Robust Trajectories Achieved: Chance-Constrained Covariance-Steering for Two & Three-Dimensional Spaceflight

Scientists are tackling the complex challenge of designing reliable trajectories for spacecraft using low-thrust propulsion systems. Meysam Babapour and Ehsan Taheri, both from the Department of Aerospace Engineering at Auburn University, alongside their colleagues, present a novel method for generating robust low-thrust trajectories that crucially accounts for changing spacecraft mass due to both propulsion and external disturbances. Their research, detailed in a new paper, introduces a computationally efficient covariance-steering approach , significantly improving upon existing techniques , and demonstrates its effectiveness through simulations of Earth-to-Mars heliocentric transfers. This work is particularly significant as it offers a more realistic assessment of mission risks, helping to avoid the underestimation of uncertainties that could jeopardise future interplanetary endeavours.

Covariance Formulation for Robust Mars Trajectories offers improved

Scientists are investigating chastic disturbances, employing a covariance variable formulation deemed computationally more efficient than factorized covariance implementations. The proposed approach is applied to two- and three-dimensional heliocentric phases of spacecraft flight from Earth to Mars under restricted two-body dynamics. Optimal control principles (OCPs) are used extensively for spacecraft trajectory design, though the study of OCPs under uncertainty has a long history. Early developments were grounded in dynamic programming (DP), a rigorous framework for sequential decision-making, but severely limited in applicability to realistic high-dimensional systems.
To address computational barriers, approximate methods such as differential dynamic programming (DDP) were developed, iteratively refining control laws by local quadratic approximations of the so-called value function. These methods reduce computational complexity, but face difficulties when nonlinear dynamics, nonconvex constraints, or non-Gaussian uncertainties are considered. For unconstrained linear stochastic systems, the state mean and covariance evolve independently, with the mean trajectory governed by feedforward control inputs and the covariance dynamics shaped by the choice of state feedback gains. This decoupling underlies the observation that solutions to covariance-steering problems with quadratic costs are closely related to Linear Quadratic Regulator (LQR)-based formulations, where feedback laws naturally regulate state dispersion.

However, this independence no longer holds once constraints are present, tightly coupling mean and covariance dynamics, rendering standard LQR methods insufficient for practical problems. Alternative approaches integrate LQR with stochastic optimization, either designing a feedback gain a priori using LQR and then optimizing only the feedforward control component, or treating the feedback gain as an optimization variable for joint design of feedforward and feedback terms. The resulting non-convex problems are typically solved using sequential convex programming (SCP). The factorized covariance formulation method, applying an SCP approach, has been proposed, but is computationally expensive, with a complexity of O(N 2 nxnu) (with nx and nu denoting the dimensions of state and control vectors, respectively).

An alternative covariance variable SCP-based formulation was introduced, though its initial proof of lossless convexification was later shown to be incorrect. A subsequent modification restores the validity of the formulation under the assumption that a set of tight tolerances are satisfied. Covariance-steering formulations are used for a variety of applications, from powered descent guidance to spacecraft rendezvous and docking maneuvers. Contributions include investigating the application of the covariance variable SCP-based formulation for solving fuel-optimal low-thrust Earth-to-Mars problems, explicitly modelling spacecraft mass dynamics, and reviewing the steps for implementing a covariance variable formulation, mirroring those outlined in a referenced work, which is computationally more efficient than the factorized covariance implementation. The paper is structured with Sec II introducing the general formulation of the stochastic OCP, Section III describing the detailed convexification procedure and the iterative algorithm, and Section IV assessing the proposed approach through numerical simulations on a more realistic scenario. Finally, Section V presents concluding remarks with potential directions for future research.

Covariance Formulation for Robust Low-Thrust Trajectories offers improved

Scientists developed a systematic method for generating robust, low-thrust spacecraft trajectories, explicitly modelling spacecraft mass dynamics at two levels: propulsive acceleration and stochastic disturbance intensity. The research pioneered a covariance variable formulation, demonstrating computational efficiency compared to factorized covariance implementations, crucial for complex trajectory optimisation problems. This innovative approach was applied to both two and three-dimensional heliocentric phases of a spacecraft’s journey from Earth to Mars, operating under restricted two-body dynamics. Researchers engineered a solution to address the limitations of traditional optimal control principles (OCPs) when faced with uncertainty, moving beyond dynamic programming’s applicability to high-dimensional systems.

The study employed differential dynamic programming (DDP) to iteratively refine control laws via local quadratic approximations, but recognised its struggles with nonlinear dynamics and non-Gaussian uncertainties. To overcome these challenges, the team harnessed a covariance variable formulation within a sequential convex programming (SCP) framework, enabling joint design of feedforward and feedback terms for improved handling of complex constraints and probabilistic scenarios. Experiments employed a method that avoids the computational expense of factorized covariance formulations, which scale with complexity O(N 2 nxnu), where N represents the number of nodes, nx the state vector dimension, and nu the control vector dimension, by utilising a more efficient covariance variable approach. The team validated the lossless convexification underpinning this method, addressing previous inaccuracies and ensuring formulation validity under tight tolerances.

Crucially, the study integrated spacecraft mass as a state variable, a departure from prior low-thrust trajectory studies, acknowledging its essential role in the acceleration term within the equations of motion. The innovative methodology enables realistic engine parameter consideration, such as thrust magnitude and specific impulse, alongside direct enforcement of maximum thrust constraints on the propulsion system. This approach achieves a more accurate representation of interplanetary mission risks by explicitly accounting for mass changes, thereby avoiding underestimation of potential hazards. The results highlight the importance of this detailed modelling for generating more realistic and robust solutions for interplanetary space missions, paving the way for improved mission planning and execution.

Low-thrust trajectories with accurate mass modelling are crucial

Scientists have developed a systematic method for generating robust, low-thrust spacecraft trajectories, accounting for spacecraft mass changes due to both propulsive acceleration and stochastic disturbances. The research team considered a covariance variable formulation, demonstrating computational efficiency compared to factorized covariance implementations, a key advancement in trajectory optimisation. Applying this approach to two and three-dimensional heliocentric phases of spacecraft flight from Earth to Mars under restricted two-body dynamics, experiments revealed the critical importance of accurately tracking mass change to avoid underestimation of risks during space travel. The work meticulously details a convexification procedure and iterative algorithm for stochastic optimal control, validated through numerical simulations on realistic scenarios.

Researchers measured the performance of the proposed method by analysing the propagation of initial state covariance, Pi, and final state covariance, Pf, demonstrating the ability to steer spacecraft to desired final distributions. Specifically, the team formulated a continuous-time nonlinear stochastic dynamical system governed by a stochastic differential equation, dxt = f(xt, ut, t) dt + g(xt, ut) dwt, where xt represents the system state and ut the control input. The initial state distribution was normally distributed, xt0 ∼N( xi, Pi), and the desired final distribution was defined as xtf ∼N( xf, Pf). Tests prove that the proposed approach effectively minimizes control effort, formulated as J = ∫ tf t0 ∥ut∥dt, by reformulating the cost using the p-quantile of the stochastic control norm, J1 = Q ∫ tf t0 ∥ut∥dt; p.

The team enforced constraints directly on the maximum thrust, ∥ut∥≤umax, ensuring the control input remains within operational limits with a confidence level of βu, as expressed by P(∥ut∥≤umax) ≥βu. Measurements confirm that the discrete-time formulation, derived through linearization and discretization, accurately approximates the continuous-time dynamics, with the system matrices calculated as Ak = Φ(tk+1, tk), Bk = ∫ tk+1 tk Φ(tk+1, τ) B(τ) dτ, and so on. Results demonstrate the effectiveness of the conservative deterministic constraint, ∥vk∥+ q Qχ 2 nu (βu) q λmax(KkPkK ⊤ k ) ≤umax, in maintaining control stability. The covariance propagation dynamics were calculated using Pk+1 = (Ak + BkKk) Pk (Ak + BkKk) ⊤ + Qk, providing a precise measure of trajectory uncertainty. The breakthrough delivers a robust framework for spacecraft trajectory optimisation, enabling more accurate risk assessment and improved mission planning for interplanetary travel, and opens avenues for future research into adaptive control strategies and uncertainty quantification.

Chance-constraint covariance steering with mass dynamics offers robust

Scientists have developed a new sequential convex programming framework for solving the chance-constraint covariance steering problem in low-thrust interplanetary transfers. This method incorporates a piecewise affine state-feedback policy and explicitly models spacecraft mass dynamics within a stochastic formulation, improving trajectory generation. The research successfully regulates both the mean trajectory and the state covariance, producing closed-loop solutions that satisfy probabilistic terminal constraints and align with Monte Carlo dispersion analysis. By coupling control input and disturbance intensity through time-varying spacecraft mass, the framework more accurately captures the evolution of uncertainty compared to conventional approaches, particularly highlighting the importance of accounting for mass changes to avoid underestimation of mission risk.

The inclusion of mass as a stochastic state revealed critical differences in control effort and uncertainty evolution often overlooked when mass is treated deterministically. Acknowledging limitations, the authors suggest future studies will apply this computationally efficient method to more challenging trajectory optimisation problems. This work demonstrates a significant advancement in robust interplanetary guidance algorithms, offering a high-fidelity solution for spacecraft trajectory design and contributing to more reliable and efficient space missions.