Rigid Hub Research Articles

Momentum exchange - Feedback control of flexible spacecraft maneuvers and vibration

The momentum exchange controller in which the time rate of change of the flexible momentum relative to the rigid-body motion is used as a part of the feedback control law for maneuvers and vibration suppression of flexible systems is introduced. This control concept is applied to a model of a rigid hub (base) with a cantilevered flexible appendage undergoing a single-axis maneuver. The feedback control on the hub includes the rigid-body motion and the time rate of change of that part of the flexible momentum resulting from flexible vibration. The lower and upper bounds of the control law for system Lyapunov stability are obtained, and the relationship of the control law with the energy Lyapunov test function is established. With the presence of this feedback control, an additional independent flexible control system acting on the flexible parts can be designed for further vibration suppression. This hybrid control system can be applied to both stationkeeping and large-angle maneuvers. Both analytical and numerical results are presented to show the theoretical and practical merit of this approach.

Near-minimum-time control of a flexible manipulator

Near-minimum-time control of flexible manipulators can be accomplished by designing a controller that tracks a reference maneuver. The control method presented here uses a near-minimum-time rigid-link reference maneuver to generate reference control torques and a Liapunov controller to make the flexible manipulator track the reference maneuver while reducing and eventually eliminating flexible motions. The near-minimumtime reference maneuver uses a smoothed bang-bang control plus a term to cancel the nonlinear dynamics of the rigid manipulator. The Liapunov function is a weighted sum of the energies of the elements of the flexible manipulator, and the control law is chosen to make the rate of change of the Liapunov function negative. The Liapunov function is bounded during a maneuver and decays asymptotically after the maneuver ends. RECISE control of a manipulator is an important aspect of robot performance. In the past, this has led to the design of very stiff links for manipulators, which also leads to very heavy links, large motors, and slow response. To avoid these problems, lighter and more flexible manipulators will require advanced control strategies. This work builds on results from the control of rigid robots1 and the control of flexible structures. Fujii and Ishijima2 developed a globally stable Liapunov controller to reorient a structure composed of a rigid hub and a flexible appendage. Junkins et al. 3 extended this control law to track a reference maneuver and showed that the Liapunov function is bounded during the maneuver and decays asymptotically to zero after the maneuver ends. Using a Gibbs phenomenon analysis, Baruh and Tadikonda4 have shown that using smoothed control torque profiles as in Ref. 3 reduces the excitation of the flexible modes during a maneuver. Experimental work in this area is also being done.5'6 In this work, we demonstrate that the control design procedure of Ref. 3 can be developed, using work-energy relations, and we extend it to a multibody case, using a planar two-link manipulator to demonstrate the procedure. A schematic diagram of the manipulator is shown in Fig. 1. The control law developed for the two-link manipulator allows the flexible motions to be stabilized without any finite-dimensional estimation of the flexible modes, thus avoiding any spillover problems.

Accommodation of kinematic disturbances during minimum-time maneuvers of flexible spacecraft

This paper is concerned with control of the perturbations experienced by a flexible spacecraft during a minimum-time maneuver. The spacecraft is modeled as a rigid hub with a flexible appendage. The perturbations consist of deviations from a rigid-body maneuver and elastic vibration. The vibration is described by a linear, time-varying set of ordinary differential equations subjected to piecewise-constant disturbances caused by inertia! forces resulting from the maneuver. The control is carried out during the maneuver period, which is relatively short, and it uses an observer to estimate the controlled state and part of the disturbance vector. The controller is divided into an optimal finite-time linear quadratic regulator for the reduced-order model and a disturbance-accommodation control that minimizes a weighted norm spanning the constant part of the full-modeled state. The controller is designed to mitigate the effects of control and observation spillover, as well as of modeling errors. The developments are illustrated by means of a numerical example.

On the Modeling, and Open-Loop Control of a Rotating Thin Flexible Beam

A set of governing differential equations is derived for the inplane motion of a rotating thin flexible beam. The beam is assumed to be linearly elastic and is connected to a rigid hub driven by a torque motor. Both flexural and extensional effects are included in the derivation. This coupling due to flexure and extension is usually neglected in studies dealing with the control of such a system. Models for typical control studies are often derived by utilizing an assumed mode approach where the mode shapes are obtained by solving the Euler-Bernoulli beam equation for flexural vibrations, with clamped-free or pinned-free boundary conditions. The coupled equations developed in this paper are used to demonstrate that typical models in control studies give satisfactory results up to a critical rotational speed. For the case where these coupled equations are specialized to simple flexure only, valid for low angular speeds, a unique feedforward control strategy can be derived. This is an open-loop control strategy that enables total elimination of an a priori specified vibratory mode from the gross motion in a finite critical time.

Real-time computer control of a flexible spacecraft emulator

A description is given of a ground-based test facility named Daisy, which was established to study control issues for large flexible spacecraft. The validation of modern algorithms for system identification and shape and attitude control and the development of new control devices can be performed using Daisy. The experimental structure consists of a radial mesh of ribs attached flexibly to a rigid hub. The structure is instrumented with position and rate sensors and controlled by a real-time computer and data acquisition system by means of torque actuators. The more important aspects of these subsystems and their integration are presented along with experimental results for a baseline controller, consisting of a simple integral-derivative feedback control on the position of the hub. >

Enhancement of data separability in multisensor-multitarget tracking problems

and Conclusions An analytical solution has been obtained for the displacement and rotation angle of a single-joint flexible robot modeled by a continuous, clamped-free beam attached to a rigid hub of negligible radius. The results for a representative-applied harmonic torque are obtained and show the effect of the beam flexibility, which can be superimposed on the rigid-beam motion. Fourier series representations can be applied in the usual way to generalize these results. The nondimensionalization for both dynamic equations permitted a simpler problem formulation. The solution also provides a quick means of testing the effect of the various physical parameters on the design of a flexible one-link robot. The solution gives the accurate location of every point of the link and, thus, changing the parameters can give the desired results for a specific task for the arm.

A study of feedback, blade and hub parameters on flap bending due to non-uniform rotor disk turbulence

The effects of non-uniform (vertical) turbulence in the rotor disk are further studied with reference to comprehensive hingeless rotorcraft during “low-lift high advance ratio” operations . The analysis is based on linear quasi-steady aerodynamics including reverse flow, and on the assumption that the flow in the rotor plane is not appreciably influenced by random vortices shed from the blades. It is adequate for such operations which are anticipated for future compound or convertible configurations. The analytical model pertains to a multi-bladed system with rigidly or elastically supported hub, and with various control feedback schemes in combination with widely varying blade characteristics. Extensive parametric studies are presented with respect to advance ratios, blade inertia numbers, feedback combinations, phase angles, and gains, flap-bending flexibility of uniform and tapered-in-thickness blades, rotor support pitching and rolling natural frequencies, and ratios of turbulence scale length over rotor radius. They demonstrate that the variation in these parameters has practically no influence on the effects of non-uniformity (consequently non-stationarity) of turbulence in the rotor plane. They formally substantiate an earlier finding based on results of a highly idealized rotor system—a single blade fixed to a rigid hub without feedback—that the non-uniformity effects are negligible. Further it is shown that for a wide range of lifting rotor configurations a rigid-hub approximation is satisfactory to evaluate steady state standard deviations of flap bending responses due to both uniform and non-uniform turbulence. These findings pertaining to the adequacy of simpler uniform turbulence and rigid hub representation are significant for the further development of approximate probabilistic models of lifting rotor systems in a stochastic environment and for the experimental correlation with test results.

A general method with shaping filters to study random vibration statistics of lifting rotors with feedback controls

This paper is concerned with the random vibration statistics of linear or perturbed linear dynamic systems with variable parameters, in particular of lifting rotors with feedback controls. By introducing shaping filters to random inputs, the response variance matrix is solved directly from another set of linear equations. Therefore, with the stipulation of stationary Gaussian random inputs, the computational scheme of response threshold crossing statistics under steady-state conditions is essentially no more involved than generating the state vector to step inputs. For certain types of feedback controls of lifting rotors having coupling effects between blades, the equations of motion studied by other means using blade coordinates are adopted. Treated for illustrative purposes are the rigid flapping oscillations of multibladed rotors at a high advance ratio, having a rigid hub with elastically restrained and centrally arranged flapping hinges. The stochastic input refers to free atmospheric vertical turbulence at the centre of the rotor disk. The numerical results of the flapping response statistics include the root-mean-square description and the expected values of threshold up-crossings per unit time. When compared to earlier related studies, the proposed method requires negligible core storage requirement, directly permits variable step sizes to guarantee preset accuracy criteria, and offers substantial saving in machine time. In the parametric studies of system and feedback constants, the available core space could be used in storing discrete values of aerodynamic damping, spring and input modulating functions which otherwise involve lengthy Fourier series approximations at high advance ratios or repeated computations. The method in general offers considerable promise in the routine analysis of lifting rotor models having blades with many degrees of freedom of motion and having other combinations of feedback control systems.

Stresses in Wire Wheels

T P U R I N G the last two or three years the * ^ writer has been carrying out a research into the stresses caused in a wire wheel under loads applied to the rim. This work has been done for the Aeronautical Research Committee and the results have been published in a series of papers which are enumerated at the end qf this note which gives a brief resume of the problem, method of attack and certain results. A wire wheel consists of a comparatively rigid hub and a rim connected to it by a large number of small section spokes which may be either radial or inclined to the radius. Since the flexural rigidity of the spokes is small it is sufficiently accurate for analytical purposes to assume that they are pin jointed to the hub and rim. With this assumption it is, of course, impossible for the radially spoked wheel to resist actions on the rim which are tangential, e.g., those due to the application of brakes: for efficient resistance to such forces the spokes must be inclined in the plane of the rim. To resist side forces on the rim the spokes must also have an inclination in a plane perpendicular to tha t of the rim. The problem to be considered, therefore, was tha t of a wheel having radial or inclined spokes, held at the hub and subjected to loads at the rim acting radially, tangentially and transversely. An attack was first made by means of a strain energy treatment but the resulting equations were too complex for a complete solution to be obtained. This analysis did, however, yield certain important relationships between the resultant actions which were required. A second line of attack was suggested to the author by Professor R. V. Southwell, which was instrumental in completing the solution of the general problem. It was assumed that the spokes could be replaced by a continuous disk which had the peculiar property of transmitting load in one direction only. If the spokes were radial, the disk could have stresses only in a radial direction; if they were inclined, the disk transmitted stresses only along the inclined lines. If more than one system of spokes were used in the wheel, as is commonly the case, each system of spokes was assumed to be replaced by an appropriate disk. The displacements of the rim under load, at any point \p from an axis of reference are denoted by u and v in the radial and tangential directions, respectively. R is the radius to the centroid of the rim, R0, the radius to the tangent circle on the hub, I, the moment of inertia of the cross section of the rim about its axis of bending, a, the cross-sectional area of the rim, £ i , the modulus of elasticity of the rim material, E2, the modulus of elasticity of the disk material, /i, the inward pull per unit length of rim of the disk acting at e to the radius, t2} the inward pull per unit length of rim of the disk acting at — e to the radius, where e is the angle of inclination of the spokes to the radii in the plane of the wheel. If the thickness of each disk at the rim is A, we have