Deployment Considerations for Spacecraft Formation at Sun-earth L2 Point

The coordination and control of a constellation of spacecraft, flying a few meters from one another, dictates several interesting design requirements, including efficient architectures and algorithms for formation acquisition, reorientation and resizing. The spacecraft must perform these transitions without interfering or colliding into each other. Furthermore position keeping is fundamental for formation efficiency. This paper presents an optimal deployment of the DARWIN formation using the potential function control technique in the vicinity of the Sun-Earth L2 point. The method hinges on defining a potential function from the geometric configuration of the constellation together with any collision avoidance requirement. A review of the fundamentals of relative motion and dynamics is presented before describing the features of the different control algorithms and validating the method using Lyapunov’s theorem. The potential function method has been used to control both translational and rotational control. Obstacles, in the shape of other satellites and constrained payload pointing directions have been included. Finally it will be shown that the attitude control algorithm can successfully used to avoid plume impingement that can have catastrophic consequences for the mission. INTRODUCTION Over the past decade, the introduction of cost reduction policy by the major space agencies caused a paradigm shift in the design of scientific satellites, as the primary metric by which spacecraft were judged switched from purely performance to specific performance or performance per unit cost [Cyrus and Miller, 1997]. Several new technologies, including multifunctional structures, microelectro-mechanical systems, nano-technology and distributed satellite systems, have the potential to revolutionise the field of satellite design. In particular in the field of distributed satellite systems, that is systems based on dividing the tasks among several light and small satellites, two approaches exist: constellations and formations. The difference between the two methods lies in the relative positioning between satellites. Constellations are positioned relative to an object, such as the Earth, while in formations spacecraft are positioned relative to each other. Each satellite communicates with the others and shares the processing, communications, and payload or mission functions. Thus the cluster of satellites forms a “virtual satellite”. This concept promises many benefits, including greater utility and flexibility by allowing the cluster to reconfigure and optimise its geometry for a given mission, enhanced survivability, and increased reliability. In general terms, the formation-flying approach has the following advantages: the opportunity of completing space observation missions without large and expensive ground infrastructures, reducing operational costs. The deeper covering of the phenomena under observation, since different instruments, under different points of view, inspect it at the same time. By substituting one large complex satellite with a group of small satellites, a better flexibility is achieved, with the chance of reconfiguring the system in case of malfunction, thus avoiding the mission failure. The failure of one spacecraft will not compromise the mission. The employment of identical platforms within the constellation allows a standardisation of the manufacturing, thus reducing production costs. The system functionality is not extremely dependent from technology: it is possible to launch a temporary formation with state-of-the-art instruments and later increasing the system performance by adding one or more spacecraft to the formation. On the other hand, the development of formation-flying presents the following technological challenges: accurate sensors are needed to allow a precise determination of the state of the system in order to control the formation. High precision in spacecraft coordination is indispensable in order to avoid troubles linked with the reciprocal distances among the elements of the formation, most of all collisions between the satellites. Current studies in spacecraft formation control vary from individual satellite control to the use of stochastic algorithms [Gurfil et al., 2002]. The main problem to be addressed in formation control is that of workload. For small, Earth centred formations, individual control is a viable options. As the satellite number and operational distance from Earth increase, methods that automate the control processes become a necessity. The method proposed here aims, to drastically reduce the workload required to control the formation. The potential function control method represents a means of both estimating the desired states of a spacecraft’s location, and autonomously correct and control these states. It is based on Lyapunov’s method for stability analysis and its efficiency in the problem of collision avoidance is due to the fact that it aims to avoid a particular condition rather than to reach a state of equilibrium. FORMATION FLIGHT DYNAMICS We will now introduce the model used for formation dynamics, including the simplifications used. We assume that the Sun-Earth system is not disturbed by the inclusion of a third infinitively small body. The whole system rotates with a constant angular velocity ω, about the barycentre G, the position of which is a function of the Earth mass mE and the Sun mass mS as shown in Figure 1. The Earth is in a circular orbit around the Sun at distance RES. Additionally, Sun and Earth are supposed to be perfect spherical bodies. The acceleration of a spacecraft M in the inertial frame R0 is: 0 0 0 2 M M M R R M R R R R a a a V ∈ = + + Ω ∧ r r r r r R [1] where denotes an acceleration , V a velocity and a r Ω r an angular velocity . In the local frame the acceleration becomes: