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Introducing Tropical Geometric Approaches to Delay Tolerant Networking Optimization

Delay Tolerant Networking (DTN) is the standard approach to the networking of space systems with the goal of supporting the Solar System Internet (SSI). Current space networks have a small scale and often depend on rigorously scheduled (pre-determined) contact opportunities; this manual approach inhibits scalability. The goal of this paper is to recast these scheduling problems in order to apply the optimization machinery of tropical geometry. Contact opportunities in space are dependent on such factors as orbital mechanics and asset availability, which induce time-varying connectivity; indeed, end-to-end connectivity might never occur. Routing optimization within this structure is classically difficult and typically utilizes Dijkstra's algorithm as applied to contact graphs. Alternatively, we follow the successes of tropical geometry in train schedule optimization, job assignments, and even traditional networking, by extending this approach to this more general (i.e. disconnected) problem space. These successes imply tropical geometry provides a useful framework in the context of DTNs, starting with applications to queuing theory and long-haul links. Recently, tropical geometry has been applied to parametric path optimization on graphs with variable edge weights. In this work, we extend these advances to account for the problem of routing in a space network, and find that tropical geometry is well-suited to the challenges offered by this new setting, including contact schedules featuring probabilities. Our approach leverages the combinatorial nature of the problem to give feasible shortest path trees in the presence of variable channel conditions and latency, evolving topologies, and uncertainty inherent in space routing. We discuss our tropical approach to DTN for two Python implementations, a Verilog Tropical ALU implementation, tropical frameworks for other parametric graph problems, and solution stability. Lastly, a program for future work is included to illuminate the path ahead.

Delay Tolerant Networking↗

NASA NTRS2022

Planting a Flag in the Tropics: The Essential Tropical Geometric Background for Networking Applications

This Technical Memorandum serves to guide future NASA researchers and the public at large in terms of how Tropical Geometry can be applied to optimization problems, namely in networking. We give a basic overview of Tropical Geometry through a collection of resources we have summarized. Lastly, we include a discussion of power diagrams viewed as stratified spaces which may be a useful tool to study tropical varieties.

Tropical Geometry↗

NASA NTRS2017

Using CYGNSS to Observe Convectively Driven Near-Surface Winds in Tropical Precipitation Systems During Madden-Julian Oscillation Events

The Cyclone Global Navigation OKLMA 1411 UTC Satellite System (CYGNSS) is a multi-satellite constellation that launched 15 December 2016. The primary objective of CYGNSS is to use bistatic Global Positioning System (GPS) reflectometry to accurately measure near-surface wind speeds within the heavily raining inner core of tropical cyclones. CYGNSS also features rapid revisit times over a given region in the tropics - ranging from several minutes to a few hours, depending on the constellation geometry at that time. Despite the focus on tropical cyclones, the ability of CYGNSS to provide rapid updates of winds, unbiased by the presence of precipitation, has many other potential applications related to general tropical convection.

Convection↗

NASA NTRS1990

Remote sensing science for the Nineties; Proceedings of IGARSS '90 - 10th Annual International Geoscience and Remote Sensing Symposium, University of Maryland, College Park, May 20-24, 1990. Vols. 1, 2, & 3

Various papers on remote sensing (RS) for the nineties are presented. The general topics addressed include: subsurface methods, radar scattering, oceanography, microwave models, atmospheric correction, passive microwave systems, RS in tropical forests, moderate resolution land analysis, SAR geometry and SNR improvement, image analysis, inversion and signal processing for geoscience, surface scattering, rain measurements, sensor calibration, wind measurements, terrestrial ecology, agriculture, geometric registration, subsurface sediment geology, radar modulation mechanisms, radar ocean scattering, SAR calibration, airborne radar systems, water vapor retrieval, forest ecosystem dynamics, land analysis, multisensor data fusion. Also considered are: geologic RS, RS sensor optical measurements, RS of snow, temperature retrieval, vegetation structure, global change, artificial intelligence, SAR processing techniques, geologic RS field experiment, stochastic modeling, topography and Digital Elevation model, SAR ocean waves, spaceborne lidar and optical, sea ice field measurements, millimeter waves, advanced spectroscopy, spatial analysis and data compression, SAR polarimetry techniques. Also discussed are: plant canopy modeling, optical RS techniques, optical and IR oceanography, soil moisture, sea ice back scattering, lightning cloud measurements, spatial textural analysis, SAR systems and techniques, active microwave sensing, lidar and optical, radar scatterometry, RS of estuaries, vegetation modeling, RS systems, EOS/SAR Alaska, applications for developing countries, SAR speckle and texture.

Source record↗

NASA NTRS2001

Use of TRMM Data to Test an Improved Parameterization of Stratiform Precipitation

By incorporating the TRMM satellite orbital information into the most recent geodesic version of the Colorado State University General Circulation Model (CSU GCM), we are able to fly a satellite in the GCM, and sample the simulated atmosphere in the same way as the TRMM sensors sample the real atmosphere. The TRMM sampling statistics of precipitation and radiative fluxes at annual, intra-seasonal, monthly-mean and composited diurnal time scales are evaluated by comparing the satellite-sampled against fully-sampled simulated atmospheres. This information provides a valuable guidance for efficient usage of TRMM data and future satellite mission planning. We have also evaluated the effects of TRMM sampling errors on the inferred tropical-mean hydrologic cycle and radiative fluxes. We have found that there are strong spurious oscillations associated with the TRMM orbital geometry, with periods of 23 days and 3-4 months, in tropical-mean daily and monthly precipitation. Caution must therefore be used when applying TRMM observations of tropical-mean precipitation to interpret climate variations at intraseasonal and interannual scales.

Randall, David A.↗

NASA NTRS1994

ARMAR: An airborne rain-mapping radar

A new airborne rain-mapping radar (ARMAR) has been developed by NASA and the Jet Propulsion Laboratory for operation on the NASA Ames DC-8 aircraft. The radar operates at 13.8 GHz, the frequency to be used by the radar on the Tropical Rainfall Measuring Mission (TRMM). ARMAR simulates the TRMM radar geometry by looking downward and scanning its antenna in the cross-track direction. This basic compatibility between ARMAR and TRMM allows ARMAR to provide information useful for the TRMM radar design, for rain retrieval algorithm development, and for postlaunch calibration. ARMAR has additional capabilities, including multiple polarization, Doppler velocity measurement, and a radiometer channel for brightness temperature measurement. The system has been tested in both ground-based and airborne configurations. This paper describes the design of the system and shows results of field tests.

Durden, S. L.↗

NASA NTRS2012

Retrieval of Ocean Surface Windspeed and Rainrate from the Hurricane Imaging Radiometer (HIRAD) Brightness Temperature Observations

The Hurricane Imaging Radiometer (HIRAD) is a new airborne synthetic aperture passive microwave radiometer capable of wide swath imaging of the ocean surface wind speed under heavy precipitation e.g. in tropical cyclones. It uses interferometric signal processing to produce upwelling brightness temperature (Tb) images at its four operating frequencies 4, 5, 6 and 6.6 GHz [1,2]. HIRAD participated in NASA s Genesis and Rapid Intensification Processes (GRIP) mission during 2010 as its first science field campaign. It produced Tb images with ~70 km swath width and 3 km resolution from a ~ 20 km altitude. From this, ocean surface wind speed and column averaged atmospheric liquid water content can be retrieved across the swath. The column averaged liquid water then could be related to an average rain rate. The retrieval algorithm (and the HIRAD instrument itself) is a direct descendant of the nadir-only Stepped Frequency Microwave Radiometer that is used operationally by the NOAA Hurricane Research Division to monitor tropical cyclones [3,4]. However, due to HIRAD s slant viewing geometry (compared to nadir viewing SFMR) a major modification is required in the algorithm. Results based on the modified algorithm from the GRIP campaign will be presented in the paper.

Biswas, Sayak K.↗

NASA NTRS1999

Analyses of On-orbit Determinations of the Clouds and the Earth Radiant Energy System (CERES) Thermistor Bolometer Sensor Zero-radiance Effects

The Clouds and Earth's Radiant Energy System (CERES) missions were designed to measure broadband earth-reflected shortwave solar (0.3 micrometers to less than 5.0 micrometers) and earth-emitted longwave (5.0 micrometers to greater than 100 micrometers) radiances as well as earth-emitted narrow-band radiances in the water vapor window region between 8 micrometers and 12 micrometers. However, the CERES scanning thermistor bolometer sensor zero-radiance offsets were found to vary as much as 1.0 Wm (exp -2) sr (exp -1) with the scan angle measurement geometry due to gravitational forces and systematic electronic noise. To minimize the gravitational effects, the Tropical Rainfall Measuring Mission (TRMM) Spacecraft CERES sensors' offsets were derived on-orbit as functions of scan elevation and azimuth angles from the January 7-8, 1998 radiometric observations of deep cold space, representative of a 3 K blackbody. In this paper, the TRMM/CERES six orbit data base of on-orbit derived offsets is presented and analyzed to define the sampling requirements for the CERES sensors located on the Earth Science Enterprise (ESE) Terra Spacecraft and on the Earth Observing System (EOS) Afternoon (PM-1) Spacecraft, scheduled for launches in 1999 and 2000, respectively. Analyses of the TRMM/CERES shortwave sensor earth radiance measurements indicate that offsets can be determined on-orbit at the plus or minus 0.02 Wm (exp -2) sr (exp -1) precision level. Offset measuring techniques and sampling requirements are discussed for the TRMM and ESE missions. Ground, pre-launch Terra CERES cross-track scan offsets are presented and described which were measured as a function of scan angle.

Lee, Robert B., III↗

NASA NTRS2012

Gravity Wave Variances and Propagation Derived from AIRS Radiances

As the first gravity wave (GW) climatology study using nadir-viewing infrared sounders, 50 Atmospheric Infrared Sounder (AIRS) radiance channels are selected to estimate GW variances at pressure levels between 2-100 hPa. The GW variance for each scan in the cross-track direction is derived from radiance perturbations in the scan, independently of adjacent scans along the orbit. Since the scanning swaths are perpendicular to the satellite orbits, which are inclined meridionally at most latitudes, the zonal component of GW propagation can be inferred by differencing the variances derived between the westmost and the eastmost viewing angles. Consistent with previous GW studies using various satellite instruments, monthly mean AIRS variance shows large enhancements over meridionally oriented mountain ranges as well as some islands at winter hemisphere high latitudes. Enhanced wave activities are also found above tropical deep convective regions. GWs prefer to propagate westward above mountain ranges, and eastward above deep convection. AIRS 90 field-of-views (FOVs), ranging from +48 deg. to −48 deg. off nadir, can detect large-amplitude GWs with a phase velocity propagating preferentially at steep angles (e.g., those from orographic and convective sources). The annual cycle dominates the GW variances and the preferred propagation directions for all latitudes. Indication of a weak two-year variation in the tropics is found, which is presumably related to the Quasi-biennial oscillation (QBO). AIRS geometry makes its out-tracks capable of detecting GWs with vertical wavelengths substantially shorter than the thickness of instrument weighting functions. The novel discovery of AIRS capability of observing shallow inertia GWs will expand the potential of satellite GW remote sensing and provide further constraints on the GW drag parameterization schemes in the general circulation models (GCMs).

AIRS↗

NASA NTRS2019

Extreme Case of Spectral Band Difference Correction Between the OSIRIS-REX-NAVCAM2 and DSCOVR-EPIC Imagers

Earth-viewed images acquired during a recent asteroid intercept mission present a unique opportunity for radiometric calibration of visible imagers onboard a space exploration probe. Measurements from the CERES consistent DSCOVR-EPIC imager act as a reference in providing spatially, temporally, and angularly matched radiance values for deriving OSIRIS-REx-NavCam sensor calibration gains. The calibration is accomplished using an optimized all-sky tropical ocean ray-matching technique, which employs complex pixel remapping, navigation correction, and angular geometry consideration. Of critical consideration in this specific inter-calibration event is the extreme difference in spectral response function (SRF) width between the NavCam and EPIC imagers, which could cause a rather large bias. The NASA-LaRC SCIAMACHY based online spectral band adjustment factor (SBAF) calculation tool provides an empirical solution to such potential spectral-difference-induced biases through a high spectral- resolution hyper spectral convolution approach. The adjustments produced from this tool can effectively reduce the calibration gain bias of NavCam2 by nearly 6%, thereby adjusting the NavCam2 sensor to within 3.2% of its prelaunch calibration. These results highlight the capability of the SBAF tool to account for exceptionally disparate SRFs.

Scarino, Benjamin↗

NASA NTRS2011

APR-2 Tropical Cyclone Observations

The Second Generation Airborne Precipitation Radar (APR-2) participated in the Genesis and Rapid Intensification Processes (GRIP) experiment in August and September of 2010, collecting a large volume of data in several tropical systems, including Hurricanes Earl and Karl. Additional measurements of tropical cyclone have been made by APR-2 in experiments prior to GRIP (namely, CAMEX-4, NAMMA, TC4); Table 1 lists all the APR-2 tropical cyclone observations. The APR-2 observations consist of the vertical structure of rain reflectivity at 13.4 and 35.6 GHz, and at both co-polarization and crosspolarization, as well as vertical Doppler measurements and crosswind measurements. APR-2 normally flies on the NASA DC-8 aircraft, as in GRIP, collecting data with a downward looking, cross-track scanning geometry. The scan limits are 25 degrees on either side of the aircraft, resulting in a roughly 10-km swath, depending on the aircraft altitude. Details of the APR-2 observation geometry and performance can be found in Sadowy et al. (2003).The multiparameter nature of the APR-2 measurements makes the collection of tropical cyclone measurements valuable for detailed studies of the processes, microphysics and dynamics of tropical cyclones, as well as weaker systems that are associated with tropical cyclone formation. In this paper, we give a brief overview of how the APR-2 data are processed. We also discuss use of the APR-2 cross-track winds to estimate various quantities of interest in in studies of storm intensification. Finally, we show examples of the standard products and derived information.

radar↗

NASA NTRS2001

Flying the TRMM Satellite in a GCM

By incorporating the Tropical Rainfall Measurement Mission (TRMM) satellite orbital information into the Colorado State University General Circulation Model (CSU GCM), we are able to 'fly' a satellite in the GCM, and sample the simulated atmosphere in the same way as the TRMM sensors sample the real atmosphere. The TRMM-sampled statistics for precipitation and radiative fluxes at annual, intraseasonal, monthly-mean and seasonal-mean diurnal time scales are evaluated by comparing the satellite-sampled against fully-sampled simulated atmospheres. The sampling rates of the TRMM sensors are significantly affected by the sensors' swath widths. The TRMM Microwave Imager (TMI) and the Visible Infrared Scanner (VIRS) sample each 2.25 x 2.25 degree grid box in the Tropics and subtropics about once per day, but at a different local time every day, while the Precipitation Radar (PR) and the Clouds and the Earth's Radiant Energy System (CERES) sensor visit each grid box about once every three days and twice per day, respectively. Besides inadequate samplings resulting from sensors' swath widths, there is a large, systematic diurnal undersampling associated with TRMM's orbital geometry for grid boxes away from the Equator. When only one month of TRMM data are used, this diurnal undersampling can lead to more daytime samples relative to nighttime samples in one hemisphere, and more nighttime samples relative to daytime samples in the other hemisphere. The resulting sampling biases (3-6 W m(exp-2)) are very pronounced in outgoing longwave radiation (OLR) over the subtropical land masses. The sampling errors in OLR monthly- and seasonal-means are less than 8 W m(exp-2) (5%) for each 2.25 x 2.25 degree grid box. The OLR monthly- and seasonal-means are not sensitive to diurnal undersamplings associated with the TRMM orbits and sensors' swath widths. However, this is not the case for total precipitation. Diurnal undersampling could produce errors as large as 20% in the Tropics and 40% in the subtropics, for the zonally averaged monthly-mean rain rates. The TRMM orbits sample each 2.25 x 2.25 degree grid box in the Tropics and subtropics 1-6 times for each hour of the day within a single season. The seasonal-mean diurnal cycles of precipitation and OLR are not well sampled for any one grid box. By either accumulating the satellite data for a long enough period, or averaging the data over a large area with a relatively uniform diurnal signal, the diurnal cycles of precipitation and OLR can be satisfactorily sampled. The effects TRMM sampling errors on the inferred tropical-mean hydrologic cycle and radiative fluxes are also evaluated. There are strong spurious oscillations associated with TRMM's orbital geometry, with periods of 23 days and 3-4 months, in tropical-mean daily and monthly precipitation. While the relative fluctuations of the sampled-OLR are negligible, the relative fluctuations of the sampled precipitation have magnitudes similar to those of the observed climate variability. Caution must therefore be used when applying TRMM observations of tropical-mean precipitation to interpret climate variations at intraseasonal and interannual scales.

Lin, Xin↗

NASA NTRS1991

Atmospheric weather regimes over tropical South America

Infrared radiance measurements by the GOES-6 satellite during April 1986 through April 1987 were used to characterize and identify distinct regimes of persistent large-scale cloudiness patterns over the Amazon Basin. It is suggested that the energetics of the tropical troposphere over the Amazon Basin can be directly related to the GOES large-scale cloudiness patterns. The geometry and persistence of the cloud patterns are influenced by shifts in general circulation features and are likely modulated by 4- to 5-day and 40- to 60-day waves. Diurnal forcing effects are more pronounced during weather regimes characterized by prominently clear skies over land areas.

Connors, Vickie S.↗

NASA NTRS1994

Observations of the far ultraviolet airflow by the Ultraviolet Limb Imaging experiment on STS-39

The Ultraviolet Limb Imaging (UVLIM) experiment flew on STS-39 in the spring of 1991 to observe the Earth's thermospheric airglow and included a far ultraviolet (1080-1800 A) spectrometer. We present first results from this spectrometer, including a spectroscopic analysis at 6-A resolution of H, O, N, and N2 dayglow emissions and modeling of the observed limb-scan profiles of dayglow emissions. The observed N2 Lyman-Birge-Hopfield (LBH) emission reflects a vibrational population distribution in the a(1 Pi)(sub g) state that differs significantly from those predicted for direct electron excitation and excitation with cascade from the a('1 Sigma)(sub u)(-) and w(1 Delta)(sub u) states. The vibrational population distribution and LBH brightness suggest a total cascade rate 45% that of direct excitation, in contrast to laboratory measurements. For the first time, pronounced limb brightening is observed in both the N I lambda 1200 limb emission profiles, as expected for emissions excited by N2 dissociation which produces kinetically fast N fragments; however, optically thick components of these features are also observed. Preliminary modeling of the OI lambda 1356, HI lambda 1216, and OI lambda 1304 and OI lambda 1641 emissions agrees to within roughly 10% of the observed limb-scan profiles, but the models underestimate the N2 LBH profiles by a factor of 1.4-1.6, consistent with the inferred cascade effect. Other findings include: an OI lambda 1152/lambda 1356 intensity ratio that is inconsistent with the large cascade contribution to OI lambda 1356 from np 5P states required by laboratory and nightglow observations; nightglow observations of the tropical ultraviolet arcs exhibit a wide range of OI lambda 1356/lambda 1304 intensity ratios and illustrate the complicated observing geometry and radiative transfer effects that must be modeled; and we find a 3-sigma upper limit of 8.5 R to the total LBH vehicle glow emission.

Budzien, S. A.↗

NASA NTRS1988

Canopy reflectance modeling in a tropical wooded grassland

The Li-Strahler canopy reflectance model, driven by LANDSAT Thematic Mapper (TM) data, provided regional estimates of tree size and density in two bioclimatic zones in Africa. This model exploits tree geometry in an inversion technique to predict average tree size and density from reflectance data using a few simple patameters measured in the field and in the imagery. Reflectance properties of the trees were measured in the study sites using a pole-mounted radiometer. The measurements showed that the assumptions of the simple Li-Strahler model are reasonable for these woodlands. The field radiometer measurements were used to calculate the normalized difference vegetation index (NDVI), and the integrated NDVI over the canopy was related to crown volume. Predictions of tree size and density from the canopy model were used with allometric equations from the literature to estimate woody biomass and potential foliar biomass for the sites and for the regions. Estimates were compared with independent measurements made in the Sahelian sites, and to typical values from the literature for these regions and for similar woodlands. In order to apply the inversion procedure regionally, an area must first be stratified into woodland cover classes, and dry-season TM data were used to generate a stratum map of the study areas with reasonable accuracy. The method used was unsupervised classification of multi-data principal components images.

Simonett, David↗

NASA NTRS1989

Ground data system architecture for precipitation determination from space-based radar

The Tropical Rain Mapping Radar (Tramar) is proposed as an attached payload as part of the Space Station Earth Observing System Program. Tramar would measure rainfall rates, rain velocity, and rain cell areal extent in the latitude band from 30 deg S to 30 deg N for use in studies of large-scale atmospheric circulation, variations of latent heating, tropical hydrologic processes, and mesoscale precipitation systems. The Tramar science requirements, radar design, and ground data system architecture are examined, including the three-dimensional scan geometry, the radar system performance parameters, the production of earth-gridded maps, and the telemetry, sensor, radiometric, and geophysical data that would be obtained by Tramar.

Hilland, Jeffrey E.↗

NASA NTRS2012

Multi-Angle Implementation of Atmospheric Correction for MODIS (MAIAC). Part 3: Atmospheric Correction

This paper describes the atmospheric correction (AC) component of the Multi-Angle Implementation of Atmospheric Correction algorithm (MAIAC) which introduces a new way to compute parameters of the Ross-Thick Li-Sparse (RTLS) Bi-directional reflectance distribution function (BRDF), spectral surface albedo and bidirectional reflectance factors (BRF) from satellite measurements obtained by the Moderate Resolution Imaging Spectroradiometer (MODIS). MAIAC uses a time series and spatial analysis for cloud detection, aerosol retrievals and atmospheric correction. It implements a moving window of up to 16 days of MODIS data gridded to 1 km resolution in a selected projection. The RTLS parameters are computed directly by fitting the cloud-free MODIS top of atmosphere (TOA) reflectance data stored in the processing queue. The RTLS retrieval is applied when the land surface is stable or changes slowly. In case of rapid or large magnitude change (as for instance caused by disturbance), MAIAC follows the MODIS operational BRDF/albedo algorithm and uses a scaling approach where the BRDF shape is assumed stable but its magnitude is adjusted based on the latest single measurement. To assess the stability of the surface, MAIAC features a change detection algorithm which analyzes relative change of reflectance in the Red and NIR bands during the accumulation period. To adjust for the reflectance variability with the sun-observer geometry and allow comparison among different days (view geometries), the BRFs are normalized to the fixed view geometry using the RTLS model. An empirical analysis of MODIS data suggests that the RTLS inversion remains robust when the relative change of geometry-normalized reflectance stays below 15%. This first of two papers introduces the algorithm, a second, companion paper illustrates its potential by analyzing MODIS data over a tropical rainforest and assessing errors and uncertainties of MAIAC compared to conventional MODIS products.

Lyapustin, A.↗

NASA NTRS2010

High-Precision Shape Control of In-Space Deployable Large Membrane/Thin-Shell Reflectors

This innovation has been developed to improve the resolutions of future spacebased active and passive microwave antennas for earth-science remote sensing missions by maintaining surface figure precisions of large membrane/thin-shell reflectors during orbiting. The intention is for these sensing instruments to be deployable at orbit altitudes one or two orders of magnitude higher than Low Earth Orbit (LEO), but still being able to acquire measurements at spatial resolution and sensitivity similar to those of LEO. Because active and passive microwave remote sensors are able to penetrate through clouds to acquire vertical profile measurements of geophysical parameters, it is desirable to elevate them to the higher orbits to obtain orbital geometries that offer large spatial coverage and more frequent observations. This capability is essential for monitoring and for detailed understanding of the life cycles of natural hazards, such as hurricanes, tropical storms, flash floods, and tsunamis. Major components of this high-precision antenna-surface-control system include a membrane/thin shell reflector, a metrology sensor, a controller, actuators, and corresponding power amplifier and signal conditioning electronics (see figure). Actuators are attached to the back of the reflector to produce contraction/ expansion forces to adjust the shape of the thin-material reflector. The wavefront-sensing metrology system continuously measures the surface figure of the reflector, converts the surface figure to digital data and feeds the data to the controller. The controller determines the control parameters and generates commands to the actuator system. The flexible, piezoelectric polymer actuators are thus activated, providing the control forces needed to correct any distortions that exist in the reflector surface. Piezoelectric polymer actuators are very thin and flexible. They can be implemented on the back of the membrane/thin-shell reflector without introducing significant amounts of mass or stiffness to the reflector. They can be rolled up or folded to accommodate the packaging needed for launch. An analytical model of the system, which includes the membrane reflector, actuator, and controller has been developed to investigate the functionality of this control system on a 35-meter-diameter membrane reflector. The performance of this system under external disturbances such as in space thermal loads and W-error due to inflation has been investigated. A subscale breadboard has been developed, and the functionality of this control concept has been demonstrated by this breadboard.

Watkins, Ronald↗