arXiv subjects

Jackson Kulik

Publications and source records attributed to Jackson Kulik.

12 recordsLinked to original sources

arXiv2026

Covariance Square Root Second-Order Mapping

In recursive state estimation, numerical error can play a major role in an algorithm's overall performance and reliability. Roundoff errors due to finite precision arithmetic can violate theoretical guarantees, leading to asymmetric and non-positive-semidefinite covariance matrices. In algorithms employing first-order covariance mappings, such as the extended Kalman filter, these issues have been mitigated by employing square root factorizations of the covariance matrix. However, existing techniques do not directly extend to higher-order moment mappings, which show great value in highly nonlinear settings. This paper presents the first known square root computation of the second-order covariance mapping. The square root computation is not only more accurate, as is shown in two distinct numerical experiments, but generally requires fewer floating point operations compared to the full covariance matrix computation.

eess.SP↗

arXiv2026

Bayesian hypergraph inference from scarce and noisy dynamical observations

Inferring higher-order interaction structure from observations of dynamics is a central challenge in complex systems, particularly when data are scarce, noisy, or concentrated in lower-dimensional regions of state space. We develop Bayes-THIS, a Bayesian extension of Taylor-based Hypergraph Inference using SINDy (THIS), which reconstructs hypergraph structure from time-series data by identifying sparse Taylor coefficients associated with pairwise and higher-order interactions. By replacing fixed-threshold sparse regression with sparse Bayesian regression using automatic relevance determination, Bayes-THIS explicitly models residual variance and applies adaptive, term-wise coefficient shrinkage, improving robustness in data-limited, high-noise, and ill-conditioned regimes. The resulting Gaussian posterior also enables an uncertainty-aware inference workflow: a posterior predictive check assesses whether the data contain sufficient higher-order signal to reliably support inference beyond a pairwise model, and credible-interval pruning selects hyperedges whose inferred coefficients are statistically distinguishable from zero. Finally, we characterize a fundamental limitation of the Taylor-based inference framework: when higher-order interactions concentrate on nodes that lack lower-order connections, the Taylor expansion systematically inflates lower-order coefficient estimates, producing spurious edges indistinguishable from genuine lower-order interactions. This structural non-identifiability cannot be resolved by either THIS or Bayes-THIS.

physics.soc-ph↗

arXiv2025

Unscented and Higher-Order Linear Covariance Fidelity Checks and Measures of Non-Gaussianity

Linear covariance (LinCov) techniques have gained widespread traction in the modeling of uncertainty, including in the preliminary study of spacecraft navigation performance. While LinCov methods offer improved computational efficiency compared to Monte Carlo based uncertainty analysis, they inherently rely on linearization approximations. Understanding the fidelity of these approximations and identifying when they are deficient is critically important for spacecraft navigation and mission planning, especially when dealing with highly nonlinear systems and large state uncertainties. This work presents a number of computational techniques for assessing linear covariance performance. These new LinCov fidelity measures are formulated using higher-order statistics, constrained optimization, and the unscented transform.

eess.SP↗

arXiv2025

Optimal Rank-1 Directional State Transition Tensors

An optimal rank-1 approximation of state transition tensors was developed as an efficient alternative to state transition tensors for nonlinear uncertainty quantification. While previous directional state transition tensors used the dominant right singular subspace of the state transition matrix to construct a reduced-dimension representation of the state transition tensors, optimal directional state transition tensors are constructed to maximize the information retained in a rank-1 approximation of the state transition tensors in the Frobenius-norm sense. The optimal rank-1 directional state transition tensor is found by solving a tensor z-eigenpair problem of the "square" of the state transition tensor. This construct leads to increased approximation accuracy of the state transition tensors and improved Gaussian moment propagation for nonlinear flight scenarios like aerocapture.

eess.SY↗

arXiv2025

Higher-Order Tensor-Based Deferral of Gaussian Splitting for Orbit Uncertainty Propagation

Accurate propagation of orbital uncertainty is essential for a range of applications within space domain awareness. Adaptive Gaussian mixture-based approaches offer tractable nonlinear uncertainty propagation through splitting mixands to increase resolution in areas of stronger nonlinearities, as well as by reducing mixands to prevent unnecessary computational effort. Recent work introduced principled heuristics that incorporate information from the system dynamics and initial uncertainty to determine optimal directions for splitting. This paper develops adaptive uncertainty propagation methods based on these robust splitting techniques. A deferred splitting algorithm tightly integrated with higher-order splitting techniques is proposed and shown to offer substantial gains in computational efficiency without sacrificing accuracy. Second-order propagation of mixand moments is also seen to improve accuracy while retaining significant computational savings from deferred splitting. Different immediate and deferred splitting methods are compared in four representative test cases, including a low Earth orbit, a geostationary orbit, a Molniya orbit, and a multi-body cislunar orbit.

eess.SP↗

arXiv2025

Robust computation of higher-dimensional invariant tori from individual trajectories

We present a method for computing invariant tori of dimension greater than one. The method uses a single short trajectory of a dynamical system without any continuation or initial guesses. No preferred coordinate system is required, meaning the method is practical for physical systems where the user does not have much \textit{a priori} knowledge. Three main tools are used to obtain the rotation vector of the invariant torus: the reduced rank extrapolation method, Bayesian maximum a posteriori estimation, and a Korkine-Zolatarev lattice basis reduction. The parameterization of the torus is found via a least-squares approach. The robustness of the algorithm is demonstrated by accurately computing many two-dimensional invariant tori of a standard map example. Examples of islands and three-dimensional invariant tori are shown as well.

math.DS↗

arXiv2025

Mass-Optimal Low-Thrust Forced Periodic Trajectories in the Earth-Moon CR3BP

In Cislunar space, spacecraft are able to exploit naturally periodic orbits, which provide operational reliability. However, these periodic orbits only exist in a limited volume. Enabled by low-thrust propulsion, spacecraft can produce a greater number of periodic trajectories in Cislunar space. We describe a methodology for producing mass-optimal trajectories that enforce periodic structure in the circular-restricted three body problem and study the thrust-limited reachable set around a reference trajectory. In this study, we find that the thrust-limited mass-optimal reachable set is a superset of the energy-limited energy-optimal reachable set in the xy-plane.

math.DS↗

arXiv2024

Nonlinearity and Uncertainty Informed Moment-Matching Gaussian Mixture Splitting

Many problems in navigation and tracking require increasingly accurate characterizations of the evolution of uncertainty in nonlinear systems. Nonlinear uncertainty propagation approaches based on Gaussian mixture density approximations offer distinct advantages over sampling based methods in their computational cost and continuous representation. State-of-the-art Gaussian mixture approaches are adaptive in that individual Gaussian mixands are selectively split into mixtures to yield better approximations of the true propagated distribution. Despite the importance of the splitting process to accuracy and computational efficiency, relatively little work has been devoted to mixand selection and splitting direction optimization. The first part of this work presents splitting methods that preserve the mean and covariance of the original distribution. Then, we present and compare a number of novel heuristics for selecting the splitting direction. The choice of splitting direction is informed by the initial uncertainty distribution, properties of the nonlinear function through which the original distribution is propagated, and a whitening based natural scaling method to avoid dependence of the splitting direction on the scaling of coordinates. We compare these novel heuristics to existing techniques in three distinct examples involving Cartesian to polar coordinate transformation, Keplerian orbital element propagation, and uncertainty propagation in the circular restricted three-body problem.

stat.ML↗

arXiv2024

Generation of Energy-Optimal Low-Thrust Forced Periodic Trajectories in the CR3BP

In this work, we investigate trajectories that require thrust to maintain periodic structure in the circular restricted three-body problem (CR3BP). We produce bounds in position and velocity space for the energy-constrained reachable set of initial conditions. Our trajectories are energy-optimal and analyzed via linear analysis. We provide validation for our technique and analyze the cost of deviating in various directions to the reference. For our given reference, we find that it is relatively expensive to decrease perilune distance for orbits in the Earth-Moon system.

math.DS↗

arXiv2024

Applications of Induced Tensor Norms to Guidance Navigation and Control

Linear methods are ubiquitous for control and estimation problems. In this work, we present a number of tensor operator norms as a means to approximately bound the error associated with linear methods and determine the situations in which that maximum error is encountered. An emphasis is placed on induced norms that can be computed in terms of matrix or tensor eigenvalues associated with coefficient tensors from higher-order Taylor series. These operator norms can be used to understand the performance and range of applicability of an algorithm exploiting linear approximations in different sets of coordinates. We examine uses of tensor operator norms in the context of linear and higher-order rendezvous guidance, coordinate selection for a filtering measurement model, and to present a unified treatment of nonlinearity indices for dynamical systems. Tensor norm computations can offer insights into these problems in one to two orders of magnitude less time than similarly accurate sampling methods while providing more general understanding of the error performance of linear or higher-order approximations.

math.DS↗

arXiv2021

Minimal Differential Lateral Acceleration Configurations for Starshade Stationkeeping in Exoplanet Direct Imaging

Exoplanet imaging missions utilizing an external occulter (starshade) for starlight suppression require precise alignment between the telescope and starshade, necessitating maintenance of the starshade orbit during observations. Differential lateral acceleration between the two spacecraft serves as a proxy for fuel use and number of required interruptions to the observation. Comparison against results from high fidelity simulations of stationkeeping validates the use of this easy-to-compute proxy. Among starshade positions constrained to the surface of a sphere centered about the telescope, minima of differential lateral acceleration lie on a great circle and its corresponding poles. We present a closed expression for telescope to star vectors requiring minimal stationkeeping for observation from a telescope at an arbitrary position

astro-ph.EP↗

arXiv2019

Construction of J2-Invariant Periodic Relative Motion in Highly Elliptical Orbits

Two satellites with mean orbital elements which differ only in terms of right ascension of the ascending node, argument of perigee, and mean anomaly are notable for having the same mean orbital element secular drift rates due to the J2 perturbation. The relative orbits which result from this configuration are discounted in the literature for not providing sufficiently many degrees of freedom with which to design relative orbit geometries suitable for real world missions. However, this paper will explore a variety of useful geometries which result from this limited design space, and provide analytical formulas for the differences between chief and deputy mean orbital elements as a function of the desired orbit geometry and inertial chief orbit. These include in-track-centered, quadrant-time-centered, offset-circular, rectilinear, boomerang-shaped, and cross-track only geometries.

astro-ph.EP↗