Research on Shape Remain of Near-Earth Radial Collinear 4-Craft Coulomb Formation

This paper investigates the non-linear relative kinetic model of the collinear 4-craft Coulomb formation in geostationary orbit. Considering the fact that the formation remains statically fixed in the radial direction, using the equilibrium conditions to linearize the dynamic model. To keep the radial static stability of the collinear 4-craft Coulomb formation,the LQR controller is designed based on the linear dynamic model,taking the Coulomb force as the only control force. Considering the unmodeled disturbance force and the model error, the improved LQR control law is designed to improve the robustness of the traditional LQR controller. Simulation is executed by Matlab/Simulink, and numerical simulation results demonstrate the effectiveness of the dynamic model and the proposed control strategy.


Introduction
Satellites formation flying has obvious advantages in synthetic aperture radar and Spatial Interferometer.For recent years, utilizing Coulomb force to control satellite formation is a hot topic.The basic idea of Coulomb propulsion of free-flying vehicles is to control the spacecraft formation shape and size using the inter-spacecraft forces.This propellant-less thrusting is an attractive solution over conventional electric propulsion or chemical thrusting.It has very little electrical power requirements (one Watt or less) with a renewable energy source.And it causes no thruster plume contamination of the neighboring spacecraft, which increases the lifetime of the equipment and the probability of mission success.
Parker and King introduced the Coulomb propulsion concept in References 1 and 2 in 2002.It was proposed to control a cluster of free-flying spacecraft.Natarajan [3][4][5] designed feedback controller for 2-craft Coulomb formation at GEO based on relative distance and attitude dynamics equations.Inampudi and Schaub [6][7] designed charge feedback controller for 2-craft Coulomb formation at libration point based on dynamics equations.

Dynamics model
A 4-craft Coulomb formation is considered as shown in the Figure 1.It is assumed that the center of mass spin around the earth in GEO and the four charged satellites are nominally aligned along the radial direction with initial angle and separate distance error.In order to describe the relative motion of the satellite with respect to the formation center of mass, a rotating Hill orbit frame is chosen as shown in Figure 1.The relative position vector of the satellite is defined as x y z  ρ .

Figure 1. 4-craft Coulomb formation
The center of mass condition is defined as: Where i m is the th i satellite mass.
Consider a coordinate frame    and i l can be expressed by: The kinetic energy of the formation is: The kinetic energy of the formation is: ρ and Eq. ( 3), theT is rewritten as Where The gravitational potential energy of the formation is: The Coulomb potential for the two-craft formation is： 34 11 Where L is the distance between the two satellites. and  are the roll angle out of the orbital plane and the pitch angle in the orbital plane.i q is the th i satellite charge.
The equations of motion are deduced from the Lagrangian function Where i Q is the generalized force of the th i satellite excluding gravity.Using Eqs.( 4), ( 5) and ( 6) in Eq. ( 7), we can get the nonlinear equations of motion governing the roll angle  ,the pitch angle  and the separation distance 1 l 、 2 l 、 3 l .The 4-craft formation remains statically fixed relative to the frame H in the radial direction, thus, the nonlinear equations satisfy the following equilibrium conditions: The roll and pitch equations of motion are linearized about small roll and pitch angles respectively.In order to get the linearized equations of motion, assuming that the separation distance equations of motion are linearized about small variations i l  and so as charge product ij Q  .
Shown as: Where if l and ijf Q means the reference values.The charge product ij Q should satisfy equations that utilize equilibrium conditions.In this paper, there are an infinite number of charge pairs can satisfy charge product requirements.Performing the necessary linearization and decouple equations about i l  yields: 23 ,  is the orbit rate.

C
The following feedback control is used to control the system with the feedback gain matrix, K , computed using either the pole placement method or the linear quadratic regulator (LQR) method.

 u = Kx
(13) Here the LQR methodology is applied to determine the optimal control, u , such that the gain vector K minimizes the performance index.

Numerical simulation
In this section, we present numerical simulation to illustrate the effectiveness of the proposed LQR controller.
In the simulations, The dynamics of Coulomb formation system is simulated using the numerical differential equation solver in Matlab/Simulink for several cases.The relative tolerance is 6 10  .The simulation figures shown as: From the simulation results we can see that the pitch angle error (Figure 2) of formation shows a amplitude attenuation and converge to zero at around 4h.The separation distances (Figure 3~ Figure 5) also converge to zero around 4h.All state variations converge to zero finally with given charge.
In each picture of Figure 2、Figure 3 and Figure 5, curves with improved LQR controller have less amplitude oscillation and converge faster than LQR controller without integral item.In Figure 4, the curve with integral part clearly shows a small amplitude variation and converge to desired distance with less oscillation frequency.It illustrates that the improved LQR controller gets better control results than the traditional LQR controller.

Conclusion
In this paper, a LQR control law for a 4-craft Coulomb formation at GEO is given.On the basis of dynamics model, considering the model error, the improved LQR control law is designed.Simulation results shows that the improved LQR controller gets better control results in controlling the attitude and separation distance to desired position in finite time.Numerical simulation results demonstrate the validity of the dynamic model and the effectiveness of the proposed control strategy.
b as shown in Figure1.i l is the relative distance between satellites.The relationship between i