feedback particle filter on riemannian manifolds and matrix lie groups this paper is concerned with the problem of continuous time nonlinear filtering of stochastic processes evolving on connected riemannian manifolds without boundary the main contribution of this paper is to derive the feedback particle filter fpf algorithm for this problem in its general form the fpf is shown to provide an intrinsic description of the filter that automatically satisfies the geometric constraints of the manifold the particle dynamics are encapsulated in a stratonovich stochastic differential equation that retains the feedback structure of the original euclidean fpf the implementation of the filter requires a solution of a poisson equation on the manifold and a numerical algorithm is described for this purpose for the special case when the manifold is a matrix lie group explicit formulae for the filter are derived using the matrix coordinates filters for two example problems are presented the attitude estimation problem on so 3 and the robot localization problem in se 3 comparisons are also provided between the fpf and popular algorithms for attitude estimation namely the multiplicative extended kalman filter ekf the invariant ekf the unscented quaternion estimator the invariant ensemble kalman filter and the bootstrap particle filter numerical simulations are presented to illustrate these comparisons

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

minimum energy filtering for attitude estimation in this work we study minimum energy filtering for attitude kinematics with vectorial measurements using mortensen s approach the exact form of a minimum energy attitude observer is derived and is shown to depend on the hessian of the value function of an associated optimal control problem a suitably chosen matrix representation of the hessian operator leads to a riccati equation that approximates a minimum energy attitude filter an extended version of the proposed approximate filter is included for a situation where there is slowly time varying bias in the gyro measurements a unit quaternion version of the proposed filter is derived and shown to outperform the multiplicative extended kalman filter mekf for situations with large initialization errors or large measurement errors

classical q deformation of su 2 and os 1 the generators of the classical suq 2 and osq 1 quantum groups are obtained from the non deformed ones by means of an appropriate mapping and by imposing the poisson bracket pb relations of the deformed algebras the classic limit of the q deformed lipkin model is obtained from a time dependent variational principle and its orbits are represented in phase space

the structure of nonlinear control systems possessing symmetries a concept of symmetry is defined for general nonlinear control systems it is shown under various technical conditions that nonlinear control systems with symmetries admit local and or global decompositions in terms of lower dimensional subsystems and feedback loops the structure of the individual subsystems is dependent on the structure of the symmetry group for example if the symmetry group is abelian one of the subsystems is a quadrature an additional feature of the decomposition is that the state space dimensions of the subsystems sum to the state space dimension of the original system

attitude estimation with intermittent measurements we propose a framework for attitude estimation on the special orthogonal group so 3 using intermittent body frame vector measurements we consider the case where the vector measurements are synchronously intermittent all measurements are received at the same time and the case where the vector measurements are asynchronously intermittent not all measurements are received at the same time the proposed observers have a measurement triggered structure where the attitude is predicted using the continuously measured angular velocity when the vector measurements are not available and adequately corrected upon the arrival of the vector measurements a hybrid framework is proposed to capture the behaviour of the closed loop system by extending the state with timers that are reset at each jump of the observer state almost global asymptotic stability is shown using rigorous lyapunov techniques for hybrid systems

optimal control for holonomic and nonholonomic mechanical systems with symmetry and lagrangian reduction in this paper we establish necessary conditions for optimal control using the ideas of lagrangian reduction in the sense of reduction under a symmetry group the techniques developed here are designed for lagrangian mechanical control systems with symmetry the benefit of such an approach is that it makes use of the special structure of the system especially its symmetry structure and thus it leads rather directly to the desired conclusions for such systems lagrangian reduction can do in one step what one can alternatively do by applying the pontryagin maximum principle followed by an application of poisson reduction the idea of using lagrangian reduction in the sense of symmetry reduction was also obtained by bloch and crouch proc 33rd cdc ieee 1994 pp 2584 2590 in a somewhat different context and the general idea is closely related to those in montgomery comm math phys 128 1990 pp 565 592 and vershik and gershkovich dynamical systems vii v arnold and s p novikov eds springer verlag 1994 here we develop this idea further and apply it to some known examples such as optimal control on lie groups and principal bundles such as the ball and plate problem and reorientation examples with zero angular momentum such as the satellite with moveable masses however one of our main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation in the context of the work of bloch krishnaprasad marsden and murray arch rational mech anal 1996 to appear the snakeboard is used to illustrate the method

practical stabilization of driftless systems on lie groups the transverse function approach a general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional lie groups is presented the approach is based on the concept of bounded transverse functions the existence of which is equivalent to the system s controllability its outcome is the practical stabilization of any trajectory i e not necessarily a solution of the control system in the state space the possibility of applying the approach to an arbitrary controllable smooth driftless system follows in turn from the fact that any controllable homogeneous approximation of this system can be lifted via a dynamic extension to a system on a lie group illustrative examples are given

symmetry preserving observers this paper presents the theory of invariant observers i e symmetry preserving observers we consider an observer to consist of a copy of the system and a correction term and we propose a constructive method based on the cartan moving frame method to find all the symmetry preserving correction terms the construction relies on an invariant frame a classical notion and on an invariant output error a less standard notion precisely defined here using the theory we build three non linear observers for three examples of engineering interest a non holonomic car a chemical reactor and an inertial navigation system for each example the design is based on physical symmetries and the convergence analysis relies on the use of invariant state errors a symmetry preserving way to define the estimation error

invariant feedback control for the kinematic car on the sphere abstract the design of an invariant tracking control law for the kinematic car driving on a sphere is discussed using a lie group framework a left invariant description on so 3 is derived basic geometric considerations allow a direct comparison of the model with the planar case using a geometrically motivated invariant tracking error on s 2 an invariant feedback design is proposed finally one possible design of an invariant asymptotic observer is sketched

singular limit and exact decay rate of a nonlinear elliptic equation abstract for any n 3 0 m n 2 n and constants 0 0 n 2 m we prove the existence of radially symmetric solution of n 1 m v m v x v 0 v 0 in r n v 0 without using the phase plane method when 0 m n 2 n n 3 we prove that v satisfies lim x x 2 v x 1 m log x 2 n 1 n 2 n m 1 m if 2 1 m 0 and lim x x v x a for some constant a 0 if 2 1 m max 0 for 0 or 0 we prove that the radially symmetric solution v m of the above elliptic equation converges uniformly on every compact subset of r n to the solution of an elliptic equation as m 0

a characterization of functional realizations of three dimensional quantum groups the authors study in an exhaustive and systematic way the functional realizations of quantum and classical algebras related with su 2 su 1 1 and the oscillator group os 1 in order to achieve this purpose they have adopted a mapping that is a generalization of an algebra expansion procedure they have not transformed the algebras themselves but their representations this method enables them not only to relate classical algebras and their q analogues but also different type algebras whether they are quantum or not they have obtained and classified all possible q realizations including as particular cases all those already known in the literature

representations of symmetric linear dynamical systems the purpose of this paper is to study static symmetries in linear time invariant differential dynamical systems the main result is a representation theorem which brings the symmetry strongly into evidence this result is then applied to a number of examples involving permutations and rotations we close by proving a general result on the representation of compact groups on the ring of unimodular polynomial matrices

degenerate neckpinches in ricci flow in earlier work 2 we derived formal matched asymptotic profiles for families of ricci flow solutions developing type ii degenerate neckpinches in the present work we prove that there do exist ricci flow solutions that develop singularities modeled on each such profile in particular we show that for each positive integer k 3 there exist compact solutions in all dimensions m 3 that become singular at the rate t t 2 2 k

observer based hamiltonian identification for quantum systems a symmetry preserving observer based parameter identification algorithm for quantum systems is proposed starting with a 2 level quantum system qubit where the unknown parameters consist of the atom laser frequency detuning and coupling constant we prove an exponential convergence result the analysis is inspired by lyapunov and adaptive control techniques and is based on averaging theory the observer is then extended to the multi level case where all the atom laser coupling constants are unknown the extension of the convergence analysis is discussed through some heuristic arguments the relevance and the robustness with respect to various noises are tested through numerical simulations

nonlinearizable single input control systems do not admit stationary symmetries the main result of the paper states that almost any analytic single input control system which is truly nonlinear that is not feedback linearizable with controllable linearization at an equilibrium point does not admit any symmetry preserving that point by almost any system we mean that we exclude a small class of odd systems that admit just one nontrivial symmetry conjugated to minus identity the obtained results are based on a recent classification of nonlinear single input systems under formal feedback we also describe symmetries of feedback linearizable systems

local observers on linear lie groups with linear estimation error dynamics this technical note proposes local exponential observers for systems on linear lie groups we study two classes of systems in the first class the full state of the system evolves on a linear lie group and is available for measurement in the second class only part of the system s state evolves on a linear lie group and this portion of the state is available for measurement in each case we propose two different observer designs we show that depending on the observer chosen local exponential stability of one of the two observation error dynamics left or right invariant error dynamics is obtained for the first class of systems these results are developed by showing that the estimation error dynamics are differentially equivalent to a stable linear differential equation on a vector space for the second class of system the estimation error dynamics are almost linear we illustrate these observer designs on an attitude estimation problem

intrinsic filtering on lie groups with applications to attitude estimation this paper proposes a probabilistic approach to the problem of intrinsic filtering of a system on a matrix lie group with invariance properties the problem of an invariant continuous time model with discrete time measurements is cast into a rigorous stochastic and geometric framework building upon the theory of continuous time invariant observers we introduce a class of simple filters and study their properties without addressing the optimal filtering problem we show that akin to the kalman filter for linear systems the error equation is a markov chain that does not depend on the state estimate thus when the filter s gains are held fixed the noisy error s distribution is proved to converge to a stationary distribution under some convergence properties of the filter with noise turned off we also introduce two novel tools of engineering interest the discrete time invariant extended kalman filter for which the trusted covariance matrix is shown to converge and the invariant ensemble kalman filter the methods are applied to attitude estimation allowing to derive novel theoretical results in this field and illustrated through simulations on synthetic data

linear observed systems on groups abstract we propose a unifying and versatile framework for a class of discrete time systems whose state is an element of a general group g that we call linear observed systems on groups those systems strictly mimic linear systems in the sense that is replaced with group multiplication and linear maps with automorphisms we argue they are the true generalization of linear systems of the form x n 1 f n x n b n u n in the context of state estimation since 1 when g r n the latter systems are recovered 2 they are proved to possess the preintegration property a characteristic property of linear systems that relates continuous time to discrete time and has recently proved extremely useful in robotics applications and 3 we can build observers that ensure the evolution between the true state and estimated state does not depend on the followed trajectory a characteristic feature of luenberger and invariant observers the theory is applied to a 3d inertial navigation example interestingly this example cannot be put in the form of an invariant system and the proposed generalization is required

the classification of locally conformally flat yamabe solitons abstract this paper addresses the classification of locally conformally flat gradient yamabe solitons in the first part it is shown that locally conformally flat gradient yamabe solitons with positive sectional curvature are rotationally symmetric in the second part the classification of all radially symmetric gradient yamabe solitons is given and their correspondence to smooth self similar solutions of the fast diffusion equation on r n is shown in the last section it is shown that any eternal solution to the yamabe flow with positive ricci curvature and with the scalar curvature attaining an interior space time maximum must be a steady yamabe soliton

symmetry based observers for some water tank problems in this paper we consider a tank containing fluid and we want to estimate the horizontal currents when the fluid surface height is measured the fluid motion is described by shallow water equations in two horizontal dimensions we build a simple nonlinear observer which takes advantage of the symmetries of fluid dynamics laws as a result its structure is based on convolutions with smooth isotropic kernels and the observer is remarkably robust to noise we prove the convergence of the observer around a steady state in numerical applications local exponential convergence is expected the observer is also applied to the problem of predicting the ocean circulation realistic simulations illustrate the relevance of the approach compared with some standard oceanography techniques

a simple intrinsic reduced observer for geodesic flow aghannan and rouchon proposed a new design method of asymptotic observers for a class of nonlinear mechanical systems lagrangian systems with configuration position measurements the position and velocity observer is based on the riemannian structure of the configuration manifold endowed with the kinetic energy metric and is intrinsic they proved local convergence when the system is conservative we propose an intrinsic reduced order velocity observer based on the jacobi metric which can be initialized such that it converges exponentially for any initial true velocity for non conservative systems the observer can be used as a complement to the one of aghannan and rouchon more generally the reduced observer provides velocity estimation for geodesic flow with position measurements thus it can be formally used as a fluid flow soft sensor in the case of a perfect incompressible fluid when the curvature is negative in all planes the geodesic flow is sensitive to initial conditions surprisingly in this case we have global exponential convergence and the more unstable the flow is faster is the convergence

symmetries stability and control in nonlinear systems and networks this paper discusses the interplay of symmetries and stability in the analysis and control of nonlinear dynamical systems and networks specifically it combines standard results on symmetries and equivariance with recent convergence analysis tools based on nonlinear contraction theory and virtual dynamical systems this synergy between structural properties symmetries and convergence properties contraction is illustrated in the contexts of network motifs arising for example in genetic networks from invariance to environmental symmetries and from imposing different patterns of synchrony in a network

non linear symmetry preserving observers on lie groups in this technical note we give a geometrical framework for the design of observers on finite dimensional lie groups for systems which possess some specific symmetries the design and the error between true and estimated state equation are explicit and intrinsic we consider also a particular case left invariant systems on lie groups with right equivariant output the theory yields a class of observers such that the error equation is autonomous the observers converge locally around any trajectory and the global behavior is independent from the trajectory which is reminiscent of the linear stationary case

rigid body attitude estimation based on the lagrange d alembert principle estimation of rigid body attitude and angular velocity without any knowledge of the attitude dynamics model is treated using the lagrange d alembert principle from variational mechanics it is shown that wahba s cost function for attitude determination from two or more non collinear vector measurements can be generalized and represented as a morse function of the attitude estimation error on the lie group of rigid body rotations with body fixed sensor measurements of direction vectors and angular velocity a lagrangian is obtained as the difference between a kinetic energy like term that is quadratic in the angular velocity estimation error and an artificial potential obtained from wahba s function an additional dissipation term that depends on the angular velocity estimation error is introduced and the lagrange d alembert principle is applied to the lagrangian with this dissipation a lyapunov analysis shows that the state estimation scheme so obtained provides stable asymptotic convergence of state estimates to actual states in the absence of measurement noise with an almost global domain of attraction these estimation schemes are discretized for computer implementation using discrete variational mechanics a first order lie group variational integrator is obtained as a discrete time implementation in the presence of bounded measurement noise numerical simulations show that the estimated states converge to a bounded neighborhood of the actual states

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

hybrid nonlinear observers for inertial navigation using landmark measurements this article considers the problem of attitude position and linear velocity estimation for rigid body systems relying on inertial measurement unit and landmark measurements we propose two hybrid nonlinear observers on the matrix lie group text se 2 3 leading to global exponential stability the first observer relies on fixed gains while the second one uses variable gains depending on the solution of a continuous riccati equation these observers are then extended to handle biased angular velocity and linear acceleration measurements both simulation and experimental results are presented to illustrate the performance of the proposed observers

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

yamabe flow steady solitons and type ii singularities abstract we study the convergence of complete non compact conformally flat solutions to the yamabe flow to yamabe steady solitons we also prove the existence of type ii singularities which develop at either a finite time t or as t

a mathcal b invariant distributions and disturbance decoupling of nonlinear systems the concept of a b invariant subspaces has resulted in a unified approach to many of the basic structural properties of time invariant linear systems w m wonham lecture notes in economics and mathematical systems vol 101 springer verlag new york 1974 the purpose of this paper is to introduce the more general notion of a mathcal b invariant distributions on differentiable manifolds and to use this idea to study the disturbance decoupling problem for a class of nonlinear systems which evolve on real analytic manifolds

convergence of the yamabe flow for arbitrary initial energy we consider the yamabe flow frac partial g partial t r g r g g where g is a riemannian metric on a compact manifold m r g denotes its scalar curvature and r g denotes the mean value of the scalar curvature we prove convergence of the yamabe flow if the dimension n satisfies 3 leq n leq 5 or the initial metric is locally conformally flat

observers for invariant systems on lie groups with biased input measurements and homogeneous outputs this paper provides a new observer design methodology for invariant systems whose state evolves on a lie group with outputs in a collection of related homogeneous spaces and where the measurement of system input is corrupted by an unknown constant bias the key contribution of the paper is to study the combined state and input bias estimation problem in the general setting of lie groups a question for which only case studies of specific lie groups are currently available we show that any candidate observer with the same state space dimension as the observed system results in non autonomous error dynamics except in the trivial case where the lie group is abelian this precludes the application of the standard non linear observer design methodologies available in the literature and leads us to propose a new design methodology based on employing invariant cost functions and general gain mappings we provide a rigorous and general stability analysis for the case where the underlying lie group allows a faithful matrix representation we demonstrate our theory in the example of rigid body pose estimation and show that the proposed approach unifies two competing pose observers published in prior literature

analysis of non linear attitude observers for time varying reference measurements this paper provides a comprehensive stability analysis of a suite of nonlinear attitude observers that have been developed over the last few years the observers considered are based on vectorial measurements of an a priori known reference direction by treating the reference direction and the measurement in the same analysis framework and allowing time variation of either we are able to define general persistency of excitation criteria that incorporate and generalize convergence criteria used in prior work a key outcome is conditions that ensure almost global asymptotic and local exponential stability of attitude observers based on a single vector measurement as long as the excitation conditions are met on the reference and system trajectory the approach generalizes stability results provided in prior work based on rank conditions that required at least two or more vector measurements

suq h 0 2 and suq h 2 the classical and quantum q deformations of the su 2 algebra by means of classical harmonic oscillators and their canonical quantization we show that the q deformation of the su 2 lie algebra can be realized at a classical level in the sense of poisson brackets denoted by suq inf not 0 2 and that the canonical quantization of the system leads to the quantum q deformation of the su 2 algebra denoted by suq 2 this means that the q deformation and the h cross quantization are in general two different concepts

higher order nonlinear complementary filtering on lie groups nonlinear observer design for systems whose state space evolves on lie groups is considered the proposed method is similar to previously developed nonlinear observers in that it involves propagating the state estimate using a process model and corrects the propagated state estimate using an innovation term on the tangent space of the lie group in the proposed method the innovation term is constructed by passing the gradient of an invariant cost function resolved in a basis of the tangent space through a linear time invariant system the introduction of the linear system completes the extension of linear complementary filters to nonlinear lie group observers by allowing higher order filtering in practice the proposed method allows for greater design freedom and with the appropriate selection of the linear filter the ability to filter bias and noise over specific bandwidths a disturbance observer that accounts for constant and harmonic disturbances in group velocity measurements is also considered local asymptotic stability about the desired equilibrium point is demonstrated a numerical example that demonstrates the desirable properties of the observer is presented in the context of pose estimation

the invariant extended kalman filter as a stable observer we analyze the convergence aspects of the invariant extended kalman filter iekf when the latter is used as a deterministic nonlinear observer on lie groups for continuous time systems with discrete observations one of the main features of invariant observers for left invariant systems on lie groups is that the estimation error is autonomous in this paper we first generalize this result by characterizing the much broader class of systems for which this property holds for those systems the lie logarithm of the error turns out to obey a linear differential equation then we leverage this log linear property of the error evolution to prove for those systems the local stability of the iekf around any trajectory under the standard conditions of the linear case one mobile robotics example and one inertial navigation example illustrate the interest of the approach simulations evidence the fact that the ekf is capable of diverging in some challenging situations where the iekf with identical tuning keeps converging

nonlinear complementary filters on the special orthogonal group this paper considers the problem of obtaining good attitude estimates from measurements obtained from typical low cost inertial measurement units the outputs of such systems are characterized by high noise levels and time varying additive biases we formulate the filtering problem as deterministic observer kinematics posed directly on the special orthogonal group so 3 driven by reconstructed attitude and angular velocity measurements lyapunov analysis results for the proposed observers are derived that ensure almost global stability of the observer error the approach taken leads to an observer that we term the direct complementary filter by exploiting the geometry of the special orthogonal group a related observer termed the passive complementary filter is derived that decouples the gyro measurements from the reconstructed attitude in the observer inputs both the direct and passive filters can be extended to estimate gyro bias online the passive filter is further developed to provide a formulation in terms of the measurement error that avoids any algebraic reconstruction of the attitude this leads to an observer on so 3 termed the explicit complementary filter that requires only accelerometer and gyro outputs is suitable for implementation on embedded hardware and provides good attitude estimates as well as estimating the gyro biases online the performance of the observers are demonstrated with a set of experiments performed on a robotic test bed and a radio controlled unmanned aerial vehicle

synchronization with partial state coupling on so n this paper studies autonomous synchronization of k agent orientations in so n by using links that each convey a priori different partial state information between two agents the latter compare the action of their states on a common link dependent vector of mathbb r n such setting appears when each pair of agents compares the expression in their respective body frames of a common external direction force field heading of moving object etc to synchronize their states agents must combine this information over time or throughout the network a gradient coupling law for synchronization is proposed extensive convergence analysis of the coupled agents is provided both for fixed and time varying reference vectors the results are compared to a similar setting with states in mathbb r n and agents coupled through scalar product of their state with a reference vector