the convergence of subspace trust region methods the trust region method is an effective approach for solving optimization problems due to its robustness and strong convergence however the subproblem in the trust region method is difficult or time consuming to solve in practical computation especially in large scale problems in this paper we consider a new class of trust region methods specifically subspace trust region methods the subproblem in these methods has an adequate initial trust region radius and can be solved in a simple subspace it is easier to solve than the original subproblem because the dimension of the subproblem in the subspace is reduced substantially we investigate the global convergence and convergence rate of these methods

almost global stabilization of fully actuated rigid bodies this paper addresses the problem of stabilizing a fully actuated rigid body the problem is formulated by considering the natural configuration space for rigid bodies the special euclidean group se 3 the proposed solution consists of a landmark based controller for force and torque actuation that guarantees almost global asymptotic stability of the desired equilibrium point as such the equilibrium point is asymptotically stable and only a nowhere dense set of measure zero lies outside its region of attraction the controller uses velocity measurements and the position coordinates of a collection of landmarks fixed in the environment as an additional feature the control law is designed so as to verify prescribed bounds on the actuation

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

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

brief paper vision based control for rigid body stabilization this paper addresses the problem of stabilizing to a desired equilibrium point an eye in hand system which consists of a single camera mounted on a rigid body free to move on se 3 it is assumed that there is a collection of landmarks fixed in the environment and that the image coordinates of those landmarks are provided to the system by an on board ccd camera the proposed method addresses not only the problem of stabilization but also that of maintaining feature visibility along the system s trajectory the resulting solution consists of a feedback control law based on the current and desired image coordinates and reconstructed attitude and depth ratio information which guarantees that i the desired equilibrium point is an almost global attractor ii a set of necessary conditions for feature visibility holds throughout the system s trajectories and iii the image of a predefined feature point is kept inside the camera s field of view

a self adaptive trust region method with line search based on a simple subproblem model in this paper based on a simple model of the trust region subproblem we propose a new self adaptive trust region method with a line search technique for solving unconstrained optimization problems by use of the simple subproblem model the new method needs less memory capacitance and computational complexity and the trust region radius is adjusted with a new self adaptive adjustment strategy which makes full use of the information at the current point when the trial step results in an increase in the objective function the method does not resolve the subproblem but it performs a line search technique from the failed point convergence properties of the method are proved under certain conditions numerical experiments show that the new method is effective and attractive for large scale optimization problems

robust attitude tracking for rigid spacecraft with prescribed transient performance abstractthis paper investigates the prescribed performance attitude tracking control problem of a rigid spacecraft with uncertain dynamics and bounded disturbances by suitable selection of the configuration error function the trajectory tracking problem on so 3 is transformed into a point stabilisation problem of an error dynamic system in the associated lie algebra the predefined transient performance indexes which include the maximum steady state error and overshoot and the minimum convergence rate of the attitude error vector are characterised as the inequality constraints for the tracking controller design the error transformation technique is utilised to transform the attitude error dynamics with inequality constraints to an equivalent unconstrained error one then a coordinate free robust tracking controller is designed for the unconstrained error dynamics to solve the prescribed performance attitude tracking control problem a rigorous mathematical stability proof is given finally n

riemannian consensus for manifolds with bounded curvature consensus algorithms are popular distributed algorithms for computing aggregate quantities such as averages in ad hoc wireless networks however existing algorithms mostly address the case where the measurements lie in euclidean space in this work we propose riemannian consensus a natural extension of existing averaging consensus algorithms to the case of riemannian manifolds unlike previous generalizations our algorithm is intrinsic and in principle can be applied to any complete riemannian manifold we give sufficient convergence conditions on riemannian manifolds with bounded curvature and we analyze the differences with respect to the euclidean case we test the proposed algorithms on synthetic data sampled from the space of rotations the sphere and the grassmann manifold

intrinsic tetrahedron formation of reduced attitude abstract in this paper formation control for reduced attitude is studied in which a regular tetrahedron formation can be achieved and shown to be asymptotically stable under a large family of gain functions in the control moreover by further restriction on the control gain almost global stability of the desired formation is obtained in addition the control proposed is an intrinsic protocol that only uses relative information and does not need to contain any information of the desired formation beforehand the constructed formation pattern is totally attributed to the geometric properties of the space and the designed inter agent connection topology besides a novel coordinates transformation is proposed to represent the relative reduced attitudes in s 2 which is shown to be an efficient approach to reduced attitude formation problems

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

a new trust region filter algorithm in this paper an adaptive trust region filter algorithm for unconstrained optimization is presented the global convergence results of the algorithm are established and the numerical results show that the new algorithm is efficient in practical computation

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 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

nonlinear estimator design on the special orthogonal group using vector measurements directly the convergence properties of popular nonlinear attitude estimators can be traced to the choice of an attitude error function this paper considers a nonlinear deterministic direction cosine matrix estimator whose form is derived from an alternate attitude error function while the resulting estimator shares several properties with those previously presented in the literature the careful selection of an attitude error function results in an estimator with superior convergence properties the attitude estimate is propagated using a rate gyroscope measurement and corrected using two or more vector measurements simulation and experimental results are presented that highlight the desirable properties of the proposed estimator

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

exact solutions to a class of feedback systems on so n this paper provides a novel approach to the problem of attitude tracking for a class of almost globally asymptotically stable feedback laws on so n the closed loop systems are solved exactly for the rotation matrices as explicit functions of time the initial conditions and the gain parameters of the control laws the exact solutions provide insight into the transient dynamics of the system and can be used to prove almost global attractiveness of the identity matrix applications of these results are found in model predictive control problems where detailed insight into the transient attitude dynamics is utilized to approximately complete a task of secondary importance knowledge of the future trajectory of the states can also be used as an alternative to the zero order hold in systems where the attitude is sampled at discrete time instances

automatic determination of an initial trust region in nonlinear programming this paper presents a simple but efficient way to find a good initial trust region radius itrr in trust region methods for nonlinear optimization the method consists of monitoring the agreement between the model and the objective function along the steepest descent direction computed at the starting point further improvements for the starting point are also derived from the information gleaned during the initializing phase numerical results on a large set of problems show the impact the initial trust region radius may have on trust region methods behavior and the usefulness of the proposed strategy

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

quasialgebra structure of the octonions abstract we show that the octonions are a twisting of the group algebra of z 2 z 2 z 2 in the quasitensor category of representations of a quasi hopf algebra associated to a group 3 cocycle in particular we show that they are quasialgebras associative up to a 3 cocycle isomorphism we show that one may make general constructions for quasialgebras exactly along the lines of the associative theory including quasilinear algebra representation theory and an automorphism quasi hopf algebra we study the algebraic properties of quasialgebras of the type which includes the octonions further examples include the higher 2n onion cayley algebras and examples associated to hadamard matrices

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

brief paper spacecraft relative rotation tracking without angular velocity measurements we present a solution to the problem of tracking relative rotation in a leader follower spacecraft formation using feedback from relative attitude only the controller incorporates an approximate differentiation filter to account for the unmeasured angular velocity we show uniform practical asymptotic stability upas of the closed loop system for simplicity we assume that the leader is controlled and that we know orbital perturbations however this assumption can be easily relaxed to boundedness without degrading the stability property we also assume that angular velocities of spacecraft relative to an inertial frame are bounded simulation results of a leader follower spacecraft formation using the proposed controller structure are also presented

surrounding control in cooperative agent networks in this paper two surrounding control problems are proposed where a team of followers is used to surround a team of leaders the problems are solved under a decentralized estimation and control framework using tools from algebraic graph theory and dynamical systems theory it is shown that the two teams involving a team of leaders and a team of followers preserve some desired convergence properties even if the geometric center of the leaders can only be obtained from estimators a simulation example is presented to verify the validity of the derived results

a second derivative sqp method global convergence sequential quadratic programming sqp methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems although second derivative information may often be calculated there is little practical theory that justifies exact hessian sqp methods in particular the resulting quadratic programming qp subproblems are often nonconvex and thus finding their global solutions may be computationally nonviable this paper presents a second derivative sqp method based on quadratic subproblems that are either convex and thus may be solved efficiently or need not be solved globally additionally an explicit descent constraint is imposed on certain qp subproblems which guides the iterates through areas in which nonconvexity is a concern global convergence of the resulting algorithm is established

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

kloosterman sums for clifford algebras and a lower bound for the positive eigenvalues of the laplacian for congruence subgroups acting on hyperbolic spaces

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

etale backstepping for control lyapunov function design on manifold abstract backstepping is one of the most popular nonlinear controller and differentiable control lyapunov function clf design techniques however for asymptotic stabilization of systems defined on noncontractible manifolds there exists no differentiable clf the semiconcave clf design problem based on the backstepping has not been discussed in this paper we propose a backstepping based controller design method for asymptotic stabilization of systems defined on noncontractible manifolds in the method we design a controller and a clf on an etale space then we obtain a semiconcave clf on the original space by the minimum projection method the effectiveness of the proposed method is confirmed by computer simulation

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

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

finding optimal algorithmic parameters using derivative free optimization the objectives of this paper are twofold we devise a general framework for identifying locally optimal algorithmic parameters algorithmic parameters are treated as decision variables in a problem for which no derivative knowledge or existence is assumed a derivative free method for optimization seeks to minimize some measure of performance of the algorithm being fine tuned this measure is treated as a black box and may be chosen by the user examples are given in the text the second objective is to illustrate this framework by specializing it to the identification of locally optimal trust region parameters in unconstrained optimization the derivative free method chosen to guide the process is the mesh adaptive direct search a generalization of pattern search methods we illustrate the flexibility of the latter and in particular make provision for surrogate objectives locally optimal parameters with respect to overall computational time on a set of test problems are identified each function call may take several hours and may not always return a predictable result a tailored surrogate function is used to guide the search towards a local solution the parameters thus identified differ from traditionally used values and allow one to solve a problem that remained otherwise unsolved in a reasonable time using traditional values

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

exponential stability of an attitude tracking control system on so 3 for large angle rotational maneuvers abstract this paper studies a tracking control system for the attitude dynamics of a rigid body by selecting an attitude error function carefully we show that the proposed control system guarantees a desirable tracking performance uniformly for rotational maneuvers involving a large initial attitude error a strict lyapunov analysis is presented to show exponential stability and a sufficient condition to avoid non differentiable points of the attitude error function is also shown the proposed control system is directly developed on the special orthogonal group to avoid complexities and ambiguities associated with other attitude representations such as euler angles or quaternions these are illustrated by numerical examples

numerical research on the sensitivity of nonmonotone trust region algorithms to their parameters in this paper a numerical research on the sensitivity of nonmonotone trust region algorithms to their parameters is presented we compare the numerical efficiency of two classes of nonmonotone trust region ntr algorithms in the context of unconstrained optimization we examine the sensitivity of the algorithms to the parameters related to the nonmonotone technique and the initial trust region radius we show that the numerical efficiency of nonmonotone trust region algorithms can be improved by choosing appropriate parameters based on extensive numerical tests some efficient ranges of these parameters for nonmonotone trust region algorithms are recommended

contraction theory on riemannian manifolds abstract contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another in this work we present the fundamental results of contraction theory in an intrinsic coordinate free setting with the presentation highlighting the underlying geometric foundation of contraction theory and the resulting stability properties we provide coordinate free proofs of the main results for autonomous vector fields and clarify the assumptions under which the results hold we state and prove several interesting corollaries to the main result study cascade and feedback interconnections of contracting systems study some simple examples and highlight how contraction theory has arisen independently in other scientific disciplines we conclude by illustrating the developed theory for the case of gradient dynamics

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

feedback stabilization of isospectral control systems on complex flag manifolds application to quantum ensembles the convex set of density operators of an n level quantum mechanical system foliated as a complex flag manifold where each leaf is identified with the adjoint unitary orbit of the eigenvalues of a density matrix for an isospectral bilinear control system evolving on such an orbit the state feedback stabilization problem admits a natural lyapunov based time varying feedback design a global description of the domain of attraction of the closed loop system can be provided based on a ldquoroot spacerdquo like structure of the cone of density operators the converging conditions are time independent but depend on the topology of the flag manifold it is shown that the closed loop must have a number of equilibria at least equal to the euler characteristic of the manifold thus imposing topological obstructions to global stabilizability

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

coordinated path following for unicycles we formulate a coordinated path following problem for n unicycle mobile robots as an instance of a nested set stabilization problem stabilization of the first set corresponds to driving the unicycles to their assigned paths stabilization of the second set a subset of the first corresponds to meeting the coordination specification the first set is stabilized for initial conditions sufficiently close to the set in a decentralized manner using feedback linearization for general coordination tasks we utilize feedback linearization to stabilize the nested set in a centralized manner again for initial conditions sufficiently close to the second set in the special case in which coordination entails making the unicycles maintain a formation along their paths we employ a semi distributed control law under less restrictive communication assumptions experimental results are provided

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

a nonmonotone trust region method with adaptive radius for unconstrained optimization problems in this paper we incorporate a nonmonotone technique with the new proposed adaptive trust region radius shi and guo 2008 4 in order to propose a new nonmonotone trust region method with an adaptive radius for unconstrained optimization both the nonmonotone techniques and adaptive trust region radius strategies can improve the trust region methods in the sense of global convergence the global convergence to first and second order critical points together with local superlinear and quadratic convergence of the new method under some suitable conditions numerical results show that the new method is very efficient and robustness for unconstrained optimization problems

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

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

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

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

almost global consensus on the n sphere this paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multiagent systems that evolve in continuous time on the n dimensional unit sphere or n sphere for any connected undirected graph and all n in mathbb n backslash lbrace 1 rbrace each protocol in said class is shown to yield almost global consensus the feedback laws are negative gradients of lyapunov functions and one instance generates the canonical intrinsic gradient descent protocol this convergence result sheds new light on the general problem of consensus on riemannian manifolds the n sphere for n in mathbb n backslash lbrace 1 rbrace differs from the circle and mathsf so 3 where the corresponding protocols fail to generate almost global consensus moreover we derive a novel consensus protocol on mathsf so 3 by combining two almost globally convergent protocols on the n sphere for n in lbrace 1 2 rbrace theoretical and simulation results suggest that the combined protocol yields almost global consensus on mathsf so 3

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

output feedback control for stabilization on se 3 this paper addresses the problem of stabilizing systems that evolve on se 3 the proposed solution consists of an output feedback controller that guarantees almost global asymptotic stability of the desired equilibrium point in the sense that the equilibrium point is stable and we have convergence for all initial conditions except for those in a nowhere dense set of measure zero the output vector is formed by the position coordinates of a collection of landmarks fixed in the environment the resulting closed loop system exhibits the following properties i the position and orientation subsystems are decoupled ii the position error is globally exponentially stable and iii the orientation error is almost globally exponentially stable results are also provided that allow one to select landmark configurations so as to control how the position and orientation of the rigid body converge to their final equilibrium values

an efficient two step trust region algorithm for exactly determined consistent systems of nonlinear equations abstract in this paper we propose a modified two step trust region algorithm for solving nonlinear systems two step trust region algorithms at every iteration use a trust region step and an approximate step by saving the jacobian matrix to have fewer computations we introduce a convex combination to modify the trust region subproblems along with an additional criterion to verify whether the first step is accepted or not we establish global and quadratic convergence of the algorithm under some mild assumptions numerical results show that the modified algorithm is efficient and promising

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

orientation control on so 3 with piecewise sinusoids this paper addresses rigid body orientation control for the case that angular velocity is restricted to a class of piecewise sinusoidal signals the orientation kinematics are represented by the group of rotation matrices so 3 we present a new closed form solution of the rotation matrix kinematics and we use this solution to prove controllability we then present kinematic level orientation feedback controllers for setpoint tracking and command following

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 new trust region method for unconstrained optimization in this paper we propose a new trust region method for unconstrained optimization problems the new trust region method can automatically adjust the trust region radius of related subproblems at each iteration and has strong global convergence under some mild conditions we also analyze the global linear convergence local superlinear and quadratic convergence rate of the new method numerical results show that the new trust region method is available and efficient in practical computation

high dimensional kuramoto models on stiefel manifolds synchronize complex networks almost globally abstract the kuramoto model of coupled phase oscillators is often used to describe synchronization phenomena in nature some applications e g quantum synchronization and rigid body attitude synchronization involve high dimensional kuramoto models where each oscillator lives on the n sphere or so n these manifolds are special cases of the compact real stiefel manifold st p n using tools from optimization and control theory we prove that the generalized kuramoto model on st p n converges to a synchronized state for any connected graph and from almost all initial conditions provided p n satisfies p 2 3 n 1 and all oscillator frequencies are equal this result could not have been predicted based on knowledge of the kuramoto model in complex networks over the circle in that case almost global synchronization is graph dependent it applies if the network is acyclic or sufficiently dense this paper hence identifies a property that distinguishes many high dimensional generalizations of the kuramoto models from the original model

multiplicative invariant lattices in r n obtained by twisting of group algebras and some explicit characterizations abstract let g be a finite group and r g be its group algebra defined over r if we define in g a 2 cochain f then we can consider the algebra r f g which is obtained from r g deforming the product x f y f x y x y x y g examples of r f z 2 n algebras are clifford algebras and cayley algebras like octonions in this paper we consider generalizations of lattices with complex multiplication in the context of these twisted group algebras we explain how these induce the natural algebraic structure to endow any arbitrary finite dimensional lattice whose real components stem from any finite algebraic field extension over q with a multiplicative closed structure furthermore we develop some fully explicit characterizations in terms of generalized trace and norm functions

recursive trust region methods for multiscale nonlinear optimization a class of trust region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems like those arising in systems governed by partial differential equations the algorithms in this class make use of the discretization level as a means of speeding up the computation of the step this use is recursive leading to true multilevel multiscale optimization methods reminiscent of multigrid methods in linear algebra and the solution of partial differential equations a simple algorithm of the class is then described and its numerical performance is shown to be numerically promising this observation then motivates a proof of global convergence to first order stationary points on the fine grid that is valid for all algorithms in the class

consensus problems on networks with antagonistic interactions in a consensus protocol an agreement among agents is achieved thanks to the collaborative efforts of all agents expresses by a communication graph with nonnegative weights the question we ask in this paper is the following is it possible to achieve a form of agreement also in presence of antagonistic interactions modeled as negative weights on the communication graph the answer to this question is affirmative on signed networks all agents can converge to a consensus value which is the same for all agents except for the sign necessary and sufficient conditions are obtained to describe cases in which this is possible these conditions have strong analogies with the theory of monotone systems linear and nonlinear laplacian feedback designs are proposed