Challenges with bearings only tracking for missile guidance systems and how to cope with them.

This paper addresses the problem of closed loop missile guidance using bearings and target angular extent information. Comparison is performed between particle filtering methods and derivative free methods. The extent information characterizes target size and we show how this can help compensate for observability problems. We demonstrate that exploiting angular extent information improves filter estimation accuracy. The performance of the filters has been studied over a testing scenario with a static target, with respect to accuracy, sensitivity to perturbations in initial conditions and in different seeker modes (active, passive and semi-active)

Multiple Target Tracking

Due to radar\u27s range measurement accuracy, Range Time plots are used to represent radar data. When objects\u27 tracks cross on a Range Time plot, it is uncertain which track belongs to which target. An analysis of the frequency and angle of these crossings was performed. Mathematical analysis concluded that in certain situations, only one type of crossing can result. Further Monte Carlo simulations were used to study these crossing statistics in other situations. In addition, it was examined how well targets could be tracked through an individual crossing. The probabilities of correct track association were calculated as a function of a variety of factors. Given our models and assumptions, sensor fusion of Range Time and Range Doppler analysis substantially improved crossing classification

Multiple Target Tracking

Due to radar\u27s range measurement accuracy, Range Time plots are used to represent radar data. When objects\u27 tracks cross on a Range Time plot, it is uncertain which track belongs to which target. An analysis of the frequency and angle of these crossings was performed. Mathematical analysis concluded that in certain situations, only one type of crossing can result. Further Monte Carlo simulations were used to study these crossing statistics in other situations. In addition, it was examined how well targets could be tracked through an individual crossing. The probabilities of correct track association were calculated as a function of a variety of factors. Given our models and assumptions, sensor fusion of Range Time and Range Doppler analysis substantially improved crossing classification

Multitarget tracking and terrain-aided navigation using square-root consider filters

Filtering is a term used to describe methods that estimate the values of partially observed states, such as the position, velocity, and attitude of a vehicle, using current observations that are corrupted due to various sources, such as measurement noise, transmission dropouts, and spurious information. The study of filtering has been an active focus of research for decades, and the resulting filters have been the cornerstone of many of humankind\u27s greatest technological achievements. However, these achievements are enabled principally by the use of specialized techniques that seek to, in some way, combat the negative impacts that processor roundoff and truncation error have on filtering. Two of these specialized techniques are known as square-root filters and consider filters. The former alleviates the fragility induced from estimating error covariance matrices by, instead, managing a factorized representation of that matrix, known as a square-root factor. The latter chooses to account for the statistical impacts a troublesome system parameter has on the overall state estimate without directly estimating it, and the result is a substantial reduction in numerical sensitivity to errors in that parameter. While both of these techniques have found widespread use in practical application, they have never been unified in a common square-root consider framework. Furthermore, consider filters are historically rooted to standard, vector-valued estimation techniques, and they have yet to be generalized to the emerging, set-valued estimation tools for multitarget tracking. In this dissertation, formulae for the square-root consider filter are derived, and the result is extended to finite set statistics-based multitarget tracking tools. These results are used to propose a terrain-aided navigation concept wherein data regarding a vehicle\u27s environment is used to improve its state estimate, and square-root consider techniques provide the numerical stability necessary for an onboard navigation application. The newly developed square-root consider techniques are shown to be much more stable than standard formulations, and the terrain-aided navigation concept is applied to a lunar landing scenario to illustrate its applicability to navigating in challenging environments --Abstract, page iii

Physically consistent boundary conditions for free-molecular satellite aerodynamics

Thesis (M.Sc.Eng.)To determine satellite trajectories in low earth orbit, engineers need to adequately estimate aerodynamic forces. But to this day, such a task su↵ers from inexact values of drag forces acting on complicated shapes that form modern spacecraft. While some of the complications arise from the uncertainty in the upper atmosphere, this work focuses on the problems in modeling the flow interaction with the satellite geometry. The only numerical approach that accurately captures e↵ects in this flow regime—like self-shadowing and multiple molecular reflections—is known as Test Particle Monte Carlo. This method executes a ray-tracing algorithm to follow particles that pass through a control volume containing the spacecraft and accumulates the momentum transfer to the body surfaces. Statistical fluctuations inherent in the approach demand particle numbers on the order of millions, often making this scheme too costly to be practical. This work presents a parallel Test Particle Monte Carlo method that takes advantage of both graphics processing units and multi-core central processing units. The speed at which this model can run with millions of particles enabled the exploration of regimes where a flaw was revealed in the model’s initial particle seeding. A new model introduces an analytical fix to this flaw—consisting of initial position distributions at the boundary of a spherical control volume and an integral for the correct number flux—which is used to seed the calculation. This thesis includes validation of the proposed model using analytical solutions for several simple geometries and demonstrates uses of the method for the aero-stabilization of the Phobos-Grunt Martian probe and pose-estimation for the ICESat mission.2031-01-0

Approaches to Evaluating Probability of Collision Uncertainty

While the two-dimensional probability of collision (Pc) calculation has served as the main input to conjunction analysis risk assessment for over a decade, it has done this mostly as a point estimate, with relatively little effort made to produce confidence intervals on the Pc value based on the uncertainties in the inputs. The present effort seeks to try to carry these uncertainties through the calculation in order to generate a probability density of Pc results rather than a single average value. Methods for assessing uncertainty in the primary and secondary objects' physical sizes and state estimate covariances, as well as a resampling approach to reveal the natural variability in the calculation, are presented; and an initial proposal for operationally-useful display and interpretation of these data for a particular conjunction is given

Probabilistic Space Weather Modeling and Forecasting for the Challenge of Orbital Drag in Space Traffic Management

In the modern space age, private companies are crowding the already-congested low Earth orbit (LEO) regime with small satellite mega constellations. With over 25,000 objects larger than 10 cm already in LEO, this rapid expansion is forcing us towards the enterprise on Space Traffic Management (STM). STM is an operational effort that focuses on conjunction assessment and collision avoidance between objects. While the equations of motion for objects in orbit are well-known, there are many uncertain parameters that result in the uncertainty of an object\u27s future position. The force that the atmosphere exerts on satellite - known as drag - is the largest source of uncertainty in LEO. This is largely due to the difficulty in predicting mass density in the thermosphere - the neutral region in Earth\u27s upper atmosphere. Presently, most thermosphere models are deterministic and the treatment of uncertainty in density is highly simplified or nonexistent in operations. In this work, four probabilistic thermospheric mass density models are developed using machine learning (ML) to enable the investigation of the impact of model uncertainty on satellite position for the first time. Of these four models, two (HASDM-ML and TIE-GCM ROPE) are reduced order models based on outputs from existing thermosphere models while the other two (CHAMP-ML and MSIS-UQ) are based on in-situ thermosphere measurements. The data and model development are described, and the models\u27 capabilities, including the robustness of their uncertainty quantification (UQ) capabilities, are thoroughly assessed. Existing thermosphere models, and the ones developed here, use different space weather drivers to estimate density. In a forecasting environment, there are algorithms and models that forecast the drivers for a given period in order for a density model to make a forecast. The driver forecast models used by the United States Space Force for the HASDM system are assessed to benchmark our current capabilities. Using the error statistics for each driver, we can perturb the deterministic forecasts. This provides an avenue to use the ML thermosphere models to study the effect of driver uncertainty on satellite position, in addition to model uncertainty, for any period with available driver forecasts. Seven periods are considered with diverse space weather conditions to study the isolated effects of the two density uncertainty sources on a 72-hour satellite orbit. This provides insight into the relative importance of density uncertainty on satellite position for various space weather scenarios. This study also functions as a motivation to reconsider our current methods for STM in order to improve our capabilities and prevent future satellite collisions with increased confidence

Covariance determination for improving uncertainty realism in orbit determination and propagation

The reliability of the uncertainty characterization, also known as uncertainty realism, is of the uttermost importance for Space Situational Awareness (SSA) services. Among the many sources of uncertainty in the space environment, the most relevant one is the inherent uncertainty of the dynamic models, which is generally not considered in the batch least-squares Orbit Determination (OD) processes in operational scenarios. A classical approach to account for these sources of uncertainty is the theory of consider parameters. In this approach, a set of uncertain parameters are included in the underlying dynamical model, in such a way that the model uncertainty is represented by the variances of these parameters. However, realistic variances of these consider parameters are not known a priori. This work introduces a methodology to infer the variance of consider parameters based on the observed distribution of the Mahalanobis distance of the orbital differences between predicted and estimated orbits, which theoretically should follow a chi-square distribution under Gaussian assumptions. Empirical Distribution Function statistics such as the Cramer-von-Mises and the Kolmogorov–Smirnov distances are used to determine optimum consider parameter variances. The methodology is presented in this paper and validated in a series of simulated scenarios emulating the complexity of operational applications.This project has received funding from the "Comunidad de Madrid" under "Ayudas destinadas a la realización de doctorados industriales" program (project IND2020/TIC-17539)

Nonlinear bayesian filtering with applications to estimation and navigation

In principle, general approaches to optimal nonlinear filtering can be described in a unified way from the recursive Bayesian approach. The central idea to this recur- sive Bayesian estimation is to determine the probability density function of the state vector of the nonlinear systems conditioned on the available measurements. However, the optimal exact solution to this Bayesian filtering problem is intractable since it requires an infinite dimensional process. For practical nonlinear filtering applications approximate solutions are required. Recently efficient and accurate approximate non- linear filters as alternatives to the extended Kalman filter are proposed for recursive nonlinear estimation of the states and parameters of dynamical systems. First, as sampling-based nonlinear filters, the sigma point filters, the unscented Kalman fil- ter and the divided difference filter are investigated. Secondly, a direct numerical nonlinear filter is introduced where the state conditional probability density is calcu- lated by applying fast numerical solvers to the Fokker-Planck equation in continuous- discrete system models. As simulation-based nonlinear filters, a universally effective algorithm, called the sequential Monte Carlo filter, that recursively utilizes a set of weighted samples to approximate the distributions of the state variables or param- eters, is investigated for dealing with nonlinear and non-Gaussian systems. Recentparticle filtering algorithms, which are developed independently in various engineer- ing fields, are investigated in a unified way. Furthermore, a new type of particle filter is proposed by integrating the divided difference filter with a particle filtering framework, leading to the divided difference particle filter. Sub-optimality of the ap- proximate nonlinear filters due to unknown system uncertainties can be compensated by using an adaptive filtering method that estimates both the state and system error statistics. For accurate identification of the time-varying parameters of dynamic sys- tems, new adaptive nonlinear filters that integrate the presented nonlinear filtering algorithms with noise estimation algorithms are derived. For qualitative and quantitative performance analysis among the proposed non- linear filters, systematic methods for measuring the nonlinearities, biasness, and op- timality of the proposed nonlinear filters are introduced. The proposed nonlinear optimal and sub-optimal filtering algorithms with applications to spacecraft orbit es- timation and autonomous navigation are investigated. Simulation results indicate that the advantages of the proposed nonlinear filters make these attractive alterna- tives to the extended Kalman filter