Study of Low-cost Orbit Determination System for Tethered Satellites

The statistics of estimated errors when conducting the orbit determination of a tethered satellite system (TSS) by only using Doppler frequencies is shown. TSS consists of two satellites connected each other by a tether. The center of mass (CM) moves like a single satellite, so this system can be utilized in case of a single satellite as well as TSS. This is considered as a low-cost orbit determination system because it can be realized with simple instruments. This kind of low-cost ground station will charm institutions that are developing a micro satellite in a small budget. The result will provide a useful reference in constructing a ground station for the orbit determination by only Doppler frequencies. INTRODUCTION Satellites were being developed by specific special institutions until some years ago. However, development of micro satellites with low-cost concept has been recently increasing, which is developed by institutions such as universities1. These kinds of non-commercial satellites are very effective in high-risk missions and examinations of brand-new devices because the cost is not much expensive compared to the conventional ones. In addition, a tethered satellite system (TSS) which consists of two satellites connected each other by a tether is an interesting topic in recent space systems, which applications cover a wide area2. In the category of micro satellites, TiPS (the Tether Physics and Survivability) was launched by the Naval Research Laboratory’s (NRL’s) Naval Center for Space Technology, and the missions were successful3. However, TSS has not been achieved to practical use and more experiments in orbit are necessary. Consequently, the experimental missions using low-cost micro tethered satellites will be expected to increase in future. After the low-cost micro satellite is launched, however, the operation is conducted in a large-scale ground station under present conditions. It will become a burden if observations of satellites are all entrusted to such limited institutions. Therefore, if there is another orbit determination system that can be easily utilized in simple instruments, institutions that developed satellites will be able to determine the orbit by its own facilities as well as manage commands and telemetries. Large-scale ground stations use high precision equipment and have the ability to determine satellites’ orbits in very high precision. The measurement sources are range, which can be obtained by such as SLR (Satellite Laser Ranging)4, range rate, azimuth, elevation, and so on. The objective of this study is to show the statistics of estimated error when conducting the orbit determination of TSS only by Doppler frequencies. This is considered as a low-cost orbit determination system because it can be realized with simple instruments. The required equipment are amateur radio antennas, direction control motors, a frequency analyzer (spectrum analyzer) and a The Journal of Space Technology and Science, Vol. 16, No. 1, April, 2002, 10 pages 2 personal computer to control these devices. In addition, the center of mass (CM) of TSS moves like a single satellite and error of CM is estimated, so this system can be utilized in case of a single satellite as well as TSS. About orbit determination of TSS, Ref. 5 and 6 are interesting and helpful to understand this study topic. The Doppler frequency can be converted into a range rate, which is a change rate of distance between a ground station and a satellite. The orbit determination by only Doppler frequencies has lower precision than the case of additionally using a range and angles. Although it may not be utilized in some missions which require the orbit determination in high precision, the system will be very useful when only the knowledge on future passes are necessary to send commands or receive telemetries. Although there are also very low-cost ways such as GPS (Global Positioning System), the service is not guaranteed to continue permanently and some law problems for the space use. Moreover, if similar facilities are placed in around the world, observed data in the same time span will increase and estimated error can be reduced. This supplements the demerit that a single ground station can only observe a partial orbital arc. Fortunately, instant data exchange is available by using the high-speed and low-cost Internet network. This kind of low-cost ground station will charm institutions that are developing a micro satellite in a small budget. The result will provide a useful reference in constructing a ground station for the orbit determination by only Doppler frequencies. An orbit determination is that an estimated orbital state at epoch time is improved by extra observed data. The non-linear sequential batch least squares method4 is used and a somewhat precise orbit generator is necessary. The orbital state used in this paper includes the position and velocity of a center of mass, tether length, a mass ratio of main and sub satellites, angles and angular velocity of swing motion. The observed data are only Doppler frequencies. In this paper, it is shown how much error of the state at epoch is estimated in case of being given specific error covariance of measured data. At first, numerical simulations are conducted at an example orbit in some parameters. The theoretical background is concretely explained in Ref. 11, which includes a simple orbit generator of TSS model, transformation from state space to observed data, and transformation of estimated error covariance matrices. Ref. 7-9 are helpful to understand basic theories and numerical methods of orbital mechanics. Secondly, the precision of commercial frequency analyzer is experimentally measured by using an amateur radio station, and the effectiveness of the system is estimated. Kyushu University’s TSS Project In recent years, there have been put particular attention on developing micro-satellite systems weighing several dozen kilograms because of low cost involved in launching and operating such satellites. Considering these latest development and challenges involved, Kyushu University has undertaken the work of developing tether system using micro-satellites. The design and fabrication are conducted under the leadership of a university-level single laboratory. The orbit determination by using only Doppler frequencies is one of the missions. For reference, missions and concepts of this project are explained as private background. The TSS mission is to be conducted in the following sequence: 1. As TSS is launched into the orbit, a boom is first deployed in order to achieve attitude stabilization of the system. 2. After having stable system, a sub satellite is deployed by using tether. The release of tether is carried out such that the velocity of sub satellite is controlled toward smooth arrival at the end of tether release and the system is still in stable motion. 3. The mission ends after the tether is cut by space debris. With a view to reduce the system cost and increase its reliability, the following systems have The Journal of Space Technology and Science, Vol. 16, No. 1, April, 2002, 10 pages 3 been considered: • Tether Deployment Control System: The simple open loop tether deployment system is used in order to reduce complexity. The retrieval of tether is not considered. • Boom System for Passive Attitude Control: Instead of attitude control system using thrusters or reaction wheels, a passive attitude control using tether and boom deployment has been considered. • Low-cost Orbit Determination System: A sequential recording of an orbital state is necessary to measure the dynamics of TSS. It may be imagined that a commercial Global Positioning System is easily utilized, however, it has a protection to restrict a use for military and does not work in over some altitude. In addition, we will need to overcome some legal and cost difficulties when using GPS system. Therefore, a substitute low-cost orbital state observation system is necessary. The orbit determination system by using only Doppler frequencies is shown in this paper. Moreover, information of on-board sensors will be helpful to improve the precision. About this project, Ref. 1 will be helpful. The image of TSS is shown in Fig. 1. NUMERICAL SIMULATIONS Procedure The objective of simulations is that standard deviations of a TSS orbital state are estimated when the orbit determination using observed data of only Doppler frequencies is conducted. A TSS orbital state is defined as follows. 2 1 / T TS i a e M m m l ω θ φ θ φ = Ω X (1) , , , , , i a e M ω Ω are Keplerian orbital elements of a center of mass. θ is an in-plane motion, φ is an out-of-plane motion, 2 1 / m m is a mass ratio, and l is a tether length. All variables are depicted in Fig. 2. In this dynamics model, following conditions are assumed. • Both satellites are particles and a mass of tether is neglected Orbit Orbit: Circular orbit Altitude: 400 ~1200 km (LEO) Size 500×500×500 mm3 Weight 50 kg Power 30 W Life time 6 months Table 1 Mission Parameters Subsatellite Marmonband Separation System (Recessed) Lower bulkhead adaptable for H-IIA marmon band Kevlar Tether Tether Reeling Mechanis m Main satellite Satellite Configuration Figure 1 Kyushu University’s TSS The Journal of Space Technology and Science, Vol. 16, No. 1, April, 2002, 10 pages 4 • A tension force in tether is attracted toward a center of mass from each satellite and force is constant • No perturbation forces, and a length of tether is sufficiently small against an orbital radius • Constant tether length and no bending The array of observed data is defined as follows. [ ] 1 2 T n ρ ρ ρ = Z (2) 1 2 , , , n ρ ρ ρ are range rates which can be easily transformed from Doppler frequencies. The each data is one of either satellite of TSS. The specific error standard deviation of observed Doppler frequencies can be assumed like 1 2 , , , f f fn σ σ σ . The each deviation fi σ is corresponds to one of a ground station that observe i ρ , which is defined by operators of simulation. They are transformed into 1 2 , , , n ρ ρ ρ σ σ σ , and these standard deviations give a covariance matrix as follows.