Formation flying for electric sails in displaced orbits. Part II: Distributed coordinated control

Highlights

• E-sail formation flying around a heliocentric displaced orbit is investigated.

• Distributed cooperative control laws are designed via consensus algorithms.

• Two scenarios with undirected and directed communication topology are analyzed.

Abstract

We analyze a cooperative control framework for electric sail formation flying around a heliocentric displaced orbit, aiming at observing the polar region of a celestial body. The chief spacecraft is assumed to move along an elliptic displaced orbit, while each deputy spacecraft adjusts its thrust vector (that is, both its sail attitude and characteristic acceleration) in order to track a prescribed relative trajectory. The relative motion of the electric sail formation system is formulated in the chief rotating frame, where the control inputs of each deputy are the relative sail attitude angles and the relative lightness number with respect to those of the chief. The information exchange among the spacecraft, characterized by the communication topology, is represented by a weighted graph. Two typical cases, according to whether the communication graph is directed or undirected, are discussed. For each case, a distributed coordinated control law is designed in such a way that each deputy not only tracks the chief state, but also makes full use of information from its neighbors, thus increasing the redundancy and robustness of the formation system in case of failure among the communication links. Illustrative examples show the effectiveness of the proposed approach.

Introduction

In recent years, much effort has been devoted to the study of an Electric Solar Wind Sail (E-sail), an interesting propulsion system that uses the natural solar wind dynamic pressure to generate a continuous low-thrust, without the need of any reaction mass (Janhunen and Sandroos, 2007, Mengali et al., 2008, Sanchez-Torres, 2016). A potential and challenging mission scenario of an E-sail-based spacecraft is to generate a heliocentric closed trajectory, usually referred to as displaced orbit (McInnes, 1997, Gong and Li, 2014, Salazar et al., 2016), in which the continuous propulsive acceleration is used to shift the spacecraft orbital plane off the Sun’s center-of-mass. A scientific application of this unusual trajectory is, for example, to continuously observe the polar region of a celestial body, such as a planet or an asteroid. However, the propulsive requirements for this kind of mission scenario, given in terms of maximum value of propulsive acceleration necessary to maintain the displaced orbit, could be beyond the technological capabilities of an E-sail propulsion system. A possible solution to this problem is to reduce the spacecraft launch mass (with a subsequent increase of the E-sail characteristic acceleration) by distributing the payload among different E-sail-based vehicles operating in a formation flight, where each functional module (spacecraft) of the formation takes the essential mass only (Mazal and Gurfil, 2013, Salazar et al., 2015).

So far, the control problem of different spacecraft flying around a heliocentric displaced orbit falls into two main categories, that is, control with single or leader-follower strategy. In the first case the concept is to distribute a number of sail-based (either photonic solar sail or E-sail) spacecraft into different displaced orbits and to control them separately, without the need of any real-time information about the position of each spacecraft with respect to the chief vehicle (Wang et al., 2016b, Wang et al., 2016a). In a companion paper (Wang et al., 2017) the same idea has been applied to a set of E-sail spacecraft in a formation flight. With such a simple control strategy, some typical formation geometries, as well as the bounds of the spacecraft relative motion, can be analytically estimated by selecting the displaced orbital elements. However, the robustness of the formation system cannot be guaranteed since no stability control is involved.

The second concept, instead, assumes the chief spacecraft to follow a prescribed displaced orbit, and the deputies to adjust their thrust vectors (that is, both the sail attitude and the characteristic acceleration) in order to track the desired relative trajectories with respect to the chief (Gong et al., 2007, Gong et al., 2011). This is the so called chief-deputy or leader-follower control strategy. Nevertheless, inherent limitations also exist in the latter system arrangement. For example, the unique chief spacecraft, which represents the only information source about the reference state for each deputy, is a single point of massive failure for the whole group (Ren, 2007). Another weakness associated with a chief-deputy strategy is the absence of a mutual feedback information flow throughout the formation structure. As a result, an unfavorable situation may arise if a fault happens in the chief-deputy communication links. A possible improvement consists in including the information exchange among the deputies into the feedback control. In addition, it has been proved that the mutual connection of agents also contributes to an accuracy enhancement during the transient motion (Ren, 2006).

Recognizing these issues, this paper concentrates on the problem of cooperative control for multiple E-sail formation flight around a heliocentric (elliptic) displaced orbit, by making full use of the measurable information among the formation structure. In particular, two qualitatively different cases involving either a “directed” or an “undirected” weighted graph of communication topology are addressed via consensus algorithms (Ren, 2006). The basic idea of information consensus is that each agent in a group updates its state on the basis of the data obtained from its local neighbors, in such a way that the final state of each agent converges to some consistent common value. In addition, the fundamental protocol of the consensus algorithm can be extended to deal with the problem that the state of each agent converges to a desired relative separation value or incorporate different group behaviors into the consensus building process (Ren and Atkins, 2007). The emphasis of this work relies on the fact that every available neighbor-to-neighbor information exchange among deputy E-sails is included into the feedback control system, thus preventing an undesirable situation in which a failure of the chief spacecraft would give rise to potential risks to the whole formation structure. In this sense, this analysis completes the results of Wang et al. (2017) where the chief-deputy relative motion is discussed without the use of a cooperative control.

The paper is organized as follows. The next section briefly summarizes the mathematical model used to calculate the E-sail propulsive requirements necessary to maintain a prescribed heliocentric displaced orbit. In particular, with the aid of the main results discussed in Wang et al. (2017), Section 2 illustrates the model for analyzing the relative motion of two spacecraft around a heliocentric, elliptic, displaced orbit. This relative dynamics is then used in Section 3 for the study of the control system. The latter, which is based on the consensus algorithm, allows the E-sail thrust vector to be oriented so as to track the desired chief-deputy relative trajectory. The control system effectiveness is then investigated in Section 4 by means of numerical simulation of some mission scenarios of particular interest. Finally, Section 5 contains the concluding remarks.