Frederick H. Lutze

Second-Order Relative Motion Equations

An approximate solution of second-order relative motion equations is presented. The equations of motion for a Keplerian orbit in spherical coordinates are expanded in Taylor series form using reference conditions consistent with that of a circular orbit. Only terms that are linear or quadratic in state variables are kept in the expansion. The method of multiple scales is employed to obtain an approximate solution of the resulting nonlinear differential equations, which are free of false secular terms. This new solution is compared with the previously known solution of the linear case to show improvement and with numerical integration of the quadratic differential equation to understand the error incurred by the approximation. In all cases, the comparison is made by computing the difference of the approximate state (analytical or numerical) from numerical integration of the full nonlinear Keplerian equations of motion. The results of two test cases show two orders of magnitude improvement in the second-order analytical solution compared with the previous linear solution over one period of the reference orbit.

Time-optimal lateral maneuvers of an aircraft

Results and analysis are presented from a study of time-optimal lateral maneuvers for an aircraft during the power-on-approach-to-landing portion of the flight, typically used for landing on an aircraft carrier. A full sixdegree-of-freedom model is used to model the motions of the aircraft. The optimal control problems of interest are formulated and a family of optimal solutions obtained for two classes of lateral maneuvers. These include an unconstrained maneuver and one with bank-angle and sideslip-angle constraints imposed on the approach trajectory. The control powers of elevator, rudder, and aileron are varied individually, and thus an estimate of the change in downrange distance to perform the lateral maneuvers due to the control power change is obtained.

Neural Network Control of Space Vehicle Intercept and Rendezvous Maneuvers

Neural networks are examined for use as optimal controllers. The effect of the addition of noise to the neural network input measurements is investigated to determine the performance robustness of the neural network controllers. These techniques are applied to the autonomous control of interceptor-to-target rendezvous missions. For this example, the target lies in a circular orbit and remains passive throughout the maneuver. The linearized Clohessy–Wiltshire equations with thrust are used to describe the relative motion of the two vehicles. Parameter optimization is used to generate the training data for the neural network designs. A combination of open-loop and closed-loop control is shown to work effectively for this problem.

Unified development of lateral-directional departure criteria

Several frequently used departure prediction indicators for both open- and closed-loop control of flight are developed using a unified, rigorous analytical approach applied to a linear version of the aircraft model. These criteria are for departure caused by aerodynamic disturbances only. It is shown that these indicators are limited in their accuracy because of restrictive assumptions and terms omitted. A second approach is presented that leads to the same results as the first, but is more applicable to the nonlinear problem. Some ideas concerning the application of the linear methods to the nonlinear problem are presented.

Fixed-trim re-entry guidance analysis

The terminal guidance problem for a fixed-trim re-entry body is formulated with the objective of synthesizing a closed-loop steering law. A transformation of variables and subsequent linearization of the motion, with the sight-line to the target as a reference, reduces the order of the state system for the guidance problem. The reduced order system, although nonlinear and time-varying, is simple enough to lend itself to synthesis of a class of guidance laws. A generalization of the feedforward device of classical control theory is successfully employed for compensation of roll autopilot lags. The proposed steering law exhibits superior miss-distance performance in a computational comparison with existing fixed-trim guidance laws.

Mixed H2/H-infinity optimal control for an elastic aircraft

A mixed //2/floo optimal control design and its application to a flight control problem of B-l aircraft is studied. The mixed #2/ 00 optimal control design is one of finding an internally stabilizing controller that minimizes the //2/#oo performance index subject to an inequality constraint on H^ norm. The application is a linear model of longitudinal motion of B-l aircraft. A standard eigenvalue problem that involves linear matrix inequalities is formulated, and efficient interior point algorithms are used to solve this problem numerically. Results show that this mixed //2/#oo optimal design leads the designer into a tradeoff between HI and HQQ objectives. Through this method, a designer can determine the controllers that result in the desired closed-loop noise rejection properties and stability robustness.

Trimmed drag considerations

The distribution of lift between the wing and tail surfaces of a conventional aircraft is examined in order to determine that which would produce the minimum drag for a given lift. Further, the center of gravity position is determined which gives the desired lift distribution and, at the same time, maintains aircraft trim. Analytical expressions are developed which clearly show the dependence of this optimal e.g. position on the various aerodynamic and geometric parameters. In particular, it is shown how large changes in position of the optimal e.g. position can occur for small changes in the tail downwash angle. Numerical results are presented for a small twin-engine aircraft.

Second-Order Equations for Rendezvous in a Circular Orbit

Introduction T HIS Note develops an approximate analytical solution of the two-body orbital boundary-value problem, known as Lambert’s problem, using a time-explicit analytical solution of the relative motion problem. Explicitly stated, the problem is to determine the initial velocity of a satellite, given two position vectors and a time of flight between them. Although the classical origin of the problem related to the determination of planetary orbits, present-day applications are in the area of intercept and rendezvous guidance, where the initial velocity of a maneuvering spacecraft for which its position will coincide with a target position at a specified time must be determined. A short history of Lambert’s problem is given in Ref. 1. Numerical solutions of the problem have been developed by Battin and Vaughan1,2 and Gooding3 for example; however, for small separation distances, such as those encountered in many intercept and rendezvous problems, analytical solutions may be determined using a relative motion formulation. A closed-form solution of Lambert’s problem has been developed previously by Clohessy and Wiltshire in Ref. 4 and has been applied to a variety of intercept and rendezvous problems, as in Refs. 5–10, for example. This solution results from a linear approximation of the equations of motion, and thus the validity range is limited. A higher-order approximation of the equations of motion may extend the range of applicability and lead to an increased level of accuracy. There have been several attempts at developing relative motion equations that take nonlinear dynamics into account.11−14 The solutions presented in Refs. 11–14 result from a straightforward expansion, a method that constructs a truncated Taylor series representation of the exact solution. Anthony and Sasaki,12 Kechichian,13 and Kelly14 show how the nonlinear relative motion equations may be used to solve Lambert’s problem. While these solutions are more accurate than the solutions resulting from the linearized model, the validity region is limited to short periods of time due to the presence of secular terms. For a fixed time, solutions derived from the straightforward expansion will converge to the value of the exact solution as the number of terms in the expansion grows. For a fixed number of terms, however, the accuracy is diminished as time increases and the secular terms begin to dominate the solution. Karlgaard and Lutze15 used a multiple-scale perturbation method to determine a solution of the relative motion problem that included the effects of second-order terms kept in the expansion of the Keplerian equations of motion about a circular reference orbit. The multiple-scale approach allowed elimination of the secular terms by determination of the appropriate differential equations that

NONLINEAR MODEL-FOLLOWING CONTROL APPLICATION TO AIRPLANE CONTROL

Nonlinear model-following control design is applied to the problem of control of the six degrees of freedom of an airplane that lacks direct control of lift and side force. The nonlinear expressions for the error dynamics of the model-following control are examined using Lyapunov stability analysis. The analysis results in nonlinear feedforward and feedback gains that are functions of the airplane and model states. As a consequence, gain scheduling requirements for the implementation of the model-following control are reduced to only those involving the estimation of stability and control derivatives of the airplane. The use of these gains is shown through an example application to the control of a nonlinear aerodynamic and engine model provided by NASA Ames-Dryden Flight Research Facility. The model being followed is based on a trajectory generation algorithm, and represents a form of dynamic inversion. HE design methodology to be used is based on the applica- tion of nonlinear model-following to the problem of the control of the six degrees of freedom of an airplane. This methodology is related to nonlinear inverse model theory. It is a more complete approach in that it provides a means for analysis of the dynamics of the errors involved in model-follow- ing. The particular approach has been successfully applied to the control of a nonlinear aerodynamic model of a high-angle- of-attack research vehicle (HARV) through large attitude and angle of attack changes. In general, model-following control attempts to make an actual airplane behave similarly to a prescribed mathematical model of an airplane with different force and moment character- istics than the actual airplane. The model behavior may be based on desirable flying qualities, and the matching of those flying qualities is taken to be the design objective. In this case the pilot controls are applied to the model (either conceptually or literally, to a simulation) and the airplane controls are deter- mined. Alternatively, the mathematical model may be a simplified representation of the actual airplane being controlled, in which case model-following control becomes a solution to the inverse problem. Here the state trajectory of the model is determined from a specification of a particular flight path or maneuver, and the airplane controls required to follow it are determined. Perfect, explicit model-following solutions to the inverse prob- lem provide more than the open-loop controls required to fly a maneuver, since this formulation allows control of the errors between the airplane and model during the maneuver. It is this application of model-following control that is used in this paper. To develop the nonlinear model-following controller, we will first review the model-following concepts used here. Initially a standard form of the airplane and model equations is presented with the conditions for perfect dynamic matching presented. Associated with the conditions for perfect dynamics matching are differential equations for the error. In many cases these error equations are linearized and standard linear control ideas applied to guarantee stability (i.e., they tend to go to zero in time). Hence one is led to a gain scheduling scheme. In the method presented, however, using an approach based on the stability theory of Lyapunov, a set of gains which insure stability of the nonlinear error dynamics can be found. These require no updates but are functions of the current state. The result of this analysis is illustrated through application to the nonlinear airplane simulation provided by NASA Ames- Dryden Flight Research Facility. In this application, the model being followed is a simplified description of the airplane being controlled. The model is not, however, directly flown by exter- nally applied (pilot) controls. Rather, it represents the states and state rates required to execute some prescribed maneuver.

Toward a theory of aircraft agility

A definition of an aircraft agility vector is given and the details for evaluating it are presented. This vector, which represents the time-rate of change of the forces acting on the aircraft, can be given in the usual axial, normal and lateral components. While the achieved value of the agility vector can be computed at any point along the flight-path, it is also possible to evaluate the set of achievable values. The construcion of such sets is demonstrated from aerodynamic and propulsive data for a modern fighter. Furthermore, by examining the limits of available controls, extreme points in the agility set can be determined. Figures showing a locus of these extremes for the representative fighter aircraft are presented with indications of the limiting control. Suggestions are given on the use of such plots in design and in comparisons of competing aircraft.