Robust Fusion of Position Data Abstract 1

Robust Fusion of Position Data Ruzena Bajcsy, Gerda Kamberova, Robert Mandelbaum, and Max Mintz GRASP Laboratory, Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19146 [email protected], [email protected], [email protected], [email protected]

Abstract

plete body of research (we call it, decision-theoretic approach to sensor fusion), is applicable: (1) where there is signi cant uncertainty in the underlying system and sensor noise behavior; (2) where there are known restrictions on the work space; and (3) where the estimation procedures are required to achieve hard probabilistic performance bounds. Further, since the estimates are generally used in accomplishing some overall goal, e.g., multiagent motion coordination, it is important that the estimation modules provide performance guarantees which are amenable to these higherlevel tasks without the imposition of strong assumptions on the underlying sensor error distributions, as well as an accurate prior distribution for the uncertain parameters. Tight performance bounds are critical components in the overall system operation and evaluation. In contrast, the popular moment-based methods such as mean squared error do not provide tight performance bounds. The decision-theoretic approach to sensor fusion was pioneered in the GRASP laboratory, at the University of Pennsylvania. The active sensing/vision paradigm has been in the main focus of the research in the laboratory [1]. The theoretical research was conducted by Max Mintz and his graduate students, Zeytinoglu [13, 14], Martin [10], McKendall [11], Kennedy [7] and Kamberova [5]. Mandelbaum [8] presents the rst application of this theory to a mobile robotic setting, where the minimax con dence set estimation (MCSE) is used for mobile robot pose estimation. In the current paper some of the experimental results are highlighted and the method is compared with other existing techniques. The empirical results accurately match those predicted by the theory, and dominate the performance of the best linear procedures which have been most often used. The ability to make these accurate predictions of system performance is of critical importance, since this makes it possible to meet tight design speci cations.

In this paper we present a decision-theoretic approach to location data fusion developed in the GRASP Laboratory. We give the statistical background, theoretical foundations and an engineering application. The theory is centered around the problem of deriving xedsize con dence intervals with guaranteed probability of capture for a location parameter. These intervals are based on (robust) minimax rules. The theoretical results are general and could be used in a variety of applications. The engineering implementation presented in this paper is the rst application of the theory to the mobile robotics domain. Given an input of mobile robot pose measurements by a sensor-based localization algorithm, the system produces a minimax risk xed-size con dence set for the pose of the agent. The approach is evaluated in terms of theoretical capture probability and empirical capture frequency during actual experiments with the mobile agent. The method is compared to several other procedures including the Kalman Filter (minimum mean squared error estimate) and the Maximum Likelihood Estimator (MLE). The minimax approach, in this application, is found to dominate all the other methods in performance.

1 Introduction We illustrate a decision-theoretic approach to location data fusion. We describe the statistical background, theoretical foundations (con dence sets based on robust minimax rules), and an engineering application (pose estimation of a mobile robot). We are interested in the of estimating parameter values from noisy sensory measurements. The measurements may be taken from multiple sensors over time, thus the problem of fusing consistent data robustly arises. The com1

Sensor errors can be modeled statistically, using both physical theory and empirical data. A sensor measurement often can be represented as a random variable Z =  + V where V is random noise (error) with a distribution (CDF) F, and  is the parameter of interest. We design optimal, in the minimax sense, decision rules for dealing with the uncertainties. These rules lead to con dence intervals for  with guaranteed probabilities of capture. here we give some of the results from [5]. In developing the noise models, i.e., F, one recognizes that a single distribution is usually an inadequate description of sensor noise behavior. It is much more realistic and much safer to identify an envelope or class of distributions, one of whose members could reasonably represent the actual statistical behavior of a given sensor. The use of an uncertainty class in distribution space protects the system designer against the inevitable unpredictable changes which occur in sensor behavior. In [5] we look at three dierent types of uncertainty classes and design robust1 minimax decision rules. The theory we address applies to a variety of uncertainty classes, which allows one to account for very general deviations from the nominal noise distribution. In this paper we present a tail-behavior uncertainty class which is used in the experiment. We develop the theory in the univariate case when  is scalar. In the multivariate case, if we assume that the individual components of the multivariate parameter are independent, we can apply the univariate analysis component-wise and then consider the con dence set which is the Cartesian product of the con dence intervals for the components [5]. The research on the general multivariate approach is currently under way [3]. Organization by Sections: (2) reviews basic concepts from Statistical Decision Theory (SDT); (3) gives theoretical results from [5] related to the deliniatation of robust minimax rules; (4) presents an application of the approach to robust pose estimation [8] and evaluates it in comparison with other approaches; (5) contains concluding remarks. For more comprehensive presentation { theory and application { see Mandelbaum, Kamberova and Mintz [9].

2 SDT { Basic Concepts

This section introduces decision-theoretic concepts and notation. See [2] and [4] for a more detailed intro1 Robustness refers to the statistical eectiveness of the decision rule when the noise distribution is uncertain.

duction to the subject of statistical-decision theory. Consider the location data model Z =  +V where Z is a random variable representing the measurement,  is the (unknown) true value of a parameter we wish to estimate, and V is random noise. We denote an observation (measurement) by z, z 2