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NASA NTRS1989

An indirect method for numerical optimization using the Kreisselmeir-Steinhauser function

A technique is described for converting a constrained optimization problem into an unconstrained problem. The technique transforms one of more objective functions into reduced objective functions, which are analogous to goal constraints used in the goal programming method. These reduced objective functions are appended to the set of constraints and an envelope of the entire function set is computed using the Kreisselmeir-Steinhauser function. This envelope function is then searched for an unconstrained minimum. The technique may be categorized as a SUMT algorithm. Advantages of this approach are the use of unconstrained optimization methods to find a constrained minimum without the draw down factor typical of penalty function methods, and that the technique may be started from the feasible or infeasible design space. In multiobjective applications, the approach has the advantage of locating a compromise minimum design without the need to optimize for each individual objective function separately.

Wrenn, Gregory A.↗

NASA NTRS2010

Cost Optimization and Technology Enablement COTSAT-1

Cost Optimized Test of Spacecraft Avionics and Technologies (COTSAT-1) is an ongoing spacecraft research and development project at NASA Ames Research Center (ARC). The space industry was a hot bed of innovation and development at its birth. Many new technologies were developed for and first demonstrated in space. In the recent past this trend has reversed with most of the new technology funding and research being driven by the private industry. Most of the recent advances in spaceflight hardware have come from the cell phone industry with a lag of about 10 to 15 years from lab demonstration to in space usage. NASA has started a project designed to address this problem. The prototype spacecraft known as Cost Optimized Test of Spacecraft Avionics and Technologies (COTSAT-1) and CheapSat work to reduce these issues. This paper highlights the approach taken by NASA Ames Research center to achieve significant subsystem cost reductions. The COSTAT-1 research system design incorporates use of COTS (Commercial Off The Shelf), MOTS (Modified Off The Shelf), and GOTS (Government Off The Shelf) hardware for a remote sensing spacecraft. The COTSAT-1 team demonstrated building a fully functional spacecraft for $500K parts and $2.0M labor. The COTSAT-1 system, including a selected science payload, is described within this paper. Many of the advancements identified in the process of cost reduction can be attributed to the use of a one-atmosphere pressurized structure to house the spacecraft components. By using COTS hardware, the spacecraft program can utilize investments already made by commercial vendors. This ambitious project development philosophy/cycle has yielded the COTSAT-1 flight hardware. This paper highlights the advancements of the COTSAT-1 spacecraft leading to the delivery of the current flight hardware that is now located at NASA Ames Research Center. This paper also addresses the plans for COTSAT-2.

Spremo, Stevan↗

NASA NTRS2012

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↗

NASA NTRSDate not supplied

Introducing Tropical Geometric Approaches to Delay Tolerant Networking Optimization

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

Delay Tolerant Networking↗

NASA NTRS2007

Multi-Objective Online Initialization of Spacecraft Formations

This paper extends a previously developed method for finding spacecraft initial conditions (ICs) that minimize the drift resulting from J2 disturbances while also minimizing the fuel required to attain those ICs. It generalizes the single spacecraft optimization to a formation-wide optimization valid for an arbitrary number of vehicles. Additionally, the desired locations of the spacecraft, separate from the starting location, can be specified, either with respect to a reference orbit, or relative to the other spacecraft in the formation. The three objectives (minimize drift, minimize fuel, and maintain a geometric template) are expressed as competing costs in a linear optimization, and are traded against one another through the use of scalar weights. By carefully selecting these weights and re-initializing the formation at regular intervals, a closed-loop, formation-wide control system is created. This control system can be used to reconfigure the formations on the fly, and creates fuel-efficient plans by placing the spacecraft in semi-invariant orbits. The overall approach is demonstrated through nonlinear simulations for two formations a GEO orbit, and an elliptical orbit.

Jeffrey, Matthew↗

NASA NTRS1981

Optimal transfer between close near-circular orbits

A nonlinear analysis is carried out for the optimal transfer between close near-circular noncoplanar orbits with no limit on time. The degeneracy of the linear analyses of Marec and Edelbaum is, thereby, removed and with the aid of a generalized Jacobi test, it is established that the minimum-impulse transfer requires, at most, three impulses. A field of extremals is generated in an orbit element space starting with the initial state and anticipated adjoint variables and integrating these variables by using the optimal thrust control law that maximizes the variational Hamiltonian. The nonoptimality of any four-impulse strategy is determined numerically by observing the reflection of every extremal from either an envelope (conjugate point test) or a switching surface on or before the application of the fourth impulse. This test is carried out in Regions I, II, and III of Marec, as well as the special transition region of Breakwell near the boundary of Region III.

Kechichian, J.↗

NASA NTRS2008

Development of a Smooth Trajectory Maneuver Method to Accommodate the Ares I Flight Control Constraints

Six degree-of-freedom (DOF) launch vehicle trajectories are designed to follow an optimized 3-DOF reference trajectory. A vehicle has a finite amount of control power that it can allocate to performing maneuvers. Therefore, the 3-DOF trajectory must be designed to refrain from using 100% of the allowable control capability to perform maneuvers, saving control power for handling off-nominal conditions, wind gusts and other perturbations. During the Ares I trajectory analysis, two maneuvers were found to be hard for the control system to implement; a roll maneuver prior to the gravity turn and an angle of attack maneuver immediately after the J-2X engine start-up. It was decided to develop an approach for creating smooth maneuvers in the optimized reference trajectories that accounts for the thrust available from the engines. A feature of this method is that no additional angular velocity in the direction of the maneuver has been added to the vehicle after the maneuver completion. This paper discusses the equations behind these new maneuvers and their implementation into the Ares I trajectory design cycle. Also discussed is a possible extension to adjusting closed-loop guidance.

Pinson, Robin M.↗

NASA NTRS1975

An algorithm for the space shuttle scheduling problem

The problem of determining launch times for a set of missions is considered. Launch times must satisfy early start time and resource constraints. The investigation reported is concerned with the minimization of the number of missions which are launched after specified late start times. An algorithm is presented which, under certain restrictive assumptions, provides an optimal solution. If the restrictive assumptions are not satisfied, the algorithm may be used to obtain a lower bound on the number of missions flown late.

Fisher, M. L.↗

NASA NTRS1991

A shooting approach to suboptimal control

The shooting method is used to solve the suboptimal control problem where the control history is assumed to be piecewise linear. Suboptimal solutions can be obtained without difficulty and can lead to accurate approximate controls and good starting multipliers for the regular shooting method by increasing the number of nodes. Optimal planar launch trajectories are presented for the advanced launch system.

Hull, David G.↗

NASA NTRS1999

Ames Optimized TCA Configuration

Configuration design at Ames was carried out with the SYN87-SB (single block) Euler code using a 193 x 49 x 65 C-H grid. The Euler solver is coupled to the constrained (NPSOL) and the unconstrained (QNMDIF) optimization packages. Since the single block grid is able to model only wing-body configurations, the nacelle/diverter effects were included in the optimization process by SYN87's option to superimpose the nacelle/diverter interference pressures on the wing. These interference pressures were calculated using the AIRPLANE code. AIRPLANE is an Euler solver that uses a unstructured tetrahedral mesh and is capable of computations about arbitrary complete configurations. In addition, the buoyancy effects of the nacelle/diverters were also included in the design process by imposing the pressure field obtained during the design process onto the triangulated surfaces of the nacelle/diverter mesh generated by AIRPLANE. The interference pressures and nacelle buoyancy effects are added to the final forces after each flow field calculation. Full details of the (recently enhanced) ghost nacelle capability are given in a related talk. The pseudo nacelle corrections were greatly improved during this design cycle. During the Ref H and Cycle 1 design activities, the nacelles were only translated and pitched. In the cycle 2 design effort the nacelles can translate vertically, and pitch to accommodate the changes in the lower surface geometry. The diverter heights (between their leading and trailing edges) were modified during design as the shape of the lower wing changed, with the drag of the diverter changing accordingly. Both adjoint and finite difference gradients were used during optimization. The adjoint-based gradients were found to give good direction in the design space for configurations near the starting point, but as the design approached a minimum, the finite difference gradients were found to be more accurate. Use of finite difference gradients was limited by the CPU time limit available on the Cray machines. A typical optimization run using finite difference gradients can use only 30 to 40 design variables and one optimization iteration within the 8 hour queue limit for the chosen grid size and convergence level. The efficiency afforded by the adjoint method allowed for 50-120 design variables and 5-10 optimization iterations in the 8 hour queue. Geometric perturbations to the wing and fuselage were made using the Hicks/Henne (HH) shape functions. The HH functions were distributed uniformly along the chords of the wing defining sections and lofted linearly. During single-surface design, constraints on thickness and volume at selected wing stations were imposed. Both fuselage camber and cross-sectional area distributions were permitted to change during design. The major disadvantage to the use of these functions is the inherent surface waviness produced by repeated use of such functions. Many smoothing operations were required following optimization runs to produce a configuration with reasonable smoothness. Wagner functions were also used on the wing sections but were never used on the fuselage. The Wagner functions are a family of increasingly oscillatory functions that have also been used extensively in airfoil design. The leading and trailing edge regions of the wing were designed by use of polynomial and monomial functions respectively. Twist was attempted but was abandoned because of little performance improvement available from changing the baseline twist.

Cliff, Susan E.↗

NASA NTRS1984

Regression analysis as a design optimization tool

The optimization concepts are described in relation to an overall design process as opposed to a detailed, part-design process where the requirements are firmly stated, the optimization criteria are well established, and a design is known to be feasible. The overall design process starts with the stated requirements. Some of the design criteria are derived directly from the requirements, but others are affected by the design concept. It is these design criteria that define the performance index, or objective function, that is to be minimized within some constraints. In general, there will be multiple objectives, some mutually exclusive, with no clear statement of their relative importance. The optimization loop that is given adjusts the design variables and analyzes the resulting design, in an iterative fashion, until the objective function is minimized within the constraints. This provides a solution, but it is only the beginning. In effect, the problem definition evolves as information is derived from the results. It becomes a learning process as we determine what the physics of the system can deliver in relation to the desirable system characteristics. As with any learning process, an interactive capability is a real attriubute for investigating the many alternatives that will be suggested as learning progresses.

Perley, R.↗

NASA NTRS1985

Evaluation of results of cell electrophoresis experiments on space shuttle STS-3 including pre-flight and post-flight laboratory experiments

The objectives of the red blood cell experiments were to provide a visual check on the electrophoretic process and especially electroosmotic flow in space as well as to provide test separations of non-degradable standard particles for comparison with the separations of the three viable cell types studied on the Apollo-Soyuz Test Project. Determination of the maximum concentrations of cells that can be separated in column electrophore was a significant goal. Two of the eight columns were available for red cell experiments, so two concentrations of human and rabbit RBC mixtures were used. The objectives of another experiment were to evaluate the reproducibility of microgravity electrophoretic separation of living kidney cells, to separate cells with highly viability despite two freeze-thaw cycles, and to optimize the physical conditions of cell separation. Owing to the uncertain heterogeneity of the starting material, the experimental design does not assess resolution in microgravity, but improved separability was sought in comparison to density-gradient electrophoresis or continuous-flow electrophoresis. Efforts were made to increase cell yield and cell viability and to assess reproducibility directly.

Todd, P. W.↗

NASA NTRS2013

What Reliability Engineers Should Know about Space Radiation Effects

Space radiation in space systems present unique failure modes and considerations for reliability engineers. Radiation effects is not a one size fits all field. Threat conditions that must be addressed for a given mission depend on the mission orbital profile, the technologies of parts used in critical functions and on application considerations, such as supply voltages, temperature, duty cycle, and redundancy. In general, the threats that must be addressed are of two types-the cumulative degradation mechanisms of total ionizing dose (TID) and displacement damage (DD). and the prompt responses of components to ionizing particles (protons and heavy ions) falling under the heading of single-event effects. Generally degradation mechanisms behave like wear-out mechanisms on any active components in a system: Total Ionizing Dose (TID) and Displacement Damage: (1) TID affects all active devices over time. Devices can fail either because of parametric shifts that prevent the device from fulfilling its application or due to device failures where the device stops functioning altogether. Since this failure mode varies from part to part and lot to lot, lot qualification testing with sufficient statistics is vital. Displacement damage failures are caused by the displacement of semiconductor atoms from their lattice positions. As with TID, failures can be either parametric or catastrophic, although parametric degradation is more common for displacement damage. Lot testing is critical not just to assure proper device fi.mctionality throughout the mission. It can also suggest remediation strategies when a device fails. This paper will look at these effects on a variety of devices in a variety of applications. This paper will look at these effects on a variety of devices in a variety of applications. (2) On the NEAR mission a functional failure was traced to a PIN diode failure caused by TID induced high leakage currents. NEAR was able to recover from the failure by reversing the current of a nearby Thermal Electric Cooler (turning the TEC into a heater). The elevated temperature caused the PIN diode to anneal and the device to recover. It was by lot qualification testing that NEAR knew the diode would recover when annealed. This paper will look at these effects on a variety of devices in a variety of applications. Single Event Effects (SEE): (1) In contrast to TID and displacement damage, Single Event Effects (SEE) resemble random failures. SEE modes can range from changes in device logic (single-event upset, or SEU). temporary disturbances (single-event transient) to catastrophic effects such as the destructive SEE modes, single-event latchup (SEL). single-event gate rupture (SEGR) and single-event burnout (SEB) (2) The consequences of nondestructive SEE modes such as SEU and SET depend critically on their application--and may range from trivial nuisance errors to catastrophic loss of mission. It is critical not just to ensure that potentially susceptible devices are well characterized for their susceptibility, but also to work with design engineers to understand the implications of each error mode. -For destructive SEE, the predominant risk mitigation strategy is to avoid susceptible parts, or if that is not possible. to avoid conditions under which the part may be susceptible. Destructive SEE mechanisms are often not well understood, and testing is slow and expensive, making rate prediction very challenging. (3) Because the consequences of radiation failure and degradation modes depend so critically on the application as well as the component technology, it is essential that radiation, component. design and system engineers work togetherpreferably starting early in the program to ensure critical applications are addressed in time to optimize the probability of mission success.

DiBari, Rebecca↗

NASA NTRS2015

Rapid Generation of Optimal Asteroid Powered Descent Trajectories Via Convex Optimization

Pinson, Robin↗

NASA NTRS2016

Trajectory Design Employing Convex Optimization for Landing on Irregularly Shaped Asteroids

Mission proposals that land on asteroids are becoming popular. However, in order to have a successful mission the spacecraft must reliably and softly land at the intended landing site. The problem under investigation is how to design a fuel-optimal powered descent trajectory that can be quickly computed on- board the spacecraft, without interaction from ground control. An optimal trajectory designed immediately prior to the descent burn has many advantages. These advantages include the ability to use the actual vehicle starting state as the initial condition in the trajectory design and the ease of updating the landing target site if the original landing site is no longer viable. For long trajectories, the trajectory can be updated periodically by a redesign of the optimal trajectory based on current vehicle conditions to improve the guidance performance. One of the key drivers for being completely autonomous is the infrequent and delayed communication between ground control and the vehicle. Challenges that arise from designing an asteroid powered descent trajectory include complicated nonlinear gravity fields, small rotating bodies and low thrust vehicles. There are two previous studies that form the background to the current investigation. The first set looked in-depth at applying convex optimization to a powered descent trajectory on Mars with promising results.1, 2 This showed that the powered descent equations of motion can be relaxed and formed into a convex optimization problem and that the optimal solution of the relaxed problem is indeed a feasible solution to the original problem. This analysis used a constant gravity field. The second area applied a successive solution process to formulate a second order cone program that designs rendezvous and proximity operations trajectories.3, 4 These trajectories included a Newtonian gravity model. The equivalence of the solutions between the relaxed and the original problem is theoretically established. The proposed solution for designing the asteroid powered descent trajectory is to use convex optimization, a gravity model with higher fidelity than Newtonian, and an iterative solution process to design the fuel optimal trajectory. The solution to the convex optimization problem is the thrust profile, magnitude and direction, that will yield the minimum fuel trajectory for a soft landing at the target site, subject to various mission and operational constraints. The equations of motion are formulated in a rotating coordinate system and includes a high fidelity gravity model. The vehicle's thrust magnitude can vary between maximum and minimum bounds during the burn. Also, constraints are included to ensure that the vehicle does not run out of propellant, or go below the asteroid's surface, and any vehicle pointing requirements. The equations of motion are discretized and propagated with the trapezoidal rule in order to produce equality constraints for the optimization problem. These equality constraints allow the optimization algorithm to solve the entire problem, without including a propagator inside the optimization algorithm.

Pinson, Robin M.↗

NASA NTRS2002

Genetic Optimization of a Tensegrity Structure

Marshall Space Flight Center (MSFC) is charged with developing advanced technologies for space telescopes. The next generation of space optics will be very large and lightweight. Tensegrity structures are built of compressive members (bars), and tensile members (strings). For most materials, the tensile strength of a longitudinal member is larger than its buckling strength; therefore a large stiffness to mass ratio can be achieved by increasing the use of tensile members. Tensegrities are the epitome of lightweight structures, since they take advantage of the larger tensile strength of materials. The compressive members of tensegrity structures are disjoint allowing compact storage of the structure. The structure has the potential to eliminate the requirement for assembly by man in space; it can be deployed by adjustments in its cable tension. A tensegrity structure can be more reliably modeled since none of the individual members experience bending moments. (Members that experience deformation in more than one dimension are much harder to model.) Structures that can be more precisely modeled can be more precisely controlled. Furthermore, an astoundingly wide variety of natural systems, including carbon atoms, water molecules, proteins, viruses, cells, tissues and even human and other living creatures are tensegrity structures. Through the process of evolution, nature continually improves the design of living creatures for the environment they live in. Since tensegrities are nature's structure of choice, it is conceivable that they have other benefits we are unaware of. A. Keane and S. Brown designed a satellite boom truss system with an enhanced vibration performance. They started with a standard truss system, then used a genetic algorithm to alter the design, while optimizing the vibration performance. An improvement of over 20,000% in frequency-averaged energy levels was obtained using this approach. In this report an introduction to tensegrity structures is given, along with a description of how to generate the nodal coordinates and connectivity of a multiple stage cylindrical tensegrity structure. A description of how finite elements can be used to develop a stiffness and mass matrix so that the modes of vibration can be determined from the eigenvalue problem is shown. A brief description of a micro genetic algorithm is then presented.

Jaime R. Taylor↗

NASA NTRS1979

Acoustic duct liner optimization using finite elements

Determination of a mathematical model which may be used for obtaining optimum acoustic liners in aeroengine ducts was the goal of this study. Starting with a baseline finite element model which consumed too much computer time to permit optimum liner evaluation for realistic ducts and source frequencies, a survey was conducted of candidate methods for reducing computational effort. A highly effective boundary exclusive decomposition method was developed which enables L-U decomposition of a major partition of the global matrix to be performed once only. Timing experiments using the technique in duct liner optimization show a reduction of between one and two orders of magnitude in the central processor unit (CPU) time of the baseline model.

Abrahamson, A. L.↗

NASA NTRS1991

High figure-of-merit n-type SiGe/GaP alloys

Improvements to the figure-of-merit of n-type SiGe have been achieved via a systematic anneal study aimed at determining the optimal P:Ga ratio. Figures-of-merit of 0.85 to 0.90 x 10-3K-1 have been routinely and reproducibly achieved starting with an initial P:Ga ratio of 3:1. These samples have carrier concentrations in excess of 4.0 x 1020/cu cm, the highest reported for SiGe. This value has been shown to be too high, resulting in unfavorably low Seebeck coefficients. The optimal carrier concentration has been shown to be 1.5-3.0 x 1020/cu cm. Thus, further improvements will be achieved by reductions in the P and Ga concentrations.

Scoville, A. N.↗