This paper presents a performance comparison of different estimation and prediction techniques applied to the problem of tracking multiple robots. The main performance criteria are the magnitude of the estimation or prediction error, the computational effort and the robustness of each method to non-Gaussian noise. Among the different techniques compared are the well known Kalman filters and their different variants (e.g. extended and unscented), and the more recent techniques relying on Sequential Monte Carlo Sampling methods, such as particle filters and Gaussian Mixture Sigma Point Particle Filter.
Filtering and prediction methods to accurately track multiple moving objects under noisy measurements, disturbances and model uncertainty have received much attention during the last decades in a wide variety of fields, such as aeronautics, defense, computer vision, robotics and economics, to name a few [1][2][3][4][5][6][7][8] . It is a well-known fact that the ability of any control or decision system to meet its targets directly depends on accuracy of the measurements and the state estimation 9 . On the other hand, when the measurements of the controlled variables are available with some delay, the ability to make accurate predictions of the measurements becomes particularly important in order to compensate the measurements' latency [10][11][12] .
In view of the large number of existing approaches (see references in table 1) and the key role of filtering and estimation techniques for accurate tracking, it is important to identify those techniques that have the best performance, not only in terms of accuracy, but also in terms of their ability to handle non-Gaussian noise and their computational cost. The aim of the present work is thus to evaluate the performance of classic filtering techniques for trajectory prediction and perception-latency compensation, such as the Kalman Filter (KF) 15 and the Extended Kalman Filter (EKF) 16 , as well as more recent ones, such as the Unscented Kalman Filter (UKF) 17 , Particle Filtering (PF) 2 and other variants in the class of the so-called Sigma-Point Kalman Filters (SPKF) 10,18,19 . The prediction strategies compared in this paper, which are summarized in table 1, can be grouped into two main families: (i) the Kalman Filter based approaches 15,16,17,19,18,10,20 and (ii) the sequential sampling or Particle Filter based approaches 2,7,21,22 Although there exist many studies and comparisons in the literature 3,10,18,24,25,26 , most of these compare just a few approaches and in general do not consider directly issues related to the robustness of the methods to non-Gaussian noise. The Gaussianity of the disturbances is a common assumption that often simplifies analytic developments, but is far from the actual characteristis of many real systems. In fact, some of the existing algorithms yield poor estimates or simply diverge when the Gaussianity assumption is not satisfied, while other require a significantly large computation time that may render them unpractical for real-time control and decision applications 10 . The KF and EKF are unarguably the most popular prediction approaches 2 . However, both rely on a first order Taylor series approximation of the nonlinear state space model of the system. Furthermore, both assume that process and measurement noises are identically and independently Gaussian distributed. Because of these assumptions, the KF or EKF perform poorly when the system dynamics is highly nonlinear and the system noises are non-Gaussian. These limitations have motivated the development of the Sigma-Point Kalman Filters (SPKF) 10 and the rebirth of the computationally intensive Sequential Monte Carlo Sampling methods 7 aided by the availability of more powerful computers.
The main contribution of this paper is the evaluation and comparison of the performance of the ten different filtering strategies listed in table 1 as applied to a non-linear system with measurements subject to non-Gaussian noise. The performance of each filter is measured in terms of the root mean squared-error of the filter’s prediction error, the filter’s effectiveness to handle non-Gaussian noise and its computational cost. Concisely, but in a precise manner, the paper also presents the general algorithm of the KF-based and the PF-based filtering approaches. The system employed as test bench is an ensemble of fast moving RoboCup F-180 robots 11 . The performance of the filters is first evaluated through numerical simulations of the robots’ trajectory prediction. The best performing approach is then tested with the real robots. In this context, the performance is discussed in terms of the ability of the estimation strategies to filter perception errors and compensate perception delays in real-time. The results are valuable not only to applications related to robotic soccer competitions, but to a wide range of fields characterized by non-linear dynamics/non-Gaussian disturbances that require accurate state trajectory predictions and the compensation of perception and computation delays. Examples of these fields include the process control industry 9 , the visual tracking applications 27,28 and the autonomous localization and navigation 10,12,13 .
The paper is organized as follows. Section 2 presents the mathematical background on which the KF-based and PF-based filtering approaches rely. The derivation of the kinematic and dynamic equations of a three-wheel omni-directional robot employed in the filters implementation is presented in section 3. The testing methodology and results
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