Florian Pfaff

Optimal Kalman gains for combined stochastic and set-membership state estimation

In state estimation theory, two directions are mainly followed in order to model disturbances and errors. Either uncertainties are modeled as stochastic quantities or they are characterized by their membership to a set. Both approaches have distinct advantages and disadvantages making each one inherently better suited to model different sources of estimation uncertainty. This paper is dedicated to the task of combining stochastic and set-membership estimation methods. A Kalman gain is derived that minimizes the mean squared error in the presence of both stochastic and additional unknown but bounded uncertainties, which are represented by Gaussian random variables and ellipsoidal sets, respectively. As a result, a generalization of the well-known Kalman filtering scheme is attained that reduces to the standard Kalman filter in the absence of set-membership uncertainty and that otherwise becomes the intersection of sets in case of vanishing stochastic uncertainty. The proposed concept also allows to prioritize either the minimization of the stochastic uncertainty or the minimization of the set-membership uncertainty.

TrackSort: Predictive tracking for sorting uncooperative bulk materials

Optical belt sorters are a versatile, state-of-the-art technology to sort bulk materials that are hard to sort based on only nonvisual properties. In this paper, we propose an extension to current optical belt sorters that involves replacing the line camera with an area camera to observe a wider field of view, allowing us to observe each particle over multiple time steps. By performing multitarget tracking, we are able to improve the prediction of each particles movement and thus enhance the performance of the utilized separation mechanism. We show that our approach will allow belt sorters to handle new classes of bulk materials while improving cost efficiency. Furthermore, we lay out additional extensions that are made possible by our new paradigm.

Discrete recursive Bayesian filtering on intervals and the unit circle

Various applications necessitate the estimation of quantities defined on intervals or the unit circle, which can also be parameterized as an interval. These applications include estimation of joint angles that are either limited to a certain range or that are 360-degree-periodic. For this purpose, we consider two approaches based on discretizing the state space that use fundamentally different density representations. We show how prediction and measurement update for systems with nonlinear dynamics and nonlinear measurement models can be performed in each representation. In particular, we discuss the choices that go into designing discrete filters, which are sometimes taken for granted. A thorough comparison and a numerical evaluation of both approaches show the advantages and disadvantages of each method.

Fast multitarget tracking via strategy switching for sensor-based sorting

State-of-the-art sensor-based sorting systems provide solutions to sort various products according to quality aspects. Such systems face the challenge of an existing delay between perception and separation of the material. To reliably predict an objects position when reaching the separation stage, information regarding its movement needs to be derived. Multitarget tracking offers approaches through which this can be achieved. However, processing time is typically limited since the sorting decision for each object needs to be derived sufficiently early before it reaches the separation stage. In this paper, an approach for multitarget tracking in sensor-based sorting is proposed which supports establishing an upper bound regarding processing time required for solving the measurement to track association problem. To demonstrate the success of the proposed method, experiments are conducted for data-sets obtained via simulation of a sorting system. This way, it is possible to not only demonstrate the impact on required runtime but also on the quality of the association.

Simulation-based evaluation of predictive tracking for sorting bulk materials

Multitarget tracking problems arise in many real-world applications. The performance of the utilized algorithm strongly depends both on how the data association problem is handled and on the suitability of the motion models employed. Especially the motion models can be hard to validate. Previously, we have proposed to use multitarget tracking to improve optical belt sorters. In this paper, we evaluate both the suitability of our model and the tracking and then of our entire system incorporating the image processing component via the use of highly realistic numerical simulations. We first assess the model using noise-free measurements generated by the simulation and then evaluate the entire system by using synthetically generated image data.

Real-time whole-body human motion tracking based on unlabeled markers

In this paper, we present a novel online approach for tracking whole-body human motion based on unlabeled measurements of markers attached to the body. For that purpose, we employ a given kinematic model of the human body including the locations of the attached markers. Based on the model, we apply a combination of constrained sample-based Kalman filtering and multi-target tracking techniques: 1) joint constraints imposed by the human body are satisfied by introducing a parameter transformation based on periodic functions, 2) a global nearest neighbor (GNN) algorithm computes the most likely one-to-one association between markers and measurements, and 3) multiple hypotheses tracking (MHT) allows for a robust initialization that only requires an upright standing user. Evaluations clearly demonstrate that the proposed tracking provides highly accurate pose estimates in realtime, even for fast and complex motions. In addition, it provides robustness to partial occlusion of markers and also handles unavoidable clutter measurements.

Application of Discrete Recursive Bayesian Estimation on Intervals and the Unit Circle to Filtering on SE(2)

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.

Information form distributed Kalman filtering (IDKF) with explicit inputs

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.

Optimal distributed combined stochastic and set-membership state estimation

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.

Real-time multitarget tracking for sensor-based sorting

Utilizing parallel algorithms is an established way of increasing performance in systems that are bound to real-time restrictions. Sensor-based sorting is a machine vision application for which firm real-time requirements need to be respected in order to reliably remove potentially harmful entities from a material feed. Recently, employing a predictive tracking approach using multitarget tracking in order to decrease the error in the physical separation in optical sorting has been proposed. For implementations that use hard associations between measurements and tracks, a linear assignment problem has to be solved for each frame recorded by a camera. The auction algorithm can be utilized for this purpose, which also has the advantage of being well suited for parallel architectures. In this paper, an improved implementation of this algorithm for a graphics processing unit (GPU) is presented. The resulting algorithm is implemented in both an OpenCL and a CUDA based environment. By using an optimized data structure, the presented algorithm outperforms recently proposed implementations in terms of speed while retaining the quality of output of the algorithm. Furthermore, memory requirements are significantly decreased, which is important for embedded systems. Experimental results are provided for two different GPUs and six datasets. It is shown that the proposed approach is of particular interest for applications dealing with comparatively large problem sizes.