Showing papers in "Journal of Guidance Control and Dynamics in 2003"

Unscented Filtering for Spacecraft Attitude Estimation

Abstract: A new spacecraft attitude estimation approach based on the unscented filter is derived. For nonlinear systems the unscented filter uses a carefully selected set of sample points to map the probability distribution more accurately than the linearization of the standard extended Kalman filter, leading to faster convergence from inaccurate initial conditions in attitude estimation problems. The filter formulation is based on standard attitude-vector measurements using a gyro-based model for attitude propagation. The global attitude parameterization is given by a quaternion, whereas a generalized three-dimensional attitude representation is used to define the local attitude error. A multiplicative quaternion-error approach is derived from the local attitude error, which guarantees that quaternion normalization is maintained in the filter. Simulation results indicate that the unscented filter is more robust than the extended Kalman filter under realistic initial attitude-error conditions.

Attitude Error Representations for Kalman Filtering

Abstract: The quaternion has the lowest dimensionality possible for a globally nonsingular attitude representation. The quaternion must obey a unit norm constraint, though, which has led to the development of an extended Kalman filter using a quaternion for the global attitude estimate and a three-component representation for attitude errors. We consider various attitude error representations for this Multiplicative Extended Kalman Filter and its second-order extension.

Nonlinear Disturbance Observer-Enhanced Dynamic Inversion Control of Missiles

Abstract: Nonlinear dynamic inversion control (NDIC) is one of the most promising nonlinear control methods in aerospace engineering. For longitudinal dynamics of a missile, a nonlinear disturbance observer-based approach is proposed to enhance its disturbance attenuation ability and performance robustness against uncertain aerodynamic coefficients. Stability of the proposed nonlinear disturbance observer-enhanced NDIC is established. Its flexibility and efficiency are demonstrated by two different choices of the nonlinear gain function in the observer. Simulation results show that the nonlinear disturbance observer-based technique can significantly improve disturbance attenuation ability and performance robustness of dynamic inversion control.

State Transition Matrix of Relative Motion for the Perturbed Noncircular Reference Orbit

Abstract: A precise analytic solution that includes the effects of the reference orbit eccentricity and differential perturbations is needed for the relative motion of formation-flying satellites. As a result of the spherical Earthand circular reference orbit assumptions, Hill's equations, which have often been used for describing relative motion, are insufficient for the long-term prediction of the relative motion. A new approach, called the geometric method, is developed to obtain the state transition matrix for the relative motion that includes the effects caused by the reference orbit eccentricity and the differential gravitational perturbations. The geometric method uses the relationship between the relative states and differential orbital elements to obtain the state transition matrix instead of directly solving the complex relative motion differential equations. The state transition matrices are derived for both mean and osculating elements with the primary gravitational perturbation that results from the equatorial bulge term J 2 . Although the results are based on the J 2 effects, the approach can be extended easily to include other perturbing forces.

Historical review of air-bearing spacecraft simulators

Abstract: An overview of air-bearing spacecraft simulators is provided. Air bearings have been used for satellite attitude determination and control hardware verie cation and software development for nearly 45 years. It is interesting to consider the history of this technology: how early systems were e rst devised and what diverse capabilities current systems provide. First a survey is given of planar systems that give a payload freedom to translate and spin. Then several classes of rotational air bearings are discussed: those which simulate three-axis satellite attitude dynamics. The subsequent section discusses perhaps the most interesting facilities: those that provide both translational and three-dimensional rotational freedom. Thediverse capabilities each styleofair-bearing testbed provides, themany settings they can be found in, and ways to improve facility performance are described.

Onboard Generation of Three-Dimensional Constrained Entry Trajectories

Abstract: A methodology for very fast design of three-degree-of-freedom (3DOF) entry trajectories subject to all common inequality and equality constraints is developed. The approach makes novel use of the quasi-equilibrium glide phenomenon in lifting entry as a centerpiece for effective and efficient enforcement of the inequality constraints. The highly constrained nonlinear trajectory planning problem is decomposed into two sequential one-parameter search problems. The algorithm is able to generate a complete and feasible 3DOF entry trajectory of 25-min flight time in about 2-3 s on a desktop computer, given the entry conditions, values of constraint parameters, and final conditions. High-fidelity numerical simulations with a reusable launch vehicle model for various entry missions and trajectory planning using the space shuttle model are presented to demonstrate the capability and effectiveness of the algorithm.

Limited authority adaptive flight control for reusable launch vehicles

Abstract: In the application of adaptive e ight control, signie cant issues arise due to limitations in the plant inputs, such as actuator displacement limits, actuator rate limits, linear input dynamics, and time delay. A method is introduced that allows an adaptive law to be designed for the system without theseinput characteristicsand then to be applied to the system with these characteristics, without affecting adaptation. This includes allowing correct adaptation while the plant input is saturated and allows the adaptation law to function when not actually in control of the plant. To apply the method, estimates of actuator positions must be found. However, the adaptation law can correct for errors in these estimates. Proof of boundedness of system signals is provided for a single hidden-layer perceptron neural network adaptive law. Simulation results utilizing the methods introduced for neural network adaptive control of a reusable launch vehicle are presented for nominal e ight and under failure cases that require considerable adaptation.

Formation Flying: Accommodating Nonlinearity and Eccentricity Perturbations

Abstract: Hill-Clohessy-Wiltshire equations describe the relative motion of one satellite with respect to another in a circular reference orbit. Initial conditions that generate periodic solutions to these equations have to be corrected to obtain bounded solutions in the presence of nonlinearity of the differential gravitational acceleration model and eccentricity of the reference orbit. The corrections to the initial conditions due to quadratic terms in the differential gravitational acceleration for circular reference orbits are established e rst by using a perturbation approach. These corrections are related to the period-matching constraint required for bounded relative motion. Next, the solution to the linear problem including the effect of eccentricity is determined, and a procedure for correcting the along-track bias is presented. The two solutions obtained are combined to produce an asymptotic solution for the quadratic eccentricity problem. The effects of nonlinearity and eccentricity on the relative orbits are characterized as functions of their initial position in the formation. HE problem of relative motion dynamics of satellites has been of interest since the 1960s. Much of the work has been per- formed in the context of the rendezvous problem. Accurate mod- eling of the relative motion dynamics for initial conditions close to the target is important for the rendezvous problem. Formation e ying requires bounded relative motion. Therefore, the solutions of interest are restricted to a certain set of initial conditions that lead to bounded relative motion. One particular formation of interest is the relative orbit that is circular when projected on the local hori- zontal plane. This solution is an exact solution to Hill- Clohessy- Wiltshire(HCW)equationsthatmodeltherelativemotiondynamics under the assumption of a circular reference orbit, spherical Earth, and linearized differential gravitational acceleration. Nonlinearity of the differential gravitational acceleration, eccentricity of the ref- erence orbit,and the Earth' s oblateness are the three most important perturbations that breakdown the circular orbit solutions to HCW equations. In this paper, we study the effects of nonlinearity and the eccentricity perturbations; the effects of J2 are ignored. The developments in this paper draw on several previous studies. Melton 1 developed a state transition matrix solution for the lin- earized relative motion dynamics by incorporating the effect of ec- centricity.Inalhan etal. 2 obtainedthe conditionforboundedrelative orbit solutions to the linearized problem with nonzero eccentricity. The effects of including quadratic gravitational acceleration terms were studied in Refs. 3- 6. Karlgaard and Lutze 7 used the method of multiple timescalesto obtain aperturbationsolutiontothe quadratic problem formulated using spherical coordinates. Alfriend et al. 8 used a geometric approach to map relative motion coordinates to orbital element differences. Mitchell and Richardson 9 developed an active nonlinear controller to accommodate quadratic nonlinearities in the zero-eccentricity problem. Broucke 10 has presented an exact state transition matrix solution for the linearized elliptic rendezvous problem. The solution is obtained by taking partial derivatives with

Singularity escape/avoidance steering logic for control moment gyro systems

Abstract: A new steering logic based on a mixed weighted two-norm and least squares optimization solution overcomes a deficiency of being trapped in the momentum saturation singularities. The new steering logic utilizes an additional weighting matrix, W, in conjunction with deterministic dither signals in a weighting matrix V. The new steering logic also provides a simple means of avoiding troublesome, internal elliptic singularities which are commonly encountered by most other pseudoinverse-based steering logic. Various CMG systems, such as a typical pyramid array of four single-gimbal CMGs, two and three parallel single-gimbal CMG configurations, and four parallel double-gimbal CMGs of the International Space Station are described as illustrative embodiments demonstrating the simplicity and effectiveness of the new steering logic.

Closed-loop endoatmospheric ascent guidance

Abstract: A comprehensive treatment to the optimal atmospheric ascent problem of launch vehicles subject to path constraints and final condition constraints is presented. The development is particularly tailored for the purpose of the eventual realization of closed-loop endoatmospheric ascent guidance. It is demonstrated that the classical finite difference method for two-point boundary-value problems is well suited for solving the optimal ascent problem onboard. The performance and execution of the guidance algorithm are assessed with a series of open-loop and closed-loop tests using the vehicle data of a reusable launch vehicle. The test results render strong supporting evidence to the belief that closed-loop optimal endoatmospheric ascent guidance is now feasible.

Optimal Two-Impulse Rendezvous Using Multiple-Revolution Lambert Solutions

Abstract: Theminimum- ¢V,e xed-time,two-impulserendezvousbetweentwo spacecraftorbitingalong two coplanarunidirectional circular orbits (moving-target rendezvous )is studied. To reach thisgoal, the minimum- ¢V, e xed-time, two-impulse transfer problem between two e xed points on two circular orbits is e rst solved. This e xed-endpoint transferisrelated to the moving-target rendezvousproblem by a simple transformation. The e xed-endpoint transfer problem is solved using the solution to the multiple-revolution Lambert problem. A solution procedure is proposed based on the study of an auxiliary transfer problem. When this procedure is used, the minimum ¢V of the moving-target rendezvous problem without initial and terminal coasting periodsis obtained for a range of separation angles and timesofe ight. Thus, a contour plot ofthecostvs separation angleand transfertime isobtained. This contour plot, along with a sliding rule, facilitates the task of e nding the optimal initial and terminal coasting periods and, hence, obtaining the globally optimal solution for the moving-target rendezvous problem. Numerical examples demonstrate the application of the methodology to multiple rendezvous of satellite constellations on circular orbits.

Formation Flight Optimization Using Extremum Seeking Feedback

Abstract: A comprehensive design procedure based on extremum seeking for minimum power demand formation e ight is presented, the e rst with performance guarantees. The procedure involves the design of a new wake robust formation hold autopilot and transformation of the closed-loop aircraft dynamics to a form in which a newly available rigorous design procedure for extremum seeking is applicable. The design procedure is applied to a formation of Lockheed C-5s, extending the use of maximum performance formation e ight to large transports. By the use of availableexperimental wake data of the C-5, a model of the aircraft in the wakeis developed that models aerodynamic interference as feedback nonlinearities. Thus, this work is also the e rst to attain stable extremum seeking for a plant with nonlinear feedback. Optimal formation e ight is attained by online minimization of an easily measurable objective, the pitch angle of the wingman.

Solution of the Elliptic Rendezvous Problem with the Time as Independent Variable

Abstract: An explicit solution is given for the linearized motion of a chaser in a close neighborhood of a target in an elliptic orbit. The solution is a direct generalization of the Clohessy-Wiltshire equations that are widely used for circular orbits. In other words, when the eccentricity is set equal to zero in the new formulas, the well-known Clohessy-Wiltshire formulas are obtained. The solution is completely explicit in the time. As a starting point, a closed-form solution is found of the de Vries equations of 1963. These are the linearized equations of elliptic motion in a rotating coordinate system, rotating with a variable angular velocity. This solution is shown to be obtained simply by taking the partial derivatives, with respect to the orbit elements, of the two-body solution in polar coordinates. When four classical elements are used, four linearly independent solutions of the de Vries equations are obtained. However, the classical orbit elements turn out to be singular for circular orbits. The singularity is removed by taking appropriate linear combinations of the four solutions. This gives a 4 by 4 fundamental solution matrix R that is nonsingular and reduces to the Clohessy-Wiltshire solution matrix when the eccentricity is set equal to zero.

Aspects of Control for a Parafoil and Payload System

Abstract: A parafoil controlled by parafoil brake dee ection offers a lightweight and space-efe cient control mechanism for autonomous placement of air-dropped payloads to specie ed ground coordinates. The work reported here investigates control issues for a parafoil and payload system with left and right parafoil brakes used as the control mechanism.Itisshownthatparafoilandpayloadsystemscanexhibittwobasicmodesoflateralcontrol,namely,roll and skidsteering.Thesetwo modesoflateralsteeringgeneratelateralresponseinoppositedirections.Forexample, a roll steer cone guration turns left when the right parafoil brake is activated, whereas a skid steer cone guration turnsright under the samecontrol input. In transition between roll and skid lateral steering, the lateral responseis zero, and the system becomes uncontrollable. Angle of incidence, canopy curvature of the parafoil, and magnitude of brake dee ections are important design parameters for a controllable parafoil and payload system and greatly effect control response, including whether the basic lateral control mode is roll or skid steering. It is shown how the steering mode switches when fundamental design parameters are altered and as the magnitude of the brake dee ection increases. The mode of directional control transitions toward roll steering as the canopy curvature decreases or the angle of incidence becomes more negative. The mode of directional control transitions away from the roll steering mode as the magnitude of the brake dee ection increases, and for “ large” brake dee ections most parafoils will always skid steer.

Reconfigurable Flight Control System Design Using Direct Adaptive Method

Abstract: A reconfigurable flight control system provides better survivability through the automatic reconfiguration of control system when faults occur during flight. The adaptive control method has been effectively applied to the reconfigurable flight control system design. However, reconfigurable flight control systems based on the indirect adaptive control method require persistent input excitation and smooth input-output data. To deal with the persistent input excitation problem and to obtain smooth control input, the system identification algorithm in reconfiguration flight control systems usually imposes some constraints on past input-output data, which may deteriorate the reconfiguration performance. Thus, a new reconfigurable flight control system based on the direct adaptive method is proposed to achieve better reconfiguration performance without the system identification process. The proposed control method uses a model following controller with direct adaptive update rules. To control the inner-loop states and the outer-loop states of the flight system simultaneously,the timescale separation principle is applied. The reconfiguration performance of the proposed control method is evaluated through numerical simulations using a six-degree-of-freedom nonlinear aircraft model.

Rotation Vector in Attitude Estimation

Abstract: An alternative derivation of the spacecraft attitude determination filter is developed to avoid questions of quater-nion normalization or attitude matrix orthogonality constraints, quaternion covariance, and subterfuges used to circumvent these problems. This derivation is based on the Bortz equation for the rotation vector. Because the rotation vector is an unconstrained representation of attitude, the aforementioned questions do not arise. Singularities in the state dynamics equation are avoided by maintaining the predicted body attitude as the inertial reference for the filter. A simple discrete solution to the Bortz equation provides accurate attitude propagation for highly maneuverable spacecraft and also in the presence of jitter.

Evaluation of Longitudinal Desired Dynamics for Dynamic-Inversion Controlled Generic Reentry Vehicles

Abstract: Dynamic inversion is a control synthesis technique in which the inherent dynamics of a dynamical system are canceledoutandreplacedbydesireddynamics,selectedbythedesigner.Theoutputofsuchaninner-loopcontroller isthecontrol input, whichproducesthedesiredclosed-loop response.Thedesireddynamicsessentially form aloopshaping compensator that affects the closed-loop response of the entire system. This paper attempts to quantify the effect of different forms of desired dynamics on the closed-loop performance and robustness of a dynamicinversion e ight controllerfor reentry vehicles. Proportional, proportional-integral, e ying-quality, and ride-quality forms of desired dynamics are evaluated using time-domain specie cations, robustness requirements on singular values, quadratic cost, and a passenger ride comfort index. Longitudinal controllers are synthesized for a generic X-38 type crew return vehicle, using a set of linear models at subsonic, transonic, and hypersonic e ight conditions. For the candidate forms of desired dynamics and inversion controller structures evaluated here, results indicate that the form used impacts closed-loop performance and robustness and more so for some inversion controller structures more than others. The ride-quality dynamics used with a two-loop angle-of-attack inversion controller provide the best overall system performance, in terms of both time-domain and frequency-domain specie cations, and the evaluation criteria.

Long-Term Third-Body Effects via Double Averaging

Abstract: Westudy thelong-term effects of a third body on a satelliteof negligiblemass. Wehavein mind to study thelunisolar effects on an Earth satellite but many other applications can be imagined. We begin with the representation of the disturbing function in an ine nite series in Legendre polynomials but we truncate it at the second-degree terms. Our approach consists of a double analytic averaging: with the period of the satellite and with the period of the third body. We concentrate on using the two important integrals (energy and angular momentum ) to discuss and classify properties of the perturbed orbits of the satellite. For inclinations below 39 deg, the perigee of the orbits always circulates. There are circular and elliptic orbits but the eccentricity does not vary much. For the inclinations above 39 deg, the circular orbits are unstable and the eccentricities can increase rapidly but we have the appearance of new stableorbits: two ellipticfrozen orbits with constant eccentricity and e xed perigee location, at either 90 or 270 deg.

Third-Body Perturbation in Orbits Around Natural Satellites

Abstract: An analytical and a numerical study of the perturbation imparted to a spacecraft by a third body is developed. There are several important applications of the present research, such as calculation of the effect of lunar and solar perturbations on high-altitude Earth satellites. The goal is to study the evolution of orbits around some important natural satellites of the solar system, such as the moon, the Galilean satellites of Jupiter, Titan, Titania, Triton, and Charon. There is special interest in learning under which conditions a near-circular orbit remains near circular. The existence of circular, equatorial, and frozen orbits are also considered for a lunar satellite, but the results are valid for any system of primaries by making a time transformation that depends on the masses of the bodies involved. Several plots will show the time histories of the Keplerian elements of the orbits involved. Then, a study is performed to estimate the lifetime of orbits around those natural satellites.

Stabilizing Motion Relative to an Unstable Orbit: Applications to Spacecraft Formation Flight

Abstract: Acontrol lawisderivedandanalyzed thatstabilizesa classof unstableperiodicorbitsin theHillrestrictedthreebody problem. The control law is derived by stabilizing the short-time dynamics of motion about a trajectory by the use of a feedback law specie ed by the instantaneous eigenvalue and eigenvector structure of the trajectory. This law naturally generalizes to a continuous control law along an orbit. By applying the control to an unstable periodic orbit, we can explicitly compute the stability of the control over long periods of time by computing the monodromy matrix of the periodic orbit with its neighborhood modie ed via the control law. For the case of an unstable halo periodic orbit in the Hill restricted three-body problem, we e nd that the entire periodic orbit can be stabilized. The resulting stable periodic orbits have three distinct oscillation modes in their center manifold. We discuss how this control can be applied to formation e ight about a halo orbit. Some practical implementation issues of the control are also considered. We show that the control acceleration can be provided by a low-thrust engine and that the total fuel cost of the control can be quite reasonable. I. Introduction T HIS paper studies the stabilization of an unstable periodic orbit in the Hill problem, which can serve as a general model for motion in the Earth‐ sun system. Results of this study will be relevant to the dynamics and control of a constellation of spacecraft in an unstable orbitalenvironment such as found near the Earth‐ sun libration points. It will also shed light on the practical control and computation of a single spacecraft trajectory in an unstable orbital environment over long time spans. We investigate the application of feedback control laws to stabilize a periodic orbit in the sense of Lyapunov (see Ref. 1) (note, not asymptotic stability). Thus, the stabilized trajectory will consist of oscillatory motions about the nominal trajectory, which in this context can be interpreted as motions in the center manifold of the stabilized periodic orbit. We show that an entire class of such control laws can be dee ned and their stability analyzed as a timeperiodic linear system. The fuel expenditure for such control laws is often quite small and scales with the mean distance between the controlled motion and the nominal trajectory (which is a periodic orbit in this application). The problem of spacecraft control in unstable orbits is not new. (See Refs. 2 and 3 for reviews.) However, these previous studies have focused on stationkeeping control for a single spacecraft and have not considered how the relative motion of a formation of spacecraft could be stabilized and their dynamics modie ed, which is what we consider here. Proposed space observatories of the future include ambitious interferometricimagersthatusebaselinesofhundredsorthousandsof kilometers between spacecraft to attain sufe ciently high resolutions to image planets around distant stars. To carry out these imaging procedures requires that the relative motion between spacecraft be known extremely accurately and that the spacecraft “ e ll in” an effective image e eld as they move relative to each other. A periodic

Automated method to compute orbital reentry trajectories with heating constraints

Abstract: An algorithm is presented that is designed to compute reentry trajectories for unpowered lifting reusable launch vehicles. The algorithm uses self-contained trajectory simulation and root finding techniques to determine the appropriate control sequences for solving reentry problems. For orbital reentries, the solution process breaks the trajectory into two distinct parts. The first part begins where the deorbited vehicle first encounters substantial atmosphere and can exercise trajectory control through the manipulation of aerodynamic lift. This phase of the flight is governed by an analytical, constant heat-rate following, bank angle control law. The second and final part of the trajectory begins where heat-rate control is no longer desired. During this time trajectory control is used to meet terminal range and altitude targets and is governed by a linear bank angle control law. The planning algorithm determines the value of the individual trajectory control parameters that shape the reentry. The planned trajectory is then used by a profile following guidance algorithm during actual flight. Test results for a variety of orbital and suborbital missions are shown.

Second-Order Relative Motion Equations

Abstract: 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.

Adaptive Neural Control of Deep-Space Formation Flying

Abstract: A novel nonlinear adaptive neural control methodology is presented for the challenging problem of deep-space spacecraft formation flying. When the framework of the circular restricted three-body problem with the sun and Earth as the primary gravitational bodies is utilized, a nonlinear model is developed that describes the relative formation dynamics. This model is not confined to the vicinity of the Lagrangian libration points but rather constitutes the most general nonlinear formulation. Then, a relative position controller is designed that consists of an approximate dynamic model inversion, linear compensation of the ideal feedback linearized model, and an adaptive neural-network-based element designed to compensate for the model inversion errors. The nominal dynamic inversion includes the gravitational forces, whereas the model inversion errors are assumed to stem from disturbances such as fourth-body gravitational effects and solar radiation pressure. The approach is illustrated by simulations, which confirm that the suggested methodology yields excellent tracking and disturbance rejection, thus, permitting submillimeter formation keeping precision.

Escaping Trajectories in the Hill Three-Body Problem and Applications

Abstract: Low-energy escaping trajectories in the Hill three-body problem are investigated numerically using a Poincare map that relates the crossing of a plane containing one of the collinear libration points back to the first periapsis passage. This set of periapsis points is confined in a small region that determines some conditions for escape from any planetary satellite. In particular, the minimum energy to escape from a given circular orbit is obtained together with restrictions on the initial conditions (inclination, argument of periapsis, and longitude of the ascending node). This leads to a new optimal transfer criterion for the class of directly escaping trajectories. Savings on the order of 130 m/s in the case of Europa are obtained when compared to a classic two-body model. The results are also extended to the problem of low-energy capture. Numerical applications are given for the cases of Miranda, Europa, Titan, and Triton.

Satellite Collision Probability for Nonlinear Relative Motion

Abstract: A method for calculating the collision probability between two space vehicles when the relative motion is nonlinear is developed using contour integration methodology. The method involves transforming the problem to a scaled frame in which the error covariance matrix is symmetric in three dimensions. This enables the calculation of probability increments as a function of time throughout the encounter. Thus, changes in space vehicle position, velocity, and error covariance matrices throughout the encounter can be included in the formulation. This method is applicable to low-velocity space vehicle encounters that involve nonlinear relative motion. A software tool was created and exercised with both hypothetical and actual satellite data to demonstrate the method. Results differed from those of a Monte Carlo simulation by only 2%. Only 6% error resulted for a stress case in which the exact solution was known.

Rapid Verification Method for the Trajectory Optimization of Reentry Vehicles

Abstract: Introduction T HE trajectory optimization of path-constrained nonlinear dynamic systems such as those arising in the guidanceand design of reentry vehicles has long been considereda difŽ cult problem.1i3 Of themany methods to solve such problems, the direct collocation method with sparse nonlinear programming (NLP) has proved to be quite effective. Because direct methods do not tie the resulting solutions to the Pontryaginmaximum principle4 (PMP), a two-step direct–indirect approach has been used successfully for Ž nding extremal solutions.5 One major drawback of the direct–indirect approach is that a signiŽ cant amount of labor is necessary to derive all of the necessary conditions including complex switching and junction conditions for state-constrainedproblems.Even with estimates of the adjoint arcs, solving the multipoint boundary-valueproblem is an elaborate task. Over the past few years, a new approach to solving complex trajectory optimizationproblems has been proposed.6i9 The Legendre pseudospectralmethod is a particularlyeffectivemethod because it providesspectrallyaccuratesolutionsfor the costatesand other covectorswithout the use of any analyticaldifferentialequationsfor the

Planning Algorithm for Multiple Satellite Clusters

Abstract: A generalized planning methodology for satellite clusters is proposed. The methodology utilizes Hamilton ‐ Jacobi‐Bellman optimality (minimum time or minimum fuel ) to generate quickly a set of maneuvers from an initial stable formation to a e nal stable formation. Maneuvers are selected from the original set based on the maneuver time, fuel, and collision proximity. The e nal maneuvers are calculated by optimizing the switch times using a realisticset of orbital dynamics. The algorithm is developed to be distributed and scaleswell as the number of satellites increases. A minimal level of communication is used because only switch times and collision proximity information are distributed from the planner. An example with four satellites maneuvering in an eccentric orbit (e=0.2) is presented. Results show that optimal cluster maneuvers (minimum time or minimum fuel ) can be generated within minutes, and most of the computational implementation can be accomplished in parallel. I. Introduction S ATELLITE clusters are envisioned as an enabling technology for defense- and science-based missions. NASA’ s Origins program is planning a series of missions that perform spaceborne interferometry to image far off planets for possible life forms. 1 The U.S. Air Force is planning a distributed space-based, synthetic aperture radar mission within the next few years, possibly followed by a full deployment. 2 In each case, clusters of satellites hold the promise of increasing performance and reliability through distribution, while decreasing cost. The latter is a key aspect that will rely on levels of autonomous control algorithms and software currently being developed.

Determining If Two Solid Ellipsoids Intersect

Abstract: An analytical method is presented for determining if two ellipsoids share the same volume. The formulation involves adding an extra dimension to the solution space and examining eigenvalues that are associated with degenerate quadric surfaces. The eigenvalue behavior is characterized and then demonstrated with an example. The same method is also used to determine if two ellipsoids appear to share the same projected area based on an observer' s viewing angle. The following approach yields direct results without approximation, iteration, or any form of numerical search. It is computationally efe cient in the sense that no dimensional distortions, coordinate rotations, transformations, or eigenvector computations are needed. S the U.S. Satellite Catalog transitions from general perturba- tions to special perturbations, the positional accuracy of each space object will be readily available in the form of a covariance matrix. These covariances can be used to determine probability of collision, radio-frequency interference, and/or incidental laser illu- mination. Because the probability calculations can be computation- allyburdensome,itis desirable to prescreen candidate objectsbased on user-dee ned thresholds. Specie cally, each object can be repre- sented by a covariance-based ellipsoid and then processed to deter- mine if its uncertainty volume shares some space in common with another' s. Solid ellipsoids (or their projections) that do not intersect can be eliminated from further processing. This paper presents a simple analytical method to perform such screening. To date, all ellipsoidal prescreening methods involve numerical searches. 1 For computational efe ciency such prescreening is often reduced to spheres or " keep-out" boxes that have much larger vol- umes but allow for quick distance comparisons. The drawback to suchscreeningisthattheselargervolumescausemanyobjectstobe- come candidates for further (albeit unnecessary) processing. These methods result in increased downstream computational processing and/or increased operator workload to further assess potential satel- lite conjunctions. The following method adds an extra dimension to the solution space. The subset of eigenvalues that are associated with inter- secting degenerate quadric surfaces are then examined. The same method is also used to determine if two ellipsoids appear to share the same projected area based on viewing angle. The approach yields direct results without approximation, iteration, or any form of numerical search. It is computationally efe cient in the sense that no dimensional distortions, coordinate rotations, transforma- tions, or eigenvector computations are needed. This method ex- pands the two-dimensional work of Hill 2 in his formulation of degenerate conics (i.e., the characteristic matrix is singular). It also furthers his work by examining the associated eigenvalue behavior.