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How to Determine Space Vehicle Orbits Using Automated Measuring Gear
HOW TO DETERMINE SPACE VEHICLE ORBITS USING AUTOMATED MEASURING GEAR
E. L. Akim, M. D. Kislik, P. E. El'yasberg, T. M. Eneev
The successful resolution of scientific problems involvingartificialcelestial bodies (space vehicles) launched into outer space depends in large measure on the accurate determination of the actual orbits of these vehicles.
ployed in astronomy, even though they still naturally hark back to the general laws of celestial mechanics in their fundamentals.
Orbit determination is a necessity to establish the fact that the vehicle has actually entered a space orbit, to effect a tie-in in intertial space for scientific measurements taken on board the vehicle, to predict the motion and deliver commands controlling the vehicle, and to deliver target destination information via observations and communication. Moreover, the actual orbit traversed by the space vehicle can be utilized in many instances to secure improved information on the forces operating in flight on the vehicle, and consequently on the various geophysical and astronomical parameters governing the magnitude of those forces. Most of the problems listed above require orbit determination of space vehicles to high order of precision, within very short time spans, and independently of season, time of day, or atmospheric conditions. The observational means applicable in astronomy in the study of the motion of natural celestial bodies fail to meet these requirements. Reliance on astronomical techniques in space-vehicle orbit determination is therefore relegated to an auxiliary plane.
The development of these techniques facilitates the solution of the following major problems: a) arriving at a suitable set of assumptions under which the mathematical description of the motion in space of the vehicle will occur (determining the form of the righthand members of the differential equations of motion of the vehicle launched); b) arriving at a suitable statistical method for estimating the unknown orbit parameters on the basis of redundant information in the measurements; c) compiling algorithms of orbit determinations which will be optimal in the sense of precision, rapid solution of problems, and reliability in programming the calculations in electronic computers. It goes without saying that these questions are intimately
related a.'1d that the distinction drawn is purely arbitrary. In the selection of the right-hand members of the dif-
Special-purpose automated measuring gear has had to be devised to facilitate determination of space-vehicle orbits. Mandatory components of this gear include the following items: a) high-precision radio electronic measuring systems capable of determining the required parameters of the vehicle's motion in space under any and all conditions; b) means for automatic telemetering of the information; c) electronic computers capable of mathematically processing the arriving information needed in orbit determination. Aside from the work of devising the needed technical devices for measurements, transmission and processing of information essential in space-vehicle orbit determination, no less important is the development of special mathematical techniques designed to take the special features of these orbits into account, not to mention the potentialities of electronic computer techniques. These methods differ from the methods em336
ferential equations of the vehicle's motion, the deciding factor is the shape of the orbit. In the case of low-orbiting artificial earth satellites, a useful rule of thumb is to restrict the problem to the influence of the earth's gravitational field and of atmospheric drag. For satellites having an apogee height exceeding 7000 to 10,000 km, the effect of the lunar and solar gravitational fields will have to be considered as well. If the perigee height of the satellite orbit exceeds 1000 km, then the atmospheric drag force may be excluded from the equations of motion. When the motion of lunar space vehicles is being calculated, it is sufficient to take into account the influence exerted by the earth, moon, and sun, while neglecting any attraction exerted by the planets. The orbits of interplanetary vehicles may usually be described by right-hand members of the differential equations which differ in their structure over different orbit segments, the structure depending on the local intensity of the gravitational fields established by the major bodies of the solar system. If a space vehicle possesses a reasonably small ratio of mass to effective reflecting area, the force contributed by solar radiation pressure should be introduced into the equations of motion. In practically all cases, the space vehicle is regarded as a mass point on the orbital flight segment (i.e., the effect
of the rotational motion on the motion of the center of mass will be ignored). It must be borne in mind that the recommendations made on the nature of the righthand members of the equations of motion of space vehicles of different types cannot be conclusive or definitive. They have to be revised to meet any major alteration of the orbit parameters or vehicle parameters, and to concur with any improvements in our lmowledge about outer space and the bodies making up the solar system.
rive at a system of equations nonlinear in the parameters Qi:
~t.
:
(J
( / - I,
t, "')
(3)
'Wt )
or (taking into account symmetry of the matrices KIjJ and P) (4)
In making a selection of a statistical techniques for
estimating the unknown orbit parameters, accotu1t must be taken of the nature of the information being processed. The usual assumption is that errors in the a posteriori measuring information are mainly of a random nature, subject to a normal distribution law of zero mathematical expectation and lmown variance. As a rule, these errors are assumed to be independent. In conjunction with the a posteriori measurement information used in determinations of the parameters of a motion, a priori information on the trajectory is also fitted into the total picture in many instances. A priori information contains expected values of specific ftu1ctions of the trajectory parameters with their corresponding probabilistic characteristics of the likely errors. This information will result, for example, from processing measurements of earlier segments of the trajectory and from estimates of the spread in the results obtained. It may be used as the totality of several correlation measurements with known a priori statistical characteristic in the form of an error correlation matrix.
The devising of an optimum algorithm for orbit determination involves detailed elaboration of a method for solving system (4). Two methods for solving this problem have found practical applications: Newton's method of generalized tangents and the method of steepest descent. The solution by Newton's method leads, as we lmow, to a series of successive approximations. The number of approximations depends on how close the zero-order approximation selected is to the exact solution. At each approximation step in determining the correction s ~Q to the parameters obtained in the preceding approximation we have to solve a normal system of equations '»>
L
M
A··'J
~
L.. ')1)1 7 I
~J.
M
Ei
(I'
"~"1..)
... >
»1 )
(5)
:
-r.
..... Q:.
?.. . ,t
~~." d~.t d~ .. aQj
db
~
dq; J..
(6)
(7)
The matrix and the right-hand members of systems (5) are formulated with the aid of formulas (6) and (7) in each approximation step for the trajectory to be determined from the parameters of the preceding approximation.
The maximum likelihood method employed in order to determine the unlmown parameters (Qi)i = 1,2, .. , m"" 6) from the measurements !P1, iii"2,·'" iii"n, ... ,iij ,.. ,iii"M leads to the necessity of minimizing the functional
In the absence of any fairly decent zero-order approxi-
mation, or when the measurements being processed leave much to be desired in the way of precision, so that Newton's method will not readily yield the solution, the method of steepest descent may fill the gap. This method reduces to a system of differential equations
where Pnl are the elements of the "weighted" matrix P expressed in terms of the correlation matrix K IjJ of the measurements by the formula
r :
"r,j
where the coefficients Aij and the right-hand members Bi exhibit the form
The presence of these two groups of measurements in the information being processed leads to the choice of one of the most effective techniques (in the sense of offering minimum dispersion) for estimating the unlmown parameters as our statistical technique - the maximum likelihood method. When only uncorrelated measurements are being processed, the maximum likelihood method reduces to the usual method of least squares, an eventuality which immeasurably speeds up the data processing.
-0
Ai;' L:J;(j
j -,
(8)
-.
k.,
(2)
where the variable s is the parameter along the descent path in the m-dimensional space Q of the variables.
As a result of mimization of the functional (1), we ar-
337
System (8) is numerically integrated by the Euler method. When initial conditions are assigned judiciously, the integration interval will be narrow. The method of steepest descent substantially improves the reliability of the mathematical data processing techniques in the statistical problem concerning processing of measurements, but is inferior in rapidity of solution to Newton's method. This dictates combining the two methods in attempting to solve the problems, preference being conferred upon the more rapid Newton method. But the method of steepest descent offers the further attraction of substituting reliably when the normal convergence of the process breaks down in the use of Newton's method.
LITERATURE CITED 1.
2.
We gather from the foregoing that at each approximation step in the useof Newton's method, or at each integration step in the use of the method of steepest descent, we are obliged to calculate out the values of Aij and Bi from formulas (6) and (7). These formulas include the derivatives a~/ a Q and the differences between theoretically predicted and actually observed values of the parameters being measured for each instant of observation and for each measured parameter. Since the total quantity of the measurements involved in the processing may be enormous, while the segment of the flight path from which these measurements were taken may be quite high, reduction of the total data processing time becomes a question of utmost seriousness. This problem is successfully solved by using the most economic techniques in computing derivatives and 'rated values of the parameters being measured, as well as reduction in the total quantity of approximations (integration steps) during the orbit determining process. For example, in computing the derivatives, account must be taken of the comparatively mild requirements imposed on accuracy, and the calculations are carried out in almost all cases on the basis of the finite formulas of Keplerian motion. A refinement of the zeroorder approximation leading to a reduction in the total number of approximation steps is achieved by resorting to a two-position method of orbit determination, smoothing, and numerical differentiation of the coordinates in order to determine the velocities, etc. Shortening of the time needed for orbit calculations in the parameters Q when the differences ~ are computed becomes possible when an optimum choice is made of the coordinate frame, method, and integration steps, in the integration of the differential equations of motion of the spacecraft. Approaches of this sort have been worked out in large number and are used in the practice of orbit determinaticn. Summarizing, it should be noted that the solution of the problem of space-vehicle orbit determinations meeting predetermined requirements became possible thanks to the use of sophisticated automatic measuring gear, to telemetering and processing of information on one hand, and thanks to the development of efficient mathematical tools on the other hand.
338
Sborniki "Iskusstvennye sputniki Zemli." USSR Academy of Sciences: No. 4 (1960); No. 13 (1962); No. 16 (1963). Kosmicheskie issledovaniya. 1, No. 1 (1963); 2, No. 4, No. 5 (1964).
E. L. Akim, M. D. Kislik, P. E. El'yasberg, T. M. Eneev
The successful resolution of scientific problems involvingartificialcelestial bodies (space vehicles) launched into outer space depends in large measure on the accurate determination of the actual orbits of these vehicles.
ployed in astronomy, even though they still naturally hark back to the general laws of celestial mechanics in their fundamentals.
Orbit determination is a necessity to establish the fact that the vehicle has actually entered a space orbit, to effect a tie-in in intertial space for scientific measurements taken on board the vehicle, to predict the motion and deliver commands controlling the vehicle, and to deliver target destination information via observations and communication. Moreover, the actual orbit traversed by the space vehicle can be utilized in many instances to secure improved information on the forces operating in flight on the vehicle, and consequently on the various geophysical and astronomical parameters governing the magnitude of those forces. Most of the problems listed above require orbit determination of space vehicles to high order of precision, within very short time spans, and independently of season, time of day, or atmospheric conditions. The observational means applicable in astronomy in the study of the motion of natural celestial bodies fail to meet these requirements. Reliance on astronomical techniques in space-vehicle orbit determination is therefore relegated to an auxiliary plane.
The development of these techniques facilitates the solution of the following major problems: a) arriving at a suitable set of assumptions under which the mathematical description of the motion in space of the vehicle will occur (determining the form of the righthand members of the differential equations of motion of the vehicle launched); b) arriving at a suitable statistical method for estimating the unknown orbit parameters on the basis of redundant information in the measurements; c) compiling algorithms of orbit determinations which will be optimal in the sense of precision, rapid solution of problems, and reliability in programming the calculations in electronic computers. It goes without saying that these questions are intimately
related a.'1d that the distinction drawn is purely arbitrary. In the selection of the right-hand members of the dif-
Special-purpose automated measuring gear has had to be devised to facilitate determination of space-vehicle orbits. Mandatory components of this gear include the following items: a) high-precision radio electronic measuring systems capable of determining the required parameters of the vehicle's motion in space under any and all conditions; b) means for automatic telemetering of the information; c) electronic computers capable of mathematically processing the arriving information needed in orbit determination. Aside from the work of devising the needed technical devices for measurements, transmission and processing of information essential in space-vehicle orbit determination, no less important is the development of special mathematical techniques designed to take the special features of these orbits into account, not to mention the potentialities of electronic computer techniques. These methods differ from the methods em336
ferential equations of the vehicle's motion, the deciding factor is the shape of the orbit. In the case of low-orbiting artificial earth satellites, a useful rule of thumb is to restrict the problem to the influence of the earth's gravitational field and of atmospheric drag. For satellites having an apogee height exceeding 7000 to 10,000 km, the effect of the lunar and solar gravitational fields will have to be considered as well. If the perigee height of the satellite orbit exceeds 1000 km, then the atmospheric drag force may be excluded from the equations of motion. When the motion of lunar space vehicles is being calculated, it is sufficient to take into account the influence exerted by the earth, moon, and sun, while neglecting any attraction exerted by the planets. The orbits of interplanetary vehicles may usually be described by right-hand members of the differential equations which differ in their structure over different orbit segments, the structure depending on the local intensity of the gravitational fields established by the major bodies of the solar system. If a space vehicle possesses a reasonably small ratio of mass to effective reflecting area, the force contributed by solar radiation pressure should be introduced into the equations of motion. In practically all cases, the space vehicle is regarded as a mass point on the orbital flight segment (i.e., the effect
of the rotational motion on the motion of the center of mass will be ignored). It must be borne in mind that the recommendations made on the nature of the righthand members of the equations of motion of space vehicles of different types cannot be conclusive or definitive. They have to be revised to meet any major alteration of the orbit parameters or vehicle parameters, and to concur with any improvements in our lmowledge about outer space and the bodies making up the solar system.
rive at a system of equations nonlinear in the parameters Qi:
~t.
:
(J
( / - I,
t, "')
(3)
'Wt )
or (taking into account symmetry of the matrices KIjJ and P) (4)
In making a selection of a statistical techniques for
estimating the unknown orbit parameters, accotu1t must be taken of the nature of the information being processed. The usual assumption is that errors in the a posteriori measuring information are mainly of a random nature, subject to a normal distribution law of zero mathematical expectation and lmown variance. As a rule, these errors are assumed to be independent. In conjunction with the a posteriori measurement information used in determinations of the parameters of a motion, a priori information on the trajectory is also fitted into the total picture in many instances. A priori information contains expected values of specific ftu1ctions of the trajectory parameters with their corresponding probabilistic characteristics of the likely errors. This information will result, for example, from processing measurements of earlier segments of the trajectory and from estimates of the spread in the results obtained. It may be used as the totality of several correlation measurements with known a priori statistical characteristic in the form of an error correlation matrix.
The devising of an optimum algorithm for orbit determination involves detailed elaboration of a method for solving system (4). Two methods for solving this problem have found practical applications: Newton's method of generalized tangents and the method of steepest descent. The solution by Newton's method leads, as we lmow, to a series of successive approximations. The number of approximations depends on how close the zero-order approximation selected is to the exact solution. At each approximation step in determining the correction s ~Q to the parameters obtained in the preceding approximation we have to solve a normal system of equations '»>
L
M
A··'J
~
L.. ')1)1 7 I
~J.
M
Ei
(I'
"~"1..)
... >
»1 )
(5)
:
-r.
..... Q:.
?.. . ,t
~~." d~.t d~ .. aQj
db
~
dq; J..
(6)
(7)
The matrix and the right-hand members of systems (5) are formulated with the aid of formulas (6) and (7) in each approximation step for the trajectory to be determined from the parameters of the preceding approximation.
The maximum likelihood method employed in order to determine the unlmown parameters (Qi)i = 1,2, .. , m"" 6) from the measurements !P1, iii"2,·'" iii"n, ... ,iij ,.. ,iii"M leads to the necessity of minimizing the functional
In the absence of any fairly decent zero-order approxi-
mation, or when the measurements being processed leave much to be desired in the way of precision, so that Newton's method will not readily yield the solution, the method of steepest descent may fill the gap. This method reduces to a system of differential equations
where Pnl are the elements of the "weighted" matrix P expressed in terms of the correlation matrix K IjJ of the measurements by the formula
r :
"r,j
where the coefficients Aij and the right-hand members Bi exhibit the form
The presence of these two groups of measurements in the information being processed leads to the choice of one of the most effective techniques (in the sense of offering minimum dispersion) for estimating the unlmown parameters as our statistical technique - the maximum likelihood method. When only uncorrelated measurements are being processed, the maximum likelihood method reduces to the usual method of least squares, an eventuality which immeasurably speeds up the data processing.
-0
Ai;' L:J;(j
j -,
(8)
-.
k.,
(2)
where the variable s is the parameter along the descent path in the m-dimensional space Q of the variables.
As a result of mimization of the functional (1), we ar-
337
System (8) is numerically integrated by the Euler method. When initial conditions are assigned judiciously, the integration interval will be narrow. The method of steepest descent substantially improves the reliability of the mathematical data processing techniques in the statistical problem concerning processing of measurements, but is inferior in rapidity of solution to Newton's method. This dictates combining the two methods in attempting to solve the problems, preference being conferred upon the more rapid Newton method. But the method of steepest descent offers the further attraction of substituting reliably when the normal convergence of the process breaks down in the use of Newton's method.
LITERATURE CITED 1.
2.
We gather from the foregoing that at each approximation step in the useof Newton's method, or at each integration step in the use of the method of steepest descent, we are obliged to calculate out the values of Aij and Bi from formulas (6) and (7). These formulas include the derivatives a~/ a Q and the differences between theoretically predicted and actually observed values of the parameters being measured for each instant of observation and for each measured parameter. Since the total quantity of the measurements involved in the processing may be enormous, while the segment of the flight path from which these measurements were taken may be quite high, reduction of the total data processing time becomes a question of utmost seriousness. This problem is successfully solved by using the most economic techniques in computing derivatives and 'rated values of the parameters being measured, as well as reduction in the total quantity of approximations (integration steps) during the orbit determining process. For example, in computing the derivatives, account must be taken of the comparatively mild requirements imposed on accuracy, and the calculations are carried out in almost all cases on the basis of the finite formulas of Keplerian motion. A refinement of the zeroorder approximation leading to a reduction in the total number of approximation steps is achieved by resorting to a two-position method of orbit determination, smoothing, and numerical differentiation of the coordinates in order to determine the velocities, etc. Shortening of the time needed for orbit calculations in the parameters Q when the differences ~ are computed becomes possible when an optimum choice is made of the coordinate frame, method, and integration steps, in the integration of the differential equations of motion of the spacecraft. Approaches of this sort have been worked out in large number and are used in the practice of orbit determinaticn. Summarizing, it should be noted that the solution of the problem of space-vehicle orbit determinations meeting predetermined requirements became possible thanks to the use of sophisticated automatic measuring gear, to telemetering and processing of information on one hand, and thanks to the development of efficient mathematical tools on the other hand.
338
Sborniki "Iskusstvennye sputniki Zemli." USSR Academy of Sciences: No. 4 (1960); No. 13 (1962); No. 16 (1963). Kosmicheskie issledovaniya. 1, No. 1 (1963); 2, No. 4, No. 5 (1964).