Discrete approximations to optimal trajectories using direct transcription and nonlinear programming

Paul J. Enright, Bruce A. Conway

Research output: Contribution to journalArticlepeer-review


A recently developed method for solving optimal trajectory problems uses a piecewise-polynomial representation of the state and control variables, enforces the equations of motion via a collocation procedure, and thus approximates the original calculus-of-variations problem with a nonlinear programming problem, which is solved numerically. This paper identifies this method as being of a general class of direct transcription methods and proceeds to investigate the relationship between the original optimal control problem and the approximating nonlinear programming problem, by comparing the optimal control necessary conditions with the optimality conditions for the discretized problem. Attention is focused on the Lagrange multipliers of the nonlinear programming problem, which are shown to be discrete approximations to the adjoint variables of the optimal control problem. This relationship is exploited to test the adequacy of the discretization and to verify optimality of assumed control structures. The discretized adjoint equation of the collocation method is found to have deficient accuracy, and an alternate scheme that discretizes the equations of motion using an explicit Runge-Kutta parallel-shooting approach is developed. Both methods are applied to finite-thrust spacecraft trajectory problems, including a low-thrust escape spiral, a three-burn rendezvous, and a low-thrust transfer to the moon.

Original languageEnglish (US)
Pages (from-to)994-1002
Number of pages9
JournalJournal of Guidance, Control, and Dynamics
Issue number4
StatePublished - 1992

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Space and Planetary Science
  • Electrical and Electronic Engineering
  • Applied Mathematics


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