Use of non-linearity as a characteristic in the selection of filtering algorithms in kinematic positioning

Abstract

Selection of an optimal filtering algorithm for kinematic positioning systems constitutes one of the most extensively studied applications in the surveyor engineering community. The ability of a filtering algorithm is often assessed through its performance. The performance of a filtering algorithm is frequently evaluated in terms of accuracy and computational time. According to the accuracy parameter, it is often determined by a comparison between true trajectory and the estimated one from an algorithm. However, the true trajectory is commonly unknown in real-life situations, and thus the accuracy of the filtering algorithm cannot be assessed in this manner. Indeed, lack of true trajectory is one of the primary obstacles in the evaluation of the performance of filtering algorithms. The non-linearity of the model, on the other hand, can be determined without any information about the true trajectory and is also associated with the abilities of algorithms. So far, however, very little attention has been paid to the role of the decision of filtering algorithms based on non-linearity. Thus, this study proposes an alternative characteristic in the assessment of the performance of filtering algorithms, which is the non-linearity of the observation model. This research aims to assess the ability of non-linear characteristic for the choice of an optimal filtering algorithm. In this research, the data are simulated by the Monte Carlo method. The abilities of filtering algorithms are investigated on the extended Kalman filter (EKF), unscented Kalman filter (UKF), and particle filter (PF). These algorithms are widely utilized in kinematic positioning, and they are appropriate for various levels of non-linearity. The current study evaluated the influence of the algorithm’s accuracy on three factors: measurement uncertainty, observation geometry, and the number of observations. These algorithms are also assessed on their computational times according to a certain scenario. Regarding measures of non-linearity, three different indicators are examined for the non-linearity of both system and observation models. The coefficient of determination, 1-R2, is utilized as a single indicator to measure the non-linearity of each function of the above models. The M and 1-MVA, known as the deviation of a non-linear function from linearity and multivariate association, respectively, can be used as indicators to quantify the non-linearity of numerous functions of the above models jointly. The 1-MVA indicator is proposed for the first time to quantify the non-linearity of models. From analyses of the accuracy and non-linearity, the relationship between them is determined with changing measurement uncertainty and observation geometry in several scenarios. Based on the established relationship between accuracy and non-linearity, the choice of an optimal algorithm is analyzed through numerical examples. These results indicate that the accuracy of these algorithms is strongly influenced by measurement uncertainty, observation geometry, and the number of observations. The accuracy obtained by PF is higher than that of UKF and EKF. Conversely, the computational time of EKF is shorter than that of UKF and PF. According to measures of non-linearity, the above-proposed indicators are suitable, and the tendency of non-linearity of a model obtained by these indicators is the same. The non-linearity of the system model is small due to the given small amount of standard deviations of the disturbance quantities. Inversely, the non-linearity of the observation model is high due to high measurement uncertainties, or poor observation geometries. The main finding of this research is that both non-linearity of the observation model and position accuracy are influenced by factors of measurement uncertainty and observation geometry. Therefore, the relationship between the position accuracy and the non-linearity of the observation model is established based on these factors. This relationship is strong, which is assessed by the goodness-of-fit value of the best fitting function. In addition, another important result from the present research is that the fitting function described for this relationship changes due to influencing factors of scenarios. The established relationship constitutes the main limitation of this characteristic in application. As a result, instead of accuracy, the non-linearity of the observation model can be employed for the assessment of algorithms when the true trajectory is not available. However, the optimal algorithm can only be selected using these factors in some special cases. For a general case of arbitrary scenarios’ factors, the non-linear characteristic cannot be used for this purpose.