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All sourcesNASA NTRSDOE OSTI
NASA NTRS2011

Chance-Constrained Guidance With Non-Convex Constraints

Missions to small bodies, such as comets or asteroids, require autonomous guidance for descent to these small bodies. Such guidance is made challenging by uncertainty in the position and velocity of the spacecraft, as well as the uncertainty in the gravitational field around the small body. In addition, the requirement to avoid collision with the asteroid represents a non-convex constraint that means finding the optimal guidance trajectory, in general, is intractable. In this innovation, a new approach is proposed for chance-constrained optimal guidance with non-convex constraints. Chance-constrained guidance takes into account uncertainty so that the probability of collision is below a specified threshold. In this approach, a new bounding method has been developed to obtain a set of decomposed chance constraints that is a sufficient condition of the original chance constraint. The decomposition of the chance constraint enables its efficient evaluation, as well as the application of the branch and bound method. Branch and bound enables non-convex problems to be solved efficiently to global optimality. Considering the problem of finite-horizon robust optimal control of dynamic systems under Gaussian-distributed stochastic uncertainty, with state and control constraints, a discrete-time, continuous-state linear dynamics model is assumed. Gaussian-distributed stochastic uncertainty is a more natural model for exogenous disturbances such as wind gusts and turbulence than the previously studied set-bounded models. However, with stochastic uncertainty, it is often impossible to guarantee that state constraints are satisfied, because there is typically a non-zero probability of having a disturbance that is large enough to push the state out of the feasible region. An effective framework to address robustness with stochastic uncertainty is optimization with chance constraints. These require that the probability of violating the state constraints (i.e., the probability of failure) is below a user-specified bound known as the risk bound. An example problem is to drive a car to a destination as fast as possible while limiting the probability of an accident to 10(exp -7). This framework allows users to trade conservatism against performance by choosing the risk bound. The more risk the user accepts, the better performance they can expect.

FROM↗
NASA NTRS2013

Powered Descent Guidance with General Thrust-Pointing Constraints

The Powered Descent Guidance (PDG) algorithm and software for generating Mars pinpoint or precision landing guidance profiles has been enhanced to incorporate thrust-pointing constraints. Pointing constraints would typically be needed for onboard sensor and navigation systems that have specific field-of-view requirements to generate valid ground proximity and terrain-relative state measurements. The original PDG algorithm was designed to enforce both control and state constraints, including maximum and minimum thrust bounds, avoidance of the ground or descent within a glide slope cone, and maximum speed limits. The thrust-bound and thrust-pointing constraints within PDG are non-convex, which in general requires nonlinear optimization methods to generate solutions. The short duration of Mars powered descent requires guaranteed PDG convergence to a solution within a finite time; however, nonlinear optimization methods have no guarantees of convergence to the global optimal or convergence within finite computation time. A lossless convexification developed for the original PDG algorithm relaxed the non-convex thrust bound constraints. This relaxation was theoretically proven to provide valid and optimal solutions for the original, non-convex problem within a convex framework. As with the thrust bound constraint, a relaxation of the thrust-pointing constraint also provides a lossless convexification that ensures the enhanced relaxed PDG algorithm remains convex and retains validity for the original nonconvex problem. The enhanced PDG algorithm provides guidance profiles for pinpoint and precision landing that minimize fuel usage, minimize landing error to the target, and ensure satisfaction of all position and control constraints, including thrust bounds and now thrust-pointing constraints.

Carson, John M., III↗
NASA NTRS2001

Cryogenic Tank Structure Sizing With Structural Optimization Method

Structural optimization methods in MSC /NASTRAN are used to size substructures and to reduce the weight of a composite sandwich cryogenic tank for future launch vehicles. Because the feasible design space of this problem is non-convex, many local minima are found. This non-convex problem is investigated in detail by conducting a series of analyses along a design line connecting two feasible designs. Strain constraint violations occur for some design points along the design line. Since MSC/NASTRAN uses gradient-based optimization procedures. it does not guarantee that the lowest weight design can be found. In this study, a simple procedure is introduced to create a new starting point based on design variable values from previous optimization analyses. Optimization analysis using this new starting point can produce a lower weight design. Detailed inputs for setting up the MSC/NASTRAN optimization analysis and final tank design results are presented in this paper. Approaches for obtaining further weight reductions are also discussed.

Wang, J. T.↗
NASA NTRS2001

Cryogenic Tank Structure Sizing With Structural Optimization Method

Structural optimization methods in MSC/NASTRAN are used to size substructures and to reduce the weight of a composite sandwich cryogenic tank for future launch vehicles. Because the feasible design space of this problem is non-convex, many local minima are found. This non-convex problem is investigated in detail by conducting a series of' analyses along a design line connecting two feasible designs. Strain constraint violations occur for some design points along the design line. Since MSC/NASTRAN uses gradient based optimization procedures, it does not guarantee that the lowest weight design can be found. In this study, a simple procedure is introduced to create a new starting point based on design variable values from previous optimization analyses. Optimization analysis using this new starting point can produce a lower weight design. Detailed inputs for setting up the MSC/NASTRAN optimization analysis and final tank design results are presented in this paper. Approaches for obtaining further weight reductions are also discussed.

Wang, J. T.↗
NASA NTRS2016

Computational Role of Tunneling in a Programmable Quantum Annealer

Quantum tunneling is a phenomenon in which a quantum state tunnels through energy barriers above the energy of the state itself. Tunneling has been hypothesized as an advantageous physical resource for optimization. Here we present the first experimental evidence of a computational role of multiqubit quantum tunneling in the evolution of a programmable quantum annealer. We developed a theoretical model based on a NIBA Quantum Master Equation to describe the multi-qubit dissipative cotunneling effects under the complex noise characteristics of such quantum devices.We start by considering a computational primitive, the simplest non-convex optimization problem consisting of just one global and one local minimum. The quantum evolutions enable tunneling to the global minimum while the corresponding classical paths are trapped in a false minimum. In our study the non-convex potentials are realized by frustrated networks of qubit clusters with strong intra-cluster coupling. We show that the collective effect of the quantum environment is suppressed in the critical phase during the evolution where quantum tunneling decides the right path to solution. In a later stage dissipation facilitates the multiqubit cotunneling leading to the solution state. The predictions of the model accurately describe the experimental data from the D-WaveII quantum annealer at NASA Ames. In our computational primitive the temperature dependence of the probability of success in the quantum model is opposite to that of the classical paths with thermal hopping. Specially, we provide an analysis of an optimization problem with sixteen qubits,demonstrating eight qubit cotunneling that increases success probabilities. Furthermore, we report results for larger problems with up to 200 qubits that contain the primitive as subproblems.

hard problems↗
NASA NTRS2013

Method and Apparatus for Powered Descent Guidance

A method and apparatus for landing a spacecraft having thrusters with non-convex constraints is described. The method first computes a solution to a minimum error landing problem for a convexified constraints, then applies that solution to a minimum fuel landing problem for convexified constraints. The result is a solution that is a minimum error and minimum fuel solution that is also a feasible solution to the analogous system with non-convex thruster constraints.

Acikmese, Behcet↗
NASA NTRS2002

Efficient Low Dissipative High Order Schemes for Multiscale MHD Flows

Accurate numerical simulations of complex multiscale compressible viscous flows, especially high speed turbulence combustion and acoustics, demand high order schemes with adaptive numerical dissipation controls. Standard high resolution shock-capturing methods are too dissipative to capture the small scales and/or long-time wave propagations without extreme grid refinements and small time steps. An integrated approach for the control of numerical dissipation in high order schemes for the compressible Euler and Navier-Stokes equations has been developed and verified by the authors and collaborators. These schemes are suitable for the problems in question. Basically, the scheme consists of sixth-order or higher non-dissipative spatial difference operators as the base scheme. To control the amount of numerical dissipation, multiresolution wavelets are used as sensors to adaptively limit the amount and to aid the selection and/or blending of the appropriate types of numerical dissipation to be used. Magnetohydrodynamics (MHD) waves play a key role in drag reduction in highly maneuverable high speed combat aircraft, in space weather forecasting, and in the understanding of the dynamics of the evolution of our solar system and the main sequence stars. Although there exist a few well-studied second and third-order high-resolution shock-capturing schemes for the MHD in the literature, these schemes are too diffusive and not practical for turbulence/combustion MHD flows. On the other hand, extension of higher than third-order high-resolution schemes to the MHD system of equations is not straightforward. Unlike the hydrodynamic equations, the inviscid MHD system is non-strictly hyperbolic with non-convex fluxes. The wave structures and shock types are different from their hydrodynamic counterparts. Many of the non-traditional hydrodynamic shocks are not fully understood. Consequently, reliable and highly accurate numerical schemes for multiscale MHD equations pose a great challenge to algorithm development. In addition, controlling the numerical error of the divergence free condition of the magnetic fields for high order methods has been a stumbling block. Lower order methods are not practical for the astrophysical problems in question. We propose to extend our hydrodynamics schemes to the MHD equations with several desired properties over commonly used MHD schemes.

Sjoegreen, Bjoern↗
NASA NTRS2002

The Delta x B = 0 Constraint Versus Minimization of Numerical Errors in MHD Simulations

The MHD equations are a system of non-strictly hyperbolic conservation laws. The non-convexity of the inviscid flux vector resulted in corresponding Jacobian matrices with undesirable properties. It has previously been shown by Powell et al. (1995) that an 'almost' equivalent MHD system in non-conservative form can be derived. This non-conservative system has a better conditioned eigensystem. Aside from Powell et al., the MHD equations can be derived from basic principles in either conservative or non-conservative form. The Delta x B = 0 constraint of the MHD equations is only an initial condition constraint, it is very different from the incompressible Navier-Stokes equations in which the divergence condition is needed to close the system (i.e., to have the same number of equations and the same number of unknown). In the MHD formulations, if Delta x B = 0 initially, all one needs is to construct appropriate numerical schemes that preserve this constraint at later time evolutions. In other words, one does not need the Delta x B condition to close the MHD system. We formulate our new scheme together with the Cargo & Gallice (1997) form of the MHD approximate Riemann solver in curvilinear grids for both versions of the MHD equations. A novel feature of our new method is that the well-conditioned eigen-decomposition of the non-conservative MHD equations is used to solve the conservative equations. This new feature of the method provides well-conditioned eigenvectors for the conservative formulation, so that correct wave speeds for discontinuities are assured. The justification for using the non-conservative eigen-decomposition to solve the conservative equations is that our scheme has a better control of the numerical error associated with the divergence of the magnetic condition. Consequently, computing both forms of the equations with the same eigen-decomposition is almost equivalent. It will be shown that this approach, using the non-conservative eigensystem when solving the conservative equations, also works well in the context of standard shock-capturing schemes.

Yee, H. C.↗
NASA NTRS2002

The Delta(dot) B = O Constraint vs. Minimization of Numerical Errors in MHD Simulations

The MHD equations are a system of non-strictly hyperbolic conservation laws. The non-convexity of the inviscid flux vector resulted in corresponding Jacobian matrices with undesirable properties. On the other hand, the MHD equations can be derived from basic principles in either conservative or non-conservative form. The non-conservative system has a better conditioned eigensystem. The Delta(dot)B = 0 constraint of the A4HD equations is only an initial condition constraint. One does not need the Delta(dot)B condition to close the MHD system. We formulate our new low dissipative high order scheme together with the Cargo & Gallice (1997) form of the MHD approximate Riemann solver in curvilinear grids for both versions of the MHD equations. A novel feature of our new method is that the well-conditioned eigen-decomposition of the non-conservative MHD equations is used to solve the conservative equations. This new feature of the method provides well-conditioned eigenvectors for the conservative formulation, so that correct wave speeds for discontinuities are assured. The justification for using the non-conservative eigen-decomposition to solve the conservative equations is that our scheme has a better control of the numerical error associated with the Delta(dot)B condition. Consequently, computing both forms of the equations with the same eigen-decomposition is almost equivalent. It will be shown that this approach, using the non-conservative eigensystem when solving the conservative equations, also works well in the context of standard shock-capturing schemes.

Yee, H. C.↗
NASA NTRSDate not supplied

A Convex Optimization Approach to Improving Suboptimal Hyperparameters of Sliced Normal Distributions

Sliced Normal (SN) distributions are a generalization of Gaussian distributions where the quadratic argument of the exponential is replaced with a sum of squares polynomial. SNs may be used to represent the distribution of a diverse set of random variables including multi-modal, non-symmetric, and skewed distributions. Unfortunately, the likelihood function of a SN includes a normalization constant and the inclusion of this normalization constant makes the likelihood a non-convex function of the hyperparameters which define the SN. In previous work, suboptimal fitting of the hyperparameters was performed by transforming the given data into a higher dimensional monomial basis and selecting the optimal hyperparameters of a Gaussian fit in this space. However, this approach did not account for the effect of lifting on the normalization constant. Indeed, it was observed that as the number of monomials is increased the likelihood of the Sliced Normal can decrease. In this paper, we increase the likelihood of Sliced Normals found using the previous method by developing a convex formulation which scales the covariance matrix of the Gaussian fit such that the likelihood of the Sliced Normal is maximized. The result is significant improvements of the log likelihood of fitted SN distributions, including a significant increase, especially for problems with 500+ monomials.

Convex optimization approach to improving suboptim↗
NASA NTRS1989

Numerical optimization in Hilbert space using inexact function and gradient evaluations

Trust region algorithms provide a robust iterative technique for solving non-convex unstrained optimization problems, but in many instances it is prohibitively expensive to compute high accuracy function and gradient values for the method. Of particular interest are inverse and parameter estimation problems, since function and gradient evaluations involve numerically solving large systems of differential equations. A global convergence theory is presented for trust region algorithms in which neither function nor gradient values are known exactly. The theory is formulated in a Hilbert space setting so that it can be applied to variational problems as well as the finite dimensional problems normally seen in trust region literature. The conditions concerning allowable error are remarkably relaxed: relative errors in the gradient error condition is automatically satisfied if the error is orthogonal to the gradient approximation. A technique for estimating gradient error and improving the approximation is also presented.

Carter, Richard G.↗
NASA NTRS1990

Large scale nonlinear numerical optimal control for finite element models of flexible structures

This paper discusses the development of large scale numerical optimal control algorithms for nonlinear systems and their application to finite element models of structures. This work is based on our expansion of the optimal control algorithm (DDP) in the following steps: improvement of convergence for initial policies in non-convex regions, development of a numerically accurate penalty function method approach for constrained DDP problems, and parallel processing on supercomputers. The expanded constrained DDP algorithm was applied to the control of a four-bay, two dimensional truss with 12 soft members, which generates geometric nonlinearities. Using an explicit finite element model to describe the structural system requires 32 state variables and 10,000 time steps. Our numerical results indicate that for constrained or unconstrained structural problems with nonlinear dynamics, the results obtained by our expanded constrained DDP are significantly better than those obtained using linear-quadratic feedback control.

Shoemaker, Christine A.↗
NASA NTRS1993

Surface acquisition through virtual milling

Surface acquisition deals with the reconstruction of three dimensional objects from a set of data points. The most straightforward techniques require human intervention, a time consuming proposition. It is desirable to develop a fully automated alternative. Such a method is proposed in this paper. It makes use of surface measurements obtained from a 3-D laser digitizer - an instrument which provides the (x,y,z) coordinates of surface data points from various viewpoints. These points are assembled into several partial surfaces using a visibility constraint and a 2-D triangulation technique. Reconstruction of the final object requires merging these partial surfaces. This is accomplished through a procedure that emulates milling, a standard machining operation. From a geometrical standpoint the problem reduces to constructing the intersection of two or more non-convex polyhedra.

Marshal L Merriam↗
NASA NTRS2002

Development of MLPG and LBIE Methods for Nonlinear Problems of Fracture

The truly meshless local Petrov-Galerkin (MLPG) method holds a great promise in solving boundary value problems, using a local symmetric weak form as a natural approach. In the present paper, in the context of MLPG and the meshless interpolation of a moving least squares (MLS) type, a method which uses primary and secondary nodes in the domain and on the global boundary is introduced, in order to improve the accuracy of solution. The secondary nodes can be placed at any location where one needs to obtain a better resolution. The sub-domains for the shape functions in the MLS approximation are defined only from the primary nodes, and the secondary nodes use the same sub-domains. The shape functions based on the MLS approximation, in an integration domain, have a single type of a rational function, which reduces the difficulty of numerical integration to evaluate the weak form. The present method is very useful in an adaptive calculation, because the secondary nodes can be easily added and/or moved without an additional mesh. The essential boundary conditions can be imposed exactly, and non-convex boundaries can be treated without special techniques. Several numerical examples are presented to illustrate the performance of the present method.

Atluri, Satya N.↗
NASA NTRS2004

A Study of Penalty Function Methods for Constraint Handling with Genetic Algorithm

COMETBOARDS (Comparative Evaluation Testbed of Optimization and Analysis Routines for Design of Structures) is a design optimization test bed that can evaluate the performance of several different optimization algorithms. A few of these optimization algorithms are the sequence of unconstrained minimization techniques (SUMT), sequential linear programming (SLP) and the sequential quadratic programming techniques (SQP). A genetic algorithm (GA) is a search technique that is based on the principles of natural selection or "survival of the fittest". Instead of using gradient information, the GA uses the objective function directly in the search. The GA searches the solution space by maintaining a population of potential solutions. Then, using evolving operations such as recombination, mutation and selection, the GA creates successive generations of solutions that will evolve and take on the positive characteristics of their parents and thus gradually approach optimal or near-optimal solutions. By using the objective function directly in the search, genetic algorithms can be effectively applied in non-convex, highly nonlinear, complex problems. The genetic algorithm is not guaranteed to find the global optimum, but it is less likely to get trapped at a local optimum than traditional gradient-based search methods when the objective function is not smooth and generally well behaved. The purpose of this research is to assist in the integration of genetic algorithm (GA) into COMETBOARDS. COMETBOARDS cast the design of structures as a constrained nonlinear optimization problem. One method used to solve constrained optimization problem with a GA to convert the constrained optimization problem into an unconstrained optimization problem by developing a penalty function that penalizes infeasible solutions. There have been several suggested penalty function in the literature each with there own strengths and weaknesses. A statistical analysis of some suggested penalty functions is performed in this study. Also, a response surface approach to robust design is used to develop a new penalty function approach. This new penalty function approach is then compared with the other existing penalty functions.

Ortiz, Francisco↗
NASA NTRS2005

Multivariate Parameter Sets for Optimal Synthesis of Compliant Mechanisms

This paper will propose the use of control maps along with discretized elements or meshes in the design parameter set for optimizing compliant mechanisms. The use of control maps will be demonstrated to encode the motion of groups of nodes or control points within a compliant mechanism design with simple mapping rules. The technique will serve as an alternative to increased mesh size or node wandering techniques that have been proposed to increase the number of alternative design shapes that may be considered. As an alternative approach, the proposed control map parameterization has the significant benefit that it minimizes the number of design parameters necessary (parameters increase linearly with the mesh size) in describing a given design making it computationally efficient. A limited number of tiles can produce a map that has a significant effect on the final shape. If the tiles are chosen appropriately, the problems such as material overlap and non-convex mesh elements are avoided automatically. This paper will describe the implementation of these control maps and provide several examples showing their implementation in the compliant mechanism topology synthesis process.

Shibakov, Alex↗
NASA NTRS2007

Two-Stage Path Planning Approach for Designing Multiple Spacecraft Reconfiguration Maneuvers

The paper presents a two-stage approach for designing optimal reconfiguration maneuvers for multiple spacecraft. These maneuvers involve well-coordinated and highly-coupled motions of the entire fleet of spacecraft while satisfying an arbitrary number of constraints. This problem is particularly difficult because of the nonlinearity of the attitude dynamics, the non-convexity of some of the constraints, and the coupling between the positions and attitudes of all spacecraft. As a result, the trajectory design must be solved as a single 6N DOF problem instead of N separate 6 DOF problems. The first stage of the solution approach quickly provides a feasible initial solution by solving a simplified version without differential constraints using a bi-directional Rapidly-exploring Random Tree (RRT) planner. A transition algorithm then augments this guess with feasible dynamics that are propagated from the beginning to the end of the trajectory. The resulting output is a feasible initial guess to the complete optimal control problem that is discretized in the second stage using a Gauss pseudospectral method (GPM) and solved using an off-the-shelf nonlinear solver. This paper also places emphasis on the importance of the initialization step in pseudospectral methods in order to decrease their computation times and enable the solution of a more complex class of problems. Several examples are presented and discussed.

Aoude, Georges S.↗
NASA NTRS2011

NASA Tech Briefs, June 2011

Topics covered include: Wind and Temperature Spectrometry of the Upper Atmosphere in Low-Earth Orbit; Health Monitor for Multitasking, Safety-Critical, Real-Time Software; Stereo Imaging Miniature Endoscope; Early Oscillation Detection Technique for Hybrid DC/DC Converters; Parallel Wavefront Analysis for a 4D Interferometer; Schottky Heterodyne Receivers With Full Waveguide Bandwidth; Carbon Nanofiber-Based, High-Frequency, High-Q, Miniaturized Mechanical Resonators; Ultracapacitor-Based Uninterrupted Power Supply System; Coaxial Cables for Martian Extreme Temperature Environments; Using Spare Logic Resources To Create Dynamic Test Points; Autonomous Coordination of Science Observations Using Multiple Spacecraft; Autonomous Phase Retrieval Calibration; EOS MLS Level 1B Data Processing Software, Version 3; Cassini Tour Atlas Automated Generation; Software Development Standard Processes (SDSP); Graphite Composite Panel Polishing Fixture; Material Gradients in Oxygen System Components Improve Safety; Ridge Waveguide Structures in Magnesium-Doped Lithium Niobate; Modifying Matrix Materials to Increase Wetting and Adhesion; Lightweight Magnetic Cooler With a Reversible Circulator; The Invasive Species Forecasting System; Method for Cleanly and Precisely Breaking Off a Rock Core Using a Radial Compressive Force; Praying Mantis Bending Core Breakoff and Retention Mechanism; Scoring Dawg Core Breakoff and Retention Mechanism; Rolling-Tooth Core Breakoff and Retention Mechanism; Vibration Isolation and Stabilization System for Spacecraft Exercise Treadmill Devices; Microgravity-Enhanced Stem Cell Selection; Diagnosis and Treatment of Neurological Disorders by Millimeter-Wave Stimulation; Passive Vaporizing Heat Sink; Remote Sensing and Quantization of Analog Sensors; Phase Retrieval for Radio Telescope and Antenna Control; Helium-Cooled Black Shroud for Subscale Cryogenic Testing; Receive Mode Analysis and Design of Microstrip Reflectarrays; and Chance-Constrained Guidance With Non-Convex Constraints.

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