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At least 505 records · Page 28

Celestial Mapping System for Lunar Surface Mapping and Analytics

Celestial Mapping System (CMS) is a software platform to generate virtual 3D globe for celestial bodies within our solar system. Various layers are built on top of the virtual globe to provide visualization of high resolution imagery, enable precise measurements, build analytical capabilities and broad range of functionalities to assist planetary scientists and mission planners. CMS is built using OpenJDK 11 and will run on a wide variety of platforms such as Linux, Windows, OSX, etc. It has a thick client with less overhead to access hardware resources. This allows features such as terrain profiling and distance calculations to be performed on the client and on the fly. The present focus of CMS is on developing lunar mapping tool kits to provide features such as - 3D first person view with zoom and navigational capabilities, realistic terrain visualization based on LRO data, measurement tools, Apollo landing site annotations, stereoscopic view, elevation profiles, line of sight analysis and many more. The application is developed to provide situational and domain awareness on Lunar surface, planning capabilities for equipment placements and traverse path optimization. As data becomes available, CMS has the capabilities to integrate data sets that change dynamically in real-time, which will be useful for monitoring satellites and remotely-sensed data on Lunar surface. CMS utilizes NASA WorldWind Java library and OpenGL to achieve high-performance rendering of data and measurements, and also adheres to OGC standards. CMS supports importing synthetic features in a variety of 3D, 2D, vector and raster formats. Nomenclature is pulled from USGS Moon IAU2000 database, and lunar parameters are based of the standardized IAU2000 Moon ellipsoid. GDAL (Geospatial Data Abstraction Library) was used to modify and test the accuracy of datasets before integrating into the application. Our high-resolution global elevation model was compared with the LRO LOLA DEM elevation values and tested to ensure accuracy. Celestial Mapping System has several potential use cases for NASA including subsurface lava tubes visualization and analysis, soil analysis, resource visualization and representation on 3D globe.

GIS system↗

Supersonic airfoil optimization

A procedure for the optimization of supersonic airfoils is presented. A nonlinear, full-potential solver (NCOREL) is coupled with a numerical optimizer (CONMIN). The NCOREL code is a three-dimensional supersonic marching method, however, only two-dimensional (conical) results are used for the airfoil problem. Each airfoil evaluated in the NCOREL aerodynamic code is composed of a linear combination of user-specified basis airfoils. The weighting factors of the basis airfoils plus the angle of attack form the design variable vector. The inviscid drag coefficient is minimized subject to lift and crossflow Mach number gradient constraints. The optimization strategy is found to be extremely important for the efficient determination of an optimum airfoil.

Pittman, J. L.↗

Wildfire Segmentation From Remotely Sensed Data Using Quantum-Compatible Conditional Vector Quantized-Variational Autoencoders

Wildfires represent a critical environmental hazard with multifaceted implications for ecosystems, communities, and public health [1]. The escalating frequency and intensity of wildfires globally have intensified the urgency for robust segmentation methodologies to facilitate effective mitigation, response, and recovery strategies [2]. Accurate wildfire segmentation is pivotal for delineating fire boundaries, assessing progression patterns, and prioritizing resource allocation during emergency scenarios. Furthermore, precise segmentation enables stakeholders, including policymakers, environmental scientists, and emergency responders, to formulate evidence-based strategies, thereby minimizing socio-economic disruptions and ecological degradation. Consequently, advancing wildfire segmentation techniques through innovative technological interventions remains a paramount research imperative. Although foundational in wildfire segmentation, traditional deterministic models exhibit inherent limitations that compromise their efficacy in dynamic and uncertain environments. These models often operate on rigid algorithms prioritizing deterministic classifications, thereby overlooking the inherent complexities and uncertainties associated with wildfire behavior and satellite data variability. Such deterministic frameworks tend to produce oversimplified representations that fail to capture the intricate nuances of evolving fire dynamics, spatial heterogeneity, and environmental interactions [1]. Consequently, the deterministic approach’s propensity for uncertainty collapsing [1, 3] hampers the accuracy, reliability, and applicability of segmentation outcomes in real-world scenarios. Contrastingly, stochastic models offer a more nuanced and adaptable framework for wildfire segmentation. By integrating probabilistic elements into the modeling paradigm, stochastic approaches, particularly probabilistic approaches such as variational auto encoders (VAEs) [4], facilitate comprehensive uncertainty assessment, enabling researchers to quantify and incorporate uncertainties into segmentation outcomes effectively. This probabilistic nature empowers stochastic models to encapsulate variability, account for data inconsistencies, and adapt to evolving environmental conditions, enhancing segmentation accuracy, reliability, and robustness. Embracing stochastic methodologies thus catalyzes advancements in wildfire science by fostering a more holistic, adaptive, and resilient segmentation framework. Despite VAEs demonstrating significant promise in various applications, they come with inherent limitations that have garnered attention within the machine learning community. One of the primary drawbacks lies in their reliance on static priors, which essentially assume a fixed distribution for latent variables, thereby limiting the model’s flexibility to capture complex data structures effectively [5]. This static nature leads to suboptimal representations, especially when dealing with complex and high-dimensional data. Additionally, VAEs often struggle with generating sharp and realistic samples, a phenomenon commonly referred to as mode collapse [5, 7, 6]. Furthermore, the optimization process in VAEs, which involves balancing the reconstruction loss and the regularization term, can sometimes be challenging to fine-tune [7]. In recent efforts to address these shortcomings, alternative approaches like Vector Quantized Variational Auto encoders(VQ-VAEs) [7], address the challenges by incorporating discrete latent variables and leveraging techniques that enhance the quality and diversity of generated samples while maintaining efficient training dynamics. VQ-VAEs propose a dynamic prior distribution generation mechanism that diverges from the static priors commonly associated with traditional VAEs. This dynamic approach allows for more adaptive and context-aware latent variable representations, thereby potentially capturing complex data structures more effectively. Unlike autoregressive prior models such as PixelCNN, which, despite their ability to model dependencies across data dimensions, suffer from significant computational inefficiencies and lack flexibility in handling diverse datasets. In our work, we propose to use a generative quantum-compatible approach to help alleviate the shortcomings of autoregressive prior model in VQ-VAEs. Restricted Boltzmann Machines (RBMs) are a viable alternative prior model that can learn prior distributions in a faster and more flexible manner. In this research endeavor, we meticulously curate a state-of-the-art dataset leveraging satellite MODIS data in conjunction with VIIRS fire masks, derived from Fire Radiative Power (FRP), thereby encapsulating diverse wildfire scenarios and environmental contexts. We developed a conditional VQ-VAE architecture with the RBM prior model that is trained in a supervised manner for segmenting wildfire masks. This innovative approach synergistically harnesses deep learning capabilities, enabling the generation of segmentation maps characterized by heightened precision, granularity, and contextual relevance. Furthermore, replacing the autoregressive prior learning method proposed by the original VQ-VAE with a prior density approximation via quantum-compatible RBM facilitates expedited inference processes, augments flexibility in prior sampling, optimizes computational efficiency and establishes a groundbreaking benchmark in wildfire segmentation methodologies.

quantum machine learning↗

Application of a vectorized particle simulation in high-speed near-continuum flow

An efficient particle simulation technique, developed for use on vector architecture based supercomputers for studying hypersonic rarefied gas flows is employed to simulate the complex wake generated by Mach six flow over a 10 deg half-angle wedge for freestream Reynolds numbers of 1780 and 3560. Data obtained are compared against higher Reynolds number experimental results. Simulations utilized as many as 10 to the 5th computational cells and 10 to the 7th simulated particles having power-law interaction potentials. A code performance of 1.8-2.4 microsec of Cray-2 CPU time to process a single particle per timestep is achieved. Diffuse adiabatic and isothermal wedge surface models are used in this investigation. Although the wedge geometry is two-dimensional, the simulation incorporates a width-wise direction, resulting in a three-dimensional computation.

Woronowicz, Michael S.↗

Optical laboratory solution and error model simulation of a linear time-varying finite element equation

The use of simplified error models to accurately simulate and evaluate the performance of an optical linear-algebra processor is described. The optical architecture used to perform banded matrix-vector products is reviewed, along with a linear dynamic finite-element case study. The laboratory hardware and ac-modulation technique used are presented. The individual processor error-source models and their simulator implementation are detailed. Several significant simplifications are introduced to ease the computational requirements and complexity of the simulations. The error models are verified with a laboratory implementation of the processor, and are used to evaluate its potential performance.

Taylor, B. K.↗

Update on the Juncture Flow PIV Results and Future Plans

A novel, embedded, 2D Particle Image Velocimetry system has been developed and implemented to obtain off-body velocity measurements in the junction of an 8% wing-body configuration in the NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel. Although the initial purpose for implementing the system during this test was to evaluate system performance and identify potential risks, a considerable amount of data were obtained in the wing-fuselage junction near the trailing edge at a Reynolds Number of 2.4 Million and angle of attack of 5 degrees. In addition to providing notable efficiencies with regard to image acquisition and test operations, the PIV system captured unique details of the flow separation to complement the extensive suite of measurement techniques applied during the test. Instantaneous PIV vector fields reveal that the flow separation is not stationary but rather highly dynamic. Mean flow statistics calculated from the PIV measurements highlight where reverse flow and Reynolds stresses are concentrated in the separated region and agree well with results from the embedded Laser Doppler Velocimeter system and Computational Fluid Dynamics. The comparisons and ional insight gained during this effort will help guide system improvements for the upcoming entry in 2020.

Jenkins, Luther N.↗

Lower-Hybrid Wave Structures and Interactions With Electrons Observed in Magnetotail Reconnection Diffusion Regions

We investigate waves close to the lower-hybrid frequency in 12 magnetotail reconnection electron diffusion region (EDR) events with guide field levels of near-zero to 30%. In about half of the events, the wave vector has a small component along the current sheet normal, consistent with known lower-hybrid drift wave properties, but the perpendicular magnetic field fluctuations can be comparable or greater than the parallel component, a feature unique to the waves inside and adjacent to EDRs. Another new wave property is that the wave vector has a significant component along the current sheet normal in some events and completely along the normal for one event. In 1/4of the events, the ∇∙𝑷(sub 𝑒) term has a significant contribution to the wave electric field, possibly a feature of lower-hybrid waves more likely to exist in the diffusion region than further away from the X-line. Electron temperature variations are correlated with the wave potential, due to wave electric field acceleration and crossings at the corrugated separatrix region with different amounts of mixing between reconnection inflowing and outflowing populations. The latter also leads to the anti-correlation between parallel and perpendicular temperature components. Using four-spacecraft measurements, the magnetic field line twisting is demonstrated by the correlated fluctuations in (∇×𝑽(sub 𝐸×𝐵))||and (∇×𝐁)||. The lower-hybrid wave in the EDR of weak guide field reconnection may be generated near separatrices and penetrate to the mid-plane or locally generated, and the latter possibility is beyond the prediction of previous reconnection simulations.

Magnetic reconnection↗

After DART: Using the first full-scale test of a kinetic impactor to inform a future planetary defense mission

NASA’s Double Asteroid Redirection Test (DART) is the first full-scale test of an asteroid deflection technology. Results from the hypervelocity kinetic impact and Earth-based observations, coupled with LICIACube and the later Hera mission, will result in measurement of the momentum transfer efficiency accurate to ~10% and characterization of the Didymos binary system. But DART is a single experiment; how could these results be used in a future planetary defense necessity involving a different asteroid? We examine what aspects of Dimorphos’s response to kinetic impact will be constrained by DART results; how these constraints will help refine knowledge of the physical properties of asteroidal materials and predictive power of impact simulations; what information about a potential Earth impactor could be acquired before a deflection effort; and how design of a deflection mission should be informed by this understanding. We generalize the momentum enhancement factor β, showing that a particular direction-specific β will be directly determined by the DART results, and that a related direction-specific β is a figure of merit for a kinetic impact mission. The DART β determination constrains the ejecta momentum vector, which, with hydrodynamic simulations, constrains the physical properties of Dimorphos’s near-surface. In a hypothetical planetary defense exigency, extrapolating these constraints to a newly discovered asteroid will require Earth-based observations and benefit from in-situ reconnaissance. We show representative predictions for momentum transfer based on different levels of reconnaissance and discuss strategic targeting to optimize the deflection and reduce the risk of a counterproductive deflection in the wrong direction.

Asteroids↗

Ares I-X Ground Diagnostic Prototype

Automating prelaunch diagnostics for launch vehicles offers three potential benefits. First, it potentially improves safety by detecting faults that might otherwise have been missed so that they can be corrected before launch. Second, it potentially reduces launch delays by more quickly diagnosing the cause of anomalies that occur during prelaunch processing. Reducing launch delays will be critical to the success of NASA's planned future missions that require in-orbit rendezvous. Third, it potentially reduces costs by reducing both launch delays and the number of people needed to monitor the prelaunch process. NASA is currently developing the Ares I launch vehicle to bring the Orion capsule and its crew of four astronauts to low-earth orbit on their way to the moon. Ares I-X will be the first unmanned test flight of Ares I. It is scheduled to launch on October 27, 2009. The Ares I-X Ground Diagnostic Prototype is a prototype ground diagnostic system that will provide anomaly detection, fault detection, fault isolation, and diagnostics for the Ares I-X first-stage thrust vector control (TVC) and for the associated ground hydraulics while it is in the Vehicle Assembly Building (VAB) at John F. Kennedy Space Center (KSC) and on the launch pad. It will serve as a prototype for a future operational ground diagnostic system for Ares I. The prototype combines three existing diagnostic tools. The first tool, TEAMS (Testability Engineering and Maintenance System), is a model-based tool that is commercially produced by Qualtech Systems, Inc. It uses a qualitative model of failure propagation to perform fault isolation and diagnostics. We adapted an existing TEAMS model of the TVC to use for diagnostics and developed a TEAMS model of the ground hydraulics. The second tool, Spacecraft Health Inference Engine (SHINE), is a rule-based expert system developed at the NASA Jet Propulsion Laboratory. We developed SHINE rules for fault detection and mode identification. The prototype uses the outputs of SHINE as inputs to TEAMS. The third tool, the Inductive Monitoring System (IMS), is an anomaly detection tool developed at NASA Ames Research Center and is currently used to monitor the International Space Station Control Moment Gyroscopes. IMS automatically "learns" a model of historical nominal data in the form of a set of clusters and signals an alarm when new data fails to match this model. IMS offers the potential to detect faults that have not been modeled. The three tools have been integrated and deployed to Hangar AE at KSC where they interface with live data from the Ares I-X vehicle and from the ground hydraulics. The outputs of the tools are displayed on a console in Hangar AE, one of the locations from which the Ares I-X launch will be monitored. The full paper will describe how the prototype performed before the launch. It will include an analysis of the prototype's accuracy, including false-positive rates, false-negative rates, and receiver operating characteristics (ROC) curves. It will also include a description of the prototype's computational requirements, including CPU usage, main memory usage, and disk usage. If the prototype detects any faults during the prelaunch period then the paper will include a description of those faults. Similarly, if the prototype has any false alarms then the paper will describe them and will attempt to explain their causes.

Schwabacher, Mark↗

Quasi drag free gradiometry

Gradiometers are instruments for measuring the gradient of the gravity vector. However, these devices are in varying degrees contaminated by gravity itself or by acceleration. To a large extent these problems are avoidable in orbit, and dramatic improvements in operating accuracy are envisioned. The considered improvements could lead to the possibility of detailed global gravity surveys of the earth, or other planets. An obstacle in the path toward the conduction of such surveys is related to the impossibility to demonstrate an instrument's potential orbital performance in the laboratory. A promising approach for orbital testing is possibly a procedure in which the Shuttle is operated as a drag free satellite. This procedure involves floating the gradiometer in the Shuttle bay. However, certain problems arise in connection with such a procedure. The present investigation is concerned with the possibility that the Shuttle's motion problems can be avoided through the use of a quasi-drag-free magnetic suspension.

Seaman, C. H.↗

Beaconless Pointing for Deep-Space Optical Communication

Free space optical communication is of interest to NASA as a complement to existing radio frequency communication methods. The potential for an increase in science data return capability over current radio-frequency communications is the primary objective. Deep space optical communication requires laser beam pointing accuracy on the order of a few microradians. The laser beam pointing approach discussed here operates without the aid of a terrestrial uplink beacon. Precision pointing is obtained from an on-board star tracker in combination with inertial rate sensors and an outgoing beam reference vector. The beaconless optical pointing system presented in this work is the current approach for the Integrated Radio and Optical Communication (iROC) project.

free-space optical communication↗

The equilibrium and stability of the gaseous component of the galaxy, 3

The stability of a self-gravitating, nonrotating, isothermal gas layer threaded by a one-dimensional equipartition magnetic field, immersed in a rigid isothermal layer of stars, is considered with respect to waves with motions in and perpendicular to the X plane, where X is the equilibrium magnetic and gravitational field vectors. When motions are perpendicular to the X plane, the magnetic field hinders gravitational instability, increasing the minimum length necessary to produce instability by the factor (1 + alpha) to the 1/2 power, where alpha is the ratio of magnetic pressure to gas pressure. When motions are in the plane, no such simple analytical solution is found. However, the resulting system of equations reduces to a single fourth order differential equation in the perturbed gas potential that defines an eigenvalue problem for the marginally unstable state when the four appropriate boundary conditions are considered.

Kellman, S. A.↗

The structures and energetics of fully nonlinear symmetric baroclinic waves

Rotating baroclinic flow for Richardson number lower than about 1 is studied by means of a finite difference Navier-Stokes model assuming no variations except in the vertical plane that completely contains the density gradient vector. The horizontally infinite channel to which attention is given further assumes periodic boundary conditions at the vertical computational boundaries and no-slip conducting horizontal boundaries. Two configurations are considered. Symmetric baroclinic waves developed in the flows in a manner consistent with linear theory, and it is noted that the structures and energetics of the fully developed waves were dependent on the Prandtl number Pr. For Pr greater than 1, the conversion from potential energy to wave kinetic energy was direct, via temperature and vertical motion correlation, while for Pr of less than 1, conversion proceeded from potential energy to average kinetic energy by means of an induced meridional flow, and then to wave kinetic energy.

Miller, T. L.↗

Actuator placement for active sound and vibration control of cylinders

Active structural acoustic control is a method in which the control inputs (used to reduce interior noise) are applied directly to a vibrating structural acoustic system. The control concept modeled in this work is the application of in-plane force inputs to piezoceramic patches bonded to the wall of a vibrating cylinder. The cylinder is excited by an exterior noise source -- an acoustic monopole -- located near the outside of the cylinder wall. The goal is to determine the force inputs and sites for the piezoelectric actuators so that (1) the interior noise is effectively damped; (2) the level of vibration of the cylinder shell is not increased; and (3) the power requirements needed to drive the actuators are not excessive. We studied external monopole excitations at two frequencies. A cylinder resonance of 100 Hz, where the interior acoustic field is driven in multiple, off-resonance cylinder cavity modes, and a cylinder resonance of 200 Hz are characterized by both near and off-resonance cylinder vibration modes which couple effectively with a single, dominant, low-order acoustic cavity mode at resonance. Previous work has focused almost exclusively on meeting objective (1) and solving a complex least-squares problem to arrive at an optimal force vector for a given set of actuator sites. In addition, it has been noted that when the cavity mode couples with cylinder vibration modes (our 200 Hz case) control spillover may occur in higher order cylinder shell vibrational modes. How to determine the best set of actuator sites to meet objectives (1)-(3) is the main contribution of our research effort. The selection of the best set of actuator sites from a set of potential sites is done via two metaheuristics -- simulated annealing and tabu search. Each of these metaheuristics partitions the set of potential actuator sites into two disjoint sets: those that are selected to control the noise (on) and those that are not (off). Next, each metaheuristic attempts to improve this initial solution by calculating the change in the objective value when one selected actuator site is turned off and one actuator site that previously was not selected is turned on. All such pairwise exchanges are performed and the exchange that improves the objective the most is made. Eventually the search is unable to improve the objective value and a local optimum (with respect to pairwise exchanges) is reached. Both simulated annealing and tabu search provide mechanisms to escape local optima and allow the search to continue until (hopefully) a global optimum is found. Our experiments with the 100 Hz and 200 Hz cases confirm that both metaheuristics are able to uncover better solutions than those selected based upon engineering judgement alone. In addition, the high quality solutions generated by these metaheuristics, when minimizing interior noise, do not further excite the cylinder shell. Thus, we are able to meet objective (2) without imposing an additional constraint or forming a multiobjective performance measure. An additional observation is that in many cases the amplitude and phase values for several chosen actuator sites were nearly identical. This natural grouping means that fewer control channels are needed and the resulting control system is simpler. Currently no power requirements have been set, so objective (3) cannot be addressed. A set of experiments is planned with a laboratory test article (a cylinder). For these experiments the transfer matrices will be generated experimentally. It is hoped that the predicted performance of the best actuator sites found by our metaheuristics will correlate well with the measured performance.

Kincaid, Rex K.↗

Growth of Compound Semiconductors in a Low Gravity Environment: Microgravity Growth of PbSnTe

The growth of the alloy compound semiconductor lead tin telluride (PbSnTe) was chosen for a microgravity flight experiment in the Advanced Automated Directional Solidification Furnace (AADSF), on the United States Microgravity Payload-3 (USNP-3) in February, 1996 and on USNW- 4 in November, 1997. The objective of these experiments was to determine the effect of the reduction in convection, during the growth process, brought about by the microgravity environment. The properties of devices made from PbSnTe, an alloy of PbTe and SnTe, are dependent on the ratio of the elemental components in the starting crystal. Compositional uniformity in the crystal is only obtained if there is no significant mixing in the liquid during growth. The technological importance of PbSnTe lies in its band gap versus composition diagram which has a zero energy crossing at approximately 40% SnTe. This facilitates the construction of long wavelength (greater than 6 gm) infrared detectors and lasers. The properties and utilization of PbSnTe are the subject of other papers. 1,2 PbSnTe is also interesting from a purely scientific point of view. It is, potentially, both solutally and thermally unstable due to the temperature and density gradients present during growth. Density gradients, through thermal expansion, are imposed in directional solidification because temperature gradients are required to extract heat. Solutal gradients occur in directional solidification of alloys due to segregation at the interface. Usually the gradients vary with both experiment design and inherent materials properties. In a simplified one dimensional analysis with the growth axis parallel to the gravity vector, only one of the two instabilities work at a time. During growth, the temperature in the liquid increases ahead of the interface. Therefore the density, due to thermal expansion, is decreasing in that direction. However, the phase diagram shows that the lighter SnTe is preferentially rejected at the interface. This causes the liquid density to increase with distance away from the interface.

Fripp, A. L.↗

A New Fast Algorithm to Completely Account for Non-Lambertian Surface Reflection of The Earth

Surface bidirectional reflectance distribution function (BRDF) influences not only radiance just about the surface, but that emerging from the top of the atmosphere (TOA). In this study we propose a new, fast and accurate, algorithm CASBIR (correction for anisotropic surface bidirectional reflection) to account for such influences on radiance measured above TOA. This new algorithm is based on a 4-stream theory that separates the radiation field into direct and diffuse components in both upwelling and downwelling directions. This is important because the direct component accounts for a substantial portion of incident radiation under a clear sky, and the BRDF effect is strongest in the reflection of the direct radiation reaching the surface. The model is validated by comparison with a full-scale, vector radiation transfer model for the atmosphere-surface system. The result demonstrates that CASBIR performs very well (with overall relative difference of less than one percent) for all solar and viewing zenith and azimuth angles considered in wavelengths from ultraviolet to near-infrared over three typical, but very different surface types. Application of this algorithm includes both accounting for non-Lambertian surface scattering on the emergent radiation above TOA and a potential approach for surface BRDF retrieval from satellite measured radiance.

Qin, Wen-Han↗

Correction and update to 'The earth's C21 and S21 gravity coefficients and the rotation of the core'

Wahr (1987) used satellite constraints on C21 and S21 (the spherical harmonic coefficients of the earth's external gravitational potential) to infer certain properties of the core and core/mantle boundary. It is shown here, contrary to the claim by Wahr, that it is not possible to use C21 and S21 to placed bounds on the core's products of inertia. As a result, Wahr's constraints on the l = 2, m = 1 components of the core/mantle boundary topography and on the angular orientation of the inner core with respect to the earth's rotation vector are not justified. On the other hand, Wahr's conclusions about the time-averaged torque between the core and mantle and the resulting implications for the l = 2, m = 1 components of fluid pressure at the top of the core can be strengthened. Wahr's conclusions about the mean rotational flow in the core are unaltered.

Wahr, John↗

Singular control in minimum time spacecraft reorientation

Spacecraft reorientation is investigated numerically for an inertially symmetric rigid spacecraft with three bounded independent control torques aligned with the principal axes. The dynamical system of the spacecraft and the framework of the optimal-control problem are established in order to identify all of the potential strategies. The investigation lists bang-bang solutions and finite-order and infinite-order singular arcs, and the conditions for the finite-order singular arcs are given. Numerical examples are developed for all of the control-logic systems, and the suboptimality of the rest-to-rest maneuvers is proven for principal-axis rotations. The most efficient control technique is the singular control of infinite order, and the vector-valued singular control can be utilized in a derivative of the switching function.

Seywald, Hans↗