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Constructing space-based continuous navigation and communication system utilizing formation flying satellites on GEO orbit in deep space exploration
Acta Astronautica 65 (2009) 1185 – 1189 www.elsevier.com/locate/actaastro
Constructing space-based continuous navigation and communication system utilizing formation flying satellites on GEO orbit in deep space exploration Hui Li∗ , Qinyu Zhang, Naitong Zhang Shenzhen Graduate School of Harbin Institute of Technology, PRC 518055, China Received 16 November 2007; accepted 4 March 2009 Available online 14 August 2009
Abstract Two clusters of formation flying satellites with a relative angle on geosynchronous Earth orbit were used to construct a spacebased continuous navigation system for deep space exploration. Adopting technique of passive navigation by multiple basestations, formation flying satellites obtained high precision of navigation according to time and frequency difference of arrival of downlink signal from deep space explorer. Enhancing system timing and carrier frequency, and increasing the length of baselines help to improve precision of navigation. The precision of this navigation system is close to that of NASA’s VLBI system. © 2009 Published by Elsevier Ltd. Keywords: Deep space exploration; Space-based; Continuous navigation; Formation flying satellite; Geostationary Earth orbit
1. Introduction In deep space exploration, it is difficulty to continuously navigate and communicate with explorer in all weather, and it is also difficult for one country to construct a global deep space network (DSN) terrestrially in its own land [1]. Generally, a single Earth-station is able to link explorer for 6–12 h, so continuous navigation and communication system depends on global deep space network as America and Russia [2]. However, DSN on ground is of lower coverage and is affected greatly by atmosphere. Space-based continuous navigation and communication (SBCNC) system may be a potential way of solving problems mentioned above.
∗ Corresponding author. Tel./fax: +86 451 8628 2013.
E-mail address: [email protected] (H. Li). 0094-5765/$ - see front matter © 2009 Published by Elsevier Ltd. doi:10.1016/j.actaastro.2009.03.072
Due to the huge distance in deep space exploration, requirement of precision of navigation is extremely high, and traditional way of satellite navigation and tracking cannot fulfill the need the ranging, velocity checking and angular measuring [3]. In 60–70 decades of last century, Earth-based radiometric tracking technique was generally adopted for navigation and tracking of probes during the cruise phase of a mission, and even during the approach phase [4]. In the 80 decades, very long baseline interferometry (VLBI) was adopted to achieve 20–30 nrad accuracy of navigation; however, it could not work in real-time. NASA’s continuous elements interferometry (CEI) technique is able to 80 nrad angular accuracy in real-time by two stations apart from 21 km and connected optical fibre [4,5]. Same beam interferometry (SBI) provides extremely accurate relative position measurements in the plane-of-thesky, complementing the line-of-sight information from
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H. Li et al. / Acta Astronautica 65 (2009) 1185 – 1189
Earth-based Doppler and range measurements. System errors that scale with angular and temporal separations are greatly reduced, allowing nearly the full precision of carrier-phase measurements to be utilized [4,6,7]. An elliptical inclined satellite and an Earth stations are to be utilized to construct a space-VLBI navigation system [8]. However, geostationary Earth orbit (GEO) satellite may be a better choice for simpler orbit and attitude control mechanism and longer baseline.
2. Space-based continuous navigation system Two GEO satellites with a minimum geocentric angle 34.634◦ can see and communicate with deep explorer continuously, avoiding the shade of Earth as shown in Fig. 1. Shelter 1 and 2 are the areas that cannot be seen by GEO satellite 1 and 2. However, this space-based system cannot navigate and position for the explorer. Two formation fly satellite (FFS) clusters located on GEO can solve this problem. A SBCNC system is proposed in Fig. 2, which utilizes two satellite formations with an angular spacing located on GEO orbit, and can navigate and communicate with the deep space explorers continuously. R S 1 and R S 2 are reference satellites in FFS1 and FFS2, and concomitant satellites combine Fig. 2. A SBCNC system on GEO orbit: (a) lateral view and (b) planform.
Fig. 1. Two GEO satellites with 34.634◦ angle receive signal from deep space explorer: (a) lateral view and (b) planform.
R S 1 and R S 2 satellites according to Hill equations (or called C–W equations). The shadow in Fig. 2(b) is the area that shades FFS1 by Earth. Satellite formation cooperates and accomplishes the navigation by way of passive allocation method. Many agencies and researching groups are carrying out the research of FFS in deep space and on near Earth orbit. The Cassini–Huygens mission to Saturn is the end of an era for NASA sending one large spacecraft equipped to carry out a multitude of scientific experiments [9]. Future NASA missions will deploy many smaller spacecrafts in highly controlled spatial configurations in what is referred to as “formation flying”. And both the Air Force and NASA have identified autonomous formation of spacecraft as key technological milestones for the 21st century [10]. The coming decade will see formation flying spacecraft (FFS) working simultaneously in deep space [11]. The deep space FFS missions include long-baseline mid-IR interferometry missions, X-ray interferometry missions and general relativity missions [12]. Moreover
H. Li et al. / Acta Astronautica 65 (2009) 1185 – 1189
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Table 1 Relationship of eccentricity of satellite formations and the minimum geocentric angle between them. E
0.001
0.005
0.01
0.05
0.1
0.2
0.3
0.4
l B (km) l1 (km) RS 1 O RS 2
84 41614 35.03◦
422 39550 35.96◦
843 37240 37.12◦
4216 25384 46.51◦
8433 18157 58.53◦
16866 11570 84.31◦
25298 8490 114.8◦
33731 6704 161.48◦
FFS has the potential to enhance space-based imaging and interferometry missions by distributing mission tasks as surveillance, SAR, magnetosphere sensing and a variety of other missions to many small spacecrafts [13,14]. Projects of space-FFS replacing signal large spacecraft for interferometry and surveillance certain celestial bodies may be positioned on halo orbit or Lagrange points in restricted three bodies systems as XEUS and Darwin missions [15–18]. According to Fig. 2(b), the minimal geocentric angle is calculated by
l1 + l2 = R E + h l1 RE = l2 lB
R S 1 O R S 2 = 2 + 2
(1) Fig. 3. TDOA positioning and navigation method.
(2) positioning (multiple transmitters and single receiver)
in which, l B = 2 · (R E + h) · e is the radius of FFS’s circular structure, and is also the baseline of this FFSSBCNC system. R E is the radius if Earth, h, is the altitude of GEO orbit, and e is the eccentricity ratio of concomitant satellites’ elliptical orbit around the Earth. And = arcsin(R E /l1 ), = arcsin(R E /(R E + h)). The relationship of eccentricity ratio of FFS and the minimum geocentric angle between two FFS is listed in Table 1. So the bigger size of FFS with longer baseline needs bigger geocentric angle between them for continuous navigation and communication.
⎧ 1 = d1 − d0 = C · (td1 − td0 ) = C · t10 ⎪ ⎪ ⎨ 2 = d2 − d0 = C · (td2 − td0 ) = C · t20 .. ⎪ ⎪ ⎩. n = dn − d0 = C · (tdn − td0 ) = C · tn0
(3)
in which td0 , td1 · · · tdn are the arrival time of signal from explorer to FFS component, and d0 , d1 · · · dn are the corresponding distance. t10 , t20 · · · tn0 are the difference of arrival time between concomitant satellites and reference satellite, and 1 , 2 · · · n are the corresponding distance difference. C is the speed of light. Precision curves of TDOA method are plotted in Fig. 3, when using this system in Mars exploration.
3. Methods of positioning and navigation 3.1. Time difference of arrival (TDOA)
3.2. Differential Doppler (DD)
Navigation and positioning by multi-spacecraft utilizing difference of signal’s arrival time simultaneously from position-known Earth stations or quasar in space is called time difference of arrival positioning as shown in Fig. 3. TDOA is also called “hyperbolic fix” in which the time of signal arrival from different receives is used to fix the location of transmitter as GPS-like active
Doppler shift due to relative movement between satellites and explorer is f i = f 0 + f di , in which f i is carrier frequency of NO. i is the satellite’s receiver, f 0 the transmitted carrier frequency, and f di Doppler shift: f di =
˙ · (Ri − R) 1 ( R˙ i − R) · Ri − R
(4)
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H. Li et al. / Acta Astronautica 65 (2009) 1185 – 1189
in which is the wavelength of carrier frequency. R˙ i , i =0, 1, . . . , n and R˙ are the velocity of satellites and explorer. Equations of DD method is expressed in Eq. (5), which can be solved by Newton iteration and other algorithms ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
f d1 − f d0 ˙ ˙ · (R1 − R) ˙ · (R0 − R) 1 ( R1 − R) ( R˙ 0 − R) = · − R1 − R R0 − R f d2 − f d0 ˙ ˙ · (R2 − R) ˙ · (R0 − R) 1 ( R2 − R) ( R˙ 0 − R) = · − R2 − R R0 − R ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ f dn − f d0 ⎪ ⎪ ⎪ ˙ · (Rn − R) ˙ · (R0 − R) ⎪ ( R˙ 0 − R) ⎩ = 1 · ( R˙ n − R) − Rn − R R0 − R
(5)
4. Simulation results and data analysis Two reference satellites of FFSs are located on GEO orbit 61◦ E and 120◦ E longitude with geocentric angle 59◦ . There are five concomitant satellites and one reference satellite in each formation as shown in Fig. 2(a). The Earth stations are located at Kashi (76◦ E, 39.5◦ N) and Qingdao (120.33◦ E, 36.07◦ N). The maximal distance between Mars and Earth is 4.013 × 108 km. is the covariance of error of time arrival. The explorer is at the second cosmic velocity. Both the error of time arrival and the frequency measuring obey the Gauss distribution with zero equalizing value. From Fig. 4, we can draw the conclusions that the precision of positioning and navigation increases with increase of the length of baseline and decrease of time measure and frequency measure. For Mars-distance exploration, 5000 m positioning and communication error is 12.46 nrad. 5. Conclusions and future works Two FFS located on the GEO orbit with a geocentric angle can be used to construct a space-based continuous navigation and communication system. FFS locates the position of explorer from one-way received signal in real-time. FFS adopted TDOA and DD methods of navigation and positioning to locate the position deep space explorer. Long baseline and small covariance of time-arrival and Doppler shift measure result in high precision of navigation. In this system, if a hybrid method of TDOA, DD and quasar difference is utilized, a higher accuracy of positioning and navigation is predictable.
Fig. 4. Mean error of positioning and navigation by TDOA in Mars exploration (upperbound of sat-position error 100 m): (a) Mean error of positioning navigation vs. covariance of error of time arrival (l B = 8433 km); (b) mean error of positioning navigation vs. radius of FFS/baseline ( = 1 × 10 − 13); (c) mean error of positioning navigation vs. covariance of error of frequency.
References [1] M. Lin, Trend of development of deep space tracking and communication systems, Journal of Spacecraft TT&C Technology 24 (3) (2005) 7–9.
H. Li et al. / Acta Astronautica 65 (2009) 1185 – 1189 [2] P. Li, J.S. Zhang, The current status and the future of NASA deep space network, Journal of Spacecraft TT&C Technology 22 (4) (2003) 10–18. [3] M.K. Reynolds, M.J. Reinhart, R.S. Bokulic, A two-way noncoherent ranging technique for deep space missions, in: IEEE Aerospace Conference Proceedings, vol. 3, Big Sky, MT, USA, 2002, pp. 1303–1312. [4] C.L. Thornton, J.S. Border, Radiometric Tracking Techniques for Deep Space Navigation, Wiley, NJ, USA, 2003, pp. 1–2. [5] J. Lauf, M. Calhoun, W. Diener, et al., Clocks and timing in the NASA deep space network, in: IEEE International Frequency Control Symposium and Exposition, 29–31 August, 2005, pp. 830–835. [6] R.S. Bokulic, J.R. Jensen, M.K. Flaherty, Transceiver-based spacecraft telecommunications systems with two-way Doppler tracking capability, Acta Astronautica 45 (4) (1999) 365–372. [7] T. Ichikawa, T. Kato, The study of delta VLBI and Doppler navigation for interplanetary spacecraft, Advances in Space Research 23 (4) (1999) 739–746. [8] Y. Asai, T. Nishimura, F. Kamijo, et al., Geometric effects on deep space orbit determination via space-VLBI observations, in: 41st SICE Annual Conference, Iwaki, Fukushima, Japan, vol. 1, 5–7 August, 2002, pp. 175–180. [9] O. Junge, S. Ober-Blobaum, Optimal reconfiguration of formation flying satellites, in: 44th IEEE Conference on Decision and Control, and the European Control Conference, Seville, Spain, 12–15 December, 2005, pp. 66–71. [10] K. Rahnamai, P. Arabshahi, A. Gray, Fuzzy supervised optimal regulator for spacecraft formation flying, in: 22nd International Conference of the NAFIPS, Chicago, IL, USA, 24–26 July, 2003, pp. 329–334.
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[11] D.J. Naffin, G.S. Sukhatme, A test bed for autonomous formation flying, Systems Technical Report IRIS-02-112, Institute for Robotics and Intelligent, 2002, p. 23. [12] P. Gurfil, M. Idan, N.J. Kasdin, Adaptive neural control of deep-space formation flying, in: American Control Conference, Anchorage, AK, USA, 8–12 May, 2002, pp. 2842–2847. [13] H. Wong, V. Kapila, Spacecraft formation flying near Sun–Earth L 2 Lagrange point: trajectory generation and adaptive output feedbackcontrol, in: American Control Conference, Portland, OR, USA, 8–10 June, 2005, pp. 2411–2418. [14] C. Kitts, J. How, Emerald: a low cost spacecraft mission for validating formation flying technologies, in: IEEE Aerospace Conference, Aspen, CO, USA, vol. 2, 6–13 March, 1999, pp. 217–226. [15] J. Leitner. A hardware-in-the-loop testbed for spacecraft formation flying applications, in: IEEE Aerospace Conference, Big Sky, Montana, USA, vol. 2, 10–17 March, 2001, pp. 615–620. [16] R.S. Smith, F.Y. Hadaegh, Parallel estimation and control architectures for deep-space formation flying spacecraft, in: IEEE Aerospace Conference, Budapest, Hungary, 4–11 March, 2006. [17] S. Mandutianu, F. Hadaegh, P. Elliot, Multi-agent system for formation flying missions, in: IEEE Aerospace Conference, Big Sky, Montana, USA, vol. 6, 10–17 March, 2001, pp. 2793–2802. [18] E. Gill, M. Steckling, P. Butz, Gemini: a milestone towards autonomous formation flying, in: ESA Workshop on On-Board Autonomy, ESTEC, Noordwijk, Holland, 17–19 October, 2001.
Constructing space-based continuous navigation and communication system utilizing formation flying satellites on GEO orbit in deep space exploration Hui Li∗ , Qinyu Zhang, Naitong Zhang Shenzhen Graduate School of Harbin Institute of Technology, PRC 518055, China Received 16 November 2007; accepted 4 March 2009 Available online 14 August 2009
Abstract Two clusters of formation flying satellites with a relative angle on geosynchronous Earth orbit were used to construct a spacebased continuous navigation system for deep space exploration. Adopting technique of passive navigation by multiple basestations, formation flying satellites obtained high precision of navigation according to time and frequency difference of arrival of downlink signal from deep space explorer. Enhancing system timing and carrier frequency, and increasing the length of baselines help to improve precision of navigation. The precision of this navigation system is close to that of NASA’s VLBI system. © 2009 Published by Elsevier Ltd. Keywords: Deep space exploration; Space-based; Continuous navigation; Formation flying satellite; Geostationary Earth orbit
1. Introduction In deep space exploration, it is difficulty to continuously navigate and communicate with explorer in all weather, and it is also difficult for one country to construct a global deep space network (DSN) terrestrially in its own land [1]. Generally, a single Earth-station is able to link explorer for 6–12 h, so continuous navigation and communication system depends on global deep space network as America and Russia [2]. However, DSN on ground is of lower coverage and is affected greatly by atmosphere. Space-based continuous navigation and communication (SBCNC) system may be a potential way of solving problems mentioned above.
∗ Corresponding author. Tel./fax: +86 451 8628 2013.
E-mail address: [email protected] (H. Li). 0094-5765/$ - see front matter © 2009 Published by Elsevier Ltd. doi:10.1016/j.actaastro.2009.03.072
Due to the huge distance in deep space exploration, requirement of precision of navigation is extremely high, and traditional way of satellite navigation and tracking cannot fulfill the need the ranging, velocity checking and angular measuring [3]. In 60–70 decades of last century, Earth-based radiometric tracking technique was generally adopted for navigation and tracking of probes during the cruise phase of a mission, and even during the approach phase [4]. In the 80 decades, very long baseline interferometry (VLBI) was adopted to achieve 20–30 nrad accuracy of navigation; however, it could not work in real-time. NASA’s continuous elements interferometry (CEI) technique is able to 80 nrad angular accuracy in real-time by two stations apart from 21 km and connected optical fibre [4,5]. Same beam interferometry (SBI) provides extremely accurate relative position measurements in the plane-of-thesky, complementing the line-of-sight information from
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H. Li et al. / Acta Astronautica 65 (2009) 1185 – 1189
Earth-based Doppler and range measurements. System errors that scale with angular and temporal separations are greatly reduced, allowing nearly the full precision of carrier-phase measurements to be utilized [4,6,7]. An elliptical inclined satellite and an Earth stations are to be utilized to construct a space-VLBI navigation system [8]. However, geostationary Earth orbit (GEO) satellite may be a better choice for simpler orbit and attitude control mechanism and longer baseline.
2. Space-based continuous navigation system Two GEO satellites with a minimum geocentric angle 34.634◦ can see and communicate with deep explorer continuously, avoiding the shade of Earth as shown in Fig. 1. Shelter 1 and 2 are the areas that cannot be seen by GEO satellite 1 and 2. However, this space-based system cannot navigate and position for the explorer. Two formation fly satellite (FFS) clusters located on GEO can solve this problem. A SBCNC system is proposed in Fig. 2, which utilizes two satellite formations with an angular spacing located on GEO orbit, and can navigate and communicate with the deep space explorers continuously. R S 1 and R S 2 are reference satellites in FFS1 and FFS2, and concomitant satellites combine Fig. 2. A SBCNC system on GEO orbit: (a) lateral view and (b) planform.
Fig. 1. Two GEO satellites with 34.634◦ angle receive signal from deep space explorer: (a) lateral view and (b) planform.
R S 1 and R S 2 satellites according to Hill equations (or called C–W equations). The shadow in Fig. 2(b) is the area that shades FFS1 by Earth. Satellite formation cooperates and accomplishes the navigation by way of passive allocation method. Many agencies and researching groups are carrying out the research of FFS in deep space and on near Earth orbit. The Cassini–Huygens mission to Saturn is the end of an era for NASA sending one large spacecraft equipped to carry out a multitude of scientific experiments [9]. Future NASA missions will deploy many smaller spacecrafts in highly controlled spatial configurations in what is referred to as “formation flying”. And both the Air Force and NASA have identified autonomous formation of spacecraft as key technological milestones for the 21st century [10]. The coming decade will see formation flying spacecraft (FFS) working simultaneously in deep space [11]. The deep space FFS missions include long-baseline mid-IR interferometry missions, X-ray interferometry missions and general relativity missions [12]. Moreover
H. Li et al. / Acta Astronautica 65 (2009) 1185 – 1189
1187
Table 1 Relationship of eccentricity of satellite formations and the minimum geocentric angle between them. E
0.001
0.005
0.01
0.05
0.1
0.2
0.3
0.4
l B (km) l1 (km) RS 1 O RS 2
84 41614 35.03◦
422 39550 35.96◦
843 37240 37.12◦
4216 25384 46.51◦
8433 18157 58.53◦
16866 11570 84.31◦
25298 8490 114.8◦
33731 6704 161.48◦
FFS has the potential to enhance space-based imaging and interferometry missions by distributing mission tasks as surveillance, SAR, magnetosphere sensing and a variety of other missions to many small spacecrafts [13,14]. Projects of space-FFS replacing signal large spacecraft for interferometry and surveillance certain celestial bodies may be positioned on halo orbit or Lagrange points in restricted three bodies systems as XEUS and Darwin missions [15–18]. According to Fig. 2(b), the minimal geocentric angle is calculated by
l1 + l2 = R E + h l1 RE = l2 lB
R S 1 O R S 2 = 2 + 2
(1) Fig. 3. TDOA positioning and navigation method.
(2) positioning (multiple transmitters and single receiver)
in which, l B = 2 · (R E + h) · e is the radius of FFS’s circular structure, and is also the baseline of this FFSSBCNC system. R E is the radius if Earth, h, is the altitude of GEO orbit, and e is the eccentricity ratio of concomitant satellites’ elliptical orbit around the Earth. And = arcsin(R E /l1 ), = arcsin(R E /(R E + h)). The relationship of eccentricity ratio of FFS and the minimum geocentric angle between two FFS is listed in Table 1. So the bigger size of FFS with longer baseline needs bigger geocentric angle between them for continuous navigation and communication.
⎧ 1 = d1 − d0 = C · (td1 − td0 ) = C · t10 ⎪ ⎪ ⎨ 2 = d2 − d0 = C · (td2 − td0 ) = C · t20 .. ⎪ ⎪ ⎩. n = dn − d0 = C · (tdn − td0 ) = C · tn0
(3)
in which td0 , td1 · · · tdn are the arrival time of signal from explorer to FFS component, and d0 , d1 · · · dn are the corresponding distance. t10 , t20 · · · tn0 are the difference of arrival time between concomitant satellites and reference satellite, and 1 , 2 · · · n are the corresponding distance difference. C is the speed of light. Precision curves of TDOA method are plotted in Fig. 3, when using this system in Mars exploration.
3. Methods of positioning and navigation 3.1. Time difference of arrival (TDOA)
3.2. Differential Doppler (DD)
Navigation and positioning by multi-spacecraft utilizing difference of signal’s arrival time simultaneously from position-known Earth stations or quasar in space is called time difference of arrival positioning as shown in Fig. 3. TDOA is also called “hyperbolic fix” in which the time of signal arrival from different receives is used to fix the location of transmitter as GPS-like active
Doppler shift due to relative movement between satellites and explorer is f i = f 0 + f di , in which f i is carrier frequency of NO. i is the satellite’s receiver, f 0 the transmitted carrier frequency, and f di Doppler shift: f di =
˙ · (Ri − R) 1 ( R˙ i − R) · Ri − R
(4)
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H. Li et al. / Acta Astronautica 65 (2009) 1185 – 1189
in which is the wavelength of carrier frequency. R˙ i , i =0, 1, . . . , n and R˙ are the velocity of satellites and explorer. Equations of DD method is expressed in Eq. (5), which can be solved by Newton iteration and other algorithms ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
f d1 − f d0 ˙ ˙ · (R1 − R) ˙ · (R0 − R) 1 ( R1 − R) ( R˙ 0 − R) = · − R1 − R R0 − R f d2 − f d0 ˙ ˙ · (R2 − R) ˙ · (R0 − R) 1 ( R2 − R) ( R˙ 0 − R) = · − R2 − R R0 − R ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ f dn − f d0 ⎪ ⎪ ⎪ ˙ · (Rn − R) ˙ · (R0 − R) ⎪ ( R˙ 0 − R) ⎩ = 1 · ( R˙ n − R) − Rn − R R0 − R
(5)
4. Simulation results and data analysis Two reference satellites of FFSs are located on GEO orbit 61◦ E and 120◦ E longitude with geocentric angle 59◦ . There are five concomitant satellites and one reference satellite in each formation as shown in Fig. 2(a). The Earth stations are located at Kashi (76◦ E, 39.5◦ N) and Qingdao (120.33◦ E, 36.07◦ N). The maximal distance between Mars and Earth is 4.013 × 108 km. is the covariance of error of time arrival. The explorer is at the second cosmic velocity. Both the error of time arrival and the frequency measuring obey the Gauss distribution with zero equalizing value. From Fig. 4, we can draw the conclusions that the precision of positioning and navigation increases with increase of the length of baseline and decrease of time measure and frequency measure. For Mars-distance exploration, 5000 m positioning and communication error is 12.46 nrad. 5. Conclusions and future works Two FFS located on the GEO orbit with a geocentric angle can be used to construct a space-based continuous navigation and communication system. FFS locates the position of explorer from one-way received signal in real-time. FFS adopted TDOA and DD methods of navigation and positioning to locate the position deep space explorer. Long baseline and small covariance of time-arrival and Doppler shift measure result in high precision of navigation. In this system, if a hybrid method of TDOA, DD and quasar difference is utilized, a higher accuracy of positioning and navigation is predictable.
Fig. 4. Mean error of positioning and navigation by TDOA in Mars exploration (upperbound of sat-position error 100 m): (a) Mean error of positioning navigation vs. covariance of error of time arrival (l B = 8433 km); (b) mean error of positioning navigation vs. radius of FFS/baseline ( = 1 × 10 − 13); (c) mean error of positioning navigation vs. covariance of error of frequency.
References [1] M. Lin, Trend of development of deep space tracking and communication systems, Journal of Spacecraft TT&C Technology 24 (3) (2005) 7–9.
H. Li et al. / Acta Astronautica 65 (2009) 1185 – 1189 [2] P. Li, J.S. Zhang, The current status and the future of NASA deep space network, Journal of Spacecraft TT&C Technology 22 (4) (2003) 10–18. [3] M.K. Reynolds, M.J. Reinhart, R.S. Bokulic, A two-way noncoherent ranging technique for deep space missions, in: IEEE Aerospace Conference Proceedings, vol. 3, Big Sky, MT, USA, 2002, pp. 1303–1312. [4] C.L. Thornton, J.S. Border, Radiometric Tracking Techniques for Deep Space Navigation, Wiley, NJ, USA, 2003, pp. 1–2. [5] J. Lauf, M. Calhoun, W. Diener, et al., Clocks and timing in the NASA deep space network, in: IEEE International Frequency Control Symposium and Exposition, 29–31 August, 2005, pp. 830–835. [6] R.S. Bokulic, J.R. Jensen, M.K. Flaherty, Transceiver-based spacecraft telecommunications systems with two-way Doppler tracking capability, Acta Astronautica 45 (4) (1999) 365–372. [7] T. Ichikawa, T. Kato, The study of delta VLBI and Doppler navigation for interplanetary spacecraft, Advances in Space Research 23 (4) (1999) 739–746. [8] Y. Asai, T. Nishimura, F. Kamijo, et al., Geometric effects on deep space orbit determination via space-VLBI observations, in: 41st SICE Annual Conference, Iwaki, Fukushima, Japan, vol. 1, 5–7 August, 2002, pp. 175–180. [9] O. Junge, S. Ober-Blobaum, Optimal reconfiguration of formation flying satellites, in: 44th IEEE Conference on Decision and Control, and the European Control Conference, Seville, Spain, 12–15 December, 2005, pp. 66–71. [10] K. Rahnamai, P. Arabshahi, A. Gray, Fuzzy supervised optimal regulator for spacecraft formation flying, in: 22nd International Conference of the NAFIPS, Chicago, IL, USA, 24–26 July, 2003, pp. 329–334.
1189
[11] D.J. Naffin, G.S. Sukhatme, A test bed for autonomous formation flying, Systems Technical Report IRIS-02-112, Institute for Robotics and Intelligent, 2002, p. 23. [12] P. Gurfil, M. Idan, N.J. Kasdin, Adaptive neural control of deep-space formation flying, in: American Control Conference, Anchorage, AK, USA, 8–12 May, 2002, pp. 2842–2847. [13] H. Wong, V. Kapila, Spacecraft formation flying near Sun–Earth L 2 Lagrange point: trajectory generation and adaptive output feedbackcontrol, in: American Control Conference, Portland, OR, USA, 8–10 June, 2005, pp. 2411–2418. [14] C. Kitts, J. How, Emerald: a low cost spacecraft mission for validating formation flying technologies, in: IEEE Aerospace Conference, Aspen, CO, USA, vol. 2, 6–13 March, 1999, pp. 217–226. [15] J. Leitner. A hardware-in-the-loop testbed for spacecraft formation flying applications, in: IEEE Aerospace Conference, Big Sky, Montana, USA, vol. 2, 10–17 March, 2001, pp. 615–620. [16] R.S. Smith, F.Y. Hadaegh, Parallel estimation and control architectures for deep-space formation flying spacecraft, in: IEEE Aerospace Conference, Budapest, Hungary, 4–11 March, 2006. [17] S. Mandutianu, F. Hadaegh, P. Elliot, Multi-agent system for formation flying missions, in: IEEE Aerospace Conference, Big Sky, Montana, USA, vol. 6, 10–17 March, 2001, pp. 2793–2802. [18] E. Gill, M. Steckling, P. Butz, Gemini: a milestone towards autonomous formation flying, in: ESA Workshop on On-Board Autonomy, ESTEC, Noordwijk, Holland, 17–19 October, 2001.