on vanishing stability regions in nonlinear systems with high gain feedback when local stabilization of nonlinear systems is achieved by linear feedback the resulting stability region may vanish as the feedback gains increase this is demonstrated by examples in which neglected nonlinearities create an unstable limit cycle around an asymptotically stable equilibrium as the feedback gains tend to infinity the unstable limit cycle shrinks to the equilibrium if a feedback linearization design is applied the same instability mechanism may occur when the nonlinearities are not precisely known

brief paper spacecraft coordination control in 6dof integrator backstepping vs passivity based control in this paper we present three nonlinear control solutions for 6dof spacecraft formation control adopted from euler lagrange system theory the quaternion parametrization results in dual equilibrium points in the closed loop system and the controllers are proved to render these uniformly asymptotically stable we also present a theoretical comparison of the three control solutions together with simulation results to visualize the performance of the controllers

asymptotic stabilization of passive systems without damping injection a sampled integral technique passivity in physical systems is a restatement of energy balancing and therefore is a ubiquitous property in engineering applications under some weak conditions the unique equilibrium state of passive systems is stable however to ensure asymptotic stability strict output passivity and a detectability property are required although strict output passivity may be enforced via a damping injection that feeds back the passive output this signal may be noisy or unmeasurable the paradigmatic example being velocity in mechanical systems in this paper a sampled integral stabilization sis technique for the asymptotic regulation of passive systems that requires only the knowledge of the time integral of the passive output i e position in mechanical systems is proposed as a generalization of the previous result it is shown that sis is applicable to cascade connections of passive systems measuring only the storage function of the second one several examples including a planar elbow manipulator and the rigid body dynamics are shown to satisfy the assumptions for the application of sis

the geometric structure of unit dual quaternion with application in kinematic control abstract in this paper the geometric structure especially the lie group related properties of unit dual quaternion is investigated the exponential form of unit dual quaternion and its approximate logarithmic mapping are derived correspondingly lie group and lie algebra on unit dual quaternions and the approximate logarithms are explored respectively afterwards error and metric based on unit dual quaternion are given which naturally result in a new kinematic control model with unit dual quaternion descriptors finally as a case study a generalized proportional control law using unit dual quaternion is developed

equilibrium problem for elastic body with delaminated t shape inclusion abstract we analyze an equilibrium problem for 2d elastic body with a t shape thin inclusion in presence of damage a part of the inclusion is elastic and the other part is a rigid one a delamination of the inclusion from the elastic body is assumed thus forming a crack between the elastic body and the inclusion nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces the damage is characterized by a positive parameter the paper provides an asymptotic analysis of the solutions as the damage parameter tends to infinity and to zero a passage to infinity of a rigidity parameter of the elastic part of the inclusion is also analyzed junction conditions are determined at the connection point between the elastic and rigid parts of the inclusion an existence theorem is proved for an inverse problem of finding displacement fields and the damage and rigidity parameters provided that a displacement of the tip point of the inclusion is known

analysis of two frictional viscoplastic contact problems with damage in this work we study two quasistatic frictional contact problems arising in viscoplasticity including the mechanical damage of the material caused by excessive stress or strain and modelled by an inclusion of parabolic type the variational formulation is provided for both problems and the existence of a unique solution is proved for each of them then a fully discrete scheme is introduced using the finite element method to approximate the spatial domain and the euler scheme to discretize the time derivatives error estimates are derived and under suitable regularity assumptions the linear convergence of the algorithm is deduced finally some numerical examples are presented to show the performance of the method

output feedback control for attitude tracking in this paper the problem of attitude tracking control for a rigid spacecraft is addressed it is assumed that only attitude measurements are available and thus spacecraft s angular velocity has to be properly estimated two alternative schemes are proposed in which the unit quaternion is adopted to represent the orientation in the first scheme a second order model based observer is adopted to estimate the angular velocity used in the control law in the second scheme an estimate of the angular velocity error is obtained through a lead filter sufficient conditions ensuring local exponential stability of the two controllers are derived via lyapunov analysis

output feedback control for rigid body attitude with constant disturbances in this article the control problem of rigid body attitude under constant disturbances without angular velocity measurement is solved by the combination of the immersion and invariance methodology and the dynamic scaling technique two observers which are respectively for estimating the angular velocity and the disturbance are constructed by utilising the immersion and invariance method the mismatched term arising from the observers is dominated by the high gain injection the control law is a simple proportional derivative controller plus a disturbance compensation term where the estimates of the angular velocity and the disturbance from observers are used for feedback directly the overall closed loop system is shown to be almost globally asymptotically stable under easy choices of some control parameters finally simulations are conducted to demonstrate the effectiveness of the proposed control scheme

hybrid certainty equivalence control of rigid bodies with quaternion measurements in this technical note we solve the control problem of rigid bodies with only quaternion measurements for all initial rotations and angular velocities the proposed solution is based on the theory of cascades using any switching certainty equivalence controller satisfying certain assumptions along with an in the large hybrid observer the equilibrium point of the proposed observer in closed loop with the rigid body dynamics is proven to be kappa exponentially stable in the large i e we prove that the equilibrium point is stable and that the error states converge exponentially fast towards the origin for all initial rotations and angular velocities until now stability results for quaternion based observers have typically only been valid for a bounded set of initial conditions to overcome this issue our observer design is based on dynamic scaling and switching logic furthermore we show that the origin of the proposed switching certainty equivalence controller in closed loop with the hybrid observer is asymptotically stable in the large for all available initial conditions associated with the quaternion space simulation results for the proposed scheme are presented with the particular case of the pd controller revealing that all states converge as expected from our theoretical findings

variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance friction and damage we consider a model for quasistatic frictional contact between a viscoelastic body and a foundation the material constitutive relation is assumed to be nonlinear the mechanical damage of the material caused by excessive stress or strain is described by the damage function the evolution of which is determined by a parabolic inclusion the contact is modeled with the normal compliance condition and the associated version of coulomb s law of dry friction we derive a variational formulation for the problem and prove the existence of its unique weak solution we then study a fully discrete scheme for the numerical solutions of the problem and obtain error estimates on the approximate solutions

on path lifting mechanisms and unwinding in quaternion based attitude control the unit quaternion is a pervasive representation of rigid body attitude used for the design and analysis of feedback control laws because the space of unit quaternions constitutes a double cover of the rigid body attitude space quaternion based control laws are often by design inconsistent i e they do not have a unique value for each rigid body attitude inconsistent quaternion based control laws require an additional mechanism that uniquely converts an attitude estimate into its quaternion representation however conversion mechanisms that are memoryless e g selecting the quaternion having positive scalar component have a limited domain where they remain injective and when used globally introduce discontinuities into the closed loop system we show through an explicit construction and lyapunov analysis that such discontinuities can be hijacked by arbitrarily small measurement disturbances to stabilize attitudes far from the desired attitude to remedy this limitation we propose a hybrid dynamic algorithm for smoothly lifting an attitude path to the unit quaternion space we show that this hybrid dynamic mechanism allows us to directly translate quaternion based controllers and their asymptotic stability properties obtained in the unit quaternion space to the actual rigid body attitude space we also show that when quaternion based controllers are not designed to account for the double covering of the rigid body attitude space by a unit quaternion parameterization they can give rise to the unwinding phenomenon which we characterize in terms of the projection of asymptotically stable sets finally we employ the main results to show that certain hybrid feedbacks can globally asymptotically stabilize the attitude of a rigid body

a topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon we show that a continuous dynamical system on a state space that has the structure of a vector bundle on a compact manifold possesses no globally asymptotically stable equilibrium this result is directly applicable to mechanical systems having rotational degrees of freedom in particular the result applies to the attitude motion of a rigid body in light of this result we explain how attitude stabilizing controllers obtained using local coordinates lead to unwinding instead of global asymptotic stability

leader following attitude consensus of multiple rigid body systems by attitude feedback control in this paper we study the leader following attitude consensus problem for multiple rigid body systems by a distributed unit quaternion based attitude feedback control law the control law is based on the certainty equivalence principle and contains a distributed observer for estimating the state of the leader and an auxiliary system compensating for the unavailability of the angular velocity it is shown that the problem is solvable under the standard assumption that each follower can access the information from the leader through a directed path

robust control for nonlinear systems using passivity based robust right coprime factorization the robust control for nonlinear feedback system with unknown perturbations is discussed based on the robust right coprime factorization and passivity property firstly based on passivity property the right factorization of the perturbed plant is realized secondly a robust control scheme is proposed considering the tracking property for the nonlinear system with nominal plant and perturbed plant finally we show the connection between the passivity and robust right coprime factorization for the obtained nonlinear system a numerical example is given to show the effectiveness of the proposed design scheme

unit dual quaternion based feedback linearization tracking problem for attitude and position dynamics abstract this paper provides a unified solution for the attitude and position tracking problem of a rigid body in 3 dimensional space using the concept of the unit dual quaternion the error dynamics described by a unit dual quaternion are deduced after the dual quaternion based dynamics of a single rigid body are given then by utilizing the feedback linearization principle a unit dual quaternion based tracker is proposed based on the error dynamics which is proven to render the equilibrium point of the closed loop system asymptotically stable and includes the attitude and position regulation problems as particular cases furthermore to solve the two equilibria problem a switching parameter is introduced to improve the tracker which causes the system to converge to the nearer equilibrium with a shorter path both the trackers uniquely deal with the rotational and translational dynamics simultaneously with non singularity and maintain the interconnection between rotation and translation finally the proposed control schemes with applications to the regulation problem and the tracking problem are simulated to illustrate the theoretical results

robust attitude stabilization of spacecraft using nonlinear quaternion feedback this paper considers the problem of three axis attitude stabilization of a rigid spacecraft a nonlinear control law which uses the feedback of the unit quaternion and the measured angular velocities is proposed and is shown to provide global asymptotic stability the control law does not require the knowledge of the system parameters and is therefore robust to modeling errors the significance of the control law is that it can be used for large angle maneuvers with guaranteed stability

quaternion based adaptive output feedback attitude control of spacecraft using chebyshev neural networks this paper investigates the problem of output feedback attitude control of an uncertain spacecraft two robust adaptive output feedback controllers based on chebyshev neural networks cnn termed adaptive neural networks nn controller i and adaptive nn controller ii are proposed for the attitude tracking control of spacecraft the four parameter representations quaternion are employed to describe the spacecraft attitude for global representation without singularities the nonlinear reduced order observer is used to estimate the derivative of the spacecraft output and the cnn is introduced to further improve the control performance through approximating the spacecraft attitude motion the implementation of the basis functions of the cnn used in the proposed controllers depends only on the desired signals and the smooth robust compensator using the hyperbolic tangent function is employed to counteract the cnn approximation errors and external disturbances the adaptive nn controller ii can efficiently avoid the over estimation problem i e the bound of the cnns output is much larger than that of the approximated unknown function and hence the control input may be very large existing in the adaptive nn controller i both adaptive output feedback controllers using cnn can guarantee that all signals in the resulting closed loop system are uniformly ultimately bounded for performance comparisons the standard adaptive controller using the linear parameterization of spacecraft attitude motion is also developed simulation studies are presented to show the advantages of the proposed cnn based output feedback approach over the standard adaptive output feedback approach

attitude control without angular velocity measurement a passivity approach it is well known that the linear feedback of the quaternion of the attitude error and the angular velocity globally stabilizes the attitude of a rigid body in this note we show that the angular velocity feedback can be replaced by a nonlinear filter of the quaternion thus removing the need for direct angular velocity measurement in contrast to other approaches this design exploits the inherent passivity of the system a model based observer reconstructing the velocity is not needed an application of the proposed scheme is illustrated for the robot control problem simulation results are included to illustrate the theoretical results

further passivity results for the attitude control problem in this paper we establish passivity for the system which describes the attitude motion of a rigid body in terms of minimal three dimensional kinematic parameters in particular we show that linear asymptotically stabilizing controllers and control laws without angular velocity measurements follow naturally from these passivity properties the results of this paper extend similar results for the case of the nonminimal euler parameters

on the stability and stabilization of quaternion equilibria of rigid bodies we study attitude control of rigid bodies on quaternion coordinates under three mathematically different perspectives depending on how the system dynamics are assumed to evolve in the first case we suppose that one equilibrium point is chosen a priori and a continuous controller is used under the assumption that the rigid body always spins in the same direction in the second case we relax the assumption that the sense of rotation is constant finally a third scenario is considered in which hybrid switching control is used to choose the direction in which to spin that is both equilibria are continuously considered with regard to less energy consumption it is showed that each of three scenarios must be treated in a different theoretical setting a comparative study in simulations is also provided

rigid body attitude stabilization with attitude and angular rate constraints abstract in this paper a solution to the problem of rest to rest three axis attitude reorientation of a fully actuated rigid body under multiple attitude constraint zones and angular velocity limits is presented based on the unit quaternion parameterized attitude constrained zones a quadratic potential function is developed with a global minimum locating at the desired attitude and high potential closing to the constrained zones in addition to limit the magnitude of the angular velocity another logarithmic potential function is also designed using these two potential functions and sliding mode control technique a nonlinear attitude control law is obtained to guarantee asymptotic convergence of the closed loop system with consideration of attitude and angular rate constraints and external disturbances the effectiveness of the constrained attitude control method is demonstrated through numerical simulation

unit quaternion based output feedback for the attitude tracking problem in this note we propose a quaternion based dynamic output feedback for the attitude tracking problem of a rigid body without velocity measurement our approach consists of introducing an auxiliary dynamical system whose output which is also a unit quaternion is used in the control law together with the unit quaternion representing the attitude tracking error roughly speaking the necessary damping that would have been achieved by the direct use of the angular velocity can be achieved in our approach by the vector part mathtilde q of the error signal between the output of the auxiliary system and the unit quaternion tracking error the resulting velocity free control scheme guarantees almost global asymptotic stability which is as strong as the topology of the motion space can permit in the regulation case our control law is a pure quaternion feedback i e consisting of two terms that are vector parts of unit quaternion and hence the control torques are naturally bounded by the control gains simulation results are provided to show the effectiveness of the proposed control scheme

passivity based adaptive attitude control of a rigid spacecraft an adaptive control scheme for the attitude control of a rigid spacecraft is derived using a linear parameterization of the equation of motion the tracking error is described with the euler parameter vector global convergence of the tracking error to zero is shown using passivity theory this allows for the use of time varying positive definite feedback gain matrices and the results can easily be extended to other passive parameter update laws

global uniqueness for an inverse boundary value problem arising in elasticity we prove that we can determine the lame parameters of an elastic isotropic inhomogeneous medium in dimensionsn 3 by making measurements of the displacements and corresponding stresses at the boundary of the medium

non standard analysis and representation of reality the aim of this paper is to show that the representation with the help of non standard analysis of a real phenomenon presenting different observation scales allows an important simplification of language indeed it is convenient to have available the concept of infinitely small and infinitely large quantities in dealing with the macroscopic effects of microscopic phenomena this is illustrated on two examples the representation of two time scales systems and the representation of noise

model free control model free control and the corresponding intelligent pid controllers ipids which already had many successful concrete applications are presented here for the first time in an unified manner where the new advances are taken into account the basics of model free control is now employing some old functional analysis and some elementary differential algebra the estimation techniques become quite straightforward via a recent online parameter identification approach the importance of ipis and especially of ips is deduced from the presence of friction the strange industrial ubiquity of classic pid s and the great difficulty for tuning them in complex situations is deduced via an elementary sampling from their connections with ipids several numerical simulations are presented which include some infinite dimensional systems they demonstrate not only the power of our intelligent controllers but also the great simplicity for tuning them

construction of synergistic potential functions on so 3 with application to velocity free hybrid attitude stabilization we propose a systematic and comprehensive procedure for the construction of synergistic potential functions which are instrumental in hybrid control design on text so 3 a new map via angular warping on text so 3 is introduced to generate a central family of potential functions allowing an explicit determination of the critical points and the synergistic gap some optimization results on the synergistic gap are also provided the proposed synergistic potential functions are used for the design of a global velocity free hybrid attitude stabilization scheme relying solely on inertial vector measurements

on the inverse problem of identifying lame coefficients in linear elasticity an output least squares type functional is employed to identify the lame parameters in linear elasticity to be able to identify even the discontinuous lame parameters the regularization is performed by the bv seminorm finite element discretization is used and convergence analysis is given numerical examples are given to show the feasibility of the approach

averaging on slow and fast cycles of a three time scale system abstract pontryagin rodygin s theorem for slow and fast systems describes the slow drift during the rolling up of the trajectories around the cycles of the fast dynamics this drift is approximated by the averaging on the cycles the calculation of this average is generally a difficult task since it requires the knowledge of the closed orbits and their periods we present two paradigms of three time scale systems where we can overcome this limitation it is the case of systems the fast dynamics of which have cycles with relaxation presenting or not a canard phenomenon we can not apply pontryagin rodygin s theorem to these systems because their fast equation is itself singularly perturbed we also investigate the extension of the results to unbounded time intervals the results are stated classically and proved within the framework of nonstandard analysis

semirigid inclusions in elastic bodies mechanical interplay and optimal control abstract the paper concerns an analysis of an equilibrium problem for 2d elastic body with two semirigid inclusions it is assumed that inclusions have a joint point and we investigate a junction problem for these inclusions the existence of solutions is proved and different equivalent formulations of the problem are proposed we investigate a convergence to infinity of a rigidity parameter of the semirigid inclusion it is proved that in the limit we obtain an equilibrium problem for the elastic body with a rigid inclusion and a semirigid one a parameter identification problem is investigated in particular the existence of a solution to a suitable optimal control problem is proved

finite time output feedback position and attitude tracking of a rigid body the position and attitude tracking of a rigid body without velocity measurements is addressed dual quaternions are used to describe the coupled rotational and translational motions of the rigid body yielding compact forms of the kinematics and dynamics suitable for control law synthesis an output feedback pose position and attitude tracking controller is then designed by integrating techniques from passivity and homogeneity more precisely a passivity enabling auxiliary system is proposed to provide necessary damping instead of velocity feedback and a homogeneous method is used to ensure finite time convergence the proposed controller guarantees uniform almost global finite time stability of the closed loop system and produces a well defined vector field on the attitude configuration manifold thus avoiding the unwinding phenomenon moreover it can be split to obtain velocity free controllers with finite time convergence for the cases of translation only or rotation only control numerical examples verify the effectiveness of the proposed method

hybrid output feedback for attitude tracking on mathbb so 3 we present two solutions to the global exponential attitude tacking control problem using only attitude information on mathbb so 3 the first approach is based on a combined hybrid observer hybrid controller structure we derive a hybrid angular velocity observer for rigid body systems designed directly on mathbb so 3 times mathbb r 3 guaranteeing global exponential stability of the zero estimation error thereafter we propose an observer based attitude tracking control scheme guaranteeing global exponential stability the second approach consists of introducing an auxiliary dynamical system that generates the necessary damping in the absence of the velocity we prove that the proposed auxiliary system based hybrid controller guarantees global exponential stability as well

a globally convergent angular velocity observer for rigid body motion the problem of obtaining the angular velocity of a rigid body from orientation and torque measurements only without noisy numerical differentiation is considered a novel angular velocity angular momentum observer for rigid body motion is presented using euler quaternions and a mechanical energy function approach it is shown that the observer estimates converge globally and that the convergence is eventually exponential it is hoped that a mechanical energy function approach to rigid body control can be combined with the observer presented to lead to a globally stable nonlinear observer based rigid body controller in which the observer and controller errors can be separated in much the same way as one can separate controller and observer poles in the output feedback controllers of linear system theory

global finite time attitude tracking via quaternion feedback abstract this paper addresses the attitude tracking of a rigid body using a quaternion description global finite time attitude controllers are designed with three types of measurements namely full states attitude plus constant biased angular velocity and attitude only in all three scenarios hybrid control techniques are utilized to overcome the well known topological constraint on the attitude manifold while coupled nonsmooth feedback inputs are designed via homogeneous theory to achieve finite time stability specially a finite time bias observer is derived in the second scenario and a quaternion filter is constructed to provide damping in the absence of velocity feedback the proposed methods ensure bounded control torques a priori and in particular include several existing attitude controllers as special cases

global stabilization of spherical orientation by synergistic hybrid feedback with application to reduced attitude tracking for rigid bodies in this paper we propose a hybrid feedback based on a synergistic potential function that achieves global asymptotic stabilization of a desired orientation on the n sphere with a nominal robustness to measurement disturbances a task that is not possible by classical feedback be it smooth nonsmooth periodic or any combination thereof due to the topological structure of the sphere we extend this basic result to a tracking controller for the reduced attitude or pointing direction of a rigid body and provide a method to remove jumps in the controlled torque by backstepping the proposed hybrid feedback is compared with a similar smooth feedback in simulation where it is illustrated that the hybrid feedback overcomes performance limitations inherent to the smooth feedback we provide two examples of a synergistic potential function one defined on a general sphere and the other on the unit circle

pd based output feedback attitude control of rigid bodies we address the problem of output feedback attitude control of a rigid body in quaternion coordinate space via a modified pd based tracking controller angular velocity is replaced by a low gain dynamic extension the controller ensures fast convergence to the desired operating point during transient maneuvers while keeping the gains small this contributes to diminishing the sensitivity to measurement noise hence energy consumption may be expected to drop along with a decrease of the residual more precisely we show uniform practical asymptotic stability of the equilibrium point for the closed loop system in the presence of unknown bounded input disturbances simulation results illustrate the performance improvement with respect to pd based output feedback control with static gains

linearization of attitude control error dynamics some useful properties of direction cosine and quaternion attitude formulations that include specification of attitude error and the inversion of rigid body rotational dynamics are reviewed the inversion procedure is then applied to a measure of attitude error to realize a new model follower control system that exhibits linear attitude error dynamics error analyses and simulation results for spacecraft attitude control systems are presented to demonstrate the more robust performance obtainable from an exact linear error formulation over that obtained from either direction cosine or quaternion formulations with simple linear feedback control laws

the attitude control problem a general framework for the analysis of the attitude tracking control problem for a rigid body is presented a large family of globally stable control laws is obtained by using the globally nonsingular unit quaternion representation in a lyapunov function candidate whose form is motivated by the consideration of the total energy of the rigid body the controllers share the common structure of a proportional derivative feedback plus some feedforward which can be zero the model independent case the coriolis torque compensation or an adaptive compensation these controller structures are compared in terms of the requirement on the a priori model information guaranteed transient performance and robustness the global stability of the luh walker paul robot end effector controller is also analyzed in this framework

damageable contact between an elastic body and a rigid foundation in this work the contact problem between an elastic body and a rigid obstacle is studied including the development of material damage which results from internal compression or tension the variational problem is formulated as a first kind variational inequality for the displacements coupled with a parabolic partial differential equation for the damage field the existence of a unique local weak solution is stated then a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an euler scheme to discretize the time derivatives error estimates are derived on the approximate solutions from which the linear convergence of the algorithm is deduced under suitable regularity conditions finally three two dimensional numerical simulations are performed to demonstrate the accuracy and the behaviour of the scheme