Dietrich Brunn

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

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

Vision and tactile sensing for real world tasks

Robotic fetch-and-carry tasks are commonly facilitated to demonstrate a number of research directions such as navigation, mobile manipulation, systems integration, etc. As a part of an integrated system in terms of a service robot framework, this paper describes a set of methods for real-world object manipulation tasks. We concentrate here on two particular parts of a manipulation sequence: i) robust visual servoing, and ii) grasping strategies. In terms of visual servoing we discuss the handling of singularities during a manipulation sequence. For grasping, we present a biologically motivated strategy using tactile feedback.

Closed-Form Prediction of Nonlinear Dynamic Systems by Means of Gaussian Mixture Approximation of the Transition Density

Recursive prediction of the state of a nonlinear stochastic dynamic system cannot be efficiently performed in general, since the complexity of the probability density function characterizing the system state increases with every prediction step. Thus, representing the density in an exact closed-form manner is too complex or even impossible. So, an appropriate approximation of the density is required. Instead of directly approximating the predicted density, we propose the approximation of the transition density by means of Gaussian mixtures. We treat the approximation task as an optimization problem that is solved offline via progressive processing to bypass initialization problems and to achieve high quality approximations. Once having calculated the transition density approximation offline, prediction can be performed efficiently resulting in a closed-form density representation with constant complexity

Nonlinear Multidimensional Bayesian Estimation with Fourier Densities

Efficiently implementing nonlinear Bayesian estimators is still an unsolved problem, especially for the multidimensional case. A trade-off between estimation quality and demand on computational resources has to be found. Using multidimensional Fourier series as representation for probability density functions, so called Fourier densities, is proposed. To ensure non-negativity, the approximation is performed indirectly via Psi-densities, of which the absolute square represent the Fourier density. It is shown that Psi-densities can be determined using the efficient fast Fourier transform algorithm and their coefficients have an ordering with respect to the Hellinger metric. Furthermore, the multidimensional Bayesian estimator based on Fourier densities is derived in closed form. That allows an efficient realization of the Bayesian estimator where the demands on computational resources are adjustable

Efficient Nonlinear Measurement Updating based on Gaussian Mixture Approximation of Conditional Densities

Filtering or measurement updating for nonlinear stochastic dynamic systems requires approximate calculations, since an exact solution is impossible to obtain in general. We propose a Gaussian mixture approximation of the conditional density, which allows performing measurement updating in closed form. The conditional density is a probabilistic representation of the nonlinear system and depends on the random variable of the measurement given the system state. Unlike the likelihood, the conditional density is independent of actual measurements, which permits determining its approximation off-line. By treating the approximation task as an optimization problem, we use progressive processing to achieve high quality results. Once having calculated the conditional density, the likelihood can be determined on-line, which, in turn, offers an efficient approximate filter step. As result, a Gaussian mixture representation of the posterior density is obtained. The exponential growth of Gaussian mixture components resulting from repeated filtering is avoided implicitly by the prediction step using the proposed techniques.

Finite-Horizon Optimal State-Feedback Control of Nonlinear Stochastic Systems Based on a Minimum Principle

In this paper, an approach to the finite-horizon optimal state-feedback control problem of nonlinear, stochastic, discrete-time systems is presented. Starting from the dynamic programming equation, the value function will be approximated by means of Taylor series expansion up to second-order derivatives. Moreover, the problem will be reformulated, such that a minimum principle can be applied to the stochastic problem. Employing this minimum principle, the optimal control problem can be rewritten as a two-point boundary-value problem to be solved at each time step of a shrinking horizon. To avoid numerical problems, the two-point boundary-value problem will be solved by means of a continuation method. Thus, the curse of dimensionality of dynamic programming is avoided, and good candidates for the optimal state-feedback controls are obtained. The proposed approach will be evaluated by means of a scalar example system

Efficient Nonlinear Bayesian Estimation based on Fourier Densities

Efficiently implementing nonlinear Bayesian estimators is still not a fully solved problem. For practical applications, a trade-off between estimation quality and demand on computational resources has to be found. In this paper, the use of nonnegative Fourier series, so-called Fourier densities, for Bayesian estimation is proposed. By using the absolute square of Fourier series for the density representation, it is ensured that the density stays nonnegative. Nonetheless, approximation of arbitrary probability density functions can be made by using the Fourier integral formula. An efficient Bayesian estimator algorithm with constant complexity for nonnegative Fourier series is derived and demonstrated by means of an example

Parameterized Joint Densities with Gaussian and Gaussian Mixture Marginals

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

Approximate Nonlinear Bayesian Estimation Based on Lower and Upper Densities

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