Optimal Space Trajectories—A Review of Published Work

Bell, D. J.

doi:10.1017/s0001924000083676

Cited by 17 publications

(4 citation statements)

References 42 publications

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“…The expression for ∆V is given in Eqs. (3) and (4). The velocities at P 1 and P 2 on the transfer orbit are given by 18…”

Section: Appendix: Derivation Of D∆v /Damentioning

confidence: 99%

“…This can be very well justified by the large amount of references and several survey papers. [1][2][3][4] The majority of the earlier works (prior to 1960s) studied orbital transfers with free transfer time. More recently, and especially since the advent of Lawden's primer vector theory, 5 attention has been paid primarily on fixed-time orbital transfer problems.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Two-Impulse Rendezvous Between Two Circular Orbits Using Multiple-Revolution Lambert's Solutions

Shen

Tsiotras

2002

AIAA Guidance, Navigation, and Control Conference and Exhibit

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In this paper, we study the optimal fixed-time, two-impulse rendezvous between two spacecraft orbiting along two coplanar circular orbits in the same direction. The fixed-time two-impulse transfer problem between two fixed points on two circular orbits, called a fixed-time fixed-endpoint transfer problem, is solved first. Our solution scheme involves first the solution to the related multiple-revolution Lambert problem. A solution procedure is proposed to reduce the calculation of an existing algorithm, thanks to the introduction of an auxiliary transfer problem. Unlike the fixed-endpoint transfer problem with a fixed time of flight, the auxiliary transfer problem studies the relationship between the transfer cost and the transfer orbit semimajor axis with the transfer time being free. The characteristics of the auxiliary problem are thoroughly explored and then applied to the fixed-time fixed-endpoint problem. As a result, the solution candidates are narrowed down from 2N max + 1 (N max is the maximum number of revolutions permitted) to at most two. Using this solution procedure, the minimum cost of the fixed-time transfer problem is easily obtained for all cases of different separation angles and times of flight. Thus a contour plot of the cost is obtained as a function of the separation angle and the transfer time. This contour plot along with a sliding rule facilitates the task of finding the optimal initial and terminal coasting periods, and thus obtaining the solutions for the original problem. Numerical examples demonstrate the application of the methodology to multiple rendezvous of satellite constellations on circular orbits.

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“…The expression for ∆V is given in Eqs. (3) and (4). The velocities at P 1 and P 2 on the transfer orbit are given by 18…”

Section: Appendix: Derivation Of D∆v /Damentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal Two-Impulse Rendezvous Between Two Circular Orbits Using Multiple-Revolution Lambert's Solutions

Shen

Tsiotras

2002

AIAA Guidance, Navigation, and Control Conference and Exhibit

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“…The following surveys and lectures are devoted to the methods of approximate and numerical solution of problemsin optimal control: Budak and Vasil'ev [57], Gabasov and Kirillov [74], Isaevand Sonin [376], Kolmanovskii and Chernous'ko [127], Krasovskii [139], Krasovskii, Gavrilov, Letov, and Pugachev [141], Krasovskii and Moiseev [144], Letov [168], Litovchenko [175], Mikhalevich, Ermol'ev, Shkurb, andShor [188], Moiseev [192][193], Moiseev and Lebedev [441], Polyak [226], Chernous'ko [274], Bell [302], Fleming [348], Kelley [393], Kopp [403], Larson [410], [411], Mangasarian [428], Miele [439], Paiewonski [449], Polak [452], Tabak [477], Tapley and Lewallen [480], and others.…”

Section: Ovmentioning

confidence: 99%

Computational and approximate methods of optimal control

Chernousko¹,

Kolmanovskii²

1979

J Sov Math

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“…Some of the earliest studies of orbital maneuvers were done by Contensou [3,4] and Lawden [5]. Surveys of much early work were done by Edelbaum [6], Bell [7], Robinson [8], and Gobetz and Doll [9]. An excellent introduction to the subject is found in the book by Marec [10].…”

Section: Introductionmentioning

confidence: 99%

A new approach to impulsive rendezvous near circular orbit

Carter

Humi

2012

Celest Mech Dyn Astr

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A new approach is presented for the problem of optimal impulsive rendezvous of a spacecraft in an inertial frame near a circular orbit in a Newtonian gravitational field. The total characteristic velocity to be minimized is replaced by a related characteristicvalue function and this related optimization problem can be solved in closed form. The solution of this problem is shown to approach the solution of the original problem in the limit as the boundary conditions approach those of a circular orbit. Using a form of primer-vector theory the problem is formulated in a way that leads to relatively easy calculation of the optimal velocity increments. A certain vector that can easily be calculated from the boundary conditions determines the number of impulses required for solution of the optimization problem and also is useful in the computation of these velocity increments.Necessary and sufficient conditions for boundary conditions to require exactly three nonsingular non-degenerate impulses for solution of the related optimal rendezvous problem, and a means of calculating these velocity increments are presented. If necessary these velocity increments could be calculated from a hand calculator containing trigonometric functions. A simple example of a three-impulse rendezvous problem is solved and the resulting trajectory is depicted.Optimal non-degenerate nonsingular two-impulse rendezvous for the related problem is found to consist of four categories of solutions depending on the four ways the primer vector locus intersects the unit circle. Necessary and sufficient conditions for each category of solutions are presented. The region of the boundary values that admit each category of solutions of the related problem are found, and in each case a closed-form solution of the optimal velocity increments is presented. Some examples are simulated. Similar results are presented for the simpler optimal rendezvous that require only one-impulse. For brevity degenerate and singular solutions are not discussed in detail, but should be presented in a following study Although this approach is thought to provide simpler computations than existing methods, its main contribution may be in establishing a new approach to the more general problem.

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Optimal Space Trajectories—A Review of Published Work

Cited by 17 publications

References 42 publications

Optimal Two-Impulse Rendezvous Between Two Circular Orbits Using Multiple-Revolution Lambert's Solutions

Optimal Two-Impulse Rendezvous Between Two Circular Orbits Using Multiple-Revolution Lambert's Solutions

Computational and approximate methods of optimal control

A new approach to impulsive rendezvous near circular orbit

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