OPTIMAL AIRCRAFT TRAJECTORIES FOR SPECIFIED RANGE

( NASA Ames Research Center )

     For an aircraft operating over a fixed range, the operating costs are
basically a sum of fuel cost and time cost. While minimum fuel and minimum
time trajectories are relatively easy to calculate, the determination of
a minimum cost trajectory can be a complex undertaking. This computer pro-
gram was developed to optimize trajectories with respect to a cost func-
tion based on a weighted sum of fuel cost and time cost. As a research tool,
the program could be used to study various characteristics of optimum tra-
jectories and their comparison to standard trajectories. It might also be
used to generate a model for the development of an airborne trajectory opti-
mization system. The program could be incorporated into an airline flight
planning system, with optimum flight plans determined at takeoff time for
the prevailing flight conditions. The use of trajectory optimization could
significantly reduce the cost for a given aircraft mission.

     The algorithm incorporated in the program assumes that a trajectory
consists of climb, cruise, and descent segments. The optimization of each
segment is not done independently, as in classical procedures, but is per-
formed in a manner which accounts for interaction between the segments.
This is accomplished by the application of optimal control theory. The
climb and descent profiles are generated by integrating a set of kinematic
and dynamic equations, where the total energy of the aircraft is the inde-
pendent variable. At each energy level of the climb and descent profiles,
the air speed and power setting necessary for an optimal trajectory are de-
termined. The variational Hamiltonian of the problem consists of the rate
of change of cost with respect to total energy and a term dependent on the
adjoint variable, which is identical to the optimum cruise cost at a speci-
fied altitude. This variable uniquely specifies the optimal cruise energy,
cruise altitude, cruise Mach number, and, indirectly, the climb and descent
profiles. If the optimum cruise cost is specified, an optimum trajectory
can easily be generated; however, the range obtained for a particular opti-
mum cruise cost is not known a priori. For short range flights, the program
iteratively varies the optimum cruise cost until the computed range con-
verges to the specified range. For long-range flights, iteration is unnec-
essary since the specified range can be divided into a cruise segment dis-
tance and full climb and descent distances.

     The user must supply the program with engine fuel flow rate coeffi-
cients and an aircraft aerodynamic model. The program currently includes
coefficients for the Pratt-Whitney JT8D-7 engine and an aerodynamic model
for the Boeing 727. Input to the program consists of the flight range to be
covered and the prevailing flight conditions including pressure, tempera-
ture, and wind profiles. Information output by the program includes: opti-
mum cruise tables at selected weights, optimal cruise quantities as a func-
tion of cruise weight and cruise distance, climb and descent profiles, and a
summary of the complete synthesized optimal trajectory.

     This program is written in FORTRAN IV for batch execution and has been
implemented on a CDC 6000 series computer with a central memory requirement



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of approximately 100K (octal) of 60 bit words. This aircraft trajectory op-
timization program was developed in 1979.

     COSMIC, and the COSMIC logo are registered trademarks of the National
Aeronautics and Space Administration. All other brands and product names
are the trademarks of their respective holders.



LANGUAGE: FORTRAN IV

MACHINE REQUIREMENTS: CDC 6000 SERIES

PROGRAM SIZE: APPROXIMATELY 2,583 SOURCE STATEMENTS

DISTRIBUTION MEDIA: 9 Track 1600 BPI EBCDIC Card Image Format Magnetic Tape

PROGRAM NUMBER: ARC-11282

DOMESTIC - DOCUMENTATION PRICE: $23.00    PROGRAM PRICE: $500.00

INTERNATIONAL - DOCUMENTATION PRICE: $46.00   PROGRAM PRICE: $1000.00



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