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We present two results on attitude estimation using vector and rate gyro measurements. The first result concerns an observer previously presented by Hamel, Mahony, and Pflimlin, with proven stability results when i) the reference vectors are stationary; or ii) the gyro measurements are unbiased. We prove semiglobal stability without either of these assumptions when a parameter projection is added, and convergence from all initial attitudes when using a resetting strategy. The second result is an algorithm for estimation of bias in the body-fixed vector measurements, which is analyzed in combination with the attitude and gyro bias observer. [ABSTRACT FROM PUBLISHER]

An unscented Kalman filter (UKF) for matrix Lie groups is proposed where the time propagation of the state is formulated on the Lie algebra. This is done with the kinematic differential equation of the logarithm, where the inverse of the right Jacobian is used. The sigma points can then be expressed as logarithms in vector form, and time propagation of the sigma points and the computation of the mean and the covariance can be done on the Lie algebra. The resulting formulation is to a large extent based on logarithms in vector form and is, therefore, closer to the UKF for systems in $\mathbb {R}^n$. This gives an elegant and well-structured formulation, which provides additional insight into the problem, and which is computationally efficient. The proposed method is in particular formulated and investigated on the matrix Lie group $SE(3)$. A discussion on right and left Jacobians is included, and a novel closed-form solution for the inverse of the right Jacobian on $SE(3)$ is derived, which gives a compact representation involving fewer matrix operations. The proposed method is validated in simulations. [ABSTRACT FROM AUTHOR]

We introduce a new hybrid control strategy, which is conceptually different from the commonly used synergistic hybrid approaches, to efficiently deal with the problem of the undesired equilibria that precludes smooth vectors fields on $SO(3)$ from achieving global stability. The key idea consists in constructing a suitable potential function on $SO(3)\times \mathbb {R}$ involving an auxiliary scalar variable, with flow and jump dynamics, which keeps the state away from the undesired critical points while, at the same time, guarantees a decrease of the potential function over the flows and jumps. Based on this new hybrid mechanism, a hybrid feedback control scheme for the attitude tracking problem on $SO(3)$ , endowed with global asymptotic stability and semiglobal exponential stability guarantees, is proposed. This control scheme is further improved through a smoothing mechanism that removes the discontinuities in the input torque. The third hybrid control scheme, proposed in this article, removes the requirement of the angular velocity measurements, while preserving the strong stability guarantees of the first hybrid control scheme. This approach has also been applied to the tracking problem on $SE(3)$ to illustrate its advantages with respect to the existing synergistic hybrid approaches. Finally, some simulation results are presented to illustrate the performance of the proposed hybrid controllers. [ABSTRACT FROM AUTHOR]

In this article, a new probability distribution, referred to as the matrix Fisher–Gaussian distribution, is proposed on the product manifold of three-dimensional special orthogonal group and Euclidean space. It is constructed by conditioning a multivariate Gaussian distribution from the ambient Euclidean space into the manifold, while imposing a certain geometric constraint on the correlation term to avoid over parameterization. The unique feature is that it may represent large uncertainties in attitudes, linear variables of an arbitrary dimension, and angular–linear correlations between them in a global fashion without singularities. Various stochastic properties and an approximate maximum likelihood estimator are developed. Furthermore, two methods are developed to propagate uncertainties through a stochastic differential equation representing attitude kinematics. Based on these, a Bayesian estimator is proposed to estimate the attitude and time-varying gyro bias concurrently. Numerical studies indicate that the proposed estimator provides more accurate estimates against the multiplicative extended Kalman filter and unscented Kalman filter for challenging cases. [ABSTRACT FROM AUTHOR]

This article deals with the simultaneous estimation of the attitude, position, and linear velocity for vision-aided inertial navigation systems. We propose a nonlinear observer on $SO(3)\times \mathbb {R}^{15}$ relying on body-frame acceleration, angular velocity, and (stereo or monocular) bearing measurements of some landmarks that are constant and known in the inertial frame. Unlike the existing local Kalman-type observers, our proposed nonlinear observer guarantees almost global asymptotic stability and local exponential stability. A detailed uniform observability analysis has been conducted and sufficient conditions are derived. Moreover, a hybrid version of the proposed observer is provided to handle the intermittent nature of the measurements in practical applications. Simulation and experimental results are provided to illustrate the effectiveness of the proposed state observer. [ABSTRACT FROM AUTHOR]

The aim of this article is to discuss the observability properties and observer design for the attitude of a rigid body, under conditions of partial inertial sensing. In particular, we introduce an observability analysis tool for the attitude dynamics when only accelerometer and gyroscope measurements are available, as in several robotics applications. In various scenarios, in fact, the measurement of the magnetic field via a magnetometer is unreliable, due to magnetic interferences. Herein, we first focus on a formal observability analysis, which reveals that the target dynamics is weakly locally observable, but not first-order observable. The lack of first-order observability prevents standard observers from achieving global convergence. Therefore, we discuss a more suitable approach for observer design to deal with this problem. The proposed approach is validated by providing numerical and experimental results. The former show that the proposed approach is able to achieve convergence (final error 0.004 $\%$). Experiments validate our inference about observability and show the improvements brought by the proposed approach concerning the error convergence (final error 0.15 $\%$). [ABSTRACT FROM AUTHOR]

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 SE2(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. [ABSTRACT FROM AUTHOR]

This paper deals with the design of globally exponentially stable invariant observers on the Special Euclidian group $SE(3)$. First, we propose a generic hybrid estimation scheme (depending on a generic potential function) evolving on $SE(3)\times \mathbb {R}^6$ for pose (orientation and position) and velocity-bias estimation. Thereafter, the proposed estimation scheme is formulated explicitly in terms of inertial vectors and landmark measurements. Interestingly, the proposed estimation scheme leads to a decoupled rotational error dynamics from the translational dynamics, which is an interesting feature in practical applications with noisy measurements and disturbances. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

An observer for simultaneous localization and mapping is considered in this paper. The proposed observer is based on recent theoretical foundations in a gradient-based observer design on Lie groups. As such, the form of the observer is similar to recently developed nonlinear attitude and pose observers. Translational and angular velocity measurements, as well as relative position measurements of nearby landmarks, are used directly within the observer structure. The case of biased translational and angular velocity measurements is also considered. Stability results are presented that demonstrate an asymptotic convergence of the pose and map estimates. The proposed algorithm is implemented successfully in experiment. [ABSTRACT FROM AUTHOR]

This paper presents a new adaptive attitude tracking controller for rigid body systems, with unknown inertia and unknown gyro-bias, using inertial vector measurements. The proposed control scheme guarantees almost global asymptotic convergence of the attitude and angular velocity to their desired values. Simulation results are provided to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper focuses on a stochastic formulation of Bayesian attitude estimation on the special orthogonal group. In particular, an exponential probability density model for random matrices, referred to as the matrix Fisher distribution, is utilized to represent the uncertainties of attitude estimates and measurements in a global fashion. Various stochastic properties of the matrix Fisher distribution are derived on the special orthogonal group, and based on these, two types of intrinsic frameworks for Bayesian attitude estimation are constructed. These avoid complexities or singularities of the attitude estimators developed in terms of quaternions. The proposed approaches are particularly useful to deal with large estimation errors or large uncertainties for complex attitude maneuvers. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper deals with the design, performance, and robustness analysis of nonlinear attitude complementary filters on \textSO(3). We derive explicit time solutions of the attitude estimation error dynamics of the filters (in the disturbance-free case) and analyze their performance and robustness with respect to measurement disturbances. The stability and performance properties of the filters can be easily deduced from the obtained closed-form solutions. A new class of attitude complementary filters on \textSO(3) with state-dependent gains is proposed and shown to exhibit improved performance and robustness properties compared to the fixed-gain traditional complementary filter on \textSO(3). [ABSTRACT FROM PUBLISHER]

This paper presents an approach for the design of globally exponentially stable hybrid attitude and gyro-bias observers on SO(3) \times \mathbb R^3 . First, we propose a hybrid observer developed in a generic manner involving a generic family of potential functions with specific properties. Thereafter, we present a design method for such potential functions via an appropriate angular warping transformation applied to a new potential function on $SO(3)$. This results in a hybrid observer that uses directly body-frame measurements of known inertial vectors. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

In this paper, we propose two hybrid feedbacks based on “synergistic” potential functions to achieve robust global asymptotic tracking of rigid-body attitude, a task that is impossible by classical state feedback—be it smooth, nonsmooth, or periodic—due to the topological structure of the special orthogonal group. Both hybrid feedbacks are based upon a proportional-derivative structure where the proportional term is generated from a finite family of control-induced artificial potential energies and the derivative term is due to a damping injection. In a patient-yet-greedy fashion, the hybrid feedback hysteretically switches to the minimum control-induced artificial potential energy in a finite family. The synergy property—which requires that for each undesirable critical point of each potential energy, there exists a lower potential energy in the family—guarantees the globality of robust asymptotic tracking. The first hybrid feedback is the natural extension of existing PD controllers and results in discontinuous jumps in the control signal due to a direct switch the potential energy. Using a backstepping procedure, we design a second hybrid feedback that smooths the jumps in the control torque by dynamically interpolating between the switching potential-energy terms. We show that while the class of “modified trace functions” is not wide enough to generate a “centrally synergistic” potential function, relaxing the centrality assumption allows one to construct a synergistic family and we provide explicit guidelines for doing so. [ABSTRACT FROM PUBLISHER]

The existing rigid body attitude controllers (without angular velocity measurements) involve explicitly the attitude in the feedback. Unfortunately, there does not exist any sensor that directly measures the orientation of a rigid body (without any estimation procedure). Therefore the attitude must be generated from the available sensors via some attitude determination (estimation) algorithms. The most recent and efficient attitude estimation algorithms rely on the body vector measurements and the angular velocity (which is assumed to be unavailable in velocity-free attitude controllers). To overcome this circular reasoning-like problem, we propose a velocity-free attitude stabilization control scheme relying solely on body vector measurements. Moreover, the proposed control law is a priori bounded and does not lead to the so-called unwinding phenomenonref refid="fn1"/ id="fn1" asterisk="no"paraUsing the unit-quaternion representation, the same attitude can be represented by two different unit-quaternion, namely Q and -Q. This fact, often leads to the so-called unwinding phenomenon, which is undesirable since some closed-loop trajectories, for some initial conditions close to the desired attitude equilibrium, can undergo an unnecessary homoclinic-like orbit refid="ref1"/, refid="ref2"/ .para encountered in some unit-quaternion based attitude control schemes. [ABSTRACT FROM PUBLISHER]

A method for designing reduced-order output-feedback controllers for linear time-invariant retarded systems with state-multiplicative Wiener-type noise is introduced, which achieves a minimum bound on the H\infty performance level. A cost function is defined that is the expected value of the standard H\infty performance cost with respect to the stochastic parameters. A solution is thus obtained, for the reduced-order H\infty output-feedback control problem, for the stationary case, via the input-output approach where the system is replaced by a non-retarded one, that contains deterministic norm-bounded uncertainties. The results achieved for the nominal case are extended to the uncertain case where the system matrices reside in a given polytope. A numerical example, taken from the field of aircraft control, is given that demonstrates the applicability of the theory developed. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This technical note presents the design, analysis, and performance evaluation of a novel globally asymptotically stable (GAS) filter for attitude estimation. The design is sensor-driven and departs from traditional solutions as no explicit representations of the attitude are considered. The proposed solution yields unique estimates and it does not suffer from drawbacks such as singularities, topological limitations for achieving global stabilization, or unwinding phenomena. The performance of the overall attitude estimation solution is evaluated with the design and implementation of an Attitude and Heading Reference System (AHRS) based on a single low-cost Inertial Measurement Unit. The performance of the proposed AHRS is assessed experimentally using a high precision motion rate table, which provides ground truth signals for comparison with the resulting estimates. [ABSTRACT FROM AUTHOR]

This technical note presents a novel recursive design approach for output-feedback control of a class of nonlinear systems with actuator quantization. The recently developed cyclic-small-gain theorem is employed to guarantee the input-to-state stability (ISS) of the closed-loop system, along with the construction of an ISS-Lyapunov function. Actuator dynamic quantization is designed based on the ISS-Lyapunov function. [ABSTRACT FROM PUBLISHER]

This technical note considers the problem of obtaining minimum-energy state estimates for a system defined on the rotation group, SO(3). The signals of the system are modeled as purely deterministic signals. We derive a nonlinear observer (“filter”) posed directly on SO(3) that respects the geometry of the group and achieves a performance that is close to optimal in the sense of minimizing an integral cost that is measuring the state energy. The performance of the proposed filter is demonstrated in simulations involving large initialization, process and measurement errors where the results are compared against a quaternion implementation of an Extended Kalman Filter. Our results indicate that the proposed filter achieves better robustness against a range of noise levels and initialization errors. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

This paper proposes a design methodology for nonlinear state observers for invariant kinematic systems posed on fiflute dimensional connected Lie groups, and studies the associated fundamental system structure. The concept of synchrony of two dynamical systems is specialized to systems on Lie groups. For invariant systems this leads to a general factorization theorem of a nonlinear observer into a synchronous (internal model) term and an innovation term. The synchronous term is fully specified by the system model. We propose a design methodology for the innovation term based on gradient-like terms derived from invariant or non-invariant cost functions. The resulting nonlinear observers have strong (almost) global convergence properties and examples are used to demonstrate the relevance of the proposed approach. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The article discusses various reports published within the issue, including "An Eigenvalue Method for Testing Positive Definiteness of a Multivariate Form," by Qin Ni, Liqun Qi and Fei Wang and "A Convex Optimization Approach to ARMA Modeling," by Tryphon T. Georgiou and Anders Lindquist.