The Invariant Unscented Kalman Filter

International audienceThis article proposes a novel approach for nonlinear state estimation. It combines both invariant observers theory and unscented filtering principles whitout requiring any compatibility condition such as proposed in the -IUKF algorithm. The resulting algorithm, named IUKF (Invariant Unscented Kalman Filter), relies on a geometrical-based constructive method for designing filters dedicated to nonlinear state estimation problems while preserving the physical invariances and systems symmetries. Within an invariant framework, this algorithm suggests a systematic approach to determine all the symmetry- preserving terms without requiring any linearization and highlighting remarkable invariant properties. As a result, the estimated covariance matrices of the IUKF converge to quasi-constant values due to the symmetry-preserving property provided by the invariant framework. This result enables the development of less conservative robust control strategies. The designed IUKF method has been successfully applied to some relevant practical problems such as the estimation of attitude for aerial vehicles using low-cost sensors reference systems. Typical experimental results using a Parrot quadrotor are provided in this pape

Lie Group Observer Design for Robotic Systems: Extensions, Synthesis, and Higher-Order Filtering

The kinematics and dynamics of many robotic systems evolve on differential manifolds, rather than strictly in Euclidean space. Lie groups, a class of differential manifold with a group structure, arise naturally in the study of rigid-body kinematics. This dissertation studies the design of state observers for systems whose state evolves on a Lie group. State observers, or state estimators, are a crucial part of the guidance, navigation, and control algorithms necessary for autonomous operation of many ground, air, and marine vehicles. The design of state observers on Lie groups is therefore a highly practical exercise. One such nonlinear observer, the gradient-based observer, has generated significant interest in the literature due to its computational simplicity and stability guarantees. The first part of this dissertation explores several applications of the gradient-based observer, including both the attitude estimation problem and the simultaneous localization and mapping (SLAM) problem. By modifying the cost function associated with the observer, several novel attitude estimators are introduced that provide faster convergence when the initial attitude error is large. Further, a SLAM algorithm with guaranteed convergence is introduced and tested in both simulation and experiment. In the second part of this dissertation, the state of the art in Lie group observer design is extended by the development of a higher-order filter on a Lie group. By analogy to the classical linear complementary filter, the proposed method can be interpreted as a nonlinear complementary filter on a Lie group. A disturbance observer that accounts for constant and harmonic disturbances in the group velocity measurements is also considered. Local asymptotic stability about the desired equilibrium point is demonstrated. In addition, an H2-optimal filter synthesis method is derived and disturbance rejection via the internal model principle is considered. A numerical example that demonstrates the desirable properties of the higher-order nonlinear complementary filter, as well as the synthesis techniques, is presented in the context of rigid-body attitude estimation.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147655/1/dzlotnik_1.pd

System theory as applied differential geometry

The invariants of input-output systems under the action of the feedback group was examined. The approach used the theory of Lie groups and concepts of modern differential geometry, and illustrated how the latter provides a basis for the discussion of the analytic structure of systems. Finite dimensional linear systems in a single independent variable are considered. Lessons of more general situations (e.g., distributed parameter and multidimensional systems) which are increasingly encountered as technology advances are presented

Controlled particle systems for nonlinear filtering and global optimization

This thesis is concerned with the development and applications of controlled interacting particle systems for nonlinear filtering and global optimization problems. These problems are important in a number of engineering domains. In nonlinear filtering, there is a growing interest to develop geometric approaches for systems that evolve on matrix Lie groups. Examples include the problem of attitude estimation and motion tracking in aerospace engineering, robotics and computer vision. In global optimization, the challenges typically arise from the presence of a large number of local minimizers as well as the computational scalability of the solution. Gradient-free algorithms are attractive because in many practical situations, evaluating the gradient of the objective function may be computationally prohibitive. The thesis comprises two parts that are devoted to theory and applications, respectively. The theoretical part consists of three chapters that describe methods and algorithms for nonlinear filtering, global optimization, and numerical solutions of the Poisson equation that arise in both filtering and optimization. For the nonlinear filtering problem, the main contribution is to extend the feedback particle filter (FPF) algorithm to connected matrix Lie groups. In its general form, the FPF is shown to provide an intrinsic coordinate-free description of the filter that automatically satisfies the manifold constraint. The properties of the original (Euclidean) FPF, especially the gain-times-error feedback structure, are preserved in the generalization. For the global optimization problem, a controlled particle filter algorithm is introduced to numerically approximate a solution of the global optimization problem. The theoretical significance of this work comes from its variational aspects: (i) the proposed particle filter is a controlled interacting particle system where the control input represents the solution of a mean-field type optimal control problem; and (ii) the associated density transport is shown to be a gradient flow (steepest descent) for the optimal value function, with respect to the Kullback--Leibler divergence. For both the nonlinear filtering and optimization problems, the numerical implementation of the proposed algorithms require a solution of a Poisson equation. Two numerical algorithms are described for this purpose. In the Galerkin scheme, the gain function is approximated using a set of pre-defined basis functions; In the kernel-based scheme, a numerical solution is obtained by solving a certain fixed-point equation. Well-posedness results for the Poisson equation are also discussed. The second part of the thesis contains applications of the proposed algorithms to specific nonlinear filtering and optimization problems. The FPF is applied to the problem of attitude estimation - a nonlinear filtering problem on the Lie group SO(3). The formulae of the filter are described using both the rotation matrix and the quaternion coordinates. A comparison is provided between FPF and the several popular attitude filters including the multiplicative EKF, the invariant EKF, the unscented Kalman filter, the invariant ensemble Kalman filter and the bootstrap particle filter. Numerical simulations are presented to illustrate the comparison. As a practical application, experimental results for a motion tracking problem are presented. The objective is to estimate the attitude of a wrist-worn motion sensor based on the motion of the arm. In the presence of motion, considered here as the swinging motion of the arm, the observability of the sensor attitude is shown to improve. The estimation problem is mathematically formulated as a nonlinear filtering problem on the product Lie group SO(3)XSO(2), and experimental results are described using data from the gyroscope and the accelerometer installed on the sensor. For the global optimization problem, the proposed controlled particle filter is compared with several model-based algorithms that also employ probabilistic models to inform the search of the global minimizer. Examples of the model-based algorithms include the model reference adaptive search, the cross entropy, the model-based evolutionary optimization, and two algorithms based on bootstrap particle filtering. Performance comparisons are provided between the control-based and the sampling-based implementation. Results of Monte-Carlo simulations are described for several benchmark optimization problems

Applied Dynamics and Geometric Mechanics

This one week workshop was organized around several central subjects in applied dynamics and geometric mechanics. The specific organization with afternoons free for discussion led to intense exchanges of ideas. Bridges were forged between researchers representing different fields. Links were established between pure mathematical ideas and applications. The meeting was not restricted to any particular application area. One of the main goals of the meeting, like most others in this series for the past twenty years, has been to facilitate cross fertilization between various areas of mathematics, physics, and engineering. New collaborative projects emerged due to this meeting. The workshop was well attended with participants from Europe, North America, and Asia. Young researchers (doctoral students, postdocs, junior faculty) formed about 30% of the participants

Programming by Demonstration on Riemannian Manifolds

This thesis presents a Riemannian approach to Programming by Demonstration (PbD). It generalizes an existing PbD method from Euclidean manifolds to Riemannian manifolds. In this abstract, we review the objectives, methods and contributions of the presented approach. OBJECTIVES PbD aims at providing a user-friendly method for skill transfer between human and robot. It enables a user to teach a robot new tasks using few demonstrations. In order to surpass simple record-and-replay, methods for PbD need to \u2018understand\u2019 what to imitate; they need to extract the functional goals of a task from the demonstration data. This is typically achieved through the application of statisticalmethods. The variety of data encountered in robotics is large. Typical manipulation tasks involve position, orientation, stiffness, force and torque data. These data are not solely Euclidean. Instead, they originate from a variety of manifolds, curved spaces that are only locally Euclidean. Elementary operations, such as summation, are not defined on manifolds. Consequently, standard statistical methods are not well suited to analyze demonstration data that originate fromnon-Euclidean manifolds. In order to effectively extract what-to-imitate, methods for PbD should take into account the underlying geometry of the demonstration manifold; they should be geometry-aware. Successful task execution does not solely depend on the control of individual task variables. By controlling variables individually, a task might fail when one is perturbed and the others do not respond. Task execution also relies on couplings among task variables. These couplings describe functional relations which are often called synergies. In order to understand what-to-imitate, PbDmethods should be able to extract and encode synergies; they should be synergetic. In unstructured environments, it is unlikely that tasks are found in the same scenario twice. The circumstances under which a task is executed\u2014the task context\u2014are more likely to differ each time it is executed. Task context does not only vary during task execution, it also varies while learning and recognizing tasks. To be effective, a robot should be able to learn, recognize and synthesize skills in a variety of familiar and unfamiliar contexts; this can be achieved when its skill representation is context-adaptive. THE RIEMANNIAN APPROACH In this thesis, we present a skill representation that is geometry-aware, synergetic and context-adaptive. The presented method is probabilistic; it assumes that demonstrations are samples from an unknown probability distribution. This distribution is approximated using a Riemannian GaussianMixtureModel (GMM). Instead of using the \u2018standard\u2019 Euclidean Gaussian, we rely on the Riemannian Gaussian\u2014 a distribution akin the Gaussian, but defined on a Riemannian manifold. A Riev mannian manifold is a manifold\u2014a curved space which is locally Euclidean\u2014that provides a notion of distance. This notion is essential for statistical methods as such methods rely on a distance measure. Examples of Riemannian manifolds in robotics are: the Euclidean spacewhich is used for spatial data, forces or torques; the spherical manifolds, which can be used for orientation data defined as unit quaternions; and Symmetric Positive Definite (SPD) manifolds, which can be used to represent stiffness and manipulability. The Riemannian Gaussian is intrinsically geometry-aware. Its definition is based on the geometry of the manifold, and therefore takes into account the manifold curvature. In robotics, the manifold structure is often known beforehand. In the case of PbD, it follows from the structure of the demonstration data. Like the Gaussian distribution, the Riemannian Gaussian is defined by a mean and covariance. The covariance describes the variance and correlation among the state variables. These can be interpreted as local functional couplings among state variables: synergies. This makes the Riemannian Gaussian synergetic. Furthermore, information encoded in multiple Riemannian Gaussians can be fused using the Riemannian product of Gaussians. This feature allows us to construct a probabilistic context-adaptive task representation. CONTRIBUTIONS In particular, this thesis presents a generalization of existing methods of PbD, namely GMM-GMR and TP-GMM. This generalization involves the definition ofMaximum Likelihood Estimate (MLE), Gaussian conditioning and Gaussian product for the Riemannian Gaussian, and the definition of ExpectationMaximization (EM) and GaussianMixture Regression (GMR) for the Riemannian GMM. In this generalization, we contributed by proposing to use parallel transport for Gaussian conditioning. Furthermore, we presented a unified approach to solve the aforementioned operations using aGauss-Newton algorithm. We demonstrated how synergies, encoded in a Riemannian Gaussian, can be transformed into synergetic control policies using standard methods for LinearQuadratic Regulator (LQR). This is achieved by formulating the LQR problem in a (Euclidean) tangent space of the Riemannian manifold. Finally, we demonstrated how the contextadaptive Task-Parameterized Gaussian Mixture Model (TP-GMM) can be used for context inference\u2014the ability to extract context from demonstration data of known tasks. Our approach is the first attempt of context inference in the light of TP-GMM. Although effective, we showed that it requires further improvements in terms of speed and reliability. The efficacy of the Riemannian approach is demonstrated in a variety of scenarios. In shared control, the Riemannian Gaussian is used to represent control intentions of a human operator and an assistive system. Doing so, the properties of the Gaussian can be employed to mix their control intentions. This yields shared-control systems that continuously re-evaluate and assign control authority based on input confidence. The context-adaptive TP-GMMis demonstrated in a Pick & Place task with changing pick and place locations, a box-taping task with changing box sizes, and a trajectory tracking task typically found in industr

Reflection positivity and invertible topological phases

We implement an extended version of reflection positivity (Wick-rotated unitarity) for invertible topological quantum field theories and compute the abelian group of deformation classes using stable homotopy theory. We apply these field theory considerations to lattice systems, assuming the existence and validity of low energy effective field theory approximations, and thereby produce a general formula for the group of Symmetry Protected Topological (SPT) phases in terms of Thom's bordism spectra; the only input is the dimension and symmetry group. We provide computations for fermionic systems in physically relevant dimensions. Other topics include symmetry in quantum field theories, a relativistic 10-fold way, the homotopy theory of relativistic free fermions, and a topological spin-statistics theorem.Comment: 136 pages, 16 figures; minor changes/corrections in version 2; v3 major revision; v4 minor revision: corrected proof of Lemma 9.55, many small changes throughout; v5 version for publication in Geometry & Topolog