Extended Object Tracking (EOT) plays a crucial role in various applications within robotics, signal processing, and control, especially with advancements in sensing technology enabling multiple measurements per target. This paper proposes a novel EOT method based on Gaussian Processes (GPs), incorporating an extension for range measurements and explicitly accounting for uncertainties in Cartesian coordinates. We introduce an efficient method for the integration of uncertainty propagation. Empirical evaluations using Root Mean Square Error (RMSE), Chamfer Distance (CD), and mean Intersection-over-Union (mIoU) demonstrate the superior performance of our method compared to existing GP-based techniques. Additionally, we assess model credibility by comparing the Average Normalized Estimation Error Squared (ANEES) of the existing approach with our proposed method.

Normalizing flows have proven their efficacy for density estimation in Euclidean space, but their application to rotational representations, crucial in various domains such as robotics or human pose modeling, remains under-explored. Probabilistic models of the human pose can benefit from approaches that rigorously consider the rotational nature of human Joints. For this purpose, we introduce HuProSO3, a normalizing flow model that operates on a high-dimensional product space of SO(3) manifolds, modeling the Joint distribution for human Joints with three degrees offreedom. HuProSO3's advantage over state-of-the-art approaches is demonstrated through its superior modeling accuracy in three different applications and its capability to evaluate the exact likelihood. This work not only addresses the technical challenge of learning densities on SO(3) manifolds, but it also has broader implications for domains where the probabilistic regression of correlated 3D rotations is of importance. Code will be available at https://github.com/odunkel/HuProSO.

With the growing availability of high-resolution sensors, processing more than one detection per target becomes increasingly critical when tracking multiple extended objects. However, contemporary sensors often generate spurious detections that need to be considered. Naively employing standard multitarget trackers may result in poor tracking performance for multitarget–multidetection tracking in cluttered environments, and the relevant extensions are nontrivial. This paper introduces a version of the kernel symmetric measurement equation (SME) filter that considers both multidetections and clutter. For a simulated scenario, our novel filter achieved a higher accuracy than the global nearest neighbor (GNN) and a fast variant of the joint probabilistic data association filter (JPDAF).

We aim to approximate the distribution of the first-passage time of a particle moving according to a Gaussian process with increasing trend, i. e., the distribution of the first time a particle described, e.g., by a state-space model such as a constant-velocity or constant-acceleration model, arrives at a fixed location. Since the known approaches from the literature either consider processes from different families or lead to highly complex approximations, we seek a fast-to-compute method for the problem. Motivated by an engineering particle transport task for which we can assume that once a particle has arrived at this 10-cation it cannot move back, we derive an analytic approximation for the first-passage time probabilities and calculate its inverse cumulative distribution function analytically and the moments numerically. Furthermore, we propose a Gaussian approximation based on a linearization approach. The strengths and limitations of our methods are discussed and by comparison with Monte Carlo simulations, we show that in particular, the first one satisfies the requirements of engineering problems in terms of accuracy and computation time.

In optical sorting of bulk material, the composition of particles may frequently change. State-of-the-art sorting approaches rely on tuning physical models of the particle motion. The aim of this work is to increase the prediction accuracy in complex fast-changing sorting scenarios with data-driven approaches. In this article, we propose two neural network (NN) experts for accurate prediction ofa prioriknown particle types. To handle the large variety of particle types that can occur in real-world sorting scenarios, we introduce a simple but effective mixture of experts’ approach that combines NNs with hand-crafted motion models. Our new method not only improves the prediction accuracy for bulk material consisting of many particle classes, but also proves to be very adaptive and robust to new particle types.

Gaussian process (GP) regressors are used in a wide variety of regression tasks, and many recent applications feature domains that are non-Euclidean manifolds or other metric spaces. In this paper, we examine formal consistency of GP regression on general metric spaces. Specifically, we consider a GP prior on an unknown real-valued function with a metric domain space and examine consistency of the resulting posterior distribution. If the kernel is continuous and the sequence of sampling points lies sufficiently dense, then the variance of the posterior GP is shown to converge to zero almost surely monotonically and in $L^p$ for all $p > 1$, uniformly on compact sets. Moreover, we prove that if the difference between the observed function and the mean function of the prior lies in the reproducing kernel Hilbert space of the prior's kernel, then the posterior mean converges pointwise in $L^2$ to the unknown function, and, under an additional assumption on the kernel, uniformly on compacts in $L^1$. This paper provides an important step towards the theoretical legitimization of GP regression on manifolds and other non-Euclidean metric spaces.

We present a novel sample reduction scheme for random variables belonging to the SE(2) group by means of Dirac mixture approximation. For this, dual quaternions are employed to represent uncertain planar transformations. The Cramér-von Mises distance is modified as a smooth metric to measure the statistical distance between Dirac mixtures on the manifold of planar dual quaternions. Samples of reduced size are then obtained by minimizing the probability divergence via Riemannian optimization while interpreting the correlation between rotation and translation. We further deploy the proposed scheme for nonparametric modeling of estimates for nonlinear SE(2) estimation. Simulations show superior tracking performance of the sample reduction-based filter compared with Monte Carlo-based as well as parametric model-based planar dual quaternion filters.

Estimating orientations of objects in Euclidean space is an omnipresent challenge in robotics and autonomous systems. A useful representation of orientations involves unit quaternions. While the space of all unit quaternions forms a three-dimensional unit hypersphere, inverting the sign of a quaternion does not change the orientation described by it. Therefore, all possible orientations can be described by considering only a hemisphere of the unit hypersphere. In this paper, we propose a grid filter for arbitrary-dimensional unit hyperhemispheres and apply it to an orientation estimation task and another evaluation scenario. Our approach outperforms previous approaches that consider densities on the entire hypersphere.

Many applications require state estimation where possible values of the state are constrained to an interval (say, the valve position in percent) or the unit circle (say, the direction a robot is facing). We present two approaches that rely on a discretization of the state space, which differ in their interpretation of the discretized density. The first option is a piecewise constant density and the second option is a Dirac-mixture density. We show how circular filters can be derived and discuss the advantages and disadvantages of both approaches. In addition, we show how to extend the Dirac-based approach to estimation on the special Euclidean group in 2D, the group of rigid body motions in the plane, using Rao-Blackwellization. All presented the methods are thoroughly evaluated in simulations.

Estimation of periodic quantities such as angles or phase values is a common problem. However, standard approaches, for example the Kalman filter and extensions thereof, have difficulties when estimating periodic quantities. To address this problem, circular filtering algorithms have been proposed but they are limited to just a single angle. In order to deal with multiple, possibly correlated angles, toroidal filtering algorithms are necessary. We have previously proposed a bivariate filtering algorithm on the torus [1] that is limited to identity system and measurement models. In this paper, we show how the algorithm can be extended to handle nonlinear system and measurement models. The novel approach relies on the bivariate wrapped normal distribution for representing the uncertainty and it makes use of a deterministic sampling scheme for the torus. We provide a thorough evaluation of the proposed method using simulations.

For distributed estimation, algorithms have to be specifically crafted to minimize communication between the sensor nodes. As an adjusted version of the regular Kalman filter, the distributed Kalman filter (DKF) allows for deriving optimal results while not requiring regular communication. To achieve this, the DKF requires that each node has full knowledge about the system model and measurement models of all nodes. However, the DKF is not sufficient if the characteristics of the errors in the system and measurement models are not purely stochastic. In this paper, we present a distributed version of a combined stochastic and set-membership Kalman filter. The proposed filter optimizes the approximations of the set-membership uncertainties and can even yield better results than the regular centralized filter.

With the ubiquity of information distributed in networks, performing recursive Bayesian estimation using distributed calculations is becoming more and more important. There are a wide variety of algorithms catering to different applications and requiring different degrees of knowledge about the other nodes involved. One recently developed algorithm is the distributed Kalman filter (DKF), which assumes that all knowledge about the measurements, except the measurements themselves, are known to all nodes. If this condition is met, the DKF allows deriving the optimal estimate if all information is combined in one node at an arbitrary time step. In this paper, we present an information form of the distributed Kalman filter (IDKF) that allows the use of explicit system inputs at the individual nodes while still yielding the same results as a centralized Kalman filter.

In this work, we propose a plenhaptic guidance function that systematically describes the haptic information for guiding the user in the target environment. The plenhaptic guidance function defines the strength of the guidance at any position in space, at any direction, and at any time, and takes the geometry of the target environment as well as all possible goals into account. The plenhaptic guidance function, which can be rendered as active and passive guidance, is sampled and displayed to the user through a haptic interface in the user environment. The benefits of the plenhaptic guidance function for guiding the user to several simultaneous goals while avoiding the obstacles in a large target environment are demonstrated in real experiments.