Robust Spacecraft Low-Thrust Trajectory Design: A Chance-Constrained Covariance-Steering Approach

This paper proposes a systematic method for generating practical and robust low-thrust spacecraft trajectories. One contribution is to consider the change in mass of the spacecraft at two levels: a) the propulsive acceleration and b) the intensity of the stochastic disturbances. A covariance variable formulation is considered, which is computationally more efficient than the factorized covariance implementation. The proposed approach is applied to two- (i.e., planar) and three-dimensional heliocentric phases of spacecraft flight from Earth to Mars under the restricted two-body dynamics. The results highlight the importance of keeping track of mass change to generate more realistic, robust trajectories for interplanetary space missions to avoid underestimation of mission risks.

💡 Research Summary

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The paper introduces a systematic, chance‑constrained covariance‑steering framework for designing low‑thrust interplanetary trajectories that explicitly accounts for spacecraft mass variation at two levels: (a) the mass‑dependent thrust‑to‑acceleration relationship, and (b) the mass‑dependent intensity of stochastic disturbances. Traditional low‑thrust trajectory optimizations often treat spacecraft mass as a constant parameter or ignore its effect on disturbance scaling, which can lead to under‑estimation of mission risk.

The authors start from a continuous‑time stochastic differential equation (SDE) describing the spacecraft dynamics, where the drift term includes gravitational acceleration and thrust divided by the instantaneous mass, and the diffusion term models a force disturbance whose magnitude is scaled by the inverse of the current mass. The goal is to steer the system from an initial Gaussian distribution to a desired final Gaussian distribution while satisfying probabilistic constraints on control effort (maximum thrust) and any state constraints.

To obtain a tractable problem, the nonlinear dynamics are linearized around a reference trajectory and discretized over N uniform time steps, yielding a time‑varying linear stochastic system. The mean dynamics follow a simple affine recursion, while the covariance dynamics are bilinear in the feedback gain K_k and the state covariance P_k. The authors introduce auxiliary variables U_k = K_k P_k and Y_k = K_k P_k K_kᵀ to replace the bilinear terms, and enforce consistency through a Schur‑complement linear matrix inequality (LMI).

The probabilistic thrust constraint is transformed into a deterministic second‑order cone constraint by bounding the largest eigenvalue of Y_k with an auxiliary scalar τ_k and linearizing the square‑root term around a reference value. The cost function, originally the p‑quantile of the integrated thrust norm, is expressed as a sum of the norm of the feed‑forward thrust, the χ²‑quantile scaled τ_k, and a small regularization term ε_Y tr(Y_k) to guarantee strict convexity. A slack variable ζ_k is added to capture linearization error, and a mixed L₁/L₂ penalty is imposed on ζ_k to keep the error small.

The resulting convex semidefinite program (SDP) is solved iteratively using Sequential Convex Programming (SCP). Each iteration consists of (1) linearizing the dynamics around the current trajectory, (2) forming the SDP with the updated matrices, (3) solving the SDP to obtain updated mean states, feed‑forward thrusts, covariances, and feedback gains, and (4) updating the reference trajectory until convergence criteria on state change and slack variables are met.

The methodology is demonstrated on two benchmark problems: a planar (2‑D) Earth‑to‑Mars low‑thrust transfer and a full three‑dimensional version of the same mission. Physical parameters are realistic: initial mass 5000 kg, maximum thrust 5 N, specific impulse 3000 s, solar gravitational parameter μ = 1.3271 × 10¹¹ km³/s², and disturbance intensity γ = 9 × 10⁻⁵ kg·km/s³·½. The horizon is discretized into N = 40 segments, and the initial reference trajectory is obtained from a deterministic minimum‑fuel solution using CasADi.

Results show that when mass variation is ignored, the optimizer underestimates the required thrust margin because the increasing disturbance effect due to decreasing mass is not captured. In the mass‑varying case, the final mass variance grows (≈ 70 kg²), but the chance‑constrained thrust limit (β_u = 0.95) is respected, and the required thrust margin is about 15 % higher. The optimal feed‑forward thrust profile naturally ramps up as mass decreases, reflecting the true engine capability constraints. In the 3‑D case, neglecting mass variation leads to trajectory deviations exceeding 5 % in the most nonlinear portions of the flight.

Key contributions are: (1) integration of mass‑dependent thrust and disturbance models into a covariance‑steering optimal control framework, (2) a covariance‑variable based convexification that avoids the computationally expensive factorized covariance approach, and (3) a practical SCP algorithm that simultaneously satisfies realistic engine thrust limits and probabilistic safety constraints. The authors suggest future extensions to non‑Gaussian disturbances, multi‑planetary mission sequences, and real‑time replanning with onboard fuel management.