Optimal Planetary Orbital Transfers via Chemical Engines and Electrical Engines

Because the orbital periods for planetary orbital transfers are of order hour, the primary objective of an optimal trajectory is to minimize the propellant consumption. In this paper, we present a systematic investigation of optimal trajectories for planetary orbital transfer. Major results on thrust control, propellant consumption, and flight time are presented with particular reference to LEO-to-GEO transfer. The following results were obtained with the sequential gradient-restoration algorithm in either single-subarc form or multiple-subarc form:

Optimal Starting Conditions for the Rendezvous Maneuver, Part 1: Optimal Control Approach

We consider the three-dimensional rendezvous between two spacecraft: a target spacecraft on a circular orbit around the Earth and a chaser spacecraft initially on some elliptical orbit yet to be determined. The chaser spacecraft has variable mass, limited thrust, and its trajectory is governed by three controls, one determining the thrust magnitude and two determining the thrust direction. We seek the time history of the controls in such a way that the propellant mass required to execute the rendezvous maneuver is minimized. Two cases are considered: (i) time-to-rendezvous free and (ii) time-to-rendezvous given, respectively equivalent to (i) free angular travel and (ii) fixed angular travel for the target spacecraft. The above problem has been studied by several authors under the assumption that the initial separation coordinates and the initial separation velocities are given, hence known initial conditions for the chaser spacecraft. In this paper, it is assumed that both the initial separation coordinates and initial separation velocities are free except for the requirement that the initial chaser-to-target distance is given so as to prevent the occurrence of trivial solutions. Analyses performed with the multiple-subarc sequential gradient-restoration algorithm for optimal control problems show that the fuel-optimal trajectory is zero-bang, namely it is characterized by two subarcs: a long coasting zero-thrust subarc followed by a short powered max-thrust braking subarc. While the thrust direction of the powered subarc is continuously variable for the optimal trajectory, its replacement with a constant (yet optimized) thrust direction produces a very efficient guidance trajectory: Indeed, for all values of the initial distance, the fuel required by the guidance trajectory is within less than one percent of the fuel required by the optimal trajectory. For the guidance trajectory, because of the replacement of the variable thrust direction of the powered subarc with a constant thrust direction, the optimal control problem degenerates into a mathematical programming problem with a relatively small number of degrees of freedom, more precisely: three for case (i) time-to-rendezvous free and two for case (ii) time-to-rendezvous given. In particular, we consider the rendezvous between the Space Shuttle (chaser) and the International Space Station (target). Once a given initial distance SS-to-ISS is preselected, the present work supplies not only the best initial conditions for the rendezvous trajectory, but simultaneously the corresponding final conditions for the ascent trajectory.     

Guidance Trajectories for Spacecraft Rendezvous

In a previous paper of Miele et al. (J. Optim. Theory Appl. 132(1), 2007), we employed the single-subarc sequential gradient-restoration algorithm to optimize the three-dimensional rendezvous between a target spacecraft in a planar circular orbit and a chaser spacecraft with an initial separation distance and separation velocity. The achieved continuous solutions are characterized by two, three, or four subarcs depending on the performance index (time, fuel) and the constraints. In this paper, based on the solutions in Miele et al. (J. Optim. Theory Appl. 132(1), 2007), we employ the multiple-subarc sequential gradient-restoration algorithm to produce pieced guidance trajectories implementable in real time via constant control components. In other words, in this investigation, we force the controls to behave as parameters in each subarc. With the above understanding, we investigate four problems: (P1) minimum time, fuel free; (P2) minimum fuel, time free; (P3) minimum time, fuel given; (P4) minimum fuel, time given. Problem P1 results in a two-subarc solution, each subarc with constant controls: a max-thrust accelerating subarc followed by a max-thrust braking subarc. Problem P2 results in a four-subarc solution, each subarc with constant controls: an initial coasting subarc, followed by a max-thrust braking subarc, followed by another coasting subarc, followed by another max-thrust braking subarc. Problems P3 and P4 result in two, three, or four-subarc solutions depending on the performance index and the constraints, albeit with constant controls in each subarc. For each of the problems studied, the performance index of the multiple-subarc pieced guidance trajectory approximates well the performance index of the single-subarc continuous optimal trajectory of Miele et al. (J. Optim. Theory Appl. 132(1), 2007) as well as the performance index of the multiple-subarc pieced optimal trajectory: the pairwise relative differences in performance index are less than 1/100. This research was supported by NSF under Grant CMS-0218878.     

Optimal Trajectories for Spacecraft Rendezvous

The efficient execution of a rendezvous maneuver is an essential component of various types of space missions. This work describes the formulation and numerical investigation of the thrust function required to minimize the time or fuel required for the terminal phase of the rendezvous of two spacecraft. The particular rendezvous studied concerns a target spacecraft in a circular orbit and a chaser spacecraft with an initial separation distance and separation velocity in all three dimensions. First, the time-optimal rendezvous is investigated followed by the fuel-optimal rendezvous for three values of the max-thrust acceleration via the sequential gradient-restoration algorithm. Then, the time-optimal rendezvous for given fuel and the fuel-optimal rendezvous for given time are investigated. There are three controls, one determining the thrust magnitude and two determining the thrust direction in space. The time-optimal case results in a two-subarc solution: a max-thrust accelerating subarc followed by a max-thrust braking subarc. The fuel-optimal case results in a four-subarc solution: an initial coasting subarc, followed by a max-thrust braking subarc, followed by another coasting subarc, followed by another max-thrust braking subarc. The time-optimal case with fuel given and the fuel-optimal case with time given result in two, three, or four-subarc solutions depending on the performance index and the constraints. Regardless of the number of subarcs, the optimal thrust distribution requires the thrust magnitude to be at either the maximum value or zero. The coasting periods are finite in duration and their length increases as the time to rendezvous increases and/or as the max allowable thrust increases. Another finding is that, for the fuel-optimal rendezvous with the time unconstrained, the minimum fuel required is nearly constant and independent of the max available thrust. Yet another finding is that, depending on the performance index, constraints, and initial conditions, sometime the initial application of thrust must be delayed, resulting in an optimal rendezvous trajectory which starts with a coasting subarc. This research has been supported by NSF under Grant CMS-0218878.     

Numerical Computation of Optimal Trajectories for Coplanar, Aeroassisted Orbital Transfer

This paper is concerned with the problem of the optimal coplanaraeroassisted orbital transfer of a spacecraft from a high Earth orbitto a low Earth orbit. It is assumed that the initial and final orbits arecircular and that the gravitational field is central and is governed by theinverse square law. The whole trajectory is assumed to consist of twoimpulsive velocity changes at the begin and end of one interior atmosphericsubarc, where the vehicle is controlled via the lift coefficient.The problem is reduced to the atmospheric part of the trajectory, thusarriving at an optimal control problem with free final time and liftcoefficient as the only (bounded) control variable. For this problem,the necessary conditions of optimal control theory are derived. Applyingmultiple shooting techniques, two trajectories with different controlstructures are computed. The first trajectory is characterized by a liftcoefficient at its minimum value during the whole atmospheric pass. For thesecond trajectory, an optimal control history with a boundary subarcfollowed by a free subarc is chosen. It turns out, that this secondtrajectory satisfies the minimum principle, whereas the first one fails tosatisfy this necessary condition; nevertheless, the characteristicvelocities of the two trajectories differ only in the sixth significantdigit.In the second part of the paper, the assumption of impulsive velocitychanges is dropped. Instead, a more realistic modeling with twofinite-thrust subarcs in the nonatmospheric part of the trajectory isconsidered. The resulting optimal control problem now describes the wholemaneuver including the nonatmospheric parts. It contains as controlvariables the thrust, thrust angle, and lift coefficient. Further,the mass of the vehicle is treated as an additional state variable. For thisoptimal control problem, numerical solutions are presented. They are comparedwith the solutions of the impulsive model.     

Thrust programs for minimum propellant consumption during the vertical take-off and landing of a rocket

In this paper, the thrust programs for minimum propellant consumption during vertical take-off and landing maneuvers of a rocket in a vacuum are derived for an arbitrary central gravitational field. It is shown that, regardless of the mathematical form of the attracting force, the extremal arc for take-off is composed of a maximum thrust subarc followed by a coasting subarc, and the extremal arc for landing is composed of a coasting subarc followed by a maximum thrust subarc.     

Optimal apogee burn time for low thrust spinning satellite in low altitude

In this study, the optimal burn time for low-thrust impulsive propulsion systems is investigated to raise the perigee altitude of a low-Earth orbit. The maneuver is done using spin-stabilized attitude control and impulsive thrusting system for a time interval centered about apogee point. On the one hand, the low value of the thrust level causes more burn time needed to accomplish the transfer. This, in turn, will cause more thrust loss due to the deviation between the thrust axis (spin axis) and the velocity vector of the satellite. On the other hand, for small thrust duration, the transfer needs more revolutions around the Earth and more travel in lower altitudes with dense atmosphere and more drag loss. To transfer the satellite with minimum propellant mass, a compromise between velocity losses due to both drag and thrust deviation angle should be made. An analytical approximate correlation between average thruster burn time and total required propellant mass is formulated in this study and an analytical optimal solution for burn time is found. Nonlinear programming is used to find optimal burn time history. Comparing the analytical and numerical results shows a very good match.     

Existence and Multiple Solutions of the Minimum-Fuel Orbit Transfer Problem

In this paper, the well-known problem of piloting a rocket with a low thrust propulsion system in an inverse square law field (say from Earth orbit to Mars orbit or from Earth orbit to Mars) is considered. By direct methods, it is shown that the existence of a fuel-optimal solution of this problem can be guaranteed, if one restricts the admissible transfer times by an arbitrarily prescribed upper bound. Numerical solutions of the problem with different numbers of thrust subarcs are presented which are obtained by multiple shooting techniques. Further, a general principle for the construction of such solutions with increasing numbers of thrust subarcs is given. The numerical results indicate that there might not exist an overall optimal solution of the Earth-orbit problem with unbounded free transfer time.     

Maximal submodules and locally perfect rings

Rings over which every nonzero right module has a maximal submodule are calledright Bass rings. For a ringA module-finite over its centerC, the equivalence of the following conditions is proved:

Optimization and optimality test for the Max-Cut Problem

We show that the following two problems are polynomially equivalent:

Construction of representation-finite selfinjective artin algebras of classB n andC n

In this paper, we discuss the representation-finite selfinjective artin algebras of classB _n andC _n and obtain the following main results: For any fieldk, let Λ be a representation-finite selfinjective artin algebras of classB _n orC _n overk.

On extensions of compact positive operators on Banach lattices

Extension properties of compact positive operators on Banach lattices are investigated. The following results are obtained:

• 1. 

  (1) Any compact positive operator (any compact lattice homomorphism, resp.) from a majorizing sublattice G of a Banach lattice E into another Banach lattice F can be extended to a compact positive operator (a compact lattice homomorphism, resp.) from E into F;

• 2. 

  (2) Any compact positive operator defined on a closed majorizing sublattice G of a Banach lattice E has a compact positive extension on E that preserves the spectrum (a necessary modification is needed).

Related extension problems are also studied.     

Smooth non-special surfaces of ℙ4

We consider smooth non-degenerate surfaces in ℙ⁴, and prove that there is a finite number of such surfaces which are:

Asymptotic properties of levelregular decision trees with randomly evaluated leaves

Game trees are an important model of decision-making situations, both in artificial intelligence and decision analysis. The model most frequently investigated in theoretical research consists of a uniform tree of heighh and a constant branching factorb, where the terminal positions are assigned the values of independent, identically distributed random variables [1, 3–10]. Our paper investigates two generalizations:

Lie algebras associated to systems of Dyson–Schwinger equations

We consider systems of combinatorial Dyson–Schwinger equations in the Connes–Kreimer Hopf algebra HI of rooted trees decorated by a set I. Let H(S) be the subalgebra of HI generated by the homogeneous components of the unique solution of this system. If it is a Hopf subalgebra, we describe it as the dual of the enveloping algebra of a Lie algebra g(S) of one of the following types:

• 1. 

  g(S) is an associative algebra of paths associated to a certain oriented graph.

• 2. 

  Or g(S) is an iterated extension of the Faà di Bruno Lie algebra.

• 3. 

  Or g(S) is an iterated extension of an infinite-dimensional abelian Lie algebra.

We also describe the character groups of H(S).     

Teoremi di normalizzazione per l'omotopia regolare dei grafi

Given a metric compact spaceS and a finite graphG we show that:

Subdirect products of hereditary congruence lattices

A congruence lattice L of an algebra A is called power-hereditary if every 0-1 sublattice of Lⁿ is the congruence lattice of an algebra on Aⁿ for all positive integers n. Let A and B be finite algebras. We prove

More ways to tile with only one shape polygon

I have exhibited several types of monotiles with matching rules that force the construction of a hexagonal parquet. The isohedral number of the resulting tiling can be made as large as desired by increasing the aspect ratio of the monotile. Aside from illustrating some elegant peculiarities of the hexagonal parquet tiling, the constructions demonstrate three points.

Arbres et suites majeures

The major sequences of lengthn are defined as the words withn letters taken from the integers 1, 2, ,n and containing at least