Accurate uncertainty quantification is critical for robust and trustworthy predictions in many real-world applications. Bayesian Neural Networks (BNNs) provide a principled approach for modeling uncertainty but are often limited by the computational complexity of Bayesian inference. In this paper, we introduce a statistical linearization approach for multilayer feedforward BNNs. We demonstrate that this statistical linearization is equivalent to the Kalman Bayesian Neural Networks (KBNN) framework. This equivalence unifies these methodologies, providing a theoretical foundation for understanding the relationship between different BNN training approaches.

Efficiently solving the continuous-time signal and discrete-time observation filtering problem for chaotic dynamical systems presents unique challenges in that the advected distribution between observations may encounter a separatrix structure that results in the prior distribution being far from the observation or the distribution may become split into multiple disjoint components. In an attempt to sense and overcome these dynamical issues, as well as approximate a non-Gaussian distribution, a nudged particle filtering approach has been introduced. In the nudged particle filter method a control term is added, but has the potential drawback of degenerating the weights of the particles. To counter this issue, we introduce an intermediate resampling approach based on the modified Cramér-von Mises distance. The new method is applied to a challenging scenario of the non-chaotic, unforced nonlinear Duffing oscillator, which possesses a separatrix structure. Our results show that it consistently outperforms the standard particle filter with resampling and original nudged particle filter.

Densities separable in spherical coordinates have two advantages:$i$) the normalization constant is easy to compute, as the cumulative distribution can be decomposed into individual scalar integrals, and ii) an orthogonal inverse transform is directly available via a simple, scalar initial value problem and can be used to compute deterministic samples. We propagate uniform low-discrepancy sequences through that orthogonal inverse transform and obtain very homogeneous and even visually appealing deterministic samples. To demonstrate this technique, we exemplarily propose some spherical-coordinate-separable densities in$\mathbb{S}^{2}, \mathbb{R}^{2}$, and$\mathbb{R}^{3}$, including a non-isotropic modification of the von Mises-Fisher distribution. The proposed densities may be used, e.g., to represent uncertain radar measurements and for directional estimation. Furthermore, the framework presented herein allows quite simple design of various more densities tailored to a given scenario.

The solution to the state estimation problem is given by the Bayesian recursive relations (BRRs). Recently, ensemble Gaussian mixture filters have shown to be an accurate and consistent solution to the state estimation problem. In this type of filters, the BRRs are solved by approximating the state probability density function (PDF) via Gaussian mixtures (GMs) and point masses (PMs). Throughout the propagation and measurement update steps, the approximated state PDF is constantly switching between GMs and PMs. Therefore, a key step for this solution involves optimally sampling PMs from GMs. For onboard applications, verifiable and computationally inexpensive sampling techniques are crucial. In previous work, a deterministic sampling technique was developed by minimizing a distance metric known as the modified Cramér-von Mises distance (MCVMD), yielding a verifiable solution. However, the computationally feasibility of this solution for onboard use was not considered. This work introduces a new sampling strategy that is both deterministic and computationally inexpensive compared to MCVMD approach. By solving the approximate optimal transport problem via an iterative Sinkhorn-Knopp algorithm, this new technique is able to sub-optimally sample from a GM, providing a computationally inexpensive filter.

Particle filters are an important class of algorithms for Bayesian estimation. One of their drawbacks is the socalled particle degeneration where only very few particles with a meaningful weight remain after the filter step. This effect is typically remedied by regularly resampling the particles, yielding a set of equally weighted particles. This paper investigates an approach to deterministically sample particles from the proposal distribution in such a way to automatically have equally weighted particles at the end of the filter step. The proposed method is first motivated and presented for the one-dimensional case. Using the Radon transform and projected cumulative distributions, the one-dimensional algorithm is extended to multivariate problems. Some examples of the usefulness of the proposed algorithm are also shown.

Gaussian mixture filters for nonlinear systems usually rely on severe approximations when calculating mixtures in the prediction and filtering step. Thus, offline approximations of noise densities by Gaussian mixture densities to reduce the approximation error have been proposed. This results in exponential growth in the number of components, requiring ongoing component reduction, which is computationally complex. In this paper, the key idea is to approximate the true transition density by an axis-aligned Gaussian mixture, where two different approaches are derived. These approximations automatically ensure a constant number of components in the posterior densities without the need for explicit reduction. In addition, they allow a trade-off between estimation quality and computational complexity.

Bayesian machine learning models-especially Bayesian neural networks (BNNs)-offer powerful black-box approaches for prediction and uncertainty quantification. However, these models frequently exhibit inconsistent prediction quality across input regions, and conventional global metrics (e.g., the mean squared error (MSE)) are inadequate for capturing such local discrepancies. To overcome this limitation, we introduce a novel kernel-based framework for local calibration testing that assesses how well predicted distributions reflect both the function to be learned and inherent uncertainties. In our approach, spherical input-space kernels are used to define relevant subsets in the neighborhood of a point to be tested. This enables the online assessment of these localized regions using calibration metrics or statistical tests. By aggregating results across multiple kernel widths, our method yields both robust binary decisions and a continuous analysis over arbitrary inputs. Numerical experiments on single- and multi-dimensional regression tasks demonstrate the efficiency and scalability of our approach, underscoring its potential for real-time and large-scale applications.

In this paper, we introduce a novel Gaussian Assumed Density Filter (GADF) for high-quality state estimation in discrete-time stochastic nonlinear dynamic systems, with a primary focus on the measurement update. Rooted in optimal transport theory, the Wasserstein distance is employed as a powerful metric for comparing probability distributions. Building on this foundation, we utilize the unique, explicit Wasserstein barycentric interpolation between Gaussian distributions to parameterize an initial Gaussian Process (GP) in the joint measurement/prior state space. Deterministic samples drawn from the true joint measurement/state density are then used with likelihood-based parameter estimation techniques to optimize the parameters of this Gaussian Process. As a result, the derived Gaussian Process provides a local non-Gaussian approximation to the true joint density. This approach eliminates the need for a second Gaussian assumption on the joint density and avoids an explicit likelihood function, making it a higher-quality plug-in replacement for the commonly used Linear Regression Kalman Filter (LRKF).

In this paper, we present novel results and insights into tracking extended objects using the Stochastic Medial Axis Transform (SMAT). Unlike conventional methods that depend on explicit shape parameterization with basic priors, SMAT employs an implicit inside-out representation by constructing maximum inscribed circles within the object. This is achieved by simultaneously fitting two Bézier curves: one that defines the medial manifold, providing the centers of the maximum inscribed circles, and the other that characterizes the scalar thickness field, assigning positive radii to these centers. This dual-curve formulation leverages the concept of inverse skeletonization and offers a flexible, parametric shape model capable of tracking diverse shapes, whether convex or non-convex, symmetric or asymmetric. Furthermore, we obtain a closed-form likelihood function in 2D space that facilitates the application of advanced recursive Bayesian state estimators. Finally, we conduct two simulation studies to demonstrate and evaluate the effectiveness of the proposed approach.

In extended object tracking, basic parametric shapes such as ellipses and rectangles or non-parametric shape representations such as Fourier series or Gaussian processes can be utilized as shape priors. However, flexible non-parametric shape representations can be disproportionately detailed and computationally intensive for many applications. Therefore, we propose to adopt deformable superellipses for a low-dimensional and flexible representation of basic parametric shapes in this paper. We present a measurement model in 2D space that can cope with boundary and interior measurements simultaneously by recursively estimating an artificial noise variance for interior measurements. We investigate and compare the model in a simulated and real-world maritime scenario with the result that the combination of deformable superellipses and artificial measurement noise estimation performs better than state-of-the-art methods.

We propose a novel update step of a Gaussian mixture particle filter for nonlinear state estimation. The update procedure works as follows: First, unweighted samples are drawn in an optimal deterministic sense from a prior Gaussian mixture. These samples are then assigned weights from the likelihood function, and we compute higher-order moments from this samplebased posterior. These moment approximations converge with $L^{-1}$ instead of $L^{-1 / 2}$ as our samples are optimal deterministic. Finally, the continuous posterior approximation is determined as the Gaussian mixture that has minimal Fisher information under the constraint of having the aforementioned moments. To achieve this, we employ a closed-form solution of the Fisher information that involves Gaussian root mixture densities.

The construction of a continuous probability density function (pdf) that fits a set of samples is a frequently occurring task in statistics. This is an inherently underdetermined problem, that can only be solved by making some assumptions about the samples or the distribution to be estimated. This paper proposes a density estimation method based on the premise that each sample represents the same amount of probability mass of the underlying density. The estimated pdf is parameterized as the square of a polynomial spline, which makes further processing of the estimated density very efficient. This pdf is inherently nonnegative, ensuring a monotone cumulative distribution function, which makes it easy to generate samples from it through inverse transform sampling. Furthermore, it is cheap to evaluate and easy to integrate, making moment calculations fast. To find the coefficients of the polynomials that make up the spline, an optimization problem is derived. The Fisher information is used as a regularizer in this problem to select the solution that contains the least amount of information. The method is shown to work on samples from a variety of different one-dimensional probability distributions.

The earth’s magnetic field along railway tracks is strongly distorted by magnetic material in the vicinity, e.g., steel in rails and reinforced concrete. The resulting magnetic distortions are persistent in time and characteristic for a certain part of the track. Thus, these distortions can be seen as fingerprints that enable localization when a map of the magnetic field is available. This is particularity interesting for areas where global navigation satellite system (GNSS) signals are not available, such as tunnels. Unfortunately, creating the magnetic map in a GNSS-denied area requires a position reference system that is most likely not available. This paper addresses this problem with a graph-based simultaneous localization and mapping (SLAM) algorithm that uses only odometer and magnetometer measurements. The key idea of the proposed algorithm is to use the magnetic field to detect loop closures and to calculate the relative transformation between different nodes in the pose-graph. The algorithm is evaluated based on a data set recorded with the advanced TrainLab of the Deutsche Bahn traveling on a track in Berlin. Results show that the graph SLAM algorithm together with the magnetic loop closure detection reduces and bounds the position error of the odometry.

Bayesian Neural Networks (BNNs) offer a sophisticated framework for extending classical neural network point estimates to encompass predictive distributions. Despite the high potential of BNNs, established BNN training methods such as Variational Inference (VI) and Markov Chain Monte Carlo (MCMC) grapple with issues such as scalability and hyperparameter dependence. In addressing these issues, our research focuses on the fundamental elements of BNNs, in particular perceptrons and their predictive capabilities. We introduce a new perspective on the closed-form solution for backward-pass computation for the Bayesian perceptron and prove that the state-of-the-art solution is equivalent to statistical linearization. To assess the efficacy of Bayesian perceptrons and provide insights into their performance in distinct input space regions, a novel methodology utilizing k-d trees as a space partitioning method is introduced to evaluate prediction quality within specific input space regions.

In the past years, algorithms for 3D shape tracking using radial functions in spherical coordinates represented with different methods have been proposed. However, we have seen that mainly measurements from the lateral surface of the target can be expected in a lot of dynamic scenarios and only few measurements from the top and bottom parts leading to an error-prone shape estimate in the top and bottom regions when using a representation in spherical coordinates. We, therefore, propose to represent the shape of the target using a radial function in cylindrical coordinates, as these only represent regions of the lateral surface, and no information from the top or bottom parts is needed. In this paper, we use a Fourier-Chebyshev double series for 3D shape representation since a mixture of Fourier and Chebyshev series is a suitable basis for expanding a radial function in cylindrical coordinates. We investigate the method in a simulated and real-world maritime scenario with a CAD model of the target boat as a reference. We have found that shape representation in cylindrical coordinates has decisive advantages compared to a shape representation in spherical coordinates and should preferably be used if no prior knowledge of the measurement distribution on the surface of the target is available.

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).

Knowledge about human intention can be beneficial in many disciplines of robotics, such as collaborative manufacturing, prosthetics, or encountered-type haptics. Existing intention estimation approaches are either traditional and rely on handcrafted features and heuristics, or learning-based and tailored to very specific conditions. This paper attempts to combine the best of both worlds by making recurrent neural networks adaptable to different scenarios. To achieve this, the intention estimation problem is formulated as a probabilistic classification problem and two new data sets with real-world motion and eye-tracking data are presented. Based on this data, three real-time capable classifiers with different features regarding situational awareness and additional outputs are designed and evaluated against two competing approaches. The results show that two out of three classifiers lead to improved or equivalent performance compared to traditional approaches, while good generalization is maintained.

We present a new deterministic sampling method for arbitrary densities, unnormalized densities, and likelihoods. Our rejection-free and kernel-free method uses dense equal sphere packing of the volume under the density function (PoVuD). In order to obtain an ensemble that is better than independent random particles, we enforce some local homogeneity.

We propose a method for optimal Bayesian filtering with deterministic particles. In order to avoid particle degeneration, the filter step is not performed at once. Instead, the particles progressively flow from prior to posterior. This is achieved by splitting the filter step into a series of sub-steps. In each sub-step, optimal resampling is done by a map that replaces non-equally weighted particles with equally weighted ones. Inversions of the maps or monotonicity constraints are not required, greatly simplifying the procedure. The parameters of the mapping network are optimized w.r.t. a particle set distance. This distance is differentiable, and compares non-equally and equally weighted particles. Composition of the sequence of maps provides a final mapping from prior to posterior particles. Radial basis function neural networks are used as maps. It is important that no intermediate continuous density representation is required. The entire flow works directly with particle representations. This avoids costly density estimation.

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.

Magnetic-field based indoor localization uses distortions of the Earth magnetic field caused by magnetic material in the surrounding of the platform that is to be localized. Existing magnetic localization methods compare measurements from a magnetometer with a magnetic field map to estimate the position of the platform, on which the sensor is mounted. For the comparison it is typically assumed that the magnetometer is calibrated. Unfortunately, for some platforms calibration is not straightforward, particularly when they are large or heavy. In this paper we therefore propose a simultaneous localization and calibration approach. The approach is based on a Rao-Blackwellized particle filter that makes use of the conditional linearity of magnetometer measurements and calibration parameters. The feasibility of the proposed particle filter is evaluated with three data sets recorded with a differential drive robot in an indoor environment.

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.

Abstract Joint Cramér‐Rao lower bound (JCRLB) is very useful for the performance evaluation of joint state and parameter estimation (JSPE) of non‐linear systems, in which the current measurement only depends on the current state. However, in reality, the non‐linear systems with two‐adjacent‐states dependent (TASD) measurements, that is, the current measurement is dependent on the current state as well as the most recent previous state, are also common. First, the recursive JCRLB for the general form of such non‐linear systems with unknown deterministic parameters is developed. Its relationships with the posterior CRLB for systems with TASD measurements and the hybrid CRLB for regular parametric systems are also provided. Then, the recursive JCRLBs for two special forms of parametric systems with TASD measurements, in which the measurement noises are autocorrelated or cross‐correlated with the process noises at one time step apart, are presented, respectively. Illustrative examples in radar target tracking show the effectiveness of the JCRLB for the performance evaluation of parametric TASD systems.

The control of brushless DC motors requires high-resolution angular position and accurate speed information. However, available sensor-based solutions only measure either the position or the speed directly, and then approximate the other numerically. In this work, a novel technique is presented to estimate both of these values simultaneously by sensing the stray magnetic field of the internal permanent magnets of the motor. However, achieving this requires the following two challenges to be addressed. First, the relationship between the magnetic field and the motor position is distorted by the rotational speed in a non-intuitive way, requiring careful modeling of these dependencies. Second, the derived model needs to consider that the angular position data is periodic by nature, but the magnetic field data and the angular speed data are linear (i.e., non-periodic). To achieve this, we introduce two different multidimensional regression models based on the Fourier series. Both models are first trained offline using reference data, and then used as a measurement function in a nonlinear estimator such as the EKF for online estimation. Evaluations show that both models outperform state-of-the-art techniques.

Nonlinear filtering is the most important aspect in state estimation with real-world systems. While the Kalman filter provides a simple though optimal estimate for linear systems, feasible filters for general systems are still subject of intensive research. The previously proposed Progressive Gaussian Filter PGF42 marked a new milestone, as it was able to efficiently compute an optimal Gaussian approximation of the posterior density in nonlinear systems [1]. However, for highly nonlinear systems where true posteriors are “banana-shaped” (e.g., cubic sensor problem) or multimodal (e.g., extended object tracking), even an optimal Gaussian approximation is an inadequate representation. Therefore, we generalize the established framework around the PGF42 from Gaussian to Gaussian mixture densities that are better able to approximate arbitrary density functions. Our filter simultaneously holds approximate Gaussian mixture and Dirac mixture representations of the same density, what we call coupled discrete and continuous densities (CoDiCo). For conversion between discrete and continuous representation, we employ deterministic sampling and the expectation-maximization (EM) algorithm, which we extend to deal with weighted particles.

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.

Networks consisting of several spatially distributed sensor nodes are useful in many applications. While distributed processing of information can be more robust and flexible than centralized filtering, it requires careful consideration of dependencies between local state estimates. This paper proposes an algorithm to keep track of dependencies in decentralized systems where no dedicated fusion center is present. Specifically, it addresses double counting of measurement information due to intermediate fusion results as well as correlations due to common process noise and common prior information. To limit the necessary amount of data, this paper introduces a method to bound correlations partially, leading to a more conservative fusion result while reducing the necessary amount of data. Simulation studies compare the performance and convergence rate of the proposed algorithm to other state-of-the-art methods.

Multilateration systems reconstruct the location of a target that transmits electromagnetic or acoustic signals. The employed measurements for localization are the times of arrival (TOAs) of the transmitted signal, measured by a number of spatially distributed receivers at known positions. We present a novel multilateration algorithm to localize multiple targets that transmit indistinguishable signals at unknown times. That is, each receiver measures merely a set of TOAs with no association to the targets. Our method does not need any prior information. Therefore, it can provide uncorrelated, static measurements to be introduced into a separate tracker subsequently, or an initialization routine for multi target trackers.

We present a novel region growing algorithm for plane extraction of organized point clouds using the spherical convex hull. Instead of explicit plane parameterization, our approach interprets potential underlying planes as a series of geometric constraints on the sphere that are refined during region growing. Unlike existing schemes relying on downsampling for sequential execution in real time, our approach enables pixelwise plane extraction that is highly parallelizable. We further test the proposed approach with a fully parallel implementation on a GPU. Evaluation based on public data sets has shown state-of-the-art extraction accuracy and superior speed compared to existing approaches, while guaranteeing real-time processing at full input resolution of a typical RGB-D camera.

In real world applications, robotic solutions remain impractical due to the challenges that arise in unknown and unstructured environments. To perform complex manipulation tasks in complex and cluttered situations, robots need to be able to identify the interaction possibilities with the scene, i.e. the affordances of the objects encountered. In unstructured environments with noisy perception, insufficient scene understanding and limited prior knowledge, this is a challenging task. In this work, we present an approach for grasping unknown objects in cluttered scenes with a humanoid robot in the context of a nuclear decommissioning task. Our approach combines the convenience and reliability of autonomous robot control with the precision and adaptability of teleoperation in a semi-autonomous selection of grasp affordances. Additionally, this allows exploiting the expert knowledge of an experienced human worker. To evaluate our approach, we conducted 75 real world experiments with more than 660 grasp executions on the humanoid robot ARMAR-6. The results demonstrate that high-level decisions made by the human operator, supported by autonomous robot control, contribute significantly to successful task execution.

This work is concerned with our ongoing research project CoCPN: Cooperative Cyber Physical Networking that has the goal to allow cooperation between control applications and the communication system that constitute cyber-physical systems. We describe the envisioned architecture of CoCPN and outline how it improves the flexibility of cyber-physical systems by cooperatively sharing a common network infrastructure. We also present our simulation tool CoCPN-Sim that we developed as a method to thoroughly investigate the interaction between control applications and the communication system. By providing the results of selected simulations using the well-known inverted pendulum, we identify potential aspects that can be exploited for cooperation and hence serve as starting points towards more flexible and adaptive cyber-physical systems.

With the advent of cheap sensor technology, multisensor data fusion algorithms have been becoming a key enabler for efficient in-network processing of sensor data. The information filter, in particular, has proven useful due to its simple additive structure of the measurement update equations. In order to exploit this structure for an efficient in-network processing, each node in the network is supposed to locally process and combine data from its neighboring nodes. The aspired in-network processing, at first glance, prohibits efficient privacy-preserving communication protocols, and encryption schemes that allow for algebraic manipulations are often computationally too expensive. Partially homomorphic encryption schemes constitute far more practical solutions but are restricted to a single algebraic operation on the corresponding ciphertexts. In this paper, an additive-homomorphic encryption scheme is used to derive a privacy-preserving implementation of the information filter where additive operations are sufficient to distribute the workload among the sensor nodes. However, the encryption scheme requires the floating-point data to be quantized, which impairs the estimation quality. The proposed filter and the implications of the necessary quantization are analyzed in a simulated multisensor tracking scenario.

The paper focuses on state estimation of discrete-time nonlinear stochastic dynamic systems with a special focus on the stochastic integration filter. The filter is an representative of the Gaussian filter and computes the state and measurement predictive moments by making use of a stochastic integration rule. As a result, the calculated values of the moments are random variables and exhibit favorable asymptotic properties. The paper analyzes theoretical consequences of using stochastic integration rules and proposes several modifications that improve the performance of the stochastic integration filter. As the filter requires multiple iterations of the stochastic rule, its computational costs are higher in comparison with other Gaussian filters. To reduce the costs, several modifications are proposed in the paper, which are also concerned with numerical stability issues. The proposed modifications are illustrated using both static and dynamic numerical examples used in target tracking.

In this paper, we present a novel nonlinear progressive filtering approach for estimatingSE(2) states represented by unit dual quaternions. Unlike previously published approaches, the measurement model no longer needs to be assumed as identity. Our solution utilizes deterministic sampling on a Bingham-like probability distribution, which has been adapted to simultaneously model orientation and translation. During the measurement update step, the estimate gets progressively updated. Our approach inherently incorporates the nonlinear structure ofSE(2) and enables a flexible measurement update step. We also give an evaluation for planar rigid body motion estimation with a case study that is close to real-world scenarios.

In this paper, we present a novel approach to perform simultaneous localization and mapping (SLAM) for planar motions based on stochastic filtering with dual quaternion particles using low-cost range and gyro sensor data. Here, SE(2) states are represented by unit dual quaternions and further get stochastically modeled by a distribution from directional statistics such that particles can be generated by random sampling. To build the full SLAM system, a novel dual quaternion particle filter based on Rao-Blackwellization is proposed for the tracking block, which is further integrated with an occupancy grid mapping block. Unlike previously proposed filtering approaches, our method can perform tracking in the presence of multi-modal noise in unknown environments while giving reasonable mapping results. The approach is further evaluated using a walking robot with on-board ultrasonic sensors and an IMU sensor navigating in an unknown environment in both simulated and real-world scenarios.

In a surveillance environment with high clutter, finding the correct measurement to track associations becomes extremely important for efficient target tracking. This study offers a novel algorithm to retrodict the data association probabilities at any past time instant, when the batch set of measurements is kept in memory. For the retrodiction procedure, the batch association cost is first written explicitly as a binary integer optimization problem with a quadratic cost function and it is shown that the relaxed form of the problem is convex. From the relaxed problem, a lower bound for the optimal association cost is derived, and this lower bound is used as the data association probabilities pertaining to that selected time instant in the past. Due to its consideration of the batch set of data in a retrospective manner, we will call this algorithm as Retrodictive Probabilistic Data Association, RPDA. For simplification of the mathematical analysis, a single point target with no missing measurements, i.e. PD= 1, is taken into account.

Optimal fusion of estimates that are computed in a distributed fashion is a challenging task. In general, the sensor nodes cannot keep track of the cross-correlations required to fuse estimates optimally. In this paper, a novel technique is presented that provides the means to reconstruct the required correlation structure. For this purpose, each node computes a set of deterministic samples that provides all the information required to reassemble the cross-covariance matrix for each pair of estimates. As the number of samples is increasing over time, a method to reduce the size of the sample set is presented and studied. In doing so, communication expenses can be reduced significantly, but approximation errors are possibly introduced by neglecting past correlation terms. In order to keep approximation errors at a minimum, an appropriate set size can be determined and a trade-off between communication expenses and estimation quality can be found.

In this paper, we present a new approach for solving the SLAM problem using the Ensemble Kalman Filter (EnKF). In contrast to other Kalman filter based approaches, the EnKF uses a small set of ensemble members to represent the state, thus circumventing the computation of the large covariance matrix traditionally used with Kalman filters, making this approach a viable application in high-dimensional state spaces. Our approach adapts techniques from the geoscientific community such as localization to the SLAM problem domain as well as using the Optimal Subpattern Assignment (OSPA) metric for data association. We then compare the results of our algorithm with an extended Kalman filter (EKF) and FastSLAM, showing that our approach yields a more robust, accurate, and computationally less demanding solution than the EKF and similar results to FastSLAM.

Source localization is of great importance for wireless sensor network applications. Locating emission sources using received signal strength (RSS) measurements is investigated in this paper. As RSS localization is a non-convex optimization problem, it is difficult to achieve global optima. Many optimization methods have been proposed to relax it to a convex optimization problem. Unlike these methods, we propose a convex combination scheme. By introducing a highly accurate linear approximation of a logarithmic function, the source location is represented by a convex combination of a set of virtual anchors. Then the original problem is relaxed to be a convex optimization problem of finding the optimal combination coefficients, which can be solved efficiently using constrained least squares. To obtain the virtual nodes, we construct parallel lines and use their intersections to form a convex polygon, which covers the source location with certain probability. The vertices of the polygon are taken as the virtual nodes. Numerical examples verify the performance of the proposed method in both localization accuracy and computational efficiency.

Transformation of a random variable is a common need in a design of many algorithms in signal processing, automatic control, and fault detection. Typically, the design is tied to an assumption on a probability density function of the random variable, often in the form of the Gaussian distribution. The assumption may be, however, difficult to be met in algorithms involving non‐linear transformation of the random variable. This paper focuses on techniques capable to ensure validity of the Gaussian assumption of the non‐linearly transformed Gaussian variable by approximating the to‐be‐transformed random variable distribution by a Gaussian mixture (GM) distribution. The stress is laid on an analysis and selection of design parameters of the approximate GM distribution to minimise the error imposed by the non‐linear transformation such as the location and number of the GM terms. A special attention is devoted to the definition of the novel GM splitting directions based on the measures of non‐Gaussianity. The proposed splitting directions are analysed and illustrated in numerical simulations.

This Special Section was inspired by the 2016 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2016), see mfi2016.org, which took place 19–21 September 2016 in Baden-Baden, Germany. The conference was sponsored by the IEEE Robotics and Automation Society (RAS) and the IEEE Industrial Electronics Society (IES). Several of the papers in this Special Section build upon papers presented at the conference. While a total of 106 papers were presented at MFI 2016, we invited only the authors of the highest-ranked papers to submit an extended version of their respective conference papers to this Special Section. Besides contacting these selected authors, an open call for papers for this Special Session was published to broaden the scope and to allow for papers not presented at MFI 2106. For extended conference papers, an important constraint was to include substantially novel aspects, such as added theoretical work or new experimental results. Authors are required to cite their original conference papers and clearly specify the novel contribution of the journal paper with respect to the conference paper. All papers, those building upon previously presented MFI 2016 papers and additional papers submitted related to the open call for papers, underwent the same rigorous review process as regular submissions strictly according to the guidelines of the IEEE Transactions on Industrial Informatics (TII). A total of 58 papers were submitted to this Special Section of the IEEE TII. After a careful review process, eight papers were finally selected for publication. We have grouped the eight accepted papers into four sections. The papers are briefly summarized here.

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.

A lot of performance evaluation metrics exist for nonlinear filters. At present, the most commonly used one is a single and incomprehensive metric of performance. This metric can continuously and quantitatively describe the performance of the nonlinear filters. But in many cases, we need to rank the performance of the filters. It is in general very hard to rank the filters just using a single metric. First, the rankings using a single metric at different times may be different. Then how to get a unique rank for all times? A typical existing solution is to average the single metric over all times. But it is easy to be dominated just by very large values at just some times. Second, a single metric is usually incomprehensive in measuring performance. To make the ranking more comprehensive, multiple metrics are usually needed. But how to get a comprehensive unique rank from the ranks, possibly conflicting with each other? In this paper, we propose a framework to rank multiple nonlinear filters using ranking vectors and voting fusion based on a single metric or multiple metrics. Illustrative examples show that this framework is very effective.

Decentralized data fusion is a challenging task. Either it is too difficult to maintain and track the information required to perform fusion optimally, or too much information is discarded to obtain informative fusion results. A well-known solution is Covariance Intersection, which may provide too conservative fusion results. A less conservative alternative is discussed in this paper, and generalizations are proposed in order to apply it to a wide class of fusion problems. The Inverse Covariance Intersection algorithm is about finding the maximum possible common information shared by the estimates to be fused. A bound on the possibly shared common information is derived and removed from the fusion result in order to guarantee consistency. It is shown that the conditions required for consistency can be significantly relaxed, and also other causes of correlations, such as common process noise, can be treated.

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.

We present a particle filter for multi-object tracking that is based on the ideas of the Approximate Bayesian Computation (ABC) paradigm. The main idea is to avoid the explicit computation of the likelihood function by means of simulation. For this purpose, a large amount of particles in the state space is simulated from the prior, transformed into measurement space, and then compared to the real measurement by using an appropriate distance function, i.e., the OSPA distance. By selecting the closest simulated measurements and their corresponding particles in state space, the posterior distribution is approximated. The algorithm is evaluated in a multi-object scenario with and without clutter and is compared to a global nearest neighbour Kalman filter.

We introduce the unscented von Mises-Fisher filter (UvMFF), a nonlinear filtering algorithm for dynamic state estimation on the $n$-dimensional unit hypersphere. Estimation problems on the unit hypersphere occur in computer vision, e.g., when using omnidirectional cameras, as well as in signal processing. As approaches in literature are limited to very simple system and measurement models, we propose a deterministic sampling scheme on the unit hypersphere, which allows us to handle nonlinear system and measurement models. The proposed approach can be seen as a hyperspherical variant of the unscented Kalman filter (UKF). The advantages of the novel method are shown by means of simulations.

The two title concepts have been evolving rather rapidly, but independent of each other. The Wasserstein barycenter, on one hand, has mostly made its appearance in image processing as it can describe a measure of similarity between images. Its minimization might, for example, suggest the best match in image alignment. On the other hand, MMOSPA estimation has been applied largely to multi-target tracking. The Optimal Sub-Pattern Assignment (OSPA) measures the distance between two sets and the Mean OSPA (MOSPA) can be minimized to give the Minimum MOPSA (MMOSPA), which improves MMSE estimation of the target locations when the labeling of the targets in the set is not important. Approximate and exact algorithms have evolved for both Wasserstein barycenters and MMOSPA estimation. Here, we draw connections between the two perspectives and elaborate how they can benefit from each other.

One of the key challenges in distributed linear estimation is the systematic fusion of estimates. While the fusion gains that minimize the mean squared error of the fused estimate for known correlations have been established, no analogous statement could be obtained so far for unknown correlations. In this contribution, we derive the gains that minimize the bound on the true covariance of the fused estimate and prove that Covariance Intersection (CI) is the optimal bounding algorithm for two estimates under completely unknown correlations. When combining three or more variables, the CI equations are not necessarily optimal, as shown by a counterexample.

In this paper, we present a novel approach for tracking objects whose movement is constrained to a compact one-dimensional manifold, for example a conveyer belt or a mobile robot whose movement is restricted to tracks. Standard approaches either ignore the constraint at first and retroactively move the estimate to lie on the manifold, or consider the tracking problem on a manifold but falsely assume a Gaussian distribution. Our method explicitly takes the actual topology into account from the beginning and relies on special types of probability distributions defined on the proper manifold. In particular, we consider objects moving along a closed one-dimensional track, for example an ellipse, a polygon, or similar closed shapes. This shape is transformed to a circle with a homeomorphism. Thus, we can apply a recursive circular filtering algorithm to the constrained tracking problem. Finally, the estimate is transformed back to the original manifold. We evaluate the proposed method in an experiment by tracking a toy train moving along a track and comparing the results to those of traditional approaches for this problem.

This paper describes a method to intelligently schedule a network of multiple RGBD sensors in a Bayesian object tracking scenario, with special focus on Microsoft KinectTM devices. These setups have issues such as the large amount of raw data generated by the sensors and interference caused by overlapping fields of view. The proposed algorithm addresses these issues by selecting and exclusively activating the sensor that yields the best measurement, as defined by a novel stochastic model that also considers hardware constraints and intrinsic parameters. In addition, as existing solutions to toggle the sensors were found to be insufficient, the development of a hardware module, especially designed for quick toggling and synchronization with the depth stream, is also discussed. The algorithm then is evaluated within the scope of a multi-Kinect object tracking scenario and compared to other scheduling strategies.

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.

This paper presents the control concept of a semi-mobile haptic interface for extended range telepresence that enables the user to explore spatially unrestricted target environments even from a small user environment. The semi-mobile haptic interface consists of a haptic manipulator mounted on a large grounded Cartesian robot, the prepositioning unit. The prepositioning unit is controlled in such a way that the haptic manipulator is kept off its workspace limits. At the same time, the control algorithm allows the optimal utilization of the available space in the user environment and guarantees the safety of the user. The proposed control method is based on the position and velocity of the end-effector and also takes the position of the user into account. Moreover, it is robust against noisy measurements of the user position or outliers due, for example, to occlusions in the tracking system. Experimental results show the suitability of the proposed control to provide haptic interaction in extended range telepresence.

Fitting conic sections, e.g., ellipses or circles, to noisy data points is a fundamental sensor data processing problem, which frequently arises in robotics. In this paper, we introduce a new procedure for deriving a recursive Gaussian state estimator for fitting conics to data corrupted by additive Gaussian noise. For this purpose, the original exact implicit measurement equation is reformulated with the help of suitable approximations as an explicit measurement equation corrupted by multiplicative noise. Based on stochastic linearization, an efficient Gaussian state estimator is derived for the explicit measurement equation. The performance of the new approach is evaluated by means of a typical ellipse fitting scenario.

In this paper, a multi-level approach to intention, activity, and motion recognition for a humanoid robot is proposed. Our system processes images from a monocular camera and combines this information with domain knowledge. The recognition works on-line and in real-time, it is independent of the test person, but limited to predefined view-points. Main contributions of this paper are the extensible, multi-level modeling of the robot's vision system, the efficient activity and motion recognition, and the asynchronous information fusion based on generic processing of mid-level recognition results. The complementarity of the activity and motion recognition renders the approach robust against misclassifications. Experimental results on a real-world data set of complex kitchen tasks, e.g., Prepare Cereals or Lay Table, prove the performance and robustness of the multi-level recognition approach.

Telepresence systems enable a user to experience virtual or distant environments by providing sensory feedback. Appropriate devices include head mounted displays (HMD) for visual perception, headphones for auditory response, or even haptic displays for tactile sensation and force feedback. While most common designs use dedicated input devices like joysticks or a space mouse, the approach followed in the present work takes the user's position and viewing direction as an input, as he walks freely in his local surroundings. This is achieved by using acoustic tracking, where the user's pose (position and orientation) is estimated on the basis of ranges measured between a set of ceiling-mounted loudspeakers and a microphone array fixed on the user's HMD. To allow for natural user motion, a wearable, fully wireless telepresence system is introduced. The increase in comfort compared to wired solutions is obvious, as the user's awareness of distracting cables is taken away during walking. Also the lightweight design and small dimensions contribute to ergonomics, as the whole assembly fits well into a small backpack.

In this paper, we present a novel use of haptic information in extended range telepresence, the wide-area haptic guidance. It consists of force and position signals applied to the user's hand in order to improve safety, accuracy, and speed in some telepresent tasks. Wide-area haptic guidance assists the user in reaching a desired position in a remote environment of arbitrary size without degrading the feeling of presence. Several methods for haptic guidance are analyzed. With active haptic guidance, the user is guided by superimposed forces that pull him into the desired direction of motion, whereas under passive haptic guidance, the movement of the user is lightened in the preferred direction and constrained in the other directions. By using closed-loop haptic guidance instead of open-loop haptic guidance, not only is the user guided to his target but also the deviation from the desired target path is reduced. The proposed guidance methods were tested with a haptic interface specifically designed for extended range telepresence.

In range-based localization, the trajectory of a mobile object is estimated based on noisy range measurements between the object and known landmarks. In order to deal with this uncertain information, a Bayesian state estimator is presented, which exploits optimal stochastic linearization. Compared to standard state estimators like the Extended or Unscented Kalman Filter, where a point-based Gaussian approximation is used, the proposed approach considers the entire Gaussian density for linearization. By employing the common assumption that the state and measurements are jointly Gaussian, the linearization can be calculated in closed form and thus analytic expressions for the range-based localization problem can be derived.

A reliable estimation of heart surface motion is an important prerequisite for the synchronization of surgical instruments in robotic beating heart surgery. In general, only an imprecise description of the heart dynamics and measurement systems is available. This means that the estimation of heart motion is corrupted by stochastic and systematic uncertainties. Without consideration of these uncertainties, the obtained results will be inaccurate and a safe robotic operation cannot be guaranteed. Until now, existing approaches for estimating the motion of the heart surface are either deterministic or treat only stochastic uncertainties. The proposed method extends the heart motion estimation to the simultaneous consideration of stochastic and unknown but bounded systematic uncertainties. It computes dynamic bounds in order to provide the surgeon with a guidance by constraining the motion of the surgical instruments and thereby protecting sensitive tissue.

Estimating a user's intention is central to close human-robot cooperation. In this paper, the problem of performing intention recognition with tree-structured Dynamic Bayesian Networks for large environments with many features is addressed. The proposed approach reduces the computational complexity of inference O(bs) for tree-structured measurement models with an average branching factor b and tree height s to O(b̃s), where b̃ ≪ b. The key idea is to switch between a finite set of reduced system and measurement models in order to restrict inference to the most important features. A model predictive approach to online switching between the reduced models is proposed that exploits an upper bound of the distances of the reduced models to the full model. The effectiveness of the proposed algorithm is validated in the intention recognition for a humanoid robot using a telepresent household scenario.

Extended range telepresence aims at enabling a user to experience virtual or remote environments, taking his own body movements as an input to define walking speed and viewing direction. Therefore, localization and tracking of the user's pose (position and orientation) is necessary to perform a body-centered scene rendering. Visual and acoustic feedback is provided to the user by a head mounted display (HMD). To allow for free movement within the user environment, the tracking system is supposed to be user-wearable and entirely wireless. Consequently, a lightweight design is presented featuring small dimensions to fit into a conventional 13″ laptop backpack, which satisfies the above stated demands for highly immersive extended range telepresence scenarios. Tracking is performed by measuring the time-of-flight (TOF) of coded acoustic signals between a set of fixed loudspeakers and a user-worn microphone array. Dedicated embedded hardware combined with off-the-shelf components is employed to form a robust, low-cost telepresence system that can be easily installed in any living room.

An algorithm for estimating the pose, i.e., translation and rotation, of an extended target object is introduced. Compared to conventional methods, where pose estimation is performed on the basis of time-of-flight (TOF) measurements between external sources and sensors attached to the object, the proposed approach directly uses the amplitude values measured at the sensors for estimation purposes without an intermediate TOF estimation step. This is achieved by modeling the wave propagation by a nonlinear dynamic system comprising a system and a measurement equation. The nonlinear system equation includes a model of the time-variant structure of the object rotation based on rotation vectors. As a result, the measured amplitude values at the sensors can be processed instantaneously in a recursive fashion. Uncertainties in the measurement process are systematically considered by employing a stochastic filter for estimating the pose, i.e., the state of the nonlinear dynamic system.

We propose an analytic moment-based filter for nonlinear stochastic dynamic systems modeled by Gaussian processes. Exact expressions for the expected value and the covariance matrix are provided for both the prediction step and the filter step, where an additional Gaussian assumption is exploited in the latter case. Our filter does not require further approximations. In particular, it avoids finite-sample approximations. We compare the filter to a variety of Gaussian filters, that is, the EKF, the UKF, and the recent GP-UKF proposed by Ko et al. (2007).

A novel heart surface motion estimation framework for a robotic surgery on a stabilized beating heart is proposed. It includes an approach for the reconstruction and prediction of heart surface motion based on a novel physical model of the intervention area described by a distributed-parameter system. Instead of conventional element methods, a meshless method is used for a spatial and temporal decomposition of this system. This leads to a finite-dimensional state-space form. Furthermore, the state of the resulting lumped-parameter system, which provides an approximation of the deflection and velocity of the heart surface, is dynamically estimated under consideration of uncertainties both occurring in the system and arising from noisy camera measurements. By using the estimation results, an accurate reconstruction of heart surface motion for the synchronisation of the surgical instruments is also achieved at occluded or non-measurement points.

In this paper, we present a novel approach to parametric density estimation from given samples. The samples are treated as a parametric density function by means of a Dirac mixture, which allows for applying analytic optimization techniques. The method is based on minimizing a distance measure between the integral of the approximation function and the empirical cumulative distribution function (EDF) of the given samples, where the EDF is represented by the integral of the Dirac mixture. Since this minimization problem cannot be solved directly in general, a progression technique is applied. Increased performance of the approach in comparison to iterative maximum likelihood approaches is shown in simulations.

In nonlinear Bayesian estimation it is generally inevitable to incorporate approximate descriptions of the exact estimation algorithm. There are two possible ways to involve approximations: Approximating the nonlinear stochastic system model or approximating the prior probability density function. The key idea of the introduced novel estimator called Hybrid Density Filter relies on approximating the nonlinear system, thus approximating conditional densities. These densities nonlinearly relate the current system state to the future system state at predictions or to potential measurements at measurement updates. A hybrid density consisting of both Dirac delta functions and Gaussian densities is used for an optimal approximation. This paper addresses the optimization problem for treating the conditional density approximation. Furthermore, efficient estimation algorithms are derived based upon the special structure of the hybrid density, which yield a Gaussian mixture representation of the system state's density.

This paper addresses the problem of model-based reconstruction and parameter identification of distributed phenomena characterized by partial differential equations. The novelty of the proposed method is the systematic approach and the integrated treatment of uncertainties, which naturally occur in the physical system and arise from noisy measurements. The main challenge of accurate reconstruction is that model parameters, i.e., diffusion coefficients, of the physical model are not known in advance and usually need to be identified. Generally, the problem of parameter identification leads to a nonlinear estimation problem. Hence, a novel efficient recursive procedure is employed. Unlike other estimators, the so-called Hybrid Density Filter not only assures accurate estimation results for nonlinear systems, but also offers an efficient processing. By this means it is possible to reconstruct and identify distributed phenomena monitored by autonomous wireless sensor networks. The performance of the proposed estimation method is demonstrated by means of simulations.

In order to assist surgeons during surgery on moving organs, e.g. minimally invasive beating heart bypass surgery, a master-slave system which synchronizes surgical instruments with the organ's motion is desired. This synchronization requires reliable estimation of the organ's motion. In this paper, we present a new approach to motion estimation based on a state motion model for a partition of the heart's surface. Its motion behavior is described by a partial differential equation whose input function is assumed to be periodic. An estimator is used on one hand to predict future model states based on reconstruction of the input function and on the other hand to incorporate noisy spatially discrete measurements in order to improve state estimation. The model-based motion estimation is evaluated using a simple heart simulator. Measurements are obtained by reconstructing 3D position of markers on a pulsating membrane by means of a stereo camera system.

For several tasks in sensor networks, such as localization, information fusion,or sensor scheduling, Bayesian estimation is of paramount importance. Due to the limited computational and memory resources of the nodes in a sensor network, evaluation of the prediction step of the Bayesian estimator has to be performed very effciently. An exact and closed-form representation of the predicted probability density function of the system state is typically impossible to obtain, since exactly solving the prediction step for non-linear discrete-time dynamic systems in closed form is unfeasible. Assuming additive noise, we propose an accurate approximation of the predicted density, that can be calculated effciently by optimally approximating the transition density using a hybrid density. A hybrid density consists of two different density types: Dirac delta functions that cover the domain of the current density of the system state, and another density type, e.g. Gaussian densities, that cover the domain of the predicted density. The freely selectable, second density type of the hybrid density depends strongly on the noise affecting the nonlinear system. So, the proposed approximation framework for nonlinear prediction is not restricted to a specific noise density. It further allows an analytical evaluation of the Chapman-Kolmogorov prediction equation and can be interpreted as a deterministic sampling estimation approach. In contrast to methods using random sampling like particle filters, a dramatic reduction in the number of components and a subsequent decrease in computation time for approximating the predicted density is gained.

Intention recognition is an important topic in human-robot cooperation that can be tackled using probabilistic model-based methods. A popular instance of such methods are Bayesian networks where the dependencies between random variables are modeled by means of a directed graph. Bayesian networks are very efficient for treating networks with conditionally independent parts. Unfortunately, such independence sometimes has to be constructed by introducing so called hidden variables with an intractably large state space. An example are human actions which depend on human intentions and on other human actions. Our goal in this paper is to find models for intention-action mapping with a reduced state space in order to allow for tractable on-line evaluation. We present a systematic derivation of the reduced model and experimental results of recognizing the intention of a real human in a virtual environment.

For the collaborative control of a team of robots, a set of well-suited high-level control algorithms, especially for path planning and measurement scheduling, is essential. The quality of these control algorithms can be significantly increased by considering uncertainties that arise, e.g. from noisy measurements or system model abstraction, by incorporating stochastic filters into the control. To develop these kinds of algorithms and to prove their effectiveness, obviously real- world experiments with real world uncertainties are mandatory. Therefore, a test-environment for evaluating algorithms for collaborative control of a team of robots is presented. This test-environment is founded on miniature walking robots with six degrees of freedom. Their novel locomotion concept not only allows them to move in a wide variety of different motion patterns far beyond the possibilities of traditionally employed wheel-based robots, but also to handle real-world conditions like uneven ground or small obstacles. These robots are embedded in a modular test-environment, comprising infrastructure and simulation modules as well as a high-level control module with submodules for pose estimation, path planning, and measurement scheduling. The interaction of the individual modules of the introduced test-environment is illustrated by an experiment from the field of cooperative localization with focus on measurement scheduling, where the robots that perform distance measurements are selected based on a novel criterion, the normalized mutual Mahalanobis distance.

Recursive calculation of the probability density function characterizing the state estimate of a nonlinear stochastic dynamic system in general cannot be performed exactly, since the type of the density changes with every processing step and the complexity increases. Hence, an approximation of the true density is required. Instead of using a single complicated approximating density, this paper is concerned with bounding the true density from below and from above by means of two simple densities. This provides a kind of guaranteed estimator with respect to the underlying true density, which requires a mechanism for ordering densities. Here, a partial ordering with respect to the cumulative distributions is employed. Based on this partial ordering, a modified Bayesian filter step is proposed, which recursively propagates lower and upper density bounds. A specific implementation for piecewise linear densities with finite support is used for demonstrating the performance of the new approach in simulations

In this paper we attempt to lay the foundation for a novel filtering technique for the fusion of two random vectors with imprecisely known stochastic dependency. This problem mainly occurs in decentralized estimation, e.g., of a distributed phenomenon, where the stochastic dependencies between the individual states are not stored. Thus, we derive parameterized joint densities with both Gaussian marginals and Gaussian mixture marginals. These parameterized joint densities contain all information about the stochastic dependencies between their marginal densities in terms of a parameter vector xi, which can be regarded as a generalized correlation parameter. Unlike the classical correlation coefficient, this parameter is a sufficient measure for the stochastic dependency even characterized by more complex density functions such as Gaussian mixtures. Once this structure and the bounds of these parameters are known, bounding densities containing all possible density functions could be found

Undirected cycles in Bayesian networks are often treated by using clustering methods. This results in networks with nodes characterized by joint probability densities instead of marginal densities. An efficient representation of these hybrid joint densities is essential especially in nonlinear hybrid net works containing continuous as well as discrete variables. In this article we present a unified representation of continuous, discrete, and hybrid joint densities. This representation is based on Gaussian and Dirac mixtures and allows for analytic evaluation of arbitrary hybrid networks without loosing structural in formation, even for networks containing clusters. Furthermore we derive update formulae for marginal and joint densities from a system theoretic point of view by treating a Bayesian network as a system of cascaded subsystems. Together with the presented mixture representation of densities this yields an exact analytic updating scheme

Extended range telepresence allows a human user to intuitively teleoperate a mobile robot through arbitrarily large remote environments by natural walking. In order to give the user the possibility to navigate the robot through an arbitrarily large remote environments, while his own environment is of limited size, motion compression is used. The motion compression framework provides a nonlinear transformation between the user's path and the robot's path, which preserves path length and turning angles. There is, however, a difference in path curvature, which is minimized in order to guarantee a high degree of immersion. A major drawback of the current system is its inability to deal with non-convex time-variant environments or environments shared by multiple users. This paper presents a systematic approach to extending motion compression to non-convex environments. This solution will then be used to cover the multi-user case

For estimating the posture, i.e., position and orientation, of an extended target based on range measurements, a new closed-form solution is proposed; it is based on decoupling position and orientation. For decoupling, any procedure for range-based localization of point targets, i.e., for mere position estimation, can be used. The new solution is suboptimal, but nevertheless provides good accuracy and is very practical from an application point of view.

Calibration is the procedure of quantifying mechanical deficiencies of machines and compensating them by appropriate adjustment. This paper introduces a model-based measurement framework for improving calibration procedures of machine tools. The goal is to precisely estimate the mechanical deficiencies based on a minimal number of measurements. For that purpose, the mechanical deficiencies of linear and rotary joints are modeled using splines. Uncertainties of the deficiency model are formulated stochastically, which allows incorporation of imprecise measurement data and prediction of optimal measurement parameters. We derive a method for optimally estimating a set of splines, i.e., joint errors, based on a set of measurements and for predicting the optimal joint configuration for new measurements.

Telepresence aims at giving a human user the impression of being present in a remote environment. However, the user is actually situated in a user environment and his motion is tracked. A mobile teleoperator in the remote environment replicates this motion. The user can thus control the mobile teleoperator's locomotion by walking. A stereo-camera system mounted on the mobile teleoperator constantly records live camera images and transfers them to the user environment, where they are presented to the user on a head-mounted display. This paper presents a long distance experiment, in which a mobile teleoperator was controlled over a standard Internet connection by natural locomotion. Without further processing of the user's motion data, however, only exploration of a remote environment of the same size or smaller than the user environment is possible. As this is not desirable, we use motion compression, an optimal nonlinear transformation of the user's path. This algorithm allows controlling free motion in an arbitrarily large target environment from a limited user environment.

In this paper, the problem of recursive robot localization based on relative bearing measurements is considered, where unknown but bounded measurement uncertainties are assumed. A common approach is to approximate the resulting set of feasible states by simple-shaped bounding sets such as, e.g., axis-aligned boxes, and calculate the optimal parameters of this approximation based on the measurements and prior knowledge. In the novel approach presented here, a nonlinear transformation of the measurement equation into a higher dimensional space is performed. This yields a tight, possibly complex-shaped, bounding set in a closed-form representation whose parameters can be determined analytically for the measurement step. It is shown that the new bound is superior to commonly used outer bounds.

A new control scheme for uncalibrated robotic visual tracking problem is proposed that compromises the computational expenses of overall system with offline modeling and online control. A nonlinear visual mapping model for the uncalibrated hand-eye coordination is first proposed with an artificial neural network implementation. An online visual tracking controller is then developed together with a real-time motion planner. To improve the system performance, the control scheme is also integrated with a feedforward controller to compensate unknown object motions. Extensive simulations and experiments demonstrate the effectiveness of the proposed control scheme.

This article introduces an efficient approach for calculating the moments of exponential densities. Usually, the desired moments are obtained by means of numerical integration, which is impractical due to its computational complexity and the underlying infinite integration intervals. The new approach relies on an exact conversion of these integrals into a system of ordinary differential equations with algebraic constraints. The desired moments are then obtained by solving this system of differential algebraic equations over a finite "time" interval. The resulting procedure is simple to implement and typically reduces the computational burden by one order of magnitude.

This article deals with specific aspects concerning the visual perception process of a humanoid walking machine. An active vision system provides the information about the environment necessary for autonomous goal-oriented locomotion. Due to errors in each stage of the perception process, ideal environment reconstruction is not possible. By modeling these errors, stochastic components can be compensated using a hybrid extended Kalman filter approach with an alternating reference frame, thus reflecting the discontinuous character of biped walking.. The perception results improved by filtering can be used for the autonomous locomotion of the robot. Experiments with the walking machine BARt-UH demonstrate the validity of our approach.

Presents results for state estimation based on noisy observations suffering from two different types of uncertainties. The first uncertainty is a stochastic process with given statistics. The second uncertainty is only known to be bounded, the exact underlying statistics are unknown. State estimation tasks of this kind typically arise in target localization, navigation, and sensor data fusion. A new estimator has been developed, that combines set theoretic and stochastic estimation in a rigorous manner. The estimator is efficient and, hence, well-suited for practical applications. It provides a continuous transition between the two classical estimation, concepts, because it converges to a set theoretic estimator, when the stochastic error goes to zero, and to a Kalman filter, when the bounded error vanishes. In the mixed noise case, the new estimator provides solution sets that are uncertain in a statistical sense.

This paper presents a new approach for estimating the state of a linear dynamic system when two different types of uncertainties are present simultaneously. The first type of uncertainty is a stochastic process with given distribution. The second type of uncertainty is only known to be bounded, the exact underlying distribution is unknown. This includes inequality constraints between state variables, geometric tolerances, and bounded noise sources which are possibly correlated. For this generalized uncertainty model, a new recursive estimator has been developed. The new estimator unifies Kalman filtering and set theoretic filtering. It converges to a Kalman filter, when the bounded uncertainty goes to zero, and it converges to a set theoretic filter, when the stochastic noise vanishes. In the case of mixed uncertainties, the new estimator provides solution sets that are uncertain in a statistical sense.

Introduces a modular all-in-one wheel system for combined steering/driving applied to service robots operating in indoor environments like homes, offices, or hospitals. The wheel system is based on standard tires, builds very compact, and provides a high pay load to dead load ratio. It is used as a building block to construct a wide variety of highly maneuverable mobile robots ranging from simple tricycles to fully holonomic locomotion platforms. Integration is simple, because each wheel system is equipped with the necessary control and communication infrastructure based on standard PC components.

Introduces fast sampling strategies for pulse-echo time-of-flight sonar using time multiplexing. As a result, multiple interlaced measurement streams are employed to achieve high sampling rates without sacrificing range accuracy. The method considers receiver blanking constraints caused by imperfect acoustic decoupling between transmitter and receivers. Furthermore, uncertainties due to measurement noise and moving targets are taken into account. The proposed approach has been experimentally validated for sampling rates of up to 450 Hz with a multi sonar system mounted on a mobile robot.

The paper describes design issues of a mobile service robot for health care applications and domestic automation. Key components required for achieving semi-autonomous operation are surveyed, including: 1. a highly maneuverable locomotion platform, 2. an anthropomorphous manipulator, 3. a reliable multisensor system, and a 4. multimodal human-robot-interface. In addition, related information processing and control methodologies are presented. Special emphasis is put on a system architecture for integration of the individual components into a full-size service robot. Performance and usefulness of the proposed approaches are demonstrated through experiments in various real-world service scenarios.

An efficient noniterative algorithm for active localization of objects is developed. It is based on intersecting elliptic curves defined by uncertain range-sum measurements between a signal source, the objects, and a number of arbitrarily located receivers. Measurement errors are modelled as being unknown but bounded in amplitude by a closed convex set. Based on this set-theoretic uncertainty model, an error propagation analysis is performed, that allows one to accurately bound estimation errors. For discriminating object primitives and for discarding erroneous measurements, a hypothesis test is derived. The algorithm's computational load is much lower than for grid-based methods and iterative techniques. A recursive formulation is provided to support real-time applications.

Considers state estimation based on observations which are simultaneously corrupted by a deterministic amplitude-bounded unknown bias and a possibly unbounded random process. This problem is solved by developing a combined set theoretic and Bayesian recursive estimator. It provides a continuous transition between both concepts in that it converges to a set theoretic estimator when the stochastic error vanishes and to a Bayesian estimator when the deterministic error vanishes. In the mixed noise case, the new estimator supplies solution sets defined by bounds that are uncertain in a statistical sense.

We study the problem of localizing a mobile robot with an onboard-device making angular measurements on the location of known but indistinguishable landmarks. Novel algorithms are proposed for: 1) efficient posture initialization based on a simple linear solution; and 2) recursive posture estimation. Derived in a set theoretic framework, the algorithms cope with non-white, non-Gaussian noise and deterministic errors. Experiments with the set theoretic posture estimator demonstrate its simplicity and effectiveness in real-world applications.

A new type of actively sensing multisonar system for obstacle avoidance, localization and map updating in indoor environments has been developed. It is composed of 24 individual intelligent transmitter and receiver elements arranged on a horizontal plane around a fast motion platform. The physical sensor elements can be electronically configured for various types of task-directed and situation-adaptive virtual sensor arrays, for example, reliable measurement of wall orientation and distance or for tracking a walking person. The combination of virtual sensor data with a priori information about the mission and the environmental situation permits the sensor system to keep up with the data rate requirements of fast mobile platforms. Experimental tests with the developed multisonar system demonstrate its improved perception capabilities compared to the state-of-the-art.>