Minimum-energy filtering on the unit circle

Abstract— We apply Mortensen’s deterministic filtering approach to derive a third order minimum-energy filter for a system defined on the unit circle. This yields the exact form of a minimum-energy filter (namely an observer plus a Riccati equation that updates the observer gain). The proposed Riccati equation is perturbed by a term depending on the third order derivative of the value function of the associated optimal control problem. The proposed filter is third order in the sense that it approximates the dynamics of the third order derivate of the value function by neglecting the fourth order derivative of the value function. Additionally, we show that the nearoptimal filter proposed by Coote et al. in prior work can indeed be derived from a second order application of Mortensen’s approach to minimum-energy filtering on the unit circle

Minimum-energy filtering for attitude estimation

n 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

Deterministic attitude and pose filtering, an embedded Lie groups approach

Attitude estimation is a core problem in many robotic systems that perform automated or semi automated navigation. The configuration space of the attitude motion is naturally modelled on the Lie group of special orthogonal matrices SO(3). Many current attitude estimation methods are based on non-matrix parameterization of attitude. Non-matrix parameterization schemes sometimes lead to modelling issues such as the singularities in the parameterization space, non-uniqueness of the attitude estimates and the undesired conversion errors such as the projection or normalization errors. Moreover, often attitude filters are designed by linearizing or approximating the nonlinear attitude kinematics followed by applying the Kalman filtering based methods that are primarily only suitable for linear Gaussian systems. In this thesis, the attitude estimation problem is considered directly on SO(3) along with nonlinear vectorial measurement models. Minimum-energy filtering is adapted to respect the geometry of the problem and in order to solve the problem avoiding linearization or Gaussian assumptions. This approach allows for obtaining a geometric approximate minimum-energy (GAME) filter whose performance is tested by means of Monte Carlo simulations. Many of the major attitude filtering methods in the literature are surveyed and included in the simulation study. The GAME filter outperforms all of the state of the art attitude filters studied, including the multiplicative extended Kalman filter (MEKF), the unscented quaternion estimator (USQUE), the right-invariant extended Kalman filter (RIEKF) and the nonlinear constant gain attitude observer, in the asymptotic estimation error. Furthermore, the proposed GAME filter is shown to be near-optimal by deriving a bound on the optimality error of the filter that is proven to be small in simulations. Moreover, similar GAME filters are derived for pose filtering on the special Euclidean group SE(3), attitude and bias filtering on the unit circle and attitude and bias filtering on the special orthogonal group. The approximation order of the proposed method can potentially be extended to arbitrary higher orders. For instance, for the case angle estimation on the unit circle an eighth-order approximate minimum-energy filter is provided

Multichannel source separation and tracking with phase differences by random sample consensus

Blind audio source separation (BASS) is a fascinating problem that has been tackled from many different angles. The use case of interest in this thesis is that of multiple moving and simultaneously-active speakers in a reverberant room. This is a common situation, for example, in social gatherings. We human beings have the remarkable ability to focus attention on a particular speaker while effectively ignoring the rest. This is referred to as the ``cocktail party effect'' and has been the holy grail of source separation for many decades. Replicating this feat in real-time with a machine is the goal of BASS. Single-channel methods attempt to identify the individual speakers from a single recording. However, with the advent of hand-held consumer electronics, techniques based on microphone array processing are becoming increasingly popular. Multichannel methods record a sound field from various locations to incorporate spatial information. If the speakers move over time, we need an algorithm capable of tracking their positions in the room. For compact arrays with 1-10 cm of separation between the microphones, this can be accomplished by applying a temporal filter on estimates of the directions-of-arrival (DOA) of the speakers. In this thesis, we review recent work on BSS with inter-channel phase difference (IPD) features and provide extensions to the case of moving speakers. It is shown that IPD features compose a noisy circular-linear dataset. This data is clustered with the RANdom SAmple Consensus (RANSAC) algorithm in the presence of strong reverberation to simultaneously localize and separate speakers. The remarkable performance of RANSAC is due to its natural tendency to reject outliers. To handle the case of non-stationary speakers, a factorial wrapped Kalman filter (FWKF) and a factorial von Mises-Fisher particle filter (FvMFPF) are proposed that track source DOAs directly on the unit circle and unit sphere, respectively. These algorithms combine directional statistics, Bayesian filtering theory, and probabilistic data association techniques to track the speakers with mixtures of directional distributions

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

Minimum-Energy Filtering on the Unit Circle Using Velocity Measurements with Bias and Vectorial state Measurements

We consider minimum-energy filtering of a system defined on the unit circle when the angular velocity measurements are contaminated with deterministic measurement error and bias. We propose a second order approximate minimum-energy filter for this syste

Minimum-Energy Filtering on the Unit Circle Using Velocity Measurements with Bias and Vectorial state Measurements

Abstract — We consider minimum-energy filtering of a system defined on the unit circle when the angular velocity measurements are contaminated with deterministic measurement error and bias. We propose a second order approximate minimumenergy filter for this system using vectorial measurements. This work extends prior work in two aspects; Firstly, by including a model for slowly time varying angular velocity measurement bias, we estimate and reject the bias in order to estimate the state of the system more accurately. Secondly, rather than using full state measurements we use vector measurements to derive the filter. Both of these two innovations make the filter more practical in real world applications. In simulations we show that the proposed filter is globally convergent and robust to different levels of measurement error and bias. I