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NASA NTRS1988

The expanded Lagrangian system for constrained optimization problems

Smooth penalty functions can be combined with numerical continuation/bifurcation techniques to produce a class of robust and fast algorithms for constrained optimization problems. The key to the development of these algorithms is the Expanded Lagrangian System which is derived and analyzed in this work. This parameterized system of nonlinear equations contains the penalty path as a solution, provides a smooth homotopy into the first-order necessary conditions, and yields a global optimization technique. Furthermore, the inevitable ill-conditioning present in a sequential optimization algorithm is removed for three penalty methods: the quadratic penalty function for equality constraints, and the logarithmic barrier function (an interior method) and the quadratic loss function (an interior method) for inequality constraints. Although these techniques apply to optimization in general and to linear and nonlinear programming, calculus of variations, optimal control and parameter identification in particular, the development is primarily within the context of nonlinear programming.

Poore, A. B.↗

NASA NTRS1989

A decoupled recursive approach for constrained flexible multibody system dynamics

A variational-vector calculus approach is employed to derive a recursive formulation for dynamic analysis of flexible multibody systems. Kinematic relationships for adjacent flexible bodies are derived in a companion paper, using a state vector notation that represents translational and rotational components simultaneously. Cartesian generalized coordinates are assigned for all body and joint reference frames, to explicitly formulate deformation kinematics under small deformation kinematics and an efficient flexible dynamics recursive algorithm is developed. Dynamic analysis of a closed loop robot is performed to illustrate efficiency of the algorithm.

Lai, Hao-Jan↗

NASA NTRS1990

Discrete approximations to optimal trajectories using direct transcription and nonlinear programming

A recently developed method for solving optimal trajectory problems uses a piecewise-polynomial representation of the state and control variables, enforces the equations of motion via a collocation procedure, and thus approximates the original calculus-of-variations problem with a nonlinear-programming problem, which is solved numerically. This paper identifies this method as a direct transcription method and proceeds to investigate the relationship between the original optimal-control problem and the nonlinear-programming problem. The discretized adjoint equation of the collocation method is found to have deficient accuracy, and an alternate scheme which discretizes the equations of motion using an explicit Runge-Kutta parallel-shooting approach is developed. Both methods are applied to finite-thrust spacecraft trajectory problems, including a low-thrust escape spiral, a three-burn rendezvous, and a low-thrust transfer to the moon.

Enright, Paul J.↗

NASA NTRS1990

The looped adhesive strip: An example of coplanar delamination interaction

The phenomenon of peeling and debonding of thin layers is a subject of interest to those concerned with adhesives, thin films, and layered materials. In recent years much attention has been focused on such problems as a result of increased interest and application of advanced composites and thin film coatings. A related problem which is of interest for its own sake but also represents a simple example of a tangled adhesive strip and of coplanar delamination interaction, is the problem of a looped adhesive strip. This is the subject of the present study. Researchers consider here the problem of an elastic strip which possesses an adherend on (at least) one of its surfaces. If the strip is deformed so that two portions of such a surface are brought into contact, a position of the strip becomes bonded and a loop is formed. Researchers are interested in determining the equilibrium configuration of such a strip and investigating the behavior of the strip when its edges are pulled apart. The problem is approached as a moving interior boundary problem in the calculus of variations with the strip modeled as an inextensible elastica and the bond strength characterized by its surface energy. A Griffith type energy criterion is employed for debonding, and solutions corresponding to the problem of interest obtained. The solution obtained will be seen to predict the interesting phenomenon of bond point propagation, as well as the more standard peeling type behavior. Numerical results demonstrating the phenomena of interest are presented as well and will be seen to reveal both stable and unstable propagation of the boundaries of the bonded portion of the strip, depending upon the loading conditions.

Bottega, W. J.↗

NASA NTRS1991

A game theoretic approach to a finite-time disturbance attenuation problem

A disturbance attenuation problem over a finite-time interval is considered by a game theoretic approach where the control, restricted to a function of the measurement history, plays against adversaries composed of the process and measurement disturbances, and the initial state. A zero-sum game, formulated as a quadratic cost criterion subject to linear time-varying dynamics and measurements, is solved by a calculus of variation technique. By first maximizing the quadratic cost criterion with respect to the process disturbance and initial state, a full information game between the control and the measurement residual subject to the estimator dynamics results. The resulting solution produces an n-dimensional compensator which expresses the controller as a linear combination of the measurement history. A disturbance attenuation problem is solved based on the results of the game problem. For time-invariant systems it is shown that under certain conditions the time-varying controller becomes time-invariant on the infinite-time interval. The resulting controller satisfies an H(infinity) norm bound.

Rhee, Ihnseok↗

NASA NTRS1991

Mission and system optimization of nuclear electric propulsion vehicles for lunar and Mars missions

The detailed mission and system optimization of low thrust electric propulsion missions is a complex, iterative process involving interaction between orbital mechanics and system performance. Through the use of appropriate approximations, initial system optimization and analysis can be performed for a range of missions. The intent of these calculations is to provide system and mission designers with simple methods to assess system design without requiring access or detailed knowledge of numerical calculus of variations optimizations codes and methods. Approximations for the mission/system optimization of Earth orbital transfer and Mars mission have been derived. Analyses include the variation of thruster efficiency with specific impulse. Optimum specific impulse, payload fraction, and power/payload ratios are calculated. The accuracy of these methods is tested and found to be reasonable for initial scoping studies. Results of optimization for Space Exploration Initiative lunar cargo and Mars missions are presented for a range of power system and thruster options.

Gilland, James H.↗

NASA NTRS1992

Minimization of vibration in elastic beams with time-variant boundary conditions

This paper presents an innovative method for minimizing the vibration of structures with time-variant boundary conditions (supports). The elastic body is modeled in two ways: (1) the first model is a letter seven type beam with a movable mass not to exceed the lower tip; (2) the second model has an arm that is a hollow beam with an inside mass with adjustable position. The complete solutions to both problems are carried out where the body is undergoing large rotation. The quasi-static procedure is used for the time-variant boundary conditions. The method developed employs partial differential equations governing the motion of the beam, including the effects of rigid-body motion, time-variant boundary conditions, and calculus of variations. The analytical solution is developed using Laplace and Fourier transforms. Examples of elastic robotic arms are given to illustrate the effectiveness of the methods developed.

Amirouche, F. M. L.↗

NASA NTRS1993

Incompressible spectral-element method: Derivation of equations

A fractional-step splitting scheme breaks the full Navier-Stokes equations into explicit and implicit portions amenable to the calculus of variations. Beginning with the functional forms of the Poisson and Helmholtz equations, we substitute finite expansion series for the dependent variables and derive the matrix equations for the unknown expansion coefficients. This method employs a new splitting scheme which differs from conventional three-step (nonlinear, pressure, viscous) schemes. The nonlinear step appears in the conventional, explicit manner, the difference occurs in the pressure step. Instead of solving for the pressure gradient using the nonlinear velocity, we add the viscous portion of the Navier-Stokes equation from the previous time step to the velocity before solving for the pressure gradient. By combining this 'predicted' pressure gradient with the nonlinear velocity in an explicit term, and the Crank-Nicholson method for the viscous terms, we develop a Helmholtz equation for the final velocity.

Deanna, Russell G.↗

NASA NTRS1993

Robust neighboring extremal guidance for the advanced launch system

With the availability of modern flight computers, realtime neighboring extremal guidance seems feasible. To overcome sensitivity to unknown system parameters and environmental uncertainties, a robust neighboring extremal guidance scheme is proposed. About the optimal trajectory, the accessory problem in the calculus of variations is formed, generating a quadratic cost criterion in the perturbed states and controls. By formulating a disturbance attenuation problem based upon the second variation cost criterion, a differential game is formulated. The game theoretic cost criterion is minimized with respect to the perturbed control but maximized with respect to the unknown parameters in the linearized dynamics. The resulting differential game problem gives rise to a two-point boundary-value problem solved using the sweep method. The sweep method solution provides a linear robust neighboring extremal guidance scheme that is applied to the Advanced Launch System.

Bain, John↗

NASA NTRS1993

Feedback control for unsteady flow and its application to the stochastic Burgers equation

The study applies mathematical methods of control theory to the problem of control of fluid flow with the long-range objective of developing effective methods for the control of turbulent flows. Model problems are employed through the formalism and language of control theory to present the procedure of how to cast the problem of controlling turbulence into a problem in optimal control theory. Methods of calculus of variations through the adjoint state and gradient algorithms are used to present a suboptimal control and feedback procedure for stationary and time-dependent problems. Two types of controls are investigated: distributed and boundary controls. Several cases of both controls are numerically simulated to investigate the performances of the control algorithm. Most cases considered show significant reductions of the costs to be minimized. The dependence of the control algorithm on the time-descretization method is discussed.

Choi, Haecheon↗

NASA NTRS1951

Torsion and transverse bending of cantilever plates

The problem of combined bending and torsion of cantilever plates of variable thickness, such as might be considered for solid thin high-speed airplane or missile wings, is considered in this paper. The deflections of the plate are assumed to vary linearly across the chord; minimization of the potential energy by means of the calculus of variations then leads to two ordinary linear differential equations for the bending deflections and the twist of the plate. Because the cantilever is analyzed as a plate rather than as a beam, the effect of constraint against axial warping in torsion is inherently included. The application of this method to specific problems involving static deflection, vibration, and buckling of cantilever plates is presented. In the static-deflection problems, taper and sweep are considered.

WINGS, SOLID↗

NASA NTRS1999

Human Mars Mission: Launch Window from Earth Orbit

The determination of orbital window characteristics is of major importance in the analysis of human interplanetary missions and systems. The orbital launch window characteristics are directly involved in the selection of mission trajectories, the development of orbit operational concepts, and the design of orbital launch systems. The orbital launch window problem arises because of the dynamic nature of the relative geometry between outgoing (departure) asymptote of the hyperbolic escape trajectory and the earth parking orbit. The orientation of the escape hyperbola asymptotic relative to the earth is a function of time. The required hyperbola energy level also varies with time. In addition, the inertial orientation of the parking orbit is a function of time because of the perturbations caused by the Earth's oblateness. Thus, a coplanar injection onto the escape hyperbola can be made only at a point in time when the outgoing escape asymptote is contained by the plane of parking orbit. Even though this condition may be planned as a nominal situation, it will not generally represent the more probable injection geometry. The general case of an escape injection maneuver performed at a time other than the coplanar time will involve both a path angle and plane change and, therefore, a delta V penalty. Usually, because of the delta V penalty the actual departure injection window is smaller in duration than that determined by energy requirement alone. This report contains the formulation, characteristics, and test cases for five different launch window modes for Earth orbit. These modes are: 1) One impulsive maneuver from a Highly Elliptical Orbit (HEO); 2) Two impulsive maneuvers from a Highly Elliptical Orbit (HEO); 3) One impulsive maneuver from a Low Earth Orbit (LEO); 4) Two impulsive maneuvers form LEO; and 5) Three impulsive maneuvers form LEO. The formulation of these five different launch window modes provides a rapid means of generating realistic parametric data for space exploration studies. Also the formulation provides vector and geometrical data sufficient for use as a good starting point in detail trajectory analysis based on calculus of variations, steepest descent, or parameter optimization program techniques.

Young, Archie↗

NASA NTRS1997

Preliminary Design of Low-Thrust Interplanetary Missions

For interplanetary missions, highly efficient electric propulsion systems can be used to increase the mass delivered to the destination and/or reduce the trip time over typical chemical propulsion systems. This technology is being demonstrated on the Deep Space 1 mission - part of NASA's New Millennium Program validating technologies which can lower the cost and risk and enhance the performance of future missions. With the successful demonstration on Deep Space 1, future missions can consider electric propulsion as a viable propulsion option. Electric propulsion systems, while highly efficient, produce only a small amount of thrust. As a result, the engines operate during a significant fraction of the trajectory. This characteristic makes it much more difficult to find optimal trajectories. The methods for optimizing low-thrust trajectories are typically categorized as either indirect, or direct. Indirect methods are based on calculus of variations, resulting in a two-point boundary value problem that is solved by satisfying terminal constraints and targeting conditions. These methods are subject to extreme sensitivity to the initial guess of the variables - some of which are not physically intuitive. Adding a gravity assist to the trajectory compounds the sensitivity. Direct methods parameterize the problem and use nonlinear programming techniques to optimize an objective function by adjusting a set of variables. A variety of methods of this type have been examined with varying results. These methods are subject to the limitations of the nonlinear programming techniques. In this paper we present a direct method intended to be used primarily for preliminary design of low-thrust interplanetary trajectories, including those with multiple gravity assists. Preliminary design implies a willingness to accept limited accuracy to achieve an efficient algorithm that executes quickly.

Sims, Jon A.↗

NASA NTRS2000

Guidance Concept for a Mars Ascent Vehicle First Stage

This paper presents a guidance concept for use on the first stage of a Mars Ascent Vehicle (MAV). The guidance is based on a calculus of variations approach similar to that used for the final phase of the Apollo Earth return guidance. A three degree-of-freedom (3DOF) Monte Carlo simulation is used to evaluate performance and robustness of the algorithm.

Queen, Eric M.↗

NASA NTRS2001

Human Exploration Missions Study Launch Window from Earth Orbit

The determination of orbital launch window characteristics is of major importance in the analysis of human interplanetary missions and systems. The orbital launch window characteristics are directly involved in the selection of mission trajectories, the development of orbit operational concepts, and the design of orbital launch systems. The orbital launch window problem arises because of the dynamic nature of the relative geometry between outgoing (departure) asymptote of the hyperbolic escape trajectory and the earth parking orbit. The orientation of the escape hyperbola asymptotic relative to earth is a function of time. The required hyperbola energy level also varies with time. In addition, the inertial orientation of the parking orbit is a function of time because of the perturbations caused by the Earth's oblateness. Thus, a coplanar injection onto the escape hyperbola can be made only at a point in time when the outgoing escape asymptote is contained by the plane of parking orbit. Even though this condition may be planned as a nominal situation, it will not generally represent the more probable injection geometry. The general case of an escape injection maneuver performed at a time other than the coplanar time will involve both a path angle and plane change and, therefore, a Delta(V) penalty. Usually, because of the Delta(V) penalty the actual departure injection window is smaller in duration than that determined by energy requirement alone. This report contains the formulation, characteristics, and test cases for five different launch window modes for Earth orbit. These modes are: (1) One impulsive maneuver from a Low Earth Orbit (LEO), (2) Two impulsive maneuvers from LEO, (3) Three impulsive maneuvers from LEO, (4) One impulsive maneuvers from a Highly Elliptical Orbit (HEO), (5) Two impulsive maneuvers from a Highly Elliptical Orbit (HEO) The formulation of these five different launch window modes provides a rapid means of generating realistic parametric data for space exploration studies. Also the formulation provides vector and geometrical data sufficient for use as a good starting point in detail trajectory analysis based on calculus of variations, steepest descent, or parameter optimization program techniques.

Young, Archie↗

NASA NTRS2002

Trajectories for High Specific Impulse High Specific Power Deep Space Exploration

Flight times and deliverable masses for electric and fusion propulsion systems are difficult to approximate. Numerical integration is required for these continuous thrust systems. Many scientists are not equipped with the tools and expertise to conduct interplanetary and interstellar trajectory analysis for their concepts. Several charts plotting the results of well-known trajectory simulation codes were developed and are contained in this paper. These charts illustrate the dependence of time of flight and payload ratio on jet power, initial mass, specific impulse and specific power. These charts are intended to be a tool by which people in the propulsion community can explore the possibilities of their propulsion system concepts. Trajectories were simulated using the tools VARITOP and IPOST. VARITOP is a well known trajectory optimization code that involves numerical integration based on calculus of variations. IPOST has several methods of trajectory simulation; the one used in this paper is Cowell's method for full integration of the equations of motion. An analytical method derived in the companion paper was also evaluated. The accuracy of this method is discussed in the paper.

Polsgrove, Tara↗

NASA NTRS2002

Simulation of Trajectories for High Specific Impulse Deep Space Exploration

Difficulties in approximating flight times and deliverable masses for continuous thrust propulsion systems have complicated comparison and evaluation of proposed propulsion concepts. These continuous thrust propulsion systems are of interest to many groups, not the least of which are the electric propulsion and fusion communities. Several charts plotting the results of well-known trajectory simulation codes were developed and are contained in this paper. These charts illustrate the dependence of time of flight and payload ratio on jet power, initial mass, specific impulse and specific power. These charts are intended to be a tool by which people in the propulsion community can explore the possibilities of their propulsion system concepts. Trajectories were simulated using the tools VARITOP and IPOST. VARITOP is a well known trajectory optimization code that involves numerical integration based on calculus of variations. IPOST has several methods of trajectory simulation; the one used in this paper is Cowell's method for full integration of the equations of motion. The analytical method derived in the companion paper was also used to simulate the trajectory. The accuracy of this method is discussed in the paper.

Polsgrove, Tara↗

NASA NTRS2002

Atmospheric Ascent Guidance for Rocket-Powered Launch Vehicles

An advanced ascent guidance algorithm for rocket- powered launch vehicles is developed. This algorithm cyclically solves the calculus-of-variations two-point boundary-value problem starting at vertical rise completion through main engine cutoff. This is different from traditional ascent guidance algorithms which operate in a simple open-loop mode until high dynamic pressure (including the critical max-Q) portion of the trajectory is over, at which time guidance operates under the assumption of negligible aerodynamic acceleration (i.e., vacuum dynamics). The initial costate guess is corrected based on errors in the terminal state constraints and the transversality conditions. Judicious approximations are made to reduce the order and complexity of the state/costate system. Results comparing guided launch vehicle trajectories with POST open-loop trajectories are given verifying the basic formulation of the algorithm. Multiple shooting is shown to be a very effective numerical technique for this application. In particular, just one intermediate shooting point, in addition to the initial shooting point, is sufficient to significantly reduce sensitivity to the guessed initial costates. Simulation results from a high-fidelity trajectory simulation are given for the case of launch to sub-orbital cutoff conditions as well as launch to orbit conditions. An abort to downrange landing site formulation of the algorithm is presented.

Dukeman, Greg A.↗