Q-Law Control Law

Editor source: Hu Min, Xiao Jinwei, Zhang Tiantian, Tao Xuefeng (2026) "Mission Planning for Orbit Transfer Vehicles Oriented Toward Batch Deployment of Medium and High Orbit Small Satellites"
Narayanaswamy S, Damaren C J. Equinoctial Lyapunov control law for low-thrust rendezvous[J]. Journal of Guidance, Control, and Dynamics, 2023, 46(4): 781-795.
Website: https://cislunarspace.cn

Definition

The Q-law is a feedback control law based on Lyapunov theory, used for orbit transfer control of low-thrust spacecraft. Its core idea is to construct a scalar Q function that describes the state error and ensure its monotonically decreasing behavior, thereby guiding the spacecraft to autonomously converge to the target orbit.

Theoretical Foundation

Lyapunov Stability

The Q-law is based on Lyapunov stability theory:

Construct a positive-definite Lyapunov function $Q(Z) \geq 0$
Design a control law such that $\dot{Q} < 0$ (Q function decreases monotonically)
When $Q \to 0$ , the system converges to the target state

Q Function Definition

In orbit transfers, the Q function is defined as a weighted quadratic form of orbital element errors:

Q(Z) = (1 + W_P P) \sum_{i=1}^{5} W_i S_i \left(\frac{\delta Z_i}{\max_L(\dot{X}_i)}\right)^2

where:

$W_i$ : weights for each orbital element
$P$ : penalty function term
$S_i$ : shaping function
$\delta Z_i$ : orbital element error
$\max_L(\dot{X}_i)$ : maximum rate of change (normalization factor)

Control Law Design

Optimal Thrust Direction

The control law aims to find the thrust direction that maximizes the rate of decrease of the Q function. By computing the extremum of $\dot{Q}$ , the optimal thrust direction can be analytically obtained:

\alpha^* = \arctan(-D_2, -D_1)

\beta^* = \arctan(-D_3, \sqrt{D_1^2 + D_2^2})

where $\alpha, \beta$ are thrust direction vector components, and $D_1, D_2, D_3$ are computational coefficients related to the Q function gradient.

Coasting Arc Mechanism

To achieve a trade-off between propellant mass and time, a coasting arc mechanism is introduced:

u = \begin{cases} [0, 0], & \eta_r \leq \eta_{rel} \text{ or } \eta_a \leq \eta_{abs} \\ T_{max}[\alpha^*, \beta^*], & \text{otherwise} \end{cases}

When thrust efficiency falls below a threshold, the engine is shut down and the spacecraft enters a coasting phase, trading time for propellant savings.

Integration with Modified Equinoctial Elements

Modified Equinoctial Elements (MEE)

Hu Min et al. (2026) adopted a Q-law based on Modified Equinoctial Elements (MEE):

\begin{cases} p = a(1 - e^2) \\ e_1 = e\cos(\omega + \Omega) \\ e_2 = e\sin(\omega + \Omega) \\ h_1 = \tan(i/2)\cos\Omega \\ h_2 = \tan(i/2)\sin\Omega \\ L = \omega + \Omega + \theta \end{cases}

Advantages

Compared to classical orbital elements, MEE offers:

Numerical stability: No singularities near near-circular orbits
Better control performance: Semi-major axis $a$ replacing semi-latus rectum $p$ is more suitable for control
Clear physical meaning: Each component has a clear geometric interpretation

Application in Batch Deployment

State-Dependent Cost Matrix Generation

Hu Min et al. (2026) used the Q-law control law to generate state-dependent transfer cost matrices offline:

For all discrete mass states $k = 0, 1, \dots, N$
For all origin-destination combinations $(i, j)$
Compute transfer costs $C(i, j, k)$ via Q-law simulation
Construct a complete three-dimensional cost matrix

Computational Efficiency

Although the Q-law provides sub-optimal solutions, it offers significant advantages:

Extremely low computational cost: Efficiency improvements of several orders of magnitude compared to direct optimization methods
Strong robustness: Inherent error correction capability
Good real-time performance: Closed-loop strategy can compensate for unmodeled perturbations in real time

Performance Analysis

Research results (Hu Min et al., 2026) show:

Optimization Objective	Propellant Consumption	Transfer Time	CPU Time
Minimize time	96.49 kg	32.87 d	0.00025 s
Minimize propellant	76.07 kg	37.55 d	0.00026 s
Propellant-time trade-off	85.12 kg	35.87 d	0.00025 s

By adjusting the efficiency parameters $\eta_{rel}$ and $\eta_{abs}$ , flexible switching between propellant and time optimization is achievable.

References

Hu Min, Xiao Jinwei, Zhang Tiantian, Tao Xuefeng. Mission Planning for Orbit Transfer Vehicles Oriented Toward Batch Deployment of Medium and High Orbit Small Satellites[J]. Spacecraft Engineering, 2026, 25(3): 634-646. [in Chinese]
Narayanaswamy S, Damaren C J. Equinoctial Lyapunov control law for low-thrust rendezvous[J]. Journal of Guidance, Control, and Dynamics, 2023, 46(4): 781-795.
Lee D, Ahn J. Optimal multitarget rendezvous using hybrid propulsion system[J]. Journal of Spacecraft and Rockets, 2023, 60(2): 456-471.