Absolute orbit determination using line-of-sight vector measurements between formation flying spacecraft
- 79 Downloads
Abstract
The purpose of this paper is to show that absolute orbit determination can be achieved based on spacecraft formation. The relative position vectors expressed in the inertial frame are used as measurements. In this scheme, the optical camera is applied to measure the relative line-of-sight (LOS) angles, i.e., the azimuth and elevation. The LIDAR (Light radio Detecting And Ranging) or radar is used to measure the range and we assume that high-accuracy inertial attitude is available. When more deputies are included in the formation, the formation configuration is optimized from the perspective of the Fisher information theory. Considering the limitation on the field of view (FOV) of cameras, the visibility of spacecraft and the installation of cameras are investigated. In simulations, an extended Kalman filter (EKF) is used to estimate the position and velocity. The results show that the navigation accuracy can be enhanced by using more deputies and the installation of cameras significantly affects the navigation performance.
Keywords
Autonomous orbit determination Formation flying spacecraft Measurement constraint Fisher information theory1 Introduction
Spacecraft Formation has been widely used in missions with respect to the earth and other planets. Missions that use spacecraft formation are less expensive, more robust and flexible than those using only one spacecraft (Alfriend et al. 2009). Moreover, some complicated missions can only be fulfilled with formation flying spacecraft (D’Amico et al. 2012). For example, the Magnetospheric Multiscale (MMS) Mission requires four satellites flying in a regular tetrahedron to study the magnetic reconnection and other properties of the earth atmosphere (Roscoe et al. 2011). NASA’s Gravity Recovery and Interior Laboratory (GRAIL) mission used two spacecraft to obtain high-quality gravitational field maps of the moon and determine its interior structure (Bandyopadhyay et al. 2016). Two spacecraft in leader–follower configuration were also applied to measure the gravity field model of the earth (Psiaki 2011). Other two-spacecraft configurations were also investigated in the application of determining the gravity field (Elsaka et al. 2012). Yang used two formation flying spacecraft to realize autonomous navigation in the proximity of an asteroid (Yang et al. 2016).
In the future, spacecraft formation will be equipped with the autonomous navigation system. Researchers have concentrated on the problem of autonomous navigation for many years. For autonomous relative navigation, the inter-spacecraft range and line-of-sight angles are used to determine the relative state (Huxel and Bishop 2009). Wang proposed to use the line-of-sight and range measurements between the chief and deputies as measurements in a novel decentralized relative navigation algorithm for spacecraft formation flying (Wang et al. 2016). Christian (2014) used flash LIDAR and range sensors to realize autonomous rendezvous and docking with a cooperative target. In contrast, absolute navigation requires that absolute information, such as inertial position or velocity, should be included in the measurements (Wang et al. 2014). Markley initially proposed the method to determine the orbits of two spacecraft when they were orbiting around a central body (Markley 1984). In this approach, the inertial attitude information of the chief was assumed to be known (Hill and Born 2007). In addition, Ou clarified the relationship between configuration description parameters and the navigation performance (Ou and Zhang 2018). However, the aforementioned authors assumed that the measurements were available all the time. Actually, the FOV of the camera is limited and the observation is available only when the object is visible. In Huang’s work, field of view limitation of cameras was considered and the relative position and attitude was estimated by a monocular camera (Huang et al. 2017). Similarly, the visibility of the object and the corresponding navigation performance should be investigated in this paper.
Sometimes, more than two spacecraft are used in the formation, and the observability of the system determines the navigation performance (Kang et al. 2009). For the dual-spacecraft formation flying system, special unobservable conditions have been outlined by Markley (1984) and Psiaki (2011). The effect of the chief’s orbital elements on the observability was also clarified by Ou (2016). Maessen (2012) found that both the number of sensors and their installation positions have an effect on the observability. However, these researchers only investigated the system in which only two spacecraft (one deputy) were considered. When more deputies are added, the configuration will be optimally designed to make the system more observable.
In this paper, the chief is orbiting around a central body and it takes the main payload as well as other instruments. Several small spacecraft can be released from the chief and they will operate as moving beacons. As the measurement scheme used by Markley (1984) and Psiaki (2011), we also assume that the chief’s attitude could be determined with high accuracy (e.g., 1 arcsec) using star sensors (Schmidt 2014). Thus, relative line-of-sight vector measured in the body frame could be transformed to the inertial frame. Hence, the relative position vectors expressed in the inertial frame are directly used as measurements in this paper. In view of aforementioned shortcomings, the measurement constraints on the FOV of optical cameras and their installation positions are investigated. In addition, an extended Kalman filter (EKF) that accommodates the measurement missing is used to estimate the state. Moreover, when more deputies are added into the system, the configuration is optimized from the perspective of Fisher information theory (Jauffret 2007).
The rest of the paper is organized as follows. Section 2 gives the dynamical model and measurement model of the navigation scheme. In Sect. 3, the system observability is optimized based on the Fisher information theory. Next, an extended Kalman filter is applied to estimate the position and velocity in Sect. 4. In Sect. 5, the navigation simulations are conducted and finally Sect. 6 concludes this paper.
2 Navigation system design
2.1 Spacecraft dynamics
Hence, the acceleration uncertainties in Eq. (1) are set as gaussian white noises with variance \(10^{ - 16}~\mbox{km}^{2}/\mbox{s}^{4}\).
2.2 Measurement model and measurement constraints
2.3 Formation configuration design
3 Configuration optimization method
Researchers have proved that the observability is mainly determined by the configuration. In this paper, we consider the situation when two or more deputies are flying around the chief in the formation. In order to enhance the observability, the formation configuration must be optimized.
Note that the relative observability among states is not investigated in this paper and it can be obtained by comparing the eigenvalues corresponding to the states. Therefore, this index can only be applied to assess the overall observability.
4 The extended Kalman filter
5 Simulations
5.1 Navigation analysis without measurement constraints
Orbital elements of three formation flying spacecraft
Orbital elements | Chief | Deputy 1 | Deputy 2: Case 1 | Deputy 2: Case 2 | Deputy 2: Case 3 |
---|---|---|---|---|---|
Semi-major axis a (km) | 3697 | 3697 | 3697 | 3697 | 3697 |
Eccentricity e | 0.001 | 0.0037 | 0.0017 | 0.0036 | 0.0024 |
Inclination i (^{∘}) | 90.00 | 90.00 | 90.00 | 90.00 | 90.00 |
RAAN Ω (^{∘}) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
Perigee ω (^{∘}) | 0.000 | 0.000 | 180.0 | −22.03 | −98.56 |
True anomaly f (^{∘}) | 0.000 | 0.000 | 180.0 | 22.18 | 98.82 |
Configuration design parameters in three cases
Parameters | Deputy 1 | Deputy 2: Case 1 | Deputy 2: Case 2 | Deputy 2: Case 3 |
---|---|---|---|---|
p (km) | 10 | 10 | 10 | 10 |
s (km) | 0 | 0 | 0 | 0 |
α (^{∘}) | 90 | 90 | 90 | 90 |
\(\theta _{0}\) (^{∘}) | 0 | 180 | 30 | 120 |
5.2 Navigation analysis with measurement constraints
Orbit determination errors for the chief under different camera installations (\(1\sigma \))
Camera installations | x (m) | y (m) | z (m) | \(v_{x}\) (m/s) | \(v_{y}\) (m/s) | \(v_{z}\) (m/s) |
---|---|---|---|---|---|---|
x(±), y(±) | 5.237 | 7.348 | 26.53 | 0.016 | 0.029 | 0.005 |
x(+) | 2080 | 1826 | 2102 | 1.5294 | 0.712 | 0.347 |
y(+) | 232.1 | 959.3 | 407.9 | 0.342 | 0.756 | 0.157 |
x(±) | 141.4 | 20.31 | 556.7 | 0.416 | 0.105 | 0.126 |
y(±) | 7.898 | 7.491 | 58.08 | 0.044 | 0.031 | 0.007 |
6 Conclusions
In this paper, autonomous navigation is achieved by using formation flying spacecraft. The inertial line-of-sight vectors between the chief and deputies are used as the measurements. The constraint on the camera is considered and the installation of cameras is investigated. Moreover, when there are two deputies in the formation, optimal configurations are obtained from the perspective of Fisher information theory and an index is proposed to evaluate the system’s observability. The extended Kalman filter is modified to tackle the problem of measurement missing. The navigation simulations reveal that the installation of cameras influences the navigation performance. The results also show that the camera installed along \(y\) axis can receive more measurement information. Thus, the cameras mounted along this axis can significantly enhance the navigation accuracy. After the configuration is optimized, the simulation confirms that the navigation accuracy can be enhanced by using more deputies. Future work may concentrate on how to reduce the computation when more deputies are included in the system.
Notes
Acknowledgement
This work is supported by the Major Program of National Natural Science Foundation of China under Grant Numbers 61690210 and 61690213.
References
- Alfriend, K.T., Vadali, S.R., Gurfil, P., How, J., Breger, L.S.: Spacecraft Formation Flying: Dynamics, Control and Navigation (2009). Butterworth-Heinemann Google Scholar
- Bandyopadhyay, S., Foust, R., Subramanian, G.P., Chung, S.J., Hadaegh, F.Y.: Review of formation flying and constellation missions using nanosatellites. J. Spacecr. Rockets 53, 567 (2016). https://doi.org/10.2514/1.A33291 ADSCrossRefGoogle Scholar
- Bar-Shalom, Y., Li, X.R., Kirubarajan, T.: Estimation with Applications to Tracking and Navigation. Wiley, New York (2002). https://doi.org/10.1002/0471221279.ch3 Google Scholar
- Brown, R.G., Hwang, P.Y.C.: Introduction to Random Signals and Applied Kalman Filtering. Wiley, New York (1997) zbMATHGoogle Scholar
- Christian, J.A., Robinson, S.B., D’Souza, C.N., Ruiz, J.P.: Cooperative relative navigation of spacecraft using flash light detection and ranging sensors. J. Guid. Control Dyn. 37, 452 (2014). https://doi.org/10.2514/1.61234 ADSCrossRefGoogle Scholar
- Crassidis, J.L., Junkins, J.L.: Optimal Estimation of Dynamic Systems, 2nd edn. Chapman & Hall, London (2011) zbMATHGoogle Scholar
- D’Amico, S., Montenbruck, O.: Proximity operations of formation-flying spacecraft using an eccentricity/inclination vector separation. J. Guid. Control Dyn. 29, 554 (2009). https://doi.org/10.2514/1.15114 CrossRefGoogle Scholar
- D’Amico, S., Ardaens, J.S., Larsson, R.: Spaceborne autonomous formation-flying experiment on the PRISMA mission. J. Guid. Control Dyn. 35, 834 (2012). https://doi.org/10.2514/1.55638 ADSCrossRefGoogle Scholar
- Elsaka, B., Kusche, J., Ilk, K.H.: Recovery of the Earth’s gravity field from formation-flying satellites: temporal aliasing issues. Adv. Space Res. 50, 1534 (2012). https://doi.org/10.1016/j.asr.2012.07.016 ADSCrossRefGoogle Scholar
- Gelb, A.: Applied Optimal Estimation. MIT Press, Cambridge (1974). Chap. 3 Google Scholar
- Hill, K., Born, G.: Autonomous interplanetary orbit determination using satellite-to-satellite tracking. J. Guid. Control Dyn. 30, 679 (2007). https://doi.org/10.2514/1.24574 ADSCrossRefGoogle Scholar
- Huang, P.F., Chen, L., Zhang, B., Meng, Z.J., Liu, Z.X.: Autonomous rendezvous and docking with nonfull field of view for tethered space robot. Int. J. Aerosp. Eng. 2017, 3162349 (2017) Google Scholar
- Huxel, P.J., Bishop, R.H.: Navigation algorithms and observability analysis for formation flying missions. J. Guid. Control Dyn. 32, 1218 (2009). https://doi.org/10.2514/1.41288 ADSCrossRefGoogle Scholar
- Jauffret, C.: Observability and Fisher information matrix in nonlinear regression. IEEE Trans. Aerosp. Electron. Syst. 43, 756 (2007). https://doi.org/10.1109/TAES.2007.4285368 ADSCrossRefGoogle Scholar
- Kang, W., Ross, I.M., Pham, K., Gong, Q.: Autonomous observability of networked multisatellite systems. J. Guid. Control Dyn. 32, 869 (2009). https://doi.org/10.2514/1.38826 ADSCrossRefGoogle Scholar
- Lemoine, F.G., Smith, D.E., Rowlands, D.D., Zuber, M.T., Neumann, G.A., Chinn, D.S., Pavlis, D.E.: An improved solution of the gravity field of Mars (GMM-2B) from Mars Global Surveyor. J. Geophys. Res. 106, 23359 (2001). https://doi.org/10.1029/2000JE001426 ADSCrossRefGoogle Scholar
- Maessen, D., Gill, E.: Relative state estimation and observability analysis for formation flying satellites. J. Guid. Control Dyn. 35, 321 (2012). https://doi.org/10.2514/1.55125 ADSCrossRefGoogle Scholar
- Markley, F.L.: Autonomous navigation using landmark and intersatellite data. In: AIAA/AAS Astrodynamics Conference, Seatle, WA, Hawaii (1984). AIAA Paper Google Scholar
- Markley, F.L.: Approximate Cartesian state transition matrix. J. Astronaut. Sci. 34, 161 (1986) ADSGoogle Scholar
- Ou, Y., Zhang, H.: Observability-based Mars autonomous navigation using formation flying spacecraft. J. Navig. 71, 21 (2018). https://doi.org/10.1017/S0373463317000510 CrossRefGoogle Scholar
- Ou, Y., Zhang, H., Xing, J.: Autonomous orbit determination and observability analysis for formation satellites. In: 2016 35th Chinese Control Conference, CCC, Chengdu (2016). https://doi.org/10.1109/ChiCC.2016.7554179 Google Scholar
- Psiaki, M.L.: Absolute orbit and gravity determination using relative position measurements between two satellites. J. Guid. Control Dyn. 34, 1285 (2011). https://doi.org/10.2514/1.47560 ADSCrossRefGoogle Scholar
- Roscoe, C.W., Vadali, S.R., Alfriend, K.T., Desai, U.P.: Optimal formation design for magnetospheric multiscale mission using differential orbital elements. J. Guid. Control Dyn. 34, 1070 (2011). https://doi.org/10.2514/1.52484 ADSCrossRefGoogle Scholar
- Schmidt, U.: ASTRO APS autonomous star sensor first year flight experience and operations on AlphaSat (invited). In: AIAA/AAS Astrodynamics Specialist Conference, vol. 701 (2014). American Institute of Aeronautics and Astronautics Google Scholar
- Sinopoli, B., Schenato, L., Franceschetti, M., Poolla, K., Jordan, M.I., Sastry, S.S.: Kalman filtering with intermittent observations. IEEE Trans. Autom. Control 49, 9 (2004). https://doi.org/10.1109/TAC.2004.834121 MathSciNetCrossRefzbMATHGoogle Scholar
- Steven, M.K.: In: Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice Hall Signal Processing Series. Prentice Hall, New York (1993) Google Scholar
- Wang, Y., Zheng, W., Sun, S., Li, L.: X-ray pulsar-based navigation using time-differenced measurement. Aerosp. Sci. Technol. 36, 27 (2014). https://doi.org/10.1016/j.ast.2014.03.007 CrossRefGoogle Scholar
- Wang, X., Qin, W., Bai, Y., Cui, N.: A novel decentralized relative navigation algorithm for spacecraft formation flying. Aerosp. Sci. Technol. 48, 28 (2016). https://doi.org/10.1016/j.ast.2015.10.014 CrossRefGoogle Scholar
- Yang, H.X., Vetrisano, M., Vasile, M., Zhang, W.: Autonomous navigation of spacecraft formation in the proximity of minor bodies. Proc. Inst. Mech. Eng., G J. Aerosp. Eng. 230, 189 (2016). https://doi.org/10.1177/0954410015590465 CrossRefGoogle Scholar