The Simplest Tether Control Law in a Small Satellite Yosuke Nakamura (Kyushu University) Designing of a small satellite based on the Satellite Design Contest is progressing in Kyushu University. This is a tethered satellite using double tether. This satellite s goal is to build up the technology of a tethered satellite in a very small system for the future uses such as in scientific investigations. A couple of tethered satellites have been launched by now, but these were rather complex and expensive. On the other hand, the satellite we are planning is a very small satellite which total mass weighs about 50 kg, and planned to be launched as a piggy-back satellite of the other satellite. Generally, a tethered satellite can use gravity gradient stabilizing system, and many applications are considered. At the same time, tether is made of so fine fiber deployed over hundreds of kilometers, that a tethered satellite has a weak point that tether could be cut easily by space debris. Therefore, by using double tether, even if one part of one side of the tether was cut, the other side of the tether remains. For this reason, the survivability of this system can be increased drastically. The dynamics of the tether is complex and is difficult to predict its behavior. Generally in deploying the tether, the satellite which is closer to the earth tends to move forward and another satellite backward. Further more in reeling up, tether swing libration increases as the tether length gets shorter because of the conservation of angular momentum. Many tether control laws have been studied to control these libration particularly considering stability during reeling up. Almost all of them were controlling tether libration by a feedback algorithm which uses tether length and tether velocity as state variables. Particularly in our mission, however, the satellite carries out only tether deployment, and retrieval is not considered. This enables us to reduce complexities of tether reeling system and its tether control law to very simple one. In this paper, any control laws which have been described in large and complex systems are not used. Instead, the control law in which only tether exit velocity and time are used as variables is presented. Because these variables can be known as definite values. In deployment, tether in-plane and out-of-plane swing angle decreases because of the conservation of angular momentum, unless tether is deployed too fast. So the tether swing angle need not be considered so seriously. One addition is that when tether deployment completed, the libration of tether length due to the force of repulsion remains. This libration can be decreased by using end dampers. The considered orbit is circular with altitude of 700 km. Main satellite weighs 35 kg and subsatellite weighs 15 kg. Tether is deployed from 0 m to 50000 m. Some simple ways to deploy tether are considered. The satellite is still under a preliminary stage now. So the way to deploy tether may be changed. But one choice is that: At first, tether exit velocity is described as a function proportional to time until tether exit velocity attains a certain value. After that, tether is deployed in a constant exit velocity. In the last few kilometers, tether exit velocity is controlled as a function of time and finally down to 0 m/s. Two models, one tether mass included and the other not, were considered in the control law. Both include elasticity of the tether. Satellites are initially given a small initial relative velocity. In this system, putting energy equations into Lagrange equation produces three-dimensional equations of motion. Numerical solutions will be performed for the equation. Details will be presented in the paper.
AAS 97-406 THE SIMPLEST TETHER CONTROL LAW IN A SMALL SATELLITE Yosuke Nakamura* Designing a micro tethered satellite based on a past work presented at the Satellite Design Contest is progressing in Kyushu University. The purpose of this activity is to build up the technology of a tethered satellite system in a very simple, small and economical way for future uses. In the mission, the satellite carries out tether deployment, but retrieval is not considered. This enables us to reduce complexities of tether reeling system and its tether control law to very simple one. In this paper, the system is briefly described and then the control law in which only tether exit velocity and time are used as variables is presented. Results of numerical simulations show an excellent behavior and the final libration amplitude of the satellite is controlled down to 0.05 rad. INTRODUCTION Designing a satellite based on a past work presented at the Satellite Design Contest is progressing in Kyushu University. This is a micro tethered satellite using double tether for life elongation, and planned to be launched as a piggy-back satellite of the other satellite. It weighs about 50 kg and the tether is deployed to a distance of 20 km. This satellite s goal is to build up the technology of a tethered satellite system in a very simple, small and economical way for the future uses such as in scientific investigations. Ideas on the possibilities of using tether in space structures have been discussed since 1960 s by several scientists. Since then, many applications of this technology have been studied, and are considered to play important roles in future space missions1. Some of applications are low altitude scientific missions, cargo transfer, rendezvous, orbit transfer, and generating electricity. On the other hand, actual applications to space structures started in 1980 s, and they are still in the experimental stage. The dynamics of the tether is complex and is difficult to predict its behavior. A tethered system will be stabilized along the local vertical. But, generally in during the deployment of the tether, the satellite that is closer to the earth tends to move forward and another satellite backward. Further more in reeling up, tether swing libration increases as the tether length gets shorter because of the conservation of angular momentum. Many tether control laws have been studied to control these librations particularly considering stability during reeling up. These control laws use a feedback algorithm which are used tether length, tether velocity, tether swing angle, and swing angular velocity as state variables2,3. In a small satellite system, these variables can not be measured. Further more, particularly in our mission, the satellite carries out only tether deployment, and retrieval is not considered. This enables us to reduce complexities of tether reeling system and its tether control law to very simple one. SYSTEM DESCRIPTION Students Micro Academic Satellite Project Developing micro satellites is progressing popularly in many universities and laboratories all over the world now. The reason is that a micro satellite has the advantage of short development period and low development cost, and this importance is recognized due to the successes of launches of UoSAT series which began in 1980 s and ARIANE piggy-back satellites in 1990 s. In these backgrounds, Students Micro Academic Satellite Project (SMAP) started in Kyushu University in January 1997. Kyushu University is participated in the Satellite Design Contest since 1995, and as a part of research * Department of Aeronautics and Astronautics, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-81, JAPAN
activities, students ideas have been developed to the designing a satellite. Different altitude observation of aurora using a tethered satellite is presented in the 4th Satellite Design Contest in 1996 and the studies have been progressed to the designing Double Tether Experimental Satellite (DTES). DTES aims to be launched as a H-IIA piggy-back satellite in 2000. There are three important points in DTES. The most important element is a tether control law. For DTES is a 50 kg class micro satellite, control laws in complex systems studied in the past are not applicable. Second important element is to demonstrate the availability of the double tether. This is a new way to extend the life of tethered satellite drastically. The tether is not a single element, but consists of parallel elements knotted at certain distance and avoids the risk that tether could be cut easily by a collision of space debris. Third important element is low cost development and fabrication of the satellite. By applying low cost and high reliable general-purpose electric parts for communication industry or public industry to the satellite, drastic cost down is expected. At the same time, those ground general-purpose electric parts employed in the satellite can be proven for space use. Hardware Description Table 1 shows summary of this satellite. The orbit of the satellite is selected as a sun synchronous circular orbit from the viewpoint of power requirement, and altitude as 700 km considering a distribution of debris. Main satellite loads CCD camera and observes the motion of the subsatellite. The subsatellite is initially arranged inside the main satellite, and deployed, as shown in figure 1 a), after the satellite is brought to the orbit. Table 1 DTES SUMMARY Orbit orbit sun synchronous circular orbit altitude 700 km orbit inclination 98.18 deg Power main satellite 20 W subsatellite 8 W Sensors main satellite rough sun sensor, CCD camera, magnetometer subsatellite magnetometer Size main satellite 500 * 500 * 500 mm subsatellite 150 * 150 * 200 mm Weight main satellite 35 kg subsatellite 15 kg a) convex tape b) Figure 1 DTES system : a) overview, b) convex tape Orbit Lifetime Satellite receives drag by atmosphere. After the deployment, mission starts till the satellite finally loses its altitude. Lifetime of DTES is estimated as 450 days from the view point of air drag, if tether length is 20 km. If tether length is 100 km, lifetime of the satellite gets shorter to 60 days. While a long tether makes the system lifetime short because of drag, another problem to be considered is that tether might be cut during its mission.
The lifetime of the satellite will be determined by collisions of debris referred later, rather than the effect of atmosphere. Deployment Procedure There were several choices of control laws of deployment. Feed-back control is a general way. But in a small satellite system, variables such as tether length or tether velocity are not measured easily. Furthermore, since the subsatellite is assumed to be used as observation platform and life span of the tethered satellite is so short because of debris, retrieval is not considered in the mission. So in this satellite, instead of existing control laws which have been described in large and complex systems, an open loop control in which only tether exit velocity versus time are used as variables is applied. These variables can be definite values determined during design phase. Two sequences of deployment are considered. One sequence shifts as acceleration-phase, constant-velocity-phase, deceleration-phase, and on-station-phase. The other shifts as constant-velocity-phase, deceleration-phase, and on-station-phase. To simplify the deployment, the latter is selected. Initial Stability Phase. At first, the main satellite and the subsatellite are deployed in a distance of 1 m using a convex tape, stem, or deployable truss mast. Convex tape deployment system is proposed in figure 1 b). In this phase, internal energy dissipation of the convex tape, stem or truss mast damps out satellite s motion. Constant Reeling Out Phase. After motion of the satellite damps, the subsatellite is given a initial velocity v 0 along the tether by a spring action. The tether exit velocity and the tether length are expressed as l 0 = v 0, (1) l 0 = v 0 t + 1. (2) Deceleration Phase. Tether exit velocity is decelerated when tether free length becomes a certain value l 1. When tether free length becomes final tether length l f, the tether exit velocity should be zero. The deceleration rate k is selected to meet this requirement. The tether exit velocity, the tether length, and the deceleration rate of the tether are expressed as l = - k t+ v 0, (3) 0 l 0 = -1/2k t 2 + v 0 t + 1 k = 2 0 ( l l1) 2 f v l, (4). (5) On Station Phase. After tether exit velocity becomes zero, tether reeling system is stopped and the satellite transfers to on station phase. Then tether exit velocity and the tether length are expressed as l 0 = 0, (6) l 0 = l f. (7) DOUBLE TETHER ANALYSIS Frequency of Collisions Frequency of collisions of debris to the tether is estimated using numerical analysis assuming that debris hit the tether straight from the side. Figure 2 shows debris flux as a function of size of debris. debris flux (/m 2 /year) 10 9 10 7 10 5 1000 10 0.01 0.1 1 10 size of debri (mm) Figure 2 Debris flux as a function of size of debris
Table 2 FREQUENCIES OF COLLISIONS Altitude Tether Diameter Tether Length Diameter of Debris Frequency of Collisions 750 km 0.1 mm 20 km 0.03 mm<diameter<10 mm 20 times/day Table 2 shows the summary of the analysis. Frequency of collision becomes so large, and the system survives only an hour or so. If tether diameter is 0.3 mm and only debris greater than a diameter of 0.1 mm are considered, frequency of collision can be decreased to seven times per day. Survivability of Double Tether Since almost all of debris are so small and two or more tethers are not cut by one debris at once, using double tether to build a redundant system is considered available. Survivability of double tether α is expressed as 2 n+ 1 q α = 1 n + 1, (8) where q is the count of the collision to the single tether in a certain period, and n is the number of knots. If (n + 1) > q, the life of double tether can be extended. Table 3 shows the result of calculations of survivability of the double tether in 10 day mission putting q = 20. Considering 90 % survivability, the number of knots should be more than 40000. Table 3 SURVIVABILITY OF THE DOUBLE TETHER IN 10-DAY MISSION THE SIMPLEST TETHER CONTROL Mathematical Model Number of knots Survivability (%) 1000 1.8 4000 36.8 40000 90.5 The system consists of two end bodies, a main satellite and a subsatellite, connected by a long tether. The orbit is circular with a radius R and a constant orbit rate ω. The orthogonal axis X, Y, and Z are defined as shown in figure 3, the X-axis along the orbital velocity vector, the Y-axis vertical to the orbit plane, the Z-axis along the local vertical. M is the mass of a satellite. The tether mass is not considered nor its flexural rigidity. The satellite s out-plane motion, in Y-axis direction, is assumed negligible. This can be realized when its initial out-plane motion is small. Main Satellite ω X Y Tether l R Z Sub Satellite Earth Figure 3 Schematic representation of the tethered satellite system
In this system, two dimensional equations of motion are produced by putting energy equations into Lagrange equation: 2 τ x x 2ωz + ω 1 x = H, M where ( ) i i i i ( )( ) i 2 z i + 2ω i + ω i 1 i =, i =1,2 (9) i z x z R H M τ i = R 3 {x 2 i + (z i - R) 2 } -3/2, (10) and H is an operator whose value is -1 if i = 1, 1 if i = 2. Tension τ is expressed as τx= EAξ + caξ ( x ) 2 x1, l τz = (EAξ + caξ ) ( z ) 2 z1, (11) l where E is tether elasticity, c is tether viscosity, A is a tether cross section, and tether stretch ξ is given by 0 ξ = l l, (12) l 0 where l is tether length and l 0 is tether free length. Selection of Constant Numbers with Numerical Analysis For each phase, numerical analyses are carried out to find out the dynamics of the satellite. The center of gravity of the satellite is assumed to follow a circular orbit with a radius of 700 km. The material of the tether is selected as Kevlar 49. Each values are set as follows: R =7078140 [m], ω = 0.00106 [rad/s], M 1 = 35 [kg], M 2 = 15 [kg], E = 5096 [Pa], c = 0.2 [Pa], A = 7.85 10-9 [m 2 ] (φ 0.1 [mm]). The main satellite and the subsatellite move symmetric with respect to the center of gravity of the satellite. To simplify a study, only the motion of the subsatellite is taken into consideration. Altitude of the satellite might be changed according to the orbit of the main payload. Further more, the confirmation of the stability in initial stabilization phase causes a error. So, each constant numbers must be selected with some flexibility. Constant Reeling out Phase. Numerical analysis is performed to determine the limit of initial velocity. Figure 4 Initial velocity as a function of initial angular velocity Figure 4 shows the minimum initial velocity which the subsatellite can be reeled out smoothly, assuming that the satellite is not fully stabilized in the initial stability phase. Under various initial angular velocities in a swing motion of the system around Y-axis, the stability of the satellite is judged changing its initial velocity. If initial velocity is too small, the tether is slacked and stability of the satellite is lost. The motion of the subsatellite is judged as unstable if its value of Z-axis gets smaller than its initial value. Since the subsatellite tends to move forward, when initial angular velocity is negative, initial velocity can be small. When initial angular velocity is positive, however, initial velocity must be large to some extent.
Left plot of figure 5 shows the motion of the satellites in the first 10000 seconds with different initial angular velocities. The satellites are stably deployed with initial angular velocities of 0.0 rad/s and -0.01 rad/s, but not stably deployed with initial angular velocities of 0.01 rad/s. Right plot of figure 5 shows tensions of the tether in the same situations. Maximum value of initial velocity is limited by the ability of the reeling system to 0.5 m/s. Initial velocity is finally selected as 0.5 m/s. Figure 5 Plots of motion and tension with different angular velocities l l Figure 6 Tension as a function of time Deceleration Phase. As l 1 gets closer to the final tether length l f, deployment completes earlier. But the amplitude of libration of tension gets larger. This is not desirable. Using numerical analysis, appropriate braking tether length is selected. The tension of the tether is shown in figure 6 as a function of time with two different l 1. Left plot shows the entire phase and right plot shows the final phase of deployment. The bold line shows the tension with l 1 =19500, and the narrow line with l 1 =19900. l 1 is finally selected as 19500 m. The maximum value of the tension is 0.75 N. This value fully satisfies the stretch strength of the tether. Motion of the Satellite Now that each constant numbers are selected, and numerical simulations are presented. The initial conditions and each constant numbers are set as follows: v 0 = 0.50 [m/s], l 1 = 19500 [m], l = 20000 [m], x 1 = 0.00 [m], ẋ 1 = 0.00 [m/s], z 1 = 0.30 [m], ż 1 = 0.15 [m/s], x 2 = 0.00 [m], ẋ 2 = 0.00 [m/s], z 2 = 0.70 [m], ż 2 = 0.35 [m/s]. f
Figure 7 The locus of the satellite Left plot of figure 7 shows the actual locus of the satellite. As shown in the figure, the satellite moves forward. This is caused by the change of orbit rate due to the huge scale of the system. Right plot of figure 6 shows the locus of the satellite relative to the center of gravity of the satellites to eliminate this effect of the change of orbit rate. From this plot, effective swing width can be seen as at most 1.0 km. This indicates that tether swing angle is 0.05 rad with the tether length of 20 km, and that this swing angle can be neglected in the final phase. In the plots, constant-reeling-out-phase starts at t = 0, deceleration-phase starts at t = 38998, and deployment completes at t = 40998 [s]. CONCLUSION The simplest tether control law in which only tether exit velocity and time are used as variables is presented. This control law is very flexible and can be applied to many simple space missions. Results of numerical simulations show an excellent behavior, and the final libration amplitude of the satellite is controlled down to 0.05 rad. This indicates that the tether control law can be very simple, if only deployment phase is considered. Deployment completes in 40998 seconds, that is 11.4 hours. REFERENCE 1. Ivan Bekey, Historical Evolution of Tethers in Space Tethers in Space, Vol. 62, pp. 27-34. 2. Charles C. Rupp, A Tether Tension Control Law for Tethered Subsatellites Deployed along Local Vertical NASA Technical Memorandum NASA-TM-X-64963 Nov. 1975. 3. Hironori Fujii and Shintaro Ishijima, Mission Function Control for Deployment and Retrieval of a Subsatellite J. Guidance, Vol. 12, No 2, March-April 1989 pp. 243-247.