Space debris is a rising concern for the sustained operation of our satellites. The population in space is continually growing, both gradually with a steady stream of new launches, and in sudden bursts, as evidenced with the recent collision between the Iridium and inactive COSMOS spacecraft. The problem is most severe in densely populated orbit regimes, where satellites face a sustained presence of close-orbiting objects. In general, the frequent occurrence of potential collisions with debris

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... ill have a negative impact on mission performance in two important ways. Firstly, repeated avoidance maneuvers diminish fuel and thus reduce mission life. Secondly, excursions from the nominal orbit during avoidance maneuvers may violate mission requirements or payload constraints. It is therefore important to consider both fuel minimization and station-keeping objectives in the avoidance planning problem. In this paper, we formulate the avoidance maneuver planning problem as a linear program (LP). Avoidance constraints and orbit station-keeping constraints are expressed as linear functions of the control input. The relative orbit dynamics are modeled as a discrete, linear time-varying system that models both circular and eccentric orbits. The original non-linear, non-convex avoidance constraints are transformed into a time-varying sequence of linear constraints, and the navigation uncertainty is applied in a worst-case sense. Finally, the minimum-fuel avoidance maneuver problem is formulated with stationkeeping constraints in a way that enables automatic relaxation of certain constraints to ensure feasibility. be replaced. The benefits of this kind of decentralized design include greater redundancy, the potential for lower launch costs and, the ability to extend mission life by replacing individual components. An immediate consequence of this paradigm, however, is that each module in the system effectively becomes another piece of debris that all other maneuverable satellites must avoid. The relative motion between spacecraft can be divided into two categories: 1) high relative velocity, such as the intersection of two non-coplanar orbits, and 2) low relative velocity, where the orbits are essentially coplanar with small differences in the orbital elements. High relative velocity objects are in close proximity only over short time scales (on the order of seconds), and therefore an avoidance maneuver would have to be implemented prior to the time of intersection. Fly-by's with low relative velocity objects are happening more and more frequently, due to the crowding of popular orbit regimes, especially sun-synchronous and geo-stationary orbits. In these scenarios, the time scales of close-proximity motion are much longer, on the order of hours or days. It is therefore physically possible in these cases for a spacecraft to detect neighboring objects and to enact avoidance maneuvers to ensure safe separation distances. Spacecraft with the capability to detect potential collisions, then plan and enact avoidance maneuvers can successfully mitigate the risk. In general, the frequent occurrence of potential collisions will have a negative impact on mission performance in two important ways. Firstly, repeated avoidance maneuvers diminish fuel and thus reduce mission life. Secondly, excursions from the nominal orbit during avoidance maneuvers may violate mission requirements or payload constraints. It is therefore important to consider both fuel minimization and station-keeping objectives in the avoidance planning problem. The topics of optimal maneuver planning and collision avoidance for close-orbiting spacecraft have been studied by numerous researchers in recent years [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] . Previous papers by Princeton Satellite Systems [16, 17, 18] have discussed the architectural considerations for large scale systems of formation flying satellites. Included in these papers are discussions of relative orbit dynamics in circular and eccentric orbits, the use of convenient parameter sets to describe desired periodic relative motion, and methods for planning reconfiguration maneuvers by formulating the problem as a linear program (LP). In 2008, a proposed LP-based method for robust avoidance maneuver planning was presented [19] , and in 2009 the approach was extended to capture both both low and high relative velocity encounters [20] . In this paper, we focus on avoidance planning for low relative velocity encounters, and consider the simultaneous objective of station-keeping during the avoidance maneuver. The maneuver planning problem is formulated as an LP, with avoidance and orbit station-keeping constraints expressed as linear functions of the control input. The relative orbit dynamics are modeled as a discrete, linear time-varying system that models both circular and eccentric orbits. The original non-linear, non-convex avoidance constraints are transformed into a time-varying sequence of linear constraints, and the navigation uncertainty is applied in a worst-case sense. The resulting maneuver can be solved efficiently as an LP with no integer constraints, and can guarantee collision avoidance with respect to bounded navigation uncertainty. Finally, the minimum-fuel avoidance maneuver problem is formulated with station-keeping constraints in a way that enables automatic relaxation of certain constraints to ensure feasibility. This method is compared to an ad hoc approach of constraint relaxation which reduces the overall dimension of the problem, but which requires the LP to be solved multiple times. The paper is organized as follows. Section II discusses how the relative orbit dynamics are modeled as a linear system. Section III presents the method of generating the LP problem formulation, including stationkeeping constraints, separation constraints, and accounting for uncertainty in the initial state. Section IV discusses the basic theory of constraint relaxation and provides a simple 2-dimensional example for illustration. Finally, Section V present simulation results of various avoidance maneuvers performed in conjunction with relaxed station-keeping constraints. The linear time-varying (LTV) dynamic equations of a satellite's relative motion in the local-vertical / local-horizon (LVLH) frame are given as follows [4]: d dt 