Control Algorithms for Libration Point Formation Flying

tion Flying

Recently, many approaches dealing with formation flying control around the Sun-

Earth L2 point have been presented in the literature. The most common control

strategies focus on the design of position keeping and reconfiguration for space- craft running on unstable but controllable orbits, including the HALO orbit and the Lissajous orbit. Due to the simplicity of the Linear Time-Invariant (LTI) model in the CTRBP, several Linear Quadratic Regulator (LQR) controllers for formation flying have been presented based on this model. Folta et al. (2000) presented a standard control technique using an LQR controller using linearization about a reference libration orbit. With the same LTI model, Hamilton et al. (2002) also developed an LQR controller for station keeping and formation manoeuvres of

the SI mission around the L2 point. Based on Floquet theory and a linear model,

Howell & Marchand (2003) developed an LQR and a linear feedback controller to maintain a constant separation and relative orientation between two spacecraft

in the vicinity of the Sun-Earth libration points. Roberts (2005) designed similar LQR controllers for the relative motion as well. All these authors consider the use of the LQR controller for LTI models. Furthermore, to minimize the noise of transmission and propulsion, Beugnon et al. (2004) used the Linear Quadratic Gaussian (LQG) method to develop a controller for formation keeping and ma- noeuvring of the DARWIN mission. Using a CW-like model, Scheeres & Vinh (2000) presented a nontraditional control law to maintain the motion on a HALO orbit which was described by a set of time-varying linearized equations, but its accuracy was limited to application in the interferometry mission. Furthermore in Hsiao & Scheeres (2002), the feedback control gain was adjusted to induce large winding numbers which satisfied the mission requirements and reduced the tight relative position control constraints that were used before. However, the LTI models in these research programmes include only the second order term of

the gravitational force for the L2point formation flying.

To improve the control accuracy and save fuel consumption, a more precise model such as a Linear Parameter-Varying (LPV) model or ephemeris model which will reduce the modeling error should be taken into account. Therefore, several authors have attempted to design gain-varying controllers by using an LPV system directly. A highly accurate LPV model for the relative motion around

the L2 point was developed in Segerman & Zedd (2003) by considering several

disturbances using series expansion. With a similar LPV model, Chabot (2005) compared the fuel-cost and control performance among several simple controllers to conclude that a sophisticated model is effective in improving the closed-loop control performance. With a Hamiltonian formulation of the equations of motion in the CRTBP, Luquette & Sanner (2001) developed a nonlinear satellite trajectory control strategy using adaptive control algorithms.

Furthermore, to meet the future mission’s stringent millimeter relative posi- tion accuracy, several authors have designed nonlinear controllers by using an ephemeris model directly. Marchand & Howell (2003) translated the continuous control of a LTI model into an n-body ephemeris successfully, and a discrete con- trol system required for flight formation was designed for target approaching and station keeping to enforce this non-natural formation. These algorithms were also applied to the formation configurations in their later papers (Marchand & Howell, 2004; Howell & Marchand, 2005). Considering the baseline requirements in the

1.2. Libration Point Formation Flying Control Overview

vicinity of libration points, they employed a decentralised control strategy based on existing linear and nonlinear control techniques, which can be applied to the CRTBP and the ephemeris model. In the same paper, they also discussed the po- tential constraints that may affect the formation control strategies, the conceptual design and the cost of the mission.

By using differential correction methods, Pernicka et al. (2005) developed dis- crete manoeuvring techniques for a formation maintained within required error tolerances. Infeld et al. (2007) cast the spacecraft formation flying control as a multi-agent, nonlinear, constrained optimal control problem and obtained the numerical solution to this problem by using a Legendre pseudospectral method implemented in DIDO (a Matlab optimization toolbox), which improved the pre- cision of the optimal solution without any linearization. Xin et al. (2007) used a new sub-optimal control technique to carry out formation control based on

nonlinear dynamics equations in deep space about the L2 point. Input feedback

linearization is another way to provide a unified frame for designing control laws for formation maintenance and reconfiguration (Vadali et al., 2004; Howell & Marchand, 2003). Using a novel nonlinear adaptive neural control method- ology, Gurfil et al. (2003) developed a controller to keep the formation in high precision via a nonlinear model of the CRTBP which includes the disturbances of solar radiation pressure and lunar gravity. These nonlinear controller algorithms achieve good control performance for the formation system. However, the heavy computation burden of the on-board computer for these controllers should be considered before implementation.

Other factors have also been taken into account during the controller design to improve robustness. For example in order to reduce the communication band- width and enhance the robustness of faulty communication links, Gaulocher &

Chretien (2006) designed decentralised H2-suboptimal controllers to minimize the

optical path difference of three-spacecraft formation flying interferometry mission (Pegase).