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NASA NTRS1992

Closed form solutions of constrained trajectories - Application in optimal ascent of aerospace plane

The present consideration of the flight trajectory of hypersonic aerospace vehicles subject to a class of path constraints notes the constrained dynamics to constitute a natural two-timescale system, so that problems of trajectory optimization and guidance can be dramatically simplified by means of the asymptotic analytical solutions thus obtained. An illustrative application in ascent trajectory optimization for an aerospace vehicle is presented.

Lu, Ping↗

NASA NTRS1993

LEWICE droplet trajectory calculations on a parallel computer

A parallel computer implementation (128 processors) of LEWICE, a NASA Lewis code used to predict the time-dependent ice accretion process for two-dimensional aerodynamic bodies of simple geometries, is described. Two-dimensional parallel droplet trajectory calculations are performed to demonstrate the potential benefits of applying parallel processing to ice accretion analysis. Parallel performance is evaluated as a function of the number of trajectories and the number of processors. For comparison, similar trajectory calculations are performed on single-processor Cray computers, and the best parallel results are found to be 33 and 23 times faster, respectively, than those of the Cray XMP and YMP.

Caruso, Steven C.↗

NASA NTRS1993

Analytical solutions to constrained hypersonic flight trajectories

The flight trajectory of aerospace vehicles subject to a class of path constraints is considered. The constrained dynamics is shown to be a natural two-time-scale system. Asymptotic analytical solutions are obtained. Problems of trajectory optimization and guidance can be dramatically simplified with these solutions. Applications in trajectory design for an aerospace plane strongly support the theoretical development.

Lu, Ping↗

NASA NTRS1993

Nominal trajectories for the aeroassisted flight experiment

The problem of generating good nominal trajectories is considered with particular reference to an AFE type spacecraft flying with constant angle of attack and variable angle of bank. A simulated GEO-to-LEO aeroassisted orbital transfer is considered. It is shown that, for fixed control, the AFE trajectory exhibits strong intrinsic instability in the longitudinal motion and near neutrality in the lateral motion, while stability can be artificially induced via feedback control, the effectiveness of a feedback control scheme depends on control margin availability. The AFE nominal trajectory must be chosen as a compromise between good performance and good control margin.

Miele, A.↗

NASA NTRS1990

Approximate minimum-time trajectories for two-link flexible manipulators

The method of recursive quadratic programming was used to generate approximate minimum-time tip trajectories for two-link semi-rigid and flexible manipulator movements in the horizontal plane. The manipulator is modeled with an efficient finite-element scheme for an n-link, m-joint system with bending only in the horizontal plane. Constraints on the trajectory include boundary conditions on position and energy for a rest-to-rest maneuver, straight-line tracking between boundary positions, and motor torque limits. Trajectory comparisons utilize a change in the link stiffness to compare a semi-rigid configuration to a flexible one. The level of bending flexibility necessary to excite significant modal behavior is demonstrated. Applied torques for minimum-time maneuvers are shown to be very similar between configurations and retain much of the qualitative character of rigid-body slewing motion.

Eisler, G. R.↗

NASA NTRS1990

An overview of the current state of the art optimization used for trajectory design in the GTS system

The high cost of delivering payloads to orbit demands that launch vehicles be flown in an optimal and reliable manner. Thus, computer simulated trajectories are optimized to obtain maximum vehicle payload capability or a maximum amount of reserve fuel, subject to design constraints. The Generalized Trajectory Simulation (GTS) system developed at The Aerospace Corporation is capable of simulating and optimizing trajectories for a wide variety of boost and reentry problems. Optimization is an integral part of the GTS system. This has resulted in progressive use of optimization. Additionally, the flexibility of the GTS system allows optimum trade studies to be conducted and complicated missions to be designed. This paper describes the integrated and flexible nature of the optimization software in the GTS system. Special features of the NLP2 and NLP3 optimization codes are described. Post optimality, parameter sensitivity analysis and the automatic scaling capability in the GTS system are also described.

Beltracchi, Todd J.↗

NASA NTRS1993

Trajectory optimization for the national aerospace plane

During the past six months the research objectives outlined in the last semi-annual report were accomplished. Specifically, these are: three-dimensional (3-D) fuel-optimal ascent trajectory of the aerospace plane and the effects of thrust vectoring control (TVC) on the fuel consumption and trajectory shaping were investigated; the maximum abort landing area (footprint) was studied; preliminary assessment of simultaneous design of the ascent trajectory and the vehicle configuration for the aerospace plane was also conducted. The work accomplished in the reporting period is summarized.

Lu, Ping↗

NASA NTRS1990

The vibro-acoustic mapping of low gravity trajectories on a Learjet aircraft

The NASA LeRC Learjet aircraft is used to fly parabolic trajectories for carrying out in-flight microgravity experiments. To derive full benefit from data obtained in such experiments, it is not only required to understand the low- to moderate-frequency accelerations present but also necessary to characterize the broadband aircraft vibro-acoustic environment. A vibro-acoustic measurement program was conducted during a series of low-gravity trajectories flown on the Learjet. The fuselage and an experiment rack were instrumented with 13 channels of accelerometers and three channels of pressure transducers. The locations of these sensors were determined in order to build a time history of the dynamics involved in the flight of such a parabolic trajectory and thus determine the typical rigid-body and vibro-acoustic loading a payload would experience. The instrumentation, data acquisition sequence, and experimental results showing residual accelerations and high-frequency dynamics are presented. As suspected, the measurements revealed that the aircraft environment cannot simply be described in terms of the static level low-gravity g-vector obtained, but that it also must account for both rigid-body and high-frequency vibro-acoustic dynamics.

Grodsinsky, C. M.↗

NASA NTRS1994

A parametric sensitivity study for single-stage-to-orbit hypersonic vehicles using trajectory optimization

The class of hypersonic vehicle configurations with single stage-to-orbit (SSTO) capability reflect highly integrated airframe and propulsion systems. These designs are also known to exhibit a large degree of interaction between the airframe and engine dynamics. Consequently, even simplified hypersonic models are characterized by tightly coupled nonlinear equations of motion. In addition, hypersonic SSTO vehicles present a major system design challenge; the vehicle's overall mission performance is a function of its subsystem efficiencies including structural, aerodynamic, propulsive, and operational. Further, all subsystem efficiencies are interrelated, hence, independent optimization of the subsystems is not likely to lead to an optimum design. Thus, it is desired to know the effect of various subsystem efficiencies on overall mission performance. For the purposes of this analysis, mission performance will be measured in terms of the payload weight inserted into orbit. In this report, a trajectory optimization problem is formulated for a generic hypersonic lifting body for a specified orbit-injection mission. A solution method is outlined, and results are detailed for the generic vehicle, referred to as the baseline model. After evaluating the performance of the baseline model, a sensitivity study is presented to determine the effect of various subsystem efficiencies on mission performance. This consists of performing a parametric analysis of the basic design parameters, generating a matrix of configurations, and determining the mission performance of each configuration. Also, the performance loss due to constraining the total head load experienced by the vehicle is evaluated. The key results from this analysis include the formulation of the sizing problem for this vehicle class using trajectory optimization, characteristics of the optimal trajectories, and the subsystem design sensitivities.

Lovell, T. Alan↗

NASA NTRS1994

Trajectories of ballistic impact ejecta on a rotating Earth

On an airless, slowly rotating planetary body like the Moon, ejecta particles from an impact follow simple ballistic trajectories. If gaseous interactions in the fireball are ignored, ejecta particles follow elliptical orbits with the center of the planetary body at one focus until they encounter the surface at the point of reimpact. The partial elliptical orbit of the ejecta particle lies in a plane in inertial (galactic) coordinates. Because of the slow rotation rate (for example, 360 degrees/28 days for the Moon), the intersection of the orbital plane and the surface remains nearly a great circle during the flight time of the ejecta. For this reason, lunar rays, representing concentrations of ejecta with the same azimuth but different velocities and/or ejecta angles, lie essentially along great circles. Ejecta from airless but more rapidly rotating bodies will follow more complicated, curving trajectories when plotted in the coordinate frame of the rotating planet or viewed as rays on the planetary surface. The curvature of trajectories of ejecta particles can be treated as a manifestation of the Coriolis effect, with the particles being accelerated by Coriolis pseudoforces. However, it is more straightforward to calculate the elliptical orbit in inertial space and then determine how far the planet rotates beneath the orbiting ejecta particle before reimpact. The Earth's eastward rotation affects ballistic ejecta in two ways: (1) the eastward velocity component increases the velocity of eastbound ejecta and reduces the velocity of westbound ejecta; and (2) the Earth turns underneath inflight ejecta, so that although the latitude of reimpact is not changed, the longitude is displaced westward, with the displacement increasing as a function of the time the ejecta remains aloft.

Alvarez, W.↗

NASA NTRS1995

Investigation of water droplet trajectories within the NASA icing research tunnel

Water droplet trajectories within the NASA Lewis Research Center's Icing Research Tunnel (IRT) were studied through computer analysis. Of interest was the influence of the wind tunnel contraction and wind tunnel model blockage on the water droplet trajectories. The computer analysis was carried out with a program package consisting of a three-dimensional potential panel code and a three-dimensional droplet trajectory code. The wind tunnel contraction was found to influence the droplet size distribution and liquid water content distribution across the test section from that at the inlet. The wind tunnel walls were found to have negligible influence upon the impingement of water droplets upon a wing model.

Reehorst, Andrew↗

NASA NTRS1993

The evolution of ClO and NO along air parcel trajectories

Back trajectory analysis of Arctic and Antarctic aircraft data reveals that high ClO concentrations are associated with predicted polar stratospheric cloud (PSCs) encounters. The ClO concentrations within the Arctic and Antarctic polar vortices vary widely but appear to be inversely related to parcel solar exposure since the last PSC interaction. These results imply that production of NO(x) from HNO3 photolysis and reaction with OH is the mechanism for the loss of chlorine radicals through the reformation of chlorine nitrate. Highly denitrified air parcels show no change in ClO with solar exposure. The recovery process is quantitatively duplicated using a model of chemistry along trajectories. Although PSC processing is the primary mechanism for producing elevated ClO amounts, back trajectories apparently unperturbed by PSC's also show slightly elevated ClO levels in 1992 compared to Arctic 1989 and Antarctic 1987 measurements presumably due to the presence of Pinatubo aerosol.

Schoeberl, M. R.↗

NASA NTRS1995

The analysis of control trajectories using symbolic and database computing

This final report comprises the formal semi-annual status reports for this grant for the periods June 30-December 31, 1993, January 1-June 30, 1994, and June 1-December 31, 1994. The research supported by this grant is broadly concerned with the symbolic computation, mixed numeric-symbolic computation, and database computation of trajectories of dynamical systems, especially control systems. A review of work during the report period covers: trajectories and approximating series, the Cayley algebra of trees, actions of differential operators, geometrically stable integration algorithms, hybrid systems, trajectory stores, PTool, and other activities. A list of publications written during the report period is attached.

Grossman, Robert↗

NASA NTRS1995

Freezing Behavior of Stratospheric Sulfate Aerosols Inferred from Trajectory Studies

Based on the trajectory analysis presented in this paper, a new mechanism is described for the freezing of the stratospheric sulfate aerosols. Temperature histories based on 10-day back trajectories for six ER-2 flights during AASE-I (1989) and AAOE (1987) are presented. The mechanism requires, as an initial step, the cooling of a H2SO4/H2O aerosol to low temperatures. If a cooling cycle is then followed up by a warming to approximately 196-198 K, the aerosols may freeze due to the growth of the crystallizing embryos formed at the colder temperature. The HNO3 absorbed at colder temperatures may increase the nucleation rate of the crystalling embryos and therefore influence the crystallization of the supercooled aerosols upon warming. Of all the ER-2 flights described, only the polar stratospheric clouds (PSC), observed on the flights of January 24, and 25, 1989 are consistent with the thermodynamics of liquid ternary solutions of H2SO4/HNO3/H2O (type Ib PSCs). For those two days, back trajectories indicate that the air mass was exposed to sulfuric acid tetrahydrate (SAT) melting temperatures about 24 hours prior to being sampled by the ER-2. Temperature histories, recent laboratory measurements, and the properties of glassy solids suggest that stratospheric H2SO4 aerosols may undergo a phase transition to SAT upon warming at approximately 198 K after going through a cooling cycle to about 194 K or lower.

Tabazadeh, A.↗

NASA NTRS1997

Departure Trajectory Synthesis and the Intercept Problem

Two areas of the departure problem in air traffic control are discussed. The first topic is the generation of climb-out trajectories to a fix. The trajectories would be utilized by a scheduling algorithm to allocate runways, sequence the proposed departures, and assign a departure time. The second area is concerned with finding horizontal trajectories to merge aircraft from the TRACON to an open slot in the en-route environment. Solutions are presented for the intercept problem for two cases: (1) the aircraft is traveling at the speed of the aircraft in the jetway; (2) the merging aircraft has to accelerate to reach the speed of the aircraft in the en-route stream. An algorithm is given regarding the computation of a solution for the latter case. For the former, a set of equations is given that allows us to numerically solve for the coordinate where the merge will occur.

Bolender, Michael A.↗

NASA NTRS1996

Optimal Trajectories for the Helicopter in One-Engine-Inoperative Terminal-Area Operations

This paper presents a summary of a series of recent analytical studies conducted to investigate One-Engine-Inoperative (OEI) optimal control strategies and the associated optimal trajectories for a twin engine helicopter in Category-A terminal-area operations. These studies also examine the associated heliport size requirements and the maximum gross weight capability of the helicopter. Using an eight states, two controls, augmented point-mass model representative of the study helicopter, Continued TakeOff (CTO), Rejected TakeOff (RTO), Balked Landing (BL), and Continued Landing (CL) are investigated for both Vertical-TakeOff-and-Landing (VTOL) and Short-TakeOff-and-Landing (STOL) terminal-area operations. The formulation of the nonlinear optimal control problems with considerations for realistic constraints, solution methods for the two-point boundary-value problem, a new real-time generation method for the optimal OEI trajectories, and the main results of this series of trajectory optimization studies are presented. In particular, a new balanced- weight concept for determining the takeoff decision point for VTOL Category-A operations is proposed, extending the balanced-field length concept used for STOL operations.

Zhao, Yiyuan↗

NASA NTRS1998

Trajectory design strategies that incorporate invariant manifolds and swingby

Libration point orbits serve as excellent platforms for scientific investigations involving the Sun as well as planetary environments. Trajectory design in support of such missions is increasingly challenging as more complex missions are envisioned in the next few decades. Software tools for trajectory design in this regime must be further developed to incorporate better understanding of the solution space and, thus, improve the efficiency and expand the capabilities of current approaches. Only recently applied to trajectory design, dynamical systems theory now offers new insights into the natural dynamics associated with the multi-body problem. The goal of this effort is the blending of analysis from dynamical systems theory with the well established NASA Goddard software program SWINGBY to enhance and expand the capabilities for mission design. Basic knowledge concerning the solution space is improved as well.

Guzman, J. J.↗

NASA NTRS1961

Effects of Some Typical Geometrical Contraints on Lunar Trajectories

A study has been made to determine the effects on lunar trajectories of some typical geometrical constraints. The constraints considered in this study are of two types, those resulting from specification of the trajectory characteristics near the earth and those associated with the specification of the approach conditions at the moon. The effects of the constraints are discussed from the standpoint of the limitations imposed by the constraints on the possible launch days during the month and also on the possible launch times during the day for three types of launch trajectories: direct-ascent, coasting-orbit, and parking-orbit launches. Application of the various constraints individually or in combination seriously restricts the allowable launch times during the month and day for the direct-ascent launch; whereas, less serious restrictions result for the coasting- and parking-orbit launches.

Tolson, Robert H.↗