Search NASA⌕ Search

SEARCH · Search NASA

Results for “Trajectory Based Operations”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 433 records · Page 24
All sourcesNASA NTRSDOE OSTI
NASA NTRS1998

Thermal Management Design for the X-33 Lifting Body

The X-33 Advantage Technology Demonstrator offers a rare and exciting opportunity in Thermal Protection System development. The experimental program incorporates the latest design innovation in re-useable, low life cycle cost, and highly dependable Thermal Protection materials and constructions into both ground based and flight test vehicle validations. The unique attributes of the X-33 demonstrator for design application validation for the full scale Reusable Launch Vehicle, (RLV), are represented by both the configuration of the stand-off aeroshell, and the extreme exposures of sub-orbital hypersonic re-entry simulation. There are several challenges of producing a sub-orbital prototype demonstrator of Single Stage to Orbit/Reusable Launch Vehicle (SSTO/RLV) operations. An aggressive schedule with budgetary constraints precludes the opportunity for an extensive verification and qualification program of vehicle flight hardware. However, taking advantage of off the shelf components with proven technologies reduces some of the requirements for additional testing. The effects of scale on thermal heating rates must also be taken into account during trajectory design and analysis. Described in this document are the unique Thermal Protection System (TPS) design opportunities that are available with the lifting body configuration of the X-33. The two principal objectives for the TPS are to shield the primary airframe structure from excessive thermal loads and to provide an aerodynamic mold line surface. With the relatively benign aeroheating capability of the lifting body, an integrated stand-off aeroshell design with minimal weight and reduced procurement and operational costs is allowed. This paper summarizes the design objectives of the X-33 TPS, the flight test requirements driven configuration, and design benefits. Comparisons are made of the X-33 flight profiles and Space Shuttle Orbiter, and lifting body Reusable Launch Vehicle aerothermal environments. The X-33 TPS is based on a design to cost configuration concept. Only RLV critical technologies are verified to conform to cost and schedule restrictions. The one-off prototype vehicle configuration has evolved to minimize the tooling costs by reducing the number of unique components. Low cost approaches such as a composite/blanket leeward aeroshell and the use of Shuttle technology are implemented where applicable. The success of the X-33 will overcome the ballistic re-entry TPS mindset. The X-33 TPS is tailored to an aircraft type mission while maintaining sufficient operational margins. The flight test program for the X-33 will demonstrate that TPS for the RLV is not simply a surface insulation but rather an integrated aeroshell system.

Bouslog, S.↗
NASA NTRS2008

In-Flight Operation of the Dawn Ion Propulsion System - The First Nine Months

The Dawn mission, part of NASA's Discovery Program, has as its goal the scientific exploration of the two most massive main-belt asteroids, Vesta and Ceres. The Dawn spacecraft was launched from Cape Canaveral Air Force Station on September 27, 2007 on a Delta-II 7925H-9.5 (Delta-II Heavy) rocket that placed the 1218 kg spacecraft into an Earth-escape trajectory. On-board the spacecraft is an ion propulsion system (IPS) which will provide most of the delta-V needed for heliocentric transfer to Vesta, orbit capture at Vesta, transfer to Vesta science orbits, departure and escape from Vesta, heliocentric transfer to Ceres, orbit capture at Ceres, and transfer to Ceres science orbits. The Dawn ion engine design is based on the design validated on NASA's Deep Space 1 mission. However, because of the very substantial (11 km/s) delta-V requirements for this mission Dawn requires two engines to complete its mission objectives. The power processor units (PPU), digital control and interface units (DCIU) slice boards and the xenon control assembly (XCA) are also based on the DS1 design. The DCIUs and thrust gimbal assemblies (TGA) were developed at the Jet Propulsion Laboratory. The spacecraft was provided by Orbital Sciences Corporation, Sterling, Virginia, and the mission is managed by and operated from the Jet Propulsion Laboratory. Dawn partnered with Germany, Italy and Los Alamos National Laboratory for the science instruments. The mission is led by the principal investigator, Dr. Christopher Russell, from the University of California, Los Angeles. The first 80 days after launch were dedicated to the initial checkout of the spacecraft prior to the initiation of long-term thrusting for the heliocentric transfer to Vesta. The IPS hardware, consisting of three ion thrusters and TGAs, two PPUs and DCIUs, xenon feed system, and spacecraft control software, was investigated extensively. Thrust measurements, roll torque measurements, pointing capabilities, control characteristics, and thermal behavior of the spacecraft and IPS were carefully evaluated. The Dawn IPS fully met all its initial checkout performance objectives. Deterministic thrusting for cruise began on December 17, 2007. Over the subsequent approximately 330 days the IPS will be operated virtually continuously at full power thrusting (approximately 91 mN) leading to a Mars flyby in February 2009. The encounter with Mars provides a gravity assist for a plane change and is the only source of post-launch delta-V apart from the IPS. Following the Mars gravity assist IPS will be operated for approximately one year at full power and for 1.3 years at throttled power levels leading to rendezvous with Vesta in August of 2011. Following nine months of orbital operations with IPS providing the propulsion needed for orbit capture, science orbit transfer and orbit maintenance and Vesta escape, Dawn will transit to Ceres with an expected arrival date of February 2015. As of June 16, 2008 the ion thrusters on Dawn have operated for close to 3,846 hours and have delivered nearly 1 km/s of delta-V to the spacecraft. Dawn IPS operation has been almost flawless during the initial checkout and six months of cruise. This paper provides an overview of Dawn's mission objectives, mission and system design, and the results of the post-launch Dawn IPS mission operations through June 2008

Garner, Charles E.↗
NASA NTRS1971

Dynamic upper atmospheric force model on stabilized vehicles for a high-precision trajectory computer program

The upper atmosphere model draws heavily on the behavior of the earth's upper atmosphere which exhibits cyclic as well as irregular variations in density profile, temperature, pressure, and composition in unison with solar activities as deduced from the more recent land-based and satellite observations. The lift and drag model is designed specifically for inertially stabilized vehicles of the Mariner class, with possible extension to gravity gradient stabilized vehicles of the GEOS class. The model considers operation in the free molecular flow regimes with large Knudsen numbers. The vehicle is considered a composite structure with basic components having well-defined shapes, each with its own surface characteristics in terms of temperature, reflectivity, and accommodation of free stream molecules. The model takes into account both the calculation of precise aerodynamic force coefficients in terms of expansion of modified Bessel functions in speed ratios and angle of attack, and approximate force coefficients when the speed ratios approach infinity. Other considerations include specular and diffused reflectivity, shielding, and shadow effects.

Khatib, A. R.↗
NASA NTRS1992

Neural joint control for Space Shuttle Remote Manipulator System

Neural networks are being used to control a robot arm in a telerobotic operation. The concept uses neural networks for both joint and inverse kinematics in a robotic control application. An upper level neural network is trained to learn inverse kinematic mappings. The output, a trajectory, is then fed to the Decentralized Adaptive Joint Controllers. This neural network implementation has shown that the controlled arm recovers from unexpected payload changes while following the reference trajectory. The neural network-based decentralized joint controller is faster, more robust and efficient than conventional approaches. Implementations of this architecture are discussed that would relax assumptions about dynamics, obstacles, and heavy loads. This system is being developed to use with the Space Shuttle Remote Manipulator System.

Atkins, Mark A.↗
NASA NTRS2008

A System Trade Study of Remote Infrared Imaging for Space Shuttle Reentry

A trade study reviewing the primary operational parameters concerning the deployment of imaging assets in support of the Hypersonic Thermodynamic Infrared Measurements (HYTHIRM) project was undertaken. The objective was to determine key variables and constraints for obtaining thermal images of the Space Shuttle orbiter during reentry. The trade study investigated the performance characteristics and operating environment of optical instrumentation that may be deployed during a HYTHIRM data collection mission, and specified contributions to the Point Spread Function. It also investigated the constraints that have to be considered in order to optimize deployment through the use of mission planning tools. These tools simulate the radiance modeling of the vehicle as well as the expected spatial resolution based on the Orbiter trajectory and placement of land based or airborne optical sensors for given Mach numbers. Lastly, this report focused on the tools and methodology that have to be in place for real-time mission planning in order to handle the myriad of variables such as trajectory ground track, weather, and instrumentation availability that may only be known in the hours prior to landing.

Schwartz, Richard J.↗
DOE OSTI2026

Parallel-in-Time Solution of Allen-Cahn Equations by Integrating Operator Learning into the Parareal Method

While recent advances in deep learning have shown promising efficiency gains in solving time-dependent partial differential equations (PDEs), matching the accuracy of conventional numerical solvers still remains a challenge. One strategy to improve the accuracy of deep learning-based solutions for time-dependent PDEs is to use the learned model as the coarse propagator in the Parareal method and a traditional numerical method as the fine solver. However, successful integration of deep learning into the Parareal method requires consistency between the coarse and fine solvers, particularly for PDEs exhibiting rapid changes such as sharp transitions. Here, to ensure this consistency, we propose using convolutional neural networks (CNNs) to learn the fully discrete time-stepping operator defined by the same numerical scheme employed as the fine solver. We demonstrate the effectiveness of the proposed method in solving the classical and mass-conservative Allen–Cahn (AC) equations. Through iterative updates in the Parareal algorithm, our approach achieves a significant computational speedup compared to traditional fine solvers while converging to high-accuracy solutions. Our results highlight that the proposed hybrid Parareal algorithm effectively accelerates simulations, particularly when implemented on multiple GPUs, and converges to the desired accuracy in only a few iterations. Another advantage of our method is that the CNN model is trained on trajectory-based data generated from random initial conditions, such that the trained model can be used to solve the AC equations with various initial conditions without retraining. This work demonstrates the potential of integrating neural network methods into parallel-in-time frameworks for efficient and accurate simulations of time-dependent PDEs.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2018

Pairwise Trajectory Management (PTM): Concept Description and Documentation

Pairwise Trajectory Management (PTM) is an Interval Management (IM) concept that utilizes airborne and ground-based capabilities to enable the implementation of airborne pairwise spacing capabilities in oceanic regions. The goal of PTM is to use airborne surveillance and tools to manage an "at or greater than" inter-aircraft spacing. Due to the accuracy of Automatic Dependent Surveillance-Broadcast (ADS-B) information and the use of airborne spacing guidance, the minimum PTM spacing distance will be less than distances a controller can support with current automation systems that support oceanic operations. Ground tools assist the controller in evaluating the traffic picture and determining appropriate PTM clearances to be issued. Avionics systems provide guidance information that allows the flight crew to conform to the PTM clearance issued by the controller. The combination of a reduced minimum distance and airborne spacing management will increase the capacity and efficiency of aircraft operations at a given altitude or volume of airspace. This document provides an overview of the proposed application, a description of several key scenarios, a high level discussion of expected air and ground equipment and procedure changes, a description of a NASA human-machine interface (HMI) prototype for the flight crew that would support PTM operations, and initial benefits analysis results. Additionally, included as appendices, are the following documents: the PTM Operational Services and Environment Definition (OSED) document and a companion "Future Considerations for the Pairwise Trajectory Management (PTM) Concept: Potential Future Updates for the PTM OSED" paper, a detailed description of the PTM algorithm and PTM Limit Mach rules, initial PTM safety requirements and safety assessment documents, a detailed description of the design, development, and initial evaluations of the proposed flight crew HMI, an overview of the methodology and results of PTM pilot training requirements focus group and human-in-the-loop testing activities, and the PTM Pilot Guide.

Jones, Kenneth M.↗
NASA NTRS2012

Automated Design of Noise-Minimal, Safe Rotorcraft Trajectories

NASA and the international community are investing in the development of a commercial transportation infrastructure that includes the increased use of rotorcraft, specifically helicopters and aircraft such as a 40-passenger civil tilt rotors. Rotorcraft have a number of advantages over fixed wing aircraft, primarily in not requiring direct access to the primary fixed wing runways. As such they can operate at an airport without directly interfering with major air carrier and commuter aircraft operations. However, there is significant concern over the impact of noise on the communities surrounding the transportation facilities. In this paper we propose to address the rotorcraft noise problem by exploiting powerful search techniques coming from artificial intelligence, coupled with simulation and field tests, to design trajectories that are expected to improve on the amount of ground noise generated. This paper investigates the use of simulation based on predictive physical models to facilitate the search for low-noise trajectories using a class of automated search algorithms called local search. A novel feature of this approach is the ability to incorporate constraints into the problem formulation that addresses passenger safety and comfort.

Morris, Robert A.↗
NASA NTRS2010

Physics-Based Methods of Failure Analysis and Diagnostics in Human Space Flight

The Integrated Health Management (IHM) for the future aerospace systems requires to interface models of multiple subsystems in an efficient and accurate information environment at the earlier stages of system design. The complexity of modern aeronautic and aircraft systems (including e.g. the power distribution, flight control, solid and liquid motors) dictates employment of hybrid models and high-level reasoners for analysing mixed continuous and discrete information flow involving multiple modes of operation in uncertain environments, unknown state variables, heterogeneous software and hardware components. To provide the information link between key design/performance parameters and high-level reasoners we rely on development of multi-physics performance models, distributed sensors networks, and fault diagnostic and prognostic (FD&P) technologies in close collaboration with system designers. The main challenges of our research are related to the in-flight assessment of the structural stability, engine performance, and trajectory control. The main goal is to develop an intelligent IHM that not only enhances components and system reliability, but also provides a post-flight feedback helping to optimize design of the next generation of aerospace systems. Our efforts are concentrated on several directions of the research. One of the key components of our strategy is an innovative approach to the diagnostics/prognostics based on the real time dynamical inference (DI) technologies extended to encompass hybrid systems with hidden state trajectories. The major investments are into the multiphysics performance modelling that provides an access of the FD&P technologies to the main performance parameters of e.g. solid and liquid rocket motors and composite materials of the nozzle and case. Some of the recent results of our research are discussed in this chapter. We begin by introducing the problem of dynamical inference of stochastic nonlinear models and reviewing earlier results. Next, we present our analytical approach to the solution of this problem based on the path integral formulation. The resulting algorithm does not require an extensive global search for the model parameters, provides optimal compensation for the effects of dynamical noise, and is robust for a broad range of dynamical models. In the following Section the strengths of the algorithm are illustrated illustrated by inferring the parameters of the stochastic Lorenz system and comparing the results with those of earlier research. Next, we discuss a number of recent results in application to the development of the IHM for aerospace system. Firstly, we apply dynamical inference approach to a solution of classical three tank problems with mixed unknown continuous and binary parameters. The problem is considered in the context of ground support system for filling fuel tanks of liquid rocket motors. It is shown that the DI algorithm is well suited for successful solution of a hybrid version of this benchmark problem even in the presence of additional periodic and stochastic perturbation of unknown strength. Secondly, we illustrate our approach by its application to an analysis of the nozzle fault in a solid rocket motor (SRM). The internal ballistics of the SRM is modelled as a set of one-dimensional partial differential equations coupled to the dynamics of the propellant regression. In this example we are specifically focussed on the inference of discrete and continuous parameters of the nozzle blocking fault and on the possibility of an application of the DI algorithm to reducing the probability of "misses" of an on-board FD&P for SRM. In the next section re-contact problem caused by first stage/upper stage separation failure is discussed. The reaction forces imposed on the nozzle of the upper stage during the re-contact and their connection to the nozzle damage and to the thrust vector control (TVC) signal are obtained. It is shown that transient impact induced torquean be modelled as a response of an effective damped oscillator. A possible application of the DI algorithm to the inference of damage parameters and predicting fault dynamics ahead of time using the actuator signal is discussed. Finally, we formulate Bayesian inferential framework for development of the IHM system for in-flight structural health monitoring (SHM) of composite materials. We consider the signal generated by piezoelectric actuator mounted on composite structure generating elastic waves in it. The signal received by the sensor is than compared with the baseline signal. The possibility of damage inference is discussed in the context of development of the SHM.

Smelyanskiy, Vadim N.↗
NASA NTRS2012

Kinetic Description of Ionospheric Outflows Based on the Exact Form of Fokker-Planck Collision Operator: Electrons

We present the results of a finite difference implementation of the kinetic Fokker-Planck model with an exact form of the nonlinear collisional operator, The model is time dependent and three-dimensional; one spatial dimension and two in velocity space. The spatial dimension is aligned with the local magnetic field, and the velocity space is defined by the magnitude of the velocity and the cosine of pitch angle. An important new feature of model, the concept of integration along the particle trajectories, is discussed in detail. Integration along the trajectories combined with the operator time splitting technique results in a solution scheme which accurately accounts for both the fast convection of the particles along the magnetic field lines and relatively slow collisional process. We present several tests of the model's performance and also discuss simulation results of the evolution of the plasma distribution for realistic conditions in Earth's plasmasphere under different scenarios.

Khazanov, George V.↗
NASA NTRS1975

Study of Permanent Magnet Focusing for Astronomical Camera Tubes

A design is developed of a permanent magnet assembly (PMA) useful as the magnetic focusing unit for the 35 and 70 mm (diagonal) format SEC tubes. Detailed PMA designs for both tubes are given, and all data on their magnetic configuration, size, weight, and structure of magnetic shields adequate to screen the camera tube from the earth's magnetic field are presented. A digital computer is used for the PMA design simulations, and the expected operational performance of the PMA is ascertained through the calculation of a series of photoelectron trajectories. A large volume where the magnetic field uniformity is greater than 0.5% appears obtainable, and the point spread function (PSF) and modulation transfer function(MTF) indicate nearly ideal performance. The MTF at 20 cycles per mm exceeds 90%. The weight and volume appear tractable for the large space telescope and ground based application.

Long, D. C.↗
NASA NTRS2005

Software for Engineering Simulations of a Spacecraft

Spacecraft Engineering Simulation II (SES II) is a C-language computer program for simulating diverse aspects of operation of a spacecraft characterized by either three or six degrees of freedom. A functional model in SES can include a trajectory flight plan; a submodel of a flight computer running navigational and flight-control software; and submodels of the environment, the dynamics of the spacecraft, and sensor inputs and outputs. SES II features a modular, object-oriented programming style. SES II supports event-based simulations, which, in turn, create an easily adaptable simulation environment in which many different types of trajectories can be simulated by use of the same software. The simulation output consists largely of flight data. SES II can be used to perform optimization and Monte Carlo dispersion simulations. It can also be used to perform simulations for multiple spacecraft. In addition to its generic simulation capabilities, SES offers special capabilities for space-shuttle simulations: for this purpose, it incorporates submodels of the space-shuttle dynamics and a C-language version of the guidance, navigation, and control components of the space-shuttle flight software.

Shireman, Kirk↗
NASA NTRS2024

Quantification of Uncertainty and Risk Sensitivity for Safety of Emerging Operations

The growing need to develop and deploy small unmanned aerial vehicles (sUAVs) for various applications in the airspace necessitates reliable tools to accurately predict the flight trajectories of the sUAVs. The knowledge of the predicted trajectories help decision makers anticipate potential conflict, assess the risk, and take appropriate risk mitigation actions. In addition, uncertainties in vehicle models, weather, and controller action further highlights the need for reliable prediction tools. In this project, the application of mixed sparse grid-based quadrature and generalized polynomial chaos(gPC) expansion method for uncertainty quantification and collision assessment in air traffic consisting of fixed-wing small unmanned aerial vehicles (sUAV) was studied. From the results obtained, it can be concluded that this provides a reliable framework to carry out quantitative conflict assessment in an unmanned air traffic, which when employed, can improve the functionalities of the unmanned traffic management system. It was observed that the results from the gPC expansion framework developed in the project can be utilized to conduct rapid probabilistic collision assessment for near real-time unmanned traffic management in the airspace. From the vehicle models, position updates, and wind-field data, a priori gPC based 3-σcon-fidence ellipses can provide estimates of potential conflict at some future instants. The computational costs scaled linearly when the uncertain inputs were fewer. Further, the largest allowable distribution of para-metric uncertainties that leads to the smallest risk of collision in traffic of small unmanned aerial vehicles could be calculated. The time of closest approach between two sUAVs can be established paving way for development of proactive mitigation strategies. The separation between the sUAVs was found to be most significantly affected by uncertainties in the maximum available thrusts, zero-lift drag coefficients, and wing planform areas of the sUAVs. The study of uncertain wind-fields indicated that a heterogeneous traffic mix resulted in an increased probability of conflict. Increased measurement update rate reduced the uncertain-ties in the trajectories of the vehicles, further reducing the probability of conflict but rapid updates of all vehicles in the airspace poses a stringent communication limitation. The gPC framework also provided the means to analyze vehicle impact (crash region) due to loss of control resulting from actuator failure in sUAS traffic, essentially to predict impact and crash zones for representative vehicles. The predicted regions when compared with non-participant density, provides a means to develop an early mitigation strategy, should the sUAV detect an imminent actuator failure.

Rajnish Bhusal↗
NASA NTRS1977

Space shuttle engineering and operations support. ALT separation reference trajectories for tailcone on orbiter forward and aft CG configurations. Mission planning, mission analysis and software formulation

A preflight analysis of the ALT separation reference trajectories for the tailcone on, forward, and aft cg orbiter configurations is documented. The ALT separation reference trajectories encompass the time from physical separation of the orbiter from the carrier to orbiter attainment of the maximum ALT interface airspeed. The trajectories include post separation roll maneuvers by both vehicles and are generated using the final preflight data base. The trajectories so generated satisfy all known separation design criteria and violate no known constraints. The requirement for this analysis is given along with the specifications, assumptions, and analytical approach used to generate the separation trajectories. The results of the analytical approach are evaluated, and conclusions and recommendations are summarized.

Glenn, G. M.↗
NASA NTRS2005

Best-Fit Conic Approximation of Spacecraft Trajectory

A computer program calculates a best conic fit of a given spacecraft trajectory. Spacecraft trajectories are often propagated as conics onboard. The conic-section parameters as a result of the best-conic-fit are uplinked to computers aboard the spacecraft for use in updating predictions of the spacecraft trajectory for operational purposes. In the initial application for which this program was written, there is a requirement to fit a single conic section (necessitated by onboard memory constraints) accurate within 200 microradians to a sequence of positions measured over a 4.7-hour interval. The present program supplants a prior one that could not cover the interval with fewer than four successive conic sections. The present program is based on formulating the best-fit conic problem as a parameter-optimization problem and solving the problem numerically, on the ground, by use of a modified steepest-descent algorithm. For the purpose of this algorithm, optimization is defined as minimization of the maximum directional propagation error across the fit interval. In the specific initial application, the program generates a single 4.7-hour conic, the directional propagation of which is accurate to within 34 microradians easily exceeding the mission constraints by a wide margin.

Singh, Gurkipal↗
NASA NTRS2017

Trajectory Specification for Terminal Air Traffic: Pairwise Conflict Detection and Resolution

Trajectory Specification is the explicit bounding and control of aircraft trajectories such that the position at any point in time is constrained to a precisely defined volume of space. The bounding space is defined by cross-track, along-track, and vertical tolerances relative to a reference trajectory that specifies position as a function of time. The tolerances are dynamic and will be based on the aircraft navigation capabilities and the current traffic situation. Assuming conformance, Trajectory Specification can guarantee safe separation for an arbitrary period of time even in the event of an air traffic control (ATC) system or datalink failure; hence it can help to achieve the high level of safety and reliability needed for ATC automation. It can also reduce the reliance on tactical backup systems during normal operation. This paper applies it to the terminal area around a major airport and presents algorithms and software for detecting and resolving conflicts. A representative set of pairwise conflicts was generated, and a fast-time simulation was run on them. All conflicts were successfully resolved in real time, demonstrating the computational feasibility of the concept.

terminal area↗
NASA NTRS1973

Multiwire proportional chamber development

The development of large area multiwire proportional chambers, to be used as high resolution spatial detectors in cosmic ray experiments is described. A readout system was developed which uses a directly coupled, lumped element delay-line whose characteristics are independent of the MWPC design. A complete analysis of the delay-line and the readout electronic system shows that a spatial resolution of about 0.1 mm can be reached with the MWPC operating in the strictly proportional region. This was confirmed by measurements with a small MWPC and Fe-55 X-rays. A simplified analysis was carried out to estimate the theoretical limit of spatial resolution due to delta-rays, spread of the discharge along the anode wire, and inclined trajectories. To calculate the gas gain of MWPC's of different geometrical configurations a method was developed which is based on the knowledge of the first Townsend coefficient of the chamber gas.

Doolittle, R. F.↗
NASA NTRS1972

Longitudinal dynamic stability of a shuttle vehicle.

Analytical study of the longitudinal dynamic stability of a nonrolling, lifting vehicle gliding at hypersonic speeds. The analysis applies to shuttle vehicles designed for operating up to the rim of a planetary atmosphere. A general nondimensional time transformation is introduced to derive a unified second-order linear differential equation for the angle of attack, valid for all types of reentry of a general type of vehicle. The stability of motion is discussed for two fundamental regimes of flight that are based on widely different assumptions. For near ballistic entry along a straight line trajectory, the equation reduces to a confluent hypergeometric equation, the solution of which can be expressed in terms of Whittaker's function. Using a theorem in the theory of stability of differential equations, criteria for damped oscillations are derived. It is shown that the aerodynamic criteria for stability are the same as for the case of ballistic entry. In addition, for each vehicle configuration, and specified planetary atmosphere, there exists an altitude range where the angle of attack frequency is nearly equal to the orbital frequency causing instability in pitch. This resonance instability is due to the ellipticity of the orbit. Criteria for eccentricity instability are derived.

Vinh, N. X.↗