Recuperación de la inversión

Many different control strategies, schemes and applications of multiple vehicle control can be found in literature. While the applications are different, e.g. multiple robots, unmanned air vehicles, unmanned underwater vehicles and spacecrafts, the fundamental approaches for formation control are similar being the objective the coordination of multiple vehicles. As mentioned in Section 1.1, at least one spacecraft of the formation must track a desired state profile relative to another spacecraft and the associated tracking control law must at the minimum depend upon the state of this other spacecraft. A control law satisfying the last condition is called a formation tracking control law.

Based on the above definition of FF, Scharf et al. [37] present a comprehensive survey of the spacecraft Formation Flying Control (FFC) literature. Specifically, FFC refers to design techniques and associated stability results for formation tracking control laws. In [38], Lawton et al. overview the FFC literature up to 2000 and define three main FFC architectures. In [37], Scharf et al. examine these architectures in more detail and added two new architectures, including the prolific research of the last few years and emphasize theoretical developments.

2.3.3.1 Control Architectures for Satellite Formation Flying

The primary distinction in the FFC literature is the type of FFC architecture used. A formation control architecture can be in general defined as a coordination scheme. In [4], Scharf et al. describe five basic formation architectures, briefly described in the following.

Leader/Follower Architecture. In the Leader/Follower (L/F) architectureone satellite is de- signed as leader tracking a predefined trajectory while the other satellites are designated as followers following transformed versions of the states of their nearest neighbors according to formation ge- ometry. This kind of architecture has also been referred to as Chief/Deputy, Master/Slave and Target/Chase4. The L/F has therefore a hierarchical organization in which only the leader vehicle knows the formation shape and goal.

The implementation of the L/F architecture is very simple, since individual spacecraft con- trollers reduces to tracking problems. However, there is no explicit feedback from the followers to the leader and the leader is a single point failure.

Multiple-Input, Multiple-Output Architecture. The formation is treated as a single multiple- input, multiple-output plant and all the methods of modem control may be applied to formation control; e.g. LQR controller, Directed Graphs and, more recently, nonlinear and constrained Model Predictive Control (MPC) strategies.

The primary advantages of the MIMO architecture are optimality and stability. However, the main disadvantages of this type of architecture are due to its high information requirement, since the entire state is used, and its poor robustness to local failure, since a local failure can have a global effect.

Cyclic. A formation controller in the Cyclic architecture is formed by connecting individual spacecraft controllers in a non-hierarchical architecture resulting in a cyclic control dependency directed graph.

The cyclic architecture basically lays between the L/F and the MIMO architectures and can perform better that L/F algorithms and distribute control effort more equally, since individual controllers are not hierarchically connected.

Virtual Structure Architecture. In Virtual Structure Architecture, the entire formation is treated as a single structure. For example, in an interferometry mission it may be desirable to have a constellation of spacecraft act as a single rigid body. In the virtual structure approach, the control is derived in three steps: first, the reference state trajectory of the structure is defined according to mission objectives; second, the motion of the virtual structure is translated into the reference motion for each satellite of the formation; and third, tracking control for each spacecraft is actuated.

Advantages of this formation control approach are that it is easy to prescribe the coordinated behavior for the group, and that the virtual structure can maintain the formation very well during the maneuvers in the sense that the virtual structure can evolve as a whole in a given direction with some orientation and maintain a rigid geometric relationship among multiple spacecrafts. Disadvantage is essentially related to the cases where formation shape is time-varying or it needs to be reconfigured frequently.

Behavioral Architecture. In the Behavioral Architecture all units work together to reach for- mation shape and goal on the basis of a given set of behavioral schemes. Possible behaviors include collision avoidance, obstacle avoidance, goal seeking and formation keeping.

Advantages of this formation control approach are that it is natural to derive control strategy when spacecraft have multiple competing objectives, explicit feedback is included through com- munication between neighbors. Disadvantages are that the group behavior cannot be explicitly defined, and it is hard to analyze the behavioral approach mathematically and guarantee its group stability.

As clearly stated in [38], these five control architectures may be viewed as special cases of a more general control architecture (block diagram is given in Figure 2.5), motivated by the existence of several levels of control in formation flying.

ˆ Spacecraft local control (Ki, i index of spacecraft). The system Si represents the ith space- craft, with control input vector ui representing control forces and torques, and output vector yirepresenting the measurable output of the spacecraft, most likely position and attitude vec- tors. The inputs toKiare the output of theithspacecraft yiand the coordination variableξ. The outputs of Ki are the control vectorui, and the performance variable zi. Starting from coordinate variables ξ (i.e. reference state trajectory, commands) and measured spacecraft

output y (i.e. absolute and/or relative position and attitude) , it computes the command for its actuators u.

ˆ Formation control (F ): it represents the primary coordination mechanism in the system. The formation control block outputs the coordination variable ξ which is broadcast to all spacecraft. In addition, the formation control block outputszF , which encapsulates the per- formance of the formation, to the supervisor. The inputs to F are the performance variables from each spacecraft zi, and the output of the supervisor uG. Starting from supervisor out- puts uG and measured i spacecraft local control output zi (performance variable, operating states), it computes the coordinate variables ξ.

ˆ Supervisor (G): it is discrete-event control that uses the measured formation control output zF to determine the input to the formation controluG.

CHAPTER 2 47 … Supervisor Formation Control Local control Spacecraft 1 Spacecraft N Local control G F Coordination variable  K1 KN S1 SN Formation Flying Control System Real World u1 y1 uN yN z1 zN uG zF

Figure 2.5: General Architecture for Spacecraft Formation Flying [38].

The formation control F is in charge of the coordination of each spacecraft present inside the formation. In general, it is in charge of the control of the absolute position control, the spacecraft relative control, the absolute attitude control and the relative attitude control in order to permits formation acquisition, formation keeping, formation maneuvers (nominal observation and contingency). The local controlKi is a slave of the formation control. It takes charge the spacecraft attitude and position control according to the outputs given by the formation control. The local control may be also a simple feed-through toward the actuators of the formation control outputs. The supervisor G is in charge of the mode transition according to the received tele-commands

coming from ground-segment, and events generated by formation control (failure detection events, convergence achieved, etc.).

2.3.3.2 Approaches for Formation Flying Control

There are basically three different ways or approaches5 to control formations in space [39]:

ˆ ground-based control; ˆ ground-in-the-loop control;

ˆ fully space-borne autonomous control.

A ground-based control is primarily restricted to formations with a large separation and control windows. GRACE may be a good example of this way (nominal separation 200 km, control win- dow about 50 km), where the required formation keeping maneuvers take place every few weeks. Avoiding the complexity of on-board autonomous systems, such maneuvers are efficiently executed by the satellite operations center.

Ground-in-the-loop control of formation might be necessary when high safety requirements have to be met, such as when a docking takes place involving manned missions, as for example the STS docking to the ISS.

A fully autonomous space-borne control of formation is a must when close formations (separation distance < 1km) with tight control windows, related to formation reconfiguration and/or collision avoidance. Fully autonomous control is required also when the required relative position control accuracy requires real-time.

Chapter 3

CONTROL STRATEGIES