a geometric characterization of c optimal designs for heteroscedastic regression we consider the common nonlinear regression model where the variance as well as the mean is a parametric function of the explanatory variables the c optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest a geometric characterization of c optimal designs in this context is presented which generalizes the classical result of elfving 1952 for c optimal designs as in elfving s famous characterization c optimal designs can be described as representations of boundary points of a convex set however in the case where there appear parameters of interest in the variance the structure of the elfving set is dierent roughly speaking the elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space the c optimal designs are characterized as representations of the points where the line in direction of the vector c intersects the boundary of the new elfving set the theory is illustrated in several examples including pharmacokinetic models with random eects

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

stability analysis of velocity aided attitude observers for accelerated vehicles an advanced nonlinear observer for rigid bodies evolving in 3d space relying on measurement data provided by an imu and linear velocity measured in the body fixed frame has been recently proposed but without specifying its domain of convergence and stability with respect to conventional attitude observers which do not involve linear velocity measurements this observer is better adapted for vehicles subjected to sustained accelerations this paper presents further modifications yielding two velocity aided attitude observers with rigorous almost global convergence and stability analysis moreover the estimation of the gravity direction is globally decoupled from magnetometer measurements

d optimum designs for heteroscedastic linear models abstract the methods of optimum experimental design are applied to models in which the variance as well as the mean is a parametric function of explanatory variables extensions to standard optimality theory lead to designs when the parameters of both the mean and the variance functions or the parameters of only one function are of interest the theory also applies whether the mean and variance are functions of the same variables or of different variables although the mathematical foundations differ the example studied is a second order two factor response surface for the mean with a parametric nonlinear variance function the theory is used both for constructing designs and for checking optimality a major potential for application is to experimental design in off line quality control

the existence of liapunov functions for some non conservative positional mechanical systems abstract in this work we establish a liapunov function for the equilibrium for almost all systems x x x y yg x g u r 0 u u r g c 2 0 g 0 0

multilayer minimum projection method for nonsmooth strict control lyapunov function design asymptotic stabilization on noncontractible manifolds is known as a difficult control problem to address this problem we had proposed the minimum projection method to design nonsmooth control lyapunov functions this method however has some problems difficult etale surjection design undesirable resulting control lyapunov functions etc in this paper we propose a new nonsmooth control lyapunov function design method called the multilayer minimum projection method for nonsmooth control lyapunov function design on general manifolds the method considers many simple structured smooth manifolds associated with the original manifold by etale mappings and then a function on the original manifold is obtained by projecting control lyapunov functions defined on the simple structured manifolds onto the original manifold in this paper we prove that the resulting function by the proposed method is a nonsmooth control lyapunov function on the original manifold moreover we prove that if all control lyapunov functions defined on simple structured manifolds are strict the control lyapunov function on the original manifold is a strict control lyapunov function finally the effectiveness of the proposed method and the advantage over the conventional minimum projection method are confirmed by an example

hybrid global exponential stabilization on so 3 we propose a central synergistic hybrid approach for global exponential stabilization on the special orthogonal group so 3 so 3 we introduce a new switching concept relying on a central family of possibly non differentiable potential functions that enjoy the following properties 1 being quadratic as well as their gradients with respect to the euclidean attitude distance and 2 being synergistic with respect to the gradient s singular and or critical points the proposed approach is used to solve the attitude tracking problem leading to global exponential stability results

optimal designs for comparing curves we consider the optimal design problem for a comparison of two regression curves which is used to establish the similarity between the dose response relationships of two groups an optimal pair of designs minimizes the width of the confidence band for the difference between the two regression functions optimal design theory equivalence theorems efficiency bounds is developed for this non standard design problem and for some commonly used dose response models optimal designs are found explicitly the results are illustrated in several examples modeling dose response relationships it is demonstrated that the optimal pair of designs for the comparison of the regression curves is not the pair of the optimal designs for the individual models in particular it is shown that the use of the optimal designs proposed in this paper instead of commonly used non optimal designs yields a reduction of the width of the confidence band by more than 50

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

minimax d optimal designs for item response theory models various different item response theory irt models can be used in educational and psychological measurement to analyze test data one of the major drawbacks of these models is that efficient parameter estimation can only be achieved with very large data sets therefore it is often worthwhile to search for designs of the test data that in some way will optimize the parameter estimates the results from the statistical theory on optimal design can be applied for efficient estimation of the parameters a major problem in finding an optimal design for irt models is that the designs are only optimal for a given set of parameters that is they are locally optimal locally optimal designs can be constructed with a sequential design procedure in this paper minimax designs are proposed for irt models to overcome the problem of local optimality minimax designs are compared to sequentially constructed designs for the two parameter logistic model and the results show that minimax design can be nearly as efficient as sequentially constructed designs

adaptive compensation of gyro bias in rigid body attitude estimation using a single vector measurement the presence of bias in measurement of rate gyros is a performance limiting factor for satellite attitude determination systems gyro bias is usually handled by kalman filtering methods which are mostly based on linearization approaches and lack global convergence properties on the other hand the existing asymptotically convergent nonlinear observers take into account the gyro bias only when multiple vector measurements are available we present an asymptotically convergent attitude estimator which uses only one vector measurement and a rate gyro whose output is contaminated with an unknown and constant bias the effect of unknown bias is compensated by means of a parameter adaptation law an explicit condition on the reference vectors for ensuring observability of the system is introduced a realistic simulation is presented to illustrate the performance of the proposed attitude estimator in face of significant bias in gyro measurement and when a magnetometer is used to provide the single vector measurement

topological properties of asymptotically stable sets topological properties of the domain of attraction for dynamical systems are investigated the main purpose of this paper is to prove that a compact asymptotically stable attractor of a dynamical system defined on a locally compact metric space is a deformation retract of its domain of attraction in a weak sense that is made precise under additional local assumptions the attractor can be shown to be a retract a deformation retract or a strong deformation retract the well known result that the domain of attraction of an asymptotically stable equilibrium is contractible follows as a corollary

global attitude estimation using single delayed vector measurement and biased gyro time delay in attitude determination systems is unavoidable and it is usually caused by low quality data sampling poor sensor synchronization momentary sensor outage or operational restrictions of sensors despite its performance degrading effect time delay has been widely neglected in the existing attitude estimation methods this paper presents a general framework for design of attitude estimators with and without so 3 manifold restriction by assuming that measurement of a single vector measurement is available with time delay also measurement of rigid body angular velocity is considered available with an unknown bias the proposed estimator guarantees global and asymptotic convergence of attitude estimate to its true value the observer gain is calculated by solving a delay dependent matrix differential equation solvability of the differential equation depends on an observability condition related to the reference vectors to illustrate performance of the proposed observer we present simulation of a satellite system where the single delayed vector measurement is provided by a magnetometer

minimax robust designs and weights for approximately specified regression models with heteroscedastic errors abstract this article addresses the problem of constructing designs for regression models in the presence of both possible heteroscedasticity and an approximately and possibly incorrectly specified response function working with very general models for both types of departure from the classical assumptions i exhibit minimax designs and correspondingly optimal weights simulation studies and a case study accompanying the theoretical results lead to the conclusions that the robust designs yield substantial gains over some common competitors in the presence of realistic departures that are sufficiently mild so as to be generally undetectable by common test procedures specifically i exhibit solutions to the following problems p1 for ordinary least squares determine a design to minimize the maximum value of the integrated mean squared error imse of the fitted values with the maximum being evaluated over both types of departure p2 for weighted least squares determine both weights and a design to m

a necessary and sufficient condition for the stability of the equilibrium abstract the aim of this work is to obtain a necessary and sufficient condition in order that the origin be a stable equilibrium for the system x xf x y yg x f 0 g 0 0

output regulation for control systems on se n a separation principle based approach this technical note considers a solution to an output regulation problem for control systems on the special euclidean group se n the proposed solution consists of a convergent observer together with a certainty equivalence implementation of an almost globally stable state feedback law the design of the observer and the state feedback are carried out independently we show that all closed loop solutions remain bounded that the steady state solution is contained in exactly one of a finite number of isolated sets which are all unstable except one which is the desired equilibrium set and which is asymptotically stable

technical communique retraction obstruction to time varying stabilization this paper addresses the problem of the global stabilization on a total space of a fiber bundle with a compact base space we prove that under mild assumptions existence of a continuous section and forward unicity of solutions no equilibrium of a continuous system defined on such a state space can be globally asymptotically uniformly stabilized using continuous time varying feedback

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

global attitude and gyro bias estimation based on set valued observers a b s t r a c t the problem of attitude and rate gyro bias estimation is addressed by resorting to measurements acquired from rate gyros and vector observations a set valued observer svo is proposed that has no singulari ties and that for any initial conditions provides a bounding set with guarantees of containing the actual unknown rotation matrix the sensor readings are assumed to be corrupted by bounded measurement noise and constant gyro bias conditions for the boundedness of the estimated sets are established and implementation details are discussed the feasibility of the technique is demonstrated in simulation

heteroscedastic g optimal designs this paper addresses the issue of constructing large sample g optimal designs when the variability of the response varies across a compact design space a useful characterization theorem is presented along with a computer algorithm for generating heteroscedastic g optimal designs to facilitate comparisons between d and g optimal designs atwood s inequality for comparing d and g efficiencies in homoscedastic models is generalized to heteroscedastic models some robustness properties of these designs are presented

modeling and control of quantum systems an introduction the scope of this work is to provide a self contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems as well as a brief survey on the main approaches to control synthesis while part of the existing theory especially in the open loop setting stems directly from classical control theory most notably geometric control and optimal control a number of tools specifically tailored for quantum systems have been developed since the 1980s in order to take into account their distinctive features the probabilistic nature of atomic scale physical systems the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback design problems the relevant dynamical models for both closed and open quantum systems are presented along with the main results on their controllability and stability a brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies

a graphical method for finding maximin efficiency designs we consider the problem of designing an experiment when there are two competing optimality criteria designs that maximize the minimum efficiencies under the two criteria are proposed along with a graphical method for finding these maximin designs

brief paper a ges attitude observer with single vector observations this paper proposes a globally exponentially stable ges observer for attitude estimation based on a single time varying reference vector in inertial coordinates and corresponding vector in body fixed coordinates in addition to angular velocity readings the proposed solution is computationally efficient and in spite of the fact that the observer does not evolve on the special orthogonal group so 3 an explicit solution on so 3 is also provided whose error is shown to converge exponentially fast to zero for all initial conditions the distinct roles of the inertial and the corresponding body fixed vectors on the observability of the system are also examined and simulation results are shown that illustrate the performance of the proposed attitude observer in the presence of low grade sensor specifications

coordinate independent adaptive attitude tracking control design for spacecraft robust to time varying system uncertainties abstractin this paper we consider the adaptive attitude control of spacecraft with time varying inertial properties we use a coordinate independent approach for the purpose of designing the control and estimation laws in terms of the rotation matrices representing the spacecraft body frame and reference tracking signals this method helps to overcome the difficulties regarding the attitude representation on so 3 such as ambiguities associated with quaternion representation and inherent singularities inside euler parameters we model the time variations in inertial parameters in two different ways and design adaptive control schemes for each case as the first uncertain dynamic model we consider a setting of spacecraft with multiple moving appendages and based on this model design an adaptive control scheme with three different versions where frobenious norm is used in measuring the deviation of the estimated inertia tensors from their actual values the proposed adaptive control scheme is later ex

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

a unified approach to the construction of minimax designs summary we present a unified approach for constructing large sample optimal designs when the optimality criterion is of the minimax type the assumed model is a general linear regression model with a known efficiency function defined on a closed and bounded space an equivalence theorem is formulated and computer algorithms for generating minimax optimal designs are proposed it is shown that this methodology is simple and rather general it can be used to construct for example minimax variance optimal designs minimax with respect to the single parameters designs and e optimal designs an application of this procedure to find an optimal design for minimizing the maximum predictive variance over a compact region in a heteroscedastic design problem is included

quaternion based hybrid feedback for robust global attitude synchronization we apply recent results on robust global asymptotic stabilization of the attitude of a single rigid body to the problem of globally synchronizing the attitude of a network of rigid bodies using a decentralized strategy the proposed hybrid feedback scheme relies on the communication of a binary logic variable between each pair of neighboring rigid bodies that determines the orientation of a torque component acting to reduce their relative error through a hysteretic switch of this logic variable the hybrid feedback achieves global synchronization under the assumption that the network is connected and acyclic the hysteresis eliminates chattering while preventing the unwinding phenomenon apparent in some quaternion based attitude control schemes the results are exercised in a numerical example

a nonlinear position and attitude observer on se 3 using landmark measurements this paper addresses the problem of position and attitude estimation based on landmark readings and velocity measurements a derivation of a nonlinear observer on se 3 is presented using a lyapunov function conveniently expressed as a function of the difference between the estimated and the measured landmark coordinates the resulting feedback laws are explicit functions of the landmark measurements and velocity readings exploiting the sensor information directly in the observer the proposed observer yields almost global asymptotic stabilization of the position and attitude errors and exponential convergence in any closed ball inside the region of attraction also it is shown that the asymptotic convergence of the estimation error trajectories is shaped by the landmark geometry and observer design parameters the problem of non ideal velocity readings is also considered and the observer is augmented to compensate for bias in the angular and linear velocity measurements the resulting position attitude and bias estimation errors are shown to converge exponentially fast to the desired equilibrium points for bounded initial estimation errors simulation results are presented to illustrate the stability and convergence properties of the observer

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

combination of lyapunov and density functions for stability of rotational motion lyapunov methods and density functions provide dual characterizations of the solutions of a nonlinear dynamic system this work exploits the idea of combining both techniques to yield stability results that are valid for almost all the solutions of the system based on the combination of lyapunov and density functions analysis methods are proposed for the derivation of almost input to state stability and of almost global stability in nonlinear systems the techniques are illustrated for an inertial attitude observer where angular velocity readings are corrupted by non idealities

an intrinsic pid controller for mechanical systems on lie groups this paper shows how integral action may be implemented intrinsically in the general setting of fully actuated mechanical systems on lie groups when combined with an intrinsic formulation of pd control integral action ensures that a suitably defined configuration tracking error will converge to zero in response to constant velocity commands and disturbance forces convergence is locally exponential and almost global and is robust to bounded parametric uncertainty for more general bounded commands and disturbances the tracking error can be made arbitrarily small by choosing sufficiently large controller gains a separation principle allows the addition of an intrinsic velocity observer to complete a geometric analog to the familiar output feedback pid controller for second order linear systems the controller extends naturally to accommodate tracking of equivariant output functions the result is explicitly shown for the lie group so 3 and supported by simulations of a 6dof quadrotor model

global set stabilisation of the spacecraft attitude using finite time control technique in this article we revisit the classical problem of attitude stabilisation for a rigid spacecraft with external disturbances a global set stabilisation method using finite time control technique is proposed in the absence of disturbances the states of the closed loop system will be stabilised in finite time to a set consisting of two equilibria in the presence of disturbances the states will be stabilised to a neighbourhood of this set by constructing a particular lyapunov function it is proved that the closed loop system satisfies global set stability the control method in this article is based on set control idea and thus is more natural and energy efficient numerical simulation results show the effectiveness of the method

output regulation for systems on matrix lie groups abstract this paper deals with the problem of output regulation for left invariant systems defined on general matrix lie groups reference trajectories to be tracked are generated by a right invariant exosystem defined on the same lie group and driven by a linear oscillator defined on the lie algebra only partial relative geometrical information of the exosystem with respect to actual system is available in the form of invariant measurements in a homogeneous space a quotient of the lie group in the spirit of the internal model principle the proposed control structure contains an embedded copy of the exosystem kinematics the dynamics associated with the mechanical systems can be handled by dynamic extension of the proposed control approach we demonstrate the approach by providing the details of this extension for the specific case of rotational dynamics on s o 3 the problem formulation considered is motivated by a wide range of real applications in robotics aerospace and computer vision where the state space of systems such as mobile robots aerial robots unmanned drones and camera homographies are naturally lie groups associated with inherent symmetry properties of the underlying physical system

an intrinsic approach to formation control of regular polyhedra for reduced attitudes this paper addresses formation control of reduced attitudes in which a continuous control protocol is proposed for achieving and stabilizing all regular polyhedra also known as platonic solids un

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

intrinsic reduced attitude formation with ring inter agent graph this paper investigates the reduced attitude formation control problem for a group of rigid body agents using feedback based on relative attitude information under both undirected and directed cyc

geometric controllability and stabilization of spherical robot dynamics geometric control of a spherical robot rolling on a horizontal plane with three independent inertia disc actuators is considered in this note the dynamic model of the spherical robot in the geometric framework is used to establish the strong accessibility and small time local controllability properties smooth stabilizability to an equilibrium fails for the nonholonomic spherical robot a novel contribution of this note is a smooth asymptotically stabilizing geometric control law for position and reduced attitude which corresponds to an equilibrium submanifold of dimension one from brockett s condition this is the best possible dimension of a smoothly stabilized equilibrium submanifold we also present a novel smooth global tracking controller for tracking position trajectories

a non linear feedback attitude regulator an asymptotically stabilizing non linear feedback law is presented for rigid body spacecraft attitude regulation the control law takes the form of a linear combination of non linear functions of attitude and of angular momentum each function being positively homogeneous of degree r 0

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

asymptotic smooth stabilization of the inverted 3 d pendulum the 3 d pendulum consists of a rigid body supported at a fixed pivot with three rotational degrees of freedom it is acted on by gravity and it is fully actuated by control forces the 3 d pendulum has two disjoint equilibrium manifolds namely a hanging equilibrium manifold and an inverted equilibrium manifold the contribution of this paper is that two fundamental stabilization problems for the inverted 3 d pendulum are posed and solved the first problem asymptotic stabilization of a specified equilibrium in the inverted equilibrium manifold is solved using smooth and globally defined feedback of angular velocity and attitude of the 3 d pendulum the second problem asymptotic stabilization of the inverted equilibrium manifold is solved using smooth and globally defined feedback of angular velocity and a reduced attitude vector of the 3 d pendulum these control problems for the 3 d pendulum exemplify attitude stabilization problems on the configuration manifold so 3 in the presence of potential forces lyapunov analysis and nonlinear geometric methods are used to assess global closed loop properties yielding a characterization of the almost global domain of attraction for each case

on the topological structure of attraction basins for differential inclusions abstract we show that when a compact set is globally asymptotically stable under the action of a differential inclusion satisfying certain regularity properties there exists a smooth differential equation rendering the same compact set globally asymptotically stable the regularity properties assumed in this work stem from the consideration of krasovskii filippov solutions to discontinuous differential equations and the robustness of asymptotic stability under perturbation in particular the results in this work show that when a compact set cannot be globally asymptotically stabilized by continuous feedback due to topological obstructions it cannot be robustly globally asymptotically stabilized by discontinuous feedback either the results follow from converse lyapunov theory and parallel what is known for the local stabilization problem

exponential convergence of a nonlinear attitude estimator a nonlinear attitude estimator is presented the estimator uses a minimum of two vector measurements and a rate gyro to estimate the attitude and the rate gyro bias the estimator can be implemented using a low cost inertial measurement unit such as those typically used on inexpensive aerial vehicles estimator design is based on an alternate attitude error function that yields an estimator with faster convergence properties than similar estimators previously considered in the literature a proof of exponential stability about the desired equilibrium point is provided simulation and experimental results demonstrate the desirable properties of the proposed estimator

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

control of a driftless bilinear vector field on n sphere in this paper we consider a multi input driftless bilinear system evolving on the n dimensional sphere s n we first provide examples drawn from rigid body mechanics that provide the motivation for the control of bilinear systems on s n for the general framework we establish the global controllability on s n and propose two linear control laws on s n that achieve asymptotic stabilization of an equilibrium point with an almost global domain of attraction further the asymptotically stable closed loop system trajectories are shown to be arcs on the geodesics of s n for a particular choice of the equilibrium point next we propose two linear time varying control laws to achieve trajectory tracking on s n and show the asymptotic stability of the tracking error a distributed control is designed for the consensus of multiagent bilinear systems on s n with an undirected tree as the communication graph the consensus manifold is shown to have an almost global domain of attraction

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

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

a computational method for the stability of a class of mechanical systems abstract in this work we obtain criteria for the stability of the origin for the system x xf x y yg x p where f g are positive in a neighbourhood of 0 r we start with abstract necessary and sufficient conditions and develop them into practical explicit conditions in terms of inequalities involving the successive derivatives of f x g x up to order 2 at x 0 some examples are worked out