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Set covering algorithm, a subprogram of the scheduling algorithm for mission planning and logistic evaluation

A computer program using Lemke, Salkin and Spielberg's Set Covering Algorithm (SCA) to optimize a traffic model problem in the Scheduling Algorithm for Mission Planning and Logistics Evaluation (SAMPLE) was documented. SCA forms a submodule of SAMPLE and provides for input and output, subroutines, and an interactive feature for performing the optimization and arranging the results in a readily understandable form for output.

Chang, H.

Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes: An Iterative Decoding Algorithm for Linear Block Codes Based on a Low-Weight Trellis Search - Part 3

For long linear block codes, maximum likelihood decoding based on full code trellises would be very hard to implement if not impossible. In this case, we may wish to trade error performance for the reduction in decoding complexity. Sub-optimum soft-decision decoding of a linear block code based on a low-weight sub-trellis can be devised to provide an effective trade-off between error performance and decoding complexity. This chapter presents such a suboptimal decoding algorithm for linear block codes. This decoding algorithm is iterative in nature and based on an optimality test. It has the following important features: (1) a simple method to generate a sequence of candidate code-words, one at a time, for test; (2) a sufficient condition for testing a candidate code-word for optimality; and (3) a low-weight sub-trellis search for finding the most likely (ML) code-word.

Lin, Shu

Obstacle Detection Algorithms for Aircraft Navigation: Performance Characterization of Obstacle Detection Algorithms for Aircraft Navigation

The research reported here is a part of NASA's Synthetic Vision System (SVS) project for the development of a High Speed Civil Transport Aircraft (HSCT). One of the components of the SVS is a module for detection of potential obstacles in the aircraft's flight path by analyzing the images captured by an on-board camera in real-time. Design of such a module includes the selection and characterization of robust, reliable, and fast techniques and their implementation for execution in real-time. This report describes the results of our research in realizing such a design. It is organized into three parts. Part I. Data modeling and camera characterization; Part II. Algorithms for detecting airborne obstacles; and Part III. Real time implementation of obstacle detection algorithms on the Datacube MaxPCI architecture. A list of publications resulting from this grant as well as a list of relevant publications resulting from prior NASA grants on this topic are presented.

Kasturi, Rangachar

SNPP VIIRS RSB on-orbit radiometric calibration algorithms Version 2.0 and the performances. Part I: The algorithms

The first VIIRS instrument is aboard the Suomi National Polar-orbiting Partnership satellite. The instrument has 14 reflective solar bands (RSBs) to passively collect photons reflected from the Earth surface in the design wavelengths from 412 to 2250 nm. The instrument uses a solar diffuser (SD) to radiometrically calibrate its RSBs. When lit by the Sun through an attenuation screen (the SD screen), the SD diffusely reflects off the incident sunlight to act as a radiance source for the calibration. An onboard solar diffuser stability monitor (SDSM) yields the on-orbit change of the SD bidirectional reflectance distribution function (BRDF) by comparing the signal strength from the SD with that from the Sun attenuated by another attenuation screen (the SDSM screen). Complications arise due to the discovery that the on-orbit change of the BRDF is angle dependent. Additionally, the SDSM does not cover the wavelengths for the short-wave infrared bands in the RSBs. Furthermore, satellite yaw maneuvers were performed in the early mission to yield data for refining the prelaunch SDSM screen relative effective transmittance and the relative product of the SD screen transmittance and the BRDF at the mission start. But the yaw maneuver data are coarse in the solar azimuth angles and thus are unable to yield accurate values between the measurement angles. Over the years of performing on-orbit radiometric calibration through the SD for the VIIRSRSBs, we have developed several highly effective calibration algorithms to address the issues mentioned above. This paper reviews these algorithms.

Ning Lei

Enhancing accessibility and usability of Algorithm Theoretical Basis Documents through the Algorithm Publication Tool

Effective communication of scientific theories is crucial for transforming raw instrument data into valuable Earth observation products. The NASA Earth science data community disseminates this knowledge through Algorithm Theoretical Basis Documents (ATBDs). Historically, these documents lacked a standardized format, were designed for human readability rather than machine interpretation, and were challenging to locate due to the absence of a centralized repository. The Algorithm Publication Tool (APT) transforms how ATBD content is presented, simplifying the processes of creating, updating, and locating these documents. APT offers authors the option to use its user-friendly cloud-based interface or standardized templates for ATBD development. The primary advantage of the interface is its capability to manage the entire ATBD creation process within a single environment, ensuring comprehensive tracking of all activities and facilitating user tasks. Conversely, the use of standardized ATBD templates allows users to create documents using familiar tools like Google Docs, Microsoft Word, or Overleaf for LaTeX. APT also provides a centralized repository, enabling easy search and discovery of published ATBDs. This presentation showcases APT's functionalities, illustrates its contributions to advancing open science, and highlights potential benefits for broader community adoption.

Bradley Baker

Scheduling Algorithm for Mission Planning and Logistics Evaluation (SAMPLE). Volume 3: The GREEDY algorithm

The functional specifications, functional design and flow, and the program logic of the GREEDY computer program are described. The GREEDY program is a submodule of the Scheduling Algorithm for Mission Planning and Logistics Evaluation (SAMPLE) program and has been designed as a continuation of the shuttle Mission Payloads (MPLS) program. The MPLS uses input payload data to form a set of feasible payload combinations; from these, GREEDY selects a subset of combinations (a traffic model) so all payloads can be included without redundancy. The program also provides the user a tutorial option so that he can choose an alternate traffic model in case a particular traffic model is unacceptable.

Dupnick, E.

Genetic Algorithm for Optimization: Preprocessor and Algorithm

Genetic algorithm (GA) inspired by Darwin's theory of evolution and employed to solve optimization problems - unconstrained or constrained - uses an evolutionary process. A GA has several parameters such the population size, search space, crossover and mutation probabilities, and fitness criterion. These parameters are not universally known/determined a priori for all problems. Depending on the problem at hand, these parameters need to be decided such that the resulting GA performs the best. We present here a preprocessor that achieves just that, i.e., it determines, for a specified problem, the foregoing parameters so that the consequent GA is a best for the problem. We stress also the need for such a preprocessor both for quality (error) and for cost (complexity) to produce the solution. The preprocessor includes, as its first step, making use of all the information such as that of nature/character of the function/system, search space, physical/laboratory experimentation (if already done/available), and the physical environment. It also includes the information that can be generated through any means - deterministic/nondeterministic/graphics. Instead of attempting a solution of the problem straightway through a GA without having/using the information/knowledge of the character of the system, we would do consciously a much better job of producing a solution by using the information generated/created in the very first step of the preprocessor. We, therefore, unstintingly advocate the use of a preprocessor to solve a real-world optimization problem including NP-complete ones before using the statistically most appropriate GA. We also include such a GA for unconstrained function optimization problems.

Sen, S. K.

Modeling in the State Flow Environment to Support Launch Vehicle Verification Testing for Mission and Fault Management Algorithms in the NASA Space Launch System

Analysis methods and testing processes are essential activities in the engineering development and verification of the National Aeronautics and Space Administration's (NASA) new Space Launch System (SLS). Central to mission success is reliable verification of the Mission and Fault Management (M&FM) algorithms for the SLS launch vehicle (LV) flight software. This is particularly difficult because M&FM algorithms integrate and operate LV subsystems, which consist of diverse forms of hardware and software themselves, with equally diverse integration from the engineering disciplines of LV subsystems. M&FM operation of SLS requires a changing mix of LV automation. During pre-launch the LV is primarily operated by the Kennedy Space Center (KSC) Ground Systems Development and Operations (GSDO) organization with some LV automation of time-critical functions, and much more autonomous LV operations during ascent that have crucial interactions with the Orion crew capsule, its astronauts, and with mission controllers at the Johnson Space Center. M&FM algorithms must perform all nominal mission commanding via the flight computer to control LV states from pre-launch through disposal and also address failure conditions by initiating autonomous or commanded aborts (crew capsule escape from the failing LV), redundancy management of failing subsystems and components, and safing actions to reduce or prevent threats to ground systems and crew. To address the criticality of the verification testing of these algorithms, the NASA M&FM team has utilized the State Flow environment6 (SFE) with its existing Vehicle Management End-to-End Testbed (VMET) platform which also hosts vendor-supplied physics-based LV subsystem models. The human-derived M&FM algorithms are designed and vetted in Integrated Development Teams composed of design and development disciplines such as Systems Engineering, Flight Software (FSW), Safety and Mission Assurance (S&MA) and major subsystems and vehicle elements such as Main Propulsion Systems (MPS), boosters, avionics, Guidance, Navigation, and Control (GN&C), Thrust Vector Control (TVC), liquid engines, and the astronaut crew office. Since the algorithms are realized using model-based engineering (MBE) methods from a hybrid of the Unified Modeling Language (UML) and Systems Modeling Language (SysML), SFE methods are a natural fit to provide an in depth analysis of the interactive behavior of these algorithms with the SLS LV subsystem models. For this, the M&FM algorithms and the SLS LV subsystem models are modeled using constructs provided by Matlab which also enables modeling of the accompanying interfaces providing greater flexibility for integrated testing and analysis, which helps forecast expected behavior in forward VMET integrated testing activities. In VMET, the M&FM algorithms are prototyped and implemented using the same C++ programming language and similar state machine architectural concepts used by the FSW group. Due to the interactive complexity of the algorithms, VMET testing thus far has verified all the individual M&FM subsystem algorithms with select subsystem vendor models but is steadily progressing to assessing the interactive behavior of these algorithms with LV subsystems, as represented by subsystem models. The novel SFE applications has proven to be useful for quick look analysis into early integrated system behavior and assessment of the M&FM algorithms with the modeled LV subsystems. This early MBE analysis generates vital insight into the integrated system behaviors, algorithm sensitivities, design issues, and has aided in the debugging of the M&FM algorithms well before full testing can begin in more expensive, higher fidelity but more arduous environments such as VMET, FSW testing, and the Systems Integration Lab7 (SIL). SFE has exhibited both expected and unexpected behaviors in nominal and off nominal test cases prior to full VMET testing. In many findings, these behavioral characteristics were used to correct the M&FM algorithms, enable better test coverage, and develop more effective test cases for each of the LV subsystems. This has improved the fidelity of testing and planning for the next generation of M&FM algorithms as the SLS program evolves from non-crewed to crewed flight, impacting subsystem configurations and the M&FM algorithms that control them. SFE analysis has improved robustness and reliability of the M&FM algorithms by revealing implementation errors and documentation inconsistencies. It is also improving planning efficiency for future VMET testing of the M&FM algorithms hosted in the LV flight computers, further reducing risk for the SLS launch infrastructure, the SLS LV, and most importantly the crew.

Trevino, Luis

A Vehicle Management End-to-End Testing and Analysis Platform for Validation of Mission and Fault Management Algorithms to Reduce Risk for NASAs Space Launch System

The engineering development of the National Aeronautics and Space Administration's (NASA) new Space Launch System (SLS) requires cross discipline teams with extensive knowledge of launch vehicle subsystems, information theory, and autonomous algorithms dealing with all operations from pre-launch through on orbit operations. The nominal and off-nominal characteristics of SLS's elements and subsystems must be understood and matched with the autonomous algorithm monitoring and mitigation capabilities for accurate control and response to abnormal conditions throughout all vehicle mission flight phases, including precipitating safing actions and crew aborts. This presents a large and complex systems engineering challenge, which is being addressed in part by focusing on the specific subsystems involved in the handling of off-nominal mission and fault tolerance with response management. Using traditional model-based system and software engineering design principles from the Unified Modeling Language (UML) and Systems Modeling Language (SysML), the Mission and Fault Management (M&FM) algorithms for the vehicle are crafted and vetted in Integrated Development Teams (IDTs) composed of multiple development disciplines such as Systems Engineering (SE), Flight Software (FSW), Safety and Mission Assurance (S&MA) and the major subsystems and vehicle elements such as Main Propulsion Systems (MPS), boosters, avionics, Guidance, Navigation, and Control (GNC), Thrust Vector Control (TVC), and liquid engines. These model-based algorithms and their development lifecycle from inception through FSW certification are an important focus of SLS's development effort to further ensure reliable detection and response to off-nominal vehicle states during all phases of vehicle operation from pre-launch through end of flight. To test and validate these M&FM algorithms a dedicated test-bed was developed for full Vehicle Management End-to-End Testing (VMET). For addressing fault management (FM) early in the development lifecycle for the SLS program, NASA formed the M&FM team as part of the Integrated Systems Health Management and Automation Branch under the Spacecraft Vehicle Systems Department at the Marshall Space Flight Center (MSFC). To support the development of the FM algorithms, the VMET developed by the M&FM team provides the ability to integrate the algorithms, perform test cases, and integrate vendor-supplied physics-based launch vehicle (LV) subsystem models. Additionally, the team has developed processes for implementing and validating the M&FM algorithms for concept validation and risk reduction. The flexibility of the VMET capabilities enables thorough testing of the M&FM algorithms by providing configurable suites of both nominal and off-nominal test cases to validate the developed algorithms utilizing actual subsystem models such as MPS, GNC, and others. One of the principal functions of VMET is to validate the M&FM algorithms and substantiate them with performance baselines for each of the target vehicle subsystems in an independent platform exterior to the flight software test and validation processes. In any software development process there is inherent risk in the interpretation and implementation of concepts from requirements and test cases into flight software compounded with potential human errors throughout the development and regression testing lifecycle. Risk reduction is addressed by the M&FM group but in particular by the Analysis Team working with other organizations such as S&MA, Structures and Environments, GNC, Orion, Crew Office, Flight Operations, and Ground Operations by assessing performance of the M&FM algorithms in terms of their ability to reduce Loss of Mission (LOM) and Loss of Crew (LOC) probabilities. In addition, through state machine and diagnostic modeling, analysis efforts investigate a broader suite of failure effects and associated detection and responses to be tested in VMET to ensure reliable failure detection, and confirm responses do not create additional risks or cause undesired states through interactive dynamic effects with other algorithms and systems. VMET further contributes to risk reduction by prototyping and exercising the M&FM algorithms early in their implementation and without any inherent hindrances such as meeting FSW processor scheduling constraints due to their target platform - the ARINC 6535-partitioned Operating System, resource limitations, and other factors related to integration with other subsystems not directly involved with M&FM such as telemetry packing and processing. The baseline plan for use of VMET encompasses testing the original M&FM algorithms coded in the same C++ language and state machine architectural concepts as that used by FSW. This enables the development of performance standards and test cases to characterize the M&FM algorithms and sets a benchmark from which to measure their effectiveness and performance in the exterior FSW development and test processes. This paper is outlined in a systematic fashion analogous to a lifecycle process flow for engineering development of algorithms into software and testing. Section I describes the NASA SLS M&FM context, presenting the current infrastructure, leading principles, methods, and participants. Section II defines the testing philosophy of the M&FM algorithms as related to VMET followed by section III, which presents the modeling methods of the algorithms to be tested and validated in VMET. Its details are then further presented in section IV followed by Section V presenting integration, test status, and state analysis. Finally, section VI addresses the summary and forward directions followed by the appendices presenting relevant information on terminology and documentation.

Trevino, Luis

Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes

A code trellis is a graphical representation of a code, block or convolutional, in which every path represents a codeword (or a code sequence for a convolutional code). This representation makes it possible to implement Maximum Likelihood Decoding (MLD) of a code with reduced decoding complexity. The most well known trellis-based MLD algorithm is the Viterbi algorithm. The trellis representation was first introduced and used for convolutional codes [23]. This representation, together with the Viterbi decoding algorithm, has resulted in a wide range of applications of convolutional codes for error control in digital communications over the last two decades. There are two major reasons for this inactive period of research in this area. First, most coding theorists at that time believed that block codes did not have simple trellis structure like convolutional codes and maximum likelihood decoding of linear block codes using the Viterbi algorithm was practically impossible, except for very short block codes. Second, since almost all of the linear block codes are constructed algebraically or based on finite geometries, it was the belief of many coding theorists that algebraic decoding was the only way to decode these codes. These two reasons seriously hindered the development of efficient soft-decision decoding methods for linear block codes and their applications to error control in digital communications. This led to a general belief that block codes are inferior to convolutional codes and hence, that they were not useful. Chapter 2 gives a brief review of linear block codes. The goal is to provide the essential background material for the development of trellis structure and trellis-based decoding algorithms for linear block codes in the later chapters. Chapters 3 through 6 present the fundamental concepts, finite-state machine model, state space formulation, basic structural properties, state labeling, construction procedures, complexity, minimality, and sectionalization of trellises. Chapter 7 discusses trellis decomposition and subtrellises for low-weight codewords. Chapter 8 first presents well known methods for constructing long powerful codes from short component codes or component codes of smaller dimensions, and then provides methods for constructing their trellises which include Shannon and Cartesian product techniques. Chapter 9 deals with convolutional codes, puncturing, zero-tail termination and tail-biting.Chapters 10 through 13 present various trellis-based decoding algorithms, old and new. Chapter 10 first discusses the application of the well known Viterbi decoding algorithm to linear block codes, optimum sectionalization of a code trellis to minimize computation complexity, and design issues for IC (integrated circuit) implementation of a Viterbi decoder. Then it presents a new decoding algorithm for convolutional codes, named Differential Trellis Decoding (DTD) algorithm. Chapter 12 presents a suboptimum reliability-based iterative decoding algorithm with a low-weight trellis search for the most likely codeword. This decoding algorithm provides a good trade-off between error performance and decoding complexity. All the decoding algorithms presented in Chapters 10 through 12 are devised to minimize word error probability. Chapter 13 presents decoding algorithms that minimize bit error probability and provide the corresponding soft (reliability) information at the output of the decoder. Decoding algorithms presented are the MAP (maximum a posteriori probability) decoding algorithm and the Soft-Output Viterbi Algorithm (SOVA) algorithm. Finally, the minimization of bit error probability in trellis-based MLD is discussed.

Lin, Shu

Highlights of TOMS Version 9 Total Ozone Algorithm

The fundamental basis of TOMS total ozone algorithm was developed some 45 years ago by Dave and Mateer. It was designed to estimate total ozone from satellite measurements of the backscattered UV radiances at few discrete wavelengths in the Huggins ozone absorption band (310-340 nm). Over the years, as the need for higher accuracy in measuring total ozone from space has increased, several improvements to the basic algorithms have been made. They include: better correction for the effects of aerosols and clouds, an improved method to account for the variation in shape of ozone profiles with season, latitude, and total ozone, and a multi-wavelength correction for remaining profile shape errors. These improvements have made it possible to retrieve total ozone with just 3 spectral channels of moderate spectral resolution (approx. 1 nm) with accuracy comparable to state-of-the-art spectral fitting algorithms like DOAS that require high spectral resolution measurements at large number of wavelengths. One of the deficiencies of the TOMS algorithm has been that it doesn't provide an error estimate. This is a particular problem in high latitudes when the profile shape errors become significant and vary with latitude, season, total ozone, and instrument viewing geometry. The primary objective of the TOMS V9 algorithm is to account for these effects in estimating the error bars. This is done by a straightforward implementation of the Rodgers optimum estimation method using a priori ozone profiles and their error covariances matrices constructed using Aura MLS and ozonesonde data. The algorithm produces a vertical ozone profile that contains 1-2.5 pieces of information (degrees of freedom of signal) depending upon solar zenith angle (SZA). The profile is integrated to obtain the total column. We provide information that shows the altitude range in which the profile is best determined by the measurements. One can use this information in data assimilation and analysis. A side benefit of this algorithm is that it is considerably simpler than the present algorithm that uses a database of 1512 profiles to retrieve total ozone. These profiles are tedious to construct and modify. Though conceptually similar to the SBUV V8 algorithm that was developed about a decade ago, the SBUV and TOMS V9 algorithms differ in detail. The TOMS algorithm uses 3 wavelengths to retrieve the profile while the SBUV algorithm uses 6-9 wavelengths, so TOMS provides less profile information. However both algorithms have comparable total ozone information and TOMS V9 can be easily adapted to use additional wavelengths from instruments like GOME, OMI and OMPS to provide better profile information at smaller SZAs. The other significant difference between the two algorithms is that while the SBUV algorithm has been optimized for deriving monthly zonal means by making an appropriate choice of the a priori error covariance matrix, the TOMS algorithm has been optimized for tracking short-term variability using month and latitude dependent covariance matrices.

Bhartia, Pawan

A Vehicle Management End-to-End Testing and Analysis Platform for Validation of Mission and Fault Management Algorithms to Reduce Risk for NASA's Space Launch System

The development of the Space Launch System (SLS) launch vehicle requires cross discipline teams with extensive knowledge of launch vehicle subsystems, information theory, and autonomous algorithms dealing with all operations from pre-launch through on orbit operations. The characteristics of these systems must be matched with the autonomous algorithm monitoring and mitigation capabilities for accurate control and response to abnormal conditions throughout all vehicle mission flight phases, including precipitating safing actions and crew aborts. This presents a large complex systems engineering challenge being addressed in part by focusing on the specific subsystems handling of off-nominal mission and fault tolerance. Using traditional model based system and software engineering design principles from the Unified Modeling Language (UML), the Mission and Fault Management (M&FM) algorithms are crafted and vetted in specialized Integrated Development Teams composed of multiple development disciplines. NASA also has formed an M&FM team for addressing fault management early in the development lifecycle. This team has developed a dedicated Vehicle Management End-to-End Testbed (VMET) that integrates specific M&FM algorithms, specialized nominal and off-nominal test cases, and vendor-supplied physics-based launch vehicle subsystem models. The flexibility of VMET enables thorough testing of the M&FM algorithms by providing configurable suites of both nominal and off-nominal test cases to validate the algorithms utilizing actual subsystem models. The intent is to validate the algorithms and substantiate them with performance baselines for each of the vehicle subsystems in an independent platform exterior to flight software test processes. In any software development process there is inherent risk in the interpretation and implementation of concepts into software through requirements and test processes. Risk reduction is addressed by working with other organizations such as S&MA, Structures and Environments, GNC, Orion, the Crew Office, Flight Operations, and Ground Operations by assessing performance of the M&FM algorithms in terms of their ability to reduce Loss of Mission and Loss of Crew probabilities. In addition, through state machine and diagnostic modeling, analysis efforts investigate a broader suite of failure effects and detection and responses that can be tested in VMET and confirm that responses do not create additional risks or cause undesired states through interactive dynamic effects with other algorithms and systems. VMET further contributes to risk reduction by prototyping and exercising the M&FM algorithms early in their implementation and without any inherent hindrances such as meeting FSW processor scheduling constraints due to their target platform - ARINC 653 partitioned OS, resource limitations, and other factors related to integration with other subsystems not directly involved with M&FM. The plan for VMET encompasses testing the original M&FM algorithms coded in the same C++ language and state machine architectural concepts as that used by Flight Software. This enables the development of performance standards and test cases to characterize the M&FM algorithms and sets a benchmark from which to measure the effectiveness of M&FM algorithms performance in the FSW development and test processes. This paper is outlined in a systematic fashion analogous to a lifecycle process flow for engineering development of algorithms into software and testing. Section I describes the NASA SLS M&FM context, presenting the current infrastructure, leading principles, methods, and participants. Section II defines the testing philosophy of the M&FM algorithms as related to VMET followed by section III, which presents the modeling methods of the algorithms to be tested and validated in VMET. Its details are then further presented in section IV followed by Section V presenting integration, test status, and state analysis. Finally, section VI addresses the summary and forward directions followed by the appendices presenting relevant information on terminology and documentation.

Trevino, Luis

A simple suboptimal least-squares algorithm for attitude determination with multiple sensors

Three-axis attitude determination is equivalent to finding a coordinate transformation matrix which transforms a set of reference vectors fixed in inertial space to a set of measurement vectors fixed in the spacecraft. The attitude determination problem can be expressed as a constrained optimization problem. The constraint is that a coordinate transformation matrix must be proper, real, and orthogonal. A transformation matrix can be thought of as optimal in the least-squares sense if it maps the measurement vectors to the reference vectors with minimal 2-norm errors and meets the above constraint. This constrained optimization problem is known as Wahba's problem. Several algorithms which solve Wahba's problem exactly have been developed and used. These algorithms, while steadily improving, are all rather complicated. Furthermore, they involve such numerically unstable or sensitive operations as matrix determinant, matrix adjoint, and Newton-Raphson iterations. This paper describes an algorithm which minimizes Wahba's loss function, but without the constraint. When the constraint is ignored, the problem can be solved by a straightforward, numerically stable least-squares algorithm such as QR decomposition. Even though the algorithm does not explicitly take the constraint into account, it still yields a nearly orthogonal matrix for most practical cases; orthogonality only becomes corrupted when the sensor measurements are very noisy, on the same order of magnitude as the attitude rotations. The algorithm can be simplified if the attitude rotations are small enough so that the approximation sin(theta) approximately equals theta holds. We then compare the computational requirements for several well-known algorithms. For the general large-angle case, the QR least-squares algorithm is competitive with all other know algorithms and faster than most. If attitude rotations are small, the least-squares algorithm can be modified to run faster, and this modified algorithm is faster than all but a similarly specialized version of the QUEST algorithm. We also introduce a novel measurement averaging technique which reduces the n-measurement case to the two measurement case for our particular application, a star tracker and earth sensor mounted on an earth-pointed geosynchronous communications satellite. Using this technique, many n-measurement problems reduce to less than or equal to 3 measurements; this reduces the amount of required calculation without significant degradation in accuracy. Finally, we present the results of some tests which compare the least-squares algorithm with the QUEST and FOAM algorithms in the two-measurement case. For our example case, all three algorithms performed with similar accuracy.

Brozenec, Thomas F.

Recursive Branching Simulated Annealing Algorithm

This innovation is a variation of a simulated-annealing optimization algorithm that uses a recursive-branching structure to parallelize the search of a parameter space for the globally optimal solution to an objective. The algorithm has been demonstrated to be more effective at searching a parameter space than traditional simulated-annealing methods for a particular problem of interest, and it can readily be applied to a wide variety of optimization problems, including those with a parameter space having both discrete-value parameters (combinatorial) and continuous-variable parameters. It can take the place of a conventional simulated- annealing, Monte-Carlo, or random- walk algorithm. In a conventional simulated-annealing (SA) algorithm, a starting configuration is randomly selected within the parameter space. The algorithm randomly selects another configuration from the parameter space and evaluates the objective function for that configuration. If the objective function value is better than the previous value, the new configuration is adopted as the new point of interest in the parameter space. If the objective function value is worse than the previous value, the new configuration may be adopted, with a probability determined by a temperature parameter, used in analogy to annealing in metals. As the optimization continues, the region of the parameter space from which new configurations can be selected shrinks, and in conjunction with lowering the annealing temperature (and thus lowering the probability for adopting configurations in parameter space with worse objective functions), the algorithm can converge on the globally optimal configuration. The Recursive Branching Simulated Annealing (RBSA) algorithm shares some features with the SA algorithm, notably including the basic principles that a starting configuration is randomly selected from within the parameter space, the algorithm tests other configurations with the goal of finding the globally optimal solution, and the region from which new configurations can be selected shrinks as the search continues. The key difference between these algorithms is that in the SA algorithm, a single path, or trajectory, is taken in parameter space, from the starting point to the globally optimal solution, while in the RBSA algorithm, many trajectories are taken; by exploring multiple regions of the parameter space simultaneously, the algorithm has been shown to converge on the globally optimal solution about an order of magnitude faster than when using conventional algorithms. Novel features of the RBSA algorithm include: 1. More efficient searching of the parameter space due to the branching structure, in which multiple random configurations are generated and multiple promising regions of the parameter space are explored; 2. The implementation of a trust region for each parameter in the parameter space, which provides a natural way of enforcing upper- and lower-bound constraints on the parameters; and 3. The optional use of a constrained gradient- search optimization, performed on the continuous variables around each branch s configuration in parameter space to improve search efficiency by allowing for fast fine-tuning of the continuous variables within the trust region at that configuration point.

Bolcar, Matthew

A Vehicle Management End-to-End Testing and Analysis Platform for Validation of Mission and Fault Management Algorithms to Reduce Risk for NASA's Space Launch System

The engineering development of the new Space Launch System (SLS) launch vehicle requires cross discipline teams with extensive knowledge of launch vehicle subsystems, information theory, and autonomous algorithms dealing with all operations from pre-launch through on orbit operations. The characteristics of these spacecraft systems must be matched with the autonomous algorithm monitoring and mitigation capabilities for accurate control and response to abnormal conditions throughout all vehicle mission flight phases, including precipitating safing actions and crew aborts. This presents a large and complex system engineering challenge, which is being addressed in part by focusing on the specific subsystems involved in the handling of off-nominal mission and fault tolerance with response management. Using traditional model based system and software engineering design principles from the Unified Modeling Language (UML) and Systems Modeling Language (SysML), the Mission and Fault Management (M&FM) algorithms for the vehicle are crafted and vetted in specialized Integrated Development Teams (IDTs) composed of multiple development disciplines such as Systems Engineering (SE), Flight Software (FSW), Safety and Mission Assurance (S&MA) and the major subsystems and vehicle elements such as Main Propulsion Systems (MPS), boosters, avionics, Guidance, Navigation, and Control (GNC), Thrust Vector Control (TVC), and liquid engines. These model based algorithms and their development lifecycle from inception through Flight Software certification are an important focus of this development effort to further insure reliable detection and response to off-nominal vehicle states during all phases of vehicle operation from pre-launch through end of flight. NASA formed a dedicated M&FM team for addressing fault management early in the development lifecycle for the SLS initiative. As part of the development of the M&FM capabilities, this team has developed a dedicated testbed that integrates specific M&FM algorithms, specialized nominal and off-nominal test cases, and vendor-supplied physics-based launch vehicle subsystem models. Additionally, the team has developed processes for implementing and validating these algorithms for concept validation and risk reduction for the SLS program. The flexibility of the Vehicle Management End-to-end Testbed (VMET) enables thorough testing of the M&FM algorithms by providing configurable suites of both nominal and off-nominal test cases to validate the developed algorithms utilizing actual subsystem models such as MPS. The intent of VMET is to validate the M&FM algorithms and substantiate them with performance baselines for each of the target vehicle subsystems in an independent platform exterior to the flight software development infrastructure and its related testing entities. In any software development process there is inherent risk in the interpretation and implementation of concepts into software through requirements and test cases into flight software compounded with potential human errors throughout the development lifecycle. Risk reduction is addressed by the M&FM analysis group working with other organizations such as S&MA, Structures and Environments, GNC, Orion, the Crew Office, Flight Operations, and Ground Operations by assessing performance of the M&FM algorithms in terms of their ability to reduce Loss of Mission and Loss of Crew probabilities. In addition, through state machine and diagnostic modeling, analysis efforts investigate a broader suite of failure effects and associated detection and responses that can be tested in VMET to ensure that failures can be detected, and confirm that responses do not create additional risks or cause undesired states through interactive dynamic effects with other algorithms and systems. VMET further contributes to risk reduction by prototyping and exercising the M&FM algorithms early in their implementation and without any inherent hindrances such as meeting FSW processor scheduling constraints due to their target platform - ARINC 653 partitioned OS, resource limitations, and other factors related to integration with other subsystems not directly involved with M&FM such as telemetry packing and processing. The baseline plan for use of VMET encompasses testing the original M&FM algorithms coded in the same C++ language and state machine architectural concepts as that used by Flight Software. This enables the development of performance standards and test cases to characterize the M&FM algorithms and sets a benchmark from which to measure the effectiveness of M&FM algorithms performance in the FSW development and test processes.

Trevino, Luis

A Linear-Complexity Tensor Butterfly Algorithm for Compressing High-Dimensional Oscillatory Integral Operators

This paper presents a multilevel tensor compression algorithm called tensor butterfly algorithm for efficiently representing large-scale and high-dimensional oscillatory integral operators, including Green's functions for wave equations and integral transforms such as Radon transforms and Fourier transforms. The proposed algorithm leverages a tensor extension of the so-called complementary low-rank property of existing matrix butterfly algorithms. The algorithm partitions the discretized integral operator tensor into subtensors of multiple levels and factorizes each subtensor at the middle level as a Tucker-type interpolative decomposition, whose factor matrices are formed in a multilevel fashion. For a d-dimensional (d > 1) integral operator discretized into a 2d-mode tensor with n2d entries, the overall CPU time and memory requirement scale as O(nd), in stark contrast to the O(nd log n) complexity of existing matrix algorithms such as matrix butterfly algorithms and fast Fourier transforms (FFTs), where n is the number of points per direction. When comparing with other tensor algorithms such as quantized tensor train (QTT), the proposed algorithm also shows superior CPU and memory performance for tensor contraction. Remarkably, the tensor butterfly algorithm can efficiently model high-frequency Green's function interactions between two unit cubes, each spanning 512 wavelengths per direction, which represents problems of scale over 512× larger than that existing butterfly algorithms can handle, with the same amount of computation resources. On the other hand, for a problem representing 64 wavelengths per direction, which is the largest size existing algebraic matrix algorithms can handle, our tensor butterfly algorithm exhibits 200x speedups and 30× memory reduction compared with existing ones. Moreover, the tensor butterfly algorithm also permits O(nd)-complexity FFTs and Radon transforms up to d = 6 dimensions.

Kielstra, P Michael

Parallel algorithms for placement and routing in VLSI design

The computational requirements for high quality synthesis, analysis, and verification of very large scale integration (VLSI) designs have rapidly increased with the fast growing complexity of these designs. Research in the past has focused on the development of heuristic algorithms, special purpose hardware accelerators, or parallel algorithms for the numerous design tasks to decrease the time required for solution. Two new parallel algorithms are proposed for two VLSI synthesis tasks, standard cell placement and global routing. The first algorithm, a parallel algorithm for global routing, uses hierarchical techniques to decompose the routing problem into independent routing subproblems that are solved in parallel. Results are then presented which compare the routing quality to the results of other published global routers and which evaluate the speedups attained. The second algorithm, a parallel algorithm for cell placement and global routing, hierarchically integrates a quadrisection placement algorithm, a bisection placement algorithm, and the previous global routing algorithm. Unique partitioning techniques are used to decompose the various stages of the algorithm into independent tasks which can be evaluated in parallel. Finally, results are presented which evaluate the various algorithm alternatives and compare the algorithm performance to other placement programs. Measurements are presented on the parallel speedups available.

Brouwer, Randall Jay