Spacecraft Formation - Design and Control
Spacecraft formations and distributed space systems in general have gained broad attention in the last decades in part due to their reliability and flexibility. These systems have a wide range of applications, including planetary science, Earth science, astrophysics, and solar system exploration. The deployment of spacecraft swarms presents several challenges. The formation must be established by designing geometrically-defined relative trajectories for each of the agents which is traditionally done using Cartesian coordinates. Although widely used, these coordinates do not provide strong insights into the geometrical properties of the relative orbits and are therefore not intuitive to work with. Another challenge is the derivation of a control law to bring each agents on its target orbit. Commonly used methods rely on trajectory propagation and optimal control theory, two processes which are computationally intensive. The close proximity of the agents in a formation drives the need for on-board algorithms able to monitor and prevent collisions in near real time. The research presented here addresses these two challenges with the development of a computationally lightweight control algorithm.
Several sets of elements have been developed to ease the design of relative trajectories by providing parameters which offer strong geometrical insights into the shape, size, and orientation of relative orbits. One such set called relative orbital elements (ROEs) has been presented by Lovell and Tragesser and is used here. This set consists of six elements describing the instantaneous relative trajectory of a deputy spacecraft as seen from the chief's local-vertical, local-horizontal (LVLH) frame and is derived from the analytical solution to the Clohessy-Wiltshire equations.
Artificial potential functions (APFs) methodologies have first been developed for the control of terrestrial robots in the 1980s. Later expanded to space applications in the 1990s, these methods are based on the definition of a scalar potential field over the space which admits a minimum at the location of a desired goal. A control law is then derived by computing the gradient of the APF and following the direction of steepest descent of the gradient. This minimization ultimately leads the spacecraft to the minimum of the APF, which by construction corresponds to the desired goal. One of the strengths of APFs is their ability to account for obstacles by adding a repulsive component to the potential.
The control algorithm derived in this research is based on an APF defined in terms of ROEs where the attractive part of the potential is a quadratic function and the repulsive part a Gaussian function. The use of ROEs, in addition to providing strong geometrical insights, has the advantage of allowing the design of relative orbits rather than relative positions. A reference trajectory is constructed by augmenting the equations describing the natural evolution of the ROEs with the partial derivatives of the APF taken with respect to the ROEs. This reference trajectory is then transformed into Cartesian coordinates and is tracked using a two-burn scheme.
This control methodology benefits from the intuitive representation of relative trajectories provided by the ROEs as well as the light computational cost resulting from the use of APFs. Its ability to monitor and prevent collisions in near real time makes it an ideal candidate for spacecraft formations where fast decision-making capability is desired.
Three simulations are presented to illustrate the use of this algorithm. In the first video, a 10-node triangular lattice formation is established through controlling the spacecraft to prescribed stations in two concentric circular relative orbits.
In the second video, a 19-node hexagonal lattice formation is established through controlling the spacecraft to prescribed stations in four concentric circular relative orbits.
The third video shows the establishment of a 12-spacecraft formation in which the spacecraft are equally distributed on two circular relative orbits. Two distinct relative orbital planes are used for this geometry.
In the last video, two hexagonal lattices are established on two distinct relative orbital planes. This formation is composed of 37 spacecraft.