Stochastic Continuation for Space Trajectory Design
Authors:   Surrey Space Center, University of Surrey, Guildford, United Kingdom, Acciarini Giacomo , Baresi Nicola , Lloyd David , Izzo Dario 
Published: January 2024

Designing and planning periodic orbits in an uncertain environment can be very challenging. Traditionally, trajectories are engineered in a two-step approach. First, nominal orbits are designed, describing the system as a deterministic dynamical model. Then, brute-force Monte Carlo simulations are used to test the initial conditions against model uncertainties and knowledge errors (e.g. imprecise estimation of the spacecraft state). While this approach constitutes the de facto standard in current space trajectory design practices, it leads to very time consuming and potentially suboptimal solutions, due to the fact that the robustness of candidate orbits is only assessed a-posteriori.
In this work, we propose a paradigm shift accounting uncertainties from the very early design phases. In particular, our aim is to study the low-order moments of the satellites’ probability density functions of the state extending numerical continuation techniques to stochastic scenarios. Numerical continuation for deterministic systems is a widely known and used method to find and continue equilibria, periodic orbits or tori in dynamical systems [1]. More recently, stochastic continuation techniques have been proposed as an alternative to these traditional methods to allow the continuation of the low-order moments of probability density functions in uncertain environments, by calculating the fixed points of the first moments (e.g. mean and covariance) of the uncertain parameters [2]. While these techniques have shown to be successful for some dynamical systems [3,6], they have never been applied to the complex and rich dynamical environments of the circular restricted three-body problem. Moreover, they still rely on traditional Monte Carlo approaches to integrate ensembles of initial conditions and estimate the moments to be continued: this can become quite challenging in terms of computational costs for highly nonlinear dynamics, long time horizons, or higher moments. With this work, our goal is to show the potential beneficial application of stochastic continuation to space mission design and how it can be leveraged to incorporate uncertainties from the early design stages.
We modify the standard differential correction procedure used to find fixed points of deterministic Poincare’ mappings by using the Newton-GMRES method: this allows to estimate the Jacobian-vector product for systems in which the analytical expression of the Jacobian is not available [6], as it is the case for the first moments of the uncertain parameters. To alleviate the computational complexity of stochastic continuation techniques, we also show two different approaches to propagate the initial state uncertainties at future times, discussing their merits and drawbacks. First, we use a fast Taylor propagator to numerically integrate the samples of the distribution and reconstruct the propagated uncertainties at future times [7]. Furthermore, we also use differential algebra techniques to build and quickly evaluate high-order Taylor polynomial expansions of Poincare’ mappings as a function of the random variables [8], thereby bypassing the use of Monte Carlo-based approaches to evolve the uncertainty at future times.
We will discuss some preliminary experiments focused on adapting and improving stochastic continuation techniques to find fixed points of the Poincare’ mappings of the moment map in the planar circular restricted three-body problem (PCR3BP). We assume that the initial conditions of the state are random variables, whose probability density function is a multivariate Gaussian (e.g. due to tracking errors). We then first find the fixed points of both the mean and covariance of the distributions and later continue their values as a function of model parameters. Considering Distant Retrograde Orbits (DROs) in the Earth-Moon system as a test case scenario, we show how these solutions find regions of the phase space where the spacecraft would stay in a bounded region for time scales that comply with mission phases and requirements, thereby enabling the design of trajectories that are robust against uncertainties. We also discuss how stochastically continued trajectories differ from their deterministic counterparts, thereby quantifying the sub-optimality of the current a-posteriori design process.
By combining dynamical system theory techniques with state-of-the-art computing and differential algebra advancements, we propose a first of its kind application of stochastic continuation techniques for mission design and analyses, showing its preliminary results for the PCR3BP of the Earth-Moon system.
[1] R Seydel, "Practical Bifurcation and Stability Analysis", Springer Science & Business Media, 2009;
[2] D Barkley, IG Kevrekidis, and AM Stuart, "The Moment Map: Nonlinear Dynamics of Density Evolution via a Few Moments", SIAM Journal of Applied Dynamical
[3] CT Kelley, "Iterative Methods for Solving Linear and Nonlinear Equations", SIAM Publications, 1995;
[4] C Willers, U Thiele, A Archer, DJB Lloyd, and O Kamps, "Adaptive stochastic continuation with a modified lifting procedure applied to complex systems", Physical Review E 102, 032210 (2020). DOI: 10.1103/PhysRevE.102.032210;
[5] SA Thomas, DJB Lloyd, and AC Skeldon, Equation-free analysis of agentbased models and systematic parameter determination. Physica A: Statistical Mechanics and its Applications, Volume 464, Pages 27–53 (2016). DOI: 10.1016/ j.physa.2016.07.043;
[6] C Willers, U Thiele, A Archer, DJB Lloyd, and O Kamps, "Adaptive stochastic continuation with a modified lifting procedure applied to complex systems", Physical Review E 102, 032210 (2020). DOI: 10.1103/PhysRevE.102.032210;
[7] Biscani, Francesco, and Dario Izzo. "Revisiting high-order Taylor methods for astrodynamics and celestial mechanics." Monthly Notices of the Royal Astronomical Society 504.2 (2021): 2614-2628.
[8] N Baresi, X Fu, R Armellin, “A high-order Taylor polynomial approach for continuing trajectories in three-body problems”, AAS/AIAA Astrodynamics Specialist Conference, 2020; Available Online;