Prasenjit Sengupta

Formation establishment and reconfiguration using impulsive control

We analyze spacecraft formation establishment and reconfiguration problems for two-body orbits. The desired formations are characterized by nonsingular orbital-elemental differences. An analytical, two-impulse solution is proposed for achieving the desired orbital-elemental differences. Gausss variational equations are used to compute the corresponding impulse magnitudes analytically and the resulting solutions can be easily implemented using onboard computational resources. It is also shown that the cost obtained from the analytical solution differs by less than 1 % from that obtained by numerical optimization.

Relative Motion and the Geometry of Formations in Keplerian Elliptic Orbits with Arbitrary Eccentricity

The well-known Hill-Clohessy-Wiltshire equations that are used for the design of formation flight relative orbits are based on a circular reference orbit Classical solutions such as the projected circular or general circular relative orbit are no longer valid in the presence of eccentricity. This paper studies the effects of eccentricity on relative motion for Keplerian orbits. A new linear condition for bounded motion in relative position coordinates is derived that is valid for arbitrary eccentricities and epoch of the reference orbit. It is shown that the solutions to the Tschauner-Hempel equations that are used for rendezvous in elliptic orbits are directly related to the description of relative motion using small orbital element differences. A meaningful geometric parameterization for relative motion near a Keplerian elliptic orbit of arbitrary eccentricity is also developed. The eccentricity-induced effects are studied and exploited to obtain desired shapes of the relative orbit. Equations relating these parameters to initial conditions and differential classical and nonsingular elements are also derived. This parameterization is very useful for the analysis of more complicated models, such as the nonlinear relative motion problem.

Periodic relative motion near a keplerian elliptic orbit with nonlinear differential gravity

This paper presents a perturbation approach for determining relative motion initial conditions for periodic motion in the vicinity of a Keplerian elliptic orbit of arbitrary eccentricity, as well as an analytical solution for the relative orbit that accounts for quadratic nonlinearities in the differential gravitational acceleration. The analytical solution is obtained in the phase space of the rotating coordinate system, centered at the reference satellite, and is developed in terms of a small parameter relating relative orbit size, and semimajor axis and eccentricity of the reference orbit. The results derived are applicable for arbitrary epoch of the reference satellite. Relative orbits generated using the methodology of this paper remain bounded over much longer periods in comparison to the results obtained using other approximations found in the literature, because the semimajor axes of the satellites are shown to be matched to the second order in the small parameter. The derived expressions thus serve as excellent guesses for initiating a numerical procedure for matching the semimajor axes of the two satellites. Several examples support the claims in this paper.

Averaged Relative Motion and Applications to Formation Flight Near Perturbed Orbits

DOI: 10.2514/1.30620 This paper presents expressions for describing averaged relative motion between two satellites in neighboring orbits around an oblate planet. The theory assumes small relative distances between the satellites, but is uniformly valid for all elliptic orbits as well as the special case of a circular reference orbit by the use of nonsingular orbital elements. These expressions are useful when the short-periodic variations in relative position and velocity are of limited interest and, instead, the time-averaged behavior of the states is sought. The averaged expressions also provide insight into the effects of an oblate planet on bounded relative motion. For example, a bias term due to oblateness effects, hitherto unreported, has been identified in the radial position, which can be accounted for in the reference trajectory. Application of these expressions is shown in the derivation of an analytical filter that removes short-periodic variations in relative states without the use of tuned numerical filters, one for each frequency of interest,whicharenormallyusedfordisturbanceaccommodationincontrolsystemdesign.Theuseofthisanalytical filter is demonstrated for formation keeping on a prescribed relative trajectory.

Optimal Design of Satellite Formation Relative Motion Orbits Using Least-Squares Methods

Formation flying relative orbit design can be achieved by determining six initial conditions in the local-vertical- local-horizontal frame or, equivalently, six differential orbital elements. In this paper, two novel approaches are proposed to design perturbed satellite formation relative motion orbits using least-squares techniques. First, it is shown that the initial conditions required to approximate a desired formation geometry can be analytically solved for by using the Gim―Alfriend state transition matrix in conjunction with a linear least-squares approach. An improvement to this method is obtained by using the Gaussian least-squares differential correction approach and numerical integration of the equations of motion of the two satellites. Numerical results are presented to demonstrate the applications of the two approaches.

Near-Optimal Feedback Rendezvous in Elliptic Orbits Accounting for Nonlinear Differential Gravity

DOI: 10.2514/1.26650 This paper presents a novel approach to the design of near-optimal feedback control laws, for minimum-fuel rendezvous between satellites in elliptic orbits of arbitrary eccentricity. The rendezvous problem for the nonlinear differential gravity model is solved by the application of neighboring optimal feedback control methodology used in conjunction with a nominal trajectory, obtained by solving the related minimum-fuel feedback control problem for the linear Tschauner–Hempel equations, analytically. This novel closed-form solution is used to determine the best values of the final true anomaly by examining its effect on the cost to go for rendezvous. The neighboring feedback controllawaccountingfornonlineardifferentialgravityisobtainedbyusingageneralizedsweepmethod,validwhen the reference solution does not satisfy the first-order necessary conditions for optimality, exactly. Several numerical examples are analyzed to demonstrate the efficacy of the method.

Robust Landing Guidance Law for Impaired Aircraft

A guidance law for landing an impaired aircraft is described. The approach involves a planning phase wherein the performance limits of the aircraft are used to design the landing pattern and the reference speed, followed by the guidance computations for generating attitude commands. Commanded attitude can be displayed the pilot on a heads-up display for manual control, or can be coupled to the autopilot. In addition to the attitude commands, the guidance system can also generate the flap deployment schedules, and auto-brake system initiation trigger in equipped aircraft. Guidance system development is based on a pointmass nonlinear model of the aircraft. Feedback linearization technique is used to transform the system dynamics into a linear, time-invariant form. Finite interval differential game formulation is then used to derive the robust optimal guidance law. Inverse transformation of the guidance commands to the original coordinate system produces the roll, pitch and yaw attitude commands. Attitude commands can be tailored for both the crabbed or sideslip flight modes, and for the transitions between them. Simulation results are given to demonstrate the performance of the proposed approach.

Fundamental Frequencies of Satellite Relative Motion and Control of Formations

Expressions for the fundamental natural frequencies associated with J 2 -perturbed relative motion of satellites in near-circular orbits are derived. Special values of the orbit inclination, dependent on initial conditions, are obtained, for which the in-plane and out-of-plane fundamental frequencies remain equal to each other over an extended period of time, resulting in nonprecessing relative orbits. This result validates and generalizes a similar finding, based on numerical investigations by other researchers. The analysis is extended to the developments of accurate prediction and control models for fuel-optimal formation maintenance and intersatellite fuel balancing. Numerical simulation results are presented to demonstrate the accuracy of the developed models and the effects of frequency matching on control requirements.

Queueing Network Models of the National Airspace System

Understanding the relationships between trajectory uncertainties due to aviation operations, precision of navigation and control, and the traffic flow efficiency are central to the design of next generation Air Transportation Systems. Monte-Carlo simulations using air traffic simulation software packages can be used to quantify these effects. However, they are generally time consuming, and do not provide explicit relationships for comparing various technology options. On the other hand, queuing models of the air traffic system can rapidly demonstrate the influence of trajectory uncertainties on traffic flow efficiency, facilitating tradeoff studies in an effective and time-efficient manner. A methodology for incorporating the trajectory uncertainty models into queuing network models of the air traffic at national, regional and local scales is discussed. Usefulness of these models in assessing the impact of uncertainties on traffic flow efficiency is illustrated.

Impaired Aircraft Performance Envelope Estimation

A methodology for estimating the flight envelope of impaired aircraft using an innovative differential vortex lattice algorithm tightly coupled with an extended Kalman filter is presented. The approach exploits prior knowledge about the undamaged aircraft parameters to reduce the order of the estimation problem. Point-mass flight dynamic model and a set of approximate analytical methods for structural analysis are used in the development. Estimated damage parameters are used to determine the aircraft aerodynamic, structural, structural dynamic, and aeroelastic parameters. These parameters are then used to estimate the aircraft performance envelope and maneuver limits. As conceptualized in the present work, this data is displayed to the pilot to aid in effectively managing the aircraft flight after impairment. The data can also be used for implementing safe maneuvering and landing guidance laws in the future.