Coordination Control of Distributed Spacecraft System

Spacecraft formation flying has received significant attention over the past decade, it has been a topic of interest because of its unique technical advantages and good application features. Several small, unconnected satellites operating in a coordinated way may achieve a better performance than a monolithic satellite, and possess advantages such as increased instrument resolution, reduced cost, reconfigurability, and overall system robustness, which can in turn enhance the scientific return (Zhang et al., 2008).Several ambitious distributed spacecraft missions are currently being put in operation or planned. The PRISMA satellite, which is an on-orbit technology demonstrator for autonomous formation flying and rendezvous, was launched on 15 June 2010 (Ardaens et al., 2011). The TanDEM-X satellite was launched on 21 June 2010 and orbited in close formation with the TerraSAR–X satellite on 15 October 2010. The twin satellites began a routine acquisition of the digital elevation model with flexible baselines on 12 December 2010 (Kahle et al., 2011). The F6 program of the Defense Advanced Research Projects Agency, the Terrestrial Planet Finder of the National Aeronautics and Space Administration, and the Darwin mission of the European Space Agency will all utilize the technology of formation flying.


Introduction
Spacecraft formation flying has received significant attention over the past decade, it has been a topic of interest because of its unique technical advantages and good application features.Several small, unconnected satellites operating in a coordinated way may achieve a better performance than a monolithic satellite, and possess advantages such as increased instrument resolution, reduced cost, reconfigurability, and overall system robustness, which can in turn enhance the scientific return (Zhang et al., 2008).Several ambitious distributed spacecraft missions are currently being put in operation or planned.The PRISMA satellite, which is an on-orbit technology demonstrator for autonomous formation flying and rendezvous, was launched on 15 June 2010 (Ardaens et al., 2011).The TanDEM-X satellite was launched on 21 June 2010 and orbited in close formation with the TerraSAR-X satellite on 15 October 2010.The twin satellites began a routine acquisition of the digital elevation model with flexible baselines on 12 December 2010 (Kahle et al., 2011).The F6 program of the Defense Advanced Research Projects Agency, the Terrestrial Planet Finder of the National Aeronautics and Space Administration, and the Darwin mission of the European Space Agency will all utilize the technology of formation flying.
The modelling of relative motion of distributed spacecraft has been extensively investigated in the past.The Hill-Clohessy-Wiltshire (HCW) equations are widely used.The equations describe the relative motion of two close formation flying satellites in near circular orbits about a spherical Earth, and no disturbances are included in the Hill equations.Using orbital elements to parameterize the relative motion is another important way (D' Amico & Montenbruck, 2006;Ardaens & D'Amico, 2009), which is extremely efficient and was successfully demonstrated during the swap of the GRACE satellites (Montenbruck et al., 2006).By a proper design of the relative orbit elements, a minimum distances in the crosstrack plane is guaranteed and the collision hazard is minimized.
In recent years, a significant amount of work has been focused on formation relative orbit estimation.Liu considered the relative navigation for formation flying using an unscented Kalman filter (UKF) and showed that the error of the relative position and velocity estimation can be estimated in the centimeter and millimeter per second scales, respectively (Liu et al., 2008).The original Kalman filter is widely used in relative navigation; however, www.intechopen.comAdvances in Spacecraft Systems and Orbit Determination 124 its inherent linearization process typically introduces significant biases in the estimation results.A particle filter (PF) achieves a recursive Bayesian estimation via a non-parametric Monte Carlo method and shows significant advantages in the nonlinear estimation problem (Rigatos, 2009).A way of generating the importance density function of a PF is essential to improve its performance.EKF and UKF are effective in generating the importance density function.Therefore, because of the strong non-linearity of the dynamics of satellites formation flying; the extended PF (EPF) is adopted to improve the precision of the relative orbit estimation for autonomous formation flying.Moreover, the nonlinear least squares method is applied to determine the relative orbit for the ground-in-the-loop control mode, which is more accurate and suit for the short-arc observation data of the ground station.
An accurate relative orbit control is also very important for the practical implementation of distributed spacecraft.A number of effective controllers are presented in recent literature, such as the linear quadratic regulator, the sliding mode control, and relative orbital elements.Scharf divided the formation flying control problem into five architectures: Leader/Follower, Multiple-Input Multiple-output, Virtual Structure, Cyclic, and Behavioral.We adopt the Leader/Follower approach for practical implementation (Scharf et al., 2004).
It is now known that finite-time stabilization of dynamical system usually demonstrate some nice features such as finite-time convergence to the equilibrium, high-precision performance, faster response as well as better disturbance rejection properties (Ding & Li, 2011).A number of effective methods to achieve the FTC are presented in recent literature (Wu et al., 2011), such as the time-optimal control, TSM control, adaptive control, homogeneous system approach and finite time stability approach.TSM control has been widely used in many applications.By designing a nonlinear switching manifold, the states reach the equilibrium in finite time and exhibit insensitive properties, such as robustness to parameter perturbations and external disturbances (Hu et al., 2008).Man proposed a robust control scheme for rigid robotic manipulators using the TSM technique (Man et al., 1994).However, the controller has a singularity problem.Feng presented a global non-singular TSM controller for a second-order nonlinear dynamic systems (Feng et al., 2002).On the one hand, TSM controllers converge to the equilibrium quickly once in the neighbourhood of the equilibrium, however, when the states are far away from the equilibrium, the system states converge slowly.On the other hand, the linear-hyperplane-based sliding mode controllers converge to the equilibrium quickly when the states are far away from the equilibrium, but they only guarantee asymptotic stability and convergence.All these controllers can not achieve global fast convergence performance in finite time.Therefore, the current study concentrates on the FTC technique to deal with this problem.Currently, the FTC approach has been applied in many fields, such as spacecraft attitude tracking control, consensus for multi-agent systems, robotic manipulators control and missile guidance law design.We will adopt the FTC approach for formation maintenance.
With an increasing number of projects in operation, a practical formation control has also become an area of concern.Relative orbital elements were demonstrated during the GRACE, PRISMA, and TanDEM-X missions.Therefore, the current study will concentrate on formation reconfiguration based on relative orbital elements.Ardaens and D'Amico proposed a dual-impulse method for the in-plane relative control and a single-impulse control for the cross-track motion (Ardaens & D'Amico, 2009); when the control period increases, the dual-impulse maneuver causes an additional along-track drift.Hence, the use of a dual-impulse maneuver for an extended control period of formation keeping may be restricted.The formation reconfiguration control is divided into fuel-optimal triple-impulse in-plane motion control and single-impulse cross-track motion control.
Safe trajectory planning methods are often employed in collision avoidance maneuver.By considering the minimum distances among the satellites as the constraints, safe trajectories can be generated using various planning algorithms, and collision avoidance can be realized by controlling the satellites along the planned trajectory.Tillerson and Richards introduced fuel-optimal trajectories for spacecraft using mixed-integer linear programming, which includes various avoidance constraints (Tillerson et al., 2002;Richards et al., 2002).The artificial potential function method for formation flying satellites has also received considerable attention in recent years (Nag et al., 2010;Bevilacqua et al., 2011).By constructing artificial fields, the goal position provides the attractive forces, whereas the collision avoidance constraints provide the repulsive forces, thereby enabling formation flying satellites to move into their target positions without colliding.Mueller used a robust linear programming technique for the collision avoidance manoeuvre of the PRISMA mission, enabling the satellites to rapidly exit the avoidance region through the application of a single impulse at a specified time (Mueller, 2009;Muelleret et al., 2010).Therefore, the linear programming algorithms are used for the collision avoidance manoeuvre of proximity operations.
The topics concerning simulation or experiment testbeds which focus on the verification of the new technologies of distributed spacecraft have been studied by many researchers in recent years.J. Leitner firstly developed a closed-loop hardware-in-the-loop simulation environment for GPS based formation flying (Leitner, 2001).The SPHERES testbed provided a verification environment for formation flying, rendezvous, docking and autonomy algorithms (Mark, 2002).Wang developed a real-time simulation framework for development and verification for formation flying satellites, which provides access of real sensor system via serial interface (Wang &Zhang, 2005).D'Amico presented an offline and hardware -in-the-loop validation of the GPS-based real-time navigation system for the PRISMA formation flying mission (D'Amico et al., 2008).D'Amico developed the TanDEM-X Autonomous Formation Flying (TAFF) system which is to support the design, implementation, testing and validation of real-time embedded GPS-based GNC system (D'Amico et al., 2009).
The organization of this chapter is as follows: In Section 2, the relative orbit dynamics are introduced, and the general formation description parameters are presented.In Section 3, the relative orbit estimation based on extended particle-filter and nonlinear least squares are presented, respectively.The different coordination control methods are proposed in Section 4. Section 5 presents the processor-in-the-loop distributed simulation system.Section 6 summarizes our conclusions.

Preliminaries
With respect to a near-circular reference orbit, and assuming the satellites are taken sufficiently close to each other, the relative motion given by several Keplerian elements differing can be treated to first order.

www.intechopen.com
Advances in Spacecraft Systems and Orbit Determination 126

Coordinate systems
The relative motion dynamics has been discussed in many papers.We consider two neighbour satellites flying in Earth orbit.The inertial reference frame used is the J2000 frame.The origin of the coordinate system is the centre of the Earth; the X I axis points toward the mean equinox of J2000.0, the Z I axis points toward the mean north celestial pole of J2000.0, and the Y I axis completes the right-handed system.The relative reference frame used is the Hill frame.The origin of the coordinate system is placed at the centre of mass of the master satellite; the x axis is aligned in the radial direction, the z axis is aligned with the angular momentum vector and the y axis completes the right-handed system (Fig. 1).

Dynamics equations
With respect to the circular reference orbit, the relative motion can be described as the following equations: where [] xyzxyz  is the relative position and velocity in Hill's frame, n is the mean orbit rate.
The relative dynamics for the circular orbits can be expressed in a linear time-invariant (LTI) system in state-space.
where x is the state vector in Hill's frame, u is the applied acceleration in Hill's frame and y is the output which is equal to the state vector(C is identity).

Kinematics equations
The Keplerian orbital elements are a , e , i ,  ,  , and u , which correspond to the semimajor axis, eccentricity, inclination, right ascension of the ascending node, argument of perigee, and mean argument of latitude (   uM  , where M is the mean anomaly, and can be obtained from the true anomaly f ), respectively.Spacecraft-1 is the master satellite, and Spacecraft-2 is the deputy satellite.For near-circular satellite orbits, the relative eccentricity vector can be defined as follows:  represents the amplitude of e and  defines the initial phase angle of the inplane motion.
The inclination vector i can be defined using the law of sines and cosines for the spherical triangle:  represents the amplitude of i , and  defines the initial phase angle of the cross-track plane motion.

General formation configuration description parameters
For a near-circular reference orbit, the relative motion of the formation flying satellites can be described by the following equations (Hu et al., 2010): cos( ) where   ,, , , are the five general formation configuration description parameters;  p ae  represents the semi-minor axis of the relative in-plane ellipse;  sa i  denotes the cross-track amplitude;     defines the relative initial phase angle between the in-plane and cross-track plane motions; and  is the initial phase angle of the in-plane motion. 21

Passively safe formation configuration
The distance in the cross-track plane can be expressed as 22  rxz (7) By substituting equation ( 6) into equation ( 7), we obtain r : where From Fig. 3(a), we can see that the radial and cross-track separations vanish at the same time, and that when the along-track distance is zero, the two satellites will collide.From Fig. 3(b), we can see that radial separation reaches its maximum when the cross-track separation vanishes, and the cross-track separation reaches its maximum when the radial separation vanishes.The safety of the formation flying satellites is guaranteed even in the presence of along-track uncertainty.

Measurement model
Formation flying satellites often operate in close proximity.Their relative measuring instruments include laser range finders and radio-frequency, infrared, and visible measurements.In the current work, we adopt the laser range finder and radio-frequency ranging equipment as the relative measurements.Thus, the high-precision relative distance, elevation, and azimuth angles can be obtained.The measurement geometry is shown in Fig. 4.
The relative range ρ, the azimuth angle A, and the elevation angle E can be calculated according to the following equations: where () hx is the measurement matrix, and x, y, and z are the coordinates of the deputy satellite in the body-fixed frame of the master satellite.As we know, the transformation matrix between the body-fixed frame and the Hill frame is a function of the attitude of the master satellite.In this paper, the attitude determination problem was not considered.Therefore, the relative measurements are defined with respect to the Hill frame.The state and measurement equations can be established as follows: where k X is the relative state vector in the Hill frame at time k t ; k Y is the relative measurements at time k t , which can be obtained using Eq. ( 12); k W is the zero mean value white Gaussian process noise with the covariance k Q ; and k V is the zero mean value white Gaussian observation noise with the covariance k R .
Five typical measurement errors, namely, the relative range and angle measurement error, the absolute position and velocity measurement error, and the attitude determination error, are considered.

EPF algorithm
The Kalman filter is the most common method of relative navigation.However, the PF shows better performance in a nonlinear relative state and measurement equations.The principle of PF is to implement the recursive Bayesian filter using Monte Carlo simulations, in which the choice of the importance density function is very important.We employ EKF to realize the importance sampling, which not only makes full use of the latest measurement information, but also avoids the particle exhaustion problem.The particle weights, which are closely associated with the observation, increase, whereas the other particle weights decrease. The We initialize the particles using: Importance sampling: a.The particles are updated using the following equations: The importance weights are calculated using the following equations: Thus, the state update is expressed as follows:

Nonlinear least squares method
The nonlinear state equation and observation equation are as follows (Hu et al., 2010): where l X is the state vector at the time l t , which includes the J2 perturbations.l Y is the observation vector at the time l t .l V is the observation noise with normal Gauss distribution.
Equation ( 16) can expanded at the approximation point * 0 X by using the Taylor series equation, the following equations can be derived by keeping the linear items: where


Therefore, the nonlinear model turns out to be the following form: By using the least square method, the estimation value of epoch time can be derived by iteration: The optimal estimation should be calculated iteratively, and usually can converge by 3-5 steps.

Numerical simulations and results analysis
A numerical simulation is conducted to verify the effectiveness of the presented EPF algorithm.The simulation conditions are as follows: the mean orbital elements of the master and deputy satellites are as shown in Table 1, and Fig. 5   The absolute orbit of the master and deputy satellites can be generated using the Satellite Tool Kit based on the initial elements given in Table 1.The observation values can be simulated by the absolute orbit information and the measurement covariance using the Gaussian distribution random number series.The measurement sampling period is 1 s, and the simulation time is 3000 s.

Coordinated control scheme
We consider an operational scenario with two formation flying satellites, and the deputy satellite performs the relative orbit correction maneuvers.Fig. 8 shows the schematic diagram of the formation flying guidance, navigation, and control (GNC) system.
The deputy satellite obtains the relative measurements and performs the relative orbit estimation to obtain the high-precision relative position and velocity.The formation control software generates control commands according to the current states and mission goals.Thrusters are used to control the geometry and phase angle of the formation, and the yaw angle maneuver commands are used to control the along-track drift.The ground station can monitor the formation flying system in autonomous mode and generate formation control commands in the ground-in-the-loop mode.

Finite-time controller
In this section, a robust sliding mode controller is proposed to improve the transient performance and to guarantee the finite-time stability and convergence.The formation flying satellites are close to each other, thus, the disturbances acting on the satellites will be almost the same, and the total relative disturbances D can generally be treated as bounded forces.Suppose that , where i F is a positive constant.
We propose the following controller where ,0    Theorem 1.For the formation flying system, the controller (25) can achieve the control objective of trajectory tracking presented in Section 4.2.1.

Proof:
Step 1: The system will reach the sliding mode 0  S in finite time.
Consider Lyapunov function V is positive, and  V is negative.Therefore, the sliding mode 0  S is achieved in finite time.
Step 2: The system will converge to the equilibrium in finite time once under the condition of 0  S .
Once 0  S , the system is transformed as =0 e is the terminal sliding attractor of system (30).By integrating Eq. ( 30), we obtain the convergence time T : where 0 e is the initial error state.
Therefore, once the system states reach the sliding mode manifold (26), the system will converge to the equilibrium in the time T .Combining step 1 and step 2 completes the proof of Theorem 1.
Remark 1.For linear controller, to increase the robustness of the closed-loop system, we can only modify the control gain; however, the control gain can not be too large considering the fuel consumption and system stability.For the FTC approach, we have an additional parameter  to modify, which exhibits better disturbance rejection performance.
Remark 2. In order to reduce chattering due to high-frequency switching, the boundary layer approach is adopted to replace the signum function of ( 25) with a continuous saturation one () sgn( ) where  denotes the thickness of the boundary layer.Therefore, the proposed controller (25) can be rewritten as follows: However, when  S  , the controller (33) can only guarantee asymptotic convergence, although chattering phenomenon can be substantially alleviated.Therefore, a new saturation function is put forward.
sgn( ) () sgn( ) where 0 1   . Then, we obtain the following controller: The theoretical proof of the finite time convergence inside the boundary layer is provided by Ding (Ding & Li, 2007).Hence, we can guarantee the finite time convergence by adopting the modified controller (35).

Numerical simulations and results analysis
In this scenario, formation keeping simulation is conducted to verify the effectiveness of the proposed controller (35).The initial orbital elements of the master satellite are as shown in The initial relative states in the Hill frame are as shown in Table 3.We design the formation with 0   , then, the projected trajectory in the cross-track plane is an ellipse, which guarantee the formation safety even in the presence of along-track uncertainty.As shown in Figs.9(b) and 9(c), when the relative distance error reaches 50 m at the time t = 15888 s, high position tracking accuracy and fast convergence are achieved, which shows that the proposed controller ( 35) is effective and robust, since finite time convergence is still obtained in the presence of model uncertainties and environment perturbations.

Triple-impulse in-plane control
We assume that the nominal configuration parameters in the orbital plane are 1 p and 1  , and the current configuration parameters in the orbital plane are 2 p and 2  .According to Eq. ( 6), the relative position in the orbital plane can be described as The problem of controlling the current configuration to achieve the nominal configuration is equivalent to the problem of setting 0 p to zero.According to Gauss variation equation, the variances in the relative orbital elements can be expressed by the along-track  T v : ( where v is the orbital velocity.
The relative orbital element and the configuration parameters have the following relationship:

  
The dual-impulse method mentioned by D'Amico equate to (D'Amico & Montenbruck, 2006;Ardaens & D'Amico, 2009) The first impulse will cause an additional along-track drift during the time span between the two impulses.The influence can be neglected if the control period is small; however, if the control period is large, the influence must be considered.
The conventional dual-impulse in-plane control method causes an additional along-track drift because of the time span between the two impulses.Hence, we implement the corrections three times.The maneuver sizes are 1 v , 2 v , and 3 v , respectively, and the respective locations are 1 u , 2 u , and 3 u .The triple-impulse locations must be equal to 0    or 0  and satisfy the following constraints: We obtain the maneuver commands when 10   u   , as expressed by and another solution when 10  u  , as expressed by the following equations: The along-track drift caused by the first impulse will be compensated by the subsequent two impulses.The maneuver sizes and locations can be easily calculated according to the initial and nominal formation parameters.Eq. ( 41) shows that the total v needed for formation control can be calculated once the initial and nominal formation parameters are provided, which is helpful in formation-flying mission design and analysis.

Single-impulse out-of-plane control
The relative inclination vector of the initial and target formation configurations is i , the argument is 0  , and the single burn can be provided by Gauss variation equation.Thus,

Numerical simulations and results analysis
In this scenario, formation reconfiguration simulation is conducted to verify the effectiveness of the proposed method.The initial orbital elements of the formation flying satellites are as shown in Table 4.The formation is reconfigurated from the initial configuration { 300

Linear programming method
The dynamic system mentioned in Section 2.2.1 can be discretized using zero-order hold as follows (Paluszek et al., 2008): where 0, , 1   kN , and the time-step is t .
The problem of optimal collision avoidance manoeuvre can be described as follows.Given the initial and the terminal states, equation ( 21) is minimized by a sequence of k u and manoeuvre time T : where  is the small error vector of the terminal state, Lb and Ub are the boundaries of the thrust.
The problem mentioned above can be converted into a standard linear programming problem: The problem of optimal collision avoidance manoeuvre can be written as We take the TanDEM-X formation as an example.When the relative measurement sensors fail, the formation satellites cannot obtain the relative states, which rapidly increases the collision probability.To minimize the collision hazard, we can manoeuvre the chaser satellite from the formation with 90   ° to a safe formation with 0   °.The safe configuration parameters are { 400 m}, and the terminal state error vector is [1 m, 1 m, 1 m, 0.1 m/s, 0.1 m/s, and 0.1 m/s].When the initial and terminal configurations are given, the control sequences can be calculated while minimizing total delta-v by the proposed linear programming method.The method is flexible and independent of the time window.The maneuver time is 600 s.Fig. 13 shows the control input for the maneuver, Fig. 14 indicates the three-dimensional collision avoidance trajectory, and Fig. 15 displays the projected trajectory in the cross-track plane.
Total delta-v is 0.646 m/s.The safe trajectory is reached within a short period.Fig. 15 shows that the trajectory reached has a minimum separation of 300 m.The two cases above illustrate that shorter maneuver time gives rise to a larger total delta-v, and that a collision avoidance strategy can be formulated by considering time urgency and residual propellant mass.

Scenario 2
The TanDEM-X formation is taken as an example.The nominal configuration is passively safe with 0   °.When the 40 mN cold gas thrusters are open for a certain period given some uncertainties, the collision hazard increases.After failure is eliminated, the chaser satellite should be controlled so that it immediately returns to safe orbit.We assume that the chaser satellite burns only in the along-track direction; thus, the crosstrack motion amplitude remains unchanged.The collision avoidance region is defined as a circle with a 200 m radius.The optimal maneuver trajectory is shown in Fig. 16.As seen in Fig. 16, the trajectory intersects with the collision avoidance region after the thrusters malfunction is eliminated.The proposed optimal collision avoidance manoeuvre is used to steer the chaser satellite toward the safe trajectory within a minimum distance of 200 m.Total delta-v is 0.126 m/s.The manoeuvre requires only two burns; hence, it is simple, effective, and suitable for on-board implementation.

5.
Processor-in-the-loop simulation system for distributed spacecraft

System architecture
In order to simulate the control architecture of distributed spacecraft, the distributed system architecture is selected.The main elements in the platform are the formation control embedded computers, which builds a VxWorks environment in a PowerPC8245 board and runs the GNC flight software.The dynamic simulation computers exchange data with the formation control embedded computers via CAN bus.The formation control embedded computer receive the high precision orbit, attitude and measurement data provided by the corresponding dynamic simulation computer real-time, and produce a series of time-tagged maneuver commands to add to the dynamic simulation environment, which forms the close-loop processor-in-the-loop simulation of the GNC system.The formation control embedded computers not only communicate with each other through wireless to emulate the communication among distributed spacecraft, but also communicate with the ground station to emulate the ground-in-the-loop communication.One workstation sets the simulation parameters and displays the simulation scenarios by a plasma displayer.One industrial control computer generates impulse to guarantee synchronization among different subsystems.Fig. 17 shows the system architecture diagram of the distributed simulation system (Hu et al., 2010).

System implementation
The dynamic simulation computers are the backbone of the close-loop simulation platform.
The software is written in C language and compute orbit, attitude, sensor models and actuators of distributed spacecraft system.The simulation computers are synchronized with the pulse generator and the real-time simulation time step can be set as 10 milliseconds.It provides the epoch time, ECI states of each spacecraft, relative states to the master satellite and attitude data to the formation control computers via CAN bus.The typical error models of the motion data as Guassian noise are also added to evaluate the control performance and the fuel consumption.The adopted dynamic models for orbit propagation include the Earth's gravity field (such as EGM96、JGM3、JGM2 or GEMT1 model), atmospheric drag(such as Harris-Priester or Jacchia70 atmospheric density model), solar radiation pressure, gravity of Sun and Moon and solid Earth tides.The dynamic simulation software also includes the attitude dynamic models based on quaternions to simulate six degree-offree motions of each spacecraft.
The dynamic simulation computers can receive the maneuver commands from the formation control computers via CAN bus.The maneuver commands include the start control time, the execution time and the delta-V of the desired impulsive maneuver.The net force error and the direction error of thrust are added to emulate the natural environment.The effect of the maneuver is then reflected to the motion data sent to the formation control computers.
The formation control embedded computers receive the absolute and relative states with typical errors from the dynamic simulation computers.The Extended Kalman Filter is used to determine the relative orbit real-time for the autonomous formation flying, and the nonlinear least squares estimation is used to determine the relative orbit for the ground-in-theloop control mode.The formation initialization, formation keeping, formation reconfiguration and collision avoidance maneuver control algorithms are realized.
The ground station is run on a workstation and developed by Visual C++ 6.0, it can produces the control commands in the ground-in-the-loop control, which is sent to the OBDH modules in the formation control embedded computers.
The simulation manager is developed by Visual C++ 6.0, it has a friendship user interface, and enabled the user to select the simulation parameters such as the control model (autonomous mode or ground-in-the-loop mode), the mission scenario (formation initialization, formation keeping, formation reconfiguration or collision avoidance maneuver), the simulation time and the time step etc.It also receives the position and velocity from the dynamic simulation computers and drive the STK VO 3D window through STK's Connect Module.

Numerical simulations and results analysis
This scenario demonstrates the autonomous formation keeping experiment.
Fig. 18 shows the relative navigation error in RTN frame.The statistical performance of relative position is 3cm respectively and the relative velocity is 0.2mm/s respectively.the along-track drift( l ), the 3st plot shows the change of the in-plane geometry( ae  ), the 4st plot shows the change of the in-plane phase angle(  ), the 5st plot shows the change of the cross-track geometry( ai  ), the 6st plot shows the change of the cross-track phase angle(  ).The relative semi-major axis and the relative eccentricity vector are controlled by three in-plane impulse maneuvers in the along-track direction separated by half an orbital period interval.The relative inclination vector is controlled by out-of-plane maneuvers only.
The relative semi-major axis and the long-track drift are affected by the execution of the three in-plane impulse maneuvers.The relative eccentricity vector and the relative inclination vector are properly moved from one perturbation side to the desired side in order to compensate their natural drift caused by J2.
Through the formation keeping test and the formation reconfiguration test, the functionalities and the performance of the process-in-the-loop simulation testbed are validated.

Conclusions
This chapter investigates several key technologies of distributed spacecraft, such as the high precision relative orbit estimation, the formation maintenance and reconfiguration strategies, the collision avoidance maneuver and the distributed simulation system.Simulation results show that the relative position estimation errors are within 210 -2 m, and that of the relative velocity estimation are within 110 -4 m/s.A robust sliding mode controller is designed to achieve formation maintenance in the presence of model uncertainties and external disturbances.The proposed controller can guarantee the convergence of tracking errors in finite time rather than in the asymptotic sense.By constructing a particular Lyapunov function, the closed-loop system is proved to be globally stable and convergent.Numerical simulations are finally presented to show the effectiveness of the developed controller.The full analytical fuel-optimal triple-impulse solutions for formation reconfiguration are then derived.The triple-impulse strategy is simple and effective.The linear programming method is suitable for collision avoidance maneuver, in which the initial and terminal states are provided.
A real-time testing system for the realistic demonstration of the GNC system for the distributed spacecraft in LEO is presented.The system allows elaborate validations of formation flying functionalities and performance for the full operation phases.The test results of autonomous formation keeping and formation reconfiguration provide good evidence to support performance and quality of the coordination control algorithms.
The key aim of this chapter is to introduce the important aspects of the distributed spacecraft, and pave the way for future distributed spacecraft.

Fig. 2 .
Fig. 2. Example of a relative motion in a near-circular reference orbit.
Fig.3shows the minimum distances in the cross-track plane with different relative phase angles.
shows the three-dimensional formation configuration.The formation configuration parameters are p = 400 m, s = 350 m,  = 0°,  = 90°, l = 0 m.The absolute position and velocity measurement precision are 10 m and 0.1 m/s, respectively; and the relative range and angle measurement precision are 0.1 m and 0.01°, respectively.The sampling interval is 1 s.Perturbations of Earth oblateness, atmospheric drag, solar radiation pressure, perturbation of the third-body of the sun and moon, and perturbation of the earth body tide are considered in the dynamics simulation.The fourth-order Runge-Kuta algorithm is employed for the numerical integration.

Fig. 5 .Fig. 7 .
Fig. 5. Three-dimensional formation configuration.The relative position and velocity estimation errors are shown in Figs.6 and 7, respectively.The estimation curves are globally convergent, and the EPF algorithm achieved much faster convergence rate in the relative orbit estimation.The relative position estimation errors converge to 210 -2 m within 500 s, and that of the relative velocity estimation are within 110 -4 m/s.

Fig. 9
Fig. 9(a) shows the three-dimensional formation configuration; Fig. 9(b) shows the variations of relative position error vs. time, Fig. 8(c) is the enlargement view of Fig. 9(b) and Fig. 9(d) shows the variations of sliding mode manifold vs. time.

Fig. 10
Fig. 10(a) shows the reconfiguration of the relative eccentricity vector, Fig. 10(b) shows the reconfiguration of the relative inclination vector.

Fig. 10 .
Fig. 10.Simulation results of the relative eccentricity and inclination vector.As shown in Figs.11 and 12, we can see that formation was successfully reconfigurated to the target configurations.

Fig. 19
Fig.19shows the key results of formation keeping scenario.The simulation time is 30 days, the in-plane control period is 7 days and the cross-track control period is 28 days.The 1st plot shows the change of the relative semi-major axis( a ),the 2st plot shows the change of

Fig. 19 .
Fig. 19.The key results of formation keeping scenario.

Table 1 .
Mean orbital elements of the master and deputy satellites.

Table 2 .
Initial orbital elements of the reference orbit.
www.intechopen.comAdvances in Spacecraft Systems and Orbit Determination 138

Table 3 .
Initial relative states in the Hill frame.
The threshold of starting formation keeping control is set as 10% of the nominal formation geometry, namely, 50 m.The orbit propagator model includes perturbations of Earth oblateness, atmospheric drag, solar radiation, third-body of Sun and Moon and Earth body tides.The Earth's gravity field adopts EGM96 model, and the atmospheric density model adopts Jacchia70.The eighth-order Runge-Kuta algorithm is employed for the numerical integration.The simulation time is 20000 s.The controller parameters are given by

Table 4 .
Initial orbital elements of the reference orbit.