Karl D. Bilimoria

Time-optimal three-axis reorientation of a rigid spacecraft

New results are presented for the minimum-time rest-to-rest reorientation control problem of a rigid spacecraft with independent three-axis control. It is shown that in general the eigenaxis rotation maneuver is not time optimal. An inertially symmetric (e.g., spherical or cubical) rigid body is examined to demonstrate that an eigenaxis rotation about a control axis of even such a simple body is not time optimal. The computed optimal solution is bang-bang in all three control components and has a significant nutational component of motion. The total number of switches is found to be a function of the specified reorientation angle.

New Approach for Modeling, Analysis, and Control of Air Traffic Flow

An Eulerian approach to modeling air traffic flow is advanced. This modeling technique spatially aggregates air traffic to generate models of air traffic flow in a network of interconnected, one-dimensional control volumes. The approach simplifies the problem of characterizing the air traffic flow because the order of the corresponding airspace model depends only on the number of spatial control volumes used to represent the air traffic environment and not on the number of aircraft operating in it. Under a quasi-steady-state assumption, this process results in linear models of the air traffic environment. It is shown that analysis and design methods from linear control theory can be applied to this model to yield useful approaches for characterizing and controlling the air traffic flow.

Integrated Development of the Equations of Motion for Elastic Hypersonic Flight Vehicles

An integrated, consistent analytical framework is developed for modeling the dynamics of elastic hypersonic flight vehicles. A Lagrangian approach is used to capture the dynamics of rigid-body motion, elastic deformation, fluid flow, rotating machinery, wind, and a spherical rotating Earth model and to account for their mutual interactions. The resulting equations of motion govern the rigid-body and elastic degrees of freedom (DOF). The elastic motion is represented in terms of modal displacement coordinates relative to the elastic mean axes system, and the rigid-body motion is represented in terms of the translational and rotational velocities of this axes system. A vector form of the force, moment, and elastic-deformation equations is developed from Lagranges equation; a usable scalar form of these equations is also presented. The appropriate kinematic equations are developed and are presented in a usable form. The characteristics of the three-DOF point-mass dynamic model are also outlined, and the corresponding equations are presented. A preliminary study of the significance of selected terms in the equations of motion is conducted. Using generic data for a single-stage-to-orbit vehicle, it was found that the Coriolis force can reach values up to 6 % of the vehicle weight and that the forces and moments attributable to fluid-flow terms can be significant.

Computer-Aided Eulerian Air Traffic Flow Modeling and Predictive Control

Eulerian models are used to represent the air traffic environment as traffic flows between interconnected control volumes representing the airspace system. While these models can be manually derived for simple air traffic patterns, computer-based approaches are essential for modeling realistic airspaces involving multiple traffic streams. A computer- aided methodology for deriving large-dimensional Eulerian models of air traffic flow is described here. Starting from the specification of a few airspace parameters, and traffic data, the modeling technique can automatically construct Eulerian models of the airspace. The synthesis of air traffic flow control algorithms using the model predictive control technique in conjunction with these models is given. It is shown that the flow control logic synthesis can be cast as a linear programming problem. The flow control methodology is illustrated using air traffic data over two regions in U.S. airspace.

Singular Trajectories in Airplane Cruise-Dash Optimization

HE problem of determining optimal flight trajectories for aircraft has been investigated by many researchers for a variety of performance indices. Time-optimal flight for a climbdash mission has been investigated for the case of symmetric flight and is presented in Ref. 1. A nontrivial extension of this problem to a turn-climb-dash mission involving three-dimensional flight is reported in Ref. 2. In contrast, fuel-optimal flight minimizes the fuel consumed while generally covering a specified range, and may require operation at an intermediate throttle setting. The optimality of steady-state cruise (operation at a fixed altitude and velocity pair) has been the subject of some debate.3 Speyer3 used the point-mass dynamic model and showed that for certain aeropropulsive models, steady-state cruise fails to satisfy a Jacobitype condition obtained by a transformation to the frequency domain, and is therefore nonoptimal. Between the two extremes of time-optimal and fuel-optimal flight (with range specified) is a spectrum of time-fuel-opt imal flight conditions that may be regarded as members of a cruisedash family. Such a situation arises when a linear combination of time and fuel is used as a performance index and can be regarded as an exercise in time-fuel tradeoff. Because of the time-scale separation found in aircraft dynamics,4-5 the reduced-order cruise-dash model can be used to obtain the steady-state solution, and the full point-mass model can then be analyzed to obtain the transient solution. A first cut at the cruise-dash optimization problem was made6 by considering the cruise-dash model that gives the steady-state or outer solution. In view of Speyers findings3 regarding the nonoptimality of steady-state cruise, this point is addressed in a later section where Speyers analysis is retraced for steady-state operation at a cruise-dash point. This study focuses on transient trajectories that lead to steady-state operating points.

Time-optimal reorientation of a rigid axisymmetric spacecraft

A rigid spacecraft with independent three-axis control is featured in a study of minimum-time reorientation about the axis of symmetry. The eigenaxis rotation maneuver was found to be nonoptimal for all inertia ratios and the two maximumtorque configurations investigated. The role of gyroscopic effects in minimum-time reorientation maneuvers has been identified. For an inertially symmetric configuration there are no gyroscopic effects, and the reduction in maneuver time (compared to eigenaxis rotation) is attributed solely to the kinematics of the optimal maneuver. Compared to the inertially symmetric reference configuration, disk-like body shapes have a lower maneuver time since the gyroscopic effects enhance the kinematic savings, whereas rod-like body shapes have a higher maneuver time since the gyroscopic effects erode the kinematic savings. maximum-torques. The inertially symmetric configuration does not have any gyroscopic effects, and the decrease in maneuver time is attributed to the kinematics of the optimal maneuver that involves significant nutational motion and provides a control component of magnitude substantially larger than 1.0 in the inertial direction of the reorientation. Using the inertially symmetric configuration as a reference, the present study investigates the effects of gyroscopic coupling on minimum-time reorientation maneuvers by studying various inertia and maximum-torque configurations.

Advisory Algorithm for Scheduling Open Sectors, Operating Positions, and Workstations

Air traffic controller supervisors configure available sector, operating position, and workstation resources to safely and efficiently control air trafficin a region of airspace. In this paper, an algorithm for assisting supervisors with this task is described and demonstrated on two sample problem instances. The algorithm produces configuration schedule advisories that minimize a cost. The cost is a weighted sum of two competing costs: one penalizing mismatches between configurations and predicted air traffic demand and another penalizing the effort associated with changing configurations. The problem considered by the algorithm is a shortest path problem that is solved with a dynamic programming value iteration algorithm. The cost function contains numerous parameters. Default values for most of these are suggested based on descriptions of air traffic control procedures and subject-matter expert feedback. The parameter determining the relative importance of the two competing costs is tuned by comparing historical configurations with corresponding algorithm advisories. Two sample problem instances for which appropriate configuration advisories are obvious were designed to illustrate characteristics of the algorithm. Results demonstrate how the algorithm suggests advisories that appropriately utilize changes in airspace configurations and changes in the number of operating positions allocated to each open sector. The results also demonstrate how the advisories suggest appropriate times for configuration changes.

Cleveland Center Airspace Redesign Analysis

The sectors of Cleveland Air Route Traffic Control Center wereredesigned in 2011 in response to changes in air traffic patterns and volume. This paper presents an analysis of that redesign by contrasting the performance of the old design to the new one using historical traffic from the summer of 2010. Multiple workloadfactors are measured for each sector including one called sector loading, which is the sector’s peak aircraft count over a 15-minute time interval divided by the sector’s capacity, as specified by its Monitor Alert Parameter value. Other factors, like the number of flights near sector boundaries, are used to measure specific controller tasks that contribute to overall workload. Several of the design changes involved splitting busy sectors and combining under-utilized sectors to address traffic load imbalances. By comparing the distributions of these workload factors, the majority of these changes are shown to make the workload in the new sectors more consistent with that of other sectors in the vicinity. Furthermore, many of the changes are shown to improve the balance of workload within areas of specialization.

Integrated intelligent control of flexible structures: analysis and experiments

This study investigates the feasibility of using distributed force actuators (in addition to hub- torque) to control the tip deflection of a flexible beam-like structure during a rapid reorientation maneuver. The major objectives of the study are to assess system performance and actuator force requirements. Three feedback control schemes are compared to an open- loop control law corresponding to the rigid-body minimum-time solution. These control schemes use (1) a rate feedback hybrid controller, (2) an LQR-type hybrid controller, and (3) an LQR-type integrated controller. Comparisons between these controllers are made on the basis of actuator force required, increase in closed-loop system damping, and time required to execute a 90 degree reorientation maneuver. It was found that the rate-feedback hybrid control scheme results in high damping, but requires large peak-value forces from the actuators. The LQR-type hybrid control scheme requires lower peak-value forces from the actuators, but yields lower damping. The LQR-type integrated control scheme requires even lower actuator forces and a tip deflection that is significant only at the beginning of the maneuver, but results in an increase in the maneuver time.