Although the volume will contain a few distinct chapters, it will render a coherent portrayal of astrodynamics. Encompasses the main constituents of the astrodynamical sciences in an elaborate, comprehensive and rigorous manner Presents recent astrodynamical advances and describes the challenges ahead The first volume of a series designed to give scientists and engineers worldwide an opportunity to publish their works in this multi-disciplinary field.
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Handbook of Accelerator Physics and Engineering. World Scientific a Google Scholar. Academic Press b Google Scholar. Bryson A. Hemisphere Publishing Co. Cruz, M. Di Lizia, P. Dissertation, Politecnico di Milano a Google Scholar. Di Lizia P. World Scientic Google Scholar. Griffith, T. Guibout V. Control Dyn. Guibout, V. In: Gurfil, P. Modern Astrodynamics. Tapley, B. Schutz, and G. GP methods use transformation methods e. The averaged potentials can in some cases lead to a complete reduction of the system to quadratures, revealing equilibrium points as saddles and extrema in the middle of circulating flows in the reduced space.
Vital information about both local and global stability of the design space can be gleaned using analytic techniques. Furthermore, the use of modern algebraic manipulation tools can provide high-order analytic theories that were impossible to produce in decades past. Such high-order methods and other analytic innovations can provide insight and benefit to a variety of future astrodynamics space situational awareness tasks. Wisdom and M. Holman, Symplectic maps for the n-body problem, The Astronomical Journal 4 , Broucke, Long-term third-body effects via double averaging, Journal of Guidance, Control, and Dynamics 26 1 , doi Palacian, M.
Lara, R. Paskowitz Possner and D. Scheeres, Control of science orbits about planetary satellites, Journal of Guidance, Control and Dynamics 32 1 , Coffey, A. Deprit, and L. Healy, Painting the phase space portrait of an integrable dynamical system, Science , Deprit, Frozen orbits for satellites close to an Earth-like planet, Celestial Mechanics and Dynamical Astronomy 59 1 , Park and D. Scheeres, Nonlinear semi-analytic methods for trajectory estimation, Journal of Guidance, Control and Dynamics 30 6 , Fujimoto, D.
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Alfriend, Analytical nonlinear propagation of uncertainty in the two-body problem, Journal of Guidance, Control and Dynamics 35 2 , doi Gim and K. Alfriend, State transition matrix of relative motion for the perturbed noncircular reference orbit, Journal of Guidance, Control, and Dynamics 26 6 , Danielson, C. Sagovac, B. Neta, and L. Weeden and P. Finding: Analytic theories will continue to be important for both operations and theoretical development in order to meet future needs.
Insights from Modern Dynamical Systems. While advances coming from several disciplines, such as physics, programming, computer technology, and others, have been incorporated into determining the dynamics of orbits, a striking omission is that this study of the dynamics of objects does not employ the significant progress that has been made in dynamical systems. Instead, the level of the analysis of the dynamics is essentially the same as known during the formative years of the s.
All of this earlier work predates the breakthroughs starting in the late s that have completely changed how nonlinear dynamics and chaotic systems are treated and analyzed. That the advantages of these new approaches have not even been explored by AFSPC was supported by responses when people testifying were queried; without exception, the answer was no. Furthermore, there is a clear lack of awareness of the experts at AFSPC even of the existence of this literature and the advantages it could provide. Other areas of astrodynamics have benefited tremendously from the incorporation of dynamical systems theory and principles.
A recent highlight includes space mission design to the Earth-Sun and Earth-Moon libration points. The connections between the abstract theory of dynamical systems and practical and applied spaceflight have yielded an expansive growth in the ability to design previously undiscovered efficient and practical transfers within a highly perturbed, multi-body environment. There are excellent reasons to accept that the dynamics of satellite systems experience chaotic effects. Evidence comes from the fact that observed behavior strongly mimics standard and expected predictions of chaotic dynamical systems.
As illustrations, orbits starting from very similar settings can experience dissimilar dynamical futures, predicted and actual orbits can diverge, and covariance and other means of predicting the likelihood of conjunction can vary in unexpected ways. These traits are expected from nonlinear systems, and are observed in satellite systems.
Adding support to the expectation that these orbits have chaotic behavior is that chaotic systems typically are characterized by dynamics that experience:.
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That satellite systems have a dynamic recurrence is obvious. The expansion can be caused by drag, the nonspherical shape of objects, inhomogeneous gravitational forces e. These features—expansion and recurrence—are common in essentially all N-body systems, and so most of these systems must be expected to exhibit these chaotic behaviors; moreover, celestial mechanics is precisely the area where the discovery of chaotic behavior was first made in the late 19th century. Value added by incorporating the newly developed ways to analyze dynamical systems can be expected to create new insights leading to different ways to understand and predict the structure of these orbits.
As just one example, characteristics of nonlinear systems, which permit orbits to spread in unexpected ways, suggest that the standard covariance approach used to determine the likelihood of conjunction must be reexamined. An unfortunate fact, which reflects a shortcoming in STEM education, is that courses and appropriate books needed to learn the fundamentals of modern dynamical systems are not readily available.
Until and unless a program is developed to solve this issue, such as AFSPC creating such courses, or placing pressure on university systems to do so, it is unrealistic to expect current AFSPC personnel to adopt these mathematical approaches. Farquhar, D. Muhonen, D. Richardson, Mission design for a halo orbiter of the Earth, Journal of Spacecraft and Rockets 14 3 , Gomez, J.
Llibre, R. Martinez, and C. Howell, B. T Barden, M. Lo, Application of dynamical systems theory to trajectory design for a libration point mission, Journal of the Astronautical Sciences 45 2 , Until then, and because something must be done to handle the escalating nature of these problems, one approach toward achieving longer-term advances would be to encourage collaboration between experts in astrodynamics and academic experts in dynamical systems.
Finding: While advances coming from a variety of disciplines, such as physics, programming, computer technology, and others, have been incorporated into the dynamics of orbit determination, a notable omission involves a use of advances that have been made in the area of dynamical systems. One challenge in the use of these techniques is to develop more efficient algorithms, e.
Knowledge of the state e. As described in the section on modern dynamics, a portion of this uncertainty can be attributed to features of the underlying dynamical system. If the uncertainty is properly characterized, one can then attempt to manage the uncertainty by tasking which sensors to view which objects within the field of view of the sensors up to the limits of the information and availability of sensor coverage. For Gaussian or nearly Gaussian processes, uncertainty is represented through a covariance matrix. In these cases, achieving covariance realism is the goal. For other objects, especially UCTs or objects for which few updates have been received due to the data-starved environment of space, a more complete description of the uncertainty in the form of the true probability density function may be needed for the non-Gaussian processes.
Drummond has identified seven reasons why covariances degrade in tracking systems. Proper characterization of the uncertainty of the input data for measurements, including the lack of whiteness, features, and nontraditional data;. Compensation for residual biases, bias drift between sensor calibrations, and time tags plus the correct representation of their covariances or uncertainty;. Correct characterization of non-Gaussian probability density functions arising from nonlinear transformations such as nonlinear dynamics and coordinate transformations;.
Correct characterization of errors in the model dynamics e. Well-designed numerical procedures that achieve robustness, thereby avoiding numerical round-off errors in the presence of simplified or incorrectly designed computations or ill-conditioned transformations; and. Other references to provide a flavor are the following: J. Saari and Z. Waugh, Metrics for evaluating track covariance consistency, Signal and Data Processing of Small Targets , BOX 2.
To provide an example of covariance degradation due to nonlinearity and to explain the difference between the extended Kalman filter EKF , unscented Kalman filter UKF , and an exact nonlinear transformation, consider the representation of uncertainty in Figure 2. The 1, particles are dispersed according to the level curves ranging from 0. The initial orbit is circular and non-inclined with a semi-major axis of Although the range error of 20 km is extreme, it is not inconsistent with some cases seen in real data.
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The true distribution, represented by the particles propagated through the eight orbital periods 13 hours and 20 minutes , is presented in the bottom portion of Figure 2. The mean and covariance of the UKF were obtained by propagating the 13 sigma points but agree with those computed from the probability density function PDF represented by the particles. The covariance from the UKF is consistent or realistic in the sense that it agrees with that computed from the definition of the true PDF.
The same is true of the mean; however, the mean is displaced from the mode of the PDF, and the covariance does not represent the uncertainty in the true PDF. The EKF, on the other hand, provides a good representation of the mode, but the covariance tends to collapse, making inflation necessary to begin to cover the uncertainty. In neither case does the covariance accurately model the uncertainty.
On the other hand, similar results occur after 7 days and orbital periods when the semi-major axis standard deviation is reduced to a more realistic 1 km. In this case, the covariance remains valid over 50 orbits or so; however, it also degrades in due course. This example illustrates the problem of using covariances to represent the uncertainty and suggests that a better representation of the probability density function is needed if one is to achieve statistically robust characterization of uncertainty, which again is fundamental to achieving a robust capability across the Space Surveillance Network.
As noted above, the covariance matrix may be sufficient to characterize the uncertainty for some objects in the space catalog, whereas a more complete approximation to the true probability density function may be needed in other cases such as during orbital propagation. An example that demonstrates the need to think beyond standard Gaussian distributions and covariance matrices in the representation of uncertainty is provided in Box 2.
This example also demonstrates the deficiencies in the extended Kalman filter and the unscented Kalman filter for space surveillance. Similar examples can be found in the literature. While uncertainty in some of the parameters in the nonlinear dynamic models is often addressed statistically through a consider analysis and process noise as stressed in issue number five 5 above, nonstatistical errors in the modeling of the dynamic forces on a body also play a part in the uncertainty characterization of space objects.
Given the correct association of sensor measurements to a track, the problem of fusing this information with the track to obtain an updated estimate of the track is generally posed as either 1 a general Bayesian nonlinear filtering problem augmented by smoothing or 2 a batch estimation in the form of a weighted nonlinear least-squares problem discussed above.
Batch estimation methods are robust if they converge, but are more computationally intensive than filtering methods. The various options for potential improvement are listed above. As indicated below, filtering methods require a complete and realistic probability density function of the state to start the sequential process, while batch estimation can be used even when the state probability density function is ill-conditioned.
Indeed, the observability of the state can be determined as part of the solution. The connection between these two approaches, namely, differential corrections or the Gauss Newton method and filtering and smoothing, is provided in the paper by Bell. For the propagation of uncertainty, the extended Kalman filter is often used; however, the unscented Kalman filter should also be considered. The example in Box 2. Ultimately, the Fokker-Planck equation is the correct formulation; however, its computational cost is too high, except in special cases. Junkins, M.
Akella, and K. Alfriend, Non-Gaussian error propagation in orbital mechanics, Journal of the Astronautical Sciences 44 4 , Giza, P. Singla, and M. Scheeres, Nonlinear mapping of Gaussian statistics: Theory and applications to spacecraft trajectory design, Journal of Guidance, Control, and Dynamics 29 6 , Julier, J.
Uhlmann, and H. Durant-Whyte, A new method for the nonlinear transformation of means and covariances in filters and estimators, IEEE Transactions on Automatic Control , Horwood, N. Aragon, and A. Poore, Gaussian sum filters for space surveillance: Theory and simulations, Journal of Guidance, Control and Dynamics 34 6 , In addition to the large number of traditional estimators, Lp-norm and polynomial chaos estimators are also applicable to problems in astrodynamics.
In the applied mathematics and statistics community the subject of uncertainty quantification and stochastic differential equations is an active area of research, both in modeling and in the development of numerical algorithms, and may impact the approaches to nonlinear estimation and modeling in astrodynamics. Given the data-starved environment of space, parameters in the dynamics may be part of the state and need to be estimated.
In space surveillance, a maneuver may not be observed; however, multiple model filtering may still be successful in detecting maneuvers. Such maneuvers may then be confirmed using optimal control. Finding: The proper characterization of uncertainty in a state estimate of a space object requires advanced nonlinear estimation techniques for the measurement models and stochastic differential equations arising in astrodynamics. Such characterizations must address Gaussian and non-Gaussian random processes beyond the Gaussian processes in the extended or unscented Kalman filters.
In the process of determining the orbit, the uncertainty covariance of the orbit estimate is also determined. Only in the past few years has the accuracy of this covariance become important. The first major use of the covariance started in the late s when NASA needed the covariance to compute the probability of collision of objects that were going to pass close to the International Space Station or the space shuttle. A factor-of-two error in the covariance can change the probability of collision by three to four orders of magnitude near the threshold where the decision is made to make a collision avoidance maneuver.
A covariance that is optimistic, or too small, could result in a decision to not make a maneuver when one should be made, and a covariance that is too large could result in the opposite, making an unnecessary maneuver and wasting fuel. As noted earlier, the user community is trying to improve the covariance using inflation where appropriate.
Thus, having a covariance that represents the real uncertainty is critical to the space community as a whole. Other uses of the covariance include correlating uncorrelated tracks, maneuver detection, and sensor resource management. Obviously these uses also require an. Alspach and H. Kotecha and P. Norgaard, N. Poulsen, and O. Ravn, New developments in state estimation for nonlinear systems, Automatica 36 11 , Ristic, S. Arulampalam, and N. Linares, J. Crassidis, M. Jah, and H. Bar-Shalom, X. Rong Li, and T. The primary factors that cause the covariance to not represent the real uncertainty in this case include three of the seven reasons by Drummond presented previously, namely:.
The nonlinear effects tend to be small when updating the orbit of objects in the catalog, but they can become dominant when 1 correlating uncorrelated tracks to obtain the initial orbits of space objects or 2 predicting uncertainty for long time periods without additional measurments. For objects not affected by atmospheric drag the primary factor causing the covariance to not be realistic is the mischaracterization of the sensor errors. The radars are calibrated by tracking objects for which the orbits already are very accurate as a result of being tracked by lasers.
Each day, each of the sensors track these objects. From the residuals, the mean and the standard deviation are computed for each sensor for each measurement type, e. One of the major factors that cause measurement errors is the ionosphere, but the ionospheric effect is a function of the elevation. Using a model of the ionosphere, its effect on the measurement is estimated and this is taken into account, but the model is not perfect and there are residual errors.
Consequently, the errors increase at lower elevations, but this effect is not taken into account in the error estimate. In addition, the ionospheric effects are a function of the time of day, and this effect is not taken into account in the sensor error estimate. Thus, the measurement errors are a function of the time of day and the elevation angle, but just one value of the sensor measurement error is used for all elevations and all day. This is just one example of how the sensor errors are not correctly estimated in the current JSpOC calibration process.
Finding: A major factor in the uncertainty e. Nonstatistical errors such as the approximation of nonlinear dynamics are also contributing factors to uncertainty. A program to better characterize these errors will improve the uncertainty, which will improve the accuracy of the probability of collision and improve space safety. As the number and the sophistication of space-faring nations increase, it is imperative that the operational and analytical capabilities of the JSpOC and AFSPC have the capacity to grow and evolve to provide adequate and necessary monitoring of the space object population.
Although it is possible to provide a notional list of what activities and capabilities should exist to better manage these activities, such a prescriptive approach cannot be complete and may neglect important capabilities that are still over the horizon. Instead, in this section the committee provides a brief overview of recent space situational awareness research, and then focuses on a key framework perceived by the committee to be particularly impactful in terms of meeting future JSpOC space situational awareness needs. Broad Research in Space Situational Awareness. Recent research in the realm of space situational awareness has cast a wide net: a survey of recent conference presentations at the yearly Advanced Maui Optical and Space Surveillance Technologies AMOS Conference available at www.
Representative areas of interest include:. Data correlation, association, and initial orbit determination to build and maintain a catalog, both in quiescent operations and in the presence of a debris-generating event;. Alternative object states and their characterization, such as attitude, color, and shape. Although these areas of interest will all benefit from the expanded capabilities and models discussed above, they are topics of research in their own right.
Any advances in these areas should be able to be integrated into the activities of the JSpOC given the more flexible architecture of the JMS system. For example, consider maneuver detection, the automatic detection and estimation of maneuvers and the computation of bounds on these maneuvers. Systematic inclusion of these capabilities into the space situational awareness catalog process will motivate the development of more flexible filtering methods with the ability to vary what is considered in the estimated state of the object.
These capabilities will also motivate additional computations associated with an object related to characterizing the apparent thrust capability including electric propulsion and other low-level continuous thrusting of an object and its propellant usage, which could be essential for making strategic decisions.
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Likewise, the other areas of research listed above also provide motivation for a flexible approach that the proposed JMS will enable. However, more fundamental than any specific advancement is the need for a systematic approach to the characterization and categorization of the object population, the topic of the next subsection. Object Characterization and a More Complete Catalog. Space objects have traditionally been characterized in the catalog by their orbit and, at most, an additional parameter representing the ballistic coefficient of the body.
This approach is only acceptable for tracking the trajectory motion of an object over short time periods, but does not provide for the incorporation of any additional information that may be used or available to represent the object and its state, writ large, more accurately. The development and acquisition of such additional information on objects has been a subject of active research, including uniquely characterizing an object based on information that supplements and is different from orbital data. The spectral signature, including both color and reflectivity, of an object can be an important component of the object and can be used for identification.
The shape and size of an object can be inferred from high-resolution imagers or from radar range-Doppler imaging of objects and confer unique information about a given object. Even with lower-resolution imaging it is possible to estimate the rotational dynamics and relative attitude of an object using light curves. Finally, significant improvements to the legacy nongravitational parameters currently used in the catalog are being developed and may have significantly better accuracy and utility as estimated parameters for both fitting orbit data and performing more effective extrapolation of these estimates to yield more precise predictions of satellite motion.
When appropriately fused together, these characterization techniques can provide a much richer picture of an object. If appropriately stored over time they can also be used to detect an evolving. Holzinger and D. Kelecy, D. Hall, K. Hamada, and M. Kelecy and M. Jah, Detection and orbit determination of a satellite executing low thrust maneuvers, Acta Astronautica 66 5 , Singh, J. Horwood, and A. To enable such a richer catalog to become real, however, requires that the current limited catalog be enhanced to enable multiple object properties to be tied to each other across a distributed database.
To be effective such a database must be flexible enough to enable inclusion of new properties without disrupting existing structures. Such relational databases have been developed to support the storage and retrieval of dynamical and contextual information for scientific missions, and have become a worldwide standard. For complex bodies, these ID numbers can be further nested, allowing for a detailed and high-dimensional model to be tracked, thereby providing a framework for tracking satellite clusters and formations in the space situational awareness population.
For active bodies, a history of events, maneuvers, and deployments can also be automatically accessed and evaluated. Such existing products provide a direction for capturing the future potential of expanding object characterization beyond the currently limited approach. Because the storage of this information is not necessarily within a single file, but can be distributed across several files, it is also feasible for sensitive information about an object to be effectively segregated by omitting sensitive files from the distribution.
Examples of such sensitive information may include attitude tracking, covariance information, and observed shape changes. The mere act of withholding that information would prevent it from being correlated with more common information such as ephemeris predictions. Adoption of such a relational database approach could also help resolve the existing issues related to the dis-. Acton, Jr. Acton, N. Bachman, L.