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NASA NTRS2018

Solar Sail Transfers from Invariant Objects to L5 Periodic Orbits

The rising interest in a mission to the Sun-Earth L5 point for heliophysics or Trojan asteroid search and the development of solar sails has opened the question of using solar radiation pressure for such a mission. Different solar sail trajectories to the Sun-Earth L5 point are constructed. Different invariant objects in the neighbourhood of Earth are considered for departure: equilibrium points, families of periodic orbits and their associated invariant manifold. Using a multi-objective genetic algorithm, differential correction and the optimal control solver PSOPT the transfers are obtained. The approach followed results in fast solar sail transfers which can be used for the preliminary design of a mission to the Sun-Earth L5 point.

Fernandez, Alvaro↗

NASA NTRS1999

Optimization of Insertion Cost for Transfer Trajectories to Libration Point Orbits

The objective of this work is the development of efficient techniques to optimize the cost associated with transfer trajectories to libration point orbits in the Sun-Earth-Moon four body problem, that may include lunar gravity assists. Initially, dynamical systems theory is used to determine invariant manifolds associated with the desired libration point orbit. These manifolds are employed to produce an initial approximation to the transfer trajectory. Specific trajectory requirements such as, transfer injection constraints, inclusion of phasing loops, and targeting of a specified state on the manifold are then incorporated into the design of the transfer trajectory. A two level differential corrections process is used to produce a fully continuous trajectory that satisfies the design constraints, and includes appropriate lunar and solar gravitational models. Based on this methodology, and using the manifold structure from dynamical systems theory, a technique is presented to optimize the cost associated with insertion onto a specified libration point orbit.

Howell, K. C.↗

NASA NTRS2017

Reachability Subspace Exploration Using Continuation Methods

Reachability manifold computation suffers from the curse of dimensionality and for large state spaces is computationally intractable. This paper examines the use of continuation methods to address this issue by formulating the reachability sub-space manifold calculation into a number of initial valued problems. As a result of computing the reachability manifold for a subspace of interest, an exponential improvement in computational cost occurs. This concept is applied to a position subspace reachability problem of a spacecraft in a Keplerian orbit under maximum thrust constraints. Future work includes a comparison of the proposed method with computing reachability manifolds using viscosity solutions of the Hamilton Jacobi Bellman partial differential equation.

orbital mechanics↗

NASA NTRS1949

Theory of Characteristics

The theory of characteristics will be presented generally for quasilinear differential equations of the second order in two variables. This is necessary because of the manifold requirements to be demanded from the theory of characteristics.

Tollmien, W.↗

NASA NTRS1971

Control of functional differential equations to target sets in function space

Optimal control of systems governed by functional differential equations of retarded and neutral type is considered. Problems with function space initial and terminal manifolds are investigated. Existence of optimal controls, regularity, and bang-bang properties are discussed. Necessary and sufficient conditions are derived, and several solved examples which illustrate the theory are presented.

Banks, H. T.↗

NASA NTRS1972

Control of functional differential equations of retarded and neutral type to target sets in function space.

Optimal control of systems governed by functional differential equations of retarded and neutral type is considered. Problems with function space initial and terminal manifolds are investigated. Existence of optimal controls, regularity, and bang-bang properties are discussed. Necessary and sufficient conditions are derived and several solved examples which illustrate the theory are presented.

Banks, H. T.↗

NASA NTRS1982

Global differential geometry: An introduction for control engineers

The basic concepts and terminology of modern global differential geometry are discussed as an introduction to the Lie theory of differential equations and to the role of Grassmannians in control systems analysis. To reach these topics, the fundamental notions of manifolds, tangent spaces, vector fields, and Lie algebras are discussed and exemplified. An appendix reviews such concepts needed for vector calculus as open and closed sets, compactness, continuity, and derivative. Although the content is mathematical, this is not a mathematical treatise but rather a text for engineers to understand geometric and nonlinear control.

Doolin, B. F.↗

NASA NTRS1998

On the Circulation Manifold for Two Adjacent Lifting Sections

The circulation functional relative to two adjacent lifting sections is studied for two cases. In the first case we consider two adjacent circles. The circulation is computed as a function of the displacement of the secondary circle along the axis joining the two centers and of the angle of attack of the secondary circle, The gradient of such functional is computed by deriving a set of elliptic functions with respect both to their argument and to their Period. In the second case studied, we considered a wing-flap configuration. The circulation is computed by some implicit mappings, whose differentials with respect to the variation of the geometrical configuration in the physical space are found by divided differences. Configurations giving rise to local maxima and minima in the circulation manifold are presented.

Zannetti, Luca↗

NASA NTRS1977

Single-axis gyroscopic motion with uncertain angular velocity about spin axis

A differential game approach is presented for studying the response of a gyro by treating the controlled angular velocity about the input axis as the evader, and the bounded but uncertain angular velocity about the spin axis as the pursuer. When the uncertain angular velocity about the spin axis desires to force the gyro to saturation a differential game problem with two terminal surfaces results, whereas when the evader desires to attain the equilibrium state the usual game with single terminal manifold arises. A barrier, delineating the capture zone (CZ) in which the gyro can attain saturation and the escape zone (EZ) in which the evader avoids saturation is obtained. The CZ is further delineated into two subregions such that the states in each subregion can be forced on a definite target manifold. The application of the game theoretic approach to Control Moment Gyro is briefly discussed.

Singh, S. N.↗

NASA NTRS1990

The design-to-cost manifold

A mathematical technique which both quantifies the design-to-cost process and the mass/complexity issue is presented. A simplified approximation of the PRICE H production-production cost is used to generate a dual set of differential equations which define the directions of maximum and minimum cost change over (mass, complexity) space. The equations are solved in closed form to obtain the one-dimensional design-to-cost and design-for-cost spaces. Preliminary results indicate that cost is relatively insensitive to changes in mass and that the reduction of complexity, both in the manufacturing process and in the spacecraft itself, is dominant in reducing cost. Two major objections to the design-to-cost method are discussed.

Dean, Edwin B.↗

NASA NTRS2002

Differential Flatness and Cooperative Tracking in the Lorenz System

In this paper the control of the Lorenz system for both stabilization and tracking problems is studied via feedback linearization and differential flatness. By using the Rayleigh number as the control, only variable physically tunable, a barrier in the controllability of the system is incidentally imposed. This is reflected in the appearance of a singularity in the state transformation. Composite controllers that overcome this difficulty are designed and evaluated. The transition through the manifold defined by such a singularity is achieved by inducing a chaotic response within a boundary layer that contains it. Outside this region, a conventional feedback nonlinear control is applied. In this fashion, the authority of the control is enlarged to the whole. state space and the need for high control efforts is mitigated. In addition, the differential parametrization of the problem is used to track nonlinear functions of one state variable (single tracking) as well as several state variables (cooperative tracking). Control tasks that lead to integrable and non-integrable differential equations for the nominal flat output in steady-state are considered. In particular, a novel numerical strategy to deal with the non-integrable case is proposed. Numerical results validate very well the control design.

Crespo, Luis G.↗

NASA NTRS1972

Perturbed nonlinear differential equations

The existence of a solution defined for all t and possessing a type of boundedness property is established for the perturbed nonlinear system y = f(t,y) + F(t,y). The unperturbed system x = f(t,x) has a dichotomy in which some solutions exist and are well behaved as t increases to infinity, and some solution exists and are well behaved as t decreases to minus infinity. A similar study is made for a perturbed nonlinear differential equation defined on a half line, R+, and the existence of a family of solutions with special boundedness properties is established. The ideas are applied to integral manifolds.

Proctor, T. G.↗

NASA NTRS1985

An intercomparison of carbon monoxide measurement techniques

Results from an intercomparison of techniques to measure tropospheric levels of carbon monoxide (CO) are discussed. The intercomparison was conducted as part of the National Aeronautics and Space Administration's Global Tropospheric Experiment (GTE) and was held at Wallops Island, VA, in July 1983. Instruments intercompared included a laser differential absorption method and three grab sample/gas chromatograph methods. The intercomparison consisted of simultaneous measurements of ambient levels of CO and controlled injections of CO from a common manifold. Results from the techniques exhibited a high degree of correlation among themselves and with changes in the CO mixing ratio. The results suggested a level of agreement among the techniques of about 15 percent. However, a day-to-day bias between the techniques was observed, which resulted in differences between techniques as large as 38 percent.

Hoell, J. M., Jr.↗

NASA NTRS1972

Prediction of pressure fluctuation in sounding rockets and manifolded recovery systems

The determination of altitude by means of barometric sensors in sounding rocket applications is discussed. A method for predicting the performance of such sensing systems is needed. A method is developed for predicting the pressure-time response of a volume subjected to subsonic air flow through from one to four passages. The pressure calculation is based on one-dimensional gas flow with friction. A computed program has been developed which solves the differential equations using a self-starting predictor-corrector integration technique. The input data required are the pressure sensing system dimensions, pressure forcing function(s) at the inlet port(s), and a trajectory over the time of analysis (altitude-velocity-time), if the forcing function is trajectory dependent. The program then computes the pressure-temperature history of the gas in the manifold over the time interval specified.

Laudadio, J. F.↗

NASA NTRS1974

Perturbed nonlinear differential equations

For perturbed nonlinear systems, a norm, other than the supremum norm, is introduced on some spaces of continuous functions. This makes possible the study of new types of behavior. A study is presented on a perturbed nonlinear differential equation defined on a half line, and the existence of a family of solutions with special boundedness properties is established. The ideas developed are applied to the study of integral manifolds, and examples are given.

Proctor, T. G.↗

NASA NTRS1992

Control and stabilization of nonholonomic dynamic systems

The paper establishes a theoretical framework for the control of nonholonomic dynamic systems, i.e., dynamic systems with nonintegrable constraints, with emphasis on control properties for nonholonomic systems that have no counterpart in holonomic systems. A model for nonholonomic dynamic systems is presented in terms of differential-algebraic equations defined on a phase space. A reduction procedure is carried out to obtain reduced-order state equations. Conditions for smooth asymptotic stabilization to an m-dimensional equilibrium manifold are presented. It is also demonstrated that a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback. An approach using geometric phases is developed as a basis for the control of Caplygin systems.

Bloch, Anthony M.↗

NASA NTRS2012

Current-State Constrained Filter Bank for Wald Testing of Spacecraft Conjunctions

We propose a filter bank consisting of an ordinary current-state extended Kalman filter, and two similar but constrained filters: one is constrained by a null hypothesis that the miss distance between two conjuncting spacecraft is inside their combined hard body radius at the predicted time of closest approach, and one is constrained by an alternative complementary hypothesis. The unconstrained filter is the basis of an initial screening for close approaches of interest. Once the initial screening detects a possibly risky conjunction, the unconstrained filter also governs measurement editing for all three filters, and predicts the time of closest approach. The constrained filters operate only when conjunctions of interest occur. The computed likelihoods of the innovations of the two constrained filters form a ratio for a Wald sequential probability ratio test. The Wald test guides risk mitigation maneuver decisions based on explicit false alarm and missed detection criteria. Since only current-state Kalman filtering is required to compute the innovations for the likelihood ratio, the present approach does not require the mapping of probability density forward to the time of closest approach. Instead, the hard-body constraint manifold is mapped to the filter update time by applying a sigma-point transformation to a projection function. Although many projectors are available, we choose one based on Lambert-style differential correction of the current-state velocity. We have tested our method using a scenario based on the Magnetospheric Multi-Scale mission, scheduled for launch in late 2014. This mission involves formation flight in highly elliptical orbits of four spinning spacecraft equipped with antennas extending 120 meters tip-to-tip. Eccentricities range from 0.82 to 0.91, and close approaches generally occur in the vicinity of perigee, where rapid changes in geometry may occur. Testing the method using two 12,000-case Monte Carlo simulations, we found the method achieved a missed detection rate of 0.1%, and a false alarm rate of 2%.

Carpenter, J. Russell↗

NASA NTRS1993

Accurate computation and continuation of homoclinic and heteroclinic orbits for singular perturbation problems

In earlier papers, Doedel and the authors have developed a numerical method and derived error estimates for the computation of branches of heteroclinic orbits for a system of autonomous ordinary differential equations in R(exp n). The idea of the method is to reduce a boundary value problem on the real line to a boundary value problem on a finite interval by using a local (linear or higher order) approximation of the stable and unstable manifolds. A practical limitation for the computation of homoclinic and heteroclinic orbits has been the difficulty in obtaining starting orbits. Typically these were obtained from a closed form solution or via a homotopy from a known solution. Here we consider extensions of our algorithm which allow us to obtain starting orbits on the continuation branch in a more systematic way as well as make the continuation algorithm more flexible. In applications, we use the continuation software package AUTO in combination with some initial value software. The examples considered include computation of homoclinic orbits in a singular perturbation problem and in a turbulent fluid boundary layer in the wall region problem.

Vaughan, William W.↗