Ryne Beeson

Nonlinear filtering with standard PF methods requires mitigative techniques to quell weight degeneracy, such as resampling. This is especially true in high-dimensional systems with sparse observations. Unfortunately, such techniques are also fragile when applied to systems with exceedingly rare events. Nonlinear systems with these properties can be assimilated effectively with a control-based PF method known as the nPF, but this method has a high computational cost burden. In this work, we aim to retain this strength of the nudged method while reducing the computational cost by introducing a variational method into the algorithm that acts as a continuous pseudo-observation path. By maintaining a PF representation, the resulting algorithm continues to capture an approximation of the filtering distribution, while reducing computational runtime and improving robustness to the "rare" event of switching phases. Preliminary testing of the new approach is demonstrated on a stochastic variant of the nonlinear and chaotic L63 model, which is used as a surrogate for mimicking "rare" events. The new approach helps to overcome difficulties in applying the nPF for realistic problems and performs favorably with respect to a standard PF with a higher number of particles.

Nonlinear filtering with standard PF methods requires mitigative techniques to quell weight degeneracy, such as resampling. This is especially true in high-dimensional systems with sparse observations. Unfortunately, such techniques are also fragile when applied to systems with exceedingly rare events. Nonlinear systems with these properties can be assimilated effectively with a control-based PF method known as the nPF, but this method has a high computational cost burden. In this work, we aim to retain this strength of the nudged method while reducing the computational cost by introducing a variational method into the algorithm that acts as a continuous pseudo-observation path. By maintaining a PF representation, the resulting algorithm continues to capture an approximation of the filtering distribution, while reducing computational runtime and improving robustness to the "rare" event of switching phases. Preliminary testing of the new approach is demonstrated on a stochastic variant of the nonlinear and chaotic L63 model, which is used as a surrogate for mimicking "rare" events. The new approach helps to overcome difficulties in applying the nPF for realistic problems and performs favorably with respect to a standard PF with a higher number of particles.

Preliminary mission design of low‑thrust spacecraft trajectories in the Circular Restricted Three‑Body Problem is a global search characterized by a complex objective landscape and numerous local minima. Formulating the problem as sampling from an unnormalized distribution supported on neighborhoods of locally optimal solutions, provides the opportunity to deploy Markov chain Monte Carlo methods and generative machine learning. In this work, we extend our previous self-supervised diffusion model fine-tuning framework to employ gradient‑informed Markov chain Monte Carlo. We compare two algorithms - the Metropolis‑Adjusted Langevin Algorithm and Hamiltonian Monte Carlo - both initialized from a distribution learned by a diffusion model. Derivatives of an objective function that balances fuel consumption, time of flight and constraint violations are computed analytically using state transition matrices. We show that incorporating the gradient drift term accelerates mixing and improves convergence of the Markov chain for a multi-revolution transfer in the Saturn-Titan system. Among the evaluated methods, MALA provides the best trade-off between performance and computational cost. Starting from samples generated by a baseline diffusion model trained on a related transfer, MALA explicitly targets Pareto-optimal solutions. Compared to a random walk Metropolis algorithm, it increases the feasibility rate from $17.34\%$ to $63.01\%$ and produces a denser, more diverse coverage of the Pareto front. By fine-tuning a diffusion model on the generated samples and associated reward values with reward-weighted likelihood maximization, we learn the global solution structure of the problem and eliminate the need for a tedious separate data generation phase.

In this paper, we present a bi-level optimal control framework for designing low-thrust spacecraft trajectories with robustness against missed-thrust-events. The upper-level (UL) problem generates a nominal trajectory assuming full control authority, while each lower-level (LL) problem computes the optimal recovery maneuver following a missed-thrust-event along the nominal solution. Under suitable regularity conditions ensuring uniqueness and smoothness of the LL response, the hierarchy admits a single-level reformulation by embedding the LL first-order optimality conditions within the UL constraints. We further establish a robustness certificate, which provides an upper bound on the maximum admissible missed-thrust-duration for which the structural assumptions remain valid for the LL problem. The bound depends explicitly on precomputable dynamical quantities along the nominal solution, enabling rapid evaluation over large ensembles without iterative solves. Numerical experiments show that while the certificate identifies when modeling assumptions are valid, it does not fully characterize recoverability after missed-thrust-events. A finite-horizon controllability-energy analysis is therefore used to interpret recovery beyond the theoretical bounds. Collectively, these results provide a deterministic, certifiable approach for incorporating robustness directly into trajectory design, replacing post-hoc margin allocation techniques with formal guarantees.

In this paper, we present a bi-level optimal control framework for designing low-thrust spacecraft trajectories with robustness against missed-thrust-events. The upper-level (UL) problem generates a nominal trajectory assuming full control authority, while each lower-level (LL) problem computes the optimal recovery maneuver following a missed-thrust-event along the nominal solution. Under suitable regularity conditions ensuring uniqueness and smoothness of the LL response, the hierarchy admits a single-level reformulation by embedding the LL first-order optimality conditions within the UL constraints. We further establish a robustness certificate, which provides an upper bound on the maximum admissible missed-thrust-duration for which the structural assumptions remain valid for the LL problem. The bound depends explicitly on precomputable dynamical quantities along the nominal solution, enabling rapid evaluation over large ensembles without iterative solves. Numerical experiments show that while the certificate identifies when modeling assumptions are valid, it does not fully characterize recoverability after missed-thrust-events. A finite-horizon controllability-energy analysis is therefore used to interpret recovery beyond the theoretical bounds. Collectively, these results provide a deterministic, certifiable approach for incorporating robustness directly into trajectory design, replacing post-hoc margin allocation techniques with formal guarantees.

The global search for optimal long time-duration, low-thrust spacecraft trajectories is a computationally expensive problem, that is characterized by clustering patterns in locally optimal solutions. During preliminary mission design, mission parameters have not been fully defined yet, necessitating that trajectory designers efficiently generate high-quality control solutions across a wide range of different scenarios. Generative machine learning models can be trained to learn how the solution structure varies with respect to an evolving mission parameter, thereby accelerating the global search for a large number of missions with varying parameters. In this work, state-of-the-art diffusion models are integrated with the indirect approach for trajectory optimization within a global search framework. The main difficulty with the indirect method lies in generating good initial guesses for the non-intuitive costate variables, which are crucial for solver convergence. By training a diffusion model to learn the structure of high-quality solutions in costate space, it can generate initial costate guesses for new mission parameters. This framework is tested on two low-thrust transfers of different complexity in the circular restricted three-body problem. By generating and analyzing a training data set, we develop mathematical relations and techniques to understand the complex structures in the costate domain of locally optimal solutions for these problems. A diffusion model is trained on this data and successfully predicts how the costate solution structure changes, based on the maximum spacecraft thrust magnitude. We warm-start a numerical solver with initial costates sampled from the diffusion model for problems with unseen thrust magnitudes and compare the number of solutions generated per minute to samples from a uniform distribution and from an adjoint control transformation. Results show that the diffusion model accelerates the global search process by one to two orders of magnitude, allowing for rapid generation of high-quality solutions.

Trajectory prediction and planning are essential for autonomous vehicles to navigate safely and efficiently in dynamic environments. Traditional approaches often treat them separately, limiting the ability for interactive planning. While recent diffusion-based generative models have shown promise in multi-agent trajectory generation, their slow sampling is less suitable for high-frequency planning tasks. In this paper, we leverage the consistency model to build a predictive planner that samples from a joint distribution of ego and surrounding agents, conditioned on the ego vehicle's navigational goal. Trained on real-world human driving datasets, our consistency model generates higher-quality trajectories with fewer sampling steps than standard diffusion models, making it more suitable for real-time deployment. To enforce multiple planning constraints simultaneously on the ego trajectory, a novel online guided sampling approach inspired by the Alternating Direction Method of Multipliers (ADMM) is introduced. Evaluated on the Waymo Open Motion Dataset (WOMD), our method enables proactive behavior such as nudging and yielding, and also demonstrates smoother, safer, and more efficient trajectories and satisfaction of multiple constraints under a limited computational budget. The project website is at https://anjianli21.github.io/projects/predictive_planner/.

The increasing strategic emphasis on long-term cislunar operations has catalyzed academic research efforts in recent years toward the development of efficient low-thrust mission architectures to key cislunar orbits. These missions, typically characterized by long thrust arcs, are particularly vulnerable to missed thrust events, which can disrupt mission performance if not adequately addressed early in the trajectory design process. Existing approaches for missed thrust design involve solving high-dimensional nonlinear programs, where constructing effective initial guesses can be challenging. Efficient global search approaches are therefore essential during the preliminary mission design phase, where rapid exploration of the solution space is necessary under evolving operational constraints. To improve the computational efficiency, solution quality, and depth of robustness of solutions from global search, we compare two initial guess generation strategies: a baseline nonconditional approach, which samples from a static distribution with global support, and a conditional approach, which generates initial guesses conditioned on solutions to simpler problems with lower depths of robustness. The conditional approach offers a sequential procedure for solving increasingly robust problems. We validate the improvements in the conditional approach using a low-thrust, minimum-fuel case study for the Lunar Gateway Power and Propulsion Element, where our results show significant improvements in feasibility ratio, convergence rate, and solution quality (measured by reduced propellant consumption), demonstrating its potential in missed thrust design.

Global search and optimization of long-duration, low-thrust spacecraft trajectories with the indirect method is challenging due to a complex solution space and the difficulty of generating good initial guesses for the costate variables. This is particularly true in multibody environments. Given data that reveals a partial Pareto optimal front, it is desirable to find a flexible manner in which the Pareto front can be completed and fronts for related trajectory problems can be found. In this work we use conditional diffusion models to represent the distribution of candidate optimal trajectory solutions. We then introduce into this framework the novel approach of using Markov Chain Monte Carlo algorithms with self-supervised fine-tuning to achieve the aforementioned goals. Specifically, a random walk Metropolis algorithm is employed to propose new data that can be used to fine-tune the diffusion model using a reward-weighted training based on efficient evaluations of constraint violations and missions objective functions. The framework removes the need for separate focused and often tedious data generation phases. Numerical experiments are presented for two problems demonstrating the ability to improve sample quality and explicitly target Pareto optimality based on the theory of Markov chains. The first problem does so for a transfer in the Jupiter-Europa circular restricted three-body problem, where the MCMC approach completes a partial Pareto front. The second problem demonstrates how a dense and superior Pareto front can be generated by the MCMC self-supervised fine-tuning method for a Saturn-Titan transfer starting from the Jupiter-Europa case versus a separate dedicated global search.

Efficiently solving the continuous-time signal and discrete-time observation filtering problem for chaotic dynamical systems presents unique challenges in that the advected distribution between observations may encounter a separatrix structure that results in the prior distribution being far from the observation or the distribution may become split into multiple disjoint components. In an attempt to sense and overcome these dynamical issues, as well as approximate a non-Gaussian distribution, a nudged particle filtering approach has been introduced. In the nudged particle filter method a control term is added, but has the potential drawback of degenerating the weights of the particles. To counter this issue, we introduce an intermediate resampling approach based on the modified Cramér-von Mises distance. The new method is applied to a challenging scenario of the non-chaotic, unforced nonlinear Duffing oscillator, which possesses a separatrix structure. Our results show that it consistently outperforms the standard particle filter with resampling and original nudged particle filter.

Space situational awareness (SSA) relies on accurate and efficient uncertainty realism and propagation (UR&P). In the cislunar domain, there is growing interest in the use of orbits near the collinear libration points, which are relative equilibria in the simplified circular restricted three-body model. These equilibria are hyperbolic and therefore possess dynamical structures in phase space that separate the flow and present difficulties for UR&P. In this paper, we build on prior efforts to define a canonical distribution on a cylindrical space given by the application of normal form theory at a collinear equilibria. The canonical distribution is constructed to have components with disjoint support on the phase space, each of which properly represents the characteristic dynamics for the specific subset of phase space. This is achieved by a product of a multivariate folded normal distribution and conditional normal and von Mises distributions. Comparisons to a regularized distribution, the generalized Bernoulli-Gauss-von Mises, are made.

As low-thrust space missions grow in prevalence, it is becoming increasingly important to design low-thrust trajectories with robustness against unforeseen thruster outages or missed thrust events. Accounting for such anomalies is particularly important in chaotic multibody systems, such as the cislunar realm, where pertinent dynamical structures constrain the dynamical flow. Yet it remains unclear how these dynamical structures influence robust trajectory design. This paper provides the first comprehensive statistical comparison between nonrobust and robust trajectories in relation to the invariant manifolds of resonant orbits in a circular restricted three-body problem. For both the nonrobust and robust solution categories, the optimal subset exhibits stronger alignment with the invariant manifolds, whereas the broader feasible set can sometimes deviate significantly. On average, the robust optimal trajectories shadow the invariant manifolds as closely as the nonrobust optimal trajectories and, in some instances, exhibit even stronger alignment than their nonrobust counterparts. By maintaining proximity to these invariant manifolds, the robust low-thrust solutions are able to efficiently leverage the global dynamical flow to achieve optimality even under operational uncertainties.

Diffusion models have recently emerged as effective generative frameworks for trajectory optimization, capable of producing high-quality and diverse solutions. However, training these models in a purely data-driven manner without explicit incorporation of constraint information often leads to violations of critical constraints, such as goal-reaching, collision avoidance, and adherence to system dynamics. To address this limitation, we propose a novel approach that aligns diffusion models explicitly with problem-specific constraints, drawing insights from the Dynamic Data-driven Application Systems (DDDAS) framework. Our approach introduces a hybrid loss function that explicitly measures and penalizes constraint violations during training. Furthermore, by statistically analyzing how constraint violations evolve throughout the diffusion steps, we develop a re-weighting strategy that aligns predicted violations to ground truth statistics at each diffusion step. Evaluated on a tabletop manipulation and a two-car reach-avoid problem, our constraint-aligned diffusion model significantly reduces constraint violations compared to traditional diffusion models, while maintaining the quality of trajectory solutions. This approach is well-suited for integration into the DDDAS framework for efficient online trajectory adaptation as new environmental data becomes available.

Efficiently solving the continuous-time signal and discrete-time observation filtering problem for chaotic dynamical systems presents unique challenges in that the advected distribution between observations may encounter a separatrix structure that results in the prior distribution being far from the observation or the distribution may become split into multiple disjoint components. In an attempt to sense and overcome these dynamical issues, as well as approximate a non-Gaussian distribution, a nudged particle filtering approach has been introduced. In the nudged particle filter method a control term is added, but has the potential drawback of degenerating the weights of the particles. To counter this issue, we introduce an intermediate resampling approach based on the modified Cramér-von Mises distance. The new method is applied to a challenging scenario of the non-chaotic, unforced nonlinear Duffing oscillator, which possesses a separatrix structure. Our results show that it consistently outperforms the standard particle filter with resampling and original nudged particle filter.

Trajectory prediction and planning are essential for autonomous vehicles to navigate safely and efficiently in dynamic environments. Traditional approaches often treat them separately, limiting the ability for interactive planning. While recent diffusion-based generative models have shown promise in multi-agent trajectory generation, their slow sampling is less suitable for high-frequency planning tasks. In this paper, we leverage the consistency model to build a predictive planner that samples from a joint distribution of ego and surrounding agents, conditioned on the ego vehicle's navigational goal. Trained on real-world human driving datasets, our consistency model generates higher-quality trajectories with fewer sampling steps than standard diffusion models, making it more suitable for real-time deployment. To enforce multiple planning constraints simultaneously on the ego trajectory, a novel online guided sampling approach inspired by the Alternating Direction Method of Multipliers (ADMM) is introduced. Evaluated on the Waymo Open Motion Dataset (WOMD), our method enables proactive behavior such as nudging and yielding, and also demonstrates smoother, safer, and more efficient trajectories and satisfaction of multiple constraints under a limited computational budget.

Efficiently solving the continuous-time signal and discrete-time observation filtering problem for chaotic dynamical systems presents unique challenges in that the advected distribution between observations may encounter a separatrix structure that results in the prior distribution being far from the observation or the distribution may become split into multiple disjoint components. In an attempt to sense and overcome these dynamical issues, as well as approximate a non-Gaussian distribution, a nudged particle filtering approach has been introduced. In the nudged particle filter method a control term is added, but has the potential drawback of degenerating the weights of the particles. To counter this issue, we introduce an intermediate resampling approach based on the modified Cramér-von Mises distance. The new method is applied to a challenging scenario of the non-chaotic, unforced nonlinear Duffing oscillator, which possesses a separatrix structure. Our results show that it consistently outperforms the standard particle filter with resampling and original nudged particle filter.

Preliminary spacecraft trajectory optimization is a parameter dependent global search problem that aims to provide a set of solutions that are of high quality and diverse. In the case of numerical solution, it is dependent on the original optimal control problem, the choice of a control transcription, and the behavior of a gradient based numerical solver. In this paper we formulate the parameterized global search problem as the task of sampling a conditional probability distribution with support on the neighborhoods of local basins of attraction to the high quality solutions. The conditional distribution is learned and represented using deep generative models that allow for prediction of how the local basins change as parameters vary. The approach is benchmarked on a low thrust spacecraft trajectory optimization problem in the circular restricted three-body problem, showing significant speed-up over a simple multi-start method and vanilla machine learning approaches. The paper also provides an in-depth analysis of the multi-modal funnel structure of a low-thrust spacecraft trajectory optimization problem.

Spacecraft trajectory design is a global search problem, where previous work has revealed specific solution structures that can be captured with data-driven methods. This paper explores two global search problems in the circular restricted three-body problem: hybrid cost function of minimum fuel/time-of-flight and transfers to energy-dependent invariant manifolds. These problems display a fundamental structure either in the optimal control profile or the use of dynamical structures. We build on our prior generative machine learning framework to apply diffusion models to learn the conditional probability distribution of the search problem and analyze the model's capability to capture these structures.

The growing interest in the cislunar domain over the past decade has led to an increasing demand for low-thrust missions to key orbits within this region. These low-thrust missions, typically characterized by long thrust arcs, are highly susceptible to operational disruptions such as unforeseen thruster outages or missed thrust events. Consequently, there is a critical need for efficient trajectory design frameworks which incorporate robustness against such anomalies. In this study, we utilize a robust trajectory design framework to explore the solution space for the Power and Propulsion Element (PPE) module to the Earth-Moon L2 Southern 9:2 Near Rectilinear Halo Orbit. We propose algorithmic enhancements to improve the global search for robust solutions, and present a comprehensive analysis of two approaches: a nonconditional approach which involves a purely random search for robust solutions versus a conditional approach which involves warm-starting the search for robust solutions using the non-robust solutions. Our results indicate that by using non-robust solutions as initial guesses for the robust solutions, it is possible to achieve significant improvements in both the rate of convergence and the robustness of the final solutions.

Most aspects of space situational awareness (SSA) rely on accurate and efficient uncertainty realism, propagation, and nonlinear filtering. A new frontier for SSA is the application to the cislunar realm, which lacks a global orbital element coordinate set. The dynamics of a representative model, the circular restricted three-body problem (CR3BP), for the cislunar domain provides the opportunity to define local orbital elements using dynamical systems techniques such as normal form theory. Motivated the structure of the CR3BP SSA problem, we construct a generalized Bernoulli Gauss von Mises distribution, that is defined on local orbit element coordinates generated from normal form theory at a saddle-center-center equilibrium point, and show its ability to capture what may be a common deformation mode of the CR3BP.

Particle flow methods are effective in resolving the particle degeneracy issue in the standard particle filtering algorithm. However, flow methods have their own difficulties, such as the necessity to solve a Poisson equation in the feedback particle filtering (FPF) method. This is computationally heavy, and we observe a numerical sensitivity and singularity issue dependent on parameter selection when applying to chaotic dynamical systems with limited particle size and coarse integration step size. In this paper, we address the numerical singularity issue by flowing particles in the unstable subspace (UAS), and we name the novel method the projected FPF. It brings the local dynamical information into the assimilation step by using the finite-time Lyapunov exponents and vectors to project observations and particle states to the UAS, where the error diverges. The projected FPF is tested against the Lorenz 1963 model – a nonlinear, low-dimensional, chaotic dynamical system.

Optimal trajectory design is computationally expensive for nonlinear and high-dimensional dynamical systems. The challenge arises from the non-convex nature of the optimization problem with multiple local optima, which usually requires a global search. Traditional numerical solvers struggle to find diverse solutions efficiently without appropriate initial guesses. In this paper, we introduce DiffuSolve, a general diffusion model-based solver for non-convex trajectory optimization. An expressive diffusion model is trained on pre-collected locally optimal solutions and efficiently samples initial guesses, which then warm-starts numerical solvers to fine-tune the feasibility and optimality. We also present DiffuSolve+, a novel constrained diffusion model with an additional loss in training that further reduces the problem constraint violations of diffusion samples. Experimental evaluations on three tasks verify the improved robustness, diversity, and a 2$\times$ to 11$\times$ increase in computational efficiency with our proposed method, which generalizes well to trajectory optimization problems of varying challenges.

With an increase in focus on the exploration and characterization of the Ice Giants, there is a growing need to develop mission architectures to perform scientific operations in the Uranian system. Science goals have been outlined by NASA in their Ice Giants Pre-Decadal Mission Study Report, but science priority and mission design are made difficult by the lack of knowledge of the Ice Giant systems. A primary architecture for conducting scientific exploration of Uranus has previously been identified as an orbiter with an atmospheric probe. In this paper, we explore the feasibility of a supplemental microsat-class spacecraft to this architecture, which would allow greater flexibility in achieving secondary science objectives with low risk and cost. We focus primarily on mission requirements for key ring science goals, leading to trajectory design guidelines and mission constraints necessary for characterizing the structure, dynamics, and composition of rings in the Uranian system. Broader compliance with the primary orbiter and probe architecture are studied, and guidance on instrumentation and subsystem-level requirements are detailed.

Preliminary trajectory design is a global search problem that seeks multiple qualitatively different solutions to a trajectory optimization problem. Due to its high dimensionality and non-convexity, and the frequent adjustment of problem parameters, the global search becomes computationally demanding. In this paper, we exploit the clustering structure in the solutions and propose an amortized global search (AmorGS) framework. We use deep generative models to predict trajectory solutions that share similar structures with previously solved problems, which accelerates the global search for unseen parameter values. Our method is evaluated using De Jong's 5th function and a low-thrust circular restricted three-body problem.