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NASA NTRS1997

Vector Observation-Aided/Attitude-Rate Estimation Using Global Positioning System Signals

A sequential filtering algorithm is presented for attitude and attitude-rate estimation from Global Positioning System (GPS) differential carrier phase measurements. A third-order, minimal-parameter method for solving the attitude matrix kinematic equation is used to parameterize the filter's state, which renders the resulting estimator computationally efficient. Borrowing from tracking theory concepts, the angular acceleration is modeled as an exponentially autocorrelated stochastic process, thus avoiding the use of the uncertain spacecraft dynamic model. The new formulation facilitates the use of aiding vector observations in a unified filtering algorithm, which can enhance the method's robustness and accuracy. Numerical examples are used to demonstrate the performance of the method.

Oshman, Yaakov↗

NASA NTRS2007

Numerical Study of Flow Augmented Thermal Management for Entry and Re-Entry Environments

The use of a flow augmented thermal management system for entry and re-entr environments is one method for reducing heat and drag loads. This concept relies on jet penetration from supersonic and hypersonic counterflowing jets that could significantly weaken and disperse the shock-wave system of the spacecraft flow field. The objective of this research effort is to conduct parametric studies of the supersonic flow over a 2.6% scale model of the Apollo capsule, with and without the counterflowing jet, using time-accurate and steady-state computational fluid dynamics simulations. The numerical studies, including different freestream Mach number angle of attack counterflowing jet mass flow rate, and nozzle configurations, were performed to examine their effect on the drag and beat loads and to explore the counternowing jet condition. The numerical results were compared with the test data obtained from transonic blow-down wind-tunnel experiments conducted independently at NASA MSFC.

Cheng, Gary C.↗

NASA NTRS2005

Thermal Insulation Test Apparatuses

The National Aeronautics and Space Administration (NASA) seeks to license its Thermal Insulation Test Apparatuses. Designed by the Cryogenics Test Laboratory at the John F. Kennedy Space Center (KSC) in Florida, these patented technologies (U.S. Patent Numbers: Cryostat 1 - 6,742,926, Cryostat 2 - 6,487,866, and Cryostat 4 - 6,824,306) allow manufacturers to fabricate and test cryogenic insulation at their production and/or laboratory facilities. These new inventions allow for the thermal performance characterization of cylindrical and flat specimens (e.g., bulk-fill, flat-panel, multilayer, or continuously rolled) over the full range of pressures, from high vacuum to no vacuum, and over the full range of temperatures from 77K to 300K. In today's world, efficient, low-maintenance, low-temperature refrigeration is taking a more significant role, from the food industry, transportation, energy, and medical applications to the Space Shuttle. Most countries (including the United States) have laws requiring commercially available insulation materials to be tested and rated by an accepted methodology. The new Cryostat methods go beyond the formal capabilities of the ASTM methods to provide testing for real systems, including full-temperature differences plus full-range vacuum conditions.

Berman, Brion↗

NASA NTRS2013

Investigation of VEGGIE Root Mat

VEGGIE is a plant growth facility that utilizes the phenomenon of capillary action as its primary watering system. A cloth made of Meta Aramid fiber, known as Nomex is used to wick water up from a reservoir to the bottom of the plants roots. This root mat system is intended to be low maintenance with no moving parts and requires minimal crew interface time. Unfortunately, the water wicking rates are inconsistent throughout the plant life cycle, thus causing plants to die. Over-wicking of water occurs toward the beginning of the cycle, while under-wicking occurs toward the middle. This inconsistency of wicking has become a major issue, drastically inhibiting plant growth. The primary objective is to determine the root cause of the inconsistent wicking through experimental testing. Suspect causes for the capillary water column to break include: a vacuum effect due to a negative pressure gradient in the water reservoir, contamination of material due to minerals in water and back wash from plant fertilizer, induced air bubbles while using syringe refill method, and material limitations of Nomex's ability to absorb and retain water. Experimental testing will be conducted to systematically determine the cause of under and over-wicking. Pressure gages will be used to determine pressure drop during the course of the plant life cycle and during the water refill process. A debubbler device will be connected to a root mat in order to equalize pressure inside the reservoir. Moisture and evaporation tests will simultaneously be implemented to observe moisture content and wicking rates over the course of a plant cycle. Water retention tests will be performed using strips of Nomex to determine materials wicking rates, porosity, and absorptivity. Through these experimental tests, we will have a better understanding of material properties of Nomex, as well as determine the root cause of water column breakage. With consistent test results, a forward plan can be achieved to resolve the issue and give valuable insight for the next generation of VEGGIE.

Subbiah, Arun M.↗

NASA NTRS2016

NASA Advanced Explorations Systems: Advancements in Life Support Systems

The NASA Advanced Exploration Systems (AES) Life Support Systems (LSS) project strives to develop reliable, energy-efficient, and low-mass spacecraft systems to provide environmental control and life support systems (ECLSS) critical to enabling long duration human missions beyond low Earth orbit (LEO). Highly reliable, closed-loop life support systems are among the capabilities required for the longer duration human space exploration missions assessed by NASA's Habitability Architecture Team (HAT). The LSS project is focused on four areas: architecture and systems engineering for life support systems, environmental monitoring, air revitalization, and wastewater processing and water management. Starting with the international space station (ISS) LSS systems as a point of departure (where applicable), the mission of the LSS project is three-fold: 1. Address discrete LSS technology gaps 2. Improve the reliability of LSS systems 3. Advance LSS systems towards integrated testing on the ISS. This paper summarized the work being done in the four areas listed above to meet these objectives. Details will be given on the following focus areas: Systems Engineering and Architecture- With so many complex systems comprising life support in space, it is important to understand the overall system requirements to define life support system architectures for different space mission classes, ensure that all the components integrate well together and verify that testing is as representative of destination environments as possible. Environmental Monitoring- In an enclosed spacecraft that is constantly operating complex machinery for its own basic functionality as well as science experiments and technology demonstrations, it's possible for the environment to become compromised. While current environmental monitors aboard the ISS will alert crew members and mission control if there is an emergency, long-duration environmental monitoring cannot be done in-orbit as current methodologies rely largely on sending environmental samples back to Earth. The LSS project is developing onboard analysis capabilities that will replace the need to return air and water samples from space for ground analysis. Air Revitalization- The air revitalization task is comprised of work in carbon dioxide removal, oxygen generation and recovery and trace contamination and particulate control. The CO2 Removal and associated air drying development efforts under the LSS project are focused both on improving the current SOA technology on the ISS and assessing and examining the viability of other sorbents and technologies available in academia and industry. The Oxygen Generation and Recovery technology development area encompasses several sub-tasks in an effort to supply O2 to the crew at the required conditions, to recover O2 from metabolic CO2, and to recycle recovered O2 back to the cabin environment. Current state-of-the-art oxygen generation systems aboard space station are capable of generating or recovering approximately 40% of required oxygen; for exploration missions this percentage needs to be greatly increased. A spacecraft cabin trace contaminant and particulate control system serves to keep the environment below the spacecraft maximum allowable concentration (SMAC) for chemicals and particulates. Both passive (filters) and active (scrubbers) methods contribute to the overall TC & PC design. Work in the area of trace contamination and particulate control under the LSS project is focused on making improvements to the SOA TC & PC systems on ISS to improve performance and reduce consumables. Wastewater Processing and Water Management- A major goal of the LSS project is the development of water recovery systems to support long duration human exploration beyond LEO. Current space station wastewater processing and water management systems distill urine and wastewater to recover water from urine and humidity condensate in the spacecraft at a approximately 74% recovery rate. For longer, farther missions into deep space, that recovery rate must be greatly increased so that astronauts can journey for months without resupply cargo ships from Earth.

Shull, Sarah A.↗

NASA NTRS2020

Squash-Box Feasibility Driven Differential Dynamic Programming

Recently, Differential Dynamic Programming (DDP) and other similar algorithms have become the solvers of choice when performing non-linear Model Predictive Control (nMPC) with modern robotic devices. The reason is that they have a lower computational cost per iteration when compared with off-the-shelf Non-Linear Programming (NLP) solvers, which enables its online operation. However, they cannot handle constraints, and are known to have poor convergence capabilities. In this paper, we propose a method to solve the optimal control problem with control bounds through a squashing function (i.e., a sigmoid, which is bounded by construction). It has been shown that a naive use of squashing functions damage the convergence rate. To tackle this, we first propose to add a quadratic barrier that avoids the difficulty of the plateau produced by the sigmoid. Second, we add an outer loop that adapts both the sigmoid and the barrier; it makes the optimal control problem with the squashing function converge to the original control-bounded problem. To validate our method, we present simulation results for different types of platforms including a multi-rotor, a biped, a quadruped and a humanoid robot.

Navarro, Angel Santamaria↗

NASA NTRS1999

Investigation of Sooting in Microgravity Droplet Combustion: Fuel-Dependent Effects

Kumagai and coworkers first performed microgravity droplet combustion experiments [Kumagai, 1957]. The primary goal of these early experiments were to validate simple 'd(sup 2)-law models [Spalding, 1954, Godsave, 1954] Inherent in the 'd(sup 2) -law' formulation and in the scope of the experimental observation is the neglect of sooting behavior. In fact, the influence of sooting has not received much attention until more recent works [Choi et al., 1990; Jackson et al., 1991; Jackson and Avedisian, 1994; Choi and Lee, 1996; Jackson and Avedisian, 1996; Lee et al., 1998]:. Choi and Lee measured soot volume fraction for microgravity droplet flames using full-field light extinction and subsequent tomographic inversion [Choi and Lee, 1996]. In this investigation, soot concentrations were measured for heptane droplets and it was reported that soot concentrations were considerably higher in microgravity compared to the normal gravity flame. It was reasoned that the absence of buoyancy and the effects of thermophoresis resulted in the higher soot concentrations. Lee et al. [1998] performed soot measurement experiments by varying the initial droplet diameter and found marked influence of sooting on the droplet burning behavior. There is growing sentiment that sooting in droplet combustion must no longer be neglected and that "perhaps one of the most important outstanding contributions of (micro)g droplet combustion is the observation that in the absence of asymmetrical forced and natural convection, a soot shell is formed between the droplet surface and the flame, exerting an influence on the droplet combustion response far greater than previously recognized." [Law and Faeth, 1994]. One of the methods that we are exploring to control the degree of sooting in microgravity is to use different fuels. The effect of fuel structure on sooting propensity has been investigated for over-ventilated concentric coflowing buoyant diffusion flames. (Glassman, 1996]. In these investigations, the fuel flowrate was increased until smoke was observed to escape from the "luminous visible flame" [Glassman, 1996]. A total of 29 fuels were used in order to characterize relative sooting propensity. The sooting propensity of a particular fuel was assessed by comparing the flowrates for soot emission from the tip of the flame. It was reported that the sooting tendency for diffusion flames increased for fuels with higher rates of pyrolysis. Randolph and Law [1986 and not 1994] also examined the effect of fuel structure on droplet sooting behavior. In their experiments the droplets were separated from the bulk gas stream and quenched with nitrogen prior to gravimetric measurements. A variety of fuels were studied, namely aromatics, phenyl-alkanes and alkanes. The results were in qualitative agreement with the work of Glassman [1986]. Vander Wal et al. [1994] performed relative soot concentration measurements using laser-induced incandescence for heptane and decane fuel droplets burning under normal-gravity conditions. It was found that soot volume fractions for decane was more than a factor of two larger than that for heptane. Although the normal-gravity investigations have provided some important insights regarding the influence of fuel structure on the sooting behavior of droplet flames, results cannot be easily extrapolated for microgravity studies since increased residence times and thermophoretic effects must be considered in greater detail. Several studies have compared sooting behavior of different fuel droplets burning under microgravity conditions [Card and Choi, 1990; Jackson et al., 1991; Jackson and Avedisian, 1994], however, detailed quantitative results were not provided. In all of these previous studies, the degree of sooting was only visually assessed from an incandescent backlighted image of the soot containing region. Such techniques can provide misleading results regarding sooting behavior [Choi, 1996].

Manzello, Samuel L.↗

NASA NTRS1977

Recovery of refractivity profiles and pressure and temperature distributions in the lower atmosphere from satellite-to-satellite radio occultation data

The feasibility of recovering parameters from one-way range rate between two earth orbiting spacecraft during occultation of the tracking signal by the earth's lower atmosphere. The tracking data is inverted by an integral transformation (Abel transform) to obtain a vertical refractivity profile above the point of closest approach of the ray connecting the satellites. Pressure and temperature distributions can be obtained from values of dry refractivity using the hydrostatic equation and perfect gas law. Two methods are investigated for recovering pressure and temperature parameters. Results show that recovery is much more sensitive to satellite velocity errors than to satellite position errors. An error analysis is performed. An example is given demonstrating recovery of parameters from radio occultation data obtained during satellite-to-satellite tracking of Nimbus 6 by the ATS 6 satellite.

Murray, C. W., Jr.↗

NASA NTRS1984

Dynamical relations for left ventricular ejection - Flow rate, momentum, force and impulse

An investigation was carried out to quantitatively evaluate left ventricular volume flow rate, momentum, force and impulse derived from application of conservation principles for mass and momentum of blood within the ventricle during the ejection phase. An automated digital image processing system was developed and applied to left ventricular angiograms which are computer processed and analyzed frame by frame to determine the dynamical relations by numerical methods. The initial experience with force and impulse has indicated that neither quantity seemed to be a sensitive indicator of coronary artery disease as evaluated by qualitative angiography for the particular patient group studied. Utilization of the dynamical relations in evaluating human left ventricular performance requires improved means of measurement and interpretation of clinical studies.

Back, L. H.↗

NASA NTRS1986

Statistical analysis of failure data on controllers and SSME turbine blade failures

The expressions for the maximum likelihood functions are given when the failure data are censored at a given point or at multiple points, or when the data come in groups. Different models applicable to failure data are presented with their characteristics. A graphical method of distinguishing different models by using cumulative hazard fucnction is discussed. For the failure data on controllers the model is determined by cumulative hazard function and chi-square goodness of fit. Using the Weibull Model the maximum likelihood estimators of the shape parameter and the failure rate parameter are obtained. The confidence intervals, meantime between failures, and B1 are determined. Similarly, for the data on Space Shuttle Main Engine (SSME) blade failures the maximum likelihood estimators are obtained for the Weibull parameters. The variances, confidence intervals, meantime between failures, and reliability are determined. The analysis is performed under assumption of grouped data as well as randomly placed data.

Patil, S. A.↗

NASA NTRS1989

Analysis of opposed jet hydrogen-air counter flow diffusion flame

A computational simulation of the opposed-jet diffusion flame is performed to study its structure and extinction limits. The present analysis concentrates on the nitrogen-diluted hydrogen-air diffusion flame, which provides the basic information for many vehicle designs such as the aerospace plane for which hydrogen is a candidate as the fuel. The computer program uses the time-marching technique to solve the energy and species equations coupled with the momentum equation solved by the collocation method. The procedure is implemented in two stages. In the first stage, a one-step forward overal chemical reaction is chosen with the gas phase chemical reaction rate determined by comparison with experimental data. In the second stage, a complete chemical reaction mechanism is introduced with detailed thermodynamic and transport property calculations. Comparison between experimental extinction data and theoretical predictions is discussed. The effects of thermal diffusion as well as Lewis number and Prandtl number variations on the diffusion flame are also presented.

Ho, Y. H.↗

NASA NTRS2004

Overcoming Communication Restrictions in Collectives

Many large distributed system are characterized by having a large number of components (eg., agents, neurons) whose actions and interactions determine a %orld utility which rates the performance of the overall system. Such collectives are often subject to communication restrictions, making it difficult for components which try to optimize their own private utilities, to take actions that also help optimize the world utility. In this article we address that coordination problem and derive four utility functions which present different compromises between how aligned a component s private utility is with the world utility and how readily that component can determine the actions that optimize its utility. The results show that the utility functions specifically derived to operate under communication restrictions outperform both traditional methods and previous collective-based methods by up to 75%.

Tumer, Kagan↗

NASA NTRS2019

Developments on Low Cost Manufacturing Methods for Cylindrical Launch Vehicle Structures

Due to the increasing international competition and growing pressure to lower the cost of launch vehicles, enhanced technologies and competences for efficient, low cost manufacturing have to be developed. In this paper, an innovative manufacturing technology for launch vehicle applications is presented that was developed over the last few years at MT Aerospace AG (MTA) in collaboration with NASA Langley Research Center, the European Space Agency (ESA), Lockheed Martin Corporation (LMCO), Leifeld Metal Spinning and Scot Forge. The Integrally Stiffened Cylinder (ISC) manufacturing method was first investigated in 2010. Early results indicated a high mass saving potential for cylindrical structures for launch vehicle systems, such as cryogenic tank barrels and intertanks, produced using the ISC method. Cost-benefit studies conducted at NASA and MTA estimated up to 50% reduction in manufacturing costs is possible compared to conventional multi-piece, welded fabrication methods. Associated reductions in manufacturing lead times have shown that a 40% faster production rate is possible by the ISC process. Following proof of concept and process development efforts, a cylindrical structure manufactured using the ISC method had its maiden flight in 2015 on a Black Brant sounding rocket at NASA Wallops Flight Facility. Further collaboration between ESA, NASA, MTA, LMCO, and Scot Forge led to manufacturing test campaigns at MT Aerospace in 2017 and 2018 that successfully demonstrated ISC fabrication at a scale representative of commercial launch vehicles. This paper summarizes the status and relevant results of the development sequence of the ISC cylinder manufacturing method. Results of cost / benefit and structural performance analyses are also summarized. Moreover, an outlook will be given on further planned activities and development steps.

Martin Stachulla↗

NASA NTRS1993

Determination of design and operation parameters for upper atmospheric research instrumentation to yield optimum resolution with deconvolution. Appendix 7

Because of the interesting science which can be performed using a satellite attached by a very long tether to a mother vehicle in orbit, such as the Space Shuttle, NASA will deploy TSS-1 (Tethered Satellite System) in 1992. A very long tether (20 km in this case) has the possibility of undergoing oscillations of several different types, or modes, and higher harmonics of these modes. The purpose of this document is to describe a method for detecting the amplitude, frequency, and phase (and predicting future motion in the steady state) of these modes, in particular, the skiprope mode, using tethered satellite dynamics measurements. Specifically the rotation rate data about two orthogonal axes, calculated from output from satellite gyroscopes, are used. The data of interest are the satellite pitch and roll rate measurements. NASA has determined to use two methods to diagnose skiprope properties and predict future values. One of these, a Fourier transform domain approach, is the subject of this notebook. The main program and all subroutines are described along with the test plan for evaluating the Frequency Domain Skiprope Observer.

Ioup, George E.↗

NASA NTRS2009

Transient Modeling and Analysis of a Metabolic Heat-Regenerated Temperature Swing Adsorption (MTSA) System for a PLSS

A Metabolic heat-regenerated Temperature Swing Adsorption (MTSA) system is being developed for carbon dioxide, water and thermal control in a lunar and martian portable life support system (PLSS). A previous system analysis was performed to evaluate the impact of MTSA on PLSS design. That effort was Mars specific and assumed liquid carbon dioxide (LCO2) coolant made from martian resources. Transient effects were not considered but rather average conditions were used throughout the analysis. This effort takes into further consideration the transient effects inherent in the cycling MTSA system as well as assesses the use of water as coolant. Standard heat transfer, thermodynamic, and heat exchanger methods are presented to conduct the analysis. Assumptions and model verification are discussed. The tool was used to perform various system studies. Coolant selection was explored and takes into account different operational scenarios as the minimum bed temperature is driven by the sublimation temperature of the coolant (water being significantly higher than LCO2). From this, coolant mass is sized coupled with sorbent bed mass because MTSA adsorption performance decreases with increasing sublimation temperature. Reduction in heat exchanger performance and even removal of certain heat exchangers, like a recuperative one between the two sorbent beds, is also investigated. Finally, the coolant flow rate is varied over the cycle to determine if there is a more optimal means of cooling the bed from a mass perspective. Results of these studies and subsequent recommendations for system design are presented.

Iacomini, Christie↗

NASA NTRS2010

Simulated Service and Stress Corrosion Cracking Testing for Friction Stir Welded Spun Form Domes

Damage tolerance testing development was required to help qualify a new spin forming dome fabrication process for the Ares 1 program at Marshall Space Flight Center (MSFC). One challenge of the testing was due to the compound curvature of the dome. The testing was developed on a sub-scale dome with a diameter of approximately 40 inches. The simulated service testing performed was based on the EQTP1102 Rev L 2195 Aluminum Lot Acceptance Simulated Service Test and Analysis Procedure generated by Lockheed Martin for the Space Shuttle External Fuel Tank. This testing is performed on a specimen with an induced flaw of elliptical shape generated by Electrical Discharge Machining (EDM) and subsequent fatigue cycling for crack propagation to a predetermined length and depth. The specimen is then loaded in tension at a constant rate of displacement at room temperature until fracture occurs while recording load and strain. An identical specimen with a similar flaw is then proof tested at room temperature to imminent failure based on the critical offset strain achieved by the previous fracture test. If the specimen survives the proof, it is then subjected to cryogenic cycling with loads that are a percentage of the proof load performed at room temperature. If all cryogenic cycles are successful, the specimen is loaded in tension to failure at the end of the test. This standard was generated for flat plate, so a method of translating this to a specimen of compound curvature was required. This was accomplished by fabricating a fixture that maintained the curvature of the specimen rigidly with the exception of approximately one-half inch in the center of the specimen containing the induced flaw. This in conjunction with placing the center of the specimen in the center of the load train allowed for successful testing with a minimal amount of bending introduced into the system. Stress corrosion cracking (SCC) tests were performed using the typical double beam assembly and with 4-point loaded specimens under alternate immersion conditions in a 3.5% NaCl environment for 90 days. In addition, experiments were conducted to determine the threshold stress intensity factor for SCC (K1SCC) of Al-Li 2195 which to our knowledge has not been determined previously. The successful simulated service and stress corrosion testing helped to provide confidence to continue to Ares 1 scale dome fabrication.

Stewart, Thomas J.↗

NASA NTRS2010

Measurement of Insulation Compaction in the Cryogenic Fuel Tanks at Kennedy Space Center by Fast/Thermal Neutron Techniques

The liquid hydrogen and oxygen cryogenic storage tanks at John F. Kennedy Space Center (KSC) use expanded perlite as thermal insulation. Th ere is evidence that some of the perlite has compacted over time, com promising the thermal performance and possibly also structural integr ity of the tanks. Therefore an Non-destructive Testing (NDT) method for measuring the perlite density or void fraction is urgently needed. Methods based on neutrons are good candidates because they can readil y penetrate through the 1.75 cm outer steel shell and through the ent ire 120 cm thickness of the perlite zone. Neutrons interact with the nuclei of materials to produce characteristic gamma rays which are the n detected. The gamma ray signal strength is proportional to the atom ic number density. Consequently, if the perlite is compacted then the count rates in the individual peaks in the gamma ray spectrum will i ncrease. Perlite is a feldspathic volcanic rock made up of the major elements Si, AI, Na, K and 0 along with some water. With commercially available portable neutron generators it is possible to produce simul taneously fluxes of neutrons in two energy ranges: fast (14 MeV) and thermal (25 meV). Fast neutrons produce gamma rays by inelastic scatt ering which is sensitive to Fe and O. Thermal neutrons produce gamma rays by radiative capture in prompt gamma neutron activation (PGNA) and this is sensitive to Si, AI, Na, Kand H. Thus the two energy ranges produce complementary information. The R&D program has three phases: numerical simulations of neutron and gamma ray transport with MCNP s oftware, evaluation of the system in the laboratory on test articles and finally mapping of the perlite density in the cryogenic tanks at KSC. The preliminary MCNP calculations have shown that the fast/therma l neutron NDT method is capable of distinguishing between expanded an d compacted perlite with excellent statistics.

Livingston, R. A.↗

NASA NTRS2019

Efficient On-Orbit Singularity-Free Geopotential Estimation

For vehicles in orbital cruise, either in orbit around a body or in a transfer orbit, gravity is the primary external force; therefore, the complexity of the geopotential model heavily influences the accuracy of the navigation state. As this type of spacecraft requires a highly accurate geopotential model and performs the computation numerous times a second, this calculation needs to be efficient. This calculation needs to include the first and second derivatives of the geopotential because navigation systems can require both the gravitational acceleration and gradient for inertial integration and state filtering. The most efficient method for calculating the geopotential, the forward column method, is discussed in detail, as well as a method to avoid the singularities that exist when using this method. This is shown to decrease the computation time of the geopotential compared to other popular methods. In addition, methods for first and second order propagation are discussed, which decrease the rate at which the full geopotential model needs to be calculated while maintaining the accuracy required of navigation systems. These estimation methods are shown to decrease the necessary computation of the gravity model by multiple orders of magnitude. This method decreases the computation time of the Exploration Upper Stage navigation system geopotential model by potentially an order of magnitude compared with the previous model without affecting the navigation error. A properly implemented geopotential model can have the accuracy of an 8 by 8 model while approaching the computational requirement as using only the J234 coefficients.

Amert, Joel↗