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At least 37 records · Page 2

Computational analysis of methods for reduction of induced drag

The purpose of this effort was to perform a computational flow analysis of a design concept centered around induced drag reduction and tip-vortex energy recovery. The flow model solves the unsteady three-dimensional Euler equations, discretized as a finite-volume method, utilizing a high-resolution approximate Riemann solver for cell interface flux definitions. The numerical scheme is an approximately-factored block LU implicit Newton iterative-refinement method. Multiblock domain decomposition is used to partition the field into an ordered arrangement of blocks. Three configurations are analyzed: a baseline fuselage-wing, a fuselage-wing-nacelle, and a fuselage-wing-nacelle-propfan. Aerodynamic force coefficients, propfan performance coefficients, and flowfield maps are used to qualitatively access design efficacy. Where appropriate, comparisons are made with available experimental data.

Janus, J. M.↗

Accumulated approximation: A new method for structural optimization by iterative improvement

A new method for the solution of non-linear mathematical programming problems in the field of structural optimization is presented. It is an iterative scheme which for each iteration refines the approximation of objective and constraint functions by accumulating the function values of previously visited design points. The method has proven to be competitive for a number of well-known examples of which one is presented here. Furthermore because of the accumulation strategy, the method produces convergence even when the sensitivity analysis is inaccurate.

Rasmussen, John↗

Integration and evaluation of a simulator designed to be used within a dynamic prototyping environment

The human computer interface (HCI) prototyping environment is designed to allow developers to rapidly prototype systems so that the interface and functionality of a system can be evaluated and iteratively refined early in the development process. This keeps development costs down by modifying the interface during the requirements definition phase, thus minimizing changes that need to be made during and after flight code development. Problems occur within a system when the user interface is not adequately developed and when designers and developers have an incomplete understanding of the system requirements. A process has been developed for prototyping on-board payload displays for Space Station Freedom. This prototyping process consists of five phases: identification of known requirements, analysis of the requirements, development of a formal design representation and specification, development of the prototype, and evaluation of the prototype. The actual development of the prototype involves prototyping the displays, developing a low fidelity simulator, building of an interface (or communication) between the displays and the simulator, integration of these components, and testing to ensure that the interface does what the developer expects. This research integrates and evaluates a software tool which has been developed to serve as a simulator within the prototyping environment. The tool is being evaluated to determine whether or not it meets the basic requirements which are needed for a low fidelity simulator within this environment. In order to evaluate the architecture and its components, a human computer interface for and a simulator of an automobile have been developed as a prototype. The individual components (i.e., the interface and simulator) have been developed, and the current research was designed to integrate and test the complete working system within the prototyping environment. The following sections will describe the architecture and components of the rapid prototyping environment, the development of a system to assess the environment, and the integration and evaluation of PERCNET.

Moore, Loretta A.↗

Cooperative solutions coupling a geometry engine and adaptive solver codes

Follow-on work has progressed in using Aero Grid and Paneling System (AGPS), a geometry and visualization system, as a dynamic real time geometry monitor, manipulator, and interrogator for other codes. In particular, AGPS has been successfully coupled with adaptive flow solvers which iterate, refining the grid in areas of interest, and continuing on to a solution. With the coupling to the geometry engine, the new grids represent the actual geometry much more accurately since they are derived directly from the geometry and do not use refits to the first-cut grids. Additional work has been done with design runs where the geometric shape is modified to achieve a desired result. Various constraints are used to point the solution in a reasonable direction which also more closely satisfies the desired results. Concepts and techniques are presented, as well as examples of sample case studies. Issues such as distributed operation of the cooperative codes versus running all codes locally and pre-calculation for performance are discussed. Future directions are considered which will build on these techniques in light of changing computer environments.

Dickens, Thomas P.↗

Finite-Strain Fractional-Order Viscoelastic (FOV) Material Models and Numerical Methods for Solving Them

Fraction-order viscoelastic (FOV) material models have been proposed and studied in 1D since the 1930's, and were extended into three dimensions in the 1970's under the assumption of infinitesimal straining. It was not until 1997 that Drozdov introduced the first finite-strain FOV constitutive equations. In our presentation, we shall continue in this tradition by extending the standard, FOV, fluid and solid, material models introduced in 1971 by Caputo and Mainardi into 3D constitutive formula applicable for finite-strain analyses. To achieve this, we generalize both the convected and co-rotational derivatives of tensor fields to fractional order. This is accomplished by defining them first as body tensor fields and then mapping them into space as objective Cartesian tensor fields. Constitutive equations are constructed using both variants for fractional rate, and their responses are contrasted in simple shear. After five years of research and development, we now possess a basic suite of numerical tools necessary to study finite-strain FOV constitutive equations and their iterative refinement into a mature collection of material models. Numerical methods still need to be developed for efficiently solving fraction al-order integrals, derivatives, and differential equations in a finite element setting where such constitutive formulae would need to be solved at each Gauss point in each element of a finite model, which can number into the millions in today's analysis.

Freed, Alan D.↗

An Algorithm for Converting Static Earth Sensor Measurements into Earth Observation Vectors

An algorithm has been developed that converts penetration angles reported by Static Earth Sensors (SESs) into Earth observation vectors. This algorithm allows compensation for variation in the horizon height including that caused by Earth oblateness. It also allows pitch and roll to be computed using any number (greater than 1) of simultaneous sensor penetration angles simplifying processing during periods of Sun and Moon interference. The algorithm computes body frame unit vectors through each SES cluster. It also computes GCI vectors from the spacecraft to the position on the Earth's limb where each cluster detects the Earth's limb. These body frame vectors are used as sensor observation vectors and the GCI vectors are used as reference vectors in an attitude solution. The attitude, with the unobservable yaw discarded, is iteratively refined to provide the Earth observation vector solution.

Harman, R.↗

Hardware Implementation of Serially Concatenated PPM Decoder

A prototype decoder for a serially concatenated pulse position modulation (SCPPM) code has been implemented in a field-programmable gate array (FPGA). At the time of this reporting, this is the first known hardware SCPPM decoder. The SCPPM coding scheme, conceived for free-space optical communications with both deep-space and terrestrial applications in mind, is an improvement of several dB over the conventional Reed-Solomon PPM scheme. The design of the FPGA SCPPM decoder is based on a turbo decoding algorithm that requires relatively low computational complexity while delivering error-rate performance within approximately 1 dB of channel capacity. The SCPPM encoder consists of an outer convolutional encoder, an interleaver, an accumulator, and an inner modulation encoder (more precisely, a mapping of bits to PPM symbols). Each code is describable by a trellis (a finite directed graph). The SCPPM decoder consists of an inner soft-in-soft-out (SISO) module, a de-interleaver, an outer SISO module, and an interleaver connected in a loop (see figure). Each SISO module applies the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm to compute a-posteriori bit log-likelihood ratios (LLRs) from apriori LLRs by traversing the code trellis in forward and backward directions. The SISO modules iteratively refine the LLRs by passing the estimates between one another much like the working of a turbine engine. Extrinsic information (the difference between the a-posteriori and a-priori LLRs) is exchanged rather than the a-posteriori LLRs to minimize undesired feedback. All computations are performed in the logarithmic domain, wherein multiplications are translated into additions, thereby reducing complexity and sensitivity to fixed-point implementation roundoff errors. To lower the required memory for storing channel likelihood data and the amounts of data transfer between the decoder and the receiver, one can discard the majority of channel likelihoods, using only the remainder in operation of the decoder. This is accomplished in the receiver by transmitting only a subset consisting of the likelihoods that correspond to time slots containing the largest numbers of observed photons during each PPM symbol period. The assumed number of observed photons in the remaining time slots is set to the mean of a noise slot. In low background noise, the selection of a small subset in this manner results in only negligible loss. Other features of the decoder design to reduce complexity and increase speed include (1) quantization of metrics in an efficient procedure chosen to incur no more than a small performance loss and (2) the use of the max-star function that allows sum of exponentials to be computed by simple operations that involve only an addition, a subtraction, and a table lookup. Another prominent feature of the design is a provision for access to interleaver and de-interleaver memory in a single clock cycle, eliminating the multiple clock-cycle latency characteristic of prior interleaver and de-interleaver designs.

Moision, Bruce↗

Reaction Wheel Disturbance Model Extraction Software - RWDMES

The RWDMES is a tool for modeling the disturbances imparted on spacecraft by spinning reaction wheels. Reaction wheels are usually the largest disturbance source on a precision pointing spacecraft, and can be the dominating source of pointing error. Accurate knowledge of the disturbance environment is critical to accurate prediction of the pointing performance. In the past, it has been difficult to extract an accurate wheel disturbance model since the forcing mechanisms are difficult to model physically, and the forcing amplitudes are filtered by the dynamics of the reaction wheel. RWDMES captures the wheel-induced disturbances using a hybrid physical/empirical model that is extracted directly from measured forcing data. The empirical models capture the tonal forces that occur at harmonics of the spin rate, and the broadband forces that arise from random effects. The empirical forcing functions are filtered by a physical model of the wheel structure that includes spin-rate-dependent moments (gyroscopic terms). The resulting hybrid model creates a highly accurate prediction of wheel-induced forces. It accounts for variation in disturbance frequency, as well as the shifts in structural amplification by the whirl modes, as the spin rate changes. This software provides a point-and-click environment for producing accurate models with minimal user effort. Where conventional approaches may take weeks to produce a model of variable quality, RWDMES can create a demonstrably high accuracy model in two hours. The software consists of a graphical user interface (GUI) that enables the user to specify all analysis parameters, to evaluate analysis results and to iteratively refine the model. Underlying algorithms automatically extract disturbance harmonics, initialize and tune harmonic models, and initialize and tune broadband noise models. The component steps are described in the RWDMES user s guide and include: converting time domain data to waterfall PSDs (power spectral densities); converting PSDs to order analysis data; extracting harmonics; initializing and simultaneously tuning a harmonic model and a wheel structural model; initializing and tuning a broadband model; and verifying the harmonic/broadband/structural model against the measurement data. Functional operation is through a MATLAB GUI that loads test data, performs the various analyses, plots evaluation data for assessment and refinement of analysis parameters, and exports the data to documentation or downstream analysis code. The harmonic models are defined as specified functions of frequency, typically speed-squared. The reaction wheel structural model is realized as mass, damping, and stiffness matrices (typically from a finite element analysis package) with the addition of a gyroscopic forcing matrix. The broadband noise model is realized as a set of speed-dependent filters. The tuning of the combined model is performed using nonlinear least squares techniques. RWDMES is implemented as a MATLAB toolbox comprising the Fit Manager for performing the model extraction, Data Manager for managing input data and output models, the Gyro Manager for modifying wheel structural models, and the Harmonic Editor for evaluating and tuning harmonic models. This software was validated using data from Goodrich E wheels, and from GSFC Lunar Reconnaissance Orbiter (LRO) wheels. The validation testing proved that RWDMES has the capability to extract accurate disturbance models from flight reaction wheels with minimal user effort.

Blaurock, Carl↗

Progressive Classification Using Support Vector Machines

An algorithm for progressive classification of data, analogous to progressive rendering of images, makes it possible to compromise between speed and accuracy. This algorithm uses support vector machines (SVMs) to classify data. An SVM is a machine learning algorithm that builds a mathematical model of the desired classification concept by identifying the critical data points, called support vectors. Coarse approximations to the concept require only a few support vectors, while precise, highly accurate models require far more support vectors. Once the model has been constructed, the SVM can be applied to new observations. The cost of classifying a new observation is proportional to the number of support vectors in the model. When computational resources are limited, an SVM of the appropriate complexity can be produced. However, if the constraints are not known when the model is constructed, or if they can change over time, a method for adaptively responding to the current resource constraints is required. This capability is particularly relevant for spacecraft (or any other real-time systems) that perform onboard data analysis. The new algorithm enables the fast, interactive application of an SVM classifier to a new set of data. The classification process achieved by this algorithm is characterized as progressive because a coarse approximation to the true classification is generated rapidly and thereafter iteratively refined. The algorithm uses two SVMs: (1) a fast, approximate one and (2) slow, highly accurate one. New data are initially classified by the fast SVM, producing a baseline approximate classification. For each classified data point, the algorithm calculates a confidence index that indicates the likelihood that it was classified correctly in the first pass. Next, the data points are sorted by their confidence indices and progressively reclassified by the slower, more accurate SVM, starting with the items most likely to be incorrectly classified. The user can halt this reclassification process at any point, thereby obtaining the best possible result for a given amount of computation time. Alternatively, the results can be displayed as they are generated, providing the user with real-time feedback about the current accuracy of classification.

Wagstaff, Kiri↗

Single-Command Approach and Instrument Placement by a Robot on a Target

AUTOAPPROACH is a computer program that enables a mobile robot to approach a target autonomously, starting from a distance of as much as 10 m, in response to a single command. AUTOAPPROACH is used in conjunction with (1) software that analyzes images acquired by stereoscopic cameras aboard the robot and (2) navigation and path-planning software that utilizes odometer readings along with the output of the image-analysis software. Intended originally for application to an instrumented, wheeled robot (rover) in scientific exploration of Mars, AUTOAPPROACH could be adapted to terrestrial applications, notably including the robotic removal of land mines and other unexploded ordnance. A human operator generates the approach command by selecting the target in images acquired by the robot cameras. The approach path consists of multiple legs. Feature points are derived from images that contain the target and are thereafter tracked to correct odometric errors and iteratively refine estimates of the position and orientation of the robot relative to the target on successive legs. The approach is terminated when the robot attains the position and orientation required for placing a scientific instrument at the target. The workspace of the robot arm is then autonomously checked for self/terrain collisions prior to the deployment of the scientific instrument onto the target.

Huntsberger, Terrance↗

Enabling Metric-Based Mesh Adaptation for Advanced Compressible Flow Simulations Using US3D

This work presents a metric-based mesh adaptation capability added to the flow solver US3D that is used to improve the accuracy of atmospheric entry simulations. Flow past an entry vehicle in free-flight is usually unsteady and characterized by anisotropic flow features like a strong bow shock and small-scale isotropic flow features like turbulent eddies in the wake. The results presented in this paper demonstrate that we are currently able to iteratively apply metric-based mesh adaptation to improve the solution accuracy by simultaneously targeting isotropic and anisotropic flow features. A simplified workflow of running CFD simulations for entry vehicles is presented where we start off with a coarse initial mesh to compute a first approximation of the flow. This coarse mesh is then iteratively refined using the newly developed metric to improve the resolution of the CFD computation. Furthermore, we demonstrate that we are able to adapt large mesh sizes in a timely fashion which allows us to start thinking about applying this adaptive framework to unsteady atmospheric entry CFD simulations.

Mesh adaptation↗

RadLab: A Comprehensive Database and Graphical and Programming Interfaces for Biologically Relevant Space Radiation Data

RadLab, a new component of the NASA Open Science Data Repository (OSDR), comprises a database of radiation measurements relevant to space biology, and visual and programmatic interfaces for interrogation and retrieval of these data. The attributes of data available through RadLab include spacecraft, types of radiation sensing instruments, locations within the spacecraft (e.g. modules of the ISS), associated celestial bodies, trajectories, and spacecraft coordinates. The application programming interface (API) implements a request syntax for retrieval of timestamped data filtered by various combinations of such attributes; the graphical user interface (GUI) extends this functionality with visualizations, such as spacecraft schematics, time series plots, geospatial visualizations, and provides easy means to iteratively refine search parameters, inspect the data on the fly, and download target subsets of these data. The release of RadLab currently available to the public contains datasets provided by US and international collaborators and focuses on data recorded on the ISS. Investigators from multiple countries, including the US, Canada, Germany, Bulgaria, Hungary, Italy, Japan, Russia and the Czech Republic, have committed to provide data from their instruments in and beyond low Earth orbit; RadLab will also soon expand to include past (e.g. Shuttle and Mir) and future (e.g. Artemis) data. RadLab will provide a comprehensive, dynamic compendium of space radiation data, enabling the scientific community to perform analyses of data from multiple detectors and to determine the radiation environment of research missions and experiments, both via programmatic retrieval of these data and through the graphical analysis toolkit. The RadLab Working Group has been formed to foster collaborations among data contributors and users, to identify data sources, to put in place standards for data harmonization, and to guide the development of the platform, with the goal to establish the use of RadLab in space radiation research and to advance our understanding of the space radiation environment in human habitats.

radiation↗

RadLab: A Comprehensive Database and Graphical and Programming Interfaces for Space Radiation Data

RadLab, a component of the NASA Open Science Data Repository (OSDR), is a database of radiation measurements from multiple instruments and spacecraft that provides visual and programmatic interfaces for interrogation and retrieval of these data. The attributes of data available through RadLab include spacecraft, types of radiation sensing instruments, locations within the spacecraft (e.g. ISS modules), associated celestial bodies, trajectories, and spacecraft coordinates; the primary type of data is the absorbed dose rate, as well as flux and dose equivalent rate where available. The application programming interface (API) implements a request syntax for retrieval of timestamped data filtered by various combinations of such attributes; the graphical user interface (GUI) extends this functionality with visualizations (time series plots, comparison plots, geospatial visualizations) which provide easy means to assess data availability, iteratively refine search parameters, interactively inspect the data, and export target data subsets. Datasets are continuously being added to the RadLab database as part of the rolling release process. Investigators from multiple countries, including the US, Canada, Germany, Bulgaria, Hungary, Italy, Japan, Russia and the Czech Republic, have committed to provide data from their instruments in and beyond low Earth orbit. The current release contains datasets provided by US and international collaborators and includes readings from multiple modules of the ISS, the BioSentinel CubeSat, Chang’e 4, the Lunar Reconnaissance Orbiter, the ExoMars Orbiter, and the Curiosity rover. Datasets are associated with respective RadLab knowledgebase articles which include instrument descriptions and provide bibliographical references. RadLab aims to provide a comprehensive, dynamic compendium of space radiation data, enabling the scientific community to perform analyses of data from multiple detectors and to determine the radiation environment of research missions and experiments. Some of its applications include inference of absorbed radiation dose for NASA GeneLab payloads, and training predictive models as part of the 2024 FDL-X challenge. The platform is actively expanding and seeking additional data, with plans to also cover past (e.g. Shuttle, Mir) and future (e.g. Artemis) missions. The RadLab Working Group has been created to aid in this process as well as to foster collaborations among data contributors and users, to develop standards for data harmonization, and to guide the development of the platform, with the goal to establish the use of RadLab in space radiation research and to advance our understanding of the radiation environment in outer space.

Kirill Grigorev↗

RadLab: A Comprehensive Database and Graphical and Programming Interfaces for Biologically Relevant Space Radiation Data

RadLab, a new component of the NASA Open Science Data Repository (OSDR), comprises a database of radiation measurements relevant to space biology, and visual and programmatic interfaces for interrogation and retrieval of these data. The attributes of data available through RadLab include spacecraft, types of radiation sensing instruments, locations within the spacecraft (e.g. modules of the ISS), associated celestial bodies, trajectories, and spacecraft coordinates. The application programming interface (API) implements a request syntax for retrieval of timestamped data filtered by various combinations of such attributes; the graphical user interface (GUI) extends this functionality with visualizations, such as spacecraft schematics, time series plots, geospatial visualizations, and provides easy means to iteratively refine search parameters, inspect the data on the fly, and download target subsets. The release of RadLab currently available to the public contains datasets provided by US and international collaborators and focuses on data recorded on the ISS. Investigators from multiple countries, including the US, Canada, Germany, Bulgaria, Hungary, Italy, Japan, Russia and the Czech Republic, have committed to provide data from their instruments in and beyond low Earth orbit; RadLab will also soon expand to include past (e.g. Shuttle and Mir) and future (e.g. Artemis) data. RadLab will provide a comprehensive, dynamic compendium of space radiation data, enabling the scientific community to perform analyses of data from multiple detectors and to determine the radiation environment of research missions and experiments. The RadLab Working Group has been formed to foster collaborations among data contributors and users, to identify data sources, to put in place standards for data harmonization, and to guide the development of the platform, with the goal to establish the use of RadLab in space radiation research and to advance our understanding of the space radiation environment in human habitats.

database↗

RadLab: A Comprehensive Database and Graphical and Programming Interfaces for Space Radiation Data

RadLab, a component of the NASA Open Science Data Repository (OSDR), is a database of radiation measurements from multiple instruments and spacecraft that provides visual and programmatic interfaces for interrogation and retrieval of these data. The attributes of data available through RadLab include spacecraft, types of radiation sensing instruments, locations within the spacecraft (e.g. ISS modules), associated celestial bodies, trajectories, and spacecraft coordinates; the primary type of data is the absorbed dose rate, as well as flux and dose equivalent rate where available. The application programming interface (API) implements a request syntax for retrieval of timestamped data filtered by various combinations of such attributes; the graphical user interface (GUI) extends this functionality with visualizations (time series plots, comparison plots, geospatial visualizations) which provide easy means to assess data availability, iteratively refine search parameters, interactively inspect the data, and export target data subsets. Datasets are continuously being added to the RadLab database as part of the rolling release process. Investigators from multiple countries, including the US, Canada, Germany, Bulgaria, Hungary, Italy, Japan, Russia and the Czech Republic, have committed to provide data from their instruments in and beyond low Earth orbit. The current release contains datasets provided by US and international collaborators and includes readings from multiple modules of the ISS, the BioSentinel CubeSat, Chang’e 4, the Lunar Reconnaissance Orbiter, the ExoMars Orbiter, and the Curiosity rover. Datasets are associated with respective RadLab knowledgebase articles which include instrument descriptions and provide bibliographical references. RadLab aims to provide a comprehensive, dynamic compendium of space radiation data, enabling the scientific community to perform analyses of data from multiple detectors and to determine the radiation environment of research missions and experiments. Some of its applications include inference of absorbed radiation dose for NASA GeneLab payloads, and training predictive models as part of the 2024 FDL-X challenge. The platform is actively expanding and seeking additional data, with plans to also cover past (e.g. Shuttle, Mir) and future (e.g. Artemis) missions. The RadLab Working Group has been created to aid in this process as well as to foster collaborations among data contributors and users, to develop standards for data harmonization, and to guide the development of the platform, with the goal to establish the use of RadLab in space radiation research and to advance our understanding of the radiation environment in outer space.

Kirill Grigorev↗

Composite panel development at JPL

Parametric computer studies can be use in a cost effective manner to determine optimized composite mirror panel designs. An InterDisciplinary computer Model (IDM) was created to aid in the development of high precision reflector panels for LDR. The materials properties, thermal responses, structural geometries, and radio/optical precision are synergistically analyzed for specific panel designs. Promising panels designs are fabricated and tested so that comparison with panel test results can be used to verify performance prediction models and accommodate design refinement. The iterative approach of computer design and model refinement with performance testing and materials optimization has shown good results for LDR panels.

Mcelroy, Paul↗

An interaction algorithm for prediction of mean and fluctuating velocities in two-dimensional aerodynamic wake flows

A theoretical analysis is presented yielding sets of partial differential equations for determination of turbulent aerodynamic flowfields in the vicinity of an airfoil trailing edge. A four phase interaction algorithm is derived to complete the analysis. Following input, the first computational phase is an elementary viscous corrected two dimensional potential flow solution yielding an estimate of the inviscid-flow induced pressure distribution. Phase C involves solution of the turbulent two dimensional boundary layer equations over the trailing edge, with transition to a two dimensional parabolic Navier-Stokes equation system describing the near-wake merging of the upper and lower surface boundary layers. An iteration provides refinement of the potential flow induced pressure coupling to the viscous flow solutions. The final phase is a complete two dimensional Navier-Stokes analysis of the wake flow in the vicinity of a blunt-bases airfoil. A finite element numerical algorithm is presented which is applicable to solution of all partial differential equation sets of inviscid-viscous aerodynamic interaction algorithm. Numerical results are discussed.

Baker, A. J.↗

Preliminary calibration plan for the Advanced Particles and Field Observatory (APAFO) magnetometer experiment

Prelaunch and postlaunch calibration plans for the APAFO magnetometer experiment are presented. A study of tradeoffs between boom length and spacecraft field is described; the results are summarized. The prelaunch plan includes: calibration of the Vector Fluxgate Magnetometer (VFM), Star Sensors, and Scalar Helium Magnetometer (SHM); optical bench integration; and acquisition of basic spacecraft field data. Postlaunch calibration has two phases. In phase one, SHM data are used to calibrate the VFM, total vector magnetic field data are used to calibrate a physical model of the spacecraft field, and both calibrations are refined by iteration. In phase two, corrected vector data are transformed into geocentric coordinates, previously undetected spacecraft fields are isolated, and initial geomagnetic field models are computed. Provided the SHM is accurate to the required 1.0 nT and can be used to calibrate the VFM to the required 3.0- nT accuracy, the tradeoff study indicates that a 12 m boom and a spacecraft field model uncertainty of 5 percent together allow the 1.0 nT spacecraft field error requirement to be met.

Voorhies, C. V.↗