Building a Better B-Dot: Fast Detumbling with Non-Monotonic Lyapunov Functions

TL;DR

The paper tackles the underactuated detumbling problem for spacecraft using magnetorquers by introducing a discrete-time non-monotonic Lyapunov controller that predicts the future magnetic field along the orbit to avoid uncontrollable configurations. The method formulates a convex optimization over a two-step control vector, with a regularizer and a causally computable prediction of $B_{k+1}$, guaranteeing average Lyapunov decrease while allowing transient momentum increases. Monte-Carlo simulations against five established detumbling controllers show substantially faster detumbling and lower final angular momentum, with robust performance across 100 random initial conditions and realistic sensor noise. The work highlights the practicality of predicting future controllability and sets the stage for integrating a geomagnetic-field derivative estimator into closed-loop operation.

Abstract

Spacecraft detumbling with magnetic torque coils is an inherently underactuated control problem. Contemporary and classical magnetorquer detumbling methods do not adequately consider this underactuation, and suffer from poor performance as a result. These controllers can get stuck on an uncontrollable manifold, resulting in long detumbling times and high power consumption. This work presents a novel detumble controller based on a non-monotonic Lyapunov function that predicts the future magnetic field along the satellite's orbit and avoids uncontrollable configurations. In comparison to other controllers in the literature, our controller detumbles a satellite in significantly less time while also converging to lower overall angular momentum. We provide a derivation and proof of convergence for our controller as well as Monte-Carlo simulation results demonstrating its performance in representative use cases.

Figures (6)

• Figure 1: Two simulation runs of the B-cross controller in \ref{['eq:bcross']} with different gains. As the gain increases, the controller convergence gets worse because it gets stuck in the uncontrollable subspace where $\omega$ and $B$ are parallel.
• Figure 2: Gain sweep study showing the effect each controller's gain has on its detumbling performance for a single initial condition. The solid green line is the gain that was used for the Monte-Carlo simulation experiment shown in \ref{['fig:detumble_time_histogram', 'fig:momentum_vs_time', 'fig:final_magnitude_histogram']}. The other lines are for gains varying from two orders of magnitude lower to three orders of magnitude higher than the chosen gain.
• Figure 3: Final momentum magnitude for the gain sweep trajectories shown in \ref{['fig:gain_sweep']}. The dotted horizontal line indicates the 1% threshold used to determine convergence. The square marker corresponds to the gain used in the Monte-Carlo simulation experiment shown in \ref{['fig:detumble_time_histogram', 'fig:momentum_vs_time', 'fig:final_magnitude_histogram']}.
• Figure 4: Momentum magnitude versus time plot for each of the controllers discussed in this paper. The Discrete Non-monotonic controller differs from the other controllers in that the system momentum increases before decreasing and converging to zero. This is the key distinction of this controller and allows it to have faster convergence times than other detumbling controllers.
• Figure 5: Cumulative distribution of detumble times for each of the controllers discussed in this paper. Detumble times are defined as the time when the satellite first reaches 1% of its initial angular momentum. Each simulation run ended at two hours, so only detumble times less than two hours are counted. Our Discrete Non-monotonic controller has the lowest average detumble time; it is also the only controller to detumble for all simulation runs.