Purdue University Graduate School
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Indirect Trajectory Optimization Using Automatic Differentiation

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posted on 2022-12-14, 13:42 authored by Winston Cheuvront LevinWinston Cheuvront Levin

Current indirect optimal control problem (IOCP) solvers, like beluga or PINs, use symbolic math to derive the necessary conditions to solve the IOCP. This limits the capability of IOCP solvers by only admitting symbolically representable functions. The purpose of this thesis is to present a framework that extends those solvers to derive the necessary conditions of an IOCP with fully numeric methods. With fully numeric methods, additional types of functions, including conditional logic functions and look-up tables can now be easily used in the IOCP solver.

This aim was achieved by implementing algorithmic differentiation (AD) as a method to derive the IOCP necessary conditions into a new solver called Giuseppe. The Brachistochrone problem was derived analytically and compared Giuseppe to validate the automatic derivation of necessary conditions. Two additional problems are compared and extended using this new solver. The first problem, the maximum cross-range problem, demonstrates a trajectory can be optimized indirectly while utilizing a conditional density function that switches as a function of height according to the 1976 U.S. atmosphere model. The second problem, the minimum time to climb problem, demonstrates a trajectory can be optimized indirectly while utilizing 6 interpolated look up tables for lift, drag, thrust, and atmospheric conditions. The AD method yields the exact same precision as the symbolic methods, rather than introducing numeric error as finite difference derivatives would with the benefit of admitting conditional switching functions and look-up tables. 


Degree Type

  • Master of Science


  • Aeronautics and Astronautics

Campus location

  • West Lafayette

Advisor/Supervisor/Committee Chair

Daniel DeLaurentis

Additional Committee Member 2

Ran Dai

Additional Committee Member 3

Sheyas Sundaram

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