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A mathematical model is presented for describing the bath temperature and composition trajectories for the decarburization of stainless steel for processes operating on an indus-trial scale. The model is based on a set of component balances, written with the aid of driving force expressions, the appropriate equilibrium relationships and the heat balance. The resultant set of ordinary differential equations were solved numerically. The predic-tions based on the model were compared with experimentally measured bath composition and temperature paths obtained for the operation of a 40 ton electric furnace. The predic-tions and measurements showed very good agreement. The mathematical model was then combined with a trajectory optimization technique to compute the optimal blowing programi .e., oxygen-argon content of the gas supplied) such that the total cost of the operation is minimized. This calculation was repeated for a series of cost factors.

Derivation of N-burn analytic solutions for propellant-optimal transfer trajectories of a vehicle in a vacuum between arbitrary boundary conditions. Variational changes in the desired boundary conditions are expressed, in general, in terms of variational changes in the control vector and in the initial state vector. All coefficient matrices are computed recursively in terms of the analytic matrices established from the subarcs of the N-burn solution. The solution is applicable to shuttle ascent (exoatmospheric), rendezvous, and deorbit problems. Consideration is also given to state-variable and control-variable inequality constraints.

A guidance method for the space shuttle's transition from hypersonic entry to subsonic cruising flight is presented. The method evolves from a numerical trajectory optimization technique in which kinetic energy and total energy (per unit weight) replace velocity and time in the dynamic equations. This allows the open end-time problem to be transformed to one of fixed terminal energy. In its ultimate form, E-Guidance obtains energy balance (including dynamic-pressure-rate damping) and path length control by angle-of-attack modulation and cross-range control by roll angle modulation. The guidance functions also form the basis for a pilot display of instantaneous maneuver limits and destination. Numerical results illustrate the E-Guidance concept and the optimal trajectories on which it is based.

A method is presented for designing control systems with uncertain plant parameters and initial states. The open-loop part of a semi-closed loop control guarantees a proper behaviour under the most likely circumstances and a neighbouring feedback is augmented so that trajectory dispersion under random perturbation is minimized. It is shown that the resulting optimization problem is similar to the optimal regulator problem with incomplete state feedback. An original numerical method based on imbedding techniques is proposed to provide a suboptimal control which avoids solving a complicated two-point boundary-value problem. An example exhibits the trajectory sensitivity reduction achieved.

This study is concerned with the development and application of a quasi-Newton descent algorithm for solving finite dimensional optimization problems subject to nonlinear equality and inequality constraints by the use of gradient projection and constraint restoration. The algorithm is applied to the positioning of a geostationary satellite which is interpreted as a problem of optimal TV-impulse orbital rendezvous.

9 Kocher, K., "Eine lineare Theorie der Optimierungsprobleme bei Raketen mit Kleinem Schub," Dissertation, 1971, Eidgenoessische Technische Hochschule, Zurich, Switzerland. 10 Sundman, K. F., "Memoire sur le Probleme des Trois Corps," Ada Mathematica, Vol. 36, 1912, pp. 105-179. 11 Sperling, H., "Computation of Keplerian Conic Sections," ARS Journal, Vol. 31, 1961, pp. 660-661; and private communications. 12 Burdet, C. A., "Le Mouvement Keplerien et les Oscillateurs Harmoniques," J fur die reine u. angewandte Mathematik, Vol. 238, 1969. 13 Bryson, A. E. and Ho, Y.-C, Applied Optimal Control, Blaisdell, Waltham, Mass., 1969. 14 Czuchry, A. J. and Pitkin, E. T., "Regularization in Optimal Trajectory Problems," AIAA Journal, Vol. 6, No. 6, June 1968, pp. 1209-1210. 15 Sperling, H. J., "The Collision Singularity in a Perturbed TwoBody Problem," Celestial Mechanics, Vol. 7, 1969, pp. 213-221. 16 Baumgarte, J., "Numerical Stabilization of the Differential Equations of the Keplerian Motion," Celestial Mechanics, Vol. 5, 1972, pp. 490-501. 17 StumpfT, K., Himmelsmechanik, VEB, Verlag, Berlin, 1959. 18 Lewallen, J. M. et al., "Coordinate System Influence on the Regularized Trajectory Optimization Problem," Journal of Spacecraft and Rockets, Vol. 8, No. 1, Jan. 1971, pp. 15-20. 19 Tapley, B. D., "Regularization and the Calculation of Optimal Trajectories," Celestial Mechanics, Vol. 2, 1970, pp. 319-333.

This paper discusses the results of employing a first-order optimization algorithm to the problem of minimizing the weight of the re-entry thermal protection system (TPS) for the space shuttle. Mathematical models of two types of thermal protection systems are derived, a metallic TPS and a reusable surface insulation (RSI) TPS. Optimal entries were generated using maximum orbiter nose temperature as a parameter. Thermal protection system weights were computed for both fixed and variable angles of attack using three-dimensional entry trajectories. Results indicated that variable angle-of-attack entries require less thermal protection system weight than entries at a constant angle of attack (35°) for both systems considered.

A retargeting procedure is developed for use as a nonlinear low thrust guidance scheme. The selection of a control program composed of a sequence of inertially fixed thrust-acceleration vectors permits all trajectory computations to be made with closed form expressions, and allows the controls to be represented by constant parameters, thrust-acceleration vectors and thrusting times. By requiring each trajectory to be time optimal, the guidance problem is transformed into a parameter optimization problem which is solved by the conjugate gradient method. The scheme is applied to a low thrust capture mission, and the results of computer simulations are presented.

Apollo Lunar-descent Guidance transfers the Lunar Module from a near-circular orbit to touchdown, traversing 17^o central angle and 15 km altitude in 11 min. A group of interactive programs in an onboard computer guide the descent, controlling altitude and the descent propulsion system throttle. A ground-based program precomputes guidance targets. This paper describes the concepts involved. Explicit and implicit guidance are discussed, guidance equations are derived, and the earlier Apollo explicit equation is shown to be an inferior special case of the later implicit equation. The paper describes interactive guidance by which the two-man crew selects a landing site in favorable terrain and directs the trajectory there. Interactive terminal-descent guidance enables the crew to control the essentially vertical descent rate in order to land in minimum time with safe contact speed. The attitude maneuver routine uses concepts that make gimbal lock inherently impossible. The throttle routine yields zero steady-state thrust-acceleration error or avoids operation within a thrust region forbidden because of hardware limitations. The ground-based program precomputes guidance targets which shape the trajectory to produce an efficient descent with adequate visibility and no transients at the final phasic interface.

Optimizing the trajectory of a low thrust space vehicle usually means solving a nonlinear two-point boundary value problem. In general, accuracy requirements necessitate extensive computation times. In celestial mechanics regularizing transformations of the equations of motion are used to eliminate computational and analytical problems that occur during close approaches to gravitational force centers. It was shown in previous investigations that regularization in the formulation of the trajectory optimization problem may reduce the computation time. In this study, a set of regularized equations describing the optimal trajectory of a continuously thrusting space vehicle is derived. The computational characteristics of the set are investigated and compared to the classical Newtonian unregularized set of equations. The comparison is made for low thrust minimum time escape trajectories. The comparison indicates that in the cases investigated for bad initial guesses of the known boundary values a remarkable reduction in the computation time was achieved. Furthermore, the investigated set of regularized equations shows high numerical stability even for long duration flights and is less sensitive to errors in the guesses of the unknown boundary values.

The determination of optimal rocket control profiles and the guidance* procedures used to approximate these profiles have received much attention during the last decade. Simple guidance procedures have sufficed for the Apollo flights. However, for the space shuttle which will have a low-thrust, orbital maneuvering system a more sophisticated guidance is required. This paper develops a guidance applicable to low-thrust orbital maneuvers and compares numerical results with those obtained using lengthy gradient and accelerated gradient methods.

Constant-thrust trajectory problem with gravitational acceleration as linear function of radius vector, obtaining minimum fuel solution in integrals

It is well-known that the thrust level used to optimize a rocket trajectory depends on the value of a "switching function" /£(t), and that the thrust can have an intermediate value only where K(t) is zero. In this paper the necessary conditions for a drag-free intermediate-thrust suharc (in a plane containing an inverse-squar e-law central force field) are pursued to the point of graphical representation. A coordinate system is presented on which can be defined regions where optimal intermediate-thrust trajectories are impossible. For this same coordinate system (the coordinates are functions of thrust direction and velocity direction) it is shown how analytical results permit plotting several families of contours, including lines of constant instantaneous eccentricity, instantaneous perigee angle, and acceleration ratio (ratio of thrust acceleration to local gravitational acceleration). It is further shown how numerical integration permits construction of other families of curves including a family of intermediate-thrust curves, one of which must be followed during an intermediate-thrust subarc. a b c

A slack variable is used to transform an optimal control problem with a scalar control and a scalar inequality constraint on the state variables into an unconstrained problem of higher dimension. It is shown that, for a p th order constraint, the p th time derivative of the slack variable becomes the new control variable. The usual Pontryagin principle or Lagrange multiplier rule gives necessary conditions of optimality. There are no discontinuities in the adjoint variables. A feature of the transformed problem is that any nominal control function produces a feasible trajectory. The optimal trajectory of the transformed problem exhibits singular arcs which correspond, in the original constrained problem, to arcs which lie along the constraint boundary.

Optimal control of dynamic control systems and geometry of reachability sets, discussing profit functions

In this paper, the definition of the primer vector is extended to include nonoptimal as well as optimal trajectories. With this definition, simple tests are developed which determine how a given trajectory can be improved (in terms of velocity requirements). This problem arose in the study of the use of impulsive trajectories to generate approximate adjoint initial conditions for finite thrust vehicles. To do this, the optimum fixed-time impulsive trajectory must be found. However, since many mission analyses are done on an impulsive basis, a wider application is foreseen. Necessary conditions are developed for when an additional impulse can improve the trajectory; how interior impulses of a multi-impulse trajectory can be moved so as to decrease the cost; and when initial and/or final coasts improve the trajectory. In the case of transfers between circular, coplanar orbits a geometric interpretation is given. For the case of an inverse-square gravitational field, the components of the primer vector can be calculated analytically. Using Floquet theory, a convenient form of this solution is presented.

The accelerated gradient projection technique is a terminally quadratically-convergent parameter optimization algorithm for use with non-linear constraints. This algorithm is used to find the minimum-propellant ascent trajectory of a multi-stage rocket vehicle.

A generalized "indirect" method of solving two-point boundary value problems is discussed in application to the problem of computing optimal trajectories in a vacuum. Improved numerical techniques make the method extremely fast when a good initial estimate of the solution is available, but it also converges, more slowly, from initial estimates that are far from the solution. Transversality conditions are combined with final-value constraints enabling the method to solve directly problems defined by constraints on arbitrary functions of final state. Section 1 defines the differential equations and initial and terminal conditions for optimal rocket trajectories ar iceesn tral gravitational field. The differential equations are given particularly simple form and transversality conditions are formulated analytically for typical orbital injection missions. Section 2 defines efficient numerical procedures for solving the initial value problem of optimal trajectories and so reduces the boundary value problem to a multidimensional zero-finding problem. Section 3 describes the generalized version of Newton's method used to solve this multidimensional zero-finding problem. Section 4 summarizes the results of an IBM 7094 implementation, giving execution times and convergence properties.

Necessary conditions for optimal lunar trajectories with discontinuous state variables and intermediate point constraints

Differential dynamic programming is a technique, based on dynamic programming rather than the calculus of variations, for determining the optimal control function of a nonlinear system. Unlike conventional dynamic programming where the optimal cost function is considered globally, differential dynamic programming applies the principle of optimality in the neighborhood of a nominal, possibly nonoptimal, trajectory. This allows the coefficients of a linear or quadratic expansion of the cost function to be computed in reverse time along the trajectory: these coefficients may then be used to yield a new improved trajectory (i.e., the algorithms are of the "successive sweep" type). A class of nonlinear control problems, linear in the control variables, is studied using differential dynamic programming. It is shown that for the free-end-point problem, the first partial derivatives of the optimal cost function are continuous throughout the state space, and the second partial derivatives experience jumps at switch points of the control function. A control problem that has an aualytic solution is used to illustrate these points. The fixed-end-point problem is converted into an equivalent free-end-point problem by adjoining the end-point constraints to the cost functional using Lagrange multipliers: a useful interpretation for Pontryagin's adjoint variables for this type of problem emerges from this treatment. The above results are used to devise new second- and first-order algorithms for determining the optimal bang-bang control by successively improving a nominal guessed control function. The usefulness of the proposed algorithms is illustrated by the computation of a number of control problem examples.

Improvements are made to the basic analogue -technique and experimental results obtained from three linear systems. A method is described for modifying the objective function contours to allow overall convergence of the iteration. Experimental results are obtained with the modified contours and compared with the results given by the basic technique.

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.

Some problems of automatic guidance of an aircraft in the horizontal plane are described. The horizontal guidance tasks, which such a flight control system should be capable of performing, were identified as being of three types: guiding the aircraft from any initial location and initial heading to (1) any final location and heading; (2) intercept and fly along a line of specified direction; and (3) a final location with arbitrary final heading. Guidance problems such as capturing an ILS beam at a specified point on the beam, intercepting a VOR radial, and point-to-point navigation can be formulated in terms of these problems. The guidance laws minimize the arc distance to fly between initial and final conditions subject to constraints on the minimum turning radius.

Pursuit problem improper solutions under existence assumptions emphasizing control parameters and pursuer trajectory

Multistage rocket trajectories optimized by second order numerical technique and digital computer program, considering coasting, vacuum flight and transition times

Optimum trajectory control for minimum heating paths during manned vehicle reentry at hyperbolic speeds

A second-order method for computing optimal control is described. The method computes control corrections from the state perturbations through feedback coefficients. On each successive trajectory a test is made for a conjugate point, and the algorithm automatically forms a control correction when such points occur. Unlike some earliermethods, terminal constraints of arbitrary dimensions are handled by the method. Numerical results for two cases of a non-linear example based on van der Pol's equation are included.

Time optimal rendezvous maneuver between neighboring elliptic orbits, discussing propulsive jet system with variable thrust

Analogue or hybrid computer methods for solving trajectory optimization problems usually require the solution of a two-point boundary value problem. A method of solving this problem is presented which does not require a good initial approximation to ensure convergence of the iteration. Analogue computer results are given with the method applied to a double integrator plant. Two different forms of performance index are considered.

This paper presents the analytical solutions of the problem of optimum maneuvering of a glide vehicle flying in the hypervelocity regime. The investigation is based on the approximation of Allen and Eggers; namely, that along the fundamental part of a reentry or ascent trajectory, the aerodynamic forces greatly exceed the components of the gravitational force in the directions tangent and normal to the flight path. The problem consists of finding an optimal control law for the lift such that the final velocity or the final altitude is maximized. This problem can be viewed as bringing the vehicle to the best condition for interception, penetration, or making an evasive maneuver. If the range is free, the optimal lift control is obtained in closed form. If the lift control is bounded, then bounded control is optimal whenever it is reached. The switching sequences for different cases are discussed, and it is shown that there are at most two switchings. Bounded lift control is always at the ends of the optimal trajectory; for the case of two switchings, the optimal trajectory has an inflection point.

The problem considered is that of trajectory optimization using step-by-step descent to minimum cost along the direction of the cost gradient with respect to the control. Using a hybrid computer, the gradient is computed di rectly as the response to nearly impulsive control perturba tions. A method is presented for computing the gradient when several terminal constraints are enforced. Examples of application of the method are presented. It is concluded that the direct gradient computation method has some significant advantages over other methods.

2 Miele, A., "A Survey of the Problems of Optimizing Flight Paths of Aircraft and Missiles," ARS preprint 1219-60. 3 Breakwell, J. V., "The Optimization of Trajectories," J. Soc. Indust. Appl. Math., vol. 7, no. 2, June 1959, pp. 215-247. 4 Kelley, H. J., "Gradient Theory of Optimal Flight Paths," ARS preprint 1230-60. 5 Kulakowski, L. J. and Stancil, R. T., "Rocket Boost Trajectories for Maximum Burn-Out Velocity," ARS JOURNAL, vol. 30, no. 7, July 1960, pp. 612-618. 6 Bliss. G. A., Lectures on the Calculus of Variations, University of Chicago Press, Chicago, First Ed., 1946, Chap. 7. 7 Cicala, C. and Miele, A., "Brachistochronic Maneuvers of a Variable Mass Aircraft in a Vertical Plane," J. Aero. Sci., vol. 22, no. 8, Aug. 1955, pp. 577-578. 8 Wheelon, A. D., "Free Flight of a Ballistic Missile," ARS JOURNAL, vol. 29, no. 12, Dec. 1959, pp. 915-926. 9 Leitmann, G., "On a Class of Variational Problems in Rocket Flight," J. Aero/Space Sci., vol. 26, no. 9, 1959. 10 Leitmann, G., "Minimum Transfer Time for a Power-Limited Rocket," XI Int. Astronautical Congress, Stockholm, 1960. 11 Fried, B. D., "On the Powered Flight Trajectory of an Earth Satellite," J E T PROPULSION, vol. 27, no. 6, June 1957, pp. 641-643.

The set of all controls that satisfy the Weierstrass necessary condition for optimality in the problem of time-open, coplanar orbit transfer via impulses is presented, along with the switching relations that must be satisfied at a corner in an optimal trajectory. This includes detailed data for eccentricities near unity. This study takes advantage of recently discovered closed-form solutions for the switching surfaces of this problem.

For a reentry glider, approximate solutions are found in closed form for the problem of maximizing a function of the terminal velocity, altitude, flight-path angle and heading angle subject to, at most, three terminal nonlinear constraints. The results given here extend the previous results in two important ways: (1) the second-order approximation of the entry dynamics of Loh is used instead of the first-order approximation of Allen and Eggers. This approximation is found to compare extremely well with an exact numerical optimal path. (2) A three-dimensional optimization problem is solved which includes Loh's second-order approximation where the roll angle as well as the lift coefficient are determined subject to constraints which include the terminal heading angle. Furthermore, closed-form solutions can also be obtained subject to in-flight constraints on the lift coefficient and roll angle.

Minimum fuel attitude control of spacecraft by Pontryagin principle and extended steepest descent method

It is assumed that the probability of impact for each maneuver is the same, and that the aspects of orbit determination and execution errors of each maneuver affect only the targeting. An approximation of the equal probability of impact contour is derived. It is assumed that the quarantine constraint is satisfied if the aim point is not inside the impact contour. A method is devised to find on each contour the optimum aim point which minimizes the so-called bias velocity which is required to bring back the spacecraft from the biased aim point to the originally desired aim point. The method is an improvement over the approach presented by Light (1965), and Craven and Wolfson (1967).

This paper discusses a means to describe and eventually to predict the response of a human or artificially intelligent controller which 1) has a constrained preview of the actual input course and which 2) observes the successive target values as being of nonuniform importance. Three examples are: driving an automobile in traffic, a blind pedestrian using a cane or electronic obstacle detector, and remote manipulation of solid objects using artificial sensors and effectors. Three models are presented which characterize constrained preview control better than can conventional transfer function techniques.

A method is presented for the systematic evaluation of two-point boundary value problems. Emphasis is placed on solving trajectory optimization problems formulated by the calculus of variations. The method presented in this paper, termed the adjoint method, iteratively converts the two-point boundary value problem into an initial value one, thus allowing a solution to be achieved in one run on a digital computer. Two sample problems that were solved by using the method are given: the case of a boost mission and that of an orbit transfer mission, both to be done in a minimum amount of time.

Recently, the notion of Differential Dynamic Programming has been used to obtain new second-order algorithms for solving non-linear optimal control problems. (Unlike conventional Dynamic Programming, the Principle of Optimality is applied in the neighborhood of a nominal, non-optimal, trajectory.) A novel feature of these algorithms is that they permit strong variations in the system trajectory. In this paper, Differential Dynamic Programming is used to develop a second-order algorithm for solving discrete-time dynamic optimization problems with terminal constraints. This algorithm also utilizes strong variations and, as a result, has certain advantages over existing discrete-time methods. A non-linear computed example is presented, and comparisons are made with the results of other researchers who have solved this problem. The experience gained during the computation has suggested some extensions to an earlier, previously published Differential Dynamic Programming algorithm for continuous time problems. These extensions, and their implications are discussed. (Author)

Exploration capabilities provided by Jupiter gravity assisted trajectories compared to direct ballistic flight trajectories

The Bolza problem of the calculus of variations in modern control notation is extended in scope to include situations in which a number of subarcs occur in a variety of ways. The subarcs are allowed to be overlapping and/or separated. This allows for several subarcs to occur in the same interval of the independent variable and also admits subarcs which are separated by jumps in the independent and state variables. In addition, the differential constraining equations and the integral quantity to be extremized are permitted to have different form from subarc to subarc. The necessary conditions for the extended Bolza problem are obtained by examining a related functional. Whereas the optimizing conditions for the state and control variables for each subarc are given by the usual Euler equations, new conditions associated with the end points of the subarcs are derived using ordinary theory of maxima and minima. The results presented here can be applied to a wide range of space trajectory problems. For some special cases, the theory reduces to results previously obtained and recorded elsewhere. A number of sample problems illustrating the theory are presented. The examples include the problem of inserting two payloads into separate orbits with one vehicle having two upper stages ignited simultaneously and a two-vehicle, dual-rendezvous problem.

This paper describes a method based upon the classical calculus of variations for solving directly Chebyshev minimax problems which arise in trajectory optimization. The close relationship between minimax problems and problems with state variable inequality constraints is used to gain insight into the minimax problem, and to define an order for minimax functions. The method is applicable to 1) afl problems in which the first time derivative of the minimax function does not contain control variables explicitly, and 2) all problems with a "flat" maximum (including problems in which the first time derivative of the minimax function contains control variables). The theory is applied to both a simple example and a formulation of the optimal re-entry heating problem whose performance index consists of a minimax term for maximum heating rate, a terminal function for maximum crossrange, and a path integral for minimum total heating. Several shooting and gradient-type numerical algorithms are suggested by the approach.

Optimum two impulse orbital transfer for arbitrary terminal conditions, discussing analytic characteristics

Closed form solution for minimum fuel constant thrust trajectories for vehicle transfer in vacuum between arbitrary boundary conditions

Analytic approximation for initial adjoint vector for optimal /minimum propellant/ space trajectories

Two position roll modulation of lift for aerodynamic braking for soft landing on Mars, solving boundary value problem by quasi- linearization