Maruthi R. Akella

Globally Stabilizing Saturated Attitude Control in the Presence of Bounded Unknown Disturbances

A smooth attitude-stabilizing control law is derived from which known limits on the control authority of the system are rigorously enforced. Unknown disturbance torques, assumed to be of lesser magnitude than the control limits, are included in the formulation. A smooth control signal containing hyperbolic tangent functions that rigorously obeys a known maximum-torque constraint is introduced. The controller can be viewed as a smooth analog of the variable-structure approach, with the degree of sharpness of the control permitted to vary with time according to a set of user-defined parameters. Lyapunov analysis is employed to ensure global stability, and asymptotic convergence of the angular velocity is guaranteed via the Barbalat lemma. Attitude errors, expressed as Euler parameters, are shown via simulation to vanish whenever certain design parameters are selected appropriately, and guidelines for selection of those parameters are provided in depth.

Rigid body attitude tracking without angular velocity feedback

Abstract In this paper, we revisit the classical problem of attitude tracking for a rigid body. The interesting difference in the formulation is the assumption that only attitude measurements are available. We proceed to construct globally stabilizing control laws in terms of a minimal set of three-dimensional kinematic parameters that enable the rigid body to track any specified trajectory without requiring angular velocity measurements. The results presented here complement and extend some recent developments available for the nonminimal case of Euler parameters (quaternions).

Coordinated Standoff Tracking of Moving Targets: Control Laws and Information Architectures

In this paper, we present work on control of autonomous vehicle formations in the context of the coordinated stando tracking problem . The objective is to use a team of unmanned aircraft to fly a circular orbit around a moving target with prescribed inter-vehicle angular spacing using only local information. We use the recently introduced Lyapunov guidance vector field approach to achieve the desired circular trajectory. The contributions of this paper involve both single vehicle path planning and multiple vehicle coordination. For single vehicle path planning, we complete a proof of heading convergence using feedback, which has thus far not been fully addressed in the literature, and also oer a novel approach for heading convergence that does not require continuous feedback in the ideal case (no wind, stationary target), taking advantage of an analytical solution to the guidance field. Further, we use a variable airspeed controller to maintain the circular trajectory despite unknown wind and unknown constant velocity target motion. Adaptive estimates of the unknown wind and target motion are introduced to ensure stability to the circular trajectory. A novel feature of our results is rigorous satisfaction of vehicle specific kinematic constraints on heading rates and airspeed variations. For multiple vehicle coordination, we again use a variable airspeed controller to achieve the prescribed angular spacing. In an eort towards a unified framework for control of autonomous vehicle formations, we make a connection with some recent work that addresses information architecture in vehicle formations using graph theory. Specifically, we utilize two types of information architectures, symmetric and asymmetric, and implement decentralized control laws. The information architectures are scalable in the sense that the number of required communication/sensing links increases linearly with the number of vehicles. The control laws are decentralized in the sense that they use only local information.

Nonlinear Adaptive Control of Spacecraft Maneuvers

A novel method is presented to carry out near-minimum-time maneuvers of a spacecraft of unknown inertia. The formulation assumes the presence of three orthogonal reaction wheels located near the system mass center and oriented arbitrarily with respect to the spacecraft principal axes. Modie ed Rodrigues parameters along with their shadow set are used as the primary attitude coordinates. Open-loop maneuver laws are designed while solving the equations of motion by inversedynamics approach. This approach permits theapproximateimposition of the maximum saturation torque constraint. The torque proe les are near-bang-bang, with the instantaneous switches replaced by cubicsplinesof specie edduration.An adaptivetrackingcontrollawisdeveloped to determine perturbations to the nominal open-loop torque commands that will ensure the actual motion to follow the nominal motion in the presence of uncertainty in the inertia matrix and errors in the initial attitude. Global stability of the overall closed-loop controller is proved analytically and demonstrated by numerical simulations.

Probability of Collision Error Analysis

The decision for the International Space Station (ISS) to maneuver to avoid a potential collision with another space object will be based on the probability of collision, PC. The calculation of PC requires the covariance of both objects at conjunction. It is well known that the covariance computed by US Space Command is optimistic (too small), especially at altitudes where atmospheric drag is the dominant perturbation, because its computation assumes there are no dynamic model errors. In this paper the effect of errors in the covariance on PC and the sensitivity of PC to the encounter geometry are investigated.

Probability of Collision Between Space Objects

The International Space Station is being designed to perform debris avoidance maneuvers based on certain criteria developed from the probability of collision Pc. Existing methods to determine the Pc are based on the dee nitionofacollision/conjunctionplanethatcontainsallofthepositionuncertaintyassociatedwiththeproblem.In thispaperwedevelopa directand more naturalway of obtaining probability ofcollision and presentanalternative but equivalent dee nition for Pc that leads to the same results obtained earlier. Because debris avoidance is crucial for every orbiting asset in low Earth orbit, a study of this nature helps to establish the equivalence of different methods for risk assessment and evaluation. HE International Space Station (ISS) shall continuously face the threat of collision with orbiting debris. Hence, there needs to be a comprehensive methodology that can assess the risks posed byindividualdebrisencountersand suggestmaneuverswhenneces- sary. Such a study shall not only benee t the ISS, but also any future orbital asset placed in low Earth orbits. Thus, although we refer to the ISS in the rest of our paper, the analysis presented here holds true for any other orbiting asset of size and orbit comparable with that of the upcoming ISS. Space shuttle (SS) maneuvers are commanded to avoid poten- tial collisions with cataloged space objects (maintained by the U.S. Space Command ) whenever the estimated conjunction with an ob- jectfalls within a box centered on the estimated SSposition. The di- mensionsofthisconjunctionboxare §5 kmin thein-track direction and §2kmintheradialandout-of-planedirections.Thedimensions of such a conjunction box are probably based on prior estimates of position error covariances. The determination was made that this simple criterion, or any other known deterministic criterion 1,2 when applied to the ISS, would result in too many maneuvers. 3 In addi- tion, unnecessary maneuvers waste fuel and hamper the micrograv- ity experiments onboard the ISS. Although the size of the conjunc- tion box could be decreased to decrease the manuever rate, such a step clearly increases the risk to unacceptable levels. Therefore, the ISS needs a more rigorous probability-based approach for collision avoidance. 4

Velocity-Free Attitude Controllers Subject to Actuator Magnitude and Rate Saturations

We consider the “velocity-free” feedback control problem associated with attitude tracking for a rigid body subject to torque-magnitude and rate-saturation limits. Whereas no angular rate measurements are utilized within the feedback structure, the controller rigorously enforces actuator-magnitude and rate-saturation constraints. The associated stability proof is constructive and accomplished by the development of a novel Lyapunov function candidate. Although control smoothness is preserved at all times, it is important to note that the results of this paper that are derived with respect to magnitude saturation place no additional restrictions on the body inertias and make no other small-angle assumptions. The closed-loop performance of the new control solution derived here is evaluated extensively through numerical simulations in which we also include realistic levels of sensor noise in the feedback signals.

Separation Property for the Rigid-Body Attitude Tracking Control Problem

Quaternion-based proportional-derivative controllers for rigid-body attitude dynamics provide globally stabilizing solutions to both set-point regulation and trajectory tracking problems. Because the quaternion vector, or for that matter, any other attitude representation, can never be directly exactly measured, proportional-derivative controllers are invariably implemented under the assumption that the attitude error is available from suitable observers whose estimates converge sufficiently fast to the corresponding true attitude. To compound the situation, given the nonlinearities within the governing dynamics, most existing attitude observers can at best be proven to provide only asymptotic (i.e., nonexponential) convergence for the attitude estimation errors. This has the serious consequence that closed-loop stability assurances provided by classical proportional-derivative control laws no longer remain valid when the true attitude errors are replaced by their corresponding estimates. In this paper, we present a new quaternion-based attitude tracking controller that guarantees global asymptotic stability for the closed-loop dynamics while adopting an observer to generate the quaternion-based attitude estimates. We show that the state feedback control law and the estimator can be independently designed so that closed-loop stability is maintained even when they are combined. Accordingly, a separation property is established for the rigid-body attitude tracking problem, the first such result to our best knowledge. The crucial step in our stability analysis involves introduction of a novel class of strict Lyapunov functions whose time derivatives contain additional negative terms that help dominate the error terms arising due to the attitude observer implementation. Detailed proofs and numerical simulation examples are presented to help illustrate all the technical aspects of this work.

Unstart Detection in a Simplified-Geometry Hypersonic Inlet-Isolator Flow

Unstart detection techniques based on high-frequency pressure measurements made in a hypersonic inlet―isolator model are investigated. In this study, data that were acquired in a previous study of backpressure-induced unstart were examined. The data were acquired in a simplified-geometry inlet―isolator model that consisted of a 6 deg compression ramp inlet followed by a constant-area duct that was 25.4 mm high by 50.8 mm wide by 227.1 mm long. Fluctuating wall pressures were measured along the length of the model. A downstream flap was used to induce unstart in the model. Beyond a certain flap angle, unstart was induced, and the shock system propagated upstream and out of the inlet. The wall-pressure data, acquired as the flap was raised, were postprocessed for spectral and statistical content to evaluate different unstart detection criteria. Three shock leading-edge detection criteria are examined based on the following observations as the shock system passes over a pressure transducer: 1) a rise in pressure, 2) an increase in standard-deviation of the pressure signal, and 3) an increase in power in the 300―400 Hz frequency band. After calibrating the algorithms based on runs with no active control, a comparison of the times detected for unstart onset and unstart arrest was made based on runs with active control. Results indicate that the power-spectrum-based algorithm implemented close to the isolator exit is more sensitive to the onset of unstart, whereas the pressure-magnitude-change criterion gives earlier detection in many cases. A combination of the two, a pressure-magnitude criterion close to the inlet entrance and a spectral-power criterion near the isolator exit, therefore seems to be the most robust scheme for unstart active control. The appropriate choice of sampling frequency significantly improved the computation speed of the spectral-power-based algorithm without delaying the unstart detection times.

Non-certainty equivalent adaptive control for robot manipulator systems

A novel adaptive control solution to robot manipulator control problems is presented, based on a non-certainty equivalent approach. The key idea involves a new filter construction for the regressor terms within the governing dynamics that enables the establishment of a stabilizing mechanism within the adaptation process for unknown parameters. Viewed in the setting of filtered signals, the technical developments share some features inherent to the immersion and invariance (I&I) adaptive control methodology while maintaining certain crucial distinctions. In particular, the results stated here are not subject to satisfaction of any integrability or manifold attractivity conditions implicit within the powerful framework of I&I adaptive control. The paper presents numerical simulation of the new control scheme to highlight potential closed-loop performance improvements compared to the application of classical certainty-equivalence based adaptive controllers.