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At least 289 records · Page 16

The Restricted Three-Body-Problem as a Perturbation of Euler's Problem of Two Fixed Centers and Its Application to Lunar Trajectories

The restricted Three-Body-Problem considers the motion of an infinitesimal mass under the gravitational attraction of two finite masses, which revolve about their common center of gravity in coplanar circles. It is well known that Euler's problem of two fixed centers, consisting of the motion of an infinitesimal mass under the gravitational attraction of two finite masses fixed in space, can be solved by elliptic functions. The idea presented here is to take the solution of Euler's problem as the solution of the restricted Three-Body-Problem by allowing the initial values to be functions of time now. Differential equations for the perturbed initial values are established. These equations can be given in closed form by using the fact that the transformation to the perturbed initial values of Euler's problem is canonical. Thus, an approximation can be obtained for the solution of the restricted Three-Body-Problem. The method can also be used to represent classes of neighboring trajectories for guidance purposes.

Euler equation↗

Training the Powered-Lift Evaluation Pilot

This poster describes a project to prepare pilots for a study assessing novel aircraft automation concepts for electric Vertical Takeoff and Landing (eVTOL) aircraft using NASA’s Vertical Motion Simulator (VMS). By exploring the operational and learning challenges related to transitioning between forward flight and vertical landing, we seek to establish baselines of pilot workload and aircraft handling qualities across varying atmospheric conditions and automation states. The simulated eVTOL design differentiates flight control allocations as a function of airspeed across four speed ranges as the vehicle transitions between fully thrust-borne lift and wing-borne lift. As speed increases, side stick controls command: translational ground speeds, vertical and lateral acceleration, vertical rate, vertical flight path angle, and bank angle. This novel approach to flight control allocation creates a significant learning challenge for pilots. Since initial eVTOL aircraft may have limitations on hover capabilities, automation and flight guidance cues also vary with airspeed to provide efficient landing profiles while still providing cues suitable for cruise flight. The NASA team prepared the study pilots to follow these flight guidance cues along curved Required Navigation Performance (RNP) approaches and along 6o and 12o glide paths to energy-efficient assistive-hover landing and goarounds. The pre-VMS preparation sought to prepare pilots from diverse levels of experience and background. To do this, NASA researchers designed and developed a fixed-based, large field-ofview simulator with terrain, structures, and air traffic. With one day of combined classroom learning and skill development in the fixedbase simulator, pilots were largely able to fly the simulated eVTOL in the VMS with sufficient mastery to provide handling quality assessments using the Cooper-Harper Handling Qualities Rating and workload assessments through the Bedford Workload Scale.

AAM↗

Development of a Crosslink Channel Simulator

Distributed Spacecraft missions are an integral part of current and future plans for NASA and other space agencies. Many of these multi-vehicle missions involve utilizing the array of spacecraft as a single, instrument requiring communication via crosslinks to achieve mission goals. NASA s Goddard Space Flight Center (GSFC) is developing the Formation Flying Test Bed (FFTB) to provide a hardware-in-the-loop simulation environment to support mission concept development and system trades with a primary focus on Guidance, Navigation, and Control (GN&C) challenges associated with spacecraft flying. The goal of the FFTB is to reduce mission risk by assisting in mission planning and analysis, provide a technology development platform that allows algorithms to be developed for mission functions such as precision formation navigation and control and time synchronization. The FFTB will provide a medium in which the various crosslink transponders being used in multi-vehicle missions can be integrated for development and test; an integral part of the FFTB is the Crosslink Channel Simulator (CCS). The CCS is placed into the communications channel between the crosslinks under test, and is used to simulate on-mission effects to the communications channel such as vehicle maneuvers, relative vehicle motion, or antenna misalignment. The CCS is based on the Starlight software programmable platform developed at General Dynamics Decision Systems and provides the CCS with the ability to be modified on the fly to adapt to new crosslink formats or mission parameters. This paper briefly describes the Formation Flying Test Bed and its potential uses. It then provides details on the current and future development of the Crosslink Channel Simulator and its capabilities.

Hunt, Chris↗

Nearly Circular Transfer Trajectories for Descending Satellites

Simplified expressions describing the transfer from a satellite bit to the point of atmospheric entry are derived. The expressions are limited to altitude changes that are small compared with the earth's radius, and velocity changes small compared with satellite velocity. They are further restricted to motion about a spherical, nonrotating earth. The transfer orbit resulting from the application of thrust in any direction at any point in an elliptic orbit is considered. Expressions for the errors in distance (miss distance) and entry angle due to an initial misalignment and magnitude error of the deflecting thrust are presented. The largest potential contributing factor towards a miss distance stems from the misalignment of the retrovelocity increment. If this velocity increment is pointed in direct opposition to the path, a 1 deg misalignment leads to a miss distance of 3.45 miles. However, it is shown that this error can be avoided by applying the velocity increment at an angle between 120 deg and 150 deg below the flight-path, direction. The guidance and accuracy requirements to establish a circular orbit, in addition to the corrections applied to to transform elliptic orbits into circular ones, are also discussed.

Low, George M.↗

Improving Efficacy and Safety of Pharmacological Treatment Through Precision Medicine and Pharmacogenomics for Human Deep Space Exploration

INTRODUCTION: Future spaceflight will require increased crew medical autonomy as exploration class missions expanding duration and distance from Earth, especially for Mars missions. As mission duration increases, it will be even more essential to have appropriate amounts of effective medication to ensure the maintenance of crew health and performance. Conversely, mass and volume constraints will become more severe as future spaceflight expands beyond low Earth orbit, where resupply is difficult or becomes impossible. These constraints thus convey an urgency to further tailor medications included in the spacecraft formulary and increased examination of appropriate dosing regimens. BACKGROUND: Precision Health is an exciting area of cutting-edge research and medicine focused on maintaining an individual’s health and performance through in-depth understanding of an individual’s unique factors and molecular profiles. This approach can be adapted to better predict, monitor, and address physiological responses to the spaceflight environment. One example is the field of pharmacogenomics (PGX),the study of how the expressed genome impacts drug responses with the goal of prescribing the right dose of the right drug at the right time. Specifically, PGX testing provides valuable information on an individual’s precise allelic variations to guide physicians in making informed decisions on pharmaceutical choice and dosing to avoid adverse drug events and maximize pharmacological efficacy. The goal of this study was to evaluate which drugs in the current space pharmacy could be evaluated using PGX testing and to understand the potential impact on the health and wellness of the astronaut population. Additionally, we sought to evaluate clinically available FDA-approved PGX testing solutions to better understand its applicability. METHODS: A complete list of drugs onboard the International Space Station (ISS) was analyzed for risk and likelihood of drug failure and PGX actionability. This analysis encompassed both personal astronaut medications, including supplements and over the counter drugs (n=151) and ISS MedKit formulary medications (n=95). Duplicate medications and different formulations were removed, which resulted in 157 total drugs used in the subsequent analysis. A 5x5 risk assessment table was produced by examining the likelihood of drug failure compared to the consequence of drug failure. Likelihood of individual drug failure was defined by whether existing processes are sufficient to prevent adverse events, as ranked from 1 (very low, can easily be prevented) to 5 (very high, cannot be prevented) during a Mars mission. In contrast, the consequence of drug failure was defined by impact to safety, schedule, cost or technical and ranked from 1 (very low) to 5 (very high).A comprehensive assessment of commercially available PGX solutions is currently underway to evaluate specimen requirements, cost/benefit analysis (cost vs. number of alleles assessed), utility of variant analysis, relevance to inflight medication usage, quality of reporting in enabling clinical application, and ease of integration into electronic medical records. RESULTS: Risk assessments(LxC 5x5 table) indicated29medicationswere in the yellow or red zone driven predominantly by drug failure or safety concerns, with the remainder(n=128)of the medications in the green zone where risk is acceptable. We found that current PGX testing results could impact 21% of the total medications in the ISS MedKit and IMAK; of these, 9 medications currently have direct clinically actionable guidance available. Results of the clinical PGX solution evaluations as related to these medications will be presented. CONCLUSION: PGX testing has demonstrated clear benefits in terrestrial medicine and clinical environments for the selection of proper medications, avoiding adverse drug reactions, and maximizing drug efficacy. We propose that similar benefits would be bestowed on the astronaut and commercial spaceflight passenger population by performing preemptive pre-flight PGX testing to reduce risk of mission failure due to ineffective or toxic medications, improve targeting drug efficacy and safety, and further open the door to countermeasure research exploring PGX-related allelic variants. For example, PGX results could allow tailoring of specific medications at optimal doses more precisely to each individual astronaut, particularly in areas of space motion sickness, sleep aids, and analgesics. An additional benefit is that PGX results could provide information for better planning of the components of a space pharmacy for deep space missions to be more cost effective and more efficient in the utilization of limited pharmaceutical resources. Finally, while PGX testing of the astronaut corps is not currently conducted, this approach could provide immediate impact in support of mission success by reducing risks, optimizing astronaut performance, and providing valuable insights into long-term astronaut health. Such advancements in clinical decision making are important next steps in building dynamic individual risk profiles for astronauts, increasing crew autonomy and providing tailored countermeasures

Alice R W Tang↗

Optimal aeroassisted coplanar orbital transfer using an energy model

The atmospheric portion of the trajectories for the aeroassisted coplanar orbit transfer was investigated. The equations of motion for the problem are expressed using reduced order model and total vehicle energy, kinetic plus potential, as the independent variable rather than time. The order reduction is achieved analytically without an approximation of the vehicle dynamics. In this model, the problem of coplanar orbit transfer is seen as one in which a given amount of energy must be transferred from the vehicle to the atmosphere during the trajectory without overheating the vehicle. An optimal control problem is posed where a linear combination of the integrated square of the heating rate and the vehicle drag is the cost function to be minimized. The necessary conditions for optimality are obtained. These result in a 4th order two-point-boundary-value problem. A parametric study of the optimal guidance trajectory in which the proportion of the heating rate term versus the drag varies is made. Simulations of the guidance trajectories are presented.

Halyo, Nesim↗

Optical flow versus retinal flow as sources of information for flight guidance

The appropriate description is considered of visual information for flight guidance, optical flow vs. retinal flow. Most descriptions in the psychological literature are based on the optical flow. However, human eyes move and this movement complicates the issues at stake, particularly when movement of the observer is involved. The question addressed is whether an observer, whose eyes register only retinal flow, use information in optical flow. It is suggested that the observer cannot and does not reconstruct the image in optical flow; instead they use retinal flow. Retinal array is defined as the projections of a three space onto a point and beyond to a movable, nearly hemispheric sensing device, like the retina. Optical array is defined as the projection of a three space environment to a point within that space. And flow is defined as global motion as a field of vectors, best placed on a spherical projection surface. Specifically, flow is the mapping of the field of changes in position of corresponding points on objects in three space onto a point, where that point has moved in position.

Cutting, James E.↗

Insights into Rotational and Translational Dynamics in Mixtures of Ethylene Glycol and Choline Chloride Using Nuclear Magnetic Resonance Techniques

This work examines molecular dynamics and interactions in ethylene glycol–choline chloride (EG–ChCl) mixtures across 0–33 mol % ChCl, spanning the true eutectic region near 17–20 mol % and the commonly used 1:2 formulation. We combine pulsed-field-gradient (PFG) diffusion, fast-field-cycling (FFC) relaxometry, temperature-dependent 13 C T 1 , and nuclear Overhauser effect spectroscopy (NOESY) to disentangle local from macroscopic dynamics. PFG and FFC show that both translational and average rotational motions largely track the strong increase in viscosity with ChCl content, with ethylene glycol consistently diffusing faster than the choline cation and no global dynamical anomaly at the eutectic composition. More subtle, site-specific composition effects nevertheless emerge. The ratio of the diffusion coefficient of the hydroxyl group of choline to the diffusion coefficient of the methyl group of choline displays a shallow minimum in the 17–25 mol % region, indicating a modest change in how the hydroxyl-bearing end of choline samples the underlying translational motion relative to the methyl groups. 13 C T 1 analysis shows that rotational correlation times at 25 °C generally increase with ChCl, reflecting viscosity-coupled slowing, while the CH 2 –N α site exhibits a small but reproducible deviation from this monotonic trend near the eutectic. NOESY spectra at similar compositions reveal enhanced cross-relaxation between EG and choline protons, consistent with increased headgroup–solvent contact density rather than a wholesale structural rearrangement. Overall, our multitechnique study demonstrates that EG–ChCl dynamics are predominantly viscosity-dominated, with the eutectic region acting as a subtle dynamical crossover where specific choline segments become maximally coupled to the hydrogen-bond network. These insights refine the structure–dynamics picture of choline-chloride DESs and provide practical guidance for tuning composition in electrochemical, separation, and catalytic applications.

diffusion↗

Simulation evaluation of display/FLIR concepts for low-altitude, terrain-following helicopter operations

A piloted simulation of three head-down display (HDD) concepts with flight-director guidance superimposed on forward-looking infrared (FLIR) imagery was performed to evaluate the task of low-level, terrain-following (TF), manual helicopter flight. The three display concepts were examined for the purpose of finding ways by which aircraft flight-attitude and command symbols and FLIR imagery could be integrated onto one instrument. In all cases, the FLIR imagery was centered on the flight-path vector of the aircraft. The three displays were then characterized by having: (1) pitch attitude conformal to the FLIR imagery; (2) pitch attitude conformal to the FLIR imagery, but with an increase in the scaling; and (3) pitch attitude nonconformal to the FLIR imagery with the same pitch scaling as in (2). The simulation was conducted on the Vertical Motion Simulator (VMS) at Ames Research Center, using NASA and Air Force test pilots. The pilots indicated that the nonconformal pitch attitude and FLIR display was the preferred way to display information because of the absence of pitch-attitude information on displays (1) and (2) during some portions of the operational flight envelope.

Swenson, H. N.↗

Using Visual Odometry to Estimate Position and Attitude

A computer program in the guidance system of a mobile robot generates estimates of the position and attitude of the robot, using features of the terrain on which the robot is moving, by processing digitized images acquired by a stereoscopic pair of electronic cameras mounted rigidly on the robot. Developed for use in localizing the Mars Exploration Rover (MER) vehicles on Martian terrain, the program can also be used for similar purposes on terrestrial robots moving in sufficiently visually textured environments: examples include low-flying robotic aircraft and wheeled robots moving on rocky terrain or inside buildings. In simplified terms, the program automatically detects visual features and tracks them across stereoscopic pairs of images acquired by the cameras. The 3D locations of the tracked features are then robustly processed into an estimate of overall vehicle motion. Testing has shown that by use of this software, the error in the estimate of the position of the robot can be limited to no more than 2 percent of the distance traveled, provided that the terrain is sufficiently rich in features. This software has proven extremely useful on the MER vehicles during driving on sandy and highly sloped terrains on Mars.

Maimone, Mark↗

Time controlled descent guidance algorithm for simulation of advanced ATC systems

Concepts and computer algorithms for generating time controlled four dimensional descent trajectories are described. The algorithms were implemented in the air traffic control simulator and used by experienced controllers in studies of advanced air traffic flow management procedures. A time controlled descent trajectory comprises a vector function of time, including position, altitude, and heading, that starts at the initial position of the aircraft and ends at touchdown. The trajectory provides a four dimensional reference path which will cause an aircraft tracking it to touchdown at a predetermined time with a minimum of fuel consumption. The problem of constructing such trajectories is divided into three subproblems involving synthesis of horizontal, vertical, and speed profiles. The horizontal profile is constructed as a sequence of turns and straight lines passing through a specified set of waypoints. The vertical profile consists of a sequence of level flight and constant descent angle segments defined by altitude waypoints. The speed profile is synthesized as a sequence of constant Mach number, constant indicated airspeed, and acceleration/deceleration legs. It is generated by integrating point mass differential equations of motion, which include the thrust and drag models of the aircraft.

Lee, H. Q.↗

Kepler Commissioning Data for Measurement of the Pixel Response Function and Focal Plane Geometry

This document describes the Kepler PRF/FPG data release. This data was taken on April 27-29, 2009, during Kepler's commissioning phase in order to measure the pixel response function (PRF) (Bryson et al., 2010a) and focal plane geometry (FPG) (Tenenbaum and Jenkins, 2010). 33,424 stellar targets were observed for 243 long cadences, each with a duration of 14.7 minutes (half the duration of a normal Kepler long cadence). During these 243 cadences the Kepler photometer was moved, pointing in a dither pattern to facilitate PRF measurement. Motion occurred during the even cadences (second, fourth, etc.), with the telescope in stable fine point at each pointing in the dither pattern during the odd cadences (first, third, etc.). The first and last cadences were at the center of the dither pattern. Motion cadences are included in this release, but they do not contain any data. For details on how this data was used to derive the Kepler PRF and FPG models, see Bryson et al. (2010a) and Tenenbaum and Jenkins (2010). Descriptions of the PRF and FPG models are found in Thompson et al. (2016), x2.3.5.17 and x2.3.5.16 respectively. The data in this release can be used to recompute the Kepler PRF and FPG. Such a reconstruction, however, would not reflect measured seasonal changes in the PRF described in Van Cleve et al. (2016b), x5.2.The dither pattern is shown in Figure 1. The crosses show the commanded pointings and the circles show the measured pointings. Measured pointings are different from the commanded pointings due to the early state of calibration of the fine guidance sensors during commissioning (Van Cleve et al., 2016a). The measured offsets from the center of the pattern are given in RADEC offsets and pixel offsets in Table 1. The order of the offsets was randomized during data collection to avoid time-dependent systematics.

Kepler↗

Guidance, Navigation, and Control Performance for the GOES-R Spacecraft

The Geostationary Operational Environmental Satellite-R Series (GOES-R) is the first of the next generation geostationary weather satellites. The series represents a dramatic increase in Earth observation capabilities, with 4 times the resolution, 5 times the observation rate, and 3 times the number of spectral bands. GOES-R also provides unprecedented availability, with less than 120 minutes per year of lost observation time. This paper presents the Guidance Navigation & Control (GN&C) requirements necessary to realize the ambitious pointing, knowledge, and Image Navigation and Registration (INR) objectives of GOES-R. Because the suite of instruments is sensitive to disturbances over a broad spectral range, a high fidelity simulation of the vehicle has been created with modal content over 500 Hz to assess the pointing stability requirements. Simulation results are presented showing acceleration, shock response spectra (SRS), and line of sight (LOS) responses for various disturbances from 0 Hz to 512 Hz. Simulation results demonstrate excellent performance relative to the pointing and pointing stability requirements, with LOS jitter for the isolated instrument platform of approximately 1 micro-rad. Attitude and attitude rate knowledge are provided directly to the instrument with an accuracy defined by the Integrated Rate Error (IRE) requirements. The data are used internally for motion compensation. The final piece of the INR performance is orbit knowledge, which GOES-R achieves with GPS navigation. Performance results are shown demonstrating compliance with the 50 to 75 m orbit position accuracy requirements. As presented in this paper, the GN&C performance supports the challenging mission objectives of GOES-R.

Control↗

Skylab task and work performance /Experiment M-151 - Time and motion study/

The primary objective of Experiment M151 was to study the inflight adaptation of Skylab crewmen to a variety of task situations involving different types of activity. A parallel objective was to examine astronaut inflight performance for any behavioral stress effects associated with the working and living conditions of the Skylab environment. Training data provided the basis for comparison of preflight and inflight performance. Efficiency was evaluated through the adaptation function, namely, the relation of performance time over task trials. The results indicate that the initial changeover from preflight to inflight was accompanied by a substantial increase in performance time for most work and task activities. Equally important was the finding that crewmen adjusted rapidly to the weightless environment and became proficient in developing techniques with which to optimize task performance. By the end of the second inflight trial, most of the activities were performed almost as efficiently as on the last preflight trial. The analysis demonstrated the sensitivity of the adaptation function to differences in task and hardware configurations. The function was found to be more regular and less variable inflight than preflight. Translation and control of masses were accomplished easily and efficiently through the rapid development of the arms and legs as subtle guidance and restraint systems.

Kubis, J. F.↗

Verification of the Generalized Aerospace Simulation in Simulink

NASA uses six-degrees-of-freedom (6-DOF) simulations tools to design, test, develop Guidance Navigation and Control (GN&C) software, and certify vehicle performance prior to flight. Therefore, it is critical that the 6-DOF tools used for vehicle design and certification are validated. The focus of this work is the validation of the NASA Marshall Space Flight Center 6-DOF “GeneraLized Aerospace Simulation in Simulink” (GLASS) framework tool. The GLASS tool framework is currently used to support NASA GN&C insight for the Human Landing System (HLS) project, simulating vehicle dynamics during lunar descent and ascent. The GLASS framework utilizes the off-the-shelf Mathworks (R) Simscape (TM) Multibody (TM) toolbox to model vehicle multi-body dynamics. NASA’s Engineering and Safety Center (NESC) provides a set of 6-DOF simulation verification “check cases” that are available to any user needing to verify 6-DOF tools. The check cases contain seventeen atmospheric and twenty-six orbital test scenarios are provided to validate equations of motion, environmental models (e.g., atmosphere, gravitation, and geodesy) and tool propagators. This paper compares GLASS 6-DOF simulation results against the NESC check cases’ results via simulation-to-simulation comparisons. The comparison demonstrates that GLASS simulation results are “in family” with the outputs of the applicable NASA NESC check-cases and verify the GLASS core framework dynamics and the implementation of the check case scenario models.

6-Dof↗

Verification of the Generalized Aerospace Simulation in Simulink (R)

NASA uses six-degrees-of-freedom (6-DOF) simulations tools to design, test, develop Guidance Navigation and Control (GN&C) software, and certify vehicle performance prior to flight. Therefore, it is critical that the 6-DOF tools used for vehicle design and certification are validated. The focus of this work is the vali-dation of the NASA Marshall Space Flight Center 6-DOF “GeneraLized Aero-space Simulation in Simulink” (GLASS) framework tool. The GLASS tool framework is currently used to support NASA GN&C insight for the Human Landing System (HLS) project, simulating vehicle dynamics during lunar descent and as-cent. The GLASS framework utilizes the off-the-shelf Mathworks ® Simscape Multibody® toolbox to model vehicle multi-body dynamics. NASA’s Engineering and Safety Center (NESC) provides a set of 6-DOF simulation verification “check cases” that are available to any user needing to verify 6-DOF tools. The check cases contain seventeen atmospheric and twenty-six orbital test scenarios are provided to validate equations of motion, environmental models (e.g., atmosphere, gravitation, and geodesy) and tool propagators. This paper compares GLASS 6-DOF simulation results against the NESC check cases’ results via simulation-to-simulation comparisons. The comparisons demonstrate that GLASS simulation results are “in family” with the outputs of the applicable NASA NESC check cases and verifies the GLASS core framework dynamics and the correct implementation of the check case scenario models.

6-Dof↗

Guidance, steering, load relief and control of an asymmetric launch vehicle

A new guidance, steering, and control concept is described and evaluated for the Third Phase of an asymmetrical configuration of the Advanced Launch System (ALS). The study also includes the consideration of trajectory shaping issues and trajectory design as well as the development of angular rate, angular acceleration, angle of attack, and dynamic pressure estimators. The Third Phase guidance, steering and control system is based on controlling the acceleration-direction of the vehicle after an initial launch maneuver. Unlike traditional concepts, the alignment of the estimated and commanded acceleration-directions is unimpaired by an add-on load relief. Instead, the acceleration-direction steering-control system features a control override that limits the product of estimated dynamic pressure and estimated angle of attack. When this product is not being limited, control is based exclusively on the commanded acceleration-direction without load relief. During limiting, control is based on nulling the error between the limited angle of attack and the estimated angle of attack. This limiting feature provides full freedom to the acceleration-direction steering and control to shape the trajectory within the limit, and also gives full priority to the limiting of angle of attack when necessary. The flight software concepts were analyzed on the basis of their effects on pitch plane motion.

Boelitz, Frederick W.↗

Predictor-Corrector Entry Guidance for Reusable Launch Vehicles

An online entry guidance algorithm has been developed using a predictor-corrector approach. The algorithm is designed for the Reusable Launch Vehicle (RLV) and is demonstrated by using, the X-33 model. The objective of the design is to handle widely dispersed entry conditions and deliver the vehicle at the Terminal Area Energy Management (TAEM) interface box within an acceptable tolerance and without violating any of the vehicle physical constraints. Combination of several control variables is used in testing the performance and computational requirement of the algorithm. The control variables are the bank angle, angle-of-attack and the time for roll reversal. The bank angle and angle-of-attack profiles are the nominal profiles plus the perturbations in each direction. The initial guess of the bank profile is a 45 degrees bank angle with reversal at 360 seconds from liftoff. A six-element state vector is propagated to the TAEM interface box through the integration of the equations of motion (EOM). Altitude, heading and range errors are computed between the desired and the achieved state at the TAEM interface. These errors are used to correct the initial guess of the control variables. This process is repeated until the errors meet an acceptable level at the TAEM interface. Several numerical optimization methods are used to evaluate the convergent property of the predictor-predictor methodology. Successful results are demonstrated using the X-33 model.

Youssef, Hussein↗