The Calculus of Variations in Applied Aerodynamics and Flight Mechanics

Miele, A.

doi:10.1016/s0076-5392(08)62092-5

Cited by 34 publications

(24 citation statements)

References 31 publications

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“…25,26 Miele showed that the most general optimal control law for planar powered vacuum flight in a uniform gravitational field is the bilinear tangent steering law. 27 In vector notation, the optimal unit thrust vector in a uniform gravitational field is opposite the direction of the λ V Lagrange vector, or "primer" vector:…”

Section: Application Of Optimal Control Theorymentioning

confidence: 99%

Analytical Dimensional Reduction of a Fuel Optimal Powered Descent Subproblem

Rea

2010

AIAA Guidance, Navigation, and Control Conference

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Current renewed interest in exploration of the moon, Mars, and other planetary objects is driving technology development in many fields of space system design. In particular, there is a desire to land both robotic and human missions on the moon and elsewhere. The landing guidance system must be able to deliver the vehicle to a desired soft landing while meeting several constraints necessary for the safety of the vehicle. Due to performance limitations of current launch vehicles, it is desired to minimize the amount of fuel used. In addition, the landing site may change in real-time in order to avoid previously undetected hazards which become apparent during the landing maneuver.This complicated maneuver can be broken into simpler subproblems that bound the full problem. One such subproblem is to find a minimum-fuel landing solution that meets constraints on the initial state, final state, and bounded thrust acceleration magnitude. With the assumptions of constant gravity and negligible atmosphere, the form of the optimal steering law is known, and the equations of motion can be integrated analytically, resulting in a system of five equations in five unknowns. It is shown that this system of equations can be reduced analytically to two equations in two unknowns. With an additional assumption of constant thrust acceleration magnitude, this system can be reduced further to one equation in one unknown. It is shown that these unknowns can be bounded analytically. An algorithm is developed to quickly and reliably solve the resulting one-dimensional bounded search, and it is used as a real-time guidance applied to a lunar landing test case.

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Section: Application Of Optimal Control Theorymentioning

confidence: 99%

Analytical Dimensional Reduction of a Fuel Optimal Powered Descent Subproblem

Rea

2010

AIAA Guidance, Navigation, and Control Conference

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“…(3) can be rewritten in the form which is used henceforth in this report. After the dimensionless coordinates = y/(d/2) (6) are introduced and after the drag coefficient and the thickness ratio are defined as CD = D/qd T = d/l (7) t Typical values of the constant a. are 0 for the idealized model in which the friction coefficient is constant, -g-for the turbulent flow model, and % for the laminar flow model. …”

Section: Two-dimensional Problemmentioning

confidence: 99%

Optimum Slender Bodies in Hypersonic Flow With a Variable Friction Coefficient

Cole

Miele

1963

AIAA Journal

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This paper considers the problem of minimizing the total drag of a nonlifting, slender, twodimensional or axisymmetric body of given length and diameter in hypersonic flow under the assumption that the distribution of pressure coefficients is Newtonian and that the distribution of friction coefficients vs the abscissa is represented by a power law. For the two-dimensional problem, the extremal arc is a wedge regardless of the amount of friction drag and the way it is distributed along the contour. For the axisymmetric problem, the extremal arc includes at most two subarcs, one of which is called the regular shape, and the other, the spike of zero thickness. Furthermore, the solution depends strongly on the friction parameter Kf, which is proportional to the cubic root of the average friction coefficient divided by the thickness ratio. If the friction parameter is less than or equal to a certain critical value K /c , the extremal arc of the axisymmetric problem consists of a regular shape only. For K f -0, the regular shape is a -f-power body. However, for all other values of K f < K fc9 the regular shape is not a power body .If Kf = K fe9 the regular shape is a power body of exponent n = 1 for the constant friction coefficient model, n = y|-for the turbulent flow model, and n = -f for the laminar flow model. Finally, if K f > K fc9 the extremal arc includes a spike of zero thickness and a regular shape. This regular shape is conical only in the constant friction coefficient case; otherwise, it is not conical but approaches a cone in the immediate neighborhood of the axis of symmetry. The difference between the shapes calculated for a constant friction coefficient and those calculated for a variable friction coefficient are of some importance in the laminar flow case but small for the turbulent flow case. For the turbulent regime, it seems entirely permissible to determine the optimum shapes assuming a constant friction coefficient but correcting the drag a posteriori to account for the variation of the friction coefficient along the contour.

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“…The general theory for minimum-fuel trajectories in a uniform gravitational field is presented by Lawden [8], Miele [9], Leitmann [10], and Marec [11]. Two-dimensional powered-flight for a pointmass model with bounded thrust was considered by Leitmann [10,12].…”

Section: Introductionmentioning

confidence: 99%