COORDINATED CONTROL OF SATELLITES : THE ATTITUDE CASE

In this paper an observer-controller structure for attitude synchronization of a satellite formation is presented. The design applies methods from mechanical synchronization to design a nonlinear observer and controller for satellites actuated by means of reaction wheels and magnetic torquers. In this approach one satellite is defined as the leader of the formation, while the rest are defined as followers which synchronize their attitude with the leader. We apply the approach to the design of an coordinated attitude control scheme for a two-satellite formation. In addition we propose a momentum dumping scheme for satellites with redundant reaction wheel assemblies. I. I NTRODUCTION The purpose of this paper is to derive a control scheme to synchronize the attitude of satellites orbiting in formation, in such a way that the satellites are able to keep their relative attitude in the presence of disturbances. The coordinated control of multiple agents in a formation environment is an area which has attracted much attention internationally in later years. The increased activity in the field, comes from a number of advantages; several small spacecraft may cooperate solving missions that would require one large spacecraft, functionality may be distributed to increase redundancy, the vehicles may be launched in stages on several launch vehicles, reducing the risk of total mission failure. The advantages do however come at the cost of a more complicated control system. Examples of current projects are Darwin, where six satellites will fly in tight formation to perform analysis of Earth-like planets, and MicroSAR, which are small low-cost SAR satellites capable of land and sea observations. Noticeable contributions on formation control may be divided in to three separate approaches; leader-follower, behavioural and virtual structure. In the leader-follower strategy, one spacecraft is defined as the leader of the formation while the rest are defined as followers. The control objective is to enable the followers to keep a fixed relative attitude with respect to the leader [1]–[4]. The behavioural strategy views each vehicle of the formation as an agent and the control action for each agent is defined by a weighted average of the controls corresponding to each desired behaviour for the agent. This approach eases the implementation of conflicting or competing control objectives, such as tracking versus avoidance. It is however difficult to enforce group behaviour, and to mathematically guarantee stability and formation convergence. In addition, unforseen behaviour may occur when goals are conflicting. This strategy is widely reported for use on mobile robots [5]– [7], and was also applied to spacecraft formations in [8]. In the virtual structure approach, the formation is defined as a virtual rigid body. In this approach the problem is how to define the desired attitude and position for each member of the formation such that the formation as a whole moves as a rigid body. In this scheme it is easy to prescribe a coordinated group behaviour and to maintain the formation during maneuvers. It is however dependent on the performance of the individual control systems of each member. This approach was used on mobile robots in [9] and more recently on spacecraft formations in [10], [11]. Stability analysis of both the observer and the synchronizing control scheme has been performed using an extension of a theorem due to Matrosov [12], given in [13]. This theorem proves to be very useful in the case of a semidefinite Lyapunov derivative when the analysed system is time-varying, as for example in tracking control with a timevarying reference. In such a case the theorem usually referred to as LaSalle’s invariance principle is not applicable. A. Mechanical synchronization The synchronization phenomenon describes the event when dynamical systems in some sense exhibit a similar behaviour in the time domain. In [14] some formal definitions were developed to describe the synchronization phenomenon, distinguishing between frequency and coordinate synchronization. Frequency synchronization describes the situation when the frequency of motion conforms to an integer multiple of a given frequency, ωs, while coordinate synchronization occurs when the outputs or some state-variables of a system, coincide with the corresponding variables of some other system for all t ≥ 0 or asymptotically as t → ∞. In [4] coordinate synchronization was used in the synchronization of robot manipulators and in [15] it was proposed to use the theory in the replenishment and rendezvous of ships. In this paper we adopt the problem formulation of [4], referred to as external synchronization. In this formulation we define a leader system, which is the dominant system, and a bounded set of follower systems. The synchronization problem consists of creating either physical interconnections or control feedback loops, which forces the outputs of the follower system to conform with those of the leader. B. Electromagnetic actuators Electromagnetic actuators are often chosen due to the independence of a limited fuel source, depending instead on power from solar arrays and batteries, thereby prolonging the lifespan of the mission. Electromagnetic actuators, often referred to as magnetic torquers, are based on two basic configurations. One is the coil based, where current is sent through a current loop which generates the magnetic moment proportional to the area of the coil and the number of windings. The other type is the magnetic rod, where a wire is wound around a rod made of a high permeability material. Both variations interact with the local geomagnetic field, generating a torque vector in the direction perpendicular to the magnetic moment and local field vectors. C. Wheel actuators A reaction wheel is essentially a torque providing motor with high rotor inertia. It is able to load and unload angular momentum internally, and is thus often referred to as a momentum exchange device, as it does not change the overall angular momentum of the satellite, but redistributes it to different parts. The amount of torque provided is dependent on the size of the rotor and motor, and is usually in the range from 0.01 Nm to 1 Nm. A wheel complete with motor and drive electronics, is usually referred to as a reaction wheel assembly (RWA). Three wheels, one along each axis, is needed for full threeaxis control. For redundancy and performance a composition of RWAs usually consists of more than three wheels. An Fig. 1. Illustration of the reference frames for a satellite in equatorial orbit. The axes not shown are pointing out of the paper. example is the tetrahedron composition. A regular tetrahedron is a pyramid composed of four equilateral triangular faces, three of which meets at each vertex. Each wheel-axis is placed orthogonal to a face, and meet at the center of the pyramid. The torque from a wheel to the body equals the torque applied to the wheel from a motor attached to the body, but with opposite sign.