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At least 307 records · Page 17

Performance Evaluation of the Approaches and Algorithms Using Hamburg Airport Operations

The German Aerospace Center (DLR) and the National Aeronautics and Space Administration (NASA) have been independently developing and testing their own concepts and tools for airport surface traffic management. Although these concepts and tools have been tested individually for European and US airports, they have never been compared or analyzed side-by-side. This paper presents the collaborative research devoted to the evaluation and analysis of two different surface management concepts. Hamburg Airport was used as a common test bed airport for the study. First, two independent simulations using the same traffic scenario were conducted; one by the DLR team using the Controller Assistance for Departure Optimization (CADEO) and the Taxi Routing for Aircraft: Creation and Controlling (TRACC) in a real-time simulation environment, and one by the NASA team based on the Spot and Runway Departure Advisor (SARDA) in a fast-time simulation environment. A set of common performance metrics was defined. The simulation results showed that both approaches produced operational benefits in efficiency, such as reducing taxi times, while maintaining runway throughput. Both approaches generated the gate pushback schedule to meet the runway schedule, such that the runway utilization was maximized. The conflict-free taxi guidance by TRACC helped avoid taxi conflicts and reduced taxiing stops, but the taxi benefit needed be assessed together with runway throughput to analyze the overall performance objective.

nasa-dlr research collaboration↗

Utilizing Advanced Air Mobility Rotorcraft Tools for Wildfire Applications

Over the past decade, due in large part to heavy investment in the field of Advanced Air Mobility (AAM), significant progress in rotorcraft-focused modeling tools has been made. Such progress has notably increased AAM rotorcraft modeling capabilities in the topics of conceptual design, preliminary design, and more recently flight dynamics. Yet, due to recent and persistent increases in extreme weather events, an emerging interest has been raised in utilizing such modeling capabilities for aiding in emergency relief efforts and other public good missions. This paper uses wildfire fighting as a representative public good mission and demonstrates the relevance of the NASA Revolutionary Vertical Lift Technology (RVLT) rotorcraft toolchain to such missions. An emphasis is placed on flight dynamics modeling and control because of the hazards and challenges associated with the atmospheric environment of wildfires. In this work, the NASA FlightCODE tool was used to analyze both a UH-60 and the NASA six-passenger quadcopter reference model hovering in an experimentally informed wildfire turbulent environment. Preliminary results of this study estimate actuator usage exceedances and disturbance rejection capabilities of the vehicles’ translational rate command systems. Leveraging the RVLT toolchain, refinement and expansion of this work could lead to handling qualities envelope estimation and design optimization for wildfire turbulent environments. This would provide pilots with additional information to make real-time decisions in high-risk scenarios and begins preparations for simulating these dangerous environments for pilot training and experimentation.

Rotorcraft↗

Utilizing Advanced Air Mobility Rotorcraft Tools for Wildfire Applications

Over the past decade, due in large part to heavy investment in the field of Advanced Air Mobility (AAM), significant progress in rotorcraft-focused modeling tools has been made. Such progress has notably increased AAM rotorcraft modeling capabilities in the topics of conceptual design, preliminary design, and more recently flight dynamics. Yet, due to recent and persistent increases in extreme weather events, an emerging interest has been raised in utilizing such modeling capabilities for aiding in emergency relief efforts and other public good missions. This paper uses wildfire fighting as a representative public good mission and demonstrates the relevance of the NASA Revolutionary Vertical Lift Technology (RVLT) rotorcraft toolchain to such missions. An emphasis is placed on flight dynamics modeling and control because of the hazards and challenges associated with the atmospheric environment of wildfires. In this work, the NASA FlightCODE tool was used to analyze both a UH-60 and the NASA six-passenger quadcopter reference model hovering in an experimentally informed wildfire turbulent environment. Preliminary results of this study estimate actuator usage exceedances and disturbance rejection capabilities of the vehicles’ translational rate command systems. Leveraging the RVLT toolchain, refinement and expansion of this work could lead to handling qualities envelope estimation and design optimization for wildfire turbulent environments. This would provide pilots with additional information to make real-time decisions in high-risk scenarios and begins preparations for simulating these dangerous environments for pilot training and experimentation.

Rotorcraft↗

Scenario Complexity for Unmanned Aircraft System Traffic

This work introduces an approach to estimate the complexity of a low-altitude air traffic scenario involving multiple UASs using mathematical programming. Given a set of multi-point UAS flight trajectories, vehicle dynamics, and a conflict resolution algorithm, an abstract model is developed such that it can be solved quickly using a mathematical programming optimization software without running high-fidelity simulations that can be computationally expensive and may not suit real-time apA quick and accurate assessment of complexity for a given traffic scenario can help plan and schedule flights to alleviate traffic bottleneck and mitigate operation risks, especially for unmanned aerial system traffic management where high traffic density or complexity is expected. This work introduces a traffic scenario complexity metric that was constructed based on the number of potential conflicts weighted by the conflict resolution cost associated. The cost associated with a conflict is calculated based on the corresponding conflict resolution maneuvers. To obtain the conflict resolution maneuvers, a MILP-based optimization was formulated with the vehicle model and conflict management parameters incorporated. To evaluate the complexity metrics, an approach of using measurements from high-fidelity simulations was proposed. The scenario complexity measurements for 920 random-generated scenarios were obtained through high-fidelity simulations and treated as the ground truth. Two statistics methods: Pearson and Alternative Conditional Expectations were applied for analysis. The results showed that the number of flights has low correlation with the scenario complexity according to the correlation coefficients calculated by both methods. The Alternative Conditional Expectations method shows that the proposed scenario complexity metric has better correlation with the ground truth than the number of potential conflicts.plications. In the abstract model, each vehicle is represented by a time-varied vector associated with position, speed, and heading information. The total extra distance that aircraft need to divert from their original routes to avoid collisions is computed and used to setup a quadratic programming formula. The metrics including the number of conflicts and extra distances travelled by all vehicles are then utilized to estimate the complexity of a given UAS flight scenario. Results and verification against high-fidelity simulations will be provided in the final draft.

traffic complexity↗

In-Situ X-Ray Microscopy of Phase and Composition Distributions in Metal Alloys During Solidification

This research applies a state of the art X-ray Transmission Microscope, to image the solidification of metallic or semiconductor alloys in real-time. By employing a hard x-ray source with sub-micron dimensions, resolutions of up to 3 gm can be obtained with magnifications of over 800 X. Specimen growth conditions were optimized and the best imaging technologies applied to maintain x-ray image resolution, contrast and sensitivity. In addition, a special furnace design is required to permit controlled growth conditions and still offer maximum resolution and image contrast. We have successfully imaged in real-time: interfacial morphologies, phase growth, coalescence, incorporation of phases into the growing interface, and the solute boundary layer in the liquid at the solid-liquid inter-face. We have also measured true local growth rates and can evaluate segregation structures in the solid; a form of in-situ metallography. Composition gradients within the specimen cause vafiations in absorption of the flux such that the final image represents a spatial integral of composition (or thickness). During this study, the growth of secondary phase fibers and lameilae from eutectic and monotectic alloys have been imaged during solidification, in real-time, for the first time in bulk metal alloys. Keywords: x-ray, microscope, solidification, microfocus, real-time, microstructure

Kaukler, William F.↗

Selected Flight Test Results for Online Learning Neural Network-Based Flight Control System

The NASA F-15 Intelligent Flight Control System project team has developed a series of flight control concepts designed to demonstrate the benefits of a neural network-based adaptive controller. The objective of the team is to develop and flight-test control systems that use neural network technology to optimize the performance of the aircraft under nominal conditions as well as stabilize the aircraft under failure conditions. Failure conditions include locked or failed control surfaces as well as unforeseen damage that might occur to the aircraft in flight. This report presents flight-test results for an adaptive controller using stability and control derivative values from an online learning neural network. A dynamic cell structure neural network is used in conjunction with a real-time parameter identification algorithm to estimate aerodynamic stability and control derivative increments to the baseline aerodynamic derivatives in flight. This set of open-loop flight tests was performed in preparation for a future phase of flights in which the learning neural network and parameter identification algorithm output would provide the flight controller with aerodynamic stability and control derivative updates in near real time. Two flight maneuvers are analyzed a pitch frequency sweep and an automated flight-test maneuver designed to optimally excite the parameter identification algorithm in all axes. Frequency responses generated from flight data are compared to those obtained from nonlinear simulation runs. An examination of flight data shows that addition of the flight-identified aerodynamic derivative increments into the simulation improved the pitch handling qualities of the aircraft.

Williams, Peggy S.↗

A Strategy for Advancing Earth Independent Medical Operations

Exploration medical operations to the Moon and Mars present unprecedented challenges for providing in-mission medical care. The greater distance from Earth is the primary hazard that drives the need for a medical operations paradigm shift from low earth orbit mission. Increasingly complex long duration and long distance missions will have resource constraints (ex: mass, power, volume, data), a paucity of resupply or evacuation opportunities, and disruptions in real-time communications. In order to advance a more autonomous medical approach, a multi-faceted strategy will need to optimize all aspects of human health and performance in space. This strategy will include: increasing onboard medical autonomy through the development of novel crew health and performance systems; decision support capabilities to augment astronauts’ abilities in preventing, diagnosing, and treating medical conditions; creating new procedures and training tools for skill maintenance and just-in-time training; and enabling rapid crew access to data from all on-board systems, leading to better-informed, real-time, autonomous decisions. Collectively, this approach can be referred to as “Earth Independent Medical Operations” (EIMO). The Exploration Medical Capability Element of NASA’s Human Research Program has undertaken a longitudinal planning process to consensus around EIMO. Technical interchange meetings will occur with key constituents in late 2022 and again in early 2023 to define EIMO in practical terms and identify key elements of an EIMO system, with greater conceptual articulation and dissemination planned for the second half of 2023. We will share insights into the strategy for internal and external consensus building around EIMO, with a particular focus on the approach to partner stakeholders within NASA as well as commercial agencies.

Jay Lemery↗

An Approach to Reasoning Service Migration in Data and Reasoning Fabric (DRF) Implementation

In this paper we consider service migration problem for Data and Reasoning Fabric (DRF) enabled airspace operations assuming a fixed cloud/edge infrastructure with allocated computing, storage and power resources, where cloud/edge servers, and communication stations are in a wired connected network, while vehicles use a wireless network for communication. The objective is to automatically select the best location for the requested service execution, which achieves minimum cost while satisfying the user quality of service (QoS) and available resources constraints. To this end, estimates of the response time, consumed energy and total cost are defined for each potential compute location. A mixed-integer linear program is then formulated and solved to identify optimal compute locations given QoS constraints, network infrastructure limitations, with worst-case vehicle positioning. The approach is applied to trajectory re-planning use case to avoid a collision with an emergency vehicle in real time.

Air mobility↗

Resource Prospector Instrumentation for Lunar Volatiles Prospecting, Sample Acquisition and Processing

Data gathered from lunar missions within the last two decades have significantly enhanced our understanding of the volatile resources available on the lunar surface, specifically focusing on the polar regions. Several orbiting missions such as Clementine and Lunar Prospector have suggested the presence of volatile ices and enhanced hydrogen concentrations in the permanently shadowed regions of the moon. The Lunar Crater Observation and Sensing Satellite (LCROSS) mission was the first to provide direct measurement of water ice in a permanently shadowed region. These missions with other orbiting assets have laid the groundwork for the next step in the exploration of the lunar surface; providing ground truth data of the volatiles by mapping the distribution and processing lunar regolith for resource extraction. This next step is the robotic mission Resource Prospector (RP). Resource Prospector is a lunar mission to investigate 'strategic knowledge gaps' (SKGs) for in-situ resource utilization (ISRU). The mission is proposed to land in the lunar south pole near a permanently shadowed crater. The landing site will be determined by the science team with input from broader international community as being near traversable landscape that has a high potential of containing elevated concentrations of volatiles such as water while maximizing mission duration. A rover will host the Regolith & Environment Science and Oxygen & Lunar Volatile Extraction (RESOLVE) payload for resource mapping and processing. The science instruments on the payload include a 1-meter drill, neutron spectrometer, a near infrared spectrometer, an operations camera, and a reactor with a gas chromatograph-mass spectrometer for volatile analysis. After the RP lander safely delivers the rover to the lunar surface, the science team will guide the rover team on the first traverse plan. The neutron spectrometer (NS) and near infrared (NIR) spectrometer instruments will be used as prospecting tools to guide the traverse path. The NS will map the water-equivalent hydrogen concentration as low as 0.5% by weight to an 80 centimeter depth as the rover traverses the lunar landscape. The NIR spectrometer will measure surficial H2O/OH as well as general mineralogy. When the prospecting instruments identify a potential volatile-rich area during the course of a traverse, the prospect is then mapped out and the most promising location identified. An augering drill capable of sampling to a depth of 100 centimeters will excavate regolith for analysis. A quick assay of the drill cuttings will be made using an operations camera and NIR spectrometer. With the water depth confirmed by this first auguring activity, a regolith sample may be extracted for processing. The drill will deliver the regolith sample to a crucible that will be sealed and heated. Evolved volatiles will be measured by a gas chromatograph-mass spectrometer and the water will be captured and photographed. RP is a solar powered mission, which given the polar location translates to a relatively short mission duration on the order of 4-15 days. This short mission duration drives the concept of operations, instrumentation, and data analysis towards critical real time analysis and decision support. Previous payload field tests have increased the fidelity of the hardware, software, and mission operations. Current activities include a mission level field test to optimize interfaces between the payload and rover as well as better understand the interaction of the science and rover teams during the mission timeline. This paper will include the current status of the science instruments on the payload as well as the integrated field test occurring in fall of 2015. The concept of operations will be discussed, including the real time science and engineering decision-making process based on the critical data from the instrumentation. The path to flight will be discussed with the approach to this ambitious low cost mission.

ISRU↗

Evaluating near-real time satellite flood mapping for humanitarian early action: a case study on the 2020 Cambodia floods

Forecast-based early action is an increasingly popular framework advocating for the use of numerical weather prediction to set objective, automatic conditions for roll-out of proactive flood risk reduction efforts. Within this field, there is a relatively young discourse regarding the potential value of satellite Earth observations and the optimal role that remote sensing should play in improving the efficacy and value of forecast-based early action for flood management. The Hydrological Remote Sensing Analysis for Floods tool (HYDRAFloods) was developed to provide automated, near real-time satellite-based flood extent maps to local governments, disaster managers, and humanitarian organizations. This research evaluates (a.) whether HYDRAFloods satisfactorily met needs identified by humanitarian partners during the October 2020 floods in Cambodia, and (b.) whether the novel additions of near real-time data and flood extent mapping resulted in demonstrable advantages over previous forecast-based flood early action systems as identified in the literature. From these results, the authors identify the technological and logistical limitations of the HYDRAFloods methodology and synthesize actionable recommendations for Earth observations practitioners seeking to better support forecast-based early action.

floods↗

Some data processing requirements for precision Nap-Of-the-Earth (NOE) guidance and control of rotorcraft

Nap-Of-the-Earth (NOE) flight in a conventional helicopter is extremely taxing for two pilots under visual conditions. Developing a single pilot all-weather NOE capability will require a fully automatic NOE navigation and flight control capability for which innovative guidance and control concepts were examined. Constrained time-optimality provides a validated criterion for automatically controlled NOE maneuvers if the pilot is to have confidence in the automated maneuvering technique. A second focus was to organize the storage and real-time updating of NOE terrain profiles and obstacles in course-oriented coordinates indexed to the mission flight plan. A method is presented for using pre-flight geodetic parameter identification to establish guidance commands for planned flight profiles and alternates. A method is then suggested for interpolating this guidance command information with the aid of forward and side looking sensors within the resolution of the stored data base, enriching the data content with real-time display, guidance, and control purposes. A third focus defined a class of automatic anticipative guidance algorithms and necessary data preview requirements to follow the vertical, lateral, and longitudinal guidance commands dictated by the updated flight profiles and to address the effects of processing delays in digital guidance and control system candidates. The results of this three-fold research effort offer promising alternatives designed to gain pilot acceptance for automatic guidance and control of rotorcraft in NOE operations.

Clement, Warren F.↗

Parallel processing and expert systems

Whether it be monitoring the thermal subsystem of Space Station Freedom, or controlling the navigation of the autonomous rover on Mars, NASA missions in the 1990s cannot enjoy an increased level of autonomy without the efficient implementation of expert systems. Merely increasing the computational speed of uniprocessors may not be able to guarantee that real-time demands are met for larger systems. Speedup via parallel processing must be pursued alongside the optimization of sequential implementations. Prototypes of parallel expert systems have been built at universities and industrial laboratories in the U.S. and Japan. The state-of-the-art research in progress related to parallel execution of expert systems is surveyed. The survey discusses multiprocessors for expert systems, parallel languages for symbolic computations, and mapping expert systems to multiprocessors. Results to date indicate that the parallelism achieved for these systems is small. The main reasons are (1) the body of knowledge applicable in any given situation and the amount of computation executed by each rule firing are small, (2) dividing the problem solving process into relatively independent partitions is difficult, and (3) implementation decisions that enable expert systems to be incrementally refined hamper compile-time optimization. In order to obtain greater speedups, data parallelism and application parallelism must be exploited.

Lau, Sonie↗

Studies in Astronomical Time Series Analysis. VI. Bayesian Block Representations

This paper addresses the problem of detecting and characterizing local variability in time series and other forms of sequential data. The goal is to identify and characterize statistically significant variations, at the same time suppressing the inevitable corrupting observational errors. We present a simple nonparametric modeling technique and an algorithm implementing it-an improved and generalized version of Bayesian Blocks [Scargle 1998]-that finds the optimal segmentation of the data in the observation interval. The structure of the algorithm allows it to be used in either a real-time trigger mode, or a retrospective mode. Maximum likelihood or marginal posterior functions to measure model fitness are presented for events, binned counts, and measurements at arbitrary times with known error distributions. Problems addressed include those connected with data gaps, variable exposure, extension to piece- wise linear and piecewise exponential representations, multivariate time series data, analysis of variance, data on the circle, other data modes, and dispersed data. Simulations provide evidence that the detection efficiency for weak signals is close to a theoretical asymptotic limit derived by [Arias-Castro, Donoho and Huo 2003]. In the spirit of Reproducible Research [Donoho et al. (2008)] all of the code and data necessary to reproduce all of the figures in this paper are included as auxiliary material.

signal detection↗

Constructing an Efficient Self-Tuning Aircraft Engine Model for Control and Health Management Applications

Self-tuning aircraft engine models can be applied for control and health management applications. The self-tuning feature of these models minimizes the mismatch between any given engine and the underlying engineering model describing an engine family. This paper provides details of the construction of a self-tuning engine model centered on a piecewise linear Kalman filter design. Starting from a nonlinear transient aerothermal model, a piecewise linear representation is first extracted. The linearization procedure creates a database of trim vectors and state-space matrices that are subsequently scheduled for interpolation based on engine operating point. A series of steady-state Kalman gains can next be constructed from a reduced-order form of the piecewise linear model. Reduction of the piecewise linear model to an observable dimension with respect to available sensed engine measurements can be achieved using either a subset or an optimal linear combination of "health" parameters, which describe engine performance. The resulting piecewise linear Kalman filter is then implemented for faster-than-real-time processing of sensed engine measurements, generating outputs appropriate for trending engine performance, estimating both measured and unmeasured parameters for control purposes, and performing on-board gas-path fault diagnostics. Computational efficiency is achieved by designing multidimensional interpolation algorithms that exploit the shared scheduling of multiple trim vectors and system matrices. An example application illustrates the accuracy of a self-tuning piecewise linear Kalman filter model when applied to a nonlinear turbofan engine simulation. Additional discussions focus on the issue of transient response accuracy and the advantages of a piecewise linear Kalman filter in the context of validation and verification. The techniques described provide a framework for constructing efficient self-tuning aircraft engine models from complex nonlinear simulations.Self-tuning aircraft engine models can be applied for control and health management applications. The self-tuning feature of these models minimizes the mismatch between any given engine and the underlying engineering model describing an engine family. This paper provides details of the construction of a self-tuning engine model centered on a piecewise linear Kalman filter design. Starting from a nonlinear transient aerothermal model, a piecewise linear representation is first extracted. The linearization procedure creates a database of trim vectors and state-space matrices that are subsequently scheduled for interpolation based on engine operating point. A series of steady-state Kalman gains can next be constructed from a reduced-order form of the piecewise linear model. Reduction of the piecewise linear model to an observable dimension with respect to available sensed engine measurements can be achieved using either a subset or an optimal linear combination of "health" parameters, which describe engine performance. The resulting piecewise linear Kalman filter is then implemented for faster-than-real-time processing of sensed engine measurements, generating outputs appropriate for trending engine performance, estimating both measured and unmeasured parameters for control purposes, and performing on-board gas-path fault diagnostics. Computational efficiency is achieved by designing multidimensional interpolation algorithms that exploit the shared scheduling of multiple trim vectors and system matrices. An example application illustrates the accuracy of a self-tuning piecewise linear Kalman filter model when applied to a nonlinear turbofan engine simulation. Additional discussions focus on the issue of transient response accuracy and the advantages of a piecewise linear Kalman filter in the context of validation and verification. The techniques described provide a framework for constructing efficient self-tuning aircraft engine models from complex nonlinear simulatns.

Armstrong, Jeffrey B.↗

Time-Varying Reference Model Modification of Adaptive Control with Multi-Objective Performance Optimization

This paper presents an adaptive control approach that modifies a reference model in order to satisfy a multi-objective optimization of two performance signals: a linear signal and a quadratic signal. The time varying reference model modification is accomplished by the real-time solutions of the time-varying Riccati and Sylvester equations coupled with the least-squares parameter estimation of the sensitivities of the performance signals to be minimized. The effectiveness of the proposed method is to be demonstrated by a flight control application of maneuver load alleviation and drag minimization.

Adaptive Control↗

Tracking capability for entry, descent and landing and its support to NASA Mars Exploration Rovers

This paper described a new capability recently deployed in the NASA Deep Space Network to support the entry, descent and landing of the two Mars Exploration Rovers - Spirit and Opportunity. This final segment of a 7-month journey was critical step in enabling successful mission operation. The EDL Data Analysis (EDA) equipment processed the open loop recording samples from the existing radio science receivers, extracted the MFSK-modulated tones, and provided information on the occurrence of associated events happened onboard spacecraft. The EDA software supported real-time as well as post pass processing. The design flexibility allowed the track to be divided into different segments, each could be processed with a configuration optimized for its signal conditions and uncertainties. The use of dynamic lock state concept enabled signal detection be done with minimum required computing power - less was spent when signal was steadily detected and more when signal fluctuated. With the server-client design, the EDA supported connections to different remote operation centers for monitor purpose. The paper also presented the field results from both Spirit and Opportunity landings. The detection were much better than anticipated. All critical event tones were seen in real time. Surprising detection of LCP signal was found with Spirit landing, resulted in a reconfiguration for Opportunity landing to maximize signal detection.

tracking↗

Guidance and Control strategies for aerospace vehicles

A neighboring optimal guidance scheme was devised for a nonlinear dynamic system with stochastic inputs and perfect measurements as applicable to fuel optimal control of an aeroassisted orbital transfer vehicle. For the deterministic nonlinear dynamic system describing the atmospheric maneuver, a nominal trajectory was determined. Then, a neighboring, optimal guidance scheme was obtained for open loop and closed loop control configurations. Taking modelling uncertainties into account, a linear, stochastic, neighboring optimal guidance scheme was devised. Finally, the optimal trajectory was approximated as the sum of the deterministic nominal trajectory and the stochastic neighboring optimal solution. Numerical results are presented for a typical vehicle. A fuel-optimal control problem in aeroassisted noncoplanar orbital transfer is also addressed. The equations of motion for the atmospheric maneuver are nonlinear and the optimal (nominal) trajectory and control are obtained. In order to follow the nominal trajectory under actual conditions, a neighboring optimum guidance scheme is designed using linear quadratic regulator theory for onboard real-time implementation. One of the state variables is used as the independent variable in reference to the time. The weighting matrices in the performance index are chosen by a combination of a heuristic method and an optimal modal approach. The necessary feedback control law is obtained in order to minimize the deviations from the nominal conditions.

Hibey, J. L.↗

Neighboring optimal guidance for aeroassisted noncoplanar orbital transfer

The fuel-optimal control problem in aeroassisted noncoplanar orbital transfer is addressed. The equations of motion for the atmospheric maneuver are nonlinear and the optimal (nominal) trajectory and control are obtained. In order to follow the nominal trajectory under actual conditions, a neighboring optimum guidance scheme is designed using linear quadratic regulator theory for onboard real-time implementation. One of the state variables is used as the independent variable in preference to the time. The weighting matrices in the performance index are chosen by a combination of a heuristic method and an optimal modal approach. The necessary feedback control law is obtained in order to minimize the deviations from the nominal conditions. The results are presented to a typical aeroassisted noncoplanar orbital transfer problem.

Naidu, D. S.↗