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Figure from:Optimal Autonomous Orbit Control of Remote Sensing Spacecraft

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Table 3. Navigation, sensors and actuators accuracy modeling

Table 3 Navigation, sensors and actuators accuracy modeling

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Table 1 MANGO and TerraSAR-X satellites nominal orbital elements A reference orbit is a target orbit representing the desired mean nominal motion of the satellite over a long time interval. Basically, the reference orbit is a trade-off. First of all it shall be as realistic as possible because the spacecraft actual orbit is desired to be as close as possible to the target. On the other hand the reference orbit shall be as simple as possible, because the desired orbit has to be completely periodic relative to an Earth-fixed coordinate system. Thus the reference orbit model should at least consider the non-spherical terms of the Earth gravity potential which cause the sun synchronous secular motion of (. Other secular, long-periodic and short periodic terms can be included in the reference orbit depending on the flight control accuracy requirements. Deviations of real from reference orbital elements leading to a violation of flight control requirements have to be corrected by orbit maneuvers.
Table 1 MANGO and TerraSAR-X satellites nominal orbital elements A reference orbit is a target orbit representing the desired mean nominal motion of the satellite over a long time interval. Basically, the reference orbit is a trade-off. First of all it shall be as realistic as possible because the spacecraft actual orbit is desired to be as close as possible to the target. On the other hand the reference orbit shall be as simple as possible, because the desired orbit has to be completely periodic relative to an Earth-fixed coordinate system. Thus the reference orbit model should at least consider the non-spherical terms of the Earth gravity potential which cause the sun synchronous secular motion of (. Other secular, long-periodic and short periodic terms can be included in the reference orbit depending on the flight control accuracy requirements. Deviations of real from reference orbital elements leading to a violation of flight control requirements have to be corrected by orbit maneuvers.
The choice of this parameterization of the state is dictated by the fact that it does not lead to singular equations as the eccentricity value tends to zero. Nevertheless this set of orbital elements leads to singular equations as the inclination angle tends to zero but this case is out of interest when dealing with remote sensing satellites orbits. As this absolute orbit control problem can be formulated as a two spacecraft for- mation flying control problem in which one of the spacecraft is virtual and not affected by non-gravitational orbit perturbations, the most appropriate parameterization to represent the relative motion of the real satellite with respect to the reference is a set of relative orbital elements, shown in Eq. (2), which are obtained as a non-linear combination of the absolute orbital elements defined in Eq. (1) (Ref. 12 and 14).
The choice of this parameterization of the state is dictated by the fact that it does not lead to singular equations as the eccentricity value tends to zero. Nevertheless this set of orbital elements leads to singular equations as the inclination angle tends to zero but this case is out of interest when dealing with remote sensing satellites orbits. As this absolute orbit control problem can be formulated as a two spacecraft for- mation flying control problem in which one of the spacecraft is virtual and not affected by non-gravitational orbit perturbations, the most appropriate parameterization to represent the relative motion of the real satellite with respect to the reference is a set of relative orbital elements, shown in Eq. (2), which are obtained as a non-linear combination of the absolute orbital elements defined in Eq. (1) (Ref. 12 and 14).
where the subscript r refers to the reference orbit. It is noteworthy the fact that in an ideal two body problem, the orbital element u is the only not invariant element because it represents the time variable. The relative orbit representation defined in Eq. (2) is based on the relative eccentricity and inclination vectors defined in cartesian and polar notations as
where the subscript r refers to the reference orbit. It is noteworthy the fact that in an ideal two body problem, the orbital element u is the only not invariant element because it represents the time variable. The relative orbit representation defined in Eq. (2) is based on the relative eccentricity and inclination vectors defined in cartesian and polar notations as
The phases of the relative e/i vectors are termed relative perigee ¢ and relative ascending node @ because they characterize the geometry of the relative orbit and determine the angular locations of the perigee and ascending node of the relative orbit (Ref. 12 and 14). The position of the satellite relative to the reference orbit in the RTN orbital frame (R pointing along the orbit radius, N pointing along the angular momentum vector and T = N x R pointing in the direction of motion for a circular orbit) can be described in means of orbital elements as
The phases of the relative e/i vectors are termed relative perigee ¢ and relative ascending node @ because they characterize the geometry of the relative orbit and determine the angular locations of the perigee and ascending node of the relative orbit (Ref. 12 and 14). The position of the satellite relative to the reference orbit in the RTN orbital frame (R pointing along the orbit radius, N pointing along the angular momentum vector and T = N x R pointing in the direction of motion for a circular orbit) can be described in means of orbital elements as
Figure 1 Real orbit elements evolution of MANGO satellite in free motion Figures | and 2 show respectively the evolution of MANGO’s real orbit absolute and relative elements with respect to their respective reference orbits. The analysis was performed by numerical orbit propagation over 30 days and using orbit propagation mode to the orbit. altitud es of MANGO (about 700 km) and TerraSAR-X satellites (about 514 and satellite physical parameters given in Table 2. The noisy pattern of the plots in Fig. 2 is due short periods gravitational perturbations not modeled by the 20x20 gravitational harmonics reference Fig. 3 shows the de and di vectors evolution of MANGO and TerraSAR-X in free motion. At the km) the evolution of de and 6i, i.e. the evolution of the relative motion of the controlled satellite with respect to the ideal reference satellite is mainly driven by Jz. As the reference satellite’s trajectory is not affected by non gravitational perturbation the atmos pheric drag plays also a fundamental role in the evolution of de as wel as the moon and sun third body gravitational perturbation in varying di. A clear understanding of the evolution of de and di in uncontrolled motion is fundamental in deciding the orbit control strategy. At each instant of time the place along the orbit of a correction maneuver of de and di can be determined with Eq. (5) t solution of the Gauss equations (Ref. 14). hat is derived from an analytical
Figure 1 Real orbit elements evolution of MANGO satellite in free motion Figures | and 2 show respectively the evolution of MANGO’s real orbit absolute and relative elements with respect to their respective reference orbits. The analysis was performed by numerical orbit propagation over 30 days and using orbit propagation mode to the orbit. altitud es of MANGO (about 700 km) and TerraSAR-X satellites (about 514 and satellite physical parameters given in Table 2. The noisy pattern of the plots in Fig. 2 is due short periods gravitational perturbations not modeled by the 20x20 gravitational harmonics reference Fig. 3 shows the de and di vectors evolution of MANGO and TerraSAR-X in free motion. At the km) the evolution of de and 6i, i.e. the evolution of the relative motion of the controlled satellite with respect to the ideal reference satellite is mainly driven by Jz. As the reference satellite’s trajectory is not affected by non gravitational perturbation the atmos pheric drag plays also a fundamental role in the evolution of de as wel as the moon and sun third body gravitational perturbation in varying di. A clear understanding of the evolution of de and di in uncontrolled motion is fundamental in deciding the orbit control strategy. At each instant of time the place along the orbit of a correction maneuver of de and di can be determined with Eq. (5) t solution of the Gauss equations (Ref. 14). hat is derived from an analytical
Figure 2 Relative orbital elements evolution of MANGO satellite in free motion with respect to its reference orbit. polar orbits (¢ = 90°) at the orbits nodes (u = 0° and u = 180° ) only Ade, or Adi, corrections are possible while at the highest latitudes (u = 90° and u = 270° ) only Ade, or Adi, corrections are possible if only along-track and cross-track maneuvers are used. These considerations are relevant for near-polar orbits.
Figure 2 Relative orbital elements evolution of MANGO satellite in free motion with respect to its reference orbit. polar orbits (¢ = 90°) at the orbits nodes (u = 0° and u = 180° ) only Ade, or Adi, corrections are possible while at the highest latitudes (u = 90° and u = 270° ) only Ade, or Adi, corrections are possible if only along-track and cross-track maneuvers are used. These considerations are relevant for near-polar orbits.
Figure 3. de and 6i vectors evolution of MANGO and TerraSAR-X satellites in free motion with respect to their respective reference orbits.
Figure 3. de and 6i vectors evolution of MANGO and TerraSAR-X satellites in free motion with respect to their respective reference orbits.
Table 2 Propagation parameters OPTIMAL ORBIT CONTROL LQR Controller
Table 2 Propagation parameters OPTIMAL ORBIT CONTROL LQR Controller
Gr, ap and ay are the perturbing accelerations respectively along the R, T and N axes. The real orbi can be modeled using Gauss’ equations as shown in Eq. (12) Gauss’ variational equations of motion adapted for near-circular orbits (see Ref. 17) provide a convenient set of equations relating the effect of a control acceleration vector u to the osculating orbital element time derivatives. After some manipulation, the Gauss equations can be used to represent the control problem in the form of Eq. (6) using mean elements. The Gauss’ variational equations for near-circular orbit are presented in Eq. (11).
Gr, ap and ay are the perturbing accelerations respectively along the R, T and N axes. The real orbi can be modeled using Gauss’ equations as shown in Eq. (12) Gauss’ variational equations of motion adapted for near-circular orbits (see Ref. 17) provide a convenient set of equations relating the effect of a control acceleration vector u to the osculating orbital element time derivatives. After some manipulation, the Gauss equations can be used to represent the control problem in the form of Eq. (6) using mean elements. The Gauss’ variational equations for near-circular orbit are presented in Eq. (11).
Vector A of Eq. (19) represents the orbit model of the satellite in free motion (cfr. Eq. (12)). A simple analytical orbit model expressed by Equations (21) and (22) has been used for the realization of the LQR controller. This model consists of the averaged equations of motion of a satellite perturbed by the gravity field zonal term J2 and the atmospheric drag perturbation (neglecting the atmospheric rotational speed). For details related to the model see Ref. 19 and 20. Vectors A,(«) and Ag(«) represent the different contributions respectively of the gravity field and the atmospheric drag. State-space Matrices A and B
Vector A of Eq. (19) represents the orbit model of the satellite in free motion (cfr. Eq. (12)). A simple analytical orbit model expressed by Equations (21) and (22) has been used for the realization of the LQR controller. This model consists of the averaged equations of motion of a satellite perturbed by the gravity field zonal term J2 and the atmospheric drag perturbation (neglecting the atmospheric rotational speed). For details related to the model see Ref. 19 and 20. Vectors A,(«) and Ag(«) represent the different contributions respectively of the gravity field and the atmospheric drag. State-space Matrices A and B
Figure 4 LQR control - Deviation of real from reference orbit of MANGO satellite in RTI frame and LAN deviation with respect to the reference. Fig. 4 is representative of some first results of the application of the LQR in the autonomous control of MANGO’s orbit.
Figure 4 LQR control - Deviation of real from reference orbit of MANGO satellite in RTI frame and LAN deviation with respect to the reference. Fig. 4 is representative of some first results of the application of the LQR in the autonomous control of MANGO’s orbit.
Table 4 MANGO controller parameters In this case the constraints on de,, de, and 62, have been relaxed and the control of da and du has been coupled. At the altitude of MANGO a number of expensive out-of-plane maneuvers are required for the attainment of a fine control of dry. Without a substantial relaxation of the accuracy control requirements of dry, the LQR controller proves in this case to be very expensive compared to the AOK controller. It is straightforward to note that the control of di, is more efficient at higher latitudes (compare cases 1 and 2 of Tables 4 and 5). It can be concluded that if the control requirements concern only an Earth fixed
Table 4 MANGO controller parameters In this case the constraints on de,, de, and 62, have been relaxed and the control of da and du has been coupled. At the altitude of MANGO a number of expensive out-of-plane maneuvers are required for the attainment of a fine control of dry. Without a substantial relaxation of the accuracy control requirements of dry, the LQR controller proves in this case to be very expensive compared to the AOK controller. It is straightforward to note that the control of di, is more efficient at higher latitudes (compare cases 1 and 2 of Tables 4 and 5). It can be concluded that if the control requirements concern only an Earth fixed
Figure 5 AOK control - Relative orbital elements of MANGO satellite and LAN deviation with respect to the reference. Fig. 5 in contrast shows the results for a classical autonomous LAN control method (AOK experiment, Ref. 6) in the same scenario. In comparing these two cases it has to be clear that the LQR controls all the relative orbital elements, while the AOK controller is required to control only the LAN, an Earth fixed reference parameter, with an accuracy of 20 m (r.m.s) and by means of along-track and anti-along-track velocity increments. Tables 4 and 5 collects controller parameters and control performances in the cases depicted in the figures. The LQR controller is activated one time per orbit in the orbit places indicated in Table 4.
Figure 5 AOK control - Relative orbital elements of MANGO satellite and LAN deviation with respect to the reference. Fig. 5 in contrast shows the results for a classical autonomous LAN control method (AOK experiment, Ref. 6) in the same scenario. In comparing these two cases it has to be clear that the LQR controls all the relative orbital elements, while the AOK controller is required to control only the LAN, an Earth fixed reference parameter, with an accuracy of 20 m (r.m.s) and by means of along-track and anti-along-track velocity increments. Tables 4 and 5 collects controller parameters and control performances in the cases depicted in the figures. The LQR controller is activated one time per orbit in the orbit places indicated in Table 4.
Table 5 MANGO control performances
Table 5 MANGO control performances
Figure 6 LQR control - Evolution of relative orbital elements with respect to the referenc orbit for TerraSAR-X during two months simulation. Fig. 6 shows the evolution of the relative orbital elements of TerraSAR-X with respect to the reference orbit during two month of simulated autonomous orbital control by the LQR controller activated once per day.
Figure 6 LQR control - Evolution of relative orbital elements with respect to the referenc orbit for TerraSAR-X during two months simulation. Fig. 6 shows the evolution of the relative orbital elements of TerraSAR-X with respect to the reference orbit during two month of simulated autonomous orbital control by the LQR controller activated once per day.
Table 6 TerraSAR-X controller parameters The dv budget of the ground based control, given in Table 7, is based on values averaged on the period from 20 June 2007 to 27 January 2009 and excluding the orbital maneuvers actuated during periods in which
Table 6 TerraSAR-X controller parameters The dv budget of the ground based control, given in Table 7, is based on values averaged on the period from 20 June 2007 to 27 January 2009 and excluding the orbital maneuvers actuated during periods in which
Figure 7 LQR control - Relative motion of TerraSAR-X real orbit with respect to the referenc (0,0) during two months simulation. Fig. 7 shows the relative motion of TerraSAR-X real orbit with respect to the reference in the RT'N frame and the Earth fixed parameter LAN.
Figure 7 LQR control - Relative motion of TerraSAR-X real orbit with respect to the referenc (0,0) during two months simulation. Fig. 7 shows the relative motion of TerraSAR-X real orbit with respect to the reference in the RT'N frame and the Earth fixed parameter LAN.
Table 7 TerraSAR-X control performances
Table 7 TerraSAR-X control performances
Gravity Field
Gravity Field