A simplified ascent trajectory optimization procedure has been developed with application to NASA's proposed Ares I launch vehicle. In the interest of minimizing bending loads and ensuring safe separation of the first-stage solid rocket motor, the vehicle is con- strained to follow a gravity-turn trajectory. This reduces the design space to just two free parameters, the pitch rate after a short vertical rise phase to clear the launch pad, and initial launch azimuth. The pitch rate primarily controls the in-plane parameters (altitude, speed, flight path angle) of the trajectory whereas the launch azimuth primarily controls the out-of-plane portion (velocity heading.) Thus, the optimization can be mechanized as two one-dimensional searches that converge quickly and reliably. The method is compared with POST-optimized trajectories to verify its optimality

Dukeman, Greg A.

Hill, Ashley D.

NASA Technical Reports Server

Rapid Trajectory Optimization For the ARES I
Launch Vehicle
Greg A. Dukeman∗
NASA Marshall Space Flight Center, Huntsville, Alabama, 35812, USA
Ashley D. Hill†
Dynamic Concepts Incorporated, Huntsville, Alabama, 35812, USA
A simplified ascent trajectory optimization procedure has been developed with appli-
cation to NASA’s proposed Ares I launch vehicle. In the interest of minimizing bending
loads and ensuring safe separation of the first-stage solid rocket motor, the vehicle is con-
strained to follow a gravity-turn trajectory. This reduces the design space to just two free
parameters, the pitch rate after a short vertical rise phase to clear the launch pad, and
initial launch azimuth. The pitch rate primarily controls the in-plane parameters (altitude,
speed, flight path angle) of the trajectory whereas the launch azimuth primarily controls
the out-of-plane portion (velocity heading.) Thus, the optimization can be mechanized as
two one-dimensional searches that converge quickly and reliably. The method is compared
with POST-optimized trajectories to verify its optimality.
Nomenclature
DOLILU Day-Of-Launch I-Load Update
ISS International Space Station
KSC Kennedy Space Center
MAV ERIC Marshall Aerospace Vehicle Representation in C
MECO Main Engine Cutoff
OTIS Optimal Trajectories by Implicit Simulation
POST Program to Optimize Simulated Trajectories
SRB Solid Rocket Booster
i orbital inclination
φ geocentric latitude
ψ launch azimuth, positive clockwise from local North
I. Introduction
An important ingredient of launch vehicle design and analysis and, ultimately, launch operations, is
the capability to optimize ascent trajectories while satisfying constraints such as bending moments. These
constraints are significantly influenced by wind conditions that exist in the region of maximum dynamic
pressure. Plenty of software packages and procedures exist for performing trajectory optimization, including
POST,1 OTIS,2 and DOLILU.3 POST and OTIS are very effective programs but are more powerful and
generic than what is needed for Ares I. DOLILU is tailored to the space shuttle vehicle which is very different
from the Ares I vehicle. The Atlas/Centaur launch vehicle trajectories are optimized using a software package
called ADDJUST,4 which was developed after several early launch delays due to the winds aloft not fitting
within the design wind envelopes. POST is the optimization package currently being used to generate Ares
I design trajectories. The resulting optimal attitude profile is written to a file which is then manipulated
∗Aerospace Engineer, Guidance, Navigation, and Mission Analysis Branch.
†Senior Aerospace Engineer
1 of 6
American Institute of Aeronautics and Astronautics
https://ntrs.nasa.gov/search.jsp?R=20080048217 2019-08-30T05:57:26+00:00Z
to extract the first-stage open-loop guidance I-loads. These I-loads are then uploaded to the computer that
hosts the high-fidelity trajectory simulation, MAVERIC, being used for Ares I design and analysis.
The optimizer described in this paper is a quick and reliable optimizer that is integrated into MAVERIC.
Such a capability enables efficient analysis of the potential benefits of day-of-launch optimization to pre-
launch measured winds, versus optimization to, say, the month-of-launch mean KSC winds. It also serves as
a pathfinder to the software and procedures to be used for Ares I day-of-launch operations.
The rest of this paper is organized as follows. The next section describes the Ares I trajectory sequence of
events. The subsequent section describes the general optimization strategy which involves explicit numerical
integration of first stage flight, while propagation and optimization of the upper stage flight phase takes
advantage of efficient exo-atmopheric trajectory optimization. Next, numerical results involving comparisons
with POST are given. The paper concludes with a summary and conclusions.
II. Trajectory Sequence of Events
On the launch pad, the vehicle is oriented so that the crew window points due East. After SRB (first-stage)
ignition, the launch vehicle flies a vertical, stationary attitude until the vehicle has cleared the launch tower.
This typically takes 6 seconds. Next, the vehicle commences a combined pitch/roll maneuver, continuing
until the crew window is in the desired plane of the ascent trajectory. The vehicle pitches at a constant rate
until the dynamic pressure builds up to 150 pounds per sq ft. Over the next 6 seconds, the vehicle transitions
to the gravity turn condition, that is, total angle of attack of zero. The rest of first stage flight, until burn out
of the SRB and separation, is flown in the gravity turn condition. This flight protocol keeps bending moments
low and sets up a benign attitude (low aerodynamic forces and moments) and low angular rates (reduces
lateral relative movement during separation) for the SRB separation sequence in which the interstage must
clear the J2X engine nozzle. During upper stage flight, the vehicle flies a linear tangent steering profile, for
optimum fuel usage, to a target orbit conic of prescribed orbit plane, cutoff altitude, apogee and perigee.
Several seconds after upper stage flight commences, the Launch Abort System is jettisoned as are panels
protecting the service module.
III. Optimization Procedure
The function to be maximized is total mass injected into the target orbit. Given a trial pair of pitch rate
and launch azimuth, the function can be evaluated by first numerically propagating the vehicle’s translational
states from liftoff through SRB separation, consistent with the trajectory sequence of events described in
the previous section. The final states from this flight phase are treated as initial conditions for upper stage
flight. These states (position, velocity and mass), along with the target orbit parameters and J2X engine
thrust and Isp, are provided as inputs to an efficient exoatmospheric guidance code.5 One of the natural
by-products of the guidance code is the injected mass, which is taken to be the optimized injected mass,
given the trial pitch rate and launch azimuth.
Moving up a level, a one-dimensional parameter optimizer is used to optimize first the pitch rate (given
a first guess at an optimal azimuth) and then the launch azimuth (with the optimized pitch rate.) From
experience, only a very narrow range of pitch rates need be considered in the optimization, say, 0.5 deg/s to
1.5 deg/s. The first guess for the launch azimuth, ψ, is obtained from spherical geometry, given the target
inclination, i, and launch (geocentric) latitude, φ:
sinψ =
cos i
cosφ
The final optimized launch azimuth will generally be offset a few degrees from this first guess due to the
constraint of flying a gravity turn in the presence of winds and due to Earth rotation.
Note that because the roll component of the (post-vertical rise) pitch-roll maneuver has no effect on
the translational trajectory, the roll need not be modeled during the iterations of the optimization. The
requirement to roll is accommodated in flight by commanding the desired roll angle as soon as the vertical
rise phase has completed. The onboard flight software will then automatically fly a combined pitch-roll
maneuver.
The one-dimensional optimization routine used here is Brent’s algorithm as described in Numerical
Recipes.6 It is guaranteed to converge once the optimum has been bracketed. An efficient steepest de-
2 of 6
American Institute of Aeronautics and Astronautics
scent approach is used whenever possible, otherwise a golden-section search is used.
IV. Numerical Results
A mission to the ISS was optimized using the (”day-of-launch”) optimization process described above.
Results illustrate that the simplified optimizer performs quite well in obtaining a solution very near the
optimized POST trajectory. Final mass to orbit was exactly the same. Figure 1 compares the resulting
angle of attack profile from POST and from the day-of-launch optimization procedure. Figure 2 compares
the altitude profiles, Figure 3 compares the ground track and Figure 4 compares the dynamic pressure
profiles. All the plots confirm that the simplified optimizer does as well as the POST program.
Next, a day-of-launch scenario was simulated in which a launch minus 2 hour (L-2hr) smoothed measured
wind profile was used in the optimization process to get a steering profile for first-stage open-loop guidance.
The vehicle was then flown to a wind measured two hours later (near launch time.) The purpose of this
is to show how day-of-launch trajectory optimization helps reduce load indicators such as the product of
dynamic pressure and total angle of attack. Figure 5 shows the angle of attack profile. Figure 6 shows the
product of dynamic pressure and total angle of attack. It can be seen that the load indicator is reduced from
2,500 lb/sq-ft to 1,500 lb/sq-ft by re-optimizing the first-stage attitude profile to the winds measured shortly
before launch, in contrast to flying an attitude profile optimized to the mean winds of the month of launch.
Updated results to include Monte Carlo results wherein first-stage attitude profile is optimized to a 2-hour
before launch smoothed measured wind, flown to ”2-hour later” wind, showing how much improvement day-
of-launch re-optimization provides compared to optimizing the first-stage attitude profile to month-of-launch
mean winds.
V. Conclusion
A simplified ascent trajectory optimization package, suitable for the Ares I launch vehicle, is presented.
It is shown that the design space can be reduced to just two free parameters, an initial pitch rate that
determines the in-plane trajectory characteristics, and the launch azimuth that determines the out-of-plane
trajectory characteristics. Each of the two parameters influence separate subsets of the trajectory space
so that they can be optimized separately in a one-two sequence of one-parameter optimizations. Each of
the optimizations have well-defined optima, which allows for quick and reliable optimization. This new
procedure is integrated into the high-fidelity trajectory simulation software being used for Ares I design
and analysis. This offers a process improvement compared to performing an oﬄine optimization in POST
and subsequent transcribing of the POST outputs into a data format useable by the first-stage guidance in
Maveric. Optimized trajectories using this new procedure compare favorably with corresponding optimized
trajectories from POST.
References
1Brauer, G., Cornick, D., Habeger, A., and Peterson, F., “Program to Optimize Simulated Trajectories (POST). Volume
3: Programmer’s manual,” Final report, Martin Marietta Corp., Denver, CO, April 1975.
2Hargraves, C. R. and Paris, S. W., “Direct Trajectory Optimization Using Nonlinear Programming and Collocation,”
Journal of Guidance, Control, and Dynamics, Vol. 10, 1987, pp. 338–348.
3“DOLILU II System Definition and Requirements Document, NSTS 083299, Volume VI, DOLILU II Quality Assurance
Rules, Day-of-Launch Function, DIVDT Program,” Tech. rep., National Aeronautics and Space Administration, Feb. 1992.
4Swanson, D., “ADDJUST-An Automated System for Steering Centaur Launch Vehicles in Mesured Winds,” Proceedings
of the Seventh Conference on Aerospace and Aeronautical meteorology, AIAA, Melbourne, FL, Nov. 1976.
5Dukeman, G. A., “Atmospheric Ascent Guidance for Rocket-Powered Launch Vehicles,” Proceedings of the AIAA Guid-
ance, Navigation, and Control Conference, AIAA, Wahington, D.C., 2002.
6Press, W., Teukolsky, S., Vetterling, W., and Flannery, B., Numerical Recipes in C: The Art of Scientific Computing,
Cambridge University Press, 2nd ed., 1992.
3 of 6
American Institute of Aeronautics and Astronautics
Figure 1. ISS Mission Angle Of Attack Profile.
Figure 2. ISS Mission Altitude Profile.
4 of 6
American Institute of Aeronautics and Astronautics
Figure 3. ISS Mission Ground Track.
Figure 4. ISS Mission Dynamic Pressure Profile.
5 of 6
American Institute of Aeronautics and Astronautics
Figure 5. Angle of Attack Profile for Launch 2-hour Wind Pair.
Figure 6. Product of Dynamic Pressure and Total Angle of Attack, Launch Wind Pair.
6 of 6
American Institute of Aeronautics and Astronautics


Rapid Trajectory Optimization for the ARES I Launch Vehicle

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server