Design of minimum time maneuvers for multispacecraft interferometric imaging systems

Abstract

In this paper, we consider the design of minimum time maneuvers for multi-spacecraft interferometric imaging systems. We show that the process of image formation in a multispacecraft interferometric imaging system is analogous to painting a "large disk" with smaller "paintbrushes," while satisfying a paint thickness constraint. We show that spiral maneuvers form the dominant set for the painting problem. Further, we frame the minimum time problem in the space of spiral maneuvers and obtain the Double Pantograph Problem. We show that the solution of a discretized version of the Double Pantograph Problem is given by the solution to two associated linear programming problems. We illustrate our results through an imaging example where the image of a fictitious exo-solar planet is formed using the maneuver prescribed by the Double Pantograph Problem.

Modeling of Image Formation in Multi-Spacecraft Interferometric Imaging Systems

“…Furthermore, it would be reasonable Tf to expect that the critical paint thickness for satisfactory imaging performance would be dependent on the application at hand, i.e., the questions that the image would be used to answer [6]. The solution to the time optimization problem posed in this paper is addressed in [7,8]. However, the problem of fuel minimization while satisfying the imaging constraints is an open question and is a subject of our current research [11,12].…”

“…It has been argued in [9,10,7] that spiral trajectories might be the most efficient way of covering the spatial frequency plane. In fact, it has beenshown in [7,8] that they are time optimal. However, care needs to be taken to ensure that these spiral trajectories are dense enough so that they cover the resolution disk without leaving gaps.…”

“…We define the probability of misclassification of an estimate of some image as: (7) We say that an MSIIS (or any imaging system) performs satisfactory imaging if the following is satisfied for all (8) where represents the maximum allowable probability of misclassification. Note that, since the MTF and the noise corrupting the image in an MSIIS are both dependent on the trajectory of the MSIIS, it follows that the estimated image and the probability of misclassification are both functions of the MSIIS trajectory.…”

“…We do not address the dynamical issues of the problem in the current work. These issues are addressed in separate papers [8,11,12].…”

“…For a detailed discussion of the issues involved, please see [7,8] and the references therein. However, in the case of an MSIIS, the end goal of satellite formation flying is the synthesis of a good imaging system.…”