Showing papers on "Angular velocity published in 2009"

Attitude Synchronization of a Group of Spacecraft Without Velocity Measurements

Abstract: We consider the coordinated attitude control problem for a group of spacecraft, without velocity measurements. Our approach is based on the introduction of auxiliary dynamical systems (playing the role of velocity observers in a certain sense) to generate the individual and relative damping terms in the absence of the actual angular velocities and relative angular velocities. Our main focus, in this technical note, is to address the following two problems: 1) Design a velocity-free attitude tracking and synchronization control scheme, that allows the team members to align their attitudes and track a time-varying reference trajectory (simultaneously). 2) Design a velocity-free synchronization control scheme, in the case where no reference attitude is specified, and all spacecraft are required to reach a consensus by aligning their attitudes with the same final time-varying attitude. In this work, one important and novel feature (besides the non-requirement of the angular velocity measurements), consists in the fact that the control torques are naturally bounded and the designer can arbitrarily assign the desired bounds on the control torques, a priori, through the control gains, regardless of the angular velocities. Throughout this technical note, the communication flow between spacecraft is assumed to be undirected. Simulation results of a scenario of four spacecraft are provided to show the effectiveness of the proposed control schemes.

Nonholonomic Source Seeking With Tuning of Angular Velocity

Abstract: We consider the problem of seeking the source of a scalar signal using an autonomous vehicle modeled as the nonholonomic unicycle. The vehicle does not have the capability of sensing its position or the position of the source but is capable of sensing the scalar signal originating from the source. The signal field is assumed to decay away from the position of the source but the vehicle does not have the knowledge of the functional form of the field. We employ extremum seeking to steer the vehicle to the source. Our control strategy keeps the forward velocity constant and tunes the angular velocity, a setting suitable for most autonomous vehicles, including aerial ones. Because of the constant forward velocity constraint, after it has converged near the source, the vehicle exhibits extremely interesting and complex motions. Using averaging theory, we prove local exponential convergence to an ldquoorbit-likerdquo attractor around the source. We also present a thorough analysis of non-local behaviors and attractors that the vehicle can exhibit near the source. The richness and complexity of behaviors makes only some of them amenable to analysis, whereas others are illustrated through a carefully laid out simulation study.

Diagnoses to unravel secular hydrodynamical processes in rotating main sequence stars

Abstract: Context. With the progress of observational constraints on stellar rotation and on the angular velocity profile in stars, it is necessary to understand how angular momentum is transported in stellar interiors during their whole evolution. In this context, more highly refined dynamical stellar evolution models have been built that take into account transport mechanisms.Aims. Internal gravity waves (IGWs) excited by convective regions constitute an efficient transport mechanism over long distances in stellar radiation zones. They are one of the mechanisms that are suspected of being responsible for the quasi-flat rotation profile of the solar radiative region up to 0.2 R⊙. Therefore, we include them in our detailed analysis started in Paper I of the main physical processes responsible for the transport of angular momentum and chemical species in stellar radiation zones. Here, we focus on the complete interaction between differential rotation, meridional circulation, shear-induced turbulence, and IGWs during the main sequence.Methods. We improved the diagnosis tools designed in Paper I to unravel angular momentum transport and chemical mixing in rotating stars by taking into account IGWs. The star’s secular hydrodynamics is treated using projection on axisymmetric spherical harmonics and appropriate horizontal averages that allow the problem to be reduced to one dimension while preserving the non-diffusive character of angular momentum transport by the meridional circulation and IGWs. Wave excitation by convective zones is computed at each time-step of the evolution track. We choose here to analyse the evolution of a 1.1 M⊙, Z⊙ star in which IGWs are known to be efficient.Results. We quantify the relative importance of the physical mechanisms that sustain meridional currents and that drive the transport of angular momentum, heat, and chemicals when IGWs are taken into account. First, angular momentum extraction, Reynolds stresses caused by IGWs, and viscous stresses sustain a large-scale multi-cellular meridional circulation. This circulation in turn advects entropy, which generates temperature fluctuations and a new rotation profile because of thermal wind.Conclusions. We have refined our diagnosis of secular transport processes in stellar interiors. We confirm that meridional circulation is sustained by applied torques, internal stresses, and structural readjustments, rather than by thermal imbalance, and we detail the impact of IGWs. These large-scale flows then modify the thermal structure of stars, their internal rotation profile, and their chemical stratification. The tools we developed in Paper I and generalised for the present analysis will be used in the near future to study secular hydrodynamics of rotating stars taking into account IGWs in the whole Hertzsprung-Russell diagram.

Transport by gravito-inertial waves in differentially rotating stellar radiation zones - I - Theoretical formulation

Abstract: Context. We examine the dynamics of low-frequency waves in differentially rotating stellar radiation zones, the angular velocity being taken as generally as possible depending both on radius and on latitude in stellar interiors. The associated induced transport of angular momentum, which plays a key role in the evolution of rotating stars, is derived. Aims. We focus on the wave-induced transport of angular momentum, taking into account the Coriolis acceleration in the case of strong radial and latitudinal differential rotation. We thus go beyond the “weak differential rotation” approximation, where rotation is almost a solid-body one plus a residual radial differential rotation. As has been shown in previous works, the Coriolis acceleration modifies such transport. Methods. We built analytically a complete formalism that allows the study of rotational transport in stellar radiation zones taking into account the wave action modified by a general strong differential rotation. Results. The different approximations possible for low-frequency waves in a differentially rotating stably stratified radiative region, namely the traditional and the JWKB approximations, are examined and discussed. The complete bidimensional structure of regular elliptic gravito-inertial waves, which verify these approximations, is derived and compared to those in the “weak differential rotation” case. Next, associated transport of energy and of angular momentum in the vertical and in the horizontal directions and the dynamical equations, respectively for the mean radial differential rotation and the latitudinal one, are obtained. Conclusions. The complete formalism, which takes into account low-frequency wave action, is derived and can be used for the study of secular hydrodynamics of radiative regions and of the associated mixing. The modification of waves due to general strong differential rotation and their feed-back on the angular momentum transport are treated rigourously. In a forthcoming paper (Paper II), this formalism will be applied to the case of solar differential rotation. However, the case of hyperbolic gravito-inertial waves should be carefully studied.

Effect of Kinematic Rotation-Translation Coupling on Relative Spacecraft Translational Dynamics

Abstract: A CCURATEmodeling of the differential translation and rotation between two spacecraft is essential for cooperative distributed space systems, spacecraft formation flying (SFF), rendezvous, and docking. High-fidelity relative motion modeling, as opposed to absolute motion modeling, is particularly important for autonomous missions [1]. Point-mass models for relative spacecraft translational motion have been extensively studied over the past 50 years, since Clohessy and Wiltshire (CW) presented a rendezvous model for a circular reference orbit and a spherical Earth [2]. Following the work of Clohessy and Wiltshire, variants on the point-mass model were developed, such as generalizations to elliptic reference orbits [3–5] and an oblate Earth [6,7]. The growing interest in SFF motivated the research of relative spacecraft motion modeling, yielding more accurate and complete equations and solutions for perturbed relative motion [8–10]. However, most of the works focused on point-mass, 3 degrees-offreedom (DOF) spacecraft. Obviously, performing a space mission that consists of several cooperative space vehicles requires modeling the relative rotational motion in addition to the relative translation, that is, 6-DOF models. Models for the relative motion of 6-DOF spacecraft have gained attention in the literature only in recent years. Among the first to suggest treating the spacecraft relative angular velocity in an SFF control problem were Pan and Kapila [11], who addressed the coupled translational and rotational dynamics of two spacecraft. By defining two body-fixed reference frames, one attached to the leader and the other attached to the follower, it was proposed [11] to use a two-part relative motion model: one that accounts for the relative translational dynamics of the body-fixed coordinate frame origins, and another that captures the relative attitude dynamics of the two body-fixed frames. A similar modeling approach was used for relative motion estimation [1]. In addition, tensorial equations of motion for a formation consisting ofN spacecraft, each modeled as a rigid body, were derived [12]. However, only the absolute equations of motion were developed [12]; a relative version of these equations was not given. Moreover, a clear mathematical relationship between the developed models and the traditional nonlinear point-mass relative motion and CW models was not provided. The coupling between the translational and rotational motion in the aforementioned models [1,11] was induced by gravity torques. The kinematic coupling, which is essentially a projection of the rotational motion about the center of mass (c.m.) onto the relative translational configuration space, was neglected. It is this kinematic coupling that the current paper is concerned with. In general, rigid-body dynamics can be represented as translation of the c.m. and rotation about the c.m. [13]. Thus, spacecraft relative motion must be composed by combining the relative translational and rotational dynamics of arbitrary points on the spacecraft. Whenever one of these points does not coincide with the spacecraft’s c.m., a kinematic coupling between the rotational and translational dynamics of these points is obtained. The purpose of this paper is to quantify the kinematic coupling effect and to show that this effect is key for high-precision modeling of tight SFF, rendezvous, and docking. This effect is also important in vision-based relative attitude and position control, where arbitrary feature points on a target vehicle are to be tracked. Given two rigidbody spacecraft, the model presented herein is formulated in a general manner that describes the motion between any two arbitrary points on the spacecraft. The relative translational motion is then generated by both the spacecraft orbitalmotion and the rotation about the c.m. In addition, this paper provides a CW-like approximation of the relative motion that includes the kinematic coupling. This new approximation is aimed at alleviating an apparent contradiction in linearized relative motion theories: to obtain linear equations of motion, the spacecraft are assumed to operate in close proximity. However, if the spacecraft are close to each other, then they can no longer be treated as point masses, because the spacecraft shape and size affects the relative translation between off-c.m. points. This effect is accentuated as the distances between spacecraft decrease. The remainder of this paper is organized as follows. First, a background on the relative position and attitude dynamics is given. Then, a new coupled relative spacecraft motion model is presented. The newly developed model is then examined in a simulation.

Image stabilization control apparatus and imaging apparatus

Abstract: An image stabilization control apparatus including a mechanism which causes a vibration when the mechanism moves is disclosed. The apparatus comprises a vibration correction unit configured to correct image shake occurring due to vibration applied to the image stabilization control apparatus. A correction value of an angular velocity of the vibration is calculated based on signals based on the angular velocity and an acceleration of the vibration, frequency bands of the signals are narrowed. During the mechanism is moving, the image shake is corrected by driving the vibration correction unit based on the angular velocity of the vibration which is corrected by the corrected value calculated before the mechanism moves.

A high fidelity ungrounded torque feedback device: The iTorqU 2.0

Abstract: This paper discusses the design and operation of the iTorqU 2.0, an ungrounded, handheld torque feedback device for haptic applications. Based upon the gyroscopic effect, the iTorqU 2.0 uses a metal flywheel inside of a two-axis actuated gimbal to create directional torques that are applied to the user's hand. The coupling of angular velocity and angular momentum creates a torque that is orthogonal to the two input angular velocities, giving the user the impression that their hand is being twisted in free air. Following a review of prior work in the field of ungrounded torque feedback devices, we first present our preliminary prototype, the iTorqU 1.0. Building on empirical observations and user feedback from a public demonstration, we revised and augmented this design to create the iTorqU 2.0. This paper covers the major mechanical, electrical, and controls design considerations that went into creating the iTorqU 2.0, along with an analysis of its torque output capabilities.

Differential rotation and convection in the Sun

Abstract: We show that the differential rotation profile of the solar convection zone, apart from inner and outer boundary layers, can be reproduced with great accuracy if the isorotation contours correspond to characteristics of the thermal wind equation. This requires that there be a formal quantitative relationship involving the entropy and the angular velocity. Earlier work has suggested that this could arise from magnetohydrodynamic stability constraints; here, we argue that purely hydrodynamical processes could also lead to such a result. Of special importance to the hydrodynamical solution is the fact that the thermal wind equation is insensitive to radial entropy gradients. This allows a much more general class of solutions to fit the solar isorotation contours, beyond just those in which the entropy itself must be a function of the angular velocity. In particular, for this expanded class, the thermal wind solution of the solar rotation profile remains valid even when large radial entropy gradients are present. A clear and explicit example of this class of solution appears to be present in published numerical simulations of the solar convective zone. Though hydrodynamical in character, the theory is not sensitive to the presence of weak magnetic fields. Thus, the identification of solar isorotation contours with the characteristics of the thermal wind equation appears to be robust, accommodating, but by no means requiring, magnetic field dynamics.

Convective dynamos in spherical wedge geometry

Abstract: Self-consistent convective dynamo simulations in wedge-shaped spherical shells are presented. Differential rotation is generated by the interaction of convection with rotation. Equatorward acceleration and dynamo action are obtained only for sufficiently rapid rotation. The angular velocity tends to be constant along cylinders. Oscillatory large-scale fields are found to migrate in the poleward direction. Comparison with earlier simulations in full spherical shells and Cartesian domains is made.

On Differential Rotation and Convection in the Sun

Abstract: We show that the differential rotation profile of the solar convection zone, apart from inner and outer boundary layers, can be reproduced with great accu- racy if the isorotation contours correspond to characteristics of the thermal wind equation. This requires that there be a formal quantitative relationship involving the entropy and the angular velocity. Earlier work has suggested that this could arise from magnetohydrodynamic stability constraints; here we argue that purely hydrodynamical processes could also lead to such a result. Of special importance to the hydrodynamical solution is the fact that the thermal wind equation is insensitive to radial entropy gradients. This allows a much more general class of solutions to fit the solar isorotation contours, beyond just those in which the entropy itself must be a function of the angular velocity. In particular, for this expanded class, the thermal wind solution of the solar rotation profile remains valid even when large radial entropy gradients are present. A clear and explicit example of this class of solution appears to be present in published numerical simulations of the solar convective zone. Though hydrodynamical in character, the theory is not sensitive to the presence of weak magnetic fields. Thus, the identification of solar isorotation contours with the characteristics of the thermal wind equation appears to be robust, accommodating, but by no means requiring, magnetic field dynamics.

Microelectromechanical gyroscope with rotary driving motion and improved electrical properties

Abstract: An integrated microelectromechanical structure is provided with: a die, having a substrate and a frame, defining inside it a detection region and having a first side extending along a first axis; a driving mass, anchored to the substrate, set in the detection region, and designed to be rotated in a plane with a movement of actuation about a vertical axis; and a first pair and a second pair of first sensing masses, suspended inside the driving mass via elastic supporting elements so as to be fixed with respect thereto in the movement of actuation and so as to perform a detection movement of rotation out of the plane in response to a first angular velocity; wherein the first sensing masses of the first pair and the first sensing masses of the second pair are aligned in respective directions, having non-zero inclinations of opposite sign with respect to the first axis.

Storage medium storing information processing program, information processing apparatus and information processing method

Abstract: A game apparatus includes a CPU, and the CPU controls a moving object within a virtual space on the basis of acceleration data and angular velocity data which are transmitted from a controller. For example, before the angular velocity data is above a predetermined magnitude, a position and an orientation of the moving object is controlled on the basis of the angular velocity data. When the angular velocity data is above the predetermined magnitude, an initial velocity of the moving object is decided on the basis of the acceleration data, and a moving direction (orientation) of the moving object is decided on the basis of the angular velocity data. Thereafter, the moving object moves within the virtual space according to a general physical behavior.

Vortices in Bose-Einstein condensates with dominant dipolar interactions

Abstract: We present full three-dimensional numerical calculations of single vortex states in rotating dipolar condensates. We consider a Bose-Einstein condensate of $^{52}\text{C}\text{r}$ atoms with dipole-dipole and $s$-wave contact interactions confined in an axially symmetric harmonic trap. We obtain the vortex states by numerically solving the Gross-Pitaevskii equation in the rotating frame with no further approximations. We investigate the properties of a single vortex and calculate the critical angular velocity for different values of the $s$-wave scattering length. We show that, whereas the standard variational approach breaks down in the limit of pure dipolar interactions, exact solutions of the Gross-Pitaevskii equation can be obtained for values of the $s$-wave scattering length down to zero. The energy barrier for the nucleation of a vortex is calculated as a function of the vortex displacement from the rotation axis for different values of the angular velocity of the rotating trap.

Is the Universe Rotating

Abstract: Models of a rotating universe have been studied widely since the work of G?del, who showed an example that is consistent with general relativity. By now, the possibility of a rotating universe has been discussed comprehensively in the framework of some types of Bianchi's models, such as Type V, VII, and IX and different approaches have been proposed to constrain the rotation. Recent discoveries of some non-Gaussian properties of the Cosmic Microwave Background Anisotropies (CMBA), such as the suppression of the quadrupole and the alignment of some multipoles draw attention to some Bianchi models with rotation. However, cosmological data, such as those of the CMBA, strongly prefer a homogeneous and isotropic model. Therefore, it is of interest to discuss the rotation of the universe as a perturbation of the Robertson-Walker metric, to constrain the rotating speed by cosmological data and to discuss whether it could be the origin of the non-Gaussian properties of the CMBA mentioned above. Here, we derive the general form of the metric (up to second-order perturbations) which is compatible with the rotation perturbation in a flat ?-CDM universe. By comparing the second-order Sachs-Wolfe effect due to rotation with the CMBA data, we constrain the angular speed of the rotation to be less than 10?9 rad?yr?1 at the last scattering surface. This provides the first constraint on the shear-free rotation of a ?CDM universe.

Flight Testing of Multiple-Spacecraft Control on SPHERES During Close-Proximity Operations

Abstract: A, B, C = state-space matrices a = acceleration due to linear-quadratic-regulatorand artificial-potential-field-determined control effort aAPF = acceleration due to artificial-potential-fielddetermined control effort aLQR = acceleration due to linear-quadratic-regulatordetermined control effort am = maximum acceleration aobs = acceleration of chaser spacecraft toward an obstacle ax;y;z = acceleration due to the control effort Do = obstacle region of influence da = goal acceleration decay constant dg = goal exponential decay constant do = stopping distance constant JLQR = linear quadratic regulator cost function KLQR = linear quadratic regulator state feedback gain ka = acceleration shaping parameter kd = docking safety parameter kg = velocity shaping function ko = obstacle function ks = safety function kv = velocity shaping parameter Lo = obstacle exterior surface N = linear quadratic regulator gain matrix Q = linear quadratic regulator state gain matrix R = linear quadratic regulator control effort gain matrix r = Euclidean norm distance or relative range r = relative distance vector rc = position vector of the chaser spacecraft rg = position vector of the chaser spacecraft from the goal rinit = initial distance of the chaser spacecraft from the goal rm = maximum allowable distance of the chaser spacecraft from the goal ro = position vector of the chaser spacecraft from the obstacle rt = position vector of the target spacecraft with respect to the Earth S = solution of the Riccati equation u = control effort vector V = potential function Vg = goal potential function Vo = obstacle potential function vm = maximum relative velocity vo = desired velocity of chaser spacecraft toward an obstacle vobs = velocity of chaser spacecraft toward an obstacle x = state vector x, y, z = positions, or states, along the Cartesian axis Q = linear quadratic regulator state performance gain R = linear quadratic regulator control effort gain t = time increment = standard deviation for the obstacle’s region of influence ! = orbital angular velocity

Global Regularity of the 3D Axi-symmetric Navier-Stokes Equations with Anisotropic Data

Abstract: In this paper, we study the 3D axi-symmetric Navier-Stokes Equations with swirl. We prove the global regularity of the 3D Navier-Stokes equations for a family of large anisotropic initial data. Moreover, we obtain a global bound of the solution in terms of its initial data in some $L^p$ norm. Our results also reveal some interesting dynamic growth behavior of the solution due to the interaction between the angular velocity and the angular vorticity fields.

Game apparatus and storage medium having stored thereon game program

Abstract: Whether or not an angular velocity detected by an angular velocity sensor is used for a game process is determined, and a difficulty level of a game to be subjected to a game process is set so as to vary depending on whether the angular velocity is determined to be used or the angular velocity is determined not to be used. When the angular velocity is determined not to be used, a process of specifying a motion direction of an input device is performed by using acceleration detected by an acceleration sensor, whereas when the angular velocity is determined to be used, at least a part of the process of specifying the motion of the input device is performed by using the angular velocity detected by the angular velocity sensor. Accordingly, based on the specified motion direction, the game process is performed while the set difficulty level is applied.

Input device and data processing system

Abstract: An input device includes a main body and a motion sensor unit. The main body has a longitudinal axis. The motion sensor unit is configured and arranged to detect rotation of the main body about the longitudinal axis. The motion sensor unit has an X-axis angular velocity sensor configured and arranged to detect an angular velocity of the main body about an X-axis in a three-dimensional orthogonal coordinate system defined by the X-axis, a Y-axis and a Z-axis. The X-axis coincides with the longitudinal axis of the main body and the Y-axis and the Z-axis being orthogonal to each other in a first plane perpendicular to the X-axis.

Dynamics of a two-module vibration-driven system moving along a rough horizontal plane

Abstract: A rectilinear motion of a system of two bodies connected by a spring on a rough horizontal plane is studied. The motion of the system is excited by two identical unbalanced rotors based on the respective bodies. Major attention is given to the steady-state motion. A nearly-resonant excitation mode, when the angular velocities of the rotor are close to the natural frequency of the system, is considered. A set of algebraic equations for determining an approximate value of the average steady-state velocity of the entire system is obtained for the case of small friction. It is shown that control of the steady-state motion can be provided by changing the phase shift between the rotations of the rotors and the sign of the resonant detuning measured by the difference between the angular velocity of the rotors and the natural frequency of the system. By varying the phase shift one can control the magnitude of the average velocity, and varying the detuning enables one to change the direction of the motion.

Single dust-particle rotation in glow-discharge plasma.

Abstract: Rotation of a single dust granule (spin) is investigated experimentally in a stratified glow discharge. We employ the technique of measurement of the angular velocity, which is based on coordinate tracing of the light scattered by a hollow transparent particle. The angular velocity measured in the experiment is about 1-2 orders of magnitude higher than observed in previous experiments. We found that the angular velocity depends linearly on the discharge current. The mechanism of rotation of the granule is also described.

Angular velocity measuring device

Abstract: An oscillator is oscillated at a predetermined oscillation frequency. A detecting unit exerts Coriolis force on the oscillator. A repetitive control system applies an external force to the oscillator so as to cancel out the Coriolis force to achieve an angular velocity measuring operation at a high sensitivity and a high S/N ratio.

Altitude Control of a Single Degree of Freedom Flapping Wing Micro Air Vehicle (Postprint)

Abstract: : A control strategy is proposed for a minimally actuated flapping wing micro air vehicle. The Harvard RoboFly vehicle accomplished the first takeoff of an insect scale flapping wing aircraft. This flight demonstrated the capability of the aircraft to accelerate vertically while being constrained by guide-wires to avoid translation and rotation in the other five degrees of freedom. The present work proposes an altitude control scheme that would enable a similar vehicle under the same constraints to hover and track altitude commands. Using a blade element-based aerodynamic model and cycle averaging, it will be shown that altitude control of such an aircraft can be achieved. The RoboFly makes use of a single bimorph piezoelectric actuator that symmetrically varies the angular displacement of the left and right wings in the stroke plane. The wing angle-of-attack variation is passive and is a function of the instantaneous angular velocity of the wing in the stroke plane. The control law is designed to vary the frequency of the wing beat oscillations to control the longitudinal body-axis force which is used to achieve force equilibrium in hover and acceleration when tracking time-varying altitude commands.

Condition monitoring system for wind turbine generator and method for operating wind turbine generator

Abstract: A wind turbine generator condition monitoring system (200) includes a plurality of rotor shaft angular velocity sensors (206/208) The system also includes at least one processor (216) coupled to the plurality of rotor shaft velocity sensors The at least one processor is programmed to determine a difference between each of the plurality of rotor shaft angular velocity sensors of at least one of an angular displacement, an angular velocity, and an angular acceleration of the rotor shaft An output of the at least one processor including at least one of a wind turbine generator yaw orientation signal (336) and a wind turbine generator blade pitch orientation signal (336)

Game program and game apparaus

Abstract: A game apparatus comprises an operation data acquiring means for acquiring operation data from a first input device including at least a first acceleration sensor and an angular velocity sensor and a second input device including at least a second acceleration sensor, and a game processing means for performing game processing on the basis of first acceleration data output from the first acceleration sensor, second acceleration data output from the second acceleration sensor, and angular velocity data from the angular velocity sensor, in the operation data

Altitude Control of a Single Degree of Freedom Flapping Wing Micro Air Vehicle

Abstract: A control strategy is proposed for a minimally actuated flapping wing micro air vehicle. The Harvard RoboFly vehicle accomplished the first takeoff of an insect scale flapping wing aircraft. This flight demonstrated the capability of the aircraft to accelerate vertically while being constrained by guide-wires to avoid translation and rotation in the other five degrees of freedom. The present work proposes an altitude control scheme that would enable a similar vehicle under the same constraints to hover and track altitude commands. Using a blade element-based aerodynamic model and cycle averaging, it will be shown that altitude control of such an aircraft can be achieved. The RoboFly makes use of a single bimorph piezoelectric actuator that symmetrically varies the angular displacement of the left and right wings in the stroke plane. The wing angle-of-attack variation is passive and is a function of the instantaneous angular velocity of the wing in the stroke plane. The control law is designed to vary the frequency of the wing beat oscillations to control the longitudinal body-axis force which is used to achieve force equilibrium in hover and acceleration when tracking time-varying altitude commands.

Time-Optimal Detumbling Control of Spacecraft

Abstract: T HE problem of designing fast detumbling maneuvers for a spacecraft with restricted actuation torques arises inmany space applications. Satellites are often equipped with an attitude control system (ACS), which usually has two modes of operation: detumbling and stabilization. In detumbling mode, the ACS controller is responsible for dumping the initial angular velocity from when the satellite was in the idle mode; this situation occurs when the satellite is released from the launch vehicle or when power is so low that ACS has to be turned off to save power [1]. Only after this phase can the stabilization mode be activated to control the orientation of the satellite to align antennas toward the Earth. Reaction wheels are commonly used as the actuators of the ACSs. Satellites frequently need fast maneuvers [2], minimum time maneuvers, while the speed is restricted owing to a low level capacity of the actuators. Hence, planning optimal maneuvers is highly desired. Alternatively, spin-stabilized spacecraft allow simple attitude maneuvers without the need for complex control systems. The spacecraft can be spun up around its axisymmetric axis to stabilize the orientation of the vehicle axis through the gyroscopic effect. This method is also widely used to stabilize the final stage of a launch vehicle [3]. However, when a spacecraft is in the tumblingmotion, its angular velocity is not parallel to the axisymmetric axis. Therefore, the objective in deployment of a spin-stabilized spacecraft [4] is to bring the spacecraft form the tumblingmotion to the statewherein the spacecraft spins around a single axis. The detumbling or passivation of a satellite is also required before on-orbit servicing of the satellite or its retrieval [5]. For such a mission, an orbital maneuvering vehicle can be used to apply torques to the target satellite for removing any relative velocity [5,6]. Also, the vehicle can be equipped by an articulating arm with a grappling device on it that could be driven to capture a tumbling target satellite and then detumble it [7,8]. After capture of an uncontrolled tumbling satellite by a spacemanipulator, the satellite should be brought to rest in minimum time. Again, the restriction of the manipulator end effector to provide “braking torques” for the fast maneuvers motivates an optimal trajectory planning. Optimal detumbling ofmultibody systems has been considered for spacecraft possessing appendages, such as a robotic manipulator, with well-controlled motion relative to the spacecraft to achieve detumbling [9–13].However, because of the complexity of dynamics of space manipulators, only a numerical solution or intensive trialand-error procedures have been found for the optimization problem. The problem of time-optimal detumbling control of rigid spacecraft is formulated as a nonlinear programming and solved numerically by using an iterative procedure in [14], while nonoptimal control approaches have been reported in [15–17]. There also exists a fair amount of research done on the time-optimal reorientation control, rest to rest, of rigid spacecraft [18–20] and a survey, for example, can be found in [21]. This paper presents a closed-form solution for time-optimal detumbling control of a rigid spacecraft with the constraint on maintaining the Euclidean norm of the braking torques below a prescribed value. Thefinal angular rate can be specified as zero or any vector parallel to the eigenaxis. The optimal control theory and Pontryagin’s principle are applied to derive the optimal solution, which is not only easy to implement but also gives a great deal of insight. First, it will be shown that for our particular optimal control problem, the system costates and states are related by a nonlinear but static function. Subsequently, the optimal control law is explicitly derived in the form of a nonlinear state feedback. The magnitude of the system angular momentum is found to be linearly deceasing with time, leading to a simple expression for the terminal time needed for control implementation. Furthermore, the time-optimal controller is extended for the case of nonzero terminal velocity. To this end, the time-optimal control technique is applied to derive a nonlinear feedback control law which can drive a tumbling axisymmertic spacecraft to a final spin-stabilized state. Finally, the time-optimal detumbling technique is illustrated through numerical examples.

Asymptotic perturbed feedback linearisation of underactuated Euler's dynamics

Abstract: A unifying methodology is introduced for smooth asymptotic stabilisation of underactuated rigid body dynamics under one and two degrees of actuation. The methodology is based on the concept of generalised inversion, and it aims to realise a perturbation from the unrealisable feedback linearising transformation. A desired linear dynamics in a norm measure of the angular velocity components about the unactuated axes is evaluated along solution trajectories of Euler's underactuated dynamical equations resulting in a linear relation in the control variables. This relation is used to assess asymptotic stabilisability of underactuated rigid bodies with arbitrary values of inertia parameters, and generalised inversion of the relation produces a control law that consists of particular and auxiliary parts. The generalised inverse in the particular part is scaled by a dynamic factor such that it uniformly converges to the Moore–Penrose inverse, and the null-control vector in the auxiliary part is chosen for asympto...