Spacecraft orbit prediction

The expanding role of positional covariance data in modern spacecraft operations leads to a need for better understanding of the time evolution of covariance by mission planners and operators. It is common to see positional covariance information presented as uncertainties in the radial, cross-track and in-track directions. While such a time history does provide some information, it tends to obscure the true directionality of the uncertainty. A methodology for improving the understanding of position covariance based on 3D visualization is presented. The methodology utilizes sequential estimation to provide insight into the evolution of the covariance in the presence of measurements and to incorporate the effects of dynamic modeling uncertainties between measurements. Display of the covariance as a triaxial ellipsoid representing a desired probability level is discussed. Various methods for interpolating positional covariance are examined resulting in a recommendation of a technique based on an eigenvalueeigenvector decomposition of the covariance.

Satellite orbit simulation and analysis based on data collected from NASA/ North American Defense Command (NORAD)
as a Two Line Elements (TLE) files is presented. General Mission Analysis Tool (GMAT) is used to simulate the orbital motion of
TIGRISAT. The analysis includes orbit determination and prediction of satellite’s position and velocity, satellite tracking, and
command summary. Orbital path of the TIGRISAT projected onto a two-dimensional world map over some time for one and two
revolution of the satellite is plotted.

This document details the procedure of calculation of orbital perturbation due to gravitational harmonics by simulation of various orbits at different altitudes and inclinations to the earth's equatorial plane. The orbits were first simulated and data of position and velocity was collected. The data was then analysed by another program to calculate the perturbations. Runge – Kutta fourth order method was used to simulate the orbits. The orbits were assumed to start from the perigee position which lied on the longitudinal plane containing the point of vernal equinox. The perturbations obtained were analysed w.r.t inclination of orbits and eccentricity of the orbits. The order of magnitude of the perturbations obtained were analysed w.r.t altitude. The earth was assumed to be symmetrical in all other terms.

As the spacefaring community is well aware, the increasingly rapid proliferation of man-made objects in space, whether active satellites or debris, threatens the safe and secure operation of spacecraft and requires that we change the way we conduct business in space. The introduction of appropriate protocols and procedures to regulate the use of space is predicated on the availability of quantifiable and timely information regarding the behavior of resident space objects (RSO): the basis of space domain awareness (SDA). Yet despite five decades of space operations, and a growing global dependence on the services provided by space-based platforms, the population of Earth orbiting space objects is still neither rigorously nor comprehensively quantified, and the behaviors of these objects, whether directed by human agency or governed by interaction with the space environment, are inadequately characterized. Key goals of advanced SDA are to develop a capability to predict RSO behavior, extending SDA beyond its present paradigm of catalog maintenance and forensic analysis, and to arrive at a comprehensive physical understanding of all of the inputs that affect the motion of RSOs. Solutions to these problems require multidisciplinary engagement that combines space surveillance data with other information, including space object databases and space environmental data, to help decision-making processes predict, detect, and quantify threatening and hazardous space domain activity. 1.0 INTRODUCTION This document presents an introductory overview of space surveillance, tracking, and information fusion for SDA. A relevant activity is the NATO Science and Technology Organization's Task Group (SCI-279-TG) is addressing the technical considerations for enabling a NATO-Centric Space Domain Common Operating Picture (COP). The impetus for this effort is the growing dependence by NATO and its member nations on space capabilities to achieve its mission responsibilities as well as the growing role that space, as an operational domain in its own right, is playing in matters concerning global security. NATO has recognized this important reality and increased the Alliance's collective attention on ensuring NATO operations maximize their leverage of space while ensuring the space capabilities provided by its member nations are preserved to the maximum extent possible. A critical element of ensuring the availability and efficacy of these space capabilities is the availability of a common operational perspective or picture of the space domain throughout the Alliance and its partners. The presumption is that NATO forces will be more efficient, protected and successful in their future missions if a common operational perspective can be achieved across all operational domains in which NATO must operate; air, land, sea, cyber AND space. The corollary is, without a common Alliance perspective of the space domain,

3 | P a g e CERTIFICATE I hereby certify that the work which is being presented in the project report entitled "Trajectory Analysis of UPESSAT" in partial fulfilment of the requirements for the satisfactory performance for Major Project, final year, submitted in the Department of Aerospace Engineering, University of Petroleum and Energy Studies, Dehradun is an authentic record of these students" own work carried out during a period from 4 | P a g e DECLARATION I hereby declare that the major project work titled "Trajectory Analysis of UPESSAT" is unique to the best of my knowledge and submitted to UPES, Dehradun. It is the bonafide record of the original work carried out by us under the exemplary mentorship of Dr. Ugur

Two-line elements are widely used for space operations to predict the orbit with a moderate accuracy for 2-3 days. Local optimization methods, such as the nonlinear least squares method with differential corrections, can estimate a TLE as long as there exists an initial estimate that provides the desired precision. Global optimization methods to estimate TLEs are computationally intensive, and estimating a large number of them is prohibitive. In this paper, the feasibility of estimating TLEs using machine learning methods is investigated. First, a Monte-Carlo approach to estimate a TLE, when there are no initial estimates that provide the desired precision, is introduced. The proposed Monte-Carlo method is shown to estimate TLEs with root mean square errors below 1 km for space objects with varying area-to-mass ratios and orbital characteristics. Second, gradient boosting decision trees and fully-connected neural networks are trained to map orbital evolution of space objects to the associated TLEs using 8 million publicly available TLEs from the US space catalog. The desired precision in the mapping to estimate a TLE is achieved for one of the three test cases, which is a low area-to-mass ratio space object.

A technique for integrating orbit trajectories influenced by attitude dependent forces is presented. The method employs an Encke type correction algorithm to account for the attitude dependence not modeled in the main orbit integration process. The formulation improves computational efficiency over the straightforward integration of the fully coupled system by eliminating the need to compute the acceleration due to the geopotential at the time step required for the attitude dependent forces. Examples are shown for the integration of coupled orbital and attitude motion and for the integration of the orbit trajectory when the attitude profile is known a priori.

A computationally efficient noise-adaptive Kalman filter is presented for the motion estimation and prediction of a free-falling tumbling satellite (target). The filter receives only noisy pose measurements from a laser-vision system aboard another satellite (chaser) at a close distance in a neighboring orbit. The filter estimates the full states, all the inertia parameters of the target satellite, and the covariance of the measurement noise. A comprehensive dynamics model that includes aspects of orbital mechanics is incorporated for accurate estimation. The discrete-time model, which involves a state-transition matrix and the covariance of process noise, is derived in closed form, thus rendering the filter suitable for real-time implementation. The statistical characteristics of the measurement noise is formulated by a state-dependent covariance matrix. This model allows additive quaternion noise, while preserving the unit-norm property of the quaternion. The convergence properties of the developed filter is demonstrated by simulation and experimental results. These results also demonstrate that the filter can continuously produce accurate estimates of pose even when the vision system is occluded for tens of seconds.

High fidelity orbit propagation requires detailed knowledge of the solar radiation pressure (SRP) on a space object. The SRP depends not only on the space object's shape and attitude, but also on the absorption and reflectance properties of each surface on the object. These properties are typically modeled in a simplistic fashion, but are here described by a surface bidirectional reflectance distribution function (BRDF). Several analytic BRDF models exist, and are typically complicated functions of illumination angle and material properties represented by parameters within the model. In general, the resulting calculation

We work out the impact that the recently determined time-dependent component of the
Pioneer Anomaly (PA), if interpreted as an additional exotic acceleration of gravitational
origin with respect to the well-known PA-like constant one, may have on the orbital
motions of some planets of the solar system. By assuming that it points towards the
Sun, it turns out that both the semi-major axis a and the eccentricity e of the orbit of
a test particle would experience secular variations. For Saturn and Uranus, for which
modern data records cover at least one full orbital revolution, such predicted anomalies
are up to 2–3 orders of magnitude larger than the present-day accuracies in empirical
determinations of their orbital parameters from the usual orbit determination procedures
in which the PA was not modeled. Given the predicted huge sizes of such hypothetical
signatures, it is unlikely that their absence from the presently available processed data
can be attributable to an “absorption” for them in the estimated parameters caused by
the fact that they were not explicitly modeled. The magnitude of a constant PA-type
acceleration at 9.5 au cannot be larger than 9 × 10−15 m s−2 according to the latest
observational results for the perihelion precession of Saturn.

This paper addresses this last approach and how to incorporate it in an unscented Kalman filter framework using a practical, easily implementable development. First, the familiar unscaled unscented Kalman filter is reviewed. Then, the unscented Schmidt-Kalman filter is developed and the algorithm is presented along-side the unscented Kalman filter algorithm. Next, a previouslydeveloped consider analysis approach is summarized. Finally, a simple example is used to illustrate the unscented Schmidt-Kalman filter and compare the results obtained using the different consider algorithms.

Knowledge of spacecraft orbit uncertainty is becoming increasing important for spacecraft operations and Space Situational Awareness. For cases where uncertainty is large, such as early orbit operations or anomaly situations, the manner in which orbit uncertainty is used and interpreted can be the difference between making correct or errant assessments of the effect of orbit uncertainty on spacecraft operations. The influence of coordinate selection on the proper utilization of orbit error covariance is examined and compared with the application of the unscented transform as a means to overcome issues of non-linearity. The use of preferential coordinates is seen to be of primary importance in the characterization of uncertainty while unscented transformations are seen to be effective at increasing the size of the uncertainty but are not able to overcome coordinate deficiencies to yield properly shaped uncertainty volumes.

In this paper we present results from our orbit prediction study using the publicly available Two-Line Element (TLE) sets. The method presented here is similar to that introduced by Levit and Marshall ; however, we also consider the non-spherical low-Earth orbit satellites Grace A and Grace B. A state vector is generated every 10 minutes in the orbit determination (OD) period using SGP4. These generated states are subsequently used as observations in an orbit determination run considering a full set of forces to determine the orbit over the 10-day time span. All information used is from the TLE data sets. Once the orbit has been determined, it is then numerically propagated to obtain a prediction of the object's position. The TLE-determined orbit is compared to highly accurate satellite laser ranging (SLR) Consolidated Prediction Format (CPF) data to assess the accuracy. We tested the technique by performing 200 independent simulations for Stella, Starlette, Grace A and Grace B and found that it resulted in better orbit predictions 98.5%, 93.4%, 97.5% and 95.5% of the time, respectively, when compared to standard SGP4 propagation. For Starlette and Stella after a 7 day prediction period the average absolute maximum along track bias was reduced by approximately 64% and 74%, respectively. For Grace A and Grace B after a 7 day prediction period the average absolute maximum along track bias was reduced by approximately 68% and 64%, respectively.

We comment on some statements in a recent paper by Paolozzi and Ciufolini concerning certain remarks raised by us on the realistic accuracy obtainable in testing the general relativistic Lense–Thirring effect in the gravitational field of the Earth with the newly launched LARES satellite together with the LAGEOS and LAGEOS II spacecraft in orbit for a long time. The orbital configuration of LARES is different from that of the originally proposed LAGEOS-3. Indeed, while the latter one should have been launched to the same altitude of LAGEOS (i.e. about hL=5890 km) in an orbital plane displaced by 180° with respect to that of LAGEOS (IL=110°, IL3=70°), LARES currently moves at a much smaller altitude (about hLR=1440 km) and at a slightly different inclination (ILR=69.5°). As independently pointed out in the literature by different authors, the overall accuracy of a LARES–LAGEOS–LAGEOS II Lense–Thirring test may be unfavorably impacted by the lower altitude of LARES with respect to the expected ≈1% level claimed by Ciufolini et al. because of an enhanced sensitivity to the low-degree even zonal geopotential coefficients inducing orbital precessions competing with the relativistic ones. Concerning the previous tests performed with the combined nodes of only LAGEOS and LAGEOS II, an independent analysis recently appeared in the literature indirectly confirms that the total uncertainty in them is likely far from being as little as 10%.