Collocation Versus Differential Inclusion in Direct Optimization

Conway, Bruce A.; Larson, Kevin

doi:10.2514/2.4306

Cited by 36 publications

(17 citation statements)

References 10 publications

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“…As a result, we reduced the original problem to a nonlinear programming problem, meaning we obtained a problem of minimization for the scalar function of several (but not tens as in Refs. [24][25][26][27][28][29][30][31][32][33][34][35][36] variables:…”

Section: E Minimization Of Multivariable Scalar Functionmentioning

confidence: 99%

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Direct Method for Rapid Prototyping of Near-Optimal Aircraft Trajectories

Yakimenko

2000

Journal of Guidance, Control, and Dynamics

163

125

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A direct method for a real-time generation of near-optimal spatial trajectories of short-term maneuvers onboard a ying vehicle with predetermined thrust history is introduced. The paper starts with a survey about the founders of the direct methods of calculus of variations and their followers in ight mechanics, both in Russia and in the United States. It then describes a new direct method based on three cues: high-order polynomials from the virtual arc as a reference function for aircraft's coordinates, a preset history of one of the controls (thrust), and a few optimization parameters. The trajectory optimization problem is transformed into a nonlinear programming problem and then solved numerically using an appropriate algorithm in accelerated scale of time. A series of examples is presented. Calculated near-optimal trajectory is compared with real ight data, and with the solution obtainedby Pontryagin's maximumprinciple. Fast convergence of the numerical algorithm,which has been already implemented and tested onboard a real aircraft, is illustrated. Nomenclature a ik= polynomial coef cients g = acceleration due to gravity J = cost function j = quantity pertaining to the j th time node m = aircraft mass N = number of nodes n = polynomial order n = relative revolutions of engine's rotor n x , n z = tangential and normal projections of load factor, respectively Sh = penalty function T,T = total thrust and relative thrust (fraction of maximum thrust), respectively CONCEPT of the onboard pilot's support system (PSS; electronic copilot or pilot associate) assumes the presenceof a sub-

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Section: E Minimization Of Multivariable Scalar Functionmentioning

confidence: 99%

“…The previously mentioned procedure can permit the reduction of the size of the parameter optimization problem. 33,34 Smaller problems can then be solved more quickly. Lu 35,36 used, for each piece, an approach similar to Taranenko's method.…”

mentioning

confidence: 99%

Direct Method for Rapid Prototyping of Near-Optimal Aircraft Trajectories

Yakimenko

2000

Journal of Guidance, Control, and Dynamics

163

125

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“…3, the more accurate integration rules such as trapezoid or Simpson are implicit integration rules that make it impossible to express the state derivatives at the i th node in terms of the state variables alone. With our formulation of the Legendre pseudospectral method, we have circumvented this dif culty and offer a method that is both accurate and adaptable to a differential inclusion formulation.…”

Section: Legendre Pseudospectral Methodsmentioning

confidence: 99%

“…3 The cart problem has an analytic solution and has a linear control and a quadratic cost function with a xed nal time. The state variables are x 1 , the displacement of the cart of unit mass, and x 2 the velocity, and the control u is the external force.…”

Section: Numerical Examplementioning

confidence: 99%

Second Look at Approximating Differential Inclusions

Fahroo

Ross

2001

Journal of Guidance, Control, and Dynamics

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“…On the other hand, direct methods convert the calculus of the variation problem into a parameter optimization problem that minimizes the performance index by using nonlinear programming (NLP). It also transcribes the states and controls through direct transcription and collocation [16,17] or differential inclusion [18,19]. The entire trajectory to be optimized by this direct method is represented in terms of nodes [17], and a large number of design variables.…”

Section: Introductionmentioning

confidence: 99%

A Study on Earth-Moon Transfer Orbit Design

No¹,

Lee²,

Jeon³

et al. 2012

International Journal of Aeronautical and Space Sciences

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Optimal transfer trajectories based on the planar circular restricted three body problem are designed by using mixed impulsive and continuous thrust. Continuous and dynamic trajectory optimization is reformulated in the form of discrete optimization problem. This is done by the method of direct transcription and collocation. It is then solved by using nonlinear programming software. Two very different transfer trajectories can be obtained by the different combinations of the design parameters. Furthermore, it was found out that all designed trajectories permit a ballistic capture by the Moon's gravity. Finally, the required thrust profiles are presented and they are analyzed in detail.

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Collocation Versus Differential Inclusion in Direct Optimization

Cited by 36 publications

References 10 publications

Direct Method for Rapid Prototyping of Near-Optimal Aircraft Trajectories

Direct Method for Rapid Prototyping of Near-Optimal Aircraft Trajectories

Second Look at Approximating Differential Inclusions

A Study on Earth-Moon Transfer Orbit Design

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