch, V remains continuous and its derivative expression is unaffected, so the Lyapunov argument carries across the switch.The switch occurs at ∥σ∥₂=1.Source: B. Stability · p. 2
  • The control is discontinuous at the shadow-set switch and may chatter; the GAS claim is in MRP coordinates under this switching.The discontinuity is confined to ∥σ∥=1, corresponding to 180° rotations farthest from the target.Source: B. Stability · p. 3

C. Safety

The regulation law analytically enforces the torque and wheel-momentum bounds under Assumption (A1). The safety argument bounds torque directly and rules out finite-time arrival at a wheel-momentum limit.

C. Safety

  • Under Assumption (A1), every wheel satisfies |hi(t)| < hmax and every torque satisfies |ui(t)| ≤ umax for all t ≥ 0.The result holds for the stated control and gain conditions.Source: C. Safety · p. 3
  • The scalar bound s ≤ s_i keeps each applied torque within ±umax, including when z_i=0.The proof bounds the torque between kpσ_i and −sign(z_i)umax; when z_i=0, kp|σ_i|≤kp<umax.Source: C. Safety · p. 3
  • As h1 approaches hmax, D11 diverges and the proof derives z1→+∞ while the other z components remain proportionate to z1.Across the possible scaling cases, u1 is negative and bounded away from zero near a hypothetical first boundary hit, contradicting arrival at hmax.Source: C. Safety · p. 3
  • For diagonal inertia, the axes decouple, allowing per-axis damping gains and saturations and relaxing the proportional-gain bound to kp < umax.The theorems also remain valid when some initial wheel momenta are outside the safe set; the absolute value in D keeps its corresponding entry positive and drives the state toward equilibrium.Source: C. Safety · p. 3

IV. EXTENSION TO ATTITUDE TRACKING

The controller is extended to track a desired attitude and angular-velocity trajectory while enforcing both hard constraints.

IV. EXTENSION TO ATTITUDE TRACKING

  • The tracking extension uses the same type of control law in error coordinates while enforcing hard momentum and torque constraints.The desired trajectory is (σd(t), ωd(t)).Source: §IV · p. 3

A. Error Coordinates and Dynamics

Tracking is formulated in attitude and body-frame velocity error coordinates, with feedforward terms incorporated into the control law. Under explicit feasibility assumptions, the resulting controller is asymptotically stable and satisfies the torque and wheel-momentum bounds.

A. Error Coordinates and Dynamics

  • The error state uses an attitude MRP and a body-frame velocity error, whose kinematics satisfy σ̇e = C(σe)ωe.The velocity error is ωe = ω−R(σe)ωd.Source: A. Error Coordinates and Dynamics · p. 3

B. Lyapunov Function and Tracking Control Law

  • The tracking Lyapunov function uses the regulation form with σe and ωe, and its derivative is nonpositive under the proposed control.LaSalle’s invariance principle establishes convergence to (σe, ωe) = (0, 0).Source: B. Lyapunov Function and Tracking… · pp. 3–4
  • The tracking law cancels the reference feedforward term ξ along with the proportional error term, leaving only scaled damping in the Lyapunov derivative.Unlike regulation, the gyroscopic term does not generally cancel against ωe and must be canceled explicitly.Source: B. Lyapunov Function and Tracking… · pp. 3–4
  • Under assumptions (A1)–(A3), the tracking equilibrium is asymptotically stable and both torque and wheel momentum remain within their specified bounds.The momentum bound is strict, while the torque bound is non-strict.Source: B. Lyapunov Function and Tracking… · p. 4
  • The torque bound follows from the positive scaling se and the a-priori bound on kpσe,i + ξi.The proof treats ze,i = 0 separately; otherwise the scaled feedback keeps ui between kpσe,i + ξi and −sign(ze,i)umax.Source: B. Lyapunov Function and Tracking… · p. 4
  • Momentum safety is proved using a bound on desired wheel momentum and a contradiction argument for any wheel reaching its limit.The desired momentum is hd = H − JR(σe)ωd, and assumption (A2) supplies the strict bound used in the proof.Source: B. Lyapunov Function and Tracking… · p. 4
  • Tracking is not globally asymptotically stable because the required feasibility condition (A3) can fail when ξ grows with large errors or aggressive references.The paper notes that the gyroscopic component of ξ grows with ∥ωe∥.Source: B. Lyapunov Function and Tracking… · p. 4
  • Condition (A3) is checkable a priori, and the paper says its bound is met for slow references and small initial errors.The regulation controller remains truly GAS because its gyroscopic term vanishes from the Lyapunov derivative without restricting the initial condition.Source: B. Lyapunov Function and Tracking… · p. 5

V. SIMULATIONS

The paper’s simulations compare the proposed controller with a classical controller and an OD-CLF-CBF-QP.

V. SIMULATIONS

  • The simulations compare the proposed controller against a classical controller and the OD-CLF-CBF-QP.The passage introduces the comparison but does not report its outcomes.Source: §V · p. 5

A. Setup

The simulations use a non-diagonal spacecraft inertia matrix, specified momentum and torque limits, and a classical MRP controller as the baseline.

A. Setup

  • The setup uses a rotated diagonal inertia matrix, wheel-momentum limit 5 N m s, and torque limit 0.3 N m.The classical MRP baseline uses the same gains; integration uses MATLAB ode45 with event-triggered shadow-set switching.Source: A. Setup · p. 5

B. Regulation: Safety Under Near-Saturated Conditions

Near saturation, the proposed controller keeps every wheel within its momentum limit, while the classical MRP controller lets one wheel exceed it during the transient.

B. Regulation: Safety Under Near-Saturated Conditions

  • At H0 = 0.96 hmax, the proposed controller keeps all wheels strictly inside their momentum limits.The classical MRP controller exceeds hmax on one wheel during the transient.Source: B. Regulation: Safety Under… · p. 5
  • The proposed controller’s attitude and angular-velocity histories remain comparable to the classical controller’s.Torque histories are also comparable, indicating little performance cost from embedded momentum protection.Source: B. Regulation: Safety Under… · p. 5
  • The classical MRP controller has no momentum management and exceeds the wheel limit during the transient.The proposed controller’s embedded momentum protection prevents that violation.Source: B. Regulation: Safety Under… · p. 5

C. Tracking: Sinusoidal Reference

On the slow sinusoidal reference, the tracking error converges to zero while momentum constraints hold and the Lyapunov function decreases monotonically.

C. Tracking: Sinusoidal Reference

  • The tracking error converges to zero, all wheel momenta remain within hmax, and Ve decreases monotonically.The reference satisfies the stated feasibility checks: ∥ωd∥max ≈ 0.06 rad/s is below the ≈ 0.32 rad/s bound, and ∥ξ∥∞ ≤ 0.161 N m is below ξ* ≈ 0.179 N m.Source: C. Tracking: Sinusoidal Reference · p. 5

D. Comparison with CLF-CBF-QP

The comparison reports that the explicit controller enforces constraints with fixed per-step cost, while QP tunings settle faster but require substantially greater control effort. The conclusion also identifies slower, though guaranteed, convergence and several directions left for future work.

D. Comparison with CLF-CBF-QP

  • Across the reported QP tuning configurations, every configuration respected both constraints and settled in 27–105 s, faster than the best explicit tuning at 234 s in the nominal case.Fig. 3 shows a safety stress-test at H0 = 0.96hmax, comparing the proposed controller with OD-CLF-CBF-QP [11].Source: D. Comparison with CLF-CBF-QP · p. 5, Figure 3 · p. 6
  • QP tunings use at least 3× larger control effort than the explicit controller.The paper attributes this structural trade-off to the explicit law capping its aggressiveness in exchange for analytic guarantees.Source: D. Comparison with CLF-CBF-QP · p. 5
  • Across 1000 random initial conditions, average step time is 0.0037 ms for the explicit controller versus 0.73 ms for the QP, a ∼200x speedup.The explicit formula has fixed O(1) cost, while the QP cost grows cubically with the number of variables.Source: D. Comparison with CLF-CBF-QP · p. 6

VI. CONCLUSION

  • The conclusion reports constraint satisfaction and a ∼200x per-step speedup over OD-CLF-CBF-QP, at the cost of slower but guaranteed convergence.The tracking extension requires explicit, a-priori checkable feasibility conditions.Source: §VI · p. 6
  • Future work includes disturbances, redundant wheels, zero-speed-crossing nonlinearities, and relaxed feasibility conditions for aggressive tracking.The law can also serve as an MPC terminal law within V ≤ c, guaranteeing recursive feasibility and constraint satisfaction while shortening the horizon.Source: §VI · p. 6
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