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NASA NTRS1988

An application of high authority/low authority control and positivity

Control Dynamics Company (CDy), in conjunction with NASA Marshall Space Flight Center (MSFC), has supported the U.S. Air Force Wright Aeronautical Laboratory (AFWAL) in conducting an investigation of the implementation of several DOD controls techniques. These techniques are to provide vibration suppression and precise attitude control for flexible space structures. AFWAL issued a contract to Control Dynamics to perform this work under the Active Control Technique Evaluation for Spacecraft (ACES) Program. The High Authority Control/Low Authority Control (HAC/LAC) and Positivity controls techniques, which were cultivated under the DARPA Active Control of Space Structures (ACOSS) Program, were applied to a structural model of the NASA/MSFC Ground Test Facility ACES configuration. The control systems design were accomplished and linear post-analyses of the closed-loop systems are provided. The control system designs take into account effects of sampling and delay in the control computer. Nonlinear simulation runs were used to verify the control system designs and implementations in the facility control computers. Finally, test results are given to verify operations of the control systems in the test facility.

Seltzer, S. M.↗

NASA NTRS1990

Active control of the forced and transient response of a finite beam

Structural vibrations from a point force are modelled on a finite beam. This research explores the theoretical limit on controlling beam vibrations utilizing another point source as an active controller. Three different types of excitation are considered, harmonic, random, and transient. For harmonic excitation, control over the entire beam length is possible only when the excitation frequency is near a resonant frequency of the beam. Control over a subregion may be obtained even between resonant frequencies at the cost of increasing the vibration outside of the control region. For random excitation, integrating the expected value of the displacement squared over the required interval, is shown to yield the identical cost function as obtained by integrating the cost function for harmonic excitation over all excitation frequencies. As a result, it is always possible to reduce the cost function for random excitation whether controlling the entire beam or just a subregion, without ever increasing the vibration outside the region in which control is desired. The last type of excitation considered is a single, transient pulse. The form of the controller is specified as either one or two delayed pulses, thus constraining the controller to be casual. The best possible control is examined while varying the region of control and the controller location. It is found that control is always possible using either one or two control pulses.

Post, John T.↗

NASA NTRS1989

Distributed neural control of a hexapod walking vehicle

There has been a long standing interest in the design of controllers for multilegged vehicles. The approach is to apply distributed control to this problem, rather than using parallel computing of a centralized algorithm. Researchers describe a distributed neural network controller for hexapod locomotion which is based on the neural control of locomotion in insects. The model considers the simplified kinematics with two degrees of freedom per leg, but the model includes the static stability constraint. Through simulation, it is demonstrated that this controller can generate a continuous range of statically stable gaits at different speeds by varying a single control parameter. In addition, the controller is extremely robust, and can continue the function even after several of its elements have been disabled. Researchers are building a small hexapod robot whose locomotion will be controlled by this network. Researchers intend to extend their model to the dynamic control of legs with more than two degrees of freedom by using data on the control of multisegmented insect legs. Another immediate application of this neural control approach is also exhibited in biology: the escape reflex. Advanced robots are being equipped with tactile sensing and machine vision so that the sensory inputs to the robot controller are vast and complex. Neural networks are ideal for a lower level safety reflex controller because of their extremely fast response time. The combination of robotics, computer modeling, and neurobiology has been remarkably fruitful, and is likely to lead to deeper insights into the problems of real time sensorimotor control.

Beer, R. D.↗

NASA NTRS1990

Flight demonstration of a self repairing flight control system in a NASA F-15 fighter aircraft

Battle damage causing loss of control capability can compromise mission objectives and even result in aircraft loss. The Self Repairing Flight Control System (SRFCS) flight development program directly addresses this issue with a flight control system design that measures the damage and immediately refines the control system commands to preserve mission potential. The system diagnostics process detects in flight the type of faults that are difficult to isolate post flight, and thus cause excessive ground maintenance time and cost. The control systems of fighter aircraft have the control power and surface displacement to maneuver the aircraft in a very large flight envelope with a wide variation in airspeed and g maneuvering conditions, with surplus force capacity available from each control surface. Digital flight control processors are designed to include built-in status of the control system components, as well as sensor information on aircraft control maneuver commands and response. In the event of failure or loss of a control surface, the SRFCS utilizes this capability to reconfigure control commands to the remaining control surfaces, thus preserving maneuvering response. Correct post-flight repair is the key to low maintainability support costs and high aircraft mission readiness. The SRFCS utilizes the large data base available with digital flight control systems to diagnose faults. Built-in-test data and sensor data are used as inputs to an Onboard Expert System process to accurately identify failed components for post-flight maintenance action. This diagnostic technique has the advantage of functioning during flight, and so is especially useful in identifying intermittent faults that are present only during maneuver g loads or high hydraulic flow requirements. A flight system was developed to test the reconfiguration and onboard maintenance diagnostics concepts on a NASA F-15 fighter aircraft.

Urnes, James M.↗

NASA NTRS1991

An informal analysis of flight control tasks

Issues important in rotorcraft flight control are discussed. A perceptual description is suggested of what is believed to be the major issues in flight control. When the task is considered of a pilot controlling a helicopter in flight, the task is decomposed in several subtasks. These subtasks include: (1) the control of altitude, (2) the control of speed, (3) the control of heading, (4) the control of orientation, (5) the control of flight over obstacles, and (6) the control of flight to specified positions in the world. The first four subtasks can be considered to be primary control tasks as they are not dependent on any other subtasks. However, the latter two subtasks can be considered hierarchical tasks as they are dependent on other subtasks. For example, the task of flight control over obstacles can be decomposed as a task requiring the control of speed, altitude, and heading. Thus, incorrect control of altitude should result in poor control of flight over an obstacle.

Andersen, George J.↗

NASA NTRS1992

An alternate method for achieving temperature control in the -130 C to 75 C range

Thermal vacuum testing often requires temperature control of chamber shrouds and heat exchangers within the -130 C to 75 C range. There are two conventional methods which are normally employed to achieve control through this intermediate temperature range: (1) single-pass flow where control is achieved by alternately pulsing hot gaseous nitrogen (GN2) and cold LN2 into the feed line to yield the setpoint temperature; and (2) closed-loop circulation where control is achieved by either electrically heating or LN2 cooling the circulating GN2 to yield the setpoint temperature. A third method, using a mass flow ratio controller along with modulating control valves on GN2 and LN2 lines, provides excellent control but equipment for this method is expensive and cost-prohibitive for all but long-term continuous processes. The single-pass method provides marginal control and can result in unexpected overcooling of the test article from even a short pulse of LN2. The closed-loop circulation method provides excellent control but requires an expensive blower capable of operating at elevated pressures and cryogenic temperatures. Where precise control is needed (plus or minus 2 C), single-pass flow systems typically have not provided the precision required, primarily because of overcooling temperature excursions. Where several individual circuits are to be controlled at different temperatures, the use of expensive cryogenic blowers for each circuit is also cost-prohibitive, especially for short duration of one-of-a-kind tests. At JPL, a variant of the single-pass method was developed that was shown to provide precise temperature control in the -130 C to 75 C range while exhibiting minimal setpoint overshoot during temperature transitions. This alternate method uses a commercially available temperature controller along with a GN2/LN2 mixer to dampen the amplitude of cold temperature spikes caused by LN2 pulsing. The design of the GN2/LN2 mixer, the overall control system configuration, the operational procedure, and the prototype system test results are described.

Johnson, Kenneth R.↗

NASA NTRS1993

Control of nonlinear systems with applications to constrained robots and spacecraft attitude stabilization

This thesis is organized in two parts. In Part 1, control systems described by a class of nonlinear differential and algebraic equations are introduced. A procedure for local stabilization based on a local state realization is developed. An alternative approach to local stabilization is developed based on a classical linearization of the nonlinear differential-algebraic equations. A theoretical framework is established for solving a tracking problem associated with the differential-algebraic system. First, a simple procedure is developed for the design of a feedback control law which ensures, at least locally, that the tracking error in the closed loop system lies within any given bound if the reference inputs are sufficiently slowly varying. Next, by imposing additional assumptions, a procedure is developed for the design of a feedback control law which ensures that the tracking error in the closed loop system approaches zero exponentially for reference inputs which are not necessarily slowly varying. The control design methodologies are used for simultaneous force and position control in constrained robot systems. The differential-algebraic equations are shown to characterize the slow dynamics of a certain nonlinear control system in nonstandard singularly perturbed form. In Part 2, the attitude stabilization (reorientation) of a rigid spacecraft using only two control torques is considered. First, the case of momentum wheel actuators is considered. The complete spacecraft dynamics are not controllable. However, the spacecraft dynamics are small time locally controllable in a reduced sense. The reduced spacecraft dynamics cannot be asymptotically stabilized using continuous feedback, but a discontinuous feedback control strategy is constructed. Next, the case of gas jet actuators is considered. If the uncontrolled principal axis is not an axis of symmetry, the complete spacecraft dynamics are small time locally controllable. However, the spacecraft attitude cannot be asymptotically stabilized using continuous feedback, but a discontinuous stabilizing feedback control strategy is constructed. If the uncontrolled principal axis is an axis of symmetry, the complete spacecraft dynamics cannot be stabilized. However, the spacecraft dynamics are small time locally controllable in a reduced sense. The reduced spacecraft dynamics cannot be asymptotically stabilized using continuous feedback, but again a discontinuous feedback control strategy is constructed.

Krishnan, Hariharan↗

NASA NTRS1992

A feasibility study regarding the addition of a fifth control to a rotorcraft in-flight simulator

The addition of a large movable horizontal tail surface to the control system of a rotorcraft in-flight simulator being developed from a Sikorsky UH-60A Black Hawk Helicopter is evaluated. The capabilities of the control surface as a trim control and as an active control are explored. The helicopter dynamics are modeled using the Generic Helicopter simulation program developed by Sikorsky Aircraft. The effect of the horizontal tail on the helicopter trim envelope is examined by plotting trim maps of the aircraft attitude and controls as a function of the flight speed and horizontal tail incidence. The control power of the tail surface relative to that of the other controls is examined by comparing control derivatives extracted from the simulation program over the flight speed envelope. The horizontal tail's contribution as an active control is evaluated using an explicit model following control synthesis involving a linear model of the helicopter in steady, level flight at a flight speed of eighty knots. The horizontal tail is found to provide additional control flexibility in the longitudinal axis. As a trim control, it provides effective control of the trim pitch attitude at mid to high forward speeds. As an active control, the horizontal tail provides useful pitching moment generating capabilities at mid to high forward speeds.

Turner, Simon↗

NASA NTRS1993

Control system design for flexible structures using data models

The dynamics and control of flexible aerospace structures exercises many of the engineering disciplines. In recent years there has been considerable research in the developing and tailoring of control system design techniques for these structures. This problem involves designing a control system for a multi-input, multi-output (MIMO) system that satisfies various performance criteria, such as vibration suppression, disturbance and noise rejection, attitude control and slewing control. Considerable progress has been made and demonstrated in control system design techniques for these structures. The key to designing control systems for these structures that meet stringent performance requirements is an accurate model. It has become apparent that theoretically and finite-element generated models do not provide the needed accuracy; almost all successful demonstrations of control system design techniques have involved using test results for fine-tuning a model or for extracting a model using system ID techniques. This paper describes past and ongoing efforts at Ohio University and NASA MSFC to design controllers using 'data models.' The basic philosophy of this approach is to start with a stabilizing controller and frequency response data that describes the plant; then, iteratively vary the free parameters of the controller so that performance measures become closer to satisfying design specifications. The frequency response data can be either experimentally derived or analytically derived. One 'design-with-data' algorithm presented in this paper is called the Compensator Improvement Program (CIP). The current CIP designs controllers for MIMO systems so that classical gain, phase, and attenuation margins are achieved. The center-piece of the CIP algorithm is the constraint improvement technique which is used to calculate a parameter change vector that guarantees an improvement in all unsatisfied, feasible performance metrics from iteration to iteration. The paper also presents a recently demonstrated CIP-type algorithm, called the Model and Data Oriented Computer-Aided Design System (MADCADS), developed for achieving H(sub infinity) type design specifications using data models. Control system design for the NASA/MSFC Single Structure Control Facility are demonstrated for both CIP and MADCADS. Advantages of design-with-data algorithms over techniques that require analytical plant models are also presented.

Irwin, R. Dennis↗

NASA NTRS1994

Feedback control laws for highly maneuverable aircraft

During the first half of the year, the investigators concentrated their efforts on completing the design of control laws for the longitudinal axis of the HARV. During the second half of the year they concentrated on the synthesis of control laws for the lateral-directional axes. The longitudinal control law design efforts can be briefly summarized as follows. Longitudinal control laws were developed for the HARV using mu synthesis design techniques coupled with dynamic inversion. An inner loop dynamic inversion controller was used to simplify the system dynamics by eliminating the aerodynamic nonlinearities and inertial cross coupling. Models of the errors resulting from uncertainties in the principal longitudinal aerodynamic terms were developed and included in the model of the HARV with the inner loop dynamic inversion controller. This resulted in an inner loop transfer function model which was an integrator with the modeling errors characterized as uncertainties in gain and phase. Outer loop controllers were then designed using mu synthesis to provide robustness to these modeling errors and give desired response to pilot inputs. Both pitch rate and angle of attack command following systems were designed. The following tasks have been accomplished for the lateral-directional controllers: inner and outer loop dynamic inversion controllers have been designed; an error model based on a linearized perturbation model of the inner loop system was derived; controllers for the inner loop system have been designed, using classical techniques, that control roll rate and Dutch roll response; the inner loop dynamic inversion and classical controllers have been implemented on the six degree of freedom simulation; and lateral-directional control allocation scheme has been developed based on minimizing required control effort.

Garrard, William L.↗

NASA NTRS1998

Using Engine Thrust for Emergency Flight Control: MD-11 and B-747 Results

With modern digital control systems, using engine thrust for emergency flight control to supplement or replace failed aircraft normal flight controls has become a practical consideration. The NASA Dryden Flight Research Center has developed a propulsion-controlled aircraft (PCA) system in which computer-controlled engine thrust provides emergency flight control. An F-15 and an MD-11 airplane have been landed without using any flight control surfaces. Preliminary studies have also been conducted that show that engines on only one wing can provide some flight control capability if the lateral center of gravity can be shifted toward the side of the airplane that has the operating engine(s). Simulator tests of several airplanes with no flight control surfaces operating and all engines out on the left wing have all shown positive control capability within the available range of lateral center-of-gravity offset. Propulsion-controlled aircraft systems that can operate without modifications to engine control systems, thus allowing PCA technology to be installed on less capable airplanes or at low cost, are also desirable. Further studies have examined simplified 'PCA Lite' and 'PCA Ultralite' concepts in which thrust control is provided by existing systems such as auto-throttles or a combination of existing systems and manual pilot control.

Burcham, Frank W., Jr.↗

NASA NTRS1958

Flight Investigation of a Normal-Acceleration Automatic Longitudinal Control System in a Fighter Airplane

A flight investigation was made to obtain experimental information on the handling qualities of a normal-acceleration type of automatic longitudinal control system. The control system was installed in a subsonic fighter-type airplane. In hands-off (stick-free) flight the normal-acceleration control system attempted to regulate the normal acceleration to a constant value which is dependent on the automatic-control-system trim setting. In maneuvering flight a given pilot's stick deflection produced a proportional change in normal acceleration, the change in acceleration being independent of flight condition. A small side-located controller stick was used by the pilot to introduce signals into the automatic control system. In the flight program emphasis was placed on the acceleration-limiting capabilities of the control system. The handling qualities were investigated in maneuvers such as slow and rapid pull-ups and turns and also in flight operations such as cruising, stalls, landings, aerobatics, and air-to-air tracking. Good acceleration limiting was obtained with the normal-acceleration control system by limiting the magnitude of the input signal that the pilot could introduce into the control system. The same values of control-system gain settings could be used from an acceleration-limiting stand-point at both 10,000 and 30,000 feet for the complete speed range of the airplane. The response characteristics of the airplane-control system combination were also satisfactory at both high and low altitude with these same values of control-system gain setting. In the pilot's opinion, the normal-acceleration control system provided good stability and control characteristics in flight operations such as cruising, stalls, landings, aerobatics, and air-to-air tracking.

Sjoberg, S. A.↗

NASA NTRS1999

Active Control of Inlet Noise on the JT15D Turbofan Engine

This report presents the key results obtained by the Vibration and Acoustics Laboratories at Virginia Tech over the year from November 1997 to December 1998 on the Active Noise Control of Turbofan Engines research project funded by NASA Langley Research Center. The concept of implementing active noise control techniques with fuselage-mounted error sensors is investigated both analytically and experimentally. The analytical part of the project involves the continued development of an advanced modeling technique to provide prediction and design guidelines for application of active noise control techniques to large, realistic high bypass engines of the type on which active control methods are expected to be applied. Results from the advanced analytical model are presented that show the effectiveness of the control strategies, and the analytical results presented for fuselage error sensors show good agreement with the experimentally observed results and provide additional insight into the control phenomena. Additional analytical results are presented for active noise control used in conjunction with a wavenumber sensing technique. The experimental work is carried out on a running JT15D turbofan jet engine in a test stand at Virginia Tech. The control strategy used in these tests was the feedforward Filtered-X LMS algorithm. The control inputs were supplied by single and multiple circumferential arrays of acoustic sources equipped with neodymium iron cobalt magnets mounted upstream of the fan. The reference signal was obtained from an inlet mounted eddy current probe. The error signals were obtained from a number of pressure transducers flush-mounted in a simulated fuselage section mounted in the engine test cell. The active control methods are investigated when implemented with the control sources embedded within the acoustically absorptive material on a passively-lined inlet. The experimental results show that the combination of active control techniques with fuselage-mounted error sensors and passive control techniques is an effective means of reducing radiated noise from turbofan engines. Strategic selection of the location of the error transducers is shown to be effective for reducing the radiation towards particular directions in the farfield. An analytical model is used to predict the behavior of the control system and to guide the experimental design configurations, and the analytical results presented show good agreement with the experimentally observed results.

Smith, Jerome P.↗

NASA NTRS1960

Experience with a three-axis side-located controller during a static and centrifuge simulation of the piloted launch of a manned multistage vehicle

An investigation was conducted to determine a human pilot's ability to control a multistage vehicle through the launch trajectory. The simulation was performed statically and dynamically by utilizing a human centrifuge. An interesting byproduct of the program was the three-axis side-located controller incorporated for pilot control inputs. This method of control proved to be acceptable for the successful completion of the tracking task during the simulation. There was no apparent effect of acceleration on the mechanical operation of the controller, but the pilot's control feel deteriorated as his dexterity decreased at high levels of acceleration. The application of control in a specific control mode was not difficult. However, coordination of more than one mode was difficult, and, in many instances, resulted in inadvertent control inputs. The acceptable control harmony at an acceleration level of 1 g became unacceptable at higher acceleration levels. Proper control-force harmony for a particular control task appears to be more critical for a three-axis controller than for conventional controllers. During simulations in which the pilot wore a pressure suit, the nature of the suit gloves further aggravated this condition.

Andrews, William H.↗

NASA NTRS1991

Dual-arm manipulators with adaptive control

The described and improved multi-arm invention of this application presents three strategies for adaptive control of cooperative multi-arm robots which coordinate control over a common load. In the position-position control strategy, the adaptive controllers ensure that the end-effector positions of both arms track desired trajectories in Cartesian space despite unknown time-varying interaction forces exerted through a load. In the position-hybrid control strategy, the adaptive controller of one arm controls end-effector motions in the free directions and applied forces in the constraint directions; while the adaptive controller of the other arm ensures that the end-effector tracks desired position trajectories. In the hybrid-hybrid control strategy, the adaptive controllers ensure that both end-effectors track reference position trajectories while simultaneously applying desired forces on the load. In all three control strategies, the cross-coupling effects between the arms are treated as disturbances which are compensated for by the adaptive controllers while following desired commands in a common frame of reference. The adaptive controllers do not require the complex mathematical model of the arm dynamics or any knowledge of the arm dynamic parameters or the load parameters such as mass and stiffness. Circuits in the adaptive feedback and feedforward controllers are varied by novel adaptation laws.

Seraji, Homayoun↗

NASA NTRS1995

Method and apparatus for adaptive force and position control of manipulators

Seraji, Homayoun↗

NASA NTRS2010

Projection Operator: A Step Towards Certification of Adaptive Controllers

One of the major barriers to wider use of adaptive controllers in commercial aviation is the lack of appropriate certification procedures. In order to be certified by the Federal Aviation Administration (FAA), an aircraft controller is expected to meet a set of guidelines on functionality and reliability while not negatively impacting other systems or safety of aircraft operations. Due to their inherent time-variant and non-linear behavior, adaptive controllers cannot be certified via the metrics used for linear conventional controllers, such as gain and phase margin. Projection Operator is a robustness augmentation technique that bounds the output of a non-linear adaptive controller while conforming to the Lyapunov stability rules. It can also be used to limit the control authority of the adaptive component so that the said control authority can be arbitrarily close to that of a linear controller. In this paper we will present the results of applying the Projection Operator to a Model-Reference Adaptive Controller (MRAC), varying the amount of control authority, and comparing controller s performance and stability characteristics with those of a linear controller. We will also show how adjusting Projection Operator parameters can make it easier for the controller to satisfy the certification guidelines by enabling a tradeoff between controller s performance and robustness.

Larchev, Gregory V.↗

NASA NTRS2013

Dynamics and Control of a Disordered System in Space

In this paper, we present some ideas regarding the modeling, dynamics and control aspects of granular spacecraft. Granular spacecraft are complex multibody systems composed of a spatially disordered distribution of a large number of elements, for instance a cloud of N grains in orbit, with N greater than 10(exp 3). These grains can be large (Cubesat-size) or small (mm-size), and can be active, i.e., a fully equipped vehicle capable sensing their own position and attitude, and enabled with propulsion means, or entirely passive. The ultimate objective would be to study the behavior of the single grains and of large ensembles of grains in orbit and to identify ways to guide and control the shape of a cloud composed of these grains so that it can perform a useful function in space, for instance, as an element of an optical imaging system for astrophysical applications. This concept, in which the aperture does not need to be continuous and monolithic, would increase the aperture size several times compared to large NASA observatories such as ATLAST, allowing for a true Terrestrial Planet Imager that would be able to resolve exo-planet details and do meaningful spectroscopy on distant world. In the paper, we address the modeling and autonomous operation of a distributed assembly (the cloud) of large numbers of highly miniaturized space-borne elements (the grains). A multi-scale, multi-physics model is proposed of the dynamics of the cloud in orbit, as well as a control law for cloud shape maintenance, and preliminary simulation studies yield an estimate of the computational effort, indicating a scale factor of approximately N(exp 1.4) as a function of the number of grains. A granular spacecraft can be defined as a collection of a large number of space-borne elements (in the 1000s) designed and controlled such that a desirable collective behavior emerges, either from the interactions among neighboring grains, and/or between the grains and the environment. In this paper, each grain is considered to be a highly miniaturized spacecraft which has limited size and mass, hence it has limited actuation, limited propulsive capability, limited power, limited sensing, limited communication, limited computational resources, limited range of motion, limited lifetime, and may be expendable. The modeling and dynamics of clouds of vehicles is more challenging than with conventional vehicles because we are faced with a probabilistic vehicle composed of a large number of physically disconnected vehicles. First, different scales of motion occur simultaneously in a cloud: translations and rotations of the cloud as a whole (macro-dynamics), relative rotation and translation of one cloud member with respect to another (meso-dynamics), and individual cloud member dynamics (micro-dynamics). Second, the control design needs to be tolerant of the system complexity, of the system architecture (centralized vs. decentralized large scale system control) as well as robust to un-modeled dynamics and noise sources. Figure 1, top left, shows the kinematic parameters of a 1000 element cloud in orbit. The motion of the system is described with respect to a local vertical-local horizontal (LV-LH) orbiting reference frame (x,y,z)=F(sub ORF) of origin O(sub ORF) which rotates with mean motion omega and orbital semi-major axis R(sub 0). The orbital geometry at the initial time is defined in terms of its six orbital elements, and the orbital dynamics equation for point O(sub ORF) is propagated forward in time under the influence of the gravitational field of the primary and other external perturbations, described below. The origin of this frame coincides with the initial position of the center of mass of the system, and the coordinate axes are z along the local vertical, x toward the flight direction, and y in the orbit normal direction. The assumptions we used to model the dynamics are as follows: 1) The inertial frame is fixed at Earth's center. 2) The orbiting Frame ORF follows Keplerian orbit. 3) the cloud system dynamics is referred to ORF. 4) the attitude of each grain uses the principal body frame as body fixed frame. 5) the atmosphere is assumed to be rigidly rotating with the Earth. Regarding the grains forming the cloud: 1) each grain is modeled as a rigid body; 2) a simple attitude estimator provides attitude estimates, 3) a simple guidance logic commands the position and attitude of each grain, 4) a simple local feedback controller based on PD control of local states is used to stabilize the attitude of the vehicle. Regarding the cloud: 1) the cloud as a whole is modeled as an equivalent rigid body in orbit, and 2) an associated graph establishes agent connectivity and enables coupling between modes of motion at the micro and macro scales; 3) a simple guidance and estimation logic is modeled to estimate and command the attitude of this equivalent rigid body; 4) a cloud shape maintenance controller is based on the dynamics of a stable virtual truss in the orbiting frame. Regarding the environmental perturbations acting on the cloud: 1) a non-spherical gravity field including JO (Earth's spherical field) zonal component, J2 (Earth's oblateness) and J3 zonal components is implemented; 2) atmospheric drag is modeled with an exponential model; 3) solar pressure is modeled assuming the Sun is inertially fixed; and 4) the Earth's magnetic field is model using an equivalent dipole model. The equations of motion are written in a referential system with respect to the origin of the orbiting frame and the state is propagated forward in time using an incremental predictor-corrector scheme. A representative cloud with varying number of grains is simulated to identify the limitations in computation time as the number of grains grows. We derive a control law to track a desired surface in the ORF (equivalently to maintain a reference cloud shape) by defining an error from a desired surface shape, and designing a control law that is exponentially stable and reduces the tracking error to zero. Figure 1 (top right) shows a comparison of various requirements for simulation of single spacecraft vs. granular spacecraft, indicating the high degree of complexity that needs to be taken into consideration. The ORF components of control force required by one of the grains is, for this particular case, in the micro-Newton range. However, no attempt has been made yet to reconfigure (or re-orient) the cloud configuration internally, for which forces in the milli-Newton level are expected, depending on the time required to do the reconfiguration. Figure 1, bottom, shows the computation time as a function of the number of grains, indicating an order N(exp 1.43) scaling on a 8 Gb, 1067 MHz RAM MacOSX computer with a 3.06 GHz Intel Core 2 Duo processor. With this metric, the same simulation for a system of N=1000 grains would take 5.4 hours, and 146 hours (i.e., 6 days) for a system with N=10,000 grains. Therefore, efficient ways to simulate this complex system, where not only the time scales of natural system dynamics, but also the sampling times of the Guidance, Navigation, and Control are included, remain to be explored. Additional details on the cloud modeling, dynamics, and control will be described in the paper.

simulation↗