Kyle J. DeMars

Probabilistic Initial Orbit Determination Using Gaussian Mixture Models

The most complete description of the state of a system at any time is given by knowledge of the probability density function, which describes the locus of possible states conditioned on any available measurement information. When employing optical data, the concept of the admissible region provides a physics-based region of the range/range-rate space that produces Earth-bound orbit solutions. This work develops a method that employs a probabilistic interpretation of the admissible region and approximates the admissible region by a Gaussian mixture to formulate an initial orbit determination solution. The Gaussian mixture representation of the probability density function is then forecast and updated with subsequent data to iteratively refine the region of uncertainty. Simulation results are presented using synthetic data over a range of orbits, in which it is shown that the new method is consistently able to initialize a probabilistic orbit solution and provide iterative refinement via follow-on tracking.

Initial Orbit Determination using Short-Arc Angle and Angle Rate Data

The population of space objects (SOs) is tracked with sparse resources and thus tracking data are only collected on these objects for a relatively small fraction of their orbit revolution (i.e., a short arc). This contributes to commonly mistagged or uncorrelated SOs and their associated trajectory uncertainties (covariances) to be less physically meaningful. The case of simply updating a catalogued SO is not treated here, but rather, the problem of reducing a set of collected short-arc data on an arbitrary deep space object without a priori information, and from the observations alone, determining its orbit to an acceptable level of accuracy. Fundamentally, this is a problem of data association and track correlation. The work presented here takes the concept of admissible regions and attributable vectors along with a multiple hypothesis filtering approach to determine how well these SO orbits can be recovered for short-arc data in near realtime and autonomously. While the methods presented here are explored with synthetic data, the basis for the simulations resides in actual data that has yet to be reduced, but whose characteristics are replicated as well as possible to yield results that can be expected using actual data.

Entropy-Based Approach for Uncertainty Propagation of Nonlinear Dynamical Systems

Uncertainty propagation of dynamical systems is a common need across many domains and disciplines. In nonlinear settings, the extended Kalman filter is the de facto standard propagation tool. Recently, a new class of propagation methods called sigma-point Kalman filters was introduced, which eliminated the need for explicit computation of tangent linear matrices. It has been shown in numerous cases that the actual uncertainty of a dynamical system cannot be accurately described by a Gaussian probability density function. This has motivated work in applying the Gaussian mixture model approach to better approximate the non-Gaussian probability density function. A limitation to existing approaches is that the number of Gaussian components of the Gaussian mixture model is fixed throughout the propagation of uncertainty. This limitation has made previous work ill-suited for nonstationary probability density functions either due to inaccurate representation of the probability density function or computational b...

Underweighting Nonlinear Measurements

T HE extended Kalman filter [1] (EKF) is a nonlinear approximation of the optimal linear Kalman filter [2,3]. In the presence of measurements that are nonlinear functions of the state, the EKF algorithm expands the filter’s residual (difference between the actual measurement and the estimated measurement) in a Taylor series centered at the a priori state estimate. The EKF truncates the series to the first order, but second-order filters also exist [4,5]. It is well known that, in the presence of highly accurate measurements, the contribution of the second-order terms is essential when the a priori estimation error covariance is large [5,6]. Possible solutions include implementing a second-order Gaussian filter [5] or an unscented Kalman filter (UKF) [7]. The UKF is a nonlinear extension to the Kalman filter, capable of retaining the secondmoments (or higher) of the estimation error distribution. Even when retaining the secondorder terms of the Taylor series, the methods still rely on an approximation; therefore, good filtering results may not always be achievable. Historically, second-order filters are not used because of their computational cost. The space shuttle, for example, uses an ad hoc technique known as underweighting [8,9]. The commonly implemented method for the underweighting of measurements for human space navigation was introduced by Lear [10] for the space shuttle navigation system. In 1966, Denham and Pines showed the possible inadequacy of the linearization approximationwhen the effect ofmeasurement nonlinearity is comparable to the measurement error [11]. To compensate for the nonlinearity, Denham and Pines proposed to increase the measurement noise covariance by a constant amount. In the early 1970s, in anticipation of shuttle flights, Lear [8] and others developed relationships that accounted for the second-order effects in the measurements. It was noted that, in situations involving large state errors and very precise measurements, application of the standard EKF mechanization lead to conditions inwhich the state estimation error covariance decreased more rapidly than the actual state errors. Consequently, the EKF began to ignore new measurements, even when the measurement residualwas relatively large.Underweightingwas introduced to slow down the convergence of the state estimation error covariance, thereby addressing the situation in which the error covariance became overly optimistic with respect to the actual state errors. The original work on the application of second-order correction terms led to the determination of the underweighting method by trial and error [10]. More recently, studies on the effects of nonlinearity in sensor fusion problems with application to relative navigation have produced a so-called bump-up factor [12–15]. While Ferguson and How [12] initiated the use of the bump-up factor, the problem of mitigating filter divergence was more fully studied by Plinval [13] and, subsequently, by Mandic [14]. Mandic generalized Plinval’s [13] bump-up factor to allow flexibility and notes that the value selected influences the steady-state convergence of the filter. In essence, it was found that a larger factor keeps the filter from converging to the level that a lower factor would permit. This finding prompted Mandic [14] to propose a two-step algorithm in which the bump-up factor was only applied for a certain number of measurements, upon which the factor was completely turned off. Finally, Perea, et al. [15] summarized the findings of the previous works and introduced several ways of computing the applied factor. In all of the cases, the bump-up factor amounts, in application, to the underweighting factor introduced in Lear [10]. Save for the two-step procedure of Mandic [14], the bump-up factor was allowed to persistently affect the Kalman gain, which directly influenced the obtainable steady-state covariance. Effectively, the ability to remove the underweighting factor autonomously and under some convergence condition was not introduced. While of great historical importance, the work of Lear is not well known, as it is only documented in internal NASA memos [8,10]. Kriegsman and Tau [9] mention underweighting in their 1975 shuttle navigation paper, without a detailed explanation of the technique. The purpose of this Note is to review the motivations behind underweighting and to document its historical introduction. Lear’s [10] schemeuses a single scalar coefficient, and tuning is necessary in order to achieve good performance. A new method for determining the underweighting factor is introduced together with an automated method for deciding when the underweighting factor should and should not be applied. By using the Gaussian approximation and bounding the second-order contributions, suggested values for the coefficient are easily obtained. The proposed technique has the advantage of Lear’s scheme’s simplicity combined with the theoretical foundation of the Gaussian second-order filter. The result yields a simple algorithm to aid the design of the underweightedEKF.

Advances in POST2 End-to-End Descent and Landing Simulation for the ALHAT Project

Program to Optimize Simulated Trajectories II (POST2) is used as a basis for an end-to- end descent and landing trajectory simulation that is essential in determining design and integration capability and system performance of the lunar descent and landing system and environment models for the Autonomous Landing and Hazard Avoidance Technology (ALHAT) project. The POST2 simulation provides a six degree-of-freedom capability necessary to test, design and operate a descent and landing system for successful lunar landing. This paper presents advances in the development and model-implementation of the POST2 simulation, as well as preliminary system performance analysis, used for the testing and evaluation of ALHAT project system models.

Collision Probability with Gaussian Mixture Orbit Uncertainty

In view of the high value of space assets and a growing space debris population increasingly caused by random collisions in a congested space environment, collision analysis is of great significance. Collisions between space objects (spacecraft and space debris) can at best be determined in a probabilistic manner because of the lack of perfect knowledge about the parameters and motions of the space objects. Collision analysis based on the nominal orbital parameters without taking into account the uncertainty in those parameters or in the orbit motions is inaccurate. The probability of collision between two space objects provides a quantitative measure of the likelihood that the space objects will collide with each other. Collision probability is closely related to close approaches, 3 collision detection, and conflict probability and has been derived for low Earth orbits (including the International Space Station), Assistant Professor, Department of Mechanical and Aerospace Engineering. Email: demarsk@mst.edu. Senior Member AIAA. Assistant Professor, Department of Aerospace Engineering. Email: cheng@ae.msstate.edu. Associate Fellow AIAA. Senior Research Engineer. Associate Fellow AIAA.

An AEGIS-FISST Algorithm for Multiple Object Tracking in Space Situational Awareness

In this paper we use Finite Set Statistics (FISST) to track an arbitrary but known number of objects in the presence of clutter for Space SituationalAwareness (SSA). The sensors are assumed faulty with possible misdetections, false alarms and/or noisy data. The measurements are unassociated and, hence, we also solve the data association problem, which is integral to the FISST algorithm. The main contribution of the paper is that, in order to reduce the computational burden entailed in FISST, we employ a Gaussian mixture approximation,not to the first-moment of the full FISST updateequations(knownasGM-PHD),butapplytheapproximationdirectlytothefullFISSTequations. The specific GM technique we employ is the Adaptive Entropy-based Gaussian-mixture Information Synthesis (AEGIS). The approachis demonstratedin two a multiple object SSA applicationexamples.

Multiple-Object Space Surveillance Tracking Using Finite-Set Statistics

The dynamic tracking of objects is, in general, concerned with state estimation using imperfect data. Multiple object tracking adds the difficulty of encountering unknown associations between the collected data and the objects. State estimation of objects necessitates the prediction of uncertainty through nonlinear (in the general case) dynamical systems and the processing of nonlinear (in the general case) measurement data in order to provide corrections that refine the system uncertainty, where the uncertainty may be non-Gaussian in nature. The sensors, which provide the measurement data, are imperfect with possible misdetections, false alarms, and noise-affected data. The resulting measurements are inherently unassociated upon reception. In this paper, a Bayesian method for tracking an arbitrary, but known, number of objects is developed. The method is based on finite-set statistics coupled with finite mixture model representations of the multiobject probability density function. Instead of relying on ...

Particle Filter Methods for Space Object Tracking

An approach for space object tracking utilizing particle filters is presented. New methods are developed and used to construct a robust constrained admissible region given a set of angles-only measurements, which is then approximated by a finite mixture distribution. This probabilistic initial orbit solution is refined using subsequent measurements through a particle filter approach. A proposal density is constructed based on an approximate Bayesian update and samples, or particles, are drawn from this proposed probability density to assign and correct weights, which form the basis for a more accurate Bayesian update. A finite mixture distribution is then fit to these weighted samples to reinitialize the cycle. This approach is compared to methods that approximate all probability densities as finite mixtures and process them as such. Both approaches utilize recursive estimation based on Bayesian statistics, but the benefits of densely sampling the support probability based on incoming measurements is weighed against remaining solely within the finite mixture approximation and performing measurement corrections there.

Bearings-Only Initial Relative Orbit Determination

A method for performing bearings-only initial relative orbit determination of a nearby space object in the absence of any information regarding the space object’s geometry and relative orbit is presented. To resolve the range ambiguity characteristic of a single optical sensor system, a second optical sensor is included at a known baseline distance on the observing spacecraft. To formulate an initial estimate of the space object’s relative orbit and its associated uncertainty, the angle measurements from both sensors are used to bound a region for all possible relative positions of the space object. A parameterized probability distribution in relative position that reflects uniform relative range uncertainty across the bounded region is constructed at two unique times. Linkage of the positional distributions is performed using a second-order relative Lambert solver to formulate a full-state probability density function in relative position and velocity, which can be further refined through processing subs...