Orientation estimation for 3D objects is a common problem that is usually tackled with traditional nonlinear filtering techniques such as the extended Kalman filter (EKF) or the unscented Kalman filter (UKF). Most of these techniques assume Gaussian distributions to account for system noise and uncertain measurements. This distributional assumption does not consider the periodic nature of pose and orientation uncertainty. We propose a filter that considers the periodicity of the orientation estimation problem in its distributional assumption. This is achieved by making use of the Bingham distribution, which is defined on the hypersphere and thus inherently more suitable to periodic problems. Furthermore, handling of non-trivial system functions is done using deterministic sampling in an efficient way. A deterministic sampling scheme reminiscent of the UKF is proposed for the nonlinear manifold of orientations. It is the first deterministic sampling scheme that truly reflects the nonlinear manifold of the orientation.
For recursive circular filtering based on circular statistics, we introduce a general framework for estimation of a circular state based on different circular distributions, specifically the wrapped normal distribution and the von Mises distribution. We propose an estimation method for circular systems with nonlinear system and measurement functions. This is achieved by relying on efficient deterministic sampling techniques. Furthermore, we show how the calculations can be simplified in a variety of important special cases, such as systems with additive noise as well as identity system or measurement functions. We introduce several novel key components, particularly a distribution-free prediction algorithm, a new and superior formula for the multiplication of wrapped normal densities, and the ability to deal with non-additive system noise. All proposed methods are thoroughly evaluated and compared to several state-of-the-art solutions. system measurement publication distribution model noise model noise Azmani, Reboul, Choquel, Benjelloun [9] von Mises identity additive identity additive Markovic, Chaumette, Petrovic [14] von Mises-Fisher identity additive identity additive Kurz, Gilitschenski, Julier, Hanebeck [21] Bingham identity additive identity additive Kurz, Gilitschenski, Hanebeck [15] wrapped normal/von Mises nonlinear additive identity additive Kurz, Gilitschenski, Hanebeck [16] wrapped normal nonlinear additive nonlinear any this paper wrapped normal/von Mises nonlinear any nonlinear anyTable 1: Circular filters based on directional statistics.There has been some work on filtering algorithms based on circular statistics by Azmani et al. [9], which was further investigated by Stienne et al. [10]. Their work is based on the von Mises distribution and allows for recursive filtering of systems with a circular state space. However, it is limited to the identity with additive noise as the system equation and the measurement equation. The filter from [9] has been applied to phase estimation of GPS signals [11], [12] as well as map matching [13]. Markovic et al. have published a similar filter [14] based on the von Mises-Fisher distribution, a generalization of the von Mises distribution to the hypersphere.We have previously published a recursive filter based on the wrapped normal distribution allowing for a nonlinear system equation [15]. The paper [16] extends this approach to make a nonlinear measurement update possible. Both papers rely on a deterministic sampling scheme, based on the first circular moment. This kind of sampling is reminiscent of the well-known unscented Kalman filter (UKF) [2]. We have extended this sampling scheme to the first two circular moments in [17], so the proposed filters are, in a sense, circular versions of the UKF.The developed filters have been applied in the context of constrained tracking [18], bearings-only sensor scheduling [19], as well as circular model predictive control [20].Furthermore, we proposed a recursive filter based on the circular Bingham distribution in [21]....
Estimation of circular quantities is a widespread problem that occurs in many tracking and control applications. Commonly used approaches such as the Kalman filter, the extended Kalman filter (EKF), and the unscented Kalman filter (UKF) do not take periodicity explicitly into account, which can result in low estimation accuracy. We present a filtering algorithm for angular quantities in nonlinear systems that is based on circular statistics. The new filter switches between three different representations of probability distributions on the circle, the wrapped normal, the von Mises, and a Dirac mixture density. It can be seen as a systematic generalization of the UKF to circular statistics. We evaluate the proposed filter in simulations and show its superiority to conventional approaches.
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In this paper, we present libDirectional, a MATLAB library for directional statistics and directional estimation. It supports a variety of commonly used distributions on the unit circle, such as the von Mises, wrapped normal, and wrapped Cauchy distributions. Furthermore, various distributions on higher-dimensional manifolds such as the unit hypersphere and the hypertorus are available. Based on these distributions, several recursive filtering algorithms in libDirectional allow estimation on these manifolds. The functionality is implemented in a clear, well-documented, and object-oriented structure that is both easy to use and easy to extend.
Optical belt sorters can be used to sort a large variety of bulk materials. By the use of sophisticated algorithms, the performance of the complex machinery can be further improved. Recently, we have proposed an extension to industrial optical belt sorters that involves tracking the individual particles on the belt using an area scan camera. If the estimated behavior of the particles matches the true behavior, the reliability of the separation process can be improved. The approach relies on multitarget tracking using hard association decisions between the tracks and the measurements. In this paper, we propose to include the orientation in the assessment of the compatibility of a track and a measurement. This allows us to achieve more reliable associations, facilitating a higher accuracy of the tracking results.
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