Atri Dutta

Egalitarian Peer-to-Peer Satellite Refueling Strategy

bi = supply/demand corresponding to a node in Vn C M = cost of an egalitarian peer-to-peer solution M CLB = optimal value of objective function of the optimization problem cij = cost of an edge in Gn cij = cost of an edge in G‘ c i; j; k = cost of a triplet i; j; k representing an egalitarian peer-to-peer maneuver d = distance between two triplets tp and tq E‘ = set of edges in G‘ Ef = set of edges in Gn representing forward trips of active satellites En = set of edges in Gn Er = set of edges in Gn representing return trips of active satellites Es = set of source arcs in Gn Et = set of sink arcs in Gn fi = maximum fuel capacity of satellite si f i = minimum fuel requirement by satellite si to remain operational fi;t = fuel content of satellite si at time t G = constellation digraph G‘ = bipartite graph used for calculating lower bound on cost of optimal e-p2p solution Gn = constellation network g0 = acceleration due to gravity at surface of the earth Isp = specific thrust of satellite s J = index set for satellites/orbital slots J a = index set for orbital slots of active satellites J p = index set for orbital slots of passive satellites J r = index set of orbital slots available for active satellites to return J d;t = index set for orbital slots of fuel-deficient satellites at time t J s;t = index set for orbital slots of fuel-sufficient satellites at time t M = egalitarian peer-to-peer solution composed of a set of triplets M = optimal egalitarian peer-to-peer solution MH = egalitarian peer-to-peer solution obtained after local search onMIP MIP = Egalitarian peer-to-peer solution yield by the optimization problem MP2P = optimal egalitarian peer-to-peer solution ms = mass of permanent structure of satellite s N2 tp; tq = two-exchange neighborhood of a triplet pair comprising tp and tq N M = neighborhood of an egalitarian peer-to-peer solution M p ij = fuel expenditure required for an orbital transfer by satellite s from slot i to slot j Q i; j; k = edge in E‘ corresponding to a triplet i; j; k si = satellite with index i 2 J T = total time allotted for refueling T = set of feasible triplets in the constellation graph tp = triplet ip; jp; kp with index p Vn = set of vertices in Gn xij = binary variable corresponding to an arc i; j 2 En or an edge hi; ji in E‘ Vij = velocity change required for a transfer from slot i to slot j = suboptimality measure

Asynchronous optimal mixed P2P satellite refueling strategies

In this paper, we study pure peer-to-peer (henceforth abbreviated as P2P) and mixed (combined single-spacecraft and P2P) satellite refueling in circular orbit constellations comprised of multiple satellites. We consider the optimization of two conflicting objectives in the refueling problem and show that the cost function we choose to determine the optimal refueling schedule reflects a reasonable compromise between these two conflicting objectives. In addition, we show that equal time distribution between the forward and return flights for each pair of P2P maneuvers does not necessarily lead to the optimum cost. Based on this idea, we propose a strategy for reducing the cost of P2P maneuvers. This strategy is applied to pure P2P refueling scenarios as well as to mixed refueling scenarios. Furthermore, for the case of a mixed scenario, we propose an asynchronous P2P strategy that also leads to more efficient refueling.

Network Flow Formulation for Cooperative Peer-to-Peer Refueling Strategies

The problem of a general peer-to-peer refueling strategy for satellites in a circular constellation is addressed. The proposed cooperative egalitarian peer-to-peer strategy allows the satellites participating in a refueling transaction to engage in a cooperative rendezvous, that is, both satellites engaging in a fuel exchange may be active. Furthermore, the active satellites are allowed to interchange their orbital positions during their respective return trips. A mathematical framework to solve this general refueling problem for a large number of satellites is proposed using ideas from network flow theory. The methodology determines the optimal set of maneuvers that achieve fuel-sufficiency for all satellites, while expending the minimum possible fuel during the ensuing orbital transfers. With the help of numerical examples it is shown that the proposed cooperative egalitarian peer-to-peer strategy is the best amongst all known peer-to-peer refueling alternatives to date.

Peer-to-Peer Refueling Strategy Using Low-Thrust Propulsion

The problem of minimum-fuel, time-fixed, low-thrust rendezvous is addressed with the particular aim of developing a solver to determine optimal low-thrust peer-to-peer (P2P) maneuvers, which will be an integral part of distributed low-thrust servicing missions for multiple satellites. We derive the necessary conditions of optimality for P2Prefueling maneuversinacircularplanarconstellationsubjecttoatwo-bodygravity field. Wedevelopthe solver based on an indirect optimization technique and use the well-known shooting method to solve the two-point boundary value problems associated with the forward andreturn trips of a P2P maneuver. The solver is formulated to consider implicitly the n thrust-arc case without requiring a predetermined thrust structure. The complete P2P refueling decision simulation requires the solution to many nonlinear programming subproblems and, therefore, benefitsfromthefastindirectformulation.TheapplicationofthetoolinthedeterminationofoptimalP2Pmaneuvers required for a low-thrust P2P mission for multiple satellites moving in a circular orbit is demonstrated. Finally, the paper demonstrates, via numerical examples, the benefits of a low-thrust P2P mission compared with the impulsive case.

A Direct Optimization Based Tool to Determine Orbit-Raising Trajectories to GEO for All-Electric Telecommunication Satellites

In this paper, we consider the development of an optimization solver that provides optimal low-thrust trajectories to the Geostationary Orbit, starting from an arbitrary orbit into which the satellite has been injected by an appropriate launch vehicle. Based on a direct optimization methodology, we formulate a minimum-time orbit-raising problem and use solvers like IPOPT and LOQO to solve the resulting non-linear programming problem. The tool allows for consideration of on-board energy storage system that helps the satellite to thrust in the Earth’s shadow during an eclipse. Furthermore, the tool allows for the investigation of new scenarios from the point of view of reducing radiation damage incurred by the satellite during its transit through the Van Allen belt. For instance, we consider the case in which the satellite starting from an inclined orbit, delays any out-of-plane maneuvers until it crosses the inner Van Allen belt. The tool enables us to analyze electric orbit-raising scenarios for a variety of injection orbits and different technology alternatives (electric engines, on-board energy storage). We illustrate with numerical examples the usage of the developed tool for different orbit-raising examples. The development of this solver is a first-step towards an elaborate study of new mission scenarios for all-electric telecommunication satellites.

Hohmann-Hohmann and Hohmann-Phasing Cooperative Rendezvous Maneuvers

We consider the problem of cooperative rendezvous between two satellites in circular orbits, given a fixed time for the rendezvous to be completed, and assuming a circular rendezvous orbit. We investigate two types of cooperative maneuvers for which analytical solutions can be obtained. One is the case of two Hohmann transfers, henceforth referred to as HHCM, while the other, henceforth referred to as HPCM, is the case of a Hohmann transfer and a phasing maneuver. For the latter case we derive conditions on the phasing angle that make a HPCM rendezvous cheaper than a cooperative rendezvous on an orbit that is different than either the original orbits of the two participating satellites. It is shown that minimizing the fuel expenditure is equivalent to minimizing a weighted sum of the ΔVs of the two orbital transfers, the weights being determined by the mass and engine characteristics of the satellites. Our results show that, if the time of rendezvous allows for a Hohmann transfer between the orbits of the satellites, the optimal rendezvous is either a noncooperative Hohmann transfer or a Hohmann-phasing cooperative maneuver. In both of these cases, the maneuver costs are determined analytically. A numerical example verifies these observations. Finally, we demonstrate the utility of this study for Peer-to-Peer (P2P) refueling of satellites residing in two different circular orbits.

Design of Next-Generation All-Electric Telecommunication Satellites

In this paper, we revisit the problem of electric orbit-raising of telecommunication satellites. The paper outlines the mathematical framework we have developed to determine trajectories that minimize the radiation damage during orbit-raising. This framework also allows us to analyze a variety of mission scenarios in order to identify the favourable designs for all-electric satellites. We outline the number and type of thrusters most suitable for the orbit-raising maneuver and the associated design of the power subsystem. We also consider the possible launch options of the satellites and demonstrate the advantages that all-electric satellites have by being stacked together to reduce the launch costs compared to their chemical or hybrid counterparts.

A Cooperative P2P Refueling Strategy for Circular Satellite Constellations

In this paper, we discuss the problem of peer-to-peer (P2P) refueling of satellites in a circular constellation. In particular, we propose a cooperative P2P (C-P2P) refueling strategy, in which the satellites involved in P2P maneuvers are allowed to engage in cooperative rendezvous. We discuss a formulation of the proposed C-P2P strategy and a methodology to determine the optimal C-P2P assignments. We show that in order to reduce the fuel expenditure in a C-P2P maneuver, the amount of fuel exchanged between the two satellites is such that the satellite performing the larger-ΔV transfer during the return trip, ends up having just enough amount of fuel to be fuel-sufficient. Finally, with the help of numerical examples, we provide a comparison of the P2P and the C-P2P refueling strategies. It is found that a C-P2P strategy is beneficial when the fuel-deficient satellites in the constellation do not have enough fuel to complete a non-cooperative rendezvous.

Minimum-fuel electric orbit-raising of telecommunication satellites subject to time and radiation damage constraints

The capability of a Geostationary satellite to perform electric orbit-raising enables the development of all-electric satellites. Key challenges to the realization of electric deployment of satellites to the Geostationary orbit are the long transfer times, high power requirement of the electric thrusters and the solar array degradation experienced by the satellite during its transit through the Van Allen radiation belts. In this paper, we consider the problem of electric orbit-raising of a telecommunication satellite to the Geostationary orbit. We propose an optimization framework to minimize the fuel expenditure during the transfer subject to upper bounds on the transfer time and the displacement damage dose. Our developed framework considers the discretization of the satellite trajectory as well as the energy spectrum of protons encountered during the transfer, and utilizes analytic models of the geomagnetic field and associated radiation flux. Finally, we illustrate with numerical examples the fuel savings that can be achieved by comparing minimum-time and minimum-fuel solution.

Quaternion-Based Coordinates for Nonsingular modeling of high-inclination orbital transfer

c = characteristic constant for satellite propulsion system, m∕s F1, F2, F3 = thrust components in reference frame (b1, b2, er), N Isp = specific impulse of the thruster, s m = mass of the spacecraft, kg q = quaternion vector of components q1 t , q2 t , q3 t , and q4 t r = radial position, km T = thrust magnitude, N Tmax = maximum thrust, N t = time variable, s tf = final time, s u = control vector u, v, w = velocity vector components, m∕s x = state vector α, β = thrust direction angles in reference frame (eφ, eθ, er), rad θ, φ, ψ = Euler angles, rad ω1, ω2, ω3 = angular velocity vector components, rad∕s