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Withering: 200 years is not enough
ISSN 10642307, Journal of Computer and Systems Sciences International, 2015, Vol. 54, No. 5, pp. 798–807. © Pleiades Publishing, Ltd., 2015. Original Russian Text © D.A. Kozorez, M.N. Krasilshchikov, D.M. Kruzhkov, K.I. Sypalo, 2015, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2015, No. 5, pp. 137–146.
CONTROL SYSTEMS OF MOVING OBJECTS
Autonomous Navigation during the Final Ascent of a Spacecraft into the Geostationary Orbit. Autonomous Integrated Navigation System Concept D. A. Kozoreza, M. N. Krasilshchikova, D. M. Kruzhkova, and K. I. Sypalob a
b
Moscow Aviation Institute, Moscow, Russia Central Aerohydrodynamic Institute, Moscow, Russia email: [email protected] Received March 19, 2015; in final form, April 28, 2015
Abstract—This paper is the first in a series of works devoted to the problems of a spacecraft in the geo stationary orbit autonomous operation. The paper presents the conceptual design of the spacecraft onboard autonomous navigation system applying to its final ascent to the geostationary orbit using the lowthrust electric propulsion system. This design is based on both stochastic and minimax uncontrol lable factors interpretation. The composition of needed hardware and the architecture of the space craft onboard integrated navigation system, which should be configured in accordance with standard sections of the final ascent trajectory, are determined. It is proposed to involve socalled interval esti mation algorithms in order to ensure the reliability of obtained estimations, taking into account unknown nature of possible navigation measurements information violations. An optional smoothing algorithm based on dynamic Bayesian measurements filtering, considering preliminarily obtained estimations of systematic errors. As an additional means of improving the accuracy of the navigation estimations it is proposed also to involve navigation measurements optimal scheduling methods and algorithms in order to generate optimal schedules of measurements directly on board the spacecraft. DOI: 10.1134/S106423071505007X
INTRODUCTION Nowadays, the most actual problem of the existing and advanced space systems improvement is to ensure their autonomous operation. In particular, with respect to the spacecraft (SC) on the geostationary orbit (GEO), this requirement is expressed in the necessity to autonomously SC devotion in the final orbit, transfer it to the required longitude, and keep it at the geostationary orbit point in accordance with inter national requirements [1]. The implementation of this idea requires advanced hardware set utilization for SC Navigation and Control systems implementation on all the above mentioned stages of its life cycle at the GEO. First of all, it is necessary to use electric propulsion systems (EPS) ,which provides significant reduction of necessary energy resources and simultaneously increase the payload mass delivered to the GEO [2, 3]. To provide necessary navigation problem solution accuracy without on ground infrastructure support, it is expedient to use onboard multichannel Global Navigation Satellite Systems signals receiver (GNSSreceiver) as well as traditional electrooptical equipment. It seems obvious, that listed above onboard advanced navigation and control hardware set is most effective in terms of the autonomous oper ation within an appropriate integrated onboard navigation and control system of SC at the GEO. Let us discuss in more detail problems arising by the development of such a system. It is known [2, 3], that from the point of view technical implementation the most challenging stage of SC life cycle at the GEO is the final ascent or, in other words, the long interorbital flight with EPS thrust (from dozens of days to a year). Due to EPS characteristics (a small thrust value and long active intervals), corresponding requirements of generated navigation estimations of both SC mass center motion and attitude, as well as EPS thrust vector accuracy and authenticity, used later on to calculate the trajectory control are of partic ular importance. Let us emphasize, that traditional sources of navigation information and the correspond ing algorithms cannot be directly used to address this problem because of either their low accuracy level (inertial navigation systems and astro navigational systems) or inability to provide the required authentic ity of the navigation solution ( by using a single GNSSreceiver). As it is mentioned above, the only pos sible way out of this situation is different navigation data fusion in order to ensure the accuracy and authenticity of the navigation estimations. Thus, there is an independent problem to develop new algo 798
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rithm for onboard navigation measurement data fusion, that would adequately considers all peculiarities of SC in the GEO life cycle various stages as well as requirements of the trajectory control. Authors of this paper proved the principled opportunity of refining the EPS vector actual value with developed integrated system without precise inertial elements usage [1]. However, the existence of such opportunity in principle does not preclude all the difficult technical problems, that arise in the case of the autonomous devotion SC into the GEO. Thus, as already noted one of the major problems is to ensure the authenticity of the estimations, obtained using measurements generated by the GNSSreceiver. This problem was discussed in the literature [4, 5]. However, until now it has no universal solution, that would be acceptable for different applications in space technology. This paper presents the basic concept, which according to the authors makes it possible to ensure the accuracy of such information at the stage of the final ascent of the SC to the GEO. 1. TECHNICAL PROBLEM STATEMENT As it was said above, the object of research, discussing at this paper, is the SC equipped with EPS, which is moving along interorbital transfer trajectory (final ascent) from an intermediate orbit to the desired final GEO with the known required (program/reference) orbit parameters. We also assume, that trajectory con trol of the final ascent is based on the generation of socalled “hit” trajectories, which are set of admissible (implemented) SC trajectories, that ensure the transfer from of the initial motion conditions area to the desired one in the GEO. Its sizes and shape, as well as the position of its “center of gravity”, are deter mined by international standards [3, 6]. Under otherwise equal conditions, these trajectories are functions of current value of the required (calculated) of EPS thrust vector attitude in the orbital frame. Thus, there is a unique relationship between EPS thrust vector and SC attitudes, because the control is implemented by changing the SC attitude parameters, considering the limitation consisting in the fact, that EPS is inac tivate in the shadow of the Earth. Since SC final ascent is carried out by EPS engage cyclogram with the control of SC attitude in order to provide the desired direction of the thrust vector in this case the terminal accuracy of the final ascent of SC to the terminal area of the GEO is determined by the accuracy of the motion’s prediction on passive flight sections (i.e., with EPS turned off) and the estimation accuracy of SC controlled motion on active flight sections (with EPS turned on). The accuracy of these estimations depends also on the estimation accuracy of EPS thrust module nominal value and EPS thrust vector attitude deviations from the required ones. As it was mentioned above, the possible onboard hardware set, that provides an autonomous navigation problem solution, includes a multichannel multisystem GNSSreceiver with the complements in form of the Satellite Based Augmentation System (SBAS), a satellite system differential correction, the Wide Area Augmentation System (WAAS), a system of transmission of wide range differential corrections, the Euro pean Geostationary Navigation Overlay Service (EGNOS), the System for Differential Correction and Mon itoring (SDCM) GLONASS, optoelectronic sensors, advanced intersatellite laser navigation and com munication system, a gyro stabilized platform and single degree of freedom gyros set [7]. The following methods are fundamental from the viewpoint of the required quality of the navigation problem solution for improved control: (1) SC mass center positioning on active flight sections of the final ascent with SC attitude control. (2) SC mass center positioning on passive flight sections of the final ascent and SC station keeping on the required longitude. Despite the fact, that the above mentioned onboard navigational equipment can provide potentially a continuous navigation solution with the required accuracy on SC entire trajectory, some features of men tioned equipment operation do not make it possible to directly use estimations, generated by this equip ment in SC control circuit. This is due to the following reasons: (1) In certain SC flight section during the final ascent it is impossible to obtain the necessary (for the desired navigation accuracy) number authenticity GNSS measurements because of the specific navigation spacecraft (NSC) visibility conditions on the hit trajectory [1, 8]. Thus, it is impossible to solve the navi gation problem statically; i.e., there is no solution by complete sample [4] and hence there is a need for recurrent (dynamic) estimation algorithms. (2) Measurements generated by the GNSSreceiver are accompanied by a wide range various unpre dictable information errors, which can lead to no authenticity measurements and, consequently, to no authenticity estimations, that are further fed to the SC motion control circuit [5]. (3) The use of recurrent (dynamic) algorithms for estimations on longterm (about a day) intervals is associated with the potential loss of stability of such algorithms in terms of the mismatch between the JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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errors of estimations and their a posterior variances as a result of the accumulation of all sorts of errors [9]: methodological (linearization errors and violation of the Gaussian distribution hypothesis concerning disturbances) and instrumental (rounding) errors. Thus, it can be said, that this paper deals with the integrated onboard navigation system at the stage of the autonomous final ascent of the SC to the GEO development, including its architecture, hardware composition and data fusion algorithm, that ensure the needed accuracy and authenticity of the generated navigation solution at the SC’s final ascent to the GEO. 2. FORMALIZATION OF THE NAVIGATION PROBLEM IN ORDER TO FORMULATE THE CONCEPT Let us assume, that SC motion is described by the system of nonlinear differential equations of the fol lowing form:
x (t ) = f (t, x (t ) , u (t, x* (t )) , ξ (t )) ,
x (t 0 ) = x0,
(2.1)
where t is a free integration variable indicating the time in the inertial frame (IF), x(t) is the system n × 1 state vector, f (⋅) is the vector function of the righthandside of the system n × 1, u (⋅) is the control vector m × 1, m ≠ n , x* (t ) is the vector of estimations of components of the state vector x (t ), n × 1, ξ (t ) is the vector of uncontrollable factors (disturbances) r × 1, x (t 0 ) = x0 is the vector of initial conditions of motion of sys tem (2.1), and n × 1. This vector can be interpreted in this problem as a stochastic one with the known a prior density of the probability P0 ( x ) or as an indefinite one with the known area of its components: x0 ∈ ∏ (t 0 ). Similarly, the vector of uncontrollable factors (disturbances) in (2.1) can be treated as deter ministic, stochastic, or indefinite, since as is known [4] all three components are in the description of the real motion because of various physical reasons. Thus, as a rule, deterministic factors are described by constants (but unknown) or slowly varying time functions or components of the SC state vector (for exam ple, the gravitational potential model inaccuracy and propulsion system nozzles misalignment). Stochas tic factors are described by stochastic processes (for example, the distribution of the atmosphere density depending on the geomagnetic and solar activity). Unidentified or incompletely specified factors are set by their admissible areas when only the lower and upper boundaries of the changes of such factors are known (for example, the EPS thrust vector module). Onboard navigation hardware make it possible to generate the vector of navigation measurements y (t ) l × 1, l ≠ n associated with the state vector of (2.1) by the ratio
y (t ) = h (t, x (t ) , η (t )),
(2.2)
where h (⋅) is the nonlinear vector function l × 1, and η(t ) is the vector of measurement errors l × 1. As in the case of model (2.1), the measurement error vector η(t ) in (2.2) can be treated as deterministic, stochastic or indefinite. Deterministic errors can be found by using any measuring instrument. At the same time, within a finite time interval of the measurement process such errors can be conventionally assumed to be constant. In this case, there is a practice of the inclusion of “certified” error values in the vector of the estimated parameters. It makes it possible to compensate deterministic errors in a ratio of 2.2 by the results of their estimation. Stochastic errors should include the pseudorange measurement errors caused by the deviation of NSC time scales with respect to the corresponding GNSS system, as well as GNSS measurement errors caused by the inaccurate NSC prediction of ephemerides. As a rule, stochastic errors are simulated as stationary stochastic processes generated using a particular spectral density func tion. Intervalunspecified errors are usually related to errors caused by, for example, the limited grid bit of analogtodigital converters or onboard calculator after recording of processed measurement values obtained for specific hardware. We will further understand as navigation problem solution algorithm the method for determining opti mal in certain sense estimations x* (t ) of the state vector x(t) based on measurement vector processing y(t)
x* (t ) = g ( y ( x, η, t )),
(2.3)
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From the definition of the hit trajectory it follows, that at some time T, hereinafter called terminal time, the SC control system should make it possible for the SC to reach the desired terminal area in SC state space Q ( x (T )) ∈ Ω CS ( R f ,T ),
(2.4)
where Q (⋅) is the vector function p × 1, p ≤ n. The features of the vector functions Q are defined by inter national requirements to SC final ascent to the GEO, taking into account existing technical limitations, R f is the vector of parameters of SC desired end state (required parameters of the orbit) at the terminal time T, and Ω CS ( R f ,T ) is the range of permissible accumulated control and navigation errors at the ter minal time point determined, depending on the method of interpretation of uncontrollable factors. In particular, for the generation of parameters of this area, as well as the area ∏ (t 0 ) , comprehensive proba bilistic and guaranteeing approaches previously developed by the authors can be used [4, 10]. Each of hit trajectories is characterized by its “quality” or, in other words, the standard terminal error relative to the “center of gravity” of the terminal area. This standard depends on the model of uncontrol lable factors and navigation measurement errors; i.e., in the case of the stochastic model, this standard is defined by the required probability of the vector x(T) to fall within the terminal area (2.4). In the case of the minimax model, i.e., under undefined uncontrollable factors and measurement errors, this standard should ensure that the vector x(T) falls within area (2.4) under the worst effects of these factors. Let’s denote by x f (T ) SC state vectors that ensure the fulfillment of condition (2.4). Let us assume, that hit trajectories are generated based on the necessary optimality conditions and define the SC program motion according to some integroterminal criterion [3]. The typical form of such trajectories projected on the axis of the absolute coordinate system is shown in Fig. 1. These trajectories are periodically recalculated on board SC with the rate determined, in particular, by the time of the nav igation data update. The use of this control method is caused by the dynamics of the motion of SC with EPS and the inabil ity to compensate the total accumulated control error at the terminal time. Regardless of the method of interpretation of uncontrollable factors, it means that (2.4) should be transformed into the following sys tem of relations:
Q(x k* (t k )) ∈ ΩCS ( Rk , t k ) ,
t k ∈ [t 0,T ] ,
(2.5)
where t k is the time of navigation definitions, x k* (t k ) is the vector of navigation estimations at time t k , Ω CS ( Rk , t k ) is the range of permissible (in terms of the fulfillment of (2.4)) total errors of control and nav igation at time tk determined depending on the method of the interpretation of uncontrollable factors, and Rk is the vector of parameters of the required SC condition at time tk. Proceeding from the above, in terms of conditions (2.5), the control vector at the time of the current navigation definitions t k has the form u(t, x k* (t k )), where t ∈ [t k , t k +1 ) ) is the time between instances of nav igation definitions. As already mentioned, information on the actual state of system (2.1) is formed by the navigation sys tem. In particular, this information includes the data on the components of the translational R (t ) ,V (t ) , p (t ) and angular Θ (t ) motions in the form of estimations of the corresponding vectors,
R* (t ) = R (t ) + ΔR* (t ) , V * ( t ) = V ( t ) + ΔV * ( t ) , p* (t ) = p (t ) + Δp* (t ) , Θ* (t ) = Θ (t ) + ΔΘ* (t ) ,
(2.6)
where R (t ) ,V (t ) , p (t ) , and Θ (t ) are the true values of the components of position, speed, EPS thrust vector, and SC attitude parameters, respectively, R* (t ) ,V * (t ) , p* (t ) , and Θ* (t ) are estimations of com ponents of position, speed, EPS thrust vector and SC attitude parameters, respectively, and ΔR* (t ) , ΔV * (t ) , Δp* (t ) , ΔΘ* (t ) are the corresponding estimation errors. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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Y coordinate in the IF, m
40000000 20000000 0 –20000000 –40000000 –60000000 –80000000
–40000000 0 X coordinate in the IF, m
40000000
–40000000 0 X coordinate in the IF, m
40000000
Z coordinate in the IF, m
20000000
10000000
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–10000000
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Fig. 1. The type of required trajectories for the final ascent to the GSO.
Given method used for the generation of hit trajectories, provides the fulfillment of condition (2.5) with respect to the estimation errors means ΔR* (t ) ∈ Ω ΔNSR (t ) ,
ΔV * (t ) ∈ Ω ΔNSV (t ),
(2.7)
Δp* (t ) ∈ Ω ΔNSp (t ) ,
ΔΘ* (t ) ∈ Ω ΔΘ NS ( t ) ,
(2.8)
where Ω ΔNSR (t ) , Ω ΔNSV (t ) , Ω ΔNSp (t ) ,, and Ω ΔΘ NS ( t ) are areas of admissible errors of the navigation system in the definition of parameters of translational and angular movements defined as mentioned above, depending on the interpretation method of uncontrollable factors based on the probabilistic or guaranteed approaches. As already noted, the onboard navigation system includes a power gyro stabilized platform, optoelec tronic sensors, and multichannel multisystem GNSSreceiver with SBAS complements (WAAS, Egnos, and SDCM), as well as advanced instruments of intersatellite laser navigation and communication sys tem. The GNSSreceiver is the basic equipment for the determination of translational motion parameters. As a rule, it provides required accuracy of navigation determination in the normal mode of operation, i.e., JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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in the mode corresponding to the requirements of technical specifications for the GNSS. As is well known [5], in the receiver operation on the aircraft of this class, different information errors can appear caused by both the conditions of the NSC visibility and the effect of uncontrollable factors of a different nature (including indefinite and fuzzy factors). This situation can be formalized by the following relationship:
ΔyGNSS = yGNSS (t ) − h (t, x (t )) ≥ ε GNSS ,
(2.9)
where yGNSS (t) is vector of measurements implemented by GNSSreceiver, h(t, x(t)) is the “true” value of vector measurements, i.e. without taking into account measurement errors, and ε GNSS is the maximum permissible error in measuring navigation parameters of the GNSS receiver in accordance with the tech nical specifications for the GNSS [11]. Note, that above mentioned information errors, that cause (2.9), can occur at any arbitrary point in time and continue for indefinitely. In particular, the above said determines the need for development of a optimal strategy for GNSS receiver measurement sessions scheduling together with measurements implemented by the advanced hardware of the intersatellite laser navigation and communication system. Here, the strategy refers to the schedule of corresponding measurements generated directly on board SC in accordance with the approach developed in [10,12]. The technical limitations imposed by the visibility conditions of NSC and SC, used to organize intersatellite measurements, should be taken into account in the generation of this schedule. An independent problem is the introduction of navigation estimations in the control loop, which requires the formation of authenticity estimations, that ensure the generation of areas (2.7) and (2.8), taking into account information errors. In order to solve this problem, the interval and spatial method is planned to be used for generating redundancy and processing measurements [5], the nature of which will be described below. As mentioned above, in terms of features of navigation and control, the final ascent includes typical repetitive sections of two types: passive sections, when SC trajectory is in the shadow of the Earth with EPS turned off, and active sections with continuous SC attitude control (EPS thrust vector direction) for the generation of the required trajectories. Based on this scenario, the onboard navigation system should solve the following problems. 1. On passive sections the onboard navigation system should generate authenticity estimations of vec tors R* (t ) ,V * (t ) ,, and Θ* (t ) that provide the fulfillment of conditions (2.7) and (2.8) at the time of emer sion from the shadow of the Earth, i.e., the beginning of the controlled motion with the firing of EPS. In particular, this means that the entire uncontrolled motion section can be used for the generation of the optimal measurement strategy in order to ensure their interval and spatial redundancy [5]. 2. On active sections the onboard navigation system should provide the generation of smoothed authenticity optimal Bayesian estimations of vectors R* (t ) ,V * (t ) ,, and Θ* (t ) , as well as the estimation of the EPS thrust vector Δp* (t ). In this case, there is an additional problem of determining the frequency of the navigation data updates and the formation of an optimal algorithm for the data fusion of GNSS receiver with the measurements using the advanced intersatellite laser equipment of the navigation and communication system. 3. BASIC FEATURES OF THE ONBOARD NAVIGATION SYSTEM FOR THE FINAL ASCENT OF AN SC TO THE GEO Let us now formulate the concept of the onboard navigation system for an SC’s final ascent to the GEO. We take the navigation system to refer to the hardware composition of the measurement tools and architecture of the navigation system, including the composition and features of algorithms for the navi gational information processing. Taking into account the formalization of the problem formulation of the navigation definitions in the final ascent to the GEO using EPS and main features of this stage, it is pos sible to propose the architecture of the onboard navigation system illustrated by its simplified diagram in Fig. 2. The architecture of the onboard navigation system consists of three main units between which the information is exchanged: (1) SC mass center position and velocity estimation unit. (2) SC attitude estimation unit. (3) EPS thrust vector estimation unit. The first two units are reconfigured in accordance with the strategy of the navigation problem solution defined for different sections of the trajectory. First of all, let us discuss SC attitude estimation unit. As it JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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Gyro stabilized platform
Optoelectronic sensors
Intersatellite communication equipment
Single degree of freedom gyros set
Measurement control strategy
Measurement vector Integrated navigation system architecture SC attitude estimation
Optimal measurement session scheduling
SC mass center positions and velocity estimation unit Active section Recursive data fusion algorithm (recursive ) algorithm Additional smoothing of estimations
SC attitude estimation unit
Passive section Smoothing and compressing of measurements Interval estimation
Active section Thrust vector estimation
Mass center position and velocity estimation Thrust vector estimation Program thrust vector
Control mode
Passive section Stabilization mode
SC attitude estimation
Thrust vector estimation unit SC attitude estimation Mass center position and velocity estimation
True SC attitude
Thrust vector estimation
Euler’s angles program values
Generation of updated required trajectories
SC attitude control system
Data exchange Satellite motion control
Fig. 2. The simplified diagram of an integrated onboard navigation system of the SC in the final ascent to the GSO.
is shown on diagram, the problem of SC attitude estimation was isolated in a separate unit because of high dynamics of angular motion and high rate of the measurement information output of the respective sen sors. Here, the reconfiguration is related to various modes of the SC attitude control system on the spec ified trajectory sections: the control mode on the active section and the stabilization mode on the passive section. In the first case, the processing algorithm is a tightlybound data fusion diagram [9] from single degree of freedom gyros set and optoelectronic sensors using the data of the clocking angles of the laser marks of the intersatellite laser navigation and communication system (at possible times of the visibility of navigated SC) based on the scalar modification of the quasilinear generalized Kalman filter (GKF) [9]. The loosely bound diagram of data fusion based on the GKF [4] operates in the stabilization mode. It esti JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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mations are basic solution errors. Such a solution is generated according to single degree of freedom gyros set by the processing SC attitude from backup system by the information obtained from the optoelectronic sensors. Note, that this unit transmits the received estimations not only in attitude control system but also in the unit of the translational motion estimation unit and the estimation unit of the EPS thrust vector. SC mass center position and velocity estimation unit is reconfigured in accordance with the same prin ciple as SC attitude estimation unit. This is due to the different tasks of the navigation system on the active and passive sections of SC trajectory. Thus, on the active trajectory section the onboard navigation system should provide the continuous generation of the smoothed authenticity optimal Bayesian estimations of vectors R* (t ) ,V * (t ) ,, and Θ* (t ) and the estimation of the thrust vector Δp* (t ). This problem can be solved by the integration of the heter ogeneous measuring information onboard the SC based on the GKF with the nonlinear prediction model that uses the most detailed model of the SC’s controlled motion, taking into account uncontrollable fac tors [7]. A certain compromise between the complexity of the model and its accuracy seems the preferred way of building such a model of the SC’s motion, in the sense, that it is necessary to minimize computa tional resources but satisfy accuracy requirements. Studies conducted by the authors [1, 8] show, that in this situation for smoothing and generating con tinuous estimations it is necessary to use an additional dynamic estimation algorithm (the GKF or the recursive least squares method). To do this, the components corresponding to the systematic measure ment errors are excluded from the original expanded phase vector and system (2.2). For the newly created dynamical system a new smoothing processing algorithm is developed with the following features: (1) A nonlinear SC motion prediction based on the generated dynamic system and taking into account the values of systematic errors of uncontrollable factors in (2.1) acts as prior estimations. (2) The difference between the predicted system state vector and posterior estimations previously obtained using the scalar GKF modification [7, 9] acts as measurements during the estimation. The main processing algorithm here is the quasilinear GKF [7] resulting in the generation of the smoothed estimate of SC state vector and estimate of the actual module and orientation of EPS thrust. In this case, the reference trajectory is constructed on every step of the filter and is adjusted, taking into account estimations of components of the thrust vector obtained in the previous step. As noted above, the purpose of the navigation for the passive section is the generation of the accurate prediction of the components of SC translational motion in the shadow of the Earth and the subsequent calculation of the most authenticity and optimal estimation, in terms of the accuracy of the component, of SC translational motion at the time of the end of the passive sections, i.e., the start of EPS. It means, in particular, that all the received measurements obtained within this interval can be used for interval and spatial redundancy by the generation of the corresponding measurement strategy [5], which, on the one hand, excludes GNSS measurements with information errors (2.9) and generates authenticity measure ments and, on the other hand, reduces the methodological and instrumental errors of the measurement processing methods. This method is based on the following assumptions: (1) The rejection of measurements carried out based on the comparison of the obtained extra trajectory measurements of GNSSreceiver with the predicted ones, i.e., generated based on accurate prediction of SC motion on the passive section, by condition (2.9). (2) The compression and smoothing of the measurements, i.e., the generation of measurement ses sions within which the measured values change little with the replacement of the session measurements with the only equivalent smoothed measurement put in the normal place [4, 5]. (3) The pulse refinement of the predicted estimation at the end of the passive section carried out with a special representation of the Kalman filter [5]. The methods of the dynamic scheduling of the measuring experiment are an additional reserve for enhancing the accuracy of this procedure [10]. A feature of dynamic scheduling algorithms [10] is that they make it possible to prior generate the optimal strategy for the measurement sessions under paragraph 1 based on prior analysis of the accuracy in both the stochastic and minimax statements [4, 10] using the terminal and interval criteria. The use of dynamic optimal scheduling makes it possible to preliminarily calculate the optimal strategy [10] with the subsequent correction, as posterior information is generated by the main estimation algo rithm for intermediate intervals. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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CONCLUSIONS SC in the GEO autonomous onboard integrated navigation system conceptual design was presented, using both stochastic and minimax options uncontrollable factors. Developed by the authors approach, makes possible generation of uncontrollable factors admissible sets basing on comprehensive probabilistic or guaranteeing approaches to the estimation problem solution. The composition of advanced hardware set ensuring SC in the GEO final ascent process autonomous implementation in accordance with international requirements for this type of SC was proposed. The hardware set includes the following equipment: (1) The multichannel multisystem GNSSreceiver ncluding complements in form of differential cor rections. (2) Optoelectronic sensors (sensors of the Earth, stars and the Sun). (3) Intersatellite communication equipment. SC in the GEO onboard integrated navigation system architecture was determined during its final ascent based on the developed concept and considering advanced hardware set. The system architecture consists of three main reconfigurable units (module). The first one is designed to generate SC mass center state vector components estimations. The second module estimates SC attitude. Finally, the third unit estimates EPS thrust vector. These modules are reconfigured in accordance with described below naviga tion measurements strategy. The strategy for the navigation problem solutions was determined using the above hardware equip ment. This strategy includes SC active (if EPS is turned on) and passive (if EPS is turned off) final ascent trajectory sections. In accordance with this sections, it is proposed various types of algorithms for SC nav igation problem solution: (1) On passive sections, interval algorithms are used that ensure the accuracy of estimations, taking into account possible unknown information errors of GNSSreceiver measurements. (2) On active sections, an additional smoothing algorithm, based on the Bayesian dynamic refiltering procedure was proposed to be used, considering the preliminarily obtained estimations of navigation mea surements systematic errors and nonlinear procedure of SC state vector prediction. Previously developed by the authors optimal navigational measurements scheduling algorithm that ensure the generation of the optimal schedules directly onboard SC proposed to be used as an additional mean of improving the accuracy of the navigation estimations on active and passive tra jectory sections. ACKNOWLEDGMENTS This work was supported by the Ministry of Education and Science of the Russian Federation, project no. RFMEFI57414X0100. REFERENCES 1. D. A. Kozorez, M. N. Krasilshchikov, D. M. Kruzhkov, and K. I. Sypalo, “A solution of the navigation problem for autonomous insertion of payload into a geostationary orbit using a lowthrust engine,” J. Comput. Syst. Sci. Int. 54, 104–115 (2015). 2. M. F. Reshetnev, A. A. Lebedev, V. A. Bartenev, M. N. Krasilshchikov, V. A. Malyshev, and V. V. Malyshev, The Control and Navigation of Satellites of the Earth in Nearcircle Orbits (Mashinostroenie, Moscow, 1988) [in Rus sian]. 3. V. G. Petukhov, “Quasioptimal control with feedback for multiorbit lowthrust transfer between noncoplanar elliptic and circular orbits,” Cosm. Res. 49, 121–130 (2011). 4. V. V. Malyshev, V. T. Bobronnikov, D. A. Kozorez, M. N. Krasil’shchikov, K. I. Sypalo, and A. V. Fedorov, Sta tistical Dynamics and Optimization of Flying Vehicles (Al’yans, Moscow, 2013) [in Russian]. 5. E. L. Mezhiritskii and V. D. Dishel’, “Intervaldimensional approach to Kalman filtering as methodology of integration of terminal guidance circuit and astroinertial satellite navigation,” Tr. FGUP NPTsAP, No. 1 (2011). 6. V. V. Malyshev, A. V. Starkov, and A. V. Fedorov, “Synthesis of optimal control in the solution of problem of spacecraft retaining in the orbital grouping,” Kosmonavt. Raketostroen., No. 4 (69), 150–158 (2012). 7. V. A. Bartenev, A. K. Grechkoseev, D. A. Kozorez, M. N. Krasil’shchikov, V. V. Pasynkov, and K. I. Sypalo, Mod ern and Perspective Informational GNSSTechnologies in High Precision Navigation Problems (Fizmatlit, Moscow, 2014) [in Russian]. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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8. D. A. Kozorez, M. N. Krasil’shchikov, D. M. Kruzhkov, and K. I. Sypalo, “Integrated navigation system for a space vehicle on a geostationary or highly elliptic orbit operating in the presence of active jam,” J. Comput. Syst. Sci. Int. 52, 468–479 (2013). 9. Modern Information Technologies in Problems of Navigation and Aiming of Unmanned Aerial Vehicles, Ed. by M. N. Krasil’shchikov and G. G. Sebryakov (Fizmatlit, Moscow, 2009) [in Russian]. 10. V. V. Malyshev, M. N. Krasil’shchikov, and V. I. Karlov, Optimization of Observations and Control of Aircraft (Mashinostroenie, Moscow, 1989) [in Russian]. 11. Global Satellite Navigation System GLONASS. The Interface Control Document, Vers. 5.1 (RNII KP, Moscow, 2008) [in Russian]. 12. M. N. Krasilshikov, V. V. Malyshev, and V. I. Karlov, Optimization of Observation and Control Processes (AIAA, Washington D.C., 1992).
Translated by O. Pismenov
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Autonomous Navigation during the Final Ascent of a Spacecraft into the Geostationary Orbit. Autonomous Integrated Navigation System Concept D. A. Kozoreza, M. N. Krasilshchikova, D. M. Kruzhkova, and K. I. Sypalob a
b
Moscow Aviation Institute, Moscow, Russia Central Aerohydrodynamic Institute, Moscow, Russia email: [email protected] Received March 19, 2015; in final form, April 28, 2015
Abstract—This paper is the first in a series of works devoted to the problems of a spacecraft in the geo stationary orbit autonomous operation. The paper presents the conceptual design of the spacecraft onboard autonomous navigation system applying to its final ascent to the geostationary orbit using the lowthrust electric propulsion system. This design is based on both stochastic and minimax uncontrol lable factors interpretation. The composition of needed hardware and the architecture of the space craft onboard integrated navigation system, which should be configured in accordance with standard sections of the final ascent trajectory, are determined. It is proposed to involve socalled interval esti mation algorithms in order to ensure the reliability of obtained estimations, taking into account unknown nature of possible navigation measurements information violations. An optional smoothing algorithm based on dynamic Bayesian measurements filtering, considering preliminarily obtained estimations of systematic errors. As an additional means of improving the accuracy of the navigation estimations it is proposed also to involve navigation measurements optimal scheduling methods and algorithms in order to generate optimal schedules of measurements directly on board the spacecraft. DOI: 10.1134/S106423071505007X
INTRODUCTION Nowadays, the most actual problem of the existing and advanced space systems improvement is to ensure their autonomous operation. In particular, with respect to the spacecraft (SC) on the geostationary orbit (GEO), this requirement is expressed in the necessity to autonomously SC devotion in the final orbit, transfer it to the required longitude, and keep it at the geostationary orbit point in accordance with inter national requirements [1]. The implementation of this idea requires advanced hardware set utilization for SC Navigation and Control systems implementation on all the above mentioned stages of its life cycle at the GEO. First of all, it is necessary to use electric propulsion systems (EPS) ,which provides significant reduction of necessary energy resources and simultaneously increase the payload mass delivered to the GEO [2, 3]. To provide necessary navigation problem solution accuracy without on ground infrastructure support, it is expedient to use onboard multichannel Global Navigation Satellite Systems signals receiver (GNSSreceiver) as well as traditional electrooptical equipment. It seems obvious, that listed above onboard advanced navigation and control hardware set is most effective in terms of the autonomous oper ation within an appropriate integrated onboard navigation and control system of SC at the GEO. Let us discuss in more detail problems arising by the development of such a system. It is known [2, 3], that from the point of view technical implementation the most challenging stage of SC life cycle at the GEO is the final ascent or, in other words, the long interorbital flight with EPS thrust (from dozens of days to a year). Due to EPS characteristics (a small thrust value and long active intervals), corresponding requirements of generated navigation estimations of both SC mass center motion and attitude, as well as EPS thrust vector accuracy and authenticity, used later on to calculate the trajectory control are of partic ular importance. Let us emphasize, that traditional sources of navigation information and the correspond ing algorithms cannot be directly used to address this problem because of either their low accuracy level (inertial navigation systems and astro navigational systems) or inability to provide the required authentic ity of the navigation solution ( by using a single GNSSreceiver). As it is mentioned above, the only pos sible way out of this situation is different navigation data fusion in order to ensure the accuracy and authenticity of the navigation estimations. Thus, there is an independent problem to develop new algo 798
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rithm for onboard navigation measurement data fusion, that would adequately considers all peculiarities of SC in the GEO life cycle various stages as well as requirements of the trajectory control. Authors of this paper proved the principled opportunity of refining the EPS vector actual value with developed integrated system without precise inertial elements usage [1]. However, the existence of such opportunity in principle does not preclude all the difficult technical problems, that arise in the case of the autonomous devotion SC into the GEO. Thus, as already noted one of the major problems is to ensure the authenticity of the estimations, obtained using measurements generated by the GNSSreceiver. This problem was discussed in the literature [4, 5]. However, until now it has no universal solution, that would be acceptable for different applications in space technology. This paper presents the basic concept, which according to the authors makes it possible to ensure the accuracy of such information at the stage of the final ascent of the SC to the GEO. 1. TECHNICAL PROBLEM STATEMENT As it was said above, the object of research, discussing at this paper, is the SC equipped with EPS, which is moving along interorbital transfer trajectory (final ascent) from an intermediate orbit to the desired final GEO with the known required (program/reference) orbit parameters. We also assume, that trajectory con trol of the final ascent is based on the generation of socalled “hit” trajectories, which are set of admissible (implemented) SC trajectories, that ensure the transfer from of the initial motion conditions area to the desired one in the GEO. Its sizes and shape, as well as the position of its “center of gravity”, are deter mined by international standards [3, 6]. Under otherwise equal conditions, these trajectories are functions of current value of the required (calculated) of EPS thrust vector attitude in the orbital frame. Thus, there is a unique relationship between EPS thrust vector and SC attitudes, because the control is implemented by changing the SC attitude parameters, considering the limitation consisting in the fact, that EPS is inac tivate in the shadow of the Earth. Since SC final ascent is carried out by EPS engage cyclogram with the control of SC attitude in order to provide the desired direction of the thrust vector in this case the terminal accuracy of the final ascent of SC to the terminal area of the GEO is determined by the accuracy of the motion’s prediction on passive flight sections (i.e., with EPS turned off) and the estimation accuracy of SC controlled motion on active flight sections (with EPS turned on). The accuracy of these estimations depends also on the estimation accuracy of EPS thrust module nominal value and EPS thrust vector attitude deviations from the required ones. As it was mentioned above, the possible onboard hardware set, that provides an autonomous navigation problem solution, includes a multichannel multisystem GNSSreceiver with the complements in form of the Satellite Based Augmentation System (SBAS), a satellite system differential correction, the Wide Area Augmentation System (WAAS), a system of transmission of wide range differential corrections, the Euro pean Geostationary Navigation Overlay Service (EGNOS), the System for Differential Correction and Mon itoring (SDCM) GLONASS, optoelectronic sensors, advanced intersatellite laser navigation and com munication system, a gyro stabilized platform and single degree of freedom gyros set [7]. The following methods are fundamental from the viewpoint of the required quality of the navigation problem solution for improved control: (1) SC mass center positioning on active flight sections of the final ascent with SC attitude control. (2) SC mass center positioning on passive flight sections of the final ascent and SC station keeping on the required longitude. Despite the fact, that the above mentioned onboard navigational equipment can provide potentially a continuous navigation solution with the required accuracy on SC entire trajectory, some features of men tioned equipment operation do not make it possible to directly use estimations, generated by this equip ment in SC control circuit. This is due to the following reasons: (1) In certain SC flight section during the final ascent it is impossible to obtain the necessary (for the desired navigation accuracy) number authenticity GNSS measurements because of the specific navigation spacecraft (NSC) visibility conditions on the hit trajectory [1, 8]. Thus, it is impossible to solve the navi gation problem statically; i.e., there is no solution by complete sample [4] and hence there is a need for recurrent (dynamic) estimation algorithms. (2) Measurements generated by the GNSSreceiver are accompanied by a wide range various unpre dictable information errors, which can lead to no authenticity measurements and, consequently, to no authenticity estimations, that are further fed to the SC motion control circuit [5]. (3) The use of recurrent (dynamic) algorithms for estimations on longterm (about a day) intervals is associated with the potential loss of stability of such algorithms in terms of the mismatch between the JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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errors of estimations and their a posterior variances as a result of the accumulation of all sorts of errors [9]: methodological (linearization errors and violation of the Gaussian distribution hypothesis concerning disturbances) and instrumental (rounding) errors. Thus, it can be said, that this paper deals with the integrated onboard navigation system at the stage of the autonomous final ascent of the SC to the GEO development, including its architecture, hardware composition and data fusion algorithm, that ensure the needed accuracy and authenticity of the generated navigation solution at the SC’s final ascent to the GEO. 2. FORMALIZATION OF THE NAVIGATION PROBLEM IN ORDER TO FORMULATE THE CONCEPT Let us assume, that SC motion is described by the system of nonlinear differential equations of the fol lowing form:
x (t ) = f (t, x (t ) , u (t, x* (t )) , ξ (t )) ,
x (t 0 ) = x0,
(2.1)
where t is a free integration variable indicating the time in the inertial frame (IF), x(t) is the system n × 1 state vector, f (⋅) is the vector function of the righthandside of the system n × 1, u (⋅) is the control vector m × 1, m ≠ n , x* (t ) is the vector of estimations of components of the state vector x (t ), n × 1, ξ (t ) is the vector of uncontrollable factors (disturbances) r × 1, x (t 0 ) = x0 is the vector of initial conditions of motion of sys tem (2.1), and n × 1. This vector can be interpreted in this problem as a stochastic one with the known a prior density of the probability P0 ( x ) or as an indefinite one with the known area of its components: x0 ∈ ∏ (t 0 ). Similarly, the vector of uncontrollable factors (disturbances) in (2.1) can be treated as deter ministic, stochastic, or indefinite, since as is known [4] all three components are in the description of the real motion because of various physical reasons. Thus, as a rule, deterministic factors are described by constants (but unknown) or slowly varying time functions or components of the SC state vector (for exam ple, the gravitational potential model inaccuracy and propulsion system nozzles misalignment). Stochas tic factors are described by stochastic processes (for example, the distribution of the atmosphere density depending on the geomagnetic and solar activity). Unidentified or incompletely specified factors are set by their admissible areas when only the lower and upper boundaries of the changes of such factors are known (for example, the EPS thrust vector module). Onboard navigation hardware make it possible to generate the vector of navigation measurements y (t ) l × 1, l ≠ n associated with the state vector of (2.1) by the ratio
y (t ) = h (t, x (t ) , η (t )),
(2.2)
where h (⋅) is the nonlinear vector function l × 1, and η(t ) is the vector of measurement errors l × 1. As in the case of model (2.1), the measurement error vector η(t ) in (2.2) can be treated as deterministic, stochastic or indefinite. Deterministic errors can be found by using any measuring instrument. At the same time, within a finite time interval of the measurement process such errors can be conventionally assumed to be constant. In this case, there is a practice of the inclusion of “certified” error values in the vector of the estimated parameters. It makes it possible to compensate deterministic errors in a ratio of 2.2 by the results of their estimation. Stochastic errors should include the pseudorange measurement errors caused by the deviation of NSC time scales with respect to the corresponding GNSS system, as well as GNSS measurement errors caused by the inaccurate NSC prediction of ephemerides. As a rule, stochastic errors are simulated as stationary stochastic processes generated using a particular spectral density func tion. Intervalunspecified errors are usually related to errors caused by, for example, the limited grid bit of analogtodigital converters or onboard calculator after recording of processed measurement values obtained for specific hardware. We will further understand as navigation problem solution algorithm the method for determining opti mal in certain sense estimations x* (t ) of the state vector x(t) based on measurement vector processing y(t)
x* (t ) = g ( y ( x, η, t )),
(2.3)
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From the definition of the hit trajectory it follows, that at some time T, hereinafter called terminal time, the SC control system should make it possible for the SC to reach the desired terminal area in SC state space Q ( x (T )) ∈ Ω CS ( R f ,T ),
(2.4)
where Q (⋅) is the vector function p × 1, p ≤ n. The features of the vector functions Q are defined by inter national requirements to SC final ascent to the GEO, taking into account existing technical limitations, R f is the vector of parameters of SC desired end state (required parameters of the orbit) at the terminal time T, and Ω CS ( R f ,T ) is the range of permissible accumulated control and navigation errors at the ter minal time point determined, depending on the method of interpretation of uncontrollable factors. In particular, for the generation of parameters of this area, as well as the area ∏ (t 0 ) , comprehensive proba bilistic and guaranteeing approaches previously developed by the authors can be used [4, 10]. Each of hit trajectories is characterized by its “quality” or, in other words, the standard terminal error relative to the “center of gravity” of the terminal area. This standard depends on the model of uncontrol lable factors and navigation measurement errors; i.e., in the case of the stochastic model, this standard is defined by the required probability of the vector x(T) to fall within the terminal area (2.4). In the case of the minimax model, i.e., under undefined uncontrollable factors and measurement errors, this standard should ensure that the vector x(T) falls within area (2.4) under the worst effects of these factors. Let’s denote by x f (T ) SC state vectors that ensure the fulfillment of condition (2.4). Let us assume, that hit trajectories are generated based on the necessary optimality conditions and define the SC program motion according to some integroterminal criterion [3]. The typical form of such trajectories projected on the axis of the absolute coordinate system is shown in Fig. 1. These trajectories are periodically recalculated on board SC with the rate determined, in particular, by the time of the nav igation data update. The use of this control method is caused by the dynamics of the motion of SC with EPS and the inabil ity to compensate the total accumulated control error at the terminal time. Regardless of the method of interpretation of uncontrollable factors, it means that (2.4) should be transformed into the following sys tem of relations:
Q(x k* (t k )) ∈ ΩCS ( Rk , t k ) ,
t k ∈ [t 0,T ] ,
(2.5)
where t k is the time of navigation definitions, x k* (t k ) is the vector of navigation estimations at time t k , Ω CS ( Rk , t k ) is the range of permissible (in terms of the fulfillment of (2.4)) total errors of control and nav igation at time tk determined depending on the method of the interpretation of uncontrollable factors, and Rk is the vector of parameters of the required SC condition at time tk. Proceeding from the above, in terms of conditions (2.5), the control vector at the time of the current navigation definitions t k has the form u(t, x k* (t k )), where t ∈ [t k , t k +1 ) ) is the time between instances of nav igation definitions. As already mentioned, information on the actual state of system (2.1) is formed by the navigation sys tem. In particular, this information includes the data on the components of the translational R (t ) ,V (t ) , p (t ) and angular Θ (t ) motions in the form of estimations of the corresponding vectors,
R* (t ) = R (t ) + ΔR* (t ) , V * ( t ) = V ( t ) + ΔV * ( t ) , p* (t ) = p (t ) + Δp* (t ) , Θ* (t ) = Θ (t ) + ΔΘ* (t ) ,
(2.6)
where R (t ) ,V (t ) , p (t ) , and Θ (t ) are the true values of the components of position, speed, EPS thrust vector, and SC attitude parameters, respectively, R* (t ) ,V * (t ) , p* (t ) , and Θ* (t ) are estimations of com ponents of position, speed, EPS thrust vector and SC attitude parameters, respectively, and ΔR* (t ) , ΔV * (t ) , Δp* (t ) , ΔΘ* (t ) are the corresponding estimation errors. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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Y coordinate in the IF, m
40000000 20000000 0 –20000000 –40000000 –60000000 –80000000
–40000000 0 X coordinate in the IF, m
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–40000000 0 X coordinate in the IF, m
40000000
Z coordinate in the IF, m
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Fig. 1. The type of required trajectories for the final ascent to the GSO.
Given method used for the generation of hit trajectories, provides the fulfillment of condition (2.5) with respect to the estimation errors means ΔR* (t ) ∈ Ω ΔNSR (t ) ,
ΔV * (t ) ∈ Ω ΔNSV (t ),
(2.7)
Δp* (t ) ∈ Ω ΔNSp (t ) ,
ΔΘ* (t ) ∈ Ω ΔΘ NS ( t ) ,
(2.8)
where Ω ΔNSR (t ) , Ω ΔNSV (t ) , Ω ΔNSp (t ) ,, and Ω ΔΘ NS ( t ) are areas of admissible errors of the navigation system in the definition of parameters of translational and angular movements defined as mentioned above, depending on the interpretation method of uncontrollable factors based on the probabilistic or guaranteed approaches. As already noted, the onboard navigation system includes a power gyro stabilized platform, optoelec tronic sensors, and multichannel multisystem GNSSreceiver with SBAS complements (WAAS, Egnos, and SDCM), as well as advanced instruments of intersatellite laser navigation and communication sys tem. The GNSSreceiver is the basic equipment for the determination of translational motion parameters. As a rule, it provides required accuracy of navigation determination in the normal mode of operation, i.e., JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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in the mode corresponding to the requirements of technical specifications for the GNSS. As is well known [5], in the receiver operation on the aircraft of this class, different information errors can appear caused by both the conditions of the NSC visibility and the effect of uncontrollable factors of a different nature (including indefinite and fuzzy factors). This situation can be formalized by the following relationship:
ΔyGNSS = yGNSS (t ) − h (t, x (t )) ≥ ε GNSS ,
(2.9)
where yGNSS (t) is vector of measurements implemented by GNSSreceiver, h(t, x(t)) is the “true” value of vector measurements, i.e. without taking into account measurement errors, and ε GNSS is the maximum permissible error in measuring navigation parameters of the GNSS receiver in accordance with the tech nical specifications for the GNSS [11]. Note, that above mentioned information errors, that cause (2.9), can occur at any arbitrary point in time and continue for indefinitely. In particular, the above said determines the need for development of a optimal strategy for GNSS receiver measurement sessions scheduling together with measurements implemented by the advanced hardware of the intersatellite laser navigation and communication system. Here, the strategy refers to the schedule of corresponding measurements generated directly on board SC in accordance with the approach developed in [10,12]. The technical limitations imposed by the visibility conditions of NSC and SC, used to organize intersatellite measurements, should be taken into account in the generation of this schedule. An independent problem is the introduction of navigation estimations in the control loop, which requires the formation of authenticity estimations, that ensure the generation of areas (2.7) and (2.8), taking into account information errors. In order to solve this problem, the interval and spatial method is planned to be used for generating redundancy and processing measurements [5], the nature of which will be described below. As mentioned above, in terms of features of navigation and control, the final ascent includes typical repetitive sections of two types: passive sections, when SC trajectory is in the shadow of the Earth with EPS turned off, and active sections with continuous SC attitude control (EPS thrust vector direction) for the generation of the required trajectories. Based on this scenario, the onboard navigation system should solve the following problems. 1. On passive sections the onboard navigation system should generate authenticity estimations of vec tors R* (t ) ,V * (t ) ,, and Θ* (t ) that provide the fulfillment of conditions (2.7) and (2.8) at the time of emer sion from the shadow of the Earth, i.e., the beginning of the controlled motion with the firing of EPS. In particular, this means that the entire uncontrolled motion section can be used for the generation of the optimal measurement strategy in order to ensure their interval and spatial redundancy [5]. 2. On active sections the onboard navigation system should provide the generation of smoothed authenticity optimal Bayesian estimations of vectors R* (t ) ,V * (t ) ,, and Θ* (t ) , as well as the estimation of the EPS thrust vector Δp* (t ). In this case, there is an additional problem of determining the frequency of the navigation data updates and the formation of an optimal algorithm for the data fusion of GNSS receiver with the measurements using the advanced intersatellite laser equipment of the navigation and communication system. 3. BASIC FEATURES OF THE ONBOARD NAVIGATION SYSTEM FOR THE FINAL ASCENT OF AN SC TO THE GEO Let us now formulate the concept of the onboard navigation system for an SC’s final ascent to the GEO. We take the navigation system to refer to the hardware composition of the measurement tools and architecture of the navigation system, including the composition and features of algorithms for the navi gational information processing. Taking into account the formalization of the problem formulation of the navigation definitions in the final ascent to the GEO using EPS and main features of this stage, it is pos sible to propose the architecture of the onboard navigation system illustrated by its simplified diagram in Fig. 2. The architecture of the onboard navigation system consists of three main units between which the information is exchanged: (1) SC mass center position and velocity estimation unit. (2) SC attitude estimation unit. (3) EPS thrust vector estimation unit. The first two units are reconfigured in accordance with the strategy of the navigation problem solution defined for different sections of the trajectory. First of all, let us discuss SC attitude estimation unit. As it JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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Gyro stabilized platform
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Measurement vector Integrated navigation system architecture SC attitude estimation
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SC mass center positions and velocity estimation unit Active section Recursive data fusion algorithm (recursive ) algorithm Additional smoothing of estimations
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Thrust vector estimation unit SC attitude estimation Mass center position and velocity estimation
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Fig. 2. The simplified diagram of an integrated onboard navigation system of the SC in the final ascent to the GSO.
is shown on diagram, the problem of SC attitude estimation was isolated in a separate unit because of high dynamics of angular motion and high rate of the measurement information output of the respective sen sors. Here, the reconfiguration is related to various modes of the SC attitude control system on the spec ified trajectory sections: the control mode on the active section and the stabilization mode on the passive section. In the first case, the processing algorithm is a tightlybound data fusion diagram [9] from single degree of freedom gyros set and optoelectronic sensors using the data of the clocking angles of the laser marks of the intersatellite laser navigation and communication system (at possible times of the visibility of navigated SC) based on the scalar modification of the quasilinear generalized Kalman filter (GKF) [9]. The loosely bound diagram of data fusion based on the GKF [4] operates in the stabilization mode. It esti JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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mations are basic solution errors. Such a solution is generated according to single degree of freedom gyros set by the processing SC attitude from backup system by the information obtained from the optoelectronic sensors. Note, that this unit transmits the received estimations not only in attitude control system but also in the unit of the translational motion estimation unit and the estimation unit of the EPS thrust vector. SC mass center position and velocity estimation unit is reconfigured in accordance with the same prin ciple as SC attitude estimation unit. This is due to the different tasks of the navigation system on the active and passive sections of SC trajectory. Thus, on the active trajectory section the onboard navigation system should provide the continuous generation of the smoothed authenticity optimal Bayesian estimations of vectors R* (t ) ,V * (t ) ,, and Θ* (t ) and the estimation of the thrust vector Δp* (t ). This problem can be solved by the integration of the heter ogeneous measuring information onboard the SC based on the GKF with the nonlinear prediction model that uses the most detailed model of the SC’s controlled motion, taking into account uncontrollable fac tors [7]. A certain compromise between the complexity of the model and its accuracy seems the preferred way of building such a model of the SC’s motion, in the sense, that it is necessary to minimize computa tional resources but satisfy accuracy requirements. Studies conducted by the authors [1, 8] show, that in this situation for smoothing and generating con tinuous estimations it is necessary to use an additional dynamic estimation algorithm (the GKF or the recursive least squares method). To do this, the components corresponding to the systematic measure ment errors are excluded from the original expanded phase vector and system (2.2). For the newly created dynamical system a new smoothing processing algorithm is developed with the following features: (1) A nonlinear SC motion prediction based on the generated dynamic system and taking into account the values of systematic errors of uncontrollable factors in (2.1) acts as prior estimations. (2) The difference between the predicted system state vector and posterior estimations previously obtained using the scalar GKF modification [7, 9] acts as measurements during the estimation. The main processing algorithm here is the quasilinear GKF [7] resulting in the generation of the smoothed estimate of SC state vector and estimate of the actual module and orientation of EPS thrust. In this case, the reference trajectory is constructed on every step of the filter and is adjusted, taking into account estimations of components of the thrust vector obtained in the previous step. As noted above, the purpose of the navigation for the passive section is the generation of the accurate prediction of the components of SC translational motion in the shadow of the Earth and the subsequent calculation of the most authenticity and optimal estimation, in terms of the accuracy of the component, of SC translational motion at the time of the end of the passive sections, i.e., the start of EPS. It means, in particular, that all the received measurements obtained within this interval can be used for interval and spatial redundancy by the generation of the corresponding measurement strategy [5], which, on the one hand, excludes GNSS measurements with information errors (2.9) and generates authenticity measure ments and, on the other hand, reduces the methodological and instrumental errors of the measurement processing methods. This method is based on the following assumptions: (1) The rejection of measurements carried out based on the comparison of the obtained extra trajectory measurements of GNSSreceiver with the predicted ones, i.e., generated based on accurate prediction of SC motion on the passive section, by condition (2.9). (2) The compression and smoothing of the measurements, i.e., the generation of measurement ses sions within which the measured values change little with the replacement of the session measurements with the only equivalent smoothed measurement put in the normal place [4, 5]. (3) The pulse refinement of the predicted estimation at the end of the passive section carried out with a special representation of the Kalman filter [5]. The methods of the dynamic scheduling of the measuring experiment are an additional reserve for enhancing the accuracy of this procedure [10]. A feature of dynamic scheduling algorithms [10] is that they make it possible to prior generate the optimal strategy for the measurement sessions under paragraph 1 based on prior analysis of the accuracy in both the stochastic and minimax statements [4, 10] using the terminal and interval criteria. The use of dynamic optimal scheduling makes it possible to preliminarily calculate the optimal strategy [10] with the subsequent correction, as posterior information is generated by the main estimation algo rithm for intermediate intervals. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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CONCLUSIONS SC in the GEO autonomous onboard integrated navigation system conceptual design was presented, using both stochastic and minimax options uncontrollable factors. Developed by the authors approach, makes possible generation of uncontrollable factors admissible sets basing on comprehensive probabilistic or guaranteeing approaches to the estimation problem solution. The composition of advanced hardware set ensuring SC in the GEO final ascent process autonomous implementation in accordance with international requirements for this type of SC was proposed. The hardware set includes the following equipment: (1) The multichannel multisystem GNSSreceiver ncluding complements in form of differential cor rections. (2) Optoelectronic sensors (sensors of the Earth, stars and the Sun). (3) Intersatellite communication equipment. SC in the GEO onboard integrated navigation system architecture was determined during its final ascent based on the developed concept and considering advanced hardware set. The system architecture consists of three main reconfigurable units (module). The first one is designed to generate SC mass center state vector components estimations. The second module estimates SC attitude. Finally, the third unit estimates EPS thrust vector. These modules are reconfigured in accordance with described below naviga tion measurements strategy. The strategy for the navigation problem solutions was determined using the above hardware equip ment. This strategy includes SC active (if EPS is turned on) and passive (if EPS is turned off) final ascent trajectory sections. In accordance with this sections, it is proposed various types of algorithms for SC nav igation problem solution: (1) On passive sections, interval algorithms are used that ensure the accuracy of estimations, taking into account possible unknown information errors of GNSSreceiver measurements. (2) On active sections, an additional smoothing algorithm, based on the Bayesian dynamic refiltering procedure was proposed to be used, considering the preliminarily obtained estimations of navigation mea surements systematic errors and nonlinear procedure of SC state vector prediction. Previously developed by the authors optimal navigational measurements scheduling algorithm that ensure the generation of the optimal schedules directly onboard SC proposed to be used as an additional mean of improving the accuracy of the navigation estimations on active and passive tra jectory sections. ACKNOWLEDGMENTS This work was supported by the Ministry of Education and Science of the Russian Federation, project no. RFMEFI57414X0100. REFERENCES 1. D. A. Kozorez, M. N. Krasilshchikov, D. M. Kruzhkov, and K. I. Sypalo, “A solution of the navigation problem for autonomous insertion of payload into a geostationary orbit using a lowthrust engine,” J. Comput. Syst. Sci. Int. 54, 104–115 (2015). 2. M. F. Reshetnev, A. A. Lebedev, V. A. Bartenev, M. N. Krasilshchikov, V. A. Malyshev, and V. V. Malyshev, The Control and Navigation of Satellites of the Earth in Nearcircle Orbits (Mashinostroenie, Moscow, 1988) [in Rus sian]. 3. V. G. Petukhov, “Quasioptimal control with feedback for multiorbit lowthrust transfer between noncoplanar elliptic and circular orbits,” Cosm. Res. 49, 121–130 (2011). 4. V. V. Malyshev, V. T. Bobronnikov, D. A. Kozorez, M. N. Krasil’shchikov, K. I. Sypalo, and A. V. Fedorov, Sta tistical Dynamics and Optimization of Flying Vehicles (Al’yans, Moscow, 2013) [in Russian]. 5. E. L. Mezhiritskii and V. D. Dishel’, “Intervaldimensional approach to Kalman filtering as methodology of integration of terminal guidance circuit and astroinertial satellite navigation,” Tr. FGUP NPTsAP, No. 1 (2011). 6. V. V. Malyshev, A. V. Starkov, and A. V. Fedorov, “Synthesis of optimal control in the solution of problem of spacecraft retaining in the orbital grouping,” Kosmonavt. Raketostroen., No. 4 (69), 150–158 (2012). 7. V. A. Bartenev, A. K. Grechkoseev, D. A. Kozorez, M. N. Krasil’shchikov, V. V. Pasynkov, and K. I. Sypalo, Mod ern and Perspective Informational GNSSTechnologies in High Precision Navigation Problems (Fizmatlit, Moscow, 2014) [in Russian]. JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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Translated by O. Pismenov
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2015