Oliver C. Schrempf

A novel approach to proactive human-robot cooperation

This paper introduces the concept of proactive execution of robot tasks in the context of human-robot cooperation with uncertain knowledge of the humans intentions. We present a system architecture that defines the necessary modules of the robot and their interactions with each other. The two key modules are the intention recognition that determines the human users intentions and the planner that executes the appropriate tasks based on those intentions. We show how planning conflicts due to the uncertainty of the intention information are resolved by proactive execution of the corresponding task that optimally reduces the systems uncertainly. Finally, we present an algorithm for selecting this task and suggest a benchmark scenario.

Tractable probabilistic models for intention recognition based on expert knowledge

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.

Density Approximation Based on Dirac Mixtures with Regard to Nonlinear Estimation and Filtering

A deterministic procedure for optimal approximation of arbitrary probability density functions by means of Dirac mixtures with equal weights is proposed. The optimality of this approximation is guaranteed by minimizing the distance of the approximation from the true density. For this purpose a distance measure is required, which is in general not well defined for Dirac mixtures. Hence, a key contribution is to compare the corresponding cumulative distribution functions. This paper concentrates on the simple and intuitive integral quadratic distance measure. For the special case of a Dirac mixture with equally weighted components, closed-form solutions for special types of densities like uniform and Gaussian densities are obtained. Closed-form solution of the given optimization problem is not possible in general. Hence, another key contribution is an efficient solution procedure for arbitrary true densities based on a homotopy continuation approach. In contrast to standard Monte Carlo techniques like particle filters that are based on random sampling, the proposed approach is deterministic and ensures an optimal approximation with respect to a given distance measure. In addition, the number of required components (particles) can easily be deduced by application of the proposed distance measure. The resulting approximations can be used as basis for recursive nonlinear filtering mechanism alternative to Monte Carlo methods

Recursive Prediction of Stochastic Nonlinear Systems Based on Optimal Dirac Mixture Approximations

This paper introduces a new approach to the recursive propagation of probability density functions through discrete-time stochastic nonlinear dynamic systems. An efficient recursive procedure is proposed that is based on the optimal approximation of the posterior densities after each prediction step by means of Dirac mixtures. The parameters of the individual components are selected by systematically minimizing a suitable distance measure in such a way that the future evolution of the approximate densities is as close to the exact densities as possible.

Dirac Mixture Density Approximation Based on Minimization of the Weighted Cramer-von Mises Distance

This paper proposes a systematic procedure for approximating arbitrary probability density functions by means of Dirac mixtures. For that purpose, a distance measure is required, which is in general not well defined for Dirac mixture densities. Hence, a distance measure comparing the corresponding cumulative distribution functions is employed. Here, we focus on the weighted Cramer-von Mises distance, a weighted integral quadratic distance measure, which is simple and intuitive. Since a closed-form solution of the given optimization problem is not possible in general, an efficient solution procedure based on a homotopy continuation approach is proposed. Compared to a standard particle approximation, the proposed procedure ensures an optimal approximation with respect to a given distance measure. Although useful in their own respect, the results also provide the basis for a recursive nonlinear filtering mechanism as an alternative to the popular particle filters

Greedy algorithms for dirac mixture approximation of arbitrary probability density functions

Greedy procedures for suboptimal Dirac mixture approximation of an arbitrary probability density function are proposed, which approach the desired density by sequentially adding one component at a time. Similar to the batch solutions proposed earlier, a distance measure between the corresponding cumulative distributions is minimized by selecting the corresponding density parameters. This is due to the fact, that a distance between the densities is typically not well defined for Dirac mixtures. This paper focuses on the Cramer-von Mises distance, a weighted integral quadratic distance measure between the true distribution and its approximation. In contrast to the batch solutions, the computational complexity is much lower and grows only linearly with the number of components. Computational savings are even greater, when the required number of components, e.g., the minimum number of components for achieving a given quality measure, is not a priori known and must be searched for as well. The performance of the proposed sequential approximation approach is compared to that of the optimal batch solution.

Optimal mixture approximation of the product of mixtures

Gaussian mixture densities are a very common tool for describing arbitrarily structured uncertainties in various applications. Many of these applications have to deal with the fusion of uncertainties, an operation that is usually performed by multiplication of these densities. The product of Gaussian mixtures can be calculated exactly, but the number of mixture components in the resulting mixture increases exponentially. Hence, it is essential to approximate the resulting mixture with less components, to keep it tractable for further processing steps. This paper introduces an approach for approximating the exact product with a mixture that uses less components. The maximum approximation error can be chosen by the user. This choice allows to trade accuracy of the approximation for the number of mixture components used. This is possible due to the usage of a progressive processing scheme that calculates the product operation by means of a system of ordinary differential equations. The solution of this system yields the parameters of the desired Gaussian mixture.

EVALUATION OF HYBRID BAYESIAN NETWORKS USING ANALYTICAL DENSITY REPRESENTATIONS

Abstract In this article, a new mechanism is described for modeling and evaluating Hybrid Dynamic Bayesian networks. The approach uses Gaussian mixtures and Dirac mixtures as messages to calculate marginal densities. As these densities are approximated by means of Gaussian mixtures, any desired precision is possible. The presented approach removes the restrictions of sample based evaluation of Bayesian networks since it uses an analytical approximation scheme for probability densities which systematically minimizes the distance between the exact and the approximate density.

Optimal parametric density estimation by minimizing an analytic distance measure

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.

Efficient Representation and Fusion of Hybrid Joint Densities for Clusters in Nonlinear Hybrid Bayesian Networks

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