International Journal of Aerospace Engineering

Volume 2016, Article ID 3518537, 7 pages

http://dx.doi.org/10.1155/2016/3518537

## A GRASP for Next Generation Sapphire Image Acquisition Scheduling

^{1}School of Management, Northwestern Polytechnical University, 127 Youyi West Road, Xi’an 710072, China^{2}Department of Mathematics, Simon Fraser University Surrey, 250-13450 102nd Avenue, Surrey, BC, Canada V3T 0A3^{3}MacDonald, Dettwiler and Associates Ltd., Richmond, BC, Canada

Received 25 April 2016; Accepted 10 October 2016

Academic Editor: Paolo Tortora

Copyright © 2016 Yang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates an image acquisition scheduling problem for a Canadian surveillance-of-space satellite named Sapphire that takes images of deep space Earth-orbiting objects. For a set of resident space objects (RSOs) that needs to be imaged within the time horizon of one day, the Sapphire image acquisition scheduling (SIAS) problem is to find a schedule that maximizes the “Figure of Merit” of all the scheduled RSO images. To address the problem, we propose an effective GRASP heuristic that alternates between a randomized greedy constructive procedure and a local search procedure. Experimental comparisons with the currently used greedy algorithm are presented to demonstrate the merit of the proposed algorithm in handling the SIAS problem.

#### 1. Introduction

Sapphire is an outward looking satellite used to take images of Earth-orbiting objects in outer space, so-called resident space objects (RSOs). An RSO can be a man-made object or a natural space object. An acquisition request is defined by an RSO that has to be imaged. Within the day, there may be multiple acquisition requests for imaging one RSO. Given a time horizon of one day, the* Sapphire image acquisition scheduling problem* (SAIS) is to find a schedule of a set of acquisition requests that maximizes the sum of the “Figure of Merit” (FoM) of all the scheduled RSO images, where imaging each RSO has time window constraints. The minimal time difference between the two consecutive images of the same RSO is not considered.

The “Figure of Merit” is an integrated measure of five items, including image priority, off-pointing angle, imaging time, manoeuver angle, and manoeuver time. The five items are combined as a product in the measure to generally favor images with high priority and small imaging time. The manoeuver angle is defined as the angle between the pointing vector of the imaging end time for the current RSO and the pointing vector of the imaging start time for the next RSO. The Sapphire manoeuver time between two acquisition requests and calculates the time needed for Sapphire to change its pointing angle from pointing towards at the end of its scheduled imaging time to pointing towards the at the start of its scheduled imaging time. The off-pointing angle measures how much Sapphire is oriented off the most desired position for its solar panels towards the Sun.

The Sapphire image acquisition scheduling problem is a hard discrete optimization problem, which can be modeled as a prize collecting selective asymmetric traveling salesman problem with multiple time windows for each customer, with service times, and with time-dependent distances between the customers. Each imaging request can be represented by a node of a directed graph G. A directed arc is added between any two imaging requests that can be taken one after another. The weight of an arc represents the duration of the Sapphire maneuver needed for imaging the two corresponding RSOs. Each node has a priority (prize) and the service time that corresponds to the imaging time.

The daily photograph scheduling of the SPOT5 satellite for specified areas on the Earth’s surface has been widely studied in the literature. Bensana et al. (1996) [1] formulated the daily management of an Earth observation satellite as a variable valued constraint satisfaction problem, which is handled by exact and approximate methods. Vasquez and Hao (2001) [2] formulated the daily photograph scheduling of an Earth observation satellite as a knapsack model and proposed a tabu search algorithm to determine the scheduling. Wolfe and Sorensen (2000) [3] defined the scheduling problem as a time window constrained packing problem and devised a fast and simple priority dispatch method, a look ahead algorithm, and a genetic algorithm. Mansour and Dessouky (2010) [4] presented a genetic algorithm to maximize a multicriteria objective including the profit and the number of acquired photographs. Ribeiro et al. (2010) [5] employed a commercial solver combined with a strengthened formulation based on valid inequalities arising from the node packing and 3-regular independence system polyhedra.

The photograph acquisition scheduling problems for an agile Earth observation satellite or for a set of agile Earth observation satellites have been solved by varying approaches. Lemaître et al. (2002) [6] developed four different methods to solve the track selection and scheduling problem for agile Earth observation satellites, including a fast greedy algorithm, a dynamic programming algorithm, a constraint programming algorithm, and a local search algorithm. Bianchessi et al. (2007) [7] considered the problem of selecting and scheduling requests for a constellation of agile satellites and devised a tabu search algorithm with solution quality evaluated by a column generation method. Habet et al. (2010) [8] addressed an agile Earth observing satellite scheduling problem with stereoscopic and time window visibility constraints by hybridizing a tabu search and a systemic search that uses partial enumerations. Upper bounds are given by a dynamic programming algorithm on a relaxed problem. Jang et al. (2013) [9] studied an image collection planning problem for a Korean satellite KOMPSAT-2 and developed a binary integer programming model and a heuristic solution procedure using the Lagrangian relaxation and subgradient methods. Tangpattanakul et al. (2015) [10] modeled the selection and scheduling of observations taken by an agile satellite as a two-objective optimization problem and solved it with an indicator-based multiobjective local search. Xu et al. (2016) [11] investigated an Earth observation scheduling problem for agile satellites under a time window constraint and resource constraints, which are solved by several constructive procedures including an ant colony optimization approach and two heuristics.

Currently used Sapphire image acquisition scheduling algorithm is a greedy heuristic (GH). In this paper, we explore a GRASP heuristic (greedy randomized adaptive search procedure) that integrates a randomized greedy constructive procedure and a simple insertion improvement procedure to enhance the scheduling performance. Our experimental study has demonstrated that the developed algorithm is not only able to improve the objective “Figure of Merit,” but also improves the number of scheduled requests as a bonus.

The paper is organized as follows. In Section 2, we present the problem model of the Sapphire image acquisition scheduling problem. Section 3 presents the proposed GRASP algorithm. Section 4 provides the results of the computational experiments. Finally, concluding remarks are presented in Section 5.

#### 2. Problem Model

The Sapphire image acquisition scheduling (SIAS) problem is to find a schedule that maximizes the sum of the “Figure of Merit” of all the scheduling requests. A solution to the SIAS problem, a schedule, is a set of the scheduled requests, where represents all the acquisition requests. The model of the SIAS problem is formulated aswhere the set contains all schedules satisfying the following constraints:(1)Within the time horizon of one day, each RSO can be observed in multiple different time windows(2)Each time the current request to image is finished, Sapphire needs manoeuver time to take the next request to image , and the minimum manoeuver time is 80 seconds and increases as a function of the manoeuver angle magnitude(3)The imaging time for each RSO is known(4)The next request cannot start earlier than the end time of the current request and must be finished by its time window(5)No imaging activity happens outside the planning time horizon

We assume the following:(1)Memory on-board is not a constraint(2)The energy for moving or maneuvering of the satellite is not an issue (the payload is rigidly attached to the satellite)(3)Collecting the solar energy is not a task that needs to be scheduled; however, the satellite solar panel orientation relative to the Sun is part of the “Figure of Merit” for the RSO selection(4)Sapphire telescope is focused at infinity and this is a constant

#### 3. GRASP: Greedy Randomized Adaptive Search Procedures

GRASP, which stands for Greedy Randomized Adaptive Search Procedure, is an effective approach to solving many optimization problems [12, 13]. It mainly consists of a randomized greedy component to produce an initial solution and a solution improvement component to refine the initial solution. The randomized greedy component integrates greedy and random features to sample solutions in the solution space so as to produce initial solutions with good solution quality and diversity. The solution improvement component can be any local search based heuristics, for example, tabu search [14, 15] and variable neighborhood search [16, 17].

Our proposed GRASP for the SIAS problem is shown in Algorithm 1. Since the purpose of this work is to improve performance of the previous greedy heuristic, our algorithm begins by generating an initial schedule using this greedy heuristic. If a “start” of this multistart algorithm finds a schedule with a better objective value than the previously best found value , the algorithm updates the best schedule and the objective value . After multistarts, that is, after the “start” variable reaches ( is a parameter), the GRASP is terminated. The following sections present details of the GRASP adaptation to the SIAS problem.