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A complete nonlinear control law is derived for guiding an aircraft in minimum time from an arbitrary initial position and heading to a prescribed terminal position and heading in the horizontal plane. The solution is obtained with the aid of the maximum principle, and is implemented by constructing a digital-computer program for the resulting switching logic.
The concepts presented are particularly valuable in sounding rocket applications, where experiment data-gathering time above the atmosphere is measured in minutes per flight. One concept is a practical nonlinear control law for slewing a payload in one axis, with performance that approaches time-optimal as the uncertainty in angular acceleration approaches zero. The second concept is a practical transformation-of-coordinates system allowing slewing about an axis not coincident with any of the three control axes of the payload, so that the payload experiment axis may move along the great circle arc containing the initial and terminal target points. Implementation of both concepts is facilitated by the availability of variable-thrust cold-gas thrustors. Both concepts can result in significant reductions in slewing times compared with more conventional systems.
Accomplishments of the past year and plans for the coming year are highlighted as they relate to five year plans and the objectives of the following technical areas: aerothermal loads; multidisciplinary analysis and optimization; unsteady aerodynamics; and configuration aeroelasticity. Areas of interest include thermal protection system concepts, active control, nonlinear aeroelastic analysis, aircraft aeroelasticity, and rotorcraft aeroelasticity and vibrations.
Spatial differencing schemes which are not enstrophy conserving nor implicitly damping require global filtering of short waves to eliminate the build-up of energy in the shortest wavelengths due to aliasing. Takacs and Balgovind (1983) have shown that filtering on a sphere with a latitude dependent damping function will cause spurious vorticity and divergence source terms to occur if care is not taken to ensure the irrotationality of the gradients of the stream function and velocity potential. Using a shallow water model with fourth-order energy-conserving spatial differencing, it is found that using a 16th-order Shapiro (1979) filter on the winds and heights to control nonlinear instability also creates spurious source terms when the winds are filtered in the meridional direction.
An evaluation is made of six diverse examples of nonlinear treatments of slender body theory for the prediction of missile aerodynamic behavior. The cases in question are the application of area rule to store carriage design in the drag rise region, the estimation of destabilizing pitching moments associated with transonic projectiles, the pressure loadings on elliptical missile airframes, nonlinear control characteristics, roll control effectiveness in canard missile configurations, and novel approaches for vortex flow modeling.
The conceptions of transformation and canonical form have been much used to analyze the structure of linear systems. A coordinate system and a corresponding canonical form are developed for general nonlinear control systems. Their usefulness is demonstrated by showing that every feedback linearizable system becomes a system with only feedback paths in the canonical form.
The present paper is concerned with two-dimensional Euler equations and with schemes which are in use of the time of this writing. Most of the development presented carries over directly to three dimensions. The characteristics of the two-dimensional Euler equations in Cartesian coordinates are considered along with generalized curvilinear coordinate transformations, metric relations, invariants of the transformation, flux Jacobian matrices and eigensystems, numerical algorithms, flux split algorithms, implicit and explicit nonlinear control (smoothing), upwind differencing in supersonic regions, unsteady and steady-state computation, the diagonal form of implicit algorithm, metric differencing and invariants, boundary conditions, geometry and mesh generation, and sample solutions.
Some recent developments in the state of the art in missile aerodynamics are reviewed. Among the subjects covered are: (1) tri-service/NASA data base, (2) wing-body interference, (3) nonlinear controls, (4) hypersonic transition, (5) vortex interference, (6) airbreathers, supersonic inlets, (7) store separation problems, (8) correlation of missile data, (9) CFD codes for complete configurations, (10) engineering prediction methods, and (11) future configurations. Suggestions are made for future research and development to advance the state of the art of missile aerodynamics.
Recent developments in the state of the art in missile aerodynamics are reviewed. Among the subjects covered are: (1) Tri-service/NASA data base, (2) wing-body interference, (3) nonlinear controls, (4) hypersonic transition, (5) vortex interference, (6) airbreathers, supersonic inlets, (7) store separation problems, (8) correlation of missile data, (9) CFD codes for complete configurations, (10) engineering prediction methods, and (11) future configurations. Suggestions are made for future research and development to advance the state of the art of missile aerodynamics.
Aids for the visualization of high-dimensional scientific or other data must be designed. Simply casting multidimensional data into a two- or three-dimensional spatial metaphor does not guarantee that the presentation will provide insight or parsimonious description of the phenomena underlying the data. Indeed, the communication of the essential meaning of some multidimensional data may be obscured by presentation in a spatially distributed format. Useful visualization is generally based on pre-existing theoretical beliefs concerning the underlying phenomena which guide selection and formatting of the plotted variables. Two examples from chaotic dynamics are used to illustrate how a visulaization may be an aid to insight. Two examples of displays to aid spatial maneuvering are described. The first, a perspective format for a commercial air traffic display, illustrates how geometric distortion may be introduced to insure that an operator can understand a depicted three-dimensional situation. The second, a display for planning small spacecraft maneuvers, illustrates how the complex counterintuitive character of orbital maneuvering may be made more tractable by removing higher-order nonlinear control dynamics, and allowing independent satisfaction of velocity and plume impingement constraints on orbital changes.
The local geometric properties of a nonlinear control system defined by vector fields E(sub 1),..., E(sub M) are determined by the algebraic properties of the iterated Lie brackets of the E(sub j)'s. The problem of rewriting expressions is examined involving the E(sub j)'s in terms of the D(sub mu)'s in such a way as to handle effectively any cancellation occurring due to commuting of the D(sub mu)'s. A data structure is introduced which allows us to organize the computation to take advantage of the symmetries in the expression and reduce the operation count.
This paper describes and illustrates two matched-filter-theory based schemes for obtaining maximized and time-correlated gust-loads for a nonlinear airplane. The first scheme is computationally fast because it uses a simple one-dimensional search procedure to obtain its answers. The second scheme is computationally slow because it uses a more complex multidimensional search procedure to obtain its answers, but it consistently provides slightly higher maximum loads than the first scheme. Both schemes are illustrated with numerical examples involving a nonlinear control system.
Two matched filter theory based schemes are described and illustrated for obtaining maximized and time correlated gust loads for a nonlinear aircraft. The first scheme is computationally fast because it uses a simple 1-D search procedure to obtain its answers. The second scheme is computationally slow because it uses a more complex multi-dimensional search procedure to obtain its answers, but it consistently provides slightly higher maximum loads than the first scheme. Both schemes are illustrated with numerical examples involving a nonlinear control system.
An approach to the control of constrained dynamic systems such as multiple arm systems, multifingered grippers, and walking vehicles is described. The basic philosophy is to utilize a minimal set of inputs to control the trajectory and the surplus input to control the constraint or interaction forces and moments in the closed chain. A dynamic control model for the closed chain is derived that is suitable for designing a controller in which the trajectory and the interaction forces and moments are explicitly controlled. Nonlinear feedback techniques derived from differential geometry are then applied to linearize and decouple the nonlinear model. These ideas are illustrated through a planar example in which two arms are used for cooperative manipulation. Results from a simulation are used to illustrate the efficacy of the method.
Precise, smooth aerodynamic models are required for implementing adaptive, nonlinear control strategies. Accurate representations of aerodynamic coefficients can be generated for the complete flight envelope by combining computational neural network models with an Estimation-Before-Modeling paradigm for on-line training information. A novel method of incorporating first-partial-derivative information is employed to estimate the weights in individual feedforward neural networks for each aerodynamic coefficient. The method is demonstrated by generating a model of the normal force coefficient of a twin-jet transport aircraft from simulated flight data, and promising results are obtained.
A synthesis technique for handling unmodeled disturbances in nonlinear control law synthesis was advanced using differential game theory. Two types of modeling inaccuracies can be included in the formulation. The first is a bias-type error, while the second is the scale-factor-type error in the control variables. The disturbances were assumed to satisfy an integral inequality constraint. Additionally, it was assumed that they act in such a way as to maximize a quadratic performance index. Expressions for optimal control and worst-case disturbance were then obtained using optimal control theory.
In the past few years, significant interest has developed in using artificial neural networks to model and control nonlinear dynamical systems. While there exists many proposed schemes for accomplishing this and a wealth of supporting empirical results, most approaches to date tend to be ad hoc in nature and rely mainly on heuristic justifications. The purpose of this project was to further develop some analytical tools for representing nonlinear discrete-time input-output systems, which when applied to neural networks would give insight on architecture selection, pruning strategies, and learning algorithms. A long term goal is to determine in what sense, if any, a neural network can be used as a universal approximator for nonliner input-output maps with memory (i.e., realized by a dynamical system). This property is well known for the case of static or memoryless input-output maps. The general architecture under consideration in this project was a single-input, single-output recurrent feedforward network.
A new approach for global path planning of redundant manipulators is proposed. It poses the path planning problem as a finite time nonlinear control problem. The solution is found by a Newton-Raphson type algorithm. This technique is capable of handling various goal task descriptions as well as incorporating both joint and task space constraints. The algorithm has shown promising preliminary results in planning joint path sequences for 3R and 4R planar robots to meet Cartesian tip tracking and goal endpoint planning. It is robust with respect to local path planning problems such as singularity considerations and local minimum problems. Repetitive joint path solutions for cyclic end-effector tasks are also generated. Eventual goals of this work include implementation on full spatial robots, as well as provision of an interface for supervisory input to aid in path planning for more complex problems.