Event-triggered cooperative unscented Kalman filtering and its application in multi-UAV systems

Abstract

This paper proposes a novel consensus-based distributed unscented Kalman filtering algorithm with event-triggered communication mechanisms. With such an algorithm, each sensor node transmits the newest measurement to the corresponding remote estimator selectively on the basis of its own event-triggering condition. Compared to the existing approaches, the proposed algorithm can significantly reduce unnecessary data transmissions and hence save communication energy consumption and alleviate the communication burden. A sufficient condition is provided to guarantee the stochastic stability of the distributed nonlinear filtering scheme. The proposed algorithm is applicable to a wide range of distributed estimation tasks, e.g., tracking a moving target with multiple unmanned aerial vehicles (UAVs). Simulation results demonstrate the feasibility and effectiveness of the proposed filtering algorithm.

Introduction

The past decades have witnessed ever-increasing research attention devoted to the sensor networks due to its extensive applications in many fields such as battlefield detection, environment monitoring, information processing, autonomous navigation, target tracking and localization. One of the most important problems in sensor networks is designing the functional filtering algorithms to estimate the state of the target process (Olfati-Saber, 2009). Compared with the centralized setting, distributed state estimation without requiring information center has several advantages including stronger fault-tolerance, less computational and communication loads. The central issue in distributed state estimation is to cooperatively estimate the states of a dynamic system via a wireless sensor network with given communication topology. Specifically, each node in such a distributed framework only needs to share information with its neighbors over networks. As a popular approach to address distributed state estimation problem, consensus-based methodologies have made significant progress in recent years (Battistelli and Chisci, 2014, Battistelli and Chisci, 2016, Battistelli et al., 2014, Ji et al., 2017, Kamal et al., 2013, Li et al., 2016, Matei and Baras, 2012, Olfati-Saber, 2007, Olfati-Saber, 2009, Shen et al., 2010). In Olfati-Saber, 2007, Olfati-Saber, 2009, Kalman consensus filters (KCF) are proposed to achieve a consensus in terms of the local state estimates by way of adding a consensus term, which is also applicable to the case with packet dropout. In Matei and Baras (2012), a Luenberger-like consensus algorithm is developed, where every sensor combines its own local estimate computed by the Luenberger observer and the other estimates obtained from its neighboring nodes in a convex manner. In addition, for the uncertain systems, $H_{\infty }$-consensus performance constraint is introduced in Shen et al. (2010) to quantify the consensus level with regard to the estimation errors.

However, when there exist some non-ideal conditions, such as noisy transmission channel, restricted communication network or limited observability (Ji et al., 2017), the state estimation may be a more troublesome and challenging problem. To overcome these difficulties, an information-weighted consensus filter (ICF) algorithm is discussed in Ji et al., 2017, Kamal et al., 2013. In Battistelli et al. (2014), the stability of the hybrid CMCI (consensus on measurement (Olfati-Saber, 2007) and consensus on information (Battistelli & Chisci, 2014)) filtering algorithm based on the collective observability and network connectivity condition is guaranteed in a linear setting, which is equivalent to ICF with particular weights. When it comes to the nonlinear systems, an alternative extended Kalman filter (EKF) argument is raised in Battistelli and Chisci, 2016, Battistelli et al., 2014, Hu et al., 2012, Li et al., 2016. Similar to Battistelli et al. (2014), the local stability analysis of distributed extended Kalman filter (DEKF) under certain conditions is provided in Battistelli and Chisci (2016). In Li et al. (2016), a variance-constrained DEKF is put forward without omitting the edge-covariances in Olfati-Saber (2009), and the filter gain is obtained by minimizing an upper bound for the estimation error covariance. However, the EKF-based algorithm suffers a number of limitations especially when the system contains high nonlinearities and even discontinuities, which facilitates the development of unscented Kalman filter (UKF)-based algorithm (Julier and Uhlmann, 2004, Li et al., 2015, Li et al., 2016, Li and Xia, 2012, Xiong et al., 2006). In Li et al. (2015), a distributed UKF algorithm based on CI method is proposed, and lately, the weighted average consensus-based UKF is developed with theoretical proof in Li et al. (2016).

In many practical applications of sensor networks, the sensors are battery-powered, which brings about an inevitable issue that replacing or recharging the worn batteries might be impossible in a complicated environment. Thus, it is of particular importance to decrease the frequency of sensor-to-estimator data transmission without compromising the expected estimation performance (Miskowicz, 2006, Wu et al., 2013). A large number of related works considering event-triggered communication mechanism have been reported in Dimarogonas et al., 2012, Li et al., 2016, Li et al., 2017, Liu et al., 2015, Shi et al., 2014, Trimpe, 2014, Zhang and Jia, 2017, Zhang et al., 2016, Zheng and Fang, 2016. Based on the KCF framework in Olfati-Saber (2009), a kind of event-triggered KCF is derived with a named send-on-delta (SoD) schedule (Miskowicz, 2006) on estimator-to-estimator channel to reduce communication energy consumption in Li et al. (2016). An extended work can be found in Zhang and Jia (2017), where the sensor-to-estimator channel is also taken into consideration. However, to the best of our knowledge, there have been very few results about event-triggered UKF algorithm except Li et al. (2017) with only one single sensor considered, even none in distributed setting.

In addition to the theoretical developments, the work presented here is applied in the moving target tracking problem for a team of UAVs equipped with onboard sensors. Mobile UAV sensing platforms have attracted increasing attention in recent years due to the distinctive advantages over their static counterparts with regard to the area coverage, estimation performance and robustness against failure (Campbell and Whitacre, 2007, Hausman et al., 2015, Morbidi and Mariottini, 2013). In Zhan, Casbeer, and Swindlehurst (2010), a centralized adaptive target-tracking algorithm based on the UAV sensors is developed in the EKF framework, which is further investigated in the case where a maneuvering target is tracked with distributed UKF (Li & Jia, 2012) and distributed high degree cubature information filter (Sun & Xin, 2015) in the multiple model environment, respectively. However, the power constraints on small UAVs will limit the practicality of the existing algorithms, in which the data transmissions between individuals and ground stations or among individuals are executed in a periodical fashion. Therefore, the proposed event-triggered cooperative algorithm will be utilized in this application to reach a balance between tracking performance and energy consumption.

The main contributions of this paper are threefold:

• an event-triggered cooperative UKF algorithm is derived, which can well balance the filtering performance and average communication rate by designing reasonable trigger thresholds.

• the stochastic stability of the proposed algorithm in terms of the bounded estimation errors is investigated based on the stochastic stability theory.

• the proposed algorithm is employed in the moving target localization problem with multiple UAVs to show the practical potentials.

The remainder of this paper is organized as follows. Section 2 provides some basic concepts in algebraic graph theory and the mathematical formulation of the considered problem. Section 3 derives the event-triggered cooperative unscented Kalman filtering algorithm, whose stochastic stability will be analyzed in Section 4. Section 5 presents the simulation results illustrating the performance of proposed algorithm for the ground moving target localization with multiple UAVs tracking system. Concluding remarks are stated in Section 6.