Reconfiguration Of 3-Craft Coulomb Formation Based on. Patched-Conic Section Trajectories

Transcription

1 Reconfiguration Of 3-Craft Coulomb Formation Based on Patched-Conic Section Trajectories Yinan Xu, and Shuquan Wang ABSTRACT This paper investigates the non-equilibrium fixed-shape three-craft Coulomb formation reconfiguration problem. Being aware of that using the feedback control approach might result in the chattering of the charges due to the non-equilibrium nature of the system dynamics, this paper proposes a trajectory planning approach to accomplish the reconfiguration. The entire maneuver trajectories are divided into multiple phases. During each phase, only two of the three craft are charged. In this way the relative trajectory of the charged spacecraft during a certain phase is a conic section. The entire trajectories are composed of patched conics and/or straight lines. The procedures determining the three-phase maneuver strategy is developed, including a preadjusting phase and two transition phases. Numerical simulations demonstrate the effectiveness of the algorithm and the elegance of the charge histories. INTRODUCTION The concept of Coulomb formation flying (CFF) was introduced by Lyon B. King in 001. [1] CFF utilizes the inter-spacecraft electrostatic force/forces to control the relative motion of the formation. The electric field is generated onboard by ejecting ions of a certain sign of charge using the ion engine. CFF has the following major advantages: 1. power efficient, it requires power level of only 10 3 Watt;. clean, it does not generate the plume impingement which may cause damages to the nearby spacecraft; 3. essentially propellant-less, which may enhance the life circle of the spacecraft. On the other side, CFF poses challenges to the electromagnetic compatibility design of the spacecraft. Theoretically, CFF is suitable for the long-period, close-proximity space mission in high earth orbit or deep space. An important character of CFF is that the Coulomb forces lie only along the line-of-sight directions between spacecraft. Another challenge of CFF is that the space plasma environment shields the strength of the electrostatic force field. This effect will reduce the amount of electrostatic force acting on the neighboring charged spacecraft. The amount of shielding is characterized by the Debye length. [] At separation distances greater than a Debye length, the intercraft Coulomb forces quickly go to zeros with exponential term. The Debye length is on the order of centimeters Graduate student, Mechanical Engineering Department, University of Michigan (Ann Arbor). Associate research fellow, Key Lab of Space Utilization, Technology and Engineering Center for Space Utilization, Chinese Academy of Sciences. shuquan.wang@csu.ac.cn. 1

2 at low Earth orbits, in which the plasma is relatively dense and cold. This results make Coulomb thrusting negligible. However, at geosynchronous Earth orbits (GEO), the Debye lengths range between m. [1,3] At 1 AU in deep space the Debye length ranges around 0-50 m. [1,3] This makes the Coulomb thrusting concept feasible for high Earth orbits and deep-space missions when the minimum separation distances are less than 100 m. Various mission scenarios of the CFF concept have been investigated. Reference 4 develops a formation feedback control strategy to achieve the virtual structure control. This control strategy is based on the thrusters capability to control the three-dimensional motions of the satellites. Schaub and Hussein analyze the stability of a spinning two-craft Coulomb tether and show that if the Debye length larger than the separation distance, then the nonlinear spinning motion is locally stable; otherwise, the motion is unstable. The perturbed out-of-plane motion is always stable, regardless of Debye length. [5] The nonlinear dynamics and closed-loop control are developed for the reconfiguration of a twosatellite Coulomb tether virtual structure near libration points, which is both valid for the robust reconfiguration and station-keeping mission [6]. Reference 7 designs a two-stage charge feedback-control strategy for a 1-D constrained Coulomb structure and analyze the condition for symmetric relative motions of Coulomb structure to be stabilizable by investigating the total energy of the system. Hussein and Schaub derive the collinear three-craft spinning family of solutions. Feedback control based on the linearized model is designed to stabilize a collinear virtual Coulomb tether system. Asymptotic stability is achieved if the system s angular momentum is equivalent to the estimated/nominal angular momentum, which is used to calculate the nominal charges. [8] Reference 9 investigates invariant shape conditions for a three-craft collinear formation and proves that for any collinear formations there always exists an infinite set of real open-loop charges for equilibrium. Reference 10 explores the solution families for collinear three-craft CFF and distinct regions where invariant shape solutions exist are analyzed to determine what range of trajectories are possible. Considering the challenges in maintaining and maneuvering inherently unstable formations, Reference 11 demonstrates in-plane perturbations can be asymptotically stabilized for the radially aligned configurations. Reference 1 extends the analysis to the out-of-plane motion and develops a simple control law which is successful to eliminate a wide variety of out-of-plane perturbations. In Reference 13, an innovative hybrid propulsion strategy is developed combining Coulomb forces and conventional electric/ion thrusters for close-proximity formation flying. Vasavada and Schaub present analytical tools to determine the charge solution for a static four-craft formation. [14] This paper focuses on the non-equilibrium reconfiguration problem of the three-craft Coulomb formation. This problem is challenging because controlling both three sides simultaneously is not applicable. Previous research on this problem uses feedback control approach. [15] Reference 15 develops a stable switched control strategy based on three Lyapunov-like functions. It achieves the stability during the reconfiguration process. But this approach results in the chattering of the charges due to the switched strategy. The chattering may cause tremendous increasement of the power consumption and might even damage the ion engine. Inspired by Reference 16 which develops a patched conic-section trajectories of a -body Coulomb Formation

3 Flying (CFF), the reconfiguration of the three-craft CFF could be achieved in a much smoother manner if the dynamical properties of the system were fully utilized. This paper proposes to develop a trajectory programming approach to accomplish the reconfiguration. The entire trajectories are divided into multiple phases. During each phase, only two of the three craft are charged such that these two craft possess the properties of a two-body-problem. In this way the trajectory of each spacecraft are composed of conic sections and/or straight lines. This is the fundamental idea of this paper. In developing the maneuver trajectories, many specific problems need to be solved, such as the number of phases necessary for the reconfiguration problem, the relationship between the individual spacecraft and the reconfiguration mission, how to choose the two spacecraft to be charged and so on. These problems are to be answered in the following sections. EQUATIONS OF MOTION The objective of the reconfiguration problem is to make the free-flying 3-craft formation formulate the prescribed configuration. The prescribed configuration is defined by the three separation distances, [l1, l3, l13]. Figure 1 shows the scenario of this three-body Coulomb virtual structure. m,q SC- r 1 r 3 m 1,q 1 SC-1 R 1 R r 13 m 3,q 3 SC-3 R 3 Inertial Frame Figure 1. Scenario of the 3-craft Coulomb formation in free space. Reference 17 investigates the modeling of Coulomb force in Earth orbit and/or in the plasma environment. In the plasma environment, assuming that the ith spacecraft is charged with a finite sphere of radius R i, the Coulomb force between this charged sphere and a point charge q j is [17] F ij = V ir i q j r 3 ij Å 1 + r ij λ D ã Å exp r ij R i λ D ã r ij (1) where V i is the voltage of the ith spacecraft, r ij is the distance between the center of mass of ith spacecraft and the point charge. The parameter λ D is the Debye length which characterizes the strength of the plasma shielding effect. It is influenced by the temperature and the ion/ electron density. For high Earth orbits, the Debye length ranges between 3

4 100 and 1000 m [1,18]. In deep space at 1 AU (astronomical unit) distance from the sun, the Debye length can vary between 30 and 50 m. CFF typically has spacecraft separation distances less than 100 m. Generally it is assumed that the Coulomb thrusting is applicable only when the separation distance is less than the local Debye length. In developing the maneuver strategy, this paper uses the point charge model for both spacecraft and assumes that λ D =, such that a much more concise expression of the Coulomb force can be obtained. In this way Eq.(1) is simplified to be: q i q j F ij = k c rij 3 r ij () where k c = Nm /C is the Coulomb s constant, q i is the charge of the ith spacecraft. By the assumption that there are no external forces acting on the three-body system, the inertial equations of motion (EoMs) are given by m 1 R 1 = k cq 1 q r 3 1 m R = k cq 1 q r 3 1 m 3 R 3 = k cq 1 q 3 r 3 13 r 1 k cq 1 q 3 r13 3 r 13 r 1 k cq q 3 r3 3 r 3 r 13 + k cq q 3 r3 3 r 3 (3a) (3b) (3c) where m i is the mass of the ith spacecraft, R i is the position vector of the ith spacecraft in the inertial frame. SINGLE-PHASE MANEUVER TRAJECTORY OF SC1 & SC3 SUBSYSTEM This section investigates the single-phase trajectory for the reconfiguration mission. The three-craft system are separated into two groups: a subsystem of two charged spacecraft; an uncharged spacecraft. Without loss of generality, SC-1 and SC-3 are chosen as the charged spacecraft. The relative trajectory of the SC-1 & SC-3 subsystem is a conic section. SC- is stationary or moving in a straight line. Substituting q = 0 into Eq. (3), the equations of motions reduce to m 1 R 1 = k cq 1 q 3 r13 3 r 13 m R =0 m 3 R 3 = k cq 1 q 3 r13 3 r 13 (4a) (4b) (4c) It is still complicated to find the connection between the motions of the individual spacecraft and the desired triangular shape. Note that there are no external forces acting on this system, the center of mass (CM) of the SC-1 & SC-3 subsystem has a simple type of motion. The investigation initiates by studying the motion of the CM of the subsystem. Figure shows the scenario of this case as seen from SC-. The CM of the subsystem is denoted by the point A and 4

5 the final position of CM is denoted by A. SC-3 r 13 (0) l 13 l 3 A A r A l 1 SC-1 SC- Figure. CM motion of the SC-1 & SC-3 subsystem viewed from SC-. Subsystem s Center of Mass Motion Requirements Because the point A is the CM of the SC-1 & SC-3 subsystem, the states of the point A satisfy: R A = m 1R 1 + m 3 R 3 m 1 + m 3 (5) Ṙ A = m 1Ṙ1 + m 3 Ṙ 3 m 1 + m 3 (6) Eqs. (5) and (6) are valid at any time during this phase. The relative position vector pointing from SC-1 to SC-3 is given by r 13 = R 3 R 1 (7) Utilizing Eqs. (5) and (7), R 1 and R 3 are expressed in terms of R A and r 13 as: m 1 R 1 = R A r 13 m 1 + m 3 (8) m 3 R 3 = R A + r 13 m 1 + m 3 (9) Because there are no external forces acting on the SC-1 & SC-3 subsystem, Ṙ A remains constant. The location of the point A at time t is given by: R A (t) = R A (0) + ṘA(0) t (10) where R A (0) and ṘA(0) are the initial position and velocity vector of the point A. Similarly, the position vector of 5

6 SC-3 l 13 l 3 A l A SC-1 l 1 SC- Figure 3. The desired triangle. SC- in inertial frame is given by: R (t) = R (0) + Ṙ(0) t (11) Now the connection between the motion of the point A and the desired triangular configuration is ready to be investigated. Figure 3 illustrates the desired triangle. Note that for the desired triangle, the three side lengths [l 1, l 3, l 13] are preset. The point A lies on the line segment S 1 S 3. The distance from SC-1 to the point A is r 1A = m 3 m 1 + m 3 l 13 (1) Applying the law of cosines to the triangle S 1 AS, the distance between the point A and SC- is obtained: r A =» r 1A + l 1 r 1A l 1 cos S 1 (13) where cos S 1 is calculated by cos S 1 = l 1 + l13 l3 l1 l 13 (14) Substituting Eqs. (1) and (14) into Eq. (13), yields r A = 1» m 1 (m 1 + m 3 )l1 m 1 + m m 1m 3 l13 + m 3(m 1 + m 3 )l3 (15) 3 In Eq. (15), ra is expressed in terms of the three side lengths of the triangle and the masses of the three craft. Eq. (15) is a property of the desired triangular formation. This is a necessary condition for the existence of the solution to the reconfiguration problem to rephrase. The necessary condition is that there exists a time t > 0 such that r A (t) = r A (16) 6

7 is satisfied. The necessary condition in Eq. (16) is a connection between the desired triangle and the relative motion of the point A and SC-. Note that r A is the magnitude of the relative position vector pointing from SC- to the point A: r A (t) = r A (t) = R A (t) R (t) (17) Substituting Eq. (17) into Eq. (16), yields R A (t) R (t) = r A (18) Squaring both sides in Eq. (18), yields R A(t) + R (t) R A (t) R (t) = r A (19) Substitute Eq (10) and (11) into (19), then Eq. (19) can be expressed in the scalar form: v A(0)t + r A (0)v A (0) cos θ 0 t + r A(0) r A = 0 (0) where v A = ṙ A, θ [0, π] is the angle between r A and ṙ A. This paper nominates Eq. (0) as the time constraint equation. The necessary condition in Eq. (16) is transformed to be the existence of at least one positive solution to the quadratic equation about the time t in Eq. (0). The coefficients of this quadratic equation are determined by the initial states, the masses and the required three side lengths of the three craft system. Generally there are two solutions to Eq. (0):» t 1 = r A(0) cos θ 0 + ra r A (0) sin θ 0 v A (0)» t = r A(0) cos θ 0 ra r A (0) sin θ 0 v A (0) (1) () Since t 1 > t, for Eq. (0) to have at least one positive solution, both of the following inequalities must be satisfied: r A ra(0) sin θ 0 0» ra r A (0) sin θ 0 >r A (0) cos θ 0 (3a) (3b) The inequality in Eq. (3a) ensures that the solutions to Eq. (0) are real. Eq. (3b) guarantees that t 1 > 0. Note that 7

8 if π < θ 0 < π (4) then cos θ 0 < 0, thus Eq. (3b) is always true if Eq. (3a) can be promised. Otherwise, for Eq. (3b) to be true, the following inequality must be satisfied: r A < r A (5) Moreover, if the following inequality is also satisfied:» r A r A (0) sin θ 0 < r A (0) cos θ 0 (6) then t 1 & t are both positive. If 0 < θ 0 < π (7) then Eq. (6) can not be satisfied. For Eq. (6) to be true, the following inequalities must be satisfied: π < θ 0 < π (8) r A > r A (9) Table 1. Cases of solutions to Eq. (0). Eq. (3a) Eq. (4) Eq. (5) Solutions to Eq. (0) True True False False True True False positive solutions 1 positive solution 1 positive solution negative solutions False True False True False True False N/A imaginary solutions with positive real part N/A imaginary solutions with negative real part Table 1 summarizes the cases of solutions to Eq. (0). Figure 4 illustrates the scenarios of the typical cases. Note that the scenarios shown in Figure 4 are the relative motion of the point A as seen from SC-. Figure 4(a) shows the case that both of the three inequalities in Eqs. (3a), (3b) and (6) are satisfied. As shown in Figure 4(b), the point A moves along the dashed straight line defined by the velocity ṙ A and the initial positive vector r A (0). Eq. (3a) 8

9 t 1 = t > 0 t < 0 <t 1 t 1 >t > 0 r A (0) sin 0 0 ṙ A r A 0 ṙ A r A A r A (0) A r A (0) SC- SC- (a) Two positive solutions. (b) One positive solution. r A r A (0) A 0 ṙ A r A r A (0) A 0 ṙ A t <t 1 < 0 SC- (c) Two real solutions, but no positive solutions. SC- (d) No real solutions. Figure 4. Cases of solutions to Eq. (0). ensures that the distance between SC- and the dashed straight line is not greater than ra. Thus the dashed straight line either has two cross points or has one tangential point with the sphere. In both cases Eq. (0) has two real solutions, distinct or identical as shown in Figure 4(a). Eq. (3b) and Eq. (6) ensures that the two solutions are positive. Figure 4(b) shows the case that the two inequalities in Eqs. (3a) and (3b) are satisfied, which means that Eq. (0) has one positive and one negative solutions. It can be seen that in this case the initial position of the point A r A (0) is within the sphere defined by ra. Figure 4(c) illustrates the case that the two solutions to Eq. (0) are both negative. In this case only Eq. (3a) is satisfied. Geometrically the point A is moving in the direction that is opposite to the sphere defined by r A. Satisfying the inequalities in Eqs. (3a) and (3b) ensures that the distance between the point A and SC- will reach the desired value. The desired value of the distance is calculated according to the expected triangular configuration. Note that the inequalities in Eqs. (3a) and (3b) can be verified based on the initial states of the three-craft system and the desired triangular configuration. The inequalities in Eqs. (3a) and (3b) are necessary for the existence of a one-phase trajectory that accomplishes the reconfiguration task. The next to be investigated is the relative motion of the SC-1 & SC-3 subsystem. 9

10 SC-1 & SC-3 Relative Motion Assuming that t is a positive solution to Eq. (0), this section investigates the motions of SC-1 and SC-3. The objective is to find a proper relative motion of SC-1 and SC-3 such that the three-craft system reaches the desired configuration at time t. Specifically, given the initial states of the system and the maneuver time t, find the charge product Q 13 q 1 q 3 such that the following equations are satisfied: [l 1 (t), l 3 (t), l 13 (t)] = [l 1, l 3, l 13] (30) t = t (31) First let us explore the connection between the geometry of the desired triangle and the corresponding states of the system. According to the maneuver time t, the expected states of SC- and the point A are given by: R =R (0) + Ṙ t (3) R A =R A (0) + ṘA t (33) Note that there are no external forces acting on the SC-1 & SC-3 subsystem. The relative motion of SC-1 and SC-3 is planar due to the conservation of the angular momentum. The normal direction of the orbit plane is defined by the specific angular momentum of the subsystem. h 13 = r 13 (0) ṙ 13 (0) (34) At time t, the relative position vector pointing from SC-1 to the point A r 1A must satisfy r 1A (t ) h 13 = 0 (35) r 1A (t ) = r 1A (36) where r 1A is calculated by Eq. (1). r 1(t) must satisfy the length requirement: r 1 (t ) = l 1 (37) The three equations in Eqs. (35), (36) and (37) are the equality constraints for the vector R 1, which is the expected position of SC-1 at time t. There are three unknown parameters in the vector R 1. Thus R 1 is solved through 10

11 Eqs. (35), (36) and (37). Once R 1 is obtained, the location of SC-3 is calculated by: R3 = RA + m 3 ( R m A R ) 1 1 (38) Note that Eqs. (36) and (37) are essentially quadratic equations. Generally the three equality constraints in Eqs. (35), (36) and (37) result in two solutions of R1. This can be explained by the two plots in Figure 5. In Figure 5(a), the line connecting SC-1 and SC-3 rotates an acute angle then reaches the desired configuration at time t. In Figure 5(b), the same line swipes an obtuse angle till it reaches the desired configuration. In both cases the three-craft formation reaches the desired configuration at time t. The locations of SC-1 and SC-3 are different in the two plots. Flipping over the final triangle in Figure 5(a) along the line ra yields the final triangle in Figure 5(b). SC-3 SC-3 l 1 l 13 l 3 A r A SC- A r A SC- l 13 l 1 SC-1 SC-1 (a) View from the point A. l 3 (b) View from SC-1. Figure 5. Illustration of the expected maneuver scenarios. Utilizing the geometrical properties of the desired triangle, the corresponding relative position vector r13 is found through Eqs. (35) (38). Once r13 is obtained, the maneuver problem is formulated as giving the initial states r 13 (0) and ṙ 13 (0), find the charge product Q 13 such that the relative position vector r 13 reaches the desired value r13 at time t. At the first glance, this problem seems similar to the Lambert problem. But the differences between this problem and the Lambert problem are significant and result in different solution path. The following deductions aim to find the effective gravitational coefficient µ such that the expected relative position vector r13 can be reached. The expected position vector r13 is expressed by the initial states [r 13 (0), ṙ 13 (0)] as r 13 = F r 13 (0) + G ṙ 13 (0) (39) 11

12 where F and G are the Lagrange coefficients. Cross multiplying both sides of Eq. (39) with ṙ 13 (0) yields F r 13 (0) ṙ 13 (0) = r 13 ṙ 13 (0) (40) Note that r 13 (0) ṙ 13 (0) h 13 is the specific angular momentum of the subsystem, which is constant by the assumption that there are no external forces acting on the subsystem. Dot multiplying both sides in Eq. (40) by h 13, F is obtained F = r 13 ṙ 13 (0) h 13 h 13 (41) The Lagrange coefficient F coefficient can be expressed in turns of the effective gravitational coefficient as F = 1 µ (1 cos γ) h r13 (4) 13 where γ is the angle between the vectors r 13 (0) and r 13. Utilizing Eqs. (41) and (4), the effective gravitational coefficient is obtained µ = (1 F )h 13 (1 cos γ)r13 = h 13 r13 ṙ 13 (0) h 13 (1 cos γ)r13 (43) Logically, given the initial states [r 13 (0), ṙ 13 (0)] and the effective gravitational coefficient, the relative trajectory is determined. The time of flight when r 13 is reached is also determined. The reciprocal of the semi-major axis is α = 1 a = r 13 (0) v 13(0) µ (44) where r 13 (0) and v 13 (0) are the magnitude of position vector r 13 (0) and velocity vector ṙ 13 (0), respectively. The eccentric vector is the eccentricity is given by c = ṙ 13 (0) h 13 µ r 13(0) r 13 (0) e = c µ (45) (46) The initial true anomaly of subsystem is ( (îc î r13(0)) î ) h f 13 (0) = arctan î c î r13(0) (47) The true anomaly of r 13 is given by: f 13 = f 13 (0) + γ (48) 1

13 Table. The calculations of t in different situations. µ a trajectory type calculation procedure (» ) E 13 (0) = arctan 1 e µ > 0 a > 0 a < 0 ellipse attractive hyperbola µ < 0 a > 0 repulsive hyperbola 1+e E 13 ( t) = arctan (» 1 e 1+e tan f13(0) ) tan f13( t) M 13 (0) = E 13 (0) e sin E 13 (0) M 13 ( t) = E 13 ( t) e sin E 13 ( t) t = M 13( t) M 13 (0) µ /a 3 (» ) H 13 (0) = arctanh e 1 e+1 H 13 ( t) = arctanh (» e 1 e+1 tan f13(0) ) tan f13( t) N 13 (0) = e sinh H 13 (0) H 13 (0) N 13 ( t) = e sinh H 13 ( t) H 13 ( t) t = N 13( t) N 13 (0) µ /a 3 (» ) H 13 (0) = arctanh e 1 e+1 H 13 ( t) = arctanh (» e 1 e+1 tan f13(0) ) tan f13( t) N 13 (0) = e sinh H 13 (0) + H 13 (0) N 13 ( t) = e sinh H 13 ( t) + H 13 ( t) t = N 13( t) N 13 (0) µ /a 3 Depending on the values of µ and a, the possible type of the relative trajectory of the subsystem includes ellipse, parabola, attractive hyperbola and repulsive hyperbola. Table illustrates the calculation procedures of the time of flight in various cases. It is verified that once the initial states [r 13 (0), ṙ 13 (0)] and the effective gravitational coefficient µ are determined, the time of flight t is also determined. Note that there are two requirements to the one-phase trajectory. First, the relative position vector will reach the desired value, which is formulated as r 13 ( t) = r13. Second, the time of flight t is equal to t. The first requirement is ensured by the value of the effective gravitational coefficient µ. Once µ is obtained, t is determined. The second requirement is not ensured. Only at certain coincidental occasions will the time requirement be satisfied. Generally, using one-phase relative trajectory is not enough to accomplish the reconfiguration problem. More delicate strategy is expected to solve the problem. TWO-PHASE MANEUVER TRAJECTORY The difficulty of the one-phase trajectory is that the trajectory has inadequate flexibilities to satisfy both requirements. This section proposes an approach to increase the degrees of freedom of the trajectory design. The relative motion of the SC-1 and SC-3 subsystem is divided into two phases. In this way two more degrees of freedom is 13

14 introduced into the trajectory design. C ṙ B ṙ C D r C SC-3 B r D O SC-1 r B Figure 6. Two-phase relative trajectory of the SC-1 and SC-3 subsystem. Figure 6 illustrates the two-phase trajectory of the SC-1 and SC-3 subsystem. For the notational convenience, BC and CD represent the Phase-I and Phase-II trajectories respectively. The entire trajectory are determined by the charge product of Phase-I Q 13,I, the Phase-I time duration t I, charge product of Phase-II Q 13,II and the Phase-II time duration t II. Note that once Q 13,I and t I are determined, the states of the point C are determined. Thus the Phase-II trajectory is obtained using the procedure developed in the section of the one-phase maneuver trajectory. So the independent variables in the two-phase maneuver trajectory design problem are Q 13,I and t I. The objective of the two-phase trajectory design is to determine the variables Q 13,I and t I such that r 13 (t) =r 13 (49) t =t (50) are satisfied. Here t t I + t II, t is a positive solution to Eq. (0). The last section shows that providing the states of the point C, a trajectory that satisfies Eq. (49) can be found. Essentially we need to determine two variables such that Eq. (50) is satisfied. Logically, there exists one degree of freedom in this problem. This section assumes that the charge product Q 13,I is given, then develops a procedure to determine t I such that both of the two requirements in Eqs. (49) and (50) are satisfied. Because Q 13,I is known, the shape of Phase-I is determined. The effective gravitational coefficient of the subsystem in Phase-I is µ 13,I = k c Q 13,I Å 1 m m 3 ã (51) 14

15 The basic orbital characters are obtained as following: α 13,I = 1 = v B (5) a 13,I r B µ 13,I c 13,I = ṙ B h 13 µ 13,I r B r B (53) e I = c 13,I µ 13,I Çîc13,I f B = arctan î å r B î h13 î c13,i î rb (54) (55) Next to be investigated is the relationship between the maneuver time t I and the states of the point C. This process involves solving the anomaly angles and solving the Kepler s equation. Different types of the conic section use different formulas. Depending on the values of by µ 13,I and a 13,I, the conic section type of the Phase-I trajectory can be elliptical, parabolic, attractive hyperbolic and repulsive hyperbolic. Here we take the repulsive hyperbola as an example to show the calculation steps. Giving t I, the following procedure shows steps to find r C and ṙ C. The hyperbolic anomaly of the point B is H B = arctanh Ç e I 1 e I + 1 tan f å B (56) The mean anomaly of the point B is N B = e I sinhh B + H B (57) The mean anomaly of the point C is N C,I = N B + µ 13,I a 3 t I (58) 13,I The hyperbolic anomaly of the point C is obtained by numerically solving the transcendental equation: N C,I = e I sinhh C,I + H C,I (59) The true anomaly of the point C in Phase I is f C,I = arctan Ç å e I + 1 e I 1 tanhh C,I (60) The radius of the point C is r C = h 13 µ 13,I (e I cos f C,I 1) (61) 15

16 The Lagrange coefficients are F C,I = 1 + a I r B (cos H I 1) (6) a3 I G C,I = t I + (sinh H I H I ) (63) µ 13,I F µ13,i a 13,I C,I = sinh H I (64) r B r C Ġ C,I = 1 + a 13,I r C (cosh H I 1) (65) where H I = H C,I H B. The states at the point C are: r C = F C,I r B + G C,I ṙ B (66) ṙ C = F C,I r B + ĠC,Iṙ B (67) Note that the states of the point C are obtained based on the provided value of t I. Thus r C and ṙ C are functions of t I. Once the states of the point C are known, the procedure developed in the last section can be utilized to obtain the Phase-II trajectory which satisfies the requirement in Eq. (49). In the following we are going to illustrate the way to find the Phase-II trajectory that satisfies Eq. (50). The Lagrange coefficient of the point D is: The effective gravitational coefficient of Phase-II is F D = r D ṙ C r C ṙ C (68) µ II = (1 F D)h 13 (1 cos f II )r D (69) where f II = γ f I = γ (f C,I f B ) and r D = r 13 is the desired radius of point D. The semi-major axis of Phase-II is r C µ II a II = µ II r C vc (70) where v C = ṙ C is the magnitude of the velocity at point C. The eccentricity vector is c II = ṙ C h 13 µ II r C r C (71) 16

17 The true anomaly of the point C in Phase-II is ã ÅîcII î rc î h f C,II = arctan î cii î rc (7) Now we are going to use Kepler s equation to calculate r D corresponding to t II = t t I. Different types of orbit have different formulas towards the Kepler s equation. In the following hyperbolic orbit type is taken as an example. The hyperbolic anomaly of the point C is ( eii 1 H C,II = arctanh e II + 1 tan f ) C,II (73) where e II = c II / µ II. The mean anomaly at the point C is N C,II = e sinh H C,II + H C,II (74) The desired mean anomaly at the point D is where n II = N D = N C,II + n t II (75)» µ II /a 3 II and t II = t t I. The hyperbolic anomaly of the point D is obtained by numerically solving the transcendental equation: N D = e II sinhh D + H D (76) The change of hyperbolic anomaly of Phase II is H II = H D H C,II (77) The F & G coefficients are determined by H II F D = 1 + a II r C (cos H II 1) (78) G D = t I + a3 II µ II (sinh H II H II ) (79) Then the position vector at point D is obtained through r D = F D r C + G D ṙ C (80) 17

18 Table. 3 gives the calculations of F & G coefficient for different types of orbit. Table 3. The calculations of F & G coefficient in different situations. µ a trajectory type F & G coefficient µ > 0 a > 0 ellipse F = 1 µr1 h (1 cos f) r0r1 sin f G = h F = a < 0 attractive hyperbola µ(1 cos f) h sin f Ġ = 1 µr0 h (1 cos f) µ(1 cos f) [ h 1 r 0 1 r 1 ] µ < 0 a > 0 repulsive hyperbola F = 1 + a r (cos H 1) G = t +» a3 µ (sinh H H) F = µa r 0r 1 sinh H Ġ = 1 + a r 1 (cosh H 1) Assuming that Q 13,I and t I are given, the procedure in Eqs. (56)-(80) finds the trajectory that satisfies the maneuver time requirement in Eq. (50). Next problem is to find the solution that satisfies both the maneuver time requirement in Eq. (50) and the final location requirement in Eq. (49). The procedure in Eqs. (56)-(80) reveals that the final position r D is determined by Q 13,I and t I : r D = f( t I, Q 13,I ) (81) This function is not a fundamental function because there is a transcendental equation when calculating the final vector r D. This paper assumes that Q 13,I is given, then numerically search the solution t I that satisfies Eq. (49). Defining a scalar error function as: r = r D r D r D (8) The objective of the numerical algorithm is to find the value of t I such that r 0. The combination of the types of the two-phase trajectory can be four cases: 1), both Ellipse; ), Ellipse and Hyperbola; 3), Hyperbola and Ellipse or; 4), both Hyperbola. During the numerical searching process, if the combination of the trajectory type remains in Case 1) or 4), the curve of t I - r is continuous and smooth, as shown in Figure 7(a). The Secant method is applicable to solve the solution t I in this case. The corresponding procedure of the numerical algorithm is given by: t I (k) = t I (k 1) α t I(k 1) t I (k ) r(k 1) (83) r(k 1) r(k ) where 0 < α < 1 is the coefficient used to reduce the sensitivity of the algorithm. 18

19 r 13 [m] r 13 [m] t 1 [s] (a) t I - r while the types of the trajectory remains consistent t 1 [s] (b) t I - r while the types of the trajectory changes. Figure 7. t I - r curve in different situations. There exists a possibility that the combination of the trajectory type changes when sweeping t I, as shown in Figure 7(b). On the left hand side of the switch point, the combination is both Ellipse, on the other side the combination is Ellipse and Hyperbola. Moreover, the solution t I is very close to the switch point. In this case the Secant method is very sensitive around the solution. It would cause severe oscillations of the numerical searching process as illustrated in Figure 8(a). In Figure 8, the x-axis is the iteration count, the y-axis is t I. The Bisection method is more stable in this situation. Once the oscillation phenomenon is detected, the Secant method is terminated and the Bisection method is engaged. The range of the solution is given by the largest range of t I during the oscillation as shown in Figure 8(b) T I T I K (a) Oscillating error function K (b) The searching range of the Bisection method. Figure 8. The oscillation phenomenon using the Secant method to solve t I. 19

20 Once t I is obtained, the flight time of the second phase is given by: t II = t t I (84) Utilizing the procedure determining the single-phase trajectory, the charge product Q 13,II is uniquely determined. So far, all of the parameters determining the entire maneuver trajectory, [t, t, t I, Q 13,I, t II, Q 13,II], are obtained. The procedure of the reconfiguration strategy to achieve control objective is summarized as follows: 1. Choose two spacecraft to be the subsystem such that there exists at least one positive solution to Eq. (0). (If there is no such subsystem, the initial conditions of the system need to be tuned. The adjustment of the initial condition will be discussed in the next section.). Solve Eq. (0) analytically and find a positive real solution t. 3. Obtain the maneuver time t of one-phase trajectory using the procedure listed in Table. 4. Check whether t = t is satisfied. If it is satisfied, stop. The control objective can be realized by using one-phase trajectory strategy. If not, go to next step. 5. Choose a value of Q 13,I, then numerically solve t I such that both Eq. (49) and Eq. (50) are satisfied. Note that there is one degree of freedom in the two-phase maneuver trajectory design. This flexibility can be utilized to find the optimal solution by a certain well-defined cost function. The optimization problem is beyond the scope of this paper. ADJUSTMENT OF THE IMPROPER INITIAL CONDITIONS The last two sections develop the maneuver trajectories based on that by choosing different combinations of the subsystem, there exists a combination such that Eq. (0) has at least one positive solution. In the case that there are no combinations of the subsystem that make Eq. (0) have positive solutions, the two-phase maneuver trajectories developed previously can not be applied to achieve the reconfiguration. The cases of solutions to Eq. (0) are summarized in Table 1 and Figure 4. In the case that there are no positive solutions to Eq. (0), there is a special situation that the solutions to Eq. (0) are a pair of imaginary numbers with positive real part. The corresponding scenario is illustrated in Figure 9(a). In this case, it is possible to adjust the initial conditions such that Eq. (0) has at least one positive solution. This adjustment strategy is investigated in this section. In Figure 9(b), ṙ A is decomposed along the radial direction and the tangential direction of r A. It can be seen that the reason why the trajectory of the point A deviates the circle centered at SC- is that the tangential component of 0

21 3 SC-3 ṙ 3 r 13 r 3 A ṙ A 1 A r A A SC- (a) Eq. (0) has imaginary solutions with positive real part. SC-1 ṙ 1 r 1 (b) Decomposition of the velocities. SC- Figure 9. Special case of the improper initial conditions. ṙ A is too large. Now let us investigate the tangential component of the point A, SC-1 and SC-3. Projecting ṙ 1 and ṙ 3 to the direction defined by τ va, yields: τ va τ v1 τ v3 =ṘA ṘA r A r A r A r A =Ṙ1 Ṙ1 r A r A r A r A =Ṙ3 Ṙ3 r A r A r A r A (85) (86) (87) Substituting Eq. (6) into Eq. (85), yields τ va = m 1τ v1 + m 3 τ v3 m 1 + m 3 (88) The magnitude of τ va satisfies τ va = m 1τ v1 + m 3 τ v3 m 1 + m 3 (89) As mentioned previously, due to that the magnitude τ va is too large, the trajectory of the point A deviates from the effective circle. One way to correct this is to introduce another phase, adjustment phase, to reduce the variable τ va. In the adjustment phase, we still propose the conic section trajectory approach to finish the job. SC- and another spacecraft are grouped as a subsystem and they are both charged. The other spacecraft in the subsystem is determined by comparing the contributions to τ va as: If τ v1 > τ va, choose SC-1 & SC- as the preadjust subsystem. If τ v3 > τ va, choose SC-3 & SC- as the preadjust subsystem. Without loss of generality, SC-1 and SC- are taken as the preadjust subsystem to show the calculation procedure. 1

22 ṙ 1 (0) SC-1 ṙ 0 1 SC-1 ṙ A (0) SC-1 r 1 (0) ṙ 00 1 SC- Figure 10. The elliptic pre-adjusting trajectory. The adjustment phase is illustrated in Figure 10. At time t ad, the velocity of SC- is Ṙ (t ad ) = Ṙ(0) + Ṙ (90) In the adjustment phase, there are no external forces acting on SC-1 & SC- preadjust subsystem, then m 1 Ṙ 1 (t ad ) + m Ṙ (t ad ) m 1 + m = m 1Ṙ1(0) + m Ṙ (0) m 1 + m (91) Substituting Eq. (90) into Eq. (91), yields Ṙ 1 (t ad ) = m 1Ṙ1(0) m Ṙ m 1 (9) Then the relative position vectors ṙ 1 (t ad ) and ṙ A (t ad ) are ṙ 1 (t ad ) = Ṙ(t ad ) Ṙ1(t ad ) = ṙ 1 (0) + m 1 + m m 1 Ṙ (93) ṙ A (t ad ) = Ṙ(t ad ) ṘA(t ad ) = ṙ A (0) + m 1 + m + m 3 m 1 + m 3 Ṙ (94) Note that ṙ A (t ad ) = ṙ A (0) + ṙ A, thus the relationship between ṙ A and ṙ 1 is ṙ A = m 1(m 1 + m + m 3 ) (m 1 + m )(m 1 + m 3 ) ṙ 1 (95) As shown in Figure 11, by properly design ṙ 1, the trajectory of the point A can be adjusted such that the point A will cross the effective sphere centered at SC-, which can be formulated as d os r A (96)

23 ṙ A(1) ṙ A(3) 0 ṙ A (0) r A A ṙ A() r A (0) SC- Figure 11. The effects of different ṙ A. The adjustment process is summarized as following: 1. Set the charge product Q ad and the adjustment time t ad of the pre-adjusting subsystem in the total pre-adjusting process. Q ad should satisfy the collision avoidance and charge saturation requirement. To minimize the time duration of the preadjusting process, it is recommended that Q ad = Q max. If τ v1 > τ va, set the charge product is Q ad = Q 1 = q 1 q. If τ v3 > τ va, set the charge product is Q ad = Q 3 = q q 3. The range of t ad is (0, t ).. Solve the parameters of the preadjusting phase, obtain ṙ A (t ad ). 3. Calculate the offset distance d os from SC to ṙ A. If d os r A, end. Otherwise, increase the value of t ad and goto step. If t ad t and d os > ra, the reconfiguration mission can not be accomplished using the trajectory programming method in this paper. t t (a) 3D figure. (b) Contour plot. Figure 1. Relationship between Q ad, t ad and d os. 3

24 Figure 1 illustrates the relationship between Q ad, t ad and d os. It can be seen that the range of d os is [0, 50+]m. As shown in Figure 1(b), which indicates the regions of solutions to Eq. (96) with green color, giving a value of Q ad, there are multiple regions of t ad that satisfy Eq. (96). From the bottom to the top, the first region corresponds to an adjusting trajectory which is a section of an ellipse. The second region represents solutions with the adjusting trajectory which is composed of a complete ellipse and a section of the ellipse. The first region is the most important. As the magnitude of Q ad increases, the region of the solutions decreases. The bottom line of the region corresponds to the solutions that satisfies: d os = ra (97) NUMERICAL SIMULATIONS Numerical simulations are presented in this section to illustrate the performance of the reconfiguration strategy. The desired configuration is a triangle defined by the three side lengths [l1, l3, l13]. The masses of the three spacecraft are: m 1 = m = m 3 = 50 kg (98) In the following simulation cases, the initial conditions and the expected configurations are set to be exactly the same as the simulation examples in Reference 15. Large Effort Maneuver The initial positions and velocities of the three spacecraft are R 1 = [9,, 0] m; R = [0, 4, 0] m; Ṙ 1 = [0, 0.01, 0] m/s; Ṙ = [0, 0, 0] m/s; (99) R 3 = [,, 0] m, Ṙ 3 = [0, 0.01, 0] m/s. The expected triangle of the virtual structure is defined by the separation distances: l = [l 1, l 13, l 3] = [6, 5, 7] m. (100) In this example, the initial errors of the formation are relatively large, which would result in the large control efforts. One may find that Eq. (0) does not have positive solutions based on the given initial conditions. So the adjustment of the initial conditions is implemented. SC-1 and SC- are chosen as the preadjusting subsystem. The two free variables of the adjustment are set to be Q ad = C, t ad = 49 s (101) 4

25 Q 13,I Q 13,II 10 r 1 r 13 r 3 Charge Product [C ] Q 1 Distance [m] Time [s] (a) Charge histories t [s] (b) The separation distances. Distance Error [m] r 1 r 13 r 3 y [m] 0 SC3-end -1 SC3-begin SC-begin SC-end SC1-begin -4-5 SC1-end t[s] (c) The distance errors x [m] (d) The trajectories of the three spacecraft in the inertial frame. Figure 13. Simulation results of the large effort maneuver. The solutions to Eq. (0) are t 1 = 85.0 s, t = 71.5 s (10) The unique positive solution is t = 71.5 s. The maneuver time of the single-phase trajectory is t = s. Because t > t, after the adjustment phase, the two-phase trajectory is chosen to achieve the reconfiguration. Setting the charge product of Phase I as Q 13,I = C, the remaining variables are t I = 194. s, Q 13,II = C, t II = 77.3 s (103) The simulation results are illustrated in Figure 13. Figure 13(a) is the history of the non-zero charge product. During the adjustment phase, Q C and q 3 = 0. The charge product of the adjustment phase is set to be large such that the system can reach the proper state in a short time. The entire history of the non-zero charge product 5

26 is composed of three horizontal lines. This type of control input is easy to be implemented in real situations. Figure 13(b) shows the distance histories and Figure 13(c) shows the error histories of the separation distances. At the moment when t = t ad + t 30.5 s, the distances of the triangle are [l 1, l 13, l 3 ] = [5.9984, 5.000, ] m (104) The small errors are due to the precision of the numerical routine solving for t I. Figure 13(d) shows the trajectories of the three spacecraft. It can be seen that the trajectories of SC-1 and SC-3 are pretty curvy, but the trajectory of SC- is almost in a straight line. This is because of that SC- is charged only in the adjustment phase and during that phase SC-1 lies almost on the initial velocity direction of SC Q 13,II 7 r 1 r 13 Charge Product [C ] Q 1 Q 13,I Distance [m] r Time [s] (a) Charge histories t [s] (b) The separation distances. Distance Error [m] r 1 r 13 r t [s] (c) The distance errors. y [m] SC3-end SC3-begin SC1-begin SC-end -3.5 SC-begin -4 SC1-end x [m] (d) The trajectories of the three spacecraft in the inertial frame. Figure 14. Simulation results of the small effort maneuver. 6

27 Small Effort Maneuver In this example the initial positions and velocities of the three spacecraft are R 1 = [, 0, 0] m; R = [0, 4, 0] m;, Ṙ 1 = [0, 0.00, 0] m/s; Ṙ = [0, 0, 0] m/s; (105) R 3 = [,, 0] m; Ṙ 3 = [0, 0.00, 0] m/s; The desired configuration is given by the separation distances: l = [l 1, l 13, l 3] = [4, 4, 4] m (106) Note that the initial errors are smaller than those in the last simulation case. Similar to the last simulation case, in this example Eq. (0) does not have positive solutions either. So the adjustment of the initial conditions is implemented. SC-1 and SC- are chosen as the pre-adjusting subsystem. The two free variables of the adjustment phase are set to be Q ad = C, t ad = 50 s (107) After the adjustment, there is only one positive solution to Eq. (0) which is t = 07.3 s. The maneuver time of the single-phase trajectory is t = s. Because t > t, after the adjustment the two-phase trajectory is required to achieve the control objective. Setting the charge product of Phase I as Q 13,I = C, the corresponding parameters of the trajectory are t I = 10. s, Q 13,II = C, t II = s (108) Figure 14 illustrates the simulation results of this case. It can be seen from Figure 14(a) that the non-zero charge product is negative during the adjustment phase and Phase-I, it is positive during Phase-II. Figure 14(c) shows that at time t = t ad + t 57.3s, the distance errors are almost zero. The final distances between each two spacecraft are [l 1, l 13, l 3 ] = [3.9997, 3.999, ] m (109) In these two simulation cases, the maneuver trajectories are composed of three phases. During each phase the charges of individual spacecraft remains constant. This strategy ensures that the control inputs are much smoother and much easier to be implemented in reality. Note that there are three degrees of freedom in the three-phase reconfiguration maneuver trajectory design. These flexibilities can be utilized to find a certain optimal solutions, which would be the future work beyond this paper. 7

28 CONCLUSION This paper develops a trajectory programming strategy for the reconfiguration of a three-craft Coulomb formation. The entire trajectories are composed of multiple phases. During each phase the charges of the individual spacecraft remain constant. In the general situation the complete strategy is composed of three phases: a preadjusting phase and two transition phases. The single-phase strategy is applicable only at certain occasional situations with strict requirements. The two-phase strategy is applicable if there exist positive solutions to the maneuver time requirement shown in the time constraint equation. If the solutions to the time constraint equation are imaginary values with positive real part, the initial conditions need to be adjusted. The preadjusting phase tunes the states of the system such that the time constraint equation has at least one positive solution. The resulting reconfiguration strategy has elegant control inputs. There are three extra degrees of freedom in determining the three-phase maneuver strategy, or one degree of freedom for the two-phase maneuver strategy if the solution exists. These flexibilities can be used to optimize the trajectories. The optimization of the trajectories is a future work of this paper. It should also be noted that in developing the strategy the Debye shielding effect is not taken into consideration. To implement the maneuver strategy in the real space environment, a feedback controller should be engaged to compensate for the modeling errors brought by various sources such as the Debye shielding effect. Designing the feedback control with a reference trajectory input for the reconfiguration is another direction of the future work of this paper. ACKNOWLEDGEMENT This research is supported by the National Natural Science Foundation of China: NSFC REFERENCES [1] L. B. King, G. G. Parker, S. Deshmukh, and J.-H. Chong, Spacecraft Formation-Flying using Inter-Vehicle Coulomb Forces, tech. rep., NASA/NIAC, January [] K. Nishikawa and M. Wakatani, Plasma Physics: Basic Theory with Fusion Applications. Springer Series on Atomic, Optical, and Plasma Physics, Springer, 000. pp [3] C. C. Romanelli, A. Natarajan, H. Schaub, G. G. Parker, and L. B. King, Coulomb Spacecraft Voltage Study Due to Differential Orbital Perturbations, AAS Space Flight Mechanics Meeting, Tampa, FL, Jan Paper No. AAS [4] W. Ren and R. W. Beard, Virtual Structure Based Spacecraft Formation Control With Formation Feedback, AIAA Guidance, Navigation and Control Conference and Exhibit, Monterey, CA, 5-8 Aug 00. Paper No. AIAA [5] H. Schaub and I. I. Hussein, Stability and Reconfiguration Analysis of a Circulary Spinning -Craft Coulomb 8

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