Steered Spacecraft Deployment Using Interspacecraft Coulomb Forces

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1 Steered Spacecrat Deployment Using Interspacecrat Coulomb Forces Gordon G. Parker, Lyon B. King and Hanspeter Schaub Abstract Recent work has shown that Coulomb orces can be used to maintain ixed-shape ormations o spacecrat at high Earth altitudes with practical charging requirements. These ormations require a careul balance between the intercrat orces and the relative orbital dynamics. While nearly propellantless, the resulting dynamic equations are nonlinear, and result in a challenging control system design problem. This paper explores a dierent Coulomb orce application where a chie satellite deploys deputy crat to speciied end states. By using multiple charge suraces on the chie, a linear control design approach can be used while guaranteeing reachability or one deputy. This is extended to simultaneous, multiple deputy deployment by modulating control authority across the ormation. A 3 deputy example is presented to illustrate the approach where Debye length plasma shielding is considered. This example shows that meter diameter crat can be deployed to a 3 meter, circular projected ormation in geostationary orbit in approximately 2 orbits with a maximum o 2 kilovolts on the charge suraces. I. INTRODUCTION The ocus o this paper is high Earth orbit (HEO) deployment o relatively small deputy spacecrat using Coulomb orces generated between a chie satellite and the deputies. It is assumed that the chie crat has its own conventional propulsion system, and is in a circular orbit about Earth. Furthermore, the chie has several controllable spherical charge suraces. The deputy crat are assumed to be spherical, with their own controllable charge capability. Deployment consists o simultaneously repositioning the deputies rom an initial coniguration near the chie to a speciied endshape, typically multiple chie radii s away. In general, the inal speed states o the deputies could be nonzero. During deployment, the deputies react against the chie, exploiting the chie s station keeping capability to generate accurate motions. A unique eature o Coulomb propulsion is that it requires considerably less propellant than a conventional gas or plasma thruster. In addition, it does not produce contaminating plumes, nor impingement orces. It has also been shown that the power requirements are much less than conventional propulsion systems, even when modulating the charge at relatively high requency (s o Hz) []. There are several possible uses or Coulomb orce deployment. One application is the initial deployment o ormation lying spacecrat, with the typical goal being Earth imaging. Related to this is the ability to periodically correct ormation G. Parker and L. King are with the Department o Mechanical Engineering - Engineering Mechanics, Michigan Technological University, Houghton, MI 4993, USA ggparker@mtu.edu H. Schaub is with the Department o Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 249, USA hschaub@vt.edu Chie with 4 active charge spheres Deputy trajectory Deputy crat Fig.. Steerable Coulomb orce deployment scenario. The chie shown has 6 charge spheres, 4 o which are active. lying spacecrat orbits disturbed by external disturbances (e.g. J 2, aerodynamic, solar pressure orces). Deployable and retractable autonomous or semi-autonomous vehicle health monitoring devices is another application. Here the deputies would be used to monitor the chie, with the goal o diagnosing a perceived problem with the chie spacecrat. Finally, because the deputies can be given an end state, including nonzero speed, they could also be used as steerable projectiles. The use o Coulomb orces or propulsion is a relatively new ield. The earliest work examined symmetric, crystallike structures where one node, the chie, had a conventional propulsion system [4], [8], [9], []. This allowed the deputy crat to react against a body that was maintaining a constant orbit. The report in Reerence [] also showed, using data rom the SCATHA [] and ATS [6] missions, that µn- µn level orces could be generated between spacecrat. In addition, charging times were estimated to be in the millesecond range, and the Coulomb thrust propulsion system power consumption was lower than or conventional propulsion methods. More recent work has examined the necessary conditions or static equilibrium where all spacecrat are held together by Coulomb orces alone [3], [2]. Free-lying ormations were considered in detail or 2 and 3 spacecrat cases []. This was ollowed by consideration o larger ormations using a genetic algorithm optimization strategy to ind static shapes or up to 9 spacecrats [2] where the plasma shielding eect, characterized by the Debye length, was considered negligible. The remainder o this paper is as ollows. The chie and deputy spacecrat dynamic equations are developed in

2 Section II. The control strategy is presented in Section III starting with deputy, and then extending it to N deputies. Conditions or chie charge sphere locations are also discussed. Section IV illustrates the approach with an example where the chie has 4 charge spheres, and there are 3 participating deputies. A ew concluding remarks are provided in Section V, along with plans or uture work. II. DYNAMIC MODEL Beore examining the ormation dynamic equations, the Coulomb charge, voltage and orce relationships should be discussed. It s well known that Coulomb orces exist between charged bodies. Considering the two homogeneously charged spheres o Fig. 2 the Coulomb orce magnitude acting on body, directed along the line between the spheres is 2 = k cq q 2 d 2 () where k c is Coulomb s constant ( Nm2 C ), q 2 and q 2 are the sphere charges in Coulombs and d is the distance between the center o the spheres in meters. ˆk d 2 2 dynamic equations or the Coulomb controlled system. Consider a system consisting o N d deputy crat, and a chie containing N c charge points. The speciic case o 3 deputies and charge sphere is shown in Fig. 3. The position vectors o the N d + N c bodies are ordered such that p through p Nd are the deputy vectors and p Nd + through p Nd +N c are the charge sphere vectors, all relative to the center o the chie satellite. Each position vector has elements x i, y i and z i. Reerring to the example o Fig. 3, the 3 deputy crat position vectors are denoted p through p 3 and the chie s charge sphere position vector is denoted p 4. The chie is assumed to be in a circular orbit about the Earth with a Hill coordinate rame at its center as shown in Fig. 3. The î unit vector is pointing radially outward rom the center o the Earth and the ĵ axis is in the direction o the chie s velocity vector. Each o the N d + N c bodies is assumed to have charge q i. ĵ charge sphere p p 4 î 2 p 3 p 2 chie deputy 2 ĵ ˆk 3 î Fig. 2. Coulomb orces between two spheres having the same charge sign. Earth An important deviation rom the ideal Coulomb orce model o () occurs when considering charged bodies in a plasma-rich environment, as ound in space. In short, the plasma shields the Coulomb orce eect exponentially with the separation distance as shown in (2). The exponential decay o the Coulomb orce is characterized by the Debye length,, which varies with plasma characteristics [3]. In general, the Debye length is small at LEO and large at HEO (roughly cm and m respectively) 2 = k cq q 2 d 2 e d/. (2) When charged spacecrat are more than 2 Debye length apart, then their Coulomb orce interaction becomes negligible or realistic charge scenarios. Finally, it should be noted that the charges can be converted to voltages or the ith crat, which may give more practical insight, according to V i = q ik c r n (3) where r n is the radius o the spherical charge device. Next, Hill s equations [7] (also known as the Clohessy- Wiltshire equations, [5]) are used to generate the ormation Fig. 3. Three virtual structure nodes orbiting the Earth illustrating the notation used in the model development. Applying the Coulomb orces to the right side o Hills Equations, yields the system dynamic equations or small motions about the chie satellite and are shown in (4) m ( N ẍ i 2nẏ i 3n 2 ) d +N c x i = kc N d +N c m (ÿ i + 2nẋ i ) = k c m ( z N i + n 2 ) d +N c z i = kc x i x j y i y j z i z j or i =... N d, j i during the summation and d = p i p j. The Hill rame angular velocity is denoted as n and the mass o each deputy is m. Equation (4) is nonlinear and results in a challenging control problem i the goal is to simultaneously move the deputies rom initial positions near the chie to inal states away rom the chie. When all the (4)

3 deputies are charged, the q i q j terms limit the ability to create arbitrary orce vector directions. Furthermore, as the crat move away rom each other and the chie, the Debye length eect exponentially reduces the orce capability. It should be noted that the Debye length can be as little as cm at low altitudes. Thereore, the ability to generate Coulomb orces between crat at large distances is impractical. However, imparting large initial velocities is still quite easible. At high altitudes, such as geostationary orbit (35,8 km, n = rad/sec), varies rom 75 m to 575 m. Thus, long-distance positioning capability is possible. III. CONTROL STRATEGY The movement o one deputy is considered irst, ollowed by the extension to multiple deputies. For the single deputy case, N d =, (4) becomes m ( +N ẍ 2nẏ 3n 2 ) c x = kc +N c m (ÿ + 2nẋ ) = k c m ( z +N + n 2 ) c z = kc x x j y y j z z j where d = p p j. Assume that the charge o the deputy, q, is held to some nonzero, constant value and that N d = 3. Next, denote the right sides o each equation in (5), speciiable orces, using the variables d,x, d,y, and d,z, or in vector orm, d. As long as the rank o the matrix P = [ (p p 2 ) (p p 3 )... (p p +Nd ) ] (6) is 3, then any desired orce, d, can be applied to the deputy. Because it is possible to generate any orce vector, it is clear that a suitable control law can now be applied depending on the application (e.g. tracking, regulation, etc.). For example, a simple proportional controller with rate eedback and orbital dynamics cancellation is suitable or moving the deputy rom some initial point to a inal point with zero inal speed 2nẏ 3n 2 x d = 2nẋ n 2 + K p (p d p ) K d v (7) z where K p and K d are 3 3 diagonal gain matrices, p d is the desired inal position o the deputy, and v is the velocity o the deputy. At this point, it is convenient to rewrite the right side o (5) as d = Bq where the j th column o the 3 N c matrix B is B j = k c (p p j ) d e d 3 and the N d vector o inputs, q, is q = q 2 q 3. q (+Nd ) (5) q (8). (9) Some comments on the number o chie charge spheres is in order. I N c = 3, the charge spheres orm a plane and deines a singular region where orces orthogonal to the plane cannot be achieved. Thus, a minimum o 4 charge spheres is needed to guarantee complete maneuverability o the deputy. This introduces redundancy in the control law solution. This could be handled by switching between sets o 3 charge spheres, or by utilizing all the charge sphere assets according to some optimization criteria. In this work, a weighted least squares solution is used. Speciically, given the desired orce vector to be applied to the debuty, d, the charge sphere charge values are computed according to q = W B T ( BW A T ) d () where W is an N c N c matrix o constants used to ocus control authority onto speciic charge spheres. The identity matrix is used in the example below. Extending this method to multiple deputies can be accomplished in several ways. One approach is to move each deputy to its inal state sequentially. In some applications, this may be appropriate. However, i the goal is to have all the deputies be at their goal points at the same time, this would not be suitable. The orbital dynamics quickly result in a loss o positioning accuracy once the Coulomb orce is removed and the deputies move according to their own Keplarian motion. The method used in this work is to ocus control sequentially between the participating deputies. For example, the ormation control period, denoted T, would be divided into N d equal pieces as δt = T/N d. During each δt only deputy is charged, and the solution to () is ormed with a B matrix where the p o (8) is replaced with i or the i th deputy. It should be noted that rom a practical perspective, this assumes that: () the deputy charges can be cycled rom zero to the maximum value at least times aster than δt, (2) the chie and deputies have a communication protocol that allows the chie to designate which deputy is under control, (3) the chie can measure the deputy position vectors. IV. EXAMPLE Three deputies are considered, N d = 3, and 6 charge spheres, N c = 6. The goal is to move the deputies rom their initial locations, near the chie, to a inal zero speed state such that they lie on a circle, as viewed rom Earth, with a radius o 3 meters. The deputies and charge sphere radii are all.5 meters, each with a mass o 5 kg. The initial and inal coordinates o the deputies are given in Table I along with the coordinates o the charge spheres where L i = 3/ 2, L = 3, and L c = 2. The Debye length is assumed to be meters, and the altitude o the chie is 35,8 km (the orbital period is 24 hours). The time that each deputy is being controlled by the chie s charge spheres is δt = 5 minutes and the piece-wise constant charge o the deputies is 2, volts. Fig. 4 shows the trajectories o all three deputies - blue or p, green or p 2 and red or p 3. The hollow circles indicate the initial positions and the illed circles the end

4 TABLE I INITIAL AND FINAL COORDINATES OF THE DEPUTY CRAFT, AND THE COORDINATES OF THE CHARGE SPHERES. Position Vector x (m) y (m) z (m) Initial p L i L i Initial p 2 L i L i Initial p 3 L i L i Final p 2 L 2 L 3 Final p 2 L 3 Final p 3 2 L 2 L 3 p 4 L c p 5 L c p 6 L c p 7 L c p 8 L c p 9 L c *&'() # " # " # " $# " # " # " # " %&'() Fig. 5. Trajectories o all three deputies projected onto the y z plane. Hollow circles are the initial positions, and the illed circles the inal positions. points. The charge spheres are not shown. The only care taken in selecting the initial and end points was to inhibit signiicant x-axis motion. This was done merely to create an example that could be viewed easily on a two-dimensional plot. As seen in Fig. 4 the motion in the x direction is small compared to the y and z motion. $ % Deputy Pos. (m) Deputy 2 Pos. (m) Deputy 3 Pos. (m) Time (hr),'()* % $ $% Fig. 6. Time histories o all three deputies where in each plot the dotted line is the x coordinate, the solid line is the y coordinate, and the dashed line is the z coordinate. # # $ +'()* $ # Fig. 4. Trajectories o all three deputies. The hollow circles are the initial coniguration, and the illed circles the inal positions. Fig. 5 shows the charge spheres, black circles, and the trajectories o the deputy spacecrat. Again, blue denotes p, green denotes p 2, and red denotes p 3. The hollow black sphere at the origin indicates the location o the p 4 and p 5 charge spheres and are outside the y z plane. As expected rom the control strategy, the motion o the deputies is in a straight line to their respective goals since the desired orce vector, d is always pointed rom the deputy toward its goal. This eature may be useul in the uture or path planning. Fig. 6 shows the x, y, and z coordinates or each deputy. The x motion is a dotted line, the y motion a solid line, and the z motion a dashed line in all cases. From this view it is clear that the deputies achieve their inal end points in approximately 2 orbits (2 days). Voltage time histories are shown in Fig. 7 and Fig. 8. These were computed rom the charges, q i, using (3). It s clear that the maximum voltage required is approximately 2 kilovolts. It should be noted that since the current requirements are "# &'()* "# extremely small, this requires very little power compared to conventional thrusters. The oscillatory nature o these time histories is due to the switching, every 5 minutes, o the control objective to a dierent deputy. This is seen more clearly in the zoomed view o the p 4 charge sphere shown in Fig. 9. It should be noted that the steady-state part o Fig. 7 and Fig. 8, ater about 5 hours, is keeping the deputies at the desired ixed positions, constantly working to overcome the orbital orces. V. CONCLUSIONS AND FUTURE WORK A. Conclusions A method was developed or Coulomb control positioning o deputy spacecrat using charged spheres attached to a chie satellite. By ocusing on one deputy at a time, a linear control solution was applied to a single deputy. This was extended to simultaneous movement o all the deputies by cycling the charge between deputies. The requirement or the existence o a control solution was given in (6). In practice one should plan the trajectories such that the deputies do not need to cross through the volume bounded by the charge spheres. This is not a very restrictive constraint, and means that the deputies should be moving outward rom the chie.

5 V 4 V 5 V 6 /23'45. Fig. 7. Charge sphere voltage time histories or spheres 4,5, and 6. V 7 V 8 V 9 /23'45. Fig. 8. Charge sphere voltage time histories or spheres 7,8, and 9. ( )* )( * V 4 ( +,-./2 * )( )* B. Future Work There are numerous areas or urther study in the general area o Coulomb controlled ormations. Related more speciically to the method described in this paper, certainly stability o the modulated control solutions needs to be examined. The ability to generate desired trajectories, not just end points, that exploit the orbital dynamics would also be helpul. Coulomb orce attitude control is another area that could beneit rom this approach. VI. ACKNOWLEDGMENTS This work was supported by the U.S. Deense Advance Research Projects Agency (DARPA), Special Projects Oice (SPO) under contract number HR-5-C-26. Approved or public release with unlimited distribution. REFERENCES [] J. Berryman and H. Schaub, Analytical charge analysis or 2- and 3-crat coulomb ormations, in AAS/AIAA Astrodynamics Specialist Conerence, Lake Tahoe, CA, August 7 25, paper No. AAS [2], Static equilibrium conigurations in geo coulomb spacecrat ormations, in AAS Spacelight Mechanics Meeting, Copper Mountain, CO, Jan , paper No. AAS 5 4. [3] F. F. Chen, Plasma Physics and Controlled Fusion Volume : Plasma Physics. Plenum Press, 984, ch.. [4] J.-H. Chong, Dynamic behavior o spacecrat ormation lying using coulomb orces, Master s thesis, Michigan Technological University, 22. [5] W. H. Clohessy and R. S. Wiltshire, Terminal guidance system or satellite rendezvous, Journal o the Aerospace Sciences, vol. 27, no. 9, pp , September 96. [6] H. B. Garrett and S. E. DeFrost, An analytical simulation o the geosynchronous plasma environment, Planetary Space Science, vol. 27, pp. 9, 979. [7] G. W. Hill, Researches in the lunar theory, American Journal o Mathematics, vol., no., pp. 5 26, 878. [8] L. B. King, G. G. Parker, and J.-H. Chong, Coulomb controlled spacecrat ormation lying, AIAA Journal o Propulsion and Power, vol. 9, no. 3, pp , 23. [9] L. B. King, G. G. Parker, S. Deshmukh, and J.-H. Chong, A study o inter-spacecrat coulomb orces and implications or ormation lying, in 38th AIAA/ASME/SAE/ASEE Joint Propulsion Conerence and Exhibit. Indianapolis, IN: AIAA Paper No , July 22. [], Spacecrat ormation-lying using inter-vehicle coulomb orces, NASA Institute or Advanced Concepts, Tech. Rep., 22. [Online]. Available: [] E. G. Mullen, M. S. Gussenhoven, and D. A. Hardy, Scatha survey o high-voltage spacecrat charging in sunlight, Journal o the Geophysical Sciences, vol. 9, pp. 74 9, 986. [2] H. Schaub, C. Hall, and J. Berryman, Necessary conditions or circularly-restricted static coulomb ormations, in AAS Malcolm D. Shuster Astronautics Symposium, Bualo, NY, June , paper No. AAS [3] H. Schaub, G. G. Parker, and L. B. King, Challenges and prospects o coulomb satellite ormation lying, in AAS John L. Junkins Astrodynamics Symposium. College Station, TX: AAS Paper No. AAS3-278, May 23. ( "#$ "#% % %# %#& %#$ %#% ' Fig. 9. Zoomed view o charge sphere 4 showing the control authority switching.