Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation
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Hence, the maximum absolute error depends upon the number of digits available for the fractional part of the mantissa t and the order of magnitude p of the binary number. As an immediate consequence, the maximum absolute error in the floating point representation of a specific physical quantity depends upon the adopted unit. For example:. However, averaging on a large enough interval of values, the two maximum absolute errors are equal, so they can be considered equivalent.
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The IEEE standard defines several basic floating-point binary formats that differ by the number of bits used to encode the sign, the exponent and the fractional part of the mantissa. The most important formats are Binary32 also called single precision , Binary64 also called double-precision , and Binary also called quadruple-precision.
The rounding error is a form of quantization error, with the LSB as the quantization step. With this model, it is possible to compute the statistical characteristics of the numerical error:. In this work the statistical model was assumed to be valid and this assumption was verified through a detailed analysis of the rounding errors, described in Sec. When performing an exact operation f x 0 , y 0 using floating point arithmetic, the result is affected by two kind of errors with respect to the theoretical value z 0 :.
Additional rounding error: even if the errors in the inputs are zero, the output could be affected by a rounding error. For non-basic operations, the error on the results could be larger and depends on how the operations are implemented. The additional rounding error may be zero if the intermediate result of the operation does not need rounding. The total error in the result is the sum of the propagation error and the additional rounding error:. ST , and the corresponding transmission time, at the electronics of the transmitting ground station, in ST:.
Neglecting second order terms, such as relativistic light-time delays, time scale transformations, electronic delays, and transmission media delays, the round-trip light-time can be written as:. In Eq. The reception time in Ephemeris Time 4 4 4 Ephemeris Time is the coordinate time of the Solar System barycentric space-time frame of reference used in the adopted celestial ephemeris.
Using the DRD formulation, the Doppler observable is computed as the difference of two round-trip light-times Eq. Qualitatively, this formulation has a high sensitivity to round-off errors because, as it is well known, the difference between two large and nearly equal values substantially increases the relative error, causing a loss of significance.
Hence, errors in the round-trip light-time have a large influence on the Doppler observables. From Eq. Representation of times: the rounding errors in time epochs t 3 , t 2 , and t 1 propagate only indirectly into the round-trip light-time. When the time variable increases relative to the reference epoch, the maximum rounding error increases with a piecewise trend, because it doubles when the binary order of magnitude p t increases by one. The difference of 12 hours in the reference epochs of the two programs is negligible. Hence, as discussed in Sec.
II , because of the different measurement units, the time round-off errors in these two OD codes are not exactly the same, but they have the same average magnitude, on a large enough time interval. For the ODP-like time representation, between July and January , the maximum round-off error is about 30 ns. Representation of distances: the rounding errors in each component of the position vectors propagate into the precision round-trip light-time directly, because the position vectors are used in the computation of the Newtonian distances r 12 and r 23 , but also indirectly, because the position vectors are used also in the computation of t 2 and t 1.
However, the indirect effects are much smaller than the direct ones and can be neglected.follow link
Additional rounding errors: the rounding errors in the result of each operation propagate into the round-trip light-time. Moreover, the position vectors r i j used in Eqs. For simplicity, the numerical errors introduced in these computational steps are neglected. This assumption was verified a posteriori through the validation of the numerical errors model. As discussed in Sec. The numerical error in two- or three-way Doppler observables with time-tag T T can be expressed as:.
Adopting the statistical model for each numerical error in Eq. Non stationary, because the coefficients multiplying every single numerical error are a function of time. However their time variation is typically very slow and, for a typical duration of a tracking pass, the process can be considered stationary. Because the different numerical errors are a finite number and not identically distributed, the expected PDF is only qualitatively bell-shaped.
Where the variance of each numerical error can be computed using the formulation presented in Sec. In fact, the count times are consecutive, so the end of a count time is the start of the next count time:. Given Eq. At present, almost all hardware and compilers implement the IEEE standard. Time: the time at which the OD problem is settled defines the order of magnitude of all time epochs, thus defining the quantization step and the amplitude of the round-off errors in each time variable.
The EP module is responsible for predicting celestial events such as eclipses, apsis passages, node crossings, solar communication interferences, and ground station visibilities. The DS and OD modules are closely tied, and responsible for orbit determination of spacecraft and tracking data simulation. The DS module includes the DSN sequential range and Doppler observation models, their noise characteristics, and media and antenna correction models.
In the OD module, a batch least squares estimation algorithm has been implemented to solve the orbit determination problem Cappellari et al. In addition, the OD module provides covariance analysis methods such as covariance propagation, projection, and transformation. There are two possible forms of event prediction results: a text-based event prediction report and instances of Event Prediction class, which are discussed in Section 4.
The user can make an orbit determination request from the OD module with an initial guess on solve-for parameters e. Tracking data includes information on the ground station, DSN measurement settings for range and Doppler observables, DSN measurements, and time tags of measurements. Note that all the range and Doppler observations include radio signal integration over the time interval, and a time tag represents the measurement time.
The OD module obtains ephemerides of the spacecraft, ground stations, and celestial bodies using the OP module with the spacecraft, ground stations, and relevant settings. Then, the OD module acquires the computed range and Doppler observables using the DS module with the ephemerides and relevant settings. The OD module then calculates the residuals between the computed and observed DSN measurements so that the OD module updates solve-for parameters to reduce the weighted least squares of residuals. If solvefor parameters are converged, the OD module delivers the orbit determination solution to the user.
Otherwise, the OD module continues updating solve-for parameters until solve-for parameters converge or the number of iterations reaches the predefined maximum iteration number. The OD report includes information on the solve-for parameter solution, the covariance of the OD solution, measurement residuals, the measurement data editing history, DSN settings, and orbit propagation settings. The user can make a DSN measurement data simulation request to the OD module with the truth data for solve-for parameters e.
Compared to event prediction and orbit determination processes, the tracking data simulation process includes the majority of event prediction and orbit determination processes, except the iterative process. In addition, in the tracking data simulation process, the DS module does not produce computed range and Doppler observables but simulated sequential range and Doppler observables, which include simulated white noise and user-defined biases. Now, the CUCL consists of six important classes and utility subroutines: epoch time, coordinate system base, Hermite ephemeris, spacecraft, transponder, and ground station base.
Among them, epoch time, coordinate system base, and spacecraft are associated with GMAT and compatible with the GMAT corresponding counterpart classes. The utility subroutines are not directly addressed here because there are a number of subroutines in DSODS. Thus, the following explanation on CUCL focuses on the classes in terms of information and functionalities.
The CUCL was developed to efficiently handle the interfaces between modules and various input and output data specifications. Time is one of the most fundamental concepts in astrodynamics, and needs to be handled with discretion. Currently, the epoch time class contains information on the epoch time, its time system type e. In addition, epoch time provides functionalities such as conversions between representation types and calculation of the elapsed seconds between different epochs. Regarding the time rate correction term, the following holds: 2. Thus, the time rate correcttion term is not equal to zero only for the TDB time system.
Note that one second in TDB is different from one second of the other time systems due to the general relativistic time dilation effect Moyer The time rate correction for TDB is realized in the DS module for the general relativistic light propagation in the solar system barycentric space-time reference system.
Both the time rate corrections and time difference is now handled by GMAT. Gregorian representation expresses a time moment by year, month, day, hour, minute, and second. Supporting various time representations is important for generalizing input and output specifications because each specification uses a different representation.
The conversion between MJD and Gregorian representations can cause numerical noise associated with dividing by 86, a day in seconds. For instance, if the user user set the time resolution to be 10 -6 , DSODS rounds a second of the Gregorian representation with respect to six digits to the right of the decimal point.
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As a result, the conversion different coordinate systems are handled entirely by GMAT. In accordance with GMAT, the coordinate system base class supports six axis types: J equatorial, J ecliptic, ICRF, object referenced, body fixed, and body inertial, and supports three types of coordinate system references: celestial point, spacecraft, and ground station.
Apart from the object reference axis type, the coordinate system has only one reference point that is the center of the coordinate. The object reference axis type constructs a local-vertical-local-horizontal LVLH axis, where the radial direction is defined by the vector from the first reference to the second reference, the normal direction is defined by the angular momentum vector, and the other direction is co-normal to the radial and normal directions.
The center of the coordinate is defined as the second reference point. Therefore, DSODS allows the user to define a variety of coordinate systems as a combination of the reference points and the axis type. Inside of DSODS, the Hermite ephemeris class is used to define any ephemeris as a set of polynomial coefficients to interpolate position and velocity vectors at any moment within the predefined interval by Hermite interpolation. Hermite interpolation guarantees consistency between position and velocity interpolations, which means that the derivative of the polynomial interpolating position is equal to the polynomial interpolating velocity Zill et al.
Although the concept of ephemeris is originally defined for a space object such as a celestial body or spacecraft, the Hermite ephemeris class of DSODS supports any object, including a ground station, with a position and velocity as a function of time within the predefined interval. The extended concept of ephemeris is used throughout DSODS, and is especially useful for handling iterative processes, such as root-finding in the EP module and light time equation solving in the DS module, because these iterative processes requires the position and velocity of an object as continuous curves rather than a set of discrete points.
Currently, the Hermite ephemeris class uses polynomials of the fifth degree for each coordinate by default, and allows the user to use polynomials ranging from the third to the seventh degree. The spacecraft class contains information on the name, epoch time, initial state vector, coordinate system, representation type, drag area and coefficient, SRP area and reflectivity, and transponder of a spacecraft.
It supports three state representation types: Cartesian, classical Keplerian, and spherical coordinate state representation based on right ascension and declination. Moreover, a transponder is represented as a class in DSODS, which contains information on the uplink and downlink bands, turn around ratio, and transponder delay in meters. In the DSN sequential range observable, transponder delay plays an important role because its magnitude is usually several hundred meters, e.
The ground station base class contains information on the location, range and Doppler biases, range and Doppler noise levels, antenna related information such as cut off angle for contact, diameter, antenna offset constant, and mount type , and mean meteorological models ODTBX for DSN complexes.
Also, the ground station base class provides the methods for conversions between Cartesian and geodetic spherical coordinates, antenna offset correction calculations, and media correction calculations based on mean meteorological models. The OP module is responsible for fundamental astrodynamics- related operations: orbit propagation, time and coordinate system conversions, and creation of spacecraft and celestial body ephemerides.
For instance, currently, the GMAT supports only gravitational potential data with a degree and order equal to or less than Table 1 explains the module-level methods provided by the OP module. Finally, there are four public classes accessible by any module: the air-drag model, the gravity potential model, the force model, and the propagator. These classes are used to deliver orbit propagation options, and are compatible with but not dependent on the GMAT.
Using these public classes, the script interface class writes a GMAT script that gives an order to the GMAT regarding how to propagate the spacecraft trajectory. Note that for orbit propagation, most of the data communication is indirect and based on text file access, except for the name of the script, which is a string type data, and is directly delivered to the GMAT base library through the MEX interface.
Moreover, Fig. The specific data contents are omitted in the data flow diagram because they are dependent on the operation or, method being applied. The EP module predicts astrodynamics-related events, and delivers information on the timing and type of event to the user or the OD module to aid the mission planning and design process.
The highest-level data flow diagram is provided in Fig. The EP module addresses various types of events such as orbital status, eclipses, ground station visibility, and solar communication outages. However, the current implementation of the EP module does not consider special relativistic light time delay and stellar aberration. Future extension of the EP module may include these perturbation effects for applications to other deep space missions beyond the Earth-Luna system.
The EP module consists of event functions, an event location algorithm, a mesh-refinement algorithm, and public classes. The event functions are defined for different types of event in that when an event occurs, the event function value is equal to zero. For instance, Eq. As a result, finding roots of the event function is equivalent to locating the events. The event location algorithm finds the location and type of events by solving the root-finding problem with the data provided i.
To find the roots of the event function, the event location algorithm utilizes the cubic Hermite spline and analytical roots of the cubic equation. The mesh-refinement algorithm estimates the quality of the event prediction solution and updates the mesh-points used in the root-finding so that the quality of the event prediction solution satisfies the userdefined tolerance. There are two public classes in the EP module: point-type event prediction and interval-type event prediction classes. These classes contain information on the location and type of events obtained by the event location algorithm.
Each functionality code i. If necessary, a functionality code can be divided into multiple functionality sub-codes i. For instance, ground station visibility is acquired only when the elevation of the spacecraft is higher than the cut-off angle of the ground station and the signal is not occulted by any celestial body.
Although functionality codes appear fairly fixed, new functionality sub-codes can be added in the future. The mathematical details of the event location process are provided here. The mesh-refinement algorithm controls the outer loop of the event prediction process while the event location algorithm controls the inner loop.
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The meshrefinement algorithm controls the overall quality of the event prediction solution. The event location algorithm consists of two components: the cubic Hermite spline and the analytical root-finding algorithm for cubic equations. The approach taken here has merits in that it does not require any iterative process in root-finding. The cubic Hermite spline utilizes the first derivatives, and it has a better accuracy than the natural cubic spline, which does not use any derivatives.
An error control scheme for instance, mesh refinement is not applied at the event location algorithm level but at the meshrefinement level. The flow chart for the event prediction process is presented in Fig. The user provides the spacecraft ephemeris and initial mesh configuration for the time domain. With the given mesh points, the OP module is asked to produce the ephemerides of celestial bodies and ground stations.
Based on this, the event function subroutine provides event function values and its first derivatives at mesh points. For each mesh interval, the cubic Hermite spline subroutine defines a thirddegree polynomial specified for the Hermite form based on event function values and its first derivatives at the end points of the interval. For each cubic function defined in an interval, the analytical root-finding subroutine for cubic polynomials analytically determines all inside roots.
By repeating root-finding for all intervals, the event location is completed. The mesh-refinement algorithm increases the time sampling rate by ten times, and compares the relative errors of the roots between two different mesh configurations. The mesh-refinement algorithm continues increasing the time sampling rate until all the relative errors satisfy the userdefined tolerance or the iteration counts reaches the userdefined maximum iteration constraint.
As the end product, the locations of the events and the event conditions e. The cubic Hermite spline defines a third-degree polynomial in Hermite form using the function value and its first derivative at the end points of an interval Zill et al. The coefficients, presented in Eq. Compared to the natural cubic spline, the cubic Hermite spline generally achieves a better interpolation accuracy using the first derivatives when constructing the cubic equation in an interval.
The absolute accuracy, as opposed to the relative accuracy, of the cubic Hermite spline is affected by both the event function and mesh point configuration. Although the analytical expressions of the roots of cubic equations are elementary, they are not presented here due to their complexity. Experience using the cubic hermite spline and analytical roots indicates that even for real roots, the analytical roots obtained in the complex space can contain numerical noise in the complex component.
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They seem to be affected by the cubic Hermite spline and the MATLAB built-in algorithm for calculating the cubic roots in the complex space. This subsection evaluates event prediction capability using a solar outage or, solar interference prediction example. Solar outage is a phenomenon whereby radio signals from a spacecraft are obscured by solar radiation. To prevent unexpected communication problems, it is necessary to predict the timing and duration of solar outage events for the entire mission duration. Note that the explanations for the example are rather brief because the objective here is not to justify the approach taken in the EP module but to give an idea how the EP module solves event prediction problems.
Here, a solar outage prediction simulation result is presented for a lunar orbiter. The outage angle defines when the outage occurs. If the angular separation between the spacecraft and the Sun is less than the outage angle, the signals from the spacecraft are expected to be obscured by solar radiation. From Eq. In the simulation, DSODS predicts the solar communication outage of a lunar orbiter observed by a nominal Daejeon station