Near optimal shuttle trajectories using accelerated gradient methods
Optimal ascent trajectories for delta wing space shuttle flight, using accelerated gradient methods
Johnson, I. L.↗
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Constrained optimal ascent-flyback shuttle trajectories.
An optimal Space Shuttle ascent-flyback trajectory shaping capability is presented which is based on the accelerated gradient parameter optimization technique. A typical atmospheric flight branched optimization problem is analyzed which required the determination of 31 parameters. This parameter set includes the description of the vehicle attitude control angles for three branches of Shuttle flight: first stage ascent, second stage ascent, and first stage flyback. The important in-flight inequality constraints required to maintain the integrity of the vehicle are considered. Results indicate that for a launch into a 55 deg inclined ellipse, a 13% increase in payload can be realized by using optimal control in the first-stage ascent rather than the conventional gravity-turn steering.
Kamm, J. L.↗
A nonlinear programming approach for optimizing two-stage lifting vehicle ascent to orbit
An optimal atmospheric flight branched trajectory-shaping capability is presented based on the Davidon-Fletcher-Powell variable metric parameter optimization technique. Gradient information is generated using finite difference methods. A typical atmospheric flight branched optimization problem is analyzed which requires the determination of 31 parameters. This parameter set includes the three-dimensional description of vehicle attitude control angles for three branches of flight: first-stage ascent, second-stage ascent, and first-stage flyback. The important inflight inequality contraints required to maintain the integrity of the vehicles are considered. Some of the numerical methods employed are discussed, along with several new auxiliary techniques developed to improve the compatibility of the numerical gradient and iterator.
Kamm, J. L.↗
Rocket ascent G-limited moment-balanced optimization program (RAGMOP)
This document describes the RAGMOP (Rocket Ascent G-limited Momentbalanced Optimization Program) computer program for parametric ascent trajectory optimization. RAGMOP computes optimum polynomial-form attitude control histories, launch azimuth, engine burn-time, and gross liftoff weight for space shuttle type vehicles using a search-accelerated, gradient projection parameter optimization technique. The trajectory model available in RAGMOP includes a rotating oblate earth model, the option of input wind tables, discrete and/or continuous throttling for the purposes of limiting the thrust acceleration and/or the maximum dynamic pressure, limitation of the structural load indicators (the product of dynamic pressure with angle-of-attack and sideslip angle), and a wide selection of intermediate and terminal equality constraints.
Lyons, J. T.↗
Traj_opt User's Guide
Trajectory optimization program Traj_opt was developed at Ames Research Center to help assess the potential benefits of ultrahigh temperature ceramic materials applied to reusable space vehicles with sharp noses and wing leading edges. Traj_opt loosely couples the Ames three-degrees-of-freedom trajectory package Traj (see NASA-TM-2004-212847) with the SNOPT optimization package (Stanford University Technical Report SOL 98-1). Traj_opt version January 22, 2003 is covered by this user guide. The program has been applied extensively to entry and ascent abort trajectory calculations for sharp and blunt crew transfer vehicles. The main optimization variables are control points for the angle of attack and bank angle time histories. No propulsion options are provided, but numerous objective functions may be specified and the nonlinear constraints implemented include a distributed surface heating constraint capability. Aero-capture calculations are also treated with an option to minimize orbital eccentricity at apoapsis. Traj_opt runs efficiently on a single processor, using forward or central differences for the gradient calculations. Results may be displayed conveniently with Gnuplot scripts. Control files recommended for five standard reentry and ascent abort trajectories are included along with detailed descriptions of the inputs and outputs.
Saunders, David A.↗
A two-level trajectory decomposition algorithm featuring optimal intermediate target selection
A decomposition algorithm is presented that optimizes complex missions by partitioning the trajectory into natural segments such as ascent or entry. Each segment defines a full-rank targeting subproblem. These are solved sequentially using the Newton-Raphson algorithm. The master problem, representing the complete mission, is to determine subproblem targets and master-problem controls that optimize the mission objective subject to intersegment constraints. The gradient projection algorithm solves this problem using derivatives obtained analytically from finite-difference subproblem sensitivities. Thus, the mission is optimized by coordinating the solution of tractible subproblems. Computational results for a synchronous equatorial mission are included.
Petersen, F. M.↗
A two-level trajectory decomposition algorithm featuring optimal intermediate target selection
A decomposition algorithm is presented which optimizes complex missions by partitioning the trajectory into natural segments such as ascent or entry. Each segment defines a full-rank targeting subproblem. These are solved sequentially using the Newton-Raphson algorithm. The master problem, representing the complete mission, is to determine subproblem targets and master-problem controls that optimize the mission objective subject to intersegment constraints. The gradient projection algorithm solves this problem using derivatives obtained analytically from finite-difference subproblem sensitivities. Thus, the mission is optimized by coordinating the solution of tractible subproblems. Computational results for a synchronous equatorial mission are included.
Petersen, F. M.↗
Transformation techniques for minimax optimal control problems and their application to optimal flight trajectories in a windshear - Optimal abort landing trajectories
The optimal-control problem of abort-landing trajectories in the presence of low-altitude wind shear is investigated analytically. The vertical-plane Newtonian motion of a point-mass aircraft in a steady wind field is modeled, and a sequential gradient-restoration algorithm is applied. Numerical results showing the effects of wind-shear intensity, initial altitude, and power-setting rate are presented in extensive graphs and discussed in detail. Optimal trajectories for strong or severe wind shears are found to begin with a descent, followed by level flight and then an ascent after leaving the shear region.
Miele, A.↗
An adaptive guidance law for single stage to low earth orbit
An adaptive guidance algorithm based on a cubic spline representation of the ascent profile and imposition of a dynamic pressure constraint is studied for a single stage to low earth orbit. The flight path is divided into initial and terminal phases. In the initial phase, fully adaptive, and in the terminal phase, semi-adaptive, guidance schemes are used. The cubic spline paqrameters are determined by gradient optimization for maximum payload to orbit. In the terminal phase, a linear quadratic regulator is used to derive the optimal feedback gains to keep the vehicle close to the nominal path. The computational aspects of the guidance algorithm are examined and criteria are developed to ensure stability and convergence.
Pamadi, Bandu N.↗
Comparison of a discrete steepest ascent method with the continuous steepest ascent method for optimal programing
A discrete steepest ascent method which allows controls which are not piecewise constant (for example, it allows all continuous piecewise linear controls) was derived for the solution of optimal programming problems. This method is based on the continuous steepest ascent method of Bryson and Denham and new concepts introduced by Kelley and Denham in their development of compatible adjoints for taking into account the effects of numerical integration. The method is a generalization of the algorithm suggested by Canon, Cullum, and Polak with the details of the gradient computation given. The discrete method was compared with the continuous method for an aerodynamics problem for which an analytic solution is given by Pontryagin's maximum principle, and numerical results are presented. The discrete method converges more rapidly than the continuous method at first, but then for some undetermined reason, loses its exponential convergence rate. A comparsion was also made for the algorithm of Canon, Cullum, and Polak using piecewise constant controls. This algorithm is very competitive with the continuous algorithm.
Childs, A. G.↗
The numerical evaluation of maximum-likelihood estimates of the parameters for a mixture of normal distributions from partially identified samples
Likelihood equations determined by the two types of samples which are necessary conditions for a maximum-likelihood estimate are considered. These equations, suggest certain successive-approximations iterative procedures for obtaining maximum-likelihood estimates. These are generalized steepest ascent (deflected gradient) procedures. It is shown that, with probability 1 as N sub 0 approaches infinity (regardless of the relative sizes of N sub 0 and N sub 1, i=1,...,m), these procedures converge locally to the strongly consistent maximum-likelihood estimates whenever the step size is between 0 and 2. Furthermore, the value of the step size which yields optimal local convergence rates is bounded from below by a number which always lies between 1 and 2.
Walker, H. F.↗
An iterative procedure for obtaining maximum-likelihood estimates of the parameters for a mixture of normal distributions, 2
The problem of obtaining numerically maximum likelihood estimates of the parameters for a mixture of normal distributions is addressed. In recent literature, a certain successive approximations procedure, based on the likelihood equations, is shown empirically to be effective in numerically approximating such maximum-likelihood estimates; however, the reliability of this procedure was not established theoretically. Here, a general iterative procedure is introduced, of the generalized steepest-ascent (deflected-gradient) type, which is just the procedure known in the literature when the step-size is taken to be 1. With probability 1 as the sample size grows large, it is shown that this procedure converges locally to the strongly consistent maximum-likelihood estimate whenever the step-size lies between 0 and 2. The step-size which yields optimal local convergence rates for large samples is determined in a sense by the separation of the component normal densities and is bounded below by a number between 1 and 2.
Peters, B. C., Jr.↗
An iterative procedure for obtaining maximum-likelihood estimates of the parameters for a mixture of normal distributions
This paper addresses the problem of obtaining numerically maximum-likelihood estimates of the parameters for a mixture of normal distributions. In recent literature, a certain successive-approximations procedure, based on the likelihood equations, was shown empirically to be effective in numerically approximating such maximum-likelihood estimates; however, the reliability of this procedure was not established theoretically. Here, we introduce a general iterative procedure, of the generalized steepest-ascent (deflected-gradient) type, which is just the procedure known in the literature when the step-size is taken to be 1. We show that, with probability 1 as the sample size grows large, this procedure converges locally to the strongly consistent maximum-likelihood estimate whenever the step-size lies between 0 and 2. We also show that the step-size which yields optimal local convergence rates for large samples is determined in a sense by the 'separation' of the component normal densities and is bounded below by a number between 1 and 2.
Peters, B. C., Jr.↗
Shuttle ascent trajectory optimization with function space quasi-Newton techniques
A Space Shuttle ascent trajectory optimization problem from lift-off to orbital insertion is solved with a function space version of a quasi-Newton parameter optimization method developed by Broyden. The problem includes five parameter and one bounded-function controls, two state-variable constraints, and four terminal conditions. The bounded controls are treated directly, while the remaining constraints are adjoined to the performance index (maximum payload) with penalty functions. The problem is formulated as a four-phase variational problem (liftoff, pitch-over, gravity-turn, linear tangent steering), and the appropriate gradients are developed by first variation theory. A projection operator is introduced to aid in the interpretation of the algorithm with mixed parameter and function controls.
Edge, E. R.↗
Trajectory optimization based on differential inclusion
A method for generating finite-dimensional approximations to the solutions of optimal control problems is introduced. By employing a description of the dynamical system in terms of its attainable sets in favor of using differential equations, the controls are completely eliminated from the system model. Besides reducing the dimensionality of the discretized problem compared to state-of-the-art collocation methods, this approach also alleviates the search for initial guesses from where standard gradient search methods are able to converge. The mechanics of the new method are illustrated on a simple double integrator problem. The performance of the new algorithm is demonstrated on a 1-D rocket ascent problem ('Goddard Problem') in presence of a dynamic pressure constraint.
Seywald, Hans↗
Trajectory optimization based on differential inclusion
A method for generating finite-dimensional approximations to the solutions of optimal control problems is introduced. By employing a description of the dynamical system in terms of its attainable sets in favor of using differential equations, the controls are completely eliminated from the system model. Besides reducing the dimensionality of the discretized problem compared to state-of-the-art collocation methods, this approach also alleviates the search for initial guesses from where standard gradient search methods are able to converge. The mechanics of the new method are illustrated on a simple double integrator problem. The performance of the new algorithm is demonstrated on a 1-D rocket ascent problem (`Goddard Problem') in presence of a dynamic pressure constraint.
Seywald, Hans↗
Trajectory optimization based on differential inclusion
A method for generating finite-dimensional approximations to the solutions of optimal control problems is introduced. By employing a description of the dynamical system in terms of its attainable sets in favor of using differential equations, the controls are completely eliminated from the system model. Besides reducing the dimensionality of the discretized problem compared to state-of-the-art collocation methods, this approach also alleviates the search for initial guesses from where standard gradient search methods are able to converge. The mechanics of the new method are illustrated on a simple double integrator problem. The performance of the new algorithm is demonstrated on a I-D rocket ascent problem ('Goddard Problem') in presence of a dynamic pressure constraint.
Seywald, Hans↗
Optimal sun-alignment techniques of large solar arrays in electric propulsion spacecraft
Optimum sun-alignment of large solar arrays in electric propulsion spacecraft operating in earth orbit requires periodic roll motions around the thrust axis, synchronized with the apparent conical motion of the sun line. This oscillation is sustained effectively with the aid of gravity gradient torques while only a small share of the total torque is being contributed by the attitude control system. Tuning the system for resonance requires an appropriate choice of moment-of-inertia characteristics. To minimize atmospheric drag at low orbital altitudes the solar array is oriented parallel, or nearly parallel, to the flight direction. This can increase the thrust-to-drag ratio by as much as an order of magnitude. Coupled with optimal roll orientation, this feathering technique will permit use of electric propulsion effectively at low altitudes in support of space shuttle or space station activities and in spiral ascent missions.
Meissinger, H. F.↗