Showing papers in "Journal of Guidance Control and Dynamics in 2006"

Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method

Abstract: A pseudospectral method, called the Gauss pseudospectral method, for solving nonlinear optimal control problems is presented. In the method presented here, orthogonal collocation of the dynamics is performed at the Legendre-Gauss points. This form of orthogonal collocation leads a nonlinear programming problem (NLP) whose Karush-Kuhn-Tucker (KKT) multipliers can be mapped to the costates of the continuous-time optimal control problem. In particular the Legendre-Gauss collocation leads to a costate mapping at the boundary points. The method is demonstrated on an example problem where it is shown that highly accurate costates are obtained. The results presented in this paper show that the Gauss pseudospectral method is a viable apprach for direct trajectory optimization and costate estimation.

Sliding-Mode Control for Integrated Missile Autopilot Guidance

Abstract: A sliding-mode controller is derived for an integrated missile autopilot and guidance loop. Motivated by a differential game formulation of the guidance problem, a single sliding surface, defined using the zero-effort miss distance, is used. The performance of the integrated controller is compared with that of two different two-loop designs. The latter use a sliding-mode controller for the inner autopilot loop and different guidance laws in the outer loop: one uses a standard differential game guidance law, and the other employs guidance logic based on the sliding-mode approach. To evaluate the performance of the various guidance and control solutions, a two-dimensional nonlinear simulation of the missile lateral dynamics and relative kinematics is used, while assuming first-order dynamics for the target evasive maneuvers. The benefits of the integrated design are studied in several endgame interception engagements. Its superiority is demonstrated especially in severe scenarios where spectral separation between guidance and flight control, implicitly assumed in any two-loop design, is less justified. The results validate the design approach of using the zero-effort miss distance to define the sliding surface.

Spacecraft navigation using x-ray pulsars

Abstract: The feasibility of determining spacecraft time and position using x-ray pulsars is explored. Pulsars are rapidly rotating neutron stars that generate pulsed electromagnetic radiation. A detailed analysis of eight x-ray pulsars is presented to quantify expected spacecraft position accuracy based on described pulsar properties, detector parameters, and pulsar observation times. In addition, a time transformation equation is developed to provide comparisons of measured and predicted pulse time of arrival for accurate time and position determination. This model is used in a new pulsar navigation approach that provides corrections to estimated spacecraft position. This approach is evaluated using recorded flight data obtained from the unconventional stellar aspect x-ray timing experiment. Results from these data provide first demonstration of position determination using the Crab pulsar.

Proximity Operations of Formation-Flying Spacecraft Using an Eccentricity/Inclination Vector Separation

Abstract: The implementation of synthetic apertures by means of a distributed satellite system requires tight control of the relative motion of the participating satellites. This paper investigates a formation-flying concept able to realize the demanding baselines for aperture synthesis, while minimizing the collision hazard associated with proximity operations. An elegant formulation of the linearized equations of relative motion is discussed and adopted for satellite formation design. The concept of eccentricity/inclination-vector separation, originally developed for geostationary satellites, is here extended to low-Earth-orbit (LEO) formations. It provides immediate insight into key aspects of the relative motion and is particularly useful for orbit control purposes and proximity analyses. The effects of the relevant differential perturbations acting on an initial nominal configuration are presented, and a fuel-efficient orbit control strategy is designed to maintain the target separation. Finally, the method is applied to a specific LEO formation (TanDEM-X/TerraSAR-X), and realistic simulations clearly show the simplicity and effectiveness of the formation-flying concept.

Adaptive Output Feedback Control for Spacecraft Rendezvous and Docking Under Measurement Uncertainty

Abstract: An output feedback structured model reference adaptive control law has been developed for spacecraft rendezvous and docking problems. The effect of bounded output errors on controller performance is studied in detail. Output errors can represent an aggregation of sensor calibration errors, systematic bias, or some stochastic disturbances present in any real sensor measurements or state estimates. The performance of the control laws for stable, bounded tracking of the relative position and attitude trajectories is evaluated, considering unmodeled external as well as parametric disturbances and realistic position and attitude measurement errors. Essential ideas and results from computer simulations are presented to illustrate the performance of the algorithm developed in the paper.

Decentralized Coordinated Attitude Control Within a Formation of Spacecraft

Abstract: The problem of controlling the attitude of spacecraft within a formation is investigated. A class of decentralized coordinated attitude control laws using behavior-based control is developed. The decentralized coordinated attitude control laws that comprise the class differ by the coordination architecture used by the spacecraft formation. The choice of behavior weights defines the coordination architecture. A corollary of Barbalat’s Lemma is used to prove that the class of control laws globally asymptotically stabilizes the spacecraft formation. Convergence of the system is shown to be a consequence of the closed-loop equations of motion. Numeric simulation is used to reinforce the analytic results, and to briefly investigate the effect of coordination architecture on performance.

Sliding Mode Disturbance Observer-Based Control for a Reusable Launch Vehicle

Abstract: The nation's goals to replace the aging Space Shuttle fleet and pursue exploration of our solar system and beyond will require more robust, less costly launch vehicles and spacecraft. This paper presents a novel Sliding Mode Control approach, Sliding Mode Control driven by Sliding Mode Disturbance Observers with Gain Adaptation, for the reusable launch vehicle (RLV) flight control system design as a way to improve robustness to many phenomena such as modeling uncertainties and disturbances, while retaining continuity of control without using high control gains. Due to the robustness to external disturbances (including wind gusts), mission guidance trajectories and modeling uncertainties, the proposed flight control system design also can reduce cost by requiring less time in design cycle and preflight analyses. This design is applied to Terminal Area Energy Management and Approach/Landing (TAL), a flight regime that has had little research effort in recent years. The multiple-loop, multiple time-scale design features low order disturbance observers that rely only on knowledge of the bounds of the disturbance. A gain adaptation algorithm is included in the disturbance observer design that provides the least gain needed for existence of the sliding mode. High fidelity 6 DOF computer simulations of the X-33 technology demonstration sub-orbital launch vehicle for nominal and severe wind-gust tests demonstrate improved performance over a more conventional, classical control system design.

Nonlinear Mapping of Gaussian Statistics: Theory and Applications to Spacecraft Trajectory Design

Abstract: This paper discusses the nonlinear propagation of spacecraft trajectory uncertainties via solutions of the Fokker– Planck equation. We first discuss the solutions of the Fokker–Planck equation for a deterministic system with a Gaussian boundary condition. Next, we derive an analytic expression of a nonlinear trajectory solution using a higher-order Taylor series approach, discuss the region of convergence for the solutions, and apply the result to spacecraft applications. Such applications consist of nonlinear propagation of the mean and covariance matrix, design of statistically correct trajectories, and nonlinear statistical targeting. The two-body and Hill three-body problems are chosen as examples and realistic initial uncertainty models are considered. The results show that the nonlinear map of the trajectory uncertainties can be approximated in an analytic form, and there exists an optimal place to perform a correction maneuver, which is not found using the linear method.

Unmanned Aerial Vehicle Path Following for Target Observation in Wind

Abstract: This work provides flight-path geometry, guidance laws, and synchronous camera angles to observe a ground target from an unmanned aerial vehicle. The observation of the target is affected by wind, aircraft performance, and camera limits. Analytic expressions are derived for paths that result in constant line-of-sight orientation of the target relative to the aircraft body frame. Using minimal heuristics, a guidance law based on “good helmsman” behavior is developed and implemented, and stability of its integration with aircraft dynamics is assessed. An observer estimates wind data, which are used to orient path geometry about the target. Results are demonstrated in high-fidelity simulation. Nomenclature a = helmsman sensitivity parameter c(·) = cos(·) d = distance Fb = body-fixed frame Fs = Serret‐Frenet frame g = gravity constant r = radius s = arclength position along desired path s(·) = sin(·) t(·) = tan(·)

Adaptive Terminal Guidance for Hypervelocity Impact in Specified Direction

Abstract: The problem of guiding a hypersonic gliding vehicle in the terminal phase to a target location is considered. In addition to the constraints on its final position coordinates, the vehicle must also impact the target from a specified direction with very high precision. The proposed 3-dimensional guidance laws take simple proportional forms. The analysis establishes that with appropriately selected guidance parameters the 3-dimensional guided trajectory will satisfy these impact requirements. We provide the conditions for the initial on-line selection of the guidance law parameters for the given impact direction requirement. The vehicle dynamics are explicitly taken into account in the realization of guidance commands. To ensure high accuracy in the impact angle conditions in an operational environment, we develop closed-loop nonlinear adaptation laws for the guidance parameters. We present the complete guidance logic and associated analysis. Simulation results are provided to demonstrate the effectiveness and accuracy of the proposed terminal guidance approach. I. Introduction Recent interests in developing on-demand global-reach payload delivery capability have brought to the forefront a number of underlying technological challenges. Such operations will involve responsive launch, autonomous entry flight, and precision terminal maneuvers. In certain scenarios the mission requirements call for the payload to impact the target location from a specific direction with supersonic speed. One example is to impact the target in a direction perpendicular to the tangent plane of the terrain at the target. The terminal guidance system will be responsible for directing the vehicle to the target and achieving the desired impact direction. The impact precision requirements under the scenarios considered are very high and stringent. For instance, the required Circular Error Probable (CEP) of the impact distance is just 3-meter. 1 The errors of the impact angles are desired to be within 0.5 deg. The very high speeds throughout the terminal phase only make it considerably more difficult to achieve these levels of precision. Yet cost considerations dictate that the terminal guidance algorithm should be relatively simple and computationally tractable for real-time operations. While a number of guidance methods can guide the vehicle to the target, not many address the unique need for impact from a specific direction. One method that can is the so-called “dive-line” guidance approach in Ref. 2. In this method one or more lines intersecting the Earth are established. The final dive-line intersects the target, and its direction can be set to the desired direction. The vehicle’s velocity vector is

Modeling, Control, and Flight Testing of a Small Ducted-Fan Aircraft

Abstract: Small ducted fan autonomous vehicles have potential for several applications, especially for mi ssions in urban environments. This paper discusses the use of dynamic inversion with neural network adaptation to provide an adaptive controller for the GTSpy, a small ducted fan autonomous vehicle based on the Micro Autonomous Systems’ Helispy. This app roach allows utilization of the entire low speed flight envelope with a relatively poorly understood vehicle. A simulator model is constructed from a force and moment analysis of the vehicle, allowing for a validation of the controller in preparation for flight testing. Data from flight testing of the system is provided. Nomenclature B A A 2 1 ˆ , ˆ , ˆ = linearized vehicle dynamics a

Generalized Vector Explicit Guidance

Abstract: This paper proposes and evaluates a new guidance law termed generalized vector explicit guidance (GENEX). This guidance law can simultaneously achieve design specifications on miss distance and final missile-target relative orientation. The latter may be used to enhance the performance of warheads the effectiveness of which is influenced by the terminal encounter geometry. The GENEX guidance law is parameterized in terms of a design coefficient that determines the degree of curvature in the trajectory. Feasibility of GENEX guidance was demonstrated by its application to two weapon scenarios. The first was an air-to-air missile terminal homing scenario. Assuming ideal sensor information and a single-lag missile response model, the guidance was shown to perform well against an air target performing evasive maneuvers. A specified zero-aspect terminal encounter angle was achieved while simultaneously minimizing miss distance. The second application involved an air-to-surface munition released from an unmanned air vehicle. The GENEX guidance law was able to produce trajectories satisfying a terminal impact angle constraint. In addition, an engagement region of sufficient size was shown to be achievable using guidance gains scheduled with target location and weapon release altitude.

Linear Covariance Techniques for Orbital Rendezvous Analysis and Autonomous Onboard Mission Planning

Abstract: A novel trajectory control and navigation analysis software approach is developed. The program quickly determines trajectory dispersions, navigation errors, and required maneuver Δv at selected key points along a nominal trajectory. It can be used for mission design and planning activities or in autonomous flight systems to help determine the best trajectories, the best maneuver locations, and the best navigation update times to ensure mission success. These features are illustrated with two simple examples. The software works by applying linear covariance analysis techniques to a closed-loop guidance, navigation, and control (GN&C) system. The nonlinear dynamics and flight software models of a closed-loop six-degree-of-freedom Monte Carlo simulation are linearized. Then, linear covariance techniques are used to produce a program that will accurately predict 3-σ trajectory dispersions, navigation errors, and Av variations. Although the application presented is orbital rendezvous, the tools, techniques, and mathematical formulations are applicable to a variety of other space missions and GN&C problems including atmospheric entry, low-thrust electric propulsion missions, translunar injection and lunar orbit insertion, lunar ascent/descent, and formation-flying missions.

Trajectory Tracking Controller for Vision-Based Probe and Drogue Autonomous Aerial Refueling

Abstract: This paper addresses autonomous aerial refueling between an unmanned tanker aircraft and an unmanned receiver aircraft using the probe-and-drogue method An important consideration is the ability to achieve successful docking in the presence of exogenous inputs such as atmospheric turbulence Practical probe and drogue autonomous aerial refueling requires a reliable sensor capable of providing accurate relative position measurements of su‐cient bandwidth, integrated with a robust relative navigation and control algorithm This paper develops a Reference Observer Based Tracking Controller that does not require a model of the drogue or presumed knowledge of its position, and integrates it with an existing vision based relative navigation sensor A trajectory generation module is used to translate the relative drogue position measured by the sensor into a smooth reference trajectory, and an output injection observer is used to estimate the states to be tracked by the receiver aircraft Accurate tracking is provided by a state feedback controller with good disturbance rejection properties A frequency domain stability analysis for the combined reference observer and controller shows that the system is robust to sensor noise, atmospheric turbulence, and high frequency unmodeled dynamics Feasibility and performance of the total system is demonstrated by simulated docking maneuvers of an unmanned receiver aircraft docking with the non-stationary drogue of an unmanned tanker, in the presence of atmospheric turbulence Performance characteristics of the vision based relative navigation sensor are also investigated, and the total system is compared to an earlier version Results presented in the paper indicate that the integrated sensor and controller enable precise aerial refueling, including consideration of realistic measurements errors, plant modeling errors, and disturbances

Circular-Navigation-Guidance Law for Precision Missile/Target Engagements

Abstract: A new precision guidance law with impact angle constraint for a two-dimensional planar intercept is presented. It is based on the principle of following a circular arc to the target, hence the name circular navigation guidance. The guidance law does not require range-to-target information. We prove that it attains a perfect intercept under certain ideal conditions. In a broader range of conditions, it is shown to perform favorably when compared to another law from the literature.

Combined Feedback Linearization and Constrained Model Predictive Control for Entry Flight

Abstract: The advantages of constrained model predictive control over proportional integral derivative (PID) control applied to a feedback-linearized entry flight-control problem are discussed. The feedback linearization is based on the full rotational equations of motion rather than on a conventional model derived from time-scale separation. Input and state constraints are applied to avoid input saturations, to guarantee a minimum level of tracking performance, and to avoid physical vehicle state constraints violation. A constraint mapping algorithm is developed to map the input and state constraints on the new inputs after feedback linearization. The performance of the constrained model predictive control design is compared with that of two PID control designs. Simulation of a complete entry flight demonstrates the advantages of the constrained model predictive control design, because control surfaces do not saturate, control actions are more smooth and efficient, no gain scheduling is required, and all performance requirements are satisfied.

New Global Navigation Satellite System Ambiguity Resolution Method Compared to Existing Approaches

Abstract: Integer carrier phase ambiguity resolution is the process of resolving unknown cycle ambiguities of doubledifferenced carrier phase data as integers, and it is a prerequisite for rapid and high-precision global navigation satellite system positioning and navigation. Besides integer estimation, integer ambiguity resolution also involves validation of the integer estimates. In this contribution a new ambiguity resolution method is presented, based on the class of integer aperture estimators, which for the first time reveals an overall approach to the combined problem of integer estimation and validation. Furthermore, it is shown how the different discrimination tests that are currently in use in practice can be cast into the framework of the new approach.

Parametric model and optimal control of solar sails with optical degradation

Abstract: Solar-sail mission analysis and design is currently performed assuming constant optical and mechanical properties of the thin metalized polymer films that are projected for solar sails. More realistically, however, these properties are likely to be affected by the damaging effects of the space environment. The standard solar-sail force models can therefore not be used to investigate the consequences of these effects on mission performance. The aim of this paper is to propose a new parametric model for describing the sail film's optical degradation with time. In particular, the sail film's optical coefficients are assumed to depend on its environmental history, that is, the radiation dose. Using the proposed model, the optimal control laws for degrading solar sails are derived using an indirect method and the effects of different degradation behaviors are investigated for an example interplanetary mission.

Aggregate Flow Model for Air-Traffic Management

Abstract: Traditionally, models used in air-traffic control and flow management are based on simulating the trajectories of individual aircraft. This approach results in models with a large number of states, which are intrinsically susceptible to errors and difficult for designing and implementing optimal strategies for traffic flow management. This paper outlines an innovative approach for the development of linear-time-variant dynamic traffic flow system models based on historical data about the behavior of air traffic. The resulting low-order, linear, robust models can be used both for the analysis and synthesis of traffic flow management techniques for current and future systems.

Modeling Air Combat by a Moving Horizon Influence Diagram Game

Abstract: Thepaperdescribesamultistageinfluencediagramgameformodelingthemaneuveringdecisionsofpilotsinoneon-one air combat. The game graphically describes the elements of the decision process, contains a model for the dynamics of the aircraft, and takes into account the pilots’ preferences under conditions of uncertainty. The pilots’ game optimal control sequences with respect to their preference models are obtained by solving the influence diagram game with a moving horizon control approach. In this approach, the time horizon of the original game is truncated, and a feedback Nash equilibrium of the dynamic game lasting only a limited planning horizon is determined and implemented at each decision stage. To demonstrate the influence diagram game and its aspects, exampleswithapointmassaircraftmodelarecomputedandanalyzed.Thegamemodelpresentedinthepaperoffers a new way to analyze optimal air combat maneuvering and to develop an automated decision making system for selecting combat maneuvers in air combat simulators.

Design and Flight Testing of an H00 Controller for a Robotic Helicopter

Abstract: Although robotic helicopters have received increasing interest from university, industry, and military research groups, their flight envelope in autonomous operations remains extremely limited. The absence of high-fidelity simulation models has prevented the use of well-established multivariable control techniques for the design of high-bandwidth control systems. Existing controllers are of low bandwidth and cover only small portions of the vehicle's flight envelope. The results of the synergic use of high-fidelity integrated modeling strategies and robust multivariable control techniques for the rapid and reliable design of a high-bandwidth controller for robotic helicopters are presented. The project implemented and flight tested an H ∞ loop shaping controller on the Carnegie Mellon University (CMU) Yamaha R-50 robotic helicopter. During the flight tests, the CMU R-50 flew moderate-speed coordinated maneuvers with a level of tracking performance that exceeds performance reported in the publicly available literature. The authors believe that the results open the road to the implementation on robotic helicopters of full-flight-envelope control systems for complex autonomous missions.

Nonlinear Model Predictive Control Technique for Unmanned Air Vehicles

Abstract: A nonlinear model predictive control strategy is developed and subsequently specialized to autonomous aircraft that can be adequately modeled with a rigid 6-degrees-of-freedom representation. Whereas the general air vehicle dynamic equations are nonlinear and nonaffine in control, a closed-form solution for the optimal control input is enabledby expandingboth the output and control in a truncatedTaylor series. The closed-form solution for control is relatively simple to calculate and well suited to the real time embedded computing environment. An interesting feature of this control law is that the number of Taylor series expansion terms can be used to indirectly penalize control action. Also, ill conditioning in the optimal control gain equation limits practical selection of the number of Taylor series expansion terms. These claims are substantiated through simulation by application of the method to a parafoil and payload aircraft as well as a glider.

Necessary and Sufficient Conditions for H-∞ Static Output-Feedback Control

Abstract: Necessary and sufficient conditions are presented for static output-feedback control of linear time-invariant systems using the H ∞ approach. Simplified conditions are derived which only require the solution of two coupled matrix design equations. It is shown that the static output-feedback H ∞ solution does not generally yield a well-defined saddle point for the zero-sum differential game; conditions are given under which it does. This paper also proposes a numerically efficient solution algorithm for the coupled design equations to determine the output-feedback gain. A major contribution is that an initial stabilizing gain is not needed. An F-16 normal acceleration design example is solved to illustrate the power of the proposed approach.

Optimal Path Planning for Unmanned Combat Aerial Vehicles to Defeat Radar Tracking

Abstract: The problem of path planning for unmanned combat aerial vehicles (UCAVs) in the presence of radar-guided surface-to-air missiles is treated The problem is formulated in the framework of the interaction between three subsystems: the aircraft, the radar, and the missile The main features of this integrated model are as follows The aircraft radar cross section (RCS) depends explicitly on both the aspect and bank angles; hence, the RCS and aircraft dynamics are coupled The probabilistic nature of radar tracking is accounted for, namely, the probability that the aircraft has been continuously tracked depends on the aircraft RCS and range Finally, the requirement to maintain tracking before missile launch and during missile flyout are also modeled Based on this model, the problem of UCAV path planning is formulated as a minimax optimal control problem, with the aircraft lateral acceleration serving as control Necessary conditions of optimality for this minimax problem are derived and used as a basis for an efficient numerical solution Illustrative examples are considered that confirm the standard flying tactics of “denying range, aspect, and aim,” by yielding flight paths that weave to avoid long exposures of aspects with large RCS HIS paper is devoted to the problem of automated path planning for unmanned combat aerial vehicles (UCAVs) in the presence of radar-guided surface-to-air missiles (SAMs) This problem features the interaction between three subsystems: the aircraft and its characteristics, the radar and its capabilities, and the missile and its lethality Therefore, the solution of the UCAV path-planning problem requires realistic models of these three subsystems Although the current literature offers models for each of them separately, there is no approach that integrates models of the three subsystems in a unified framework The purpose of this paper is to propose such an integrated model and present results on its use

Linear Dynamics and Stability Analysis of a Two-Craft Coulomb Tether Formation

Abstract: The linearized dynamics and stability of a two-craft Coulomb tether formation is investigated. With a Coulomb tether the relative distance between two satellites is controlled using electrostatic Coulomb forces. A charge feedback law is introduced to stabilize the relative distance between the satellites to a constant value. Compared to previous Coulomb thrusting research, this is the first feedback control law that stabilizes a particular formation shape. The two craft are connected by an electrostatic tether that is capable of both tensile and compressive forces. As a result, the two-craft formation will essentially act as a long, slender near-rigid body. Interspacecraft Coulomb forces cannot influence the inertial angular momentum of this formation. However, the differential gravitational attraction can be exploited to stabilize the attitude of this Coulomb tether formation about an orbit nadir direction. Stabilizing the separation distance will also stabilize the in-plane rotation angle, whereas the out-of-plane rotational motion remains unaffected. The Coulomb tether has been modeled as a massless, elastic component. The elastic strength of this connection is controlled through a spacecraft charge control law.

Solving Optimal Continuous Thrust Rendezvous Problems with Generating Functions

Abstract: The optimal control of a spacecraft as it transitions between specified states using continuous thrust in a fixed amount of time is studied using a recently developed technique based on Hamilton-Jacobi theory. Started from the first-order necessary conditions for optimality, a Hamiltonian system is derived for the state and adjoints with split boundary conditions. Then, with recognition of the two-point boundary-value problem as a canonical transformation, generating functions are employed to find the optimal feedback control, as well as the optimal trajectory. Although the optimal control problem is formulated in the context of the necessary conditions for optimality, our closed-loop solution also formally satisfies the sufficient conditions for optimality via the fundamental connection between the optimal cost function and generating functions. A solution procedure for these generating functions is posed and numerically tested on a nonlinear optimal rendezvous problem in the vicinity of a circular orbit. Generating functions are developed as series expansions, and the optimal trajectories obtained from them are compared favorably with those of a numerical solution to the two-point boundary-value problem using a forward-shooting method.

Tensor-product model-based control of two-dimensional aeroelastic system

Abstract: Use of a recently introduced numerical robust-control design method to stabilize aeroelastic systems according to different control specifications is studied. This numerical design is based on the tensor-product model transformation and the parallel-distributed-compensation design framework. An alternative description of aeroelastic models is also proposed as a gateway to various recent linear-matrix-inequality-based control theories. This study is conducted through an example that focuses attention on the state-variable-feedback controller design to the prototypical aeroelastic wing section with structural nonlinearity. This type of model has been traditionally used for the theoretical as well as experimental analysis of two-dimensional aeroelastic behavior and exhibits limit-cycle oscillation without control effort. Numerical simulations to provide empirical validation of the resulting controllers are presented. Comparison to former alternative control solutions is also presented.

Optimal Fighter Pursuit-Evasion Maneuvers Found via Two-Sided Optimization

Abstract: An optimal pursuit-evasion fighter maneuver is formulated as a differential game and then solved by a recently developed numerical method, semidirect collocation with nonlinear programming. In this method, the optimal control for one player is found numerically, that is, by the optimizer, but that for the other player is based on the analytical necessary conditions of the problem. Because this requires costate variables for one player, the method is not a direct method. However, the problem can be placed in the form of conventional collocation with nonlinear programming. Thus, it is referred to it as a semidirect method. A genetic algorithm is used to provide an approximate solution, an initial guess, for the nonlinear programming problem solver. The method is applied to the challenging problem of optimal fighter aircraft pursuit-evasion in three dimensions. The obtained optimal trajectories are identified as having two phases: first a rapid change, primarily in direction, followed by a period of primarily vertical maneuvering. Solutions for various initial positions and velocities of the evader aircraft with respect to the pursuer are determined.

Precision free-inertial navigation with gravity compensation by an onboard gradiometer

Abstract: Precise inertial navigation depends on external aiding to remove various systematic errors in the sensed accelerations and rotations that cause an accumulation of position error up to many hundreds of meters per hour. Technological developments in inertial sensors are underway to eliminate this dependence at the level of a few meters uncertainty over one hour of unaided inertial navigation. However, no matter how precise the inertial measurement units are, the systematic effect of unknown gravitation can cause navigation errors up to several hundred meters per hour. Compensation for this effect can take several forms, but always requires updates or information from systems external to the inertial navigator. This report analyzes the unique capabilities provided by an onboard integrated gravity gradiometer that senses the gravitational gradients, thus enabling in situ compensation. Particular attention is given to the coupling of observed gradients with angular rates and accelerations. It is shown that errors from conventional precision gyros negate the effect of gradiometer aiding on the cross-track position error if only the essential gradients are measured [accuracy of 0.1 Eotvos (E)]. With additional measurements of appropriate symmetric gradient components, both along-track and cross-track errors caused by gravitation can be controlled at the 5-m level after one hour of free-inertial navigation.