©2001 CRC Press LLC The results from the first strategy (no data distribution) are shown in Figure 12.7. As expected, the system behaves poorly. Because each node operates in isolation, only Node 1 (which measures x) is fully observable. The position variance increases without bound for the three remaining nodes. Similarly, the velocity is observable for Nodes 1, 2, and 4, but it is not observable for Node 3. The results of the second strategy (all nodes are assumed independent) are shown in Figure 12.8. The effect of assumed independence observations is obvious: all of the estimates for all of the states in all of the nodes (apart from x for Node 3) are inconsistent. This clearly illustrates the problem of double counting. Finally, the results from the CI distribution scheme are shown in Figure 12.9. Unlike the other two approaches, all the nodes are consistent and observable. Furthermore, as the results in Table 12.2 indicate, the steady-state covariances of all of the states in all of the nodes are smaller than those for case 1. In other words, this example shows that this data distribution scheme successfully and usefully propagates data through an apparently degenerate data network. FIGURE 12.7 Disconnected nodes. (A) Mean squared error in x. (B) Mean squared error in · x. (C) Mean squared error in ·· x. Mean squared errors and estimated covariances for all states in each of the four nodes. The curves for Node 1 are solid, Node 2 are dashed, Node 3 are dotted, and Node 4 are dash-dotted. The mean squared error is the rougher of the two lines for each node. 0 10 20 30 40 (A) 50 60 70 80 90 100 0 100 200 300 400 500 600 700 800 900 1000 Average MSE x(1) estimate 0 10 20 30 40 (B) 50 60 70 80 90 100 0 2 4 6 8 10 12 Average MSE x(2) estimate ©2001 CRC Press LLC 13 Data Fusion in Nonlinear Systems 13.1 Introduction 13.2 Estimation in Nonlinear Systems Problem Statement • The Transformation of Uncertainty 13.3 The Unscented Transformation (UT) The Basic Idea • An Example Set of Sigma Points • Properties of the Unscented Transformation 13.4 Uses of the Transformation Polar to Cartesian Coordinates • A Discontinuous Transformation 13.5 The Unscented Filter (UF) 13.6 Case Study: Using the UF with Linearization Errors 13.7 Case Study: Using the UF with a High-Order Nonlinear System 13.8 Multilevel Sensor Fusion 13.9 Conclusions Acknowledgments References 13.1 Introduction The extended Kalman filter (EKF) has been one of the most widely used methods for tracking and estimation based on its apparent simplicity, optimality, tractability, and robustness. However, after more than 30 years of experience with it, the tracking and control community has concluded that the EKF is difficult to implement, difficult to time, and only reliable for systems that are almost linear on the time scale of the update intervals. This chapter reviews the unscented transformation (UT), a mechanism for propagating mean and covariance information through nonlinear transformations, and describes its implications for data fusion. This method is more accurate, is easier to implement, and uses the same order of calculations as the EKF. Furthermore, the UT permits the use of Kalman-type filters in appli- cations where, traditionally, their use was not possible. For example, the UT can be used to rigorously integrate artificial intelligence-based systems with Kalman-based systems. Performing data fusion requires estimates of the state of a system to be converted to a common representation. The mean and covariance representation is the lingua franca of modern systems engi- neering. In particular, the covariance intersection (CI) 1 and Kalman filter (KF) 2 algorithms provide mechanisms for fusing state estimates defined in terms of means and covariances, where each mean vector defines the nominal state of the system and its associated error covariance matrix defines a lower bound on the squared error. However, most data fusion applications require the fusion of mean and covariance estimates defining the state of a system in different coordinate frames. For example, a tracking Simon Julier IDAK Industries Jeffrey K. Uhlmann University of Missouri ©2001 CRC Press LLC 14 Random Set Theory for Target Tracking and Identification 14.1 Introduction The “Bayesian Iceberg”: Models, Optimality, Computability • Why Multisource, Multitarget, Multi-Evidence Problems Are Tricky • Finite-Set Statistics (FISST) • Why Random Sets? 14.2 Basic Statistics for Tracking and Identification Bayes Recursive Filtering • Constructing Likelihood Functions from Sensor Models • Constructing Markov Densities from Motion Models • Optimal State Estimators 14.3 Multitarget Sensor Models Case I: No Missed Detections, No False Alarms • Case II: Missed Detections • Case III: Missed Detection and False Alarms • Case IV: Multiple Sensors 14.4 Multitarget Motion Models Case I: Target Number Does Not Change • Case II: Target Number Can Decrease • Case III: Target Number Can Increase and Decrease 14.5 The FISST Multisource-Multitarget Calculus The Belief-Mass Function of a Sensor Model • The Belief- Mass Function of a Motion Model • The Set Integral and Set Derivative • “Turn-the-Crank” Rules for the FISST Calculus 14.6 FISST Multisource-Multitarget Statistics Constructing True Multitarget Likelihood Functions • Constructing True Multitarget Markov Densities • Multitarget Prior Distributions • Multitarget Posterior Distributions • Expected Values and Covariances • The Failure of the Classical State Estimators • Optimal Multitarget State Estimators • Cramér-Rao Bounds for Multitarget State Estimators • Multitarget Miss Distance 14.7 Optimal-Bayes Fusion, Tracking, ID Multisensor-Multitarget Filtering Equations • A Short History of Multitarget Filtering • Computational Issues in Multitarget Filtering • Optimal Sensor Management 14.8 Robust-Bayes Fusion, Tracking, ID Random Set Models of “Ambiguous” Data • Forms of Ambiguous Data • True Likelihood Functions for Ambiguous Data • Generalized Likelihood Functions • Posteriors Conditioned on Ambiguous Data • Practical Generalized Likelihood Functions • Unified Multisource-Multitarget Data Fusion Ronald Mahler Lockheed Martin . Set Models of “Ambiguous” Data • Forms of Ambiguous Data • True Likelihood Functions for Ambiguous Data • Generalized Likelihood Functions • Posteriors Conditioned on Ambiguous Data • Practical. systems. Performing data fusion requires estimates of the state of a system to be converted to a common representation. The mean and covariance representation is the lingua franca of modern systems. covariances of all of the states in all of the nodes are smaller than those for case 1. In other words, this example shows that this data distribution scheme successfully and usefully propagates data