Results for “Maximum independent set problem”

NASA NTRS2017

Applying Graph Theory to Problems in Air Traffic Management

Graph theory is used to investigate three different problems arising in air traffic management. First, using a polynomial reduction from a graph partitioning problem, it is shown that both the airspace sectorization problem and its incremental counterpart, the sector combination problem are NP-hard, in general, under several simple workload models. Second, using a polynomial time reduction from maximum independent set in graphs, it is shown that for any fixed e, the problem of finding a solution to the minimum delay scheduling problem in traffic flow management that is guaranteed to be within n1-e of the optimal, where n is the number of aircraft in the problem instance, is NP-hard. Finally, a problem arising in precision arrival scheduling is formulated and solved using graph reachability. These results demonstrate that graph theory provides a powerful framework for modeling, reasoning about, and devising algorithmic solutions to diverse problems arising in air traffic management.

precision arrival scheduling↗

NASA NTRS2017

Applying Graph Theory to Problems in Air Traffic Management

Graph theory is used to investigate three different problems arising in air traffic management. First, using a polynomial reduction from a graph partitioning problem, it isshown that both the airspace sectorization problem and its incremental counterpart, the sector combination problem are NP-hard, in general, under several simple workload models. Second, using a polynomial time reduction from maximum independent set in graphs, it is shown that for any fixed e, the problem of finding a solution to the minimum delay scheduling problem in traffic flow management that is guaranteed to be within n1-e of the optimal, where n is the number of aircraft in the problem instance, is NP-hard. Finally, a problem arising in precision arrival scheduling is formulated and solved using graph reachability. These results demonstrate that graph theory provides a powerful framework for modeling, reasoning about, and devising algorithmic solutions to diverse problems arising in air traffic management.

computational complexity↗

NASA NTRS1974

Optimal control problems with mixed control-phase variable equality and inequality constraints

In this paper, necessary conditions are obtained for optimal control problems containing equality constraints defined in terms of functions of the control and phase variables. The control system is assumed to be characterized by an ordinary differential equation, and more conventional constraints, including phase inequality constraints, are also assumed to be present. Because the first-mentioned equality constraint must be satisfied for all t (the independent variable of the differential equation) belonging to an arbitrary (prescribed) measurable set, this problem gives rise to infinite-dimensional equality constraints. To obtain the necessary conditions, which are in the form of a maximum principle, an implicit-function-type theorem in Banach spaces is derived.

Makowski, K.↗

NASA NTRS1996

Upper Limit of Weights in TAI Computation

The international reference time scale International Atomic Time (TAI) computed by the Bureau International des Poids et Mesures (BIPM) relies on a weighted average of data from a large number of atomic clocks. In it, the weight attributed to a given clock depends on its long-term stability. In this paper the TAI algorithm is used as the basis for a discussion of how to implement an upper limit of weight for clocks contributing to the ensemble time. This problem is approached through the comparison of two different techniques. In one case, a maximum relative weight is fixed: no individual clock can contribute more than a given fraction to the resulting time scale. The weight of each clock is then adjusted according to the qualities of the whole set of contributing elements. In the other case, a parameter characteristic of frequency stability is chosen: no individual clock can appear more stable than the stated limit. This is equivalent to choosing an absolute limit of weight and attributing this to to the most stable clocks independently of the other elements of the ensemble. The first technique is more robust than the second and automatically optimizes the stability of the resulting time scale, but leads to a more complicated computatio. The second technique has been used in the TAI algorithm since the very beginning. Careful analysis of tests on real clock data shows that improvement of the stability of the time scale requires revision from time to time of the fixed value chosen for the upper limit of absolute weight. In particular, we present results which confirm the decision of the CCDS Working Group on TAI to increase the absolute upper limit by a factor of 2.5. We also show that the use of an upper relative contribution further helps to improve the stability and may be a useful step towards better use of the massive ensemble of HP 507IA clocks now contributing to TAI.

Thomas, Claudine↗

NASA NTRS1975

Fixed-range optimum trajectories for short-haul aircraft

An algorithm, based on the energy-state method, is derived for calculating optimum trajectories with a range constraint. The basis of the algorithm is the assumption that optimum trajectories consist of, at most, three segments: an increasing energy segment (climb); a constant energy segment (cruise); and a decreasing energy segment (descent). This assumption allows energy to be used as the independent variable in the increasing and decreasing energy segments, thereby eliminating the integration of a separate adjoint differential equation and simplifying the calculus of variations problem to one requiring only pointwise extremization of algebraic functions. The algorithm is used to compute minimum fuel, minimum time, and minimum direct-operating-cost trajectories, with range as a parameter, for an in-service CTOL aircraft and for an advanced STOL aircraft. For the CTOL aircraft and the minimum-fuel performance function, the optimum controls, consisting of air-speed and engine power setting, are continuous functions of the energy in both climb and descent as well as near the maximum or cruise energy. This is also true for the STOL aircraft except in the descent where at one energy level a nearly constant energy dive segment occurs, yielding a discontinuity in the airspeed at that energy. The reason for this segment appears to be the relatively high fuel flow at idle power of the engines used by this STOL aircraft. Use of a simplified trajectory which eliminates the dive increases the fuel consumption of the total descent trajectory by about 10 percent and the time to fly the descent by about 19 percent compared to the optimum.

Erzberger, H.↗

NASA NTRS1984

Finding maximum on an array processor with a global bus

The problem of finding the maximum of a set of values stored one/processor on an n x n array of processors is analyzed. The array has a time-shared global bus in addition to conventional processor-processor links. A two-phase algorithm for finding the maximum is presented that uses conventional links during the first phase and the global bus during the second. This algorithm is faster than algorithms that use either only the global bus or only the conventional links. Two types of interconnection patterns (the eighth nearest neighbor and the fourth nearest neighbor) are analyzed. In both cases it is shown that the time required to find the maximum using the two-phase algorithm is 0(n to the 2/3-power) assuming the propagation speed of the global bus to be a constant independent of the size of the array. In the case where the propagation speed is logarithmic in the number of processors, the time to find the maximum is 0(/n-squared log n/1/3), for both types of arrays. Extensions to q-dimensional arrays show that the two-phase algorithm is superior for any fixed value of q.

Bokhari, S. H.↗

NASA NTRS1973

Multiple regression technique for Pth degree polynominals with and without linear cross products

A multiple regression technique was developed by which the nonlinear behavior of specified independent variables can be related to a given dependent variable. The polynomial expression can be of Pth degree and can incorporate N independent variables. Two cases are treated such that mathematical models can be studied both with and without linear cross products. The resulting surface fits can be used to summarize trends for a given phenomenon and provide a mathematical relationship for subsequent analysis. To implement this technique, separate computer programs were developed for the case without linear cross products and for the case incorporating such cross products which evaluate the various constants in the model regression equation. In addition, the significance of the estimated regression equation is considered and the standard deviation, the F statistic, the maximum absolute percent error, and the average of the absolute values of the percent of error evaluated. The computer programs and their manner of utilization are described. Sample problems are included to illustrate the use and capability of the technique which show the output formats and typical plots comparing computer results to each set of input data.

Davis, J. W.↗

NASA NTRS1999

Monitoring and Modeling Astronaut Occupational Radiation Exposures in Space: Recent Advances

In 1982 astronauts were declared to be radiation workers by OSHA, and as such were subject to the rules and regulations applied to that group. NASA was already aware that space radiation was a hazard to crewmembers and had been studying and monitoring astronaut doses since 1962 at the Johnson Space Center. It was quickly realized NASA would not be able to accomplish all of its goals if the astronauts were subject to the ground based radiation worker limits, and thus received a waiver from OSHA to establish independent limits. As part of the stipulation attached to setting new limits, OSHA included a requirement to perform preflight dose projections for each crew and inform them of the associated risks. Additional requirements included measuring doses from various sources during the flight, making every effort to prevent a crewmember from exceeding the new limits, and keeping all exposures As Low As Reasonably Achievable (a.k.a. ALARA - a common health physics principle). The assembly of the International Space Station (ISS) and its initial manned operations will coincide with the 4-5 year period of high space weather activity at the next maximum in the solar cycle. For the first time in NASA's manned program, US astronauts will be in orbit continuously throughout a solar maximum period. During this period, crews are at risk of significantly increased radiation exposures due to solar particle events and trapped electron belt enhancements following geomagnetic storms. The problem of protecting crews is compounded by the difficulty of providing continuous real-time monitoring over a period of a decade in an era of tightly constrained budgets. In order to prepare for ISS radiological support needs, the NASA Space Radiation Analysis Group and the NOAA Space Environment Center have undertaken a multiyear effort to improve and automate ground-based space weather monitoring systems and real-time radiation analysis tools. These improvements include a coupled, automated space weather monitoring and alarm system--SPE exposure analysis system, an advanced space weather data distribution and display system, and a high-fidelity space weather simulation system. In addition, significant new real-time space weather data sets, which will enhance the forecasting and now-casting of near-Earth space environment conditions, are being made available through unique NASA-NOAA-USAF collaborations. These new data sets include coronal mass ejection monitoring by the Solar and Heliospheric Observatory (SOHO) and in-situ plasma and particle monitoring at the L1 libration point by the Solar Wind Monitor (SWIM) and Advanced Composition Explorer (ACE) spacecraft. Advanced real-time radiation monitoring data from charged particle telescopes and tissue equivalent proportional counters will also be available to assist crew and flight controllers in monitoring the external and intravehicular radiation environment.

Weyland, Mark↗

NASA NTRS1958

Design and experimental investigation of a single-stage turbine with a rotor entering relative Mach number of 2

The design and experimental investigation of a single-stage supersonic turbine are presented herein. The turbine was designed for a rotor entering relative Mach number of 2. The maximum equivalent specific work output of the turbine at design speed and approximately design over-all pressure ratio was 32.9 BTU per pound at a static efficiency of 0.414. This static efficiency gave good verification to an independent reference that indicated theoretical static efficiencies for similar single-stage turbines within the range 0.40 to 0.45. An experimental ratio of effective rotor blade momentum thickness to mean camber length was determined to be 0.0014, which compares favorably with the results obtained from several transonic and subsonic turbines. The design procedure for this turbine would have been improved by allowing for more rotor losses by assuming a value of this momentum parameter comparable with those obtained from transonic turbines. Removing a large portion of the rotor suction surface enabled a lower static pressure to be felt at the stator exit, at the expense of higher rotor losses. The net result was an improvement in turbine work output of about 3 percent at design setting conditions. No problems associated with supersonic starting were encountered even under the worst conditions of turbine operation with respect to this problem.

Moffitt, Thomas P↗

NASA NTRS2007

Uncertainty Analysis of Historical Hurricane Data

An analysis of variance (ANOVA) study was conducted for historical hurricane data dating back to 1851 that was obtained from the U. S. Department of Commerce National Oceanic and Atmospheric Administration (NOAA). The data set was chosen because it is a large, publicly available collection of information, exhibiting great variability which has made the forecasting of future states, from current and previous states, difficult. The availability of substantial, high-fidelity validation data, however, made for an excellent uncertainty assessment study. Several factors (independent variables) were identified from the data set, which could potentially influence the track and intensity of the storms. The values of these factors, along with the values of responses of interest (dependent variables) were extracted from the data base, and provided to a commercial software package for processing via the ANOVA technique. The primary goal of the study was to document the ANOVA modeling uncertainty and predictive errors in making predictions about hurricane location and intensity 24 to 120 hours beyond known conditions, as reported by the data set. A secondary goal was to expose the ANOVA technique to a broader community within NASA. The independent factors considered to have an influence on the hurricane track included the current and starting longitudes and latitudes (measured in degrees), and current and starting maximum sustained wind speeds (measured in knots), and the storm starting date, its current duration from its first appearance, and the current year fraction of each reading, all measured in years. The year fraction and starting date were included in order to attempt to account for long duration cyclic behaviors, such as seasonal weather patterns, and years in which the sea or atmosphere were unusually warm or cold. The effect of short duration weather patterns and ocean conditions could not be examined with the current data set. The responses analyzed were the storm latitude, longitude and intensity, as recorded in the data set, 24 or 120 hours beyond the current state. Several ANOVA modeling schemes were examined. Two forms of validation were used: 1) comparison with official hurricane prediction performance metrics and 2) cases studies conducted on hurricanes from the 2005 season, which were not included within the model construction and ANOVA assessment. In general, the ANOVA technique did not perform as well as the established official prediction performance metrics published by NOAA; still, the technique did remarkably well in this demonstration with a difficult data set and could probably be made to perform better with more knowledge of hurricane development and dynamics applied to the problem. The technique provides a repeatable prediction process that eliminates the need for judgment in the forecast.

Green, Lawrence L.↗

NASA NTRS2017

Models of Sector Flows Under Local, Regional and Airport Weather Constraints

Recently, the ATM community has made important progress in collaborative trajectory management through the introduction of a new FAA traffic management initiative called a Collaborative Trajectory Options Program (CTOP). FAA can use CTOPs to manage air traffic under multiple constraints (manifested as flow constrained areas or FCAs) in the system, and it allows flight operators to indicate their preferences for routing and delay options. CTOPs also permits better management of the overall trajectory of flights by considering both routing and departure delay options simultaneously. However, adoption of CTOPs in airspace has been hampered by many factors that include challenges in how to identify constrained areas and how to set rates for the FCAs. Decision support tools providing assistance would be particularly helpful in effective use of CTOPs. Such DSTs tools would need models of demand and capacity in the presence of multiple constraints. This study examines different approaches to using historical data to create and validate models of maximum flows in sectors and other airspace regions in the presence of multiple constraints. A challenge in creating an empirical model of flows under multiple constraints is a lack of sufficient historical data that captures diverse situations involving combinations of multiple constraints especially those with severe weather. The approach taken here to deal with this is two-fold. First, we create a generalized sector model encompassing multiple sectors rather than individual sectors in order to increase the amount of data used for creating the model by an order of magnitude. Secondly, we decompose the problem so that the amount of data needed is reduced. This involves creating a baseline demand model plus a separate weather constrained flow reduction model and then composing these into a single integrated model. A nominal demand model is a flow model (gdem) in the presence of clear local weather. This defines the flow as a function of weather constraints in neighboring regions, airport constraints and weather in locations that can cause re-routes to the location of interest. A weather constrained flow reduction model (fwx-red) is a model of reduction in baseline counts as a function of local weather. Because the number of independent variables associated with each of the two decomposed models is smaller than that with a single model, need for amount of data is reduced. Finally, a composite model that combines these two can be represented as fwx-red (gdem(e), l) where e represents non-local constraints and l represents local weather. The approaches studied to developing these models are divided into three categories: (1) Point estimation models (2) Empirical models (3) Theoretical models. Errors in predictions of these different types of models have been estimated. In situations when there is abundant data, point estimation models tend to be very accurate. In contrast, empirical models do better than theoretical models when there is some data available. The biggest benefit of theoretical models is their general applicability in wider range situations once the degree of accuracy of these has been established.

Kulkarni, Deepak↗

NASA NTRS2017

Models of Sector Aircraft Counts in the Presence of Local, Regional and Airport Constraints

Recently, the ATM community has made important progress in collaborative trajectory management through the introduction of a new FAA traffic management initiative called a Collaborative Trajectory Options Program (CTOP). FAA can use CTOPs to manage air traffic under multiple constraints (manifested as flow constrained areas or FCAs) in the system, and it allows flight operators to indicate their preferences for routing and delay options. CTOPs also permits better management of the overall trajectory of flights by considering both routing and departure delay options simultaneously. However, adoption of CTOPs in airspace has been hampered by many factors that include challenges in how to identify constrained areas and how to set rates for the FCAs. Decision support tools providing assistance would be particularly helpful in effective use of CTOPs. Such DSTs tools would need models of demand and capacity in the presence of multiple constraints. This study examines different approaches to using historical data to create and validate models of maximum flows in sectors and other airspace regions in the presence of multiple constraints. A challenge in creating an empirical model of flows under multiple constraints is a lack of sufficient historical data that captures diverse situations involving combinations of multiple constraints especially those with severe weather. The approach taken here to deal with this is two-fold. First, we create a generalized sector model encompassing multiple sectors rather than individual sectors in order to increase the amount of data used for creating the model by an order of magnitude. Secondly, we decompose the problem so that the amount of data needed is reduced. This involves creating a baseline demand model plus a separate weather constrained flow reduction model and then composing these into a single integrated model. A nominal demand model is a flow model (gdem) in the presence of clear local weather. This defines the flow as a function of weather constraints in neighboring regions, airport constraints and weather in locations that can cause re-routes to the location of interest. A weather constrained flow reduction model (fwx-red) is a model of reduction in baseline counts as a function of local weather. Because the number of independent variables associated with each of the two decomposed models is smaller than that with a single model, need for amount of data is reduced. Finally, a composite model that combines these two can be represented as fwx-red (gdem(e), l) where e represents non-local constraints and l represents local weather. The approaches studied to developing these models are divided into three categories: (1) Point estimation models (2) Empirical models (3) Theoretical models. Errors in predictions of these different types of models have been estimated. In situations when there is abundant data, point estimation models tend to be very accurate. In contrast, empirical models do better than theoretical models when there is some data available. The biggest benefit of theoretical models is their general applicability in wider range situations once the degree of accuracy of these has been established.

Kulkarni, Deepak↗

NASA NTRSDate not supplied

Lander Lighting Solution to Reduce Pilot & Autonomous Approach Errors

The south pole lighting environment will have harsh low inclination sunlight, making overhead judgement of surface features difficult. Autonomous solutions are great, however, the need for visual monitoring and independent go/no-go decisions remain. Our project proposes that lunar landing systems will be better served by including a powerful landing light system that improves visibility of surfaces from overhead by illuminating the ground at critical distances for the crew to make last minute decisions regarding an approach. The project utilized computer-based optical modeling software to predict requirements for a potential landing light system. The analysis based the lamp prediction from commercially available LED chip sets and lamp optics. The goal was to illustrate a method to raise the surface contrast of a landing site within an acceptable contrast threshold for most camera systems and human observers to recognize hazards that would not be noticed with low inclination sunlight alone. The Apollo lunar landings benefitted from overhead sun or dark conditions. The surface lighting at the Lunar South Pole is a harsh environment where surfaces are lit from a low inclination angle by the sun (from the side). This change in lighting condition precipitates a need for updated lunar landing systems that facilitate improved recognition of landing sites, and thereby increase pilot awareness of landing hazards. The reliance on LIDAR and other autonomous mechanisms alone is risky given the known usage of visual monitoring for operator concurrence on current spacecraft programs and present-day autonomous land-based vehicles. Visual monitoring via cameras or windows requires the surface contrast to be within 3 orders of magnitude for reasonable recognition of objects. Artificial overhead illumination, when sufficiently sized, provides a means to even out contrast problems created by low inclination sunlight, potentially reducing piloting errors. Current vehicle requirements do not specify this type of guidance for the purpose of increasing mission success. An optical ray-trace simulation model was developed in Zemax Optics Studio to predict the best combination of LED power, LED optics, lamp quantity, and lamp location to raise the surface contrast to within 2 orders of magnitude from 3 orders required to further increased visibility and reduce risk. The project considered the following design constraints: potential base diameter of lander, approach distance(s) for a go-no-go decision point (200 meter), solar inclination angle (2-7), lunar surface reflectance (10%), LED chip sets, LED focusing optics, LED power, lamp quantity, lamp locations, and illumination diameter of lunar surface landing zone. The results can be used to establish minimum design constraints for vehicle landing light systems. With a solar inclination angle ranging from 2-7 degrees, the horizontal illumination of the lunar surface is attenuated by about 10% when compared to overhead illumination from the Sun. This modifies the sun's maximum of 130,000 lux to 13,000 lux horizontal illuminance. The artificial lighting system was designed to provide an 18-meter-wide illumination zone, to create viewing clearances around a 6-meter-wide lander. The system provides an average illuminance of 300 lux, meeting the 2 orders of magnitude criteria. The solution utilized modern Chip On Board LEDs, that each utilized 17 watts, with focusing Total Internal Reflection (TIR) lenses. A lighting system of 300 LEDs was arrayed along the "bottom" of a “lander”. With 17 watts per LED, the system is estimated to require 5100 watts. This is a large amount of power, but it would only be needed during critical phases during the landing. LED lighting systems can be dimmed, and it is assumed that as the lander arrives closer to the landing site, the lighting system power can be adjusted as needed to produce the necessary surface illuminance. The designed reduction of contrast improves reliability of safety assessments using real time visible light camera systems and out the window viewing by the crew.

T A Clark↗