Universal Keplerian state transition matrix

Shepperd, Stanley W.

doi:10.1007/bf01227666

Cited by 37 publications

(19 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The same labeling is used in this paper to ease comparison. The orbital state and covariance matrix are propagated using Keplerian dynamics (Shepperd, 1985) for the standard Monte Carlo method and Alfano's method. In this example, the Taylor expansion of the DCA given in Eq.…”

Section: Validation Of Da-based Methodsmentioning

confidence: 99%

A high order method for orbital conjunctions analysis: Monte Carlo collision probability computation

Morselli

Armellín

Lizia

et al. 2015

Advances in Space Research

View full text Add to dashboard Cite

Three methods for the computation of the probability of collision between two space objects are presented. These methods are based on the high order Taylor expansion of the time of closest approach (TCA) and distance of closest approach (DCA) of the two orbiting objects with respect to their initial conditions. The identification of close approaches is first addressed using the nominal objects states. When a close approach is identified, the dependence of the TCA and DCA on the uncertainties in the initial states is efficiently computed with differential algebra (DA) techniques. In the first method the collision probability is estimated via fast DA-based Monte Carlo simulation, in which, for each pair of virtual objects, the DCA is obtained via the fast evaluation of its Taylor expansion. The second and the third methods are the DA version of Line Sampling and Subset Simulation algorithms, respectively. These are introduced to further improve the efficiency and accuracy of Monte Carlo collision probability computation, in particular for cases of very low collision probabilities. The performances of the methods are assessed on orbital conjunctions occurring in different orbital regimes and dynamical models. The probabilities obtained and the associated computational times are compared against standard (i.e. not DA-based) version of the algorithms and analytical methods. The dependence of the collision probability on the initial orbital state covariance is investigated as well.

show abstract

Section: Validation Of Da-based Methodsmentioning

confidence: 99%

A high order method for orbital conjunctions analysis: Monte Carlo collision probability computation

Morselli

Armellín

Lizia

et al. 2015

Advances in Space Research

View full text Add to dashboard Cite

show abstract

“…To propagate the probability distribution p y t−1 , every particle x j t−1 is propagated alongside its orbit to a new value x j t through a simplified propagation model, using the simple two-body problem via the Shepperd matrix [28,29] implementation. We havẽ…”

Section: Prediction Step and Orbit Propagationmentioning

confidence: 99%

Novel Multi-Object Filtering Approach for Space Situational Awareness

Delande

Frueh

Franco

et al. 2018

Journal of Guidance, Control, and Dynamics

View full text Add to dashboard Cite

Surveillance activities with ground-based assets in the context of space situational awareness are particularly challenging. The observation process is indeed hindered by short observation arcs, limited observability, missed detections, measurement noise, and contamination by clutter. This paper exploits a recent estimation framework for stochastic populations for space situational awareness surveillance scenarios. This framework shares the flexibility of the finite set statistics framework in the modeling of a dynamic population of objects and the representation of all the sources of uncertainty in a single coherent probabilistic framework and the intuitive approach of traditional trackbased techniques to describe individual objects and maintain track continuity. We present a recent multi-object filtering solution derived from this framework, the filter for distinguishable and independent stochastic populations, and propose a bespoke implementation of the multitarget tracking algorithm for a space situational awareness surveillance activity. The distinguishable and independent stochastic populations filter is tested on a surveillance scenario involving two ground-based Doppler radars in a challenging environment with significant measurement noise, limited observability, missed detections, false alarms, and no a priori knowledge about the number and the initial states of the objects in the scene. The tracking algorithm shows good performance in initiating tracks from object-generated observations and in maintaining track custody throughout the scenario, even when the objects are outside of the sensors' fields of view, despite the challenging conditions of the surveillance scenario.Nomenclature α y = probability of existence of previously detected target with observation path y cH = probability of existence of hypothesis H H = hypothesis, i.e., subset of compatible tracks in Y (H, n) = multitarget configuration, describing a possible composition of population X H tjt−1 ; H t = set of all hypotheses, propagated by the filter for distinguishable and independent stochastic populations (DISP) lz; x = likelihood of association between target with state x and observation with state z P tjt−1 , P t = law of whole population of targets, propagated by the distinguishable and independent stochastic populations filter P a = law of population of appearing targets p d x = probability of detection of target with state x p fa z = probability that observation with state z is false alarm p y = probability distribution of previously detected target with obs. path y p ϕ = probability distribution of each yet-to-be-detected target wH; n = joint probability of existence of targets in configuration (H, n) X = population of targets X = target state space X = target state space augmented with empty state ψ x = target state (e.g., position and velocity coordinates) Y tjt−1 , Y t = set of all observation paths, propagated by distinguishable and independent stochastic populations filter y = observation path or track characterized by said o...

show abstract

“…that can be determined according to [24]. The matrix M can be determined as linear combinations of the f , g functions and its derivatives, together with the orbital parameters.…”

Section: Modelling Of Orbital Scenarios 41 Target State Transitionmentioning

confidence: 99%

“…Utilizing the universal variables of Goodyear, the implementation of Shepperd has been chosen, for the computational efficient determination of the transition matrix [12,24]:…”

Section: Modelling Of Orbital Scenarios 41 Target State Transitionmentioning

confidence: 99%