Computation of optimal singular controls
Optimal singular controls made nonsingular for decreasing epsilon values by adding integral quadratic functional of control to cost functional
Gershwin, S. B.↗
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Management of analytical redundancy in digital flight control systems for aircraft
This paper presents a design method for optimal redundancy management for nonlinear systems with application to highly maneuvering aircraft. The approach taken is based on selecting the failure states to be covered by the system design and constructing a cost function that represents the cost of making an incorrect decision. The decision logic which minimizes the cost requires a bank of extended Kalman filters running in parallel. This produces a severe computational requirement. To reduce this requirement, a suboptimal logic is developed based on using a nonlinear single-stage prediction algorithm in the filters with filter gains and decision logic selected using steady-state results obtained from a linearization of the vehicle and sensor dynamics. The design process is then applied to designing a redundancy management system for the F8-C aircraft. Results indicate that the system is superior in failure detection to a system using the same structure but using a linear single-stage prediction algorithm in the filters.
Montgomery, R. C.↗
Order reduction for models of space structures using modal cost analysis
Modal cost analysis furnishes a promising methodology for developing dynamical models of space structures for use in control systems analysis. Economy and accuracy can be attained by only retaining vibration modes that contribute significantly to an appropriately defined cost function. Expressions for modal costs (especially simple for 'lightly damped' structures) are derived for attitude control, vibration suppression, and shape control. These techniques are illustrated through application to a high-order finite element model of a large platform-type structure.
Skelton, R. E.↗
Developing integrated parametric planning models for budgeting and managing complex projects
The applicability of integrated parametric models for the budgeting and management of complex projects is investigated. Methods for building a very flexible, interactive prototype for a project planning system, and software resources available for this purpose, are discussed and evaluated. The prototype is required to be sensitive to changing objectives, changing target dates, changing costs relationships, and changing budget constraints. To achieve the integration of costs and project and task durations, parametric cost functions are defined by a process of trapezoidal segmentation, where the total cost for the project is the sum of the various project cost segments, and each project cost segment is the integral of a linearly segmented cost loading function over a specific interval. The cost can thus be expressed algebraically. The prototype was designed using Lotus-123 as the primary software tool. This prototype implements a methodology for interactive project scheduling that provides a model of a system that meets most of the goals for the first phase of the study and some of the goals for the second phase.
Etnyre, Vance A.↗
Joint Chance-Constrained Dynamic Programming
This paper presents a novel dynamic programming algorithm with a joint chance constraint, which explicitly bounds the risk of failure in order to maintain the state within a specified feasible region. A joint chance constraint cannot be handled by existing constrained dynamic programming approaches since their application is limited to constraints in the same form as the cost function, that is, an expectation over a sum of one-stage costs. We overcome this challenge by reformulating the joint chance constraint into a constraint on an expectation over a sum of indicator functions, which can be incorporated into the cost function by dualizing the optimization problem. As a result, the primal variables can be optimized by a standard dynamic programming, while the dual variable is optimized by a root-finding algorithm that converges exponentially. Error bounds on the primal and dual objective values are rigorously derived. We demonstrate the algorithm on a path planning problem, as well as an optimal control problem for Mars entry, descent and landing. The simulations are conducted using a real terrain data of Mars, with four million discrete states at each time step.
Ono, Masahiro↗
Impact of Airspace Charges on Transatlantic Aircraft Trajectories
Aircraft flying over the airspace of different countries are subject to over-flight charges. These charges vary from country to country. Airspace charges, while necessary to support the communication, navigation and surveillance services, may lead to aircraft flying routes longer than wind-optimal routes and produce additional carbon dioxide and other gaseous emissions. This paper develops an optimal route between city pairs by modifying the cost function to include an airspace cost whenever an aircraft flies through a controlled airspace without landing or departing from that airspace. It is assumed that the aircraft will fly the trajectory at a constant cruise altitude and constant speed. The computationally efficient optimal trajectory is derived by solving a non-linear optimal control problem. The operational strategies investigated in this study for minimizing aircraft fuel burn and emissions include flying fuel-optimal routes and flying cost-optimal routes that may completely or partially reduce airspace charges en route. The results in this paper use traffic data for transatlantic flights during July 2012. The mean daily savings in over-flight charges, fuel cost and total operation cost during the period are 17.6 percent, 1.6 percent, and 2.4 percent respectively, along the cost- optimal trajectories. The transatlantic flights can potentially save $600,000 in fuel cost plus $360,000 in over-flight charges daily by flying the cost-optimal trajectories. In addition, the aircraft emissions can be potentially reduced by 2,070 metric tons each day. The airport pairs and airspace regions that have the highest potential impacts due to airspace charges are identified for possible reduction of fuel burn and aircraft emissions for the transatlantic flights. The results in the paper show that the impact of the variation in fuel price on the optimal routes is to reduce the difference between wind-optimal and cost-optimal routes as the fuel price increases. The additional fuel consumption is quantified using the 30 percent variation in fuel prices during March 2014 to March 2015.
aircraft scheduling↗
A six degree of freedom, plume-fuel optimal trajectory planner for spacecraft proximity operations using an A* node search
Spacecraft proximity operations are complicated by the fact that exhaust plume impingement from the reaction control jets of space vehicles can cause structural damage, contamination of sensitive arrays and instruments, or attitude misalignment during docking. The occurrence and effect of jet plume impingement can be reduced by planning approach trajectories with plume effects considered. An A* node search is used to find plume-fuel optimal trajectories through a discretized six dimensional attitude-translation space. A plume cost function which approximates jet plume isopressure envelopes is presented. The function is then applied to find relative costs for predictable 'trajectory altering' firings and unpredictable 'deadbanding' firings. Trajectory altering firings are calculated by running the spacecraft jet selection algorithm and summing the cost contribution from each jet fired. A 'deadbanding effects' function is defined and integrated to determine the potential for deadbanding impingement along candidate trajectories. Plume costs are weighed against fuel costs in finding the optimal solution. A* convergence speed is improved by solving approach trajectory problems in reverse time. Results are obtained on a high fidelity space shuttle/space station simulation. Trajectory following is accomplished by a six degree of freedom autopilot. Trajectories planned with, and without, plume costs are compared in terms of force applied to the target structure.
Jackson, Mark Charles↗
Designer's unified cost model
A conceptual and preliminary designers' cost prediction model has been initiated. The model will provide a technically sound method for evaluating the relative cost of different composite structural designs, fabrication processes, and assembly methods that can be compared to equivalent metallic parts or assemblies. The feasibility of developing cost prediction software in a modular form for interfacing with state of the art preliminary design tools and computer aided design programs is being evaluated. The goal of this task is to establish theoretical cost functions that relate geometric design features to summed material cost and labor content in terms of process mechanics and physics. The output of the designers' present analytical tools will be input for the designers' cost prediction model to provide the designer with a data base and deterministic cost methodology that allows one to trade and synthesize designs with both cost and weight as objective functions for optimization. The approach, goals, plans, and progress is presented for development of COSTADE (Cost Optimization Software for Transport Aircraft Design Evaluation).
Freeman, William T.↗
Designers' unified cost model
The Structures Technology Program Office (STPO) at NASA LaRC has initiated development of a conceptual and preliminary designers' cost prediction model. The model will provide a technically sound method for evaluating the relative cost of different composite structural designs, fabrication processes, and assembly methods that can be compared to equivalent metallic parts or assemblies. The feasibility of developing cost prediction software in a modular form for interfacing with state-of-the-art preliminary design tools and computer aided design programs is being evaluated. The goal of this task is to establish theoretical cost functions that relate geometric design features to summed material cost and labor content in terms of process mechanics and physics. The output of the designers' present analytical tools will be input for the designers' cost prediction model to provide the designer with a database and deterministic cost methodology that allows one to trade and synthesize designs with both cost and weight as objective functions for optimization. This paper presents the team members, approach, goals, plans, and progress to date for development of COSTADE (Cost Optimization Software for Transport Aircraft Design Evaluation).
Freeman, W.↗
Control of a serpentine manipulator with collision avoidance
The robotics lab at the Kennedy Space Center is investigating the possibility of using a 'serpentine' manipulator for Shuttle inspection and payload processing. Serpentine manipulators are characterized by a large number of degrees of freedom giving them a high degree of redundancy. This redundancy allows them to be used to reach confined areas while avoiding collisions with their environment. In this paper, the author describes a new approach to controlling the joint rates for an n degree of freedom robot such that it moves its end effector to a desired position while simultaneously avoiding collision of any part of the robot arm with obstacles. Joint rates which move the end effector toward the target are found via a Lyapunov stability function. The gradient of an obstacle cost function indicates the direction toward obstacle collision in the joint space. The component of the end effector joint rates orthogonal to the obstacle gradient becomes the commanded joint rates. A notional eleven DOF model is used to numerically demonstrate the efficacy of the control law.
Byers, Robert M.↗
Risk-Constrained Dynamic Programming for Optimal Mars Entry, Descent, and Landing
A chance-constrained dynamic programming algorithm was developed that is capable of making optimal sequential decisions within a user-specified risk bound. This work handles stochastic uncertainties over multiple stages in the CEMAT (Combined EDL-Mobility Analyses Tool) framework. It was demonstrated by a simulation of Mars entry, descent, and landing (EDL) using real landscape data obtained from the Mars Reconnaissance Orbiter. Although standard dynamic programming (DP) provides a general framework for optimal sequential decisionmaking under uncertainty, it typically achieves risk aversion by imposing an arbitrary penalty on failure states. Such a penalty-based approach cannot explicitly bound the probability of mission failure. A key idea behind the new approach is called risk allocation, which decomposes a joint chance constraint into a set of individual chance constraints and distributes risk over them. The joint chance constraint was reformulated into a constraint on an expectation over a sum of an indicator function, which can be incorporated into the cost function by dualizing the optimization problem. As a result, the chance-constraint optimization problem can be turned into an unconstrained optimization over a Lagrangian, which can be solved efficiently using a standard DP approach.
Ono, Masahiro↗
Algorithm for fixed-range optimal trajectories
An algorithm for synthesizing optimal aircraft trajectories for specified range was developed and implemented in a computer program written in FORTRAN IV. The algorithm, its computer implementation, and a set of example optimum trajectories for the Boeing 727-100 aircraft are described. The algorithm optimizes trajectories with respect to a cost function that is the weighted sum of fuel cost and time cost. The optimum trajectory consists at most of a three segments: climb, cruise, and descent. The climb and descent profiles are generated by integrating a simplified set of kinematic and dynamic equations wherein the total energy of the aircraft is the independent or time like variable. At each energy level the optimum airspeeds and thrust settings are obtained as the values that minimize the variational Hamiltonian. Although the emphasis is on an off-line, open-loop computation, eventually the most important application will be in an on-board flight management system.
Lee, H. Q.↗
Mixed Integer Programming and Heuristic Scheduling for Space Communication
Optimal planning and scheduling for a communication network was created where the nodes within the network are communicating at the highest possible rates while meeting the mission requirements and operational constraints. The planning and scheduling problem was formulated in the framework of Mixed Integer Programming (MIP) to introduce a special penalty function to convert the MIP problem into a continuous optimization problem, and to solve the constrained optimization problem using heuristic optimization. The communication network consists of space and ground assets with the link dynamics between any two assets varying with respect to time, distance, and telecom configurations. One asset could be communicating with another at very high data rates at one time, and at other times, communication is impossible, as the asset could be inaccessible from the network due to planetary occultation. Based on the network's geometric dynamics and link capabilities, the start time, end time, and link configuration of each view period are selected to maximize the communication efficiency within the network. Mathematical formulations for the constrained mixed integer optimization problem were derived, and efficient analytical and numerical techniques were developed to find the optimal solution. By setting up the problem using MIP, the search space for the optimization problem is reduced significantly, thereby speeding up the solution process. The ratio of the dimension of the traditional method over the proposed formulation is approximately an order N (single) to 2*N (arraying), where N is the number of receiving antennas of a node. By introducing a special penalty function, the MIP problem with non-differentiable cost function and nonlinear constraints can be converted into a continuous variable problem, whose solution is possible.
Lee, Charles H.↗
Determination of navigation FDI thresholds using a Markov model
A method for determining time-varying Failure Detection and Identification (FDI) thresholds for single sample decision functions is described in the context of a triplex system of inertial platforms. A cost function consisting of the probability of vehicle loss due to FDI decision errors is minimized. A discrete Markov model is constructed from which this cost can be determined as a function of the decision thresholds employed to detect and identify the first and second failures. Optimal thresholds are determined through the use of parameter optimization techniques. The application of this approach to threshold determination is illustrated for the Space Shuttle's inertial measurement instruments.
Walker, B. K.↗
Efficiency of N use by wheat as a function of influx and efflux of NO3
Since N assimilation is one of the most costly functions of a plant, its efflux before assimilation results in a serious energy cost and loss in efficiency which could decrease yields. Efficient crop production is critical to the Closed Ecology Life Support System (CELSS). The objective is to determine the extent of efflux of the N species NO3(-), NH4(+), NO2(-), and urea after uptake, and possible means of regulation. Researchers found that NO3 efflux became serious as its substrate level increased. Efflux/Influx (E/I) of NO3(-) was greater in darkness (35 percent) than in light (14 percent), and the ratio greatly increased with substrate NO3 (-), (up to 45 percent at 10 mM). It seems advantageous to use the lowest possible nutrient concentration of NO3(-). The feasibility of using ClO3(-) was assessed and its toxicity determined.
Huffaker, R. C.↗
Cost-minimized aircraft trajectories
For aircraft operating over fixed range, operating costs are basically sum of fuel cost and time cost; but determining minimum cost trajectory can be complex. Program optimizes trajectories with respect to cost function that is based on weighed sum of fuel cost and time cost. Minimum fuel, minimum time, and various delay trajectories are obtained by specifying particular values for fuel and time cost factors.
Lee, H.↗
Fuel-Efficient Descent and Landing Guidance Logic for a Safe Lunar Touchdown
The landing of a crewed lunar lander on the surface of the Moon will be the climax of any Moon mission. At touchdown, the landing mechanism must absorb the load imparted on the lander due to the vertical component of the lander's touchdown velocity. Also, a large horizontal velocity must be avoided because it could cause the lander to tip over, risking the life of the crew. To be conservative, the worst-case lander's touchdown velocity is always assumed in designing the landing mechanism, making it very heavy. Fuel-optimal guidance algorithms for soft planetary landing have been studied extensively. In most of these studies, the lander is constrained to touchdown with zero velocity. With bounds imposed on the magnitude of the engine thrust, the optimal control solutions typically have a "bang-bang" thrust profile: the thrust magnitude "bangs" instantaneously between its maximum and minimum magnitudes. But the descent engine might not be able to throttle between its extremes instantaneously. There is also a concern about the acceptability of "bang-bang" control to the crew. In our study, the optimal control of a lander is formulated with a cost function that penalizes both the touchdown velocity and the fuel cost of the descent engine. In this formulation, there is not a requirement to achieve a zero touchdown velocity. Only a touchdown velocity that is consistent with the capability of the landing gear design is required. Also, since the nominal throttle level for the terminal descent sub-phase is well below the peak engine thrust, no bound on the engine thrust is used in our formulated problem. Instead of bangbang type solution, the optimal thrust generated is a continuous function of time. With this formulation, we can easily derive analytical expressions for the optimal thrust vector, touchdown velocity components, and other system variables. These expressions provide insights into the "physics" of the optimal landing and terminal descent maneuver. These insights could help engineers to achieve a better "balance" between the conflicting needs of achieving a safe touchdown velocity, a low-weight landing mechanism, low engine fuel cost, and other design goals. In comparing the computed optimal control results with the preflight landing trajectory design of the Apollo-11 mission, we noted interesting similarities between the two missions.
guidance logic↗
Efficiency of N use by wheat as a function of influx and efflux of NO sub 3
Since N assimilation is one of the most costly functions of a plant, its efflux before assimilation results in a serious energy cost and loss in efficiency which could decrease yields. Efficient crop production is critical to the Controlled Ecological Life-Support System (CELSS). The objective is to determine the extent of efflux of the N species NO3(-), NH4(+), NO2(-), and urea after uptake, and possible means of regulation. It was found that NO3(-) efflux became serious as its substrate level increased. Efflux/Influx (E/I) of 3NO3(-) was greater in darkness (35 pct) than in light (14 pct) and the ratio greatly increased with increased substrate NO3(-), (up to 45 pct at 10 mM). It seems advantageous to use the lowest possible nutrient concentration of NO3(-). The feasibility of using ClO3(-) as a trapping agent (competitive inhibitor of NO3(-) uptake) for effluxed NO3(-) was assessed and its toxicity determined.
Huffaker, R. C.↗