## 1 Introduction
Optimization algorithms have significant influence on computational performance. Recently, some interesting and efficient naturally-inspired optimization algorithms have been proposed. Krill herd (KH) was proposed by [PERSON] and [PERSON] in 2012 [31]. Based on the innovation, other algorithms, such as chaotic KH [32], stud KH [33], hybrid KH [34], chaotic particle-swarm KH [35], multi-stage KH [36], and opposition-based KH [37], etc., are successively developed. These algorithms are so efficient and flexible that they can be improved either by internal improvement, such as introducing a new migration operator [38] and a new information exchange process [39], or by combining other algorithms together, such as quantum-behaved particle swarm algorithms [40] and cuckoo search algorithms [41], etc. Monarch butterfly optimization (MBO) [42] and MBO-based algorithms [43; 44] are also recently developed meta-heuristic algorithms, which simulate the migration of monarch butterflies. Other similar algorithms, moth search algorithms [45], earthworm optimization algorithms [46], cuckoo search algorithms [47; 48], bat algorithms [49; 50] and firefly algorithms [51; 52], etc., also simulate the natural motion process and have contributed greatly to intelligent optimization algorithms in recent years. Among these intelligent optimization algorithms, the fruit fly optimization algorithm (FOA) is a brand new intelligent optimization algorithm, which has a rapid convergence rate and easy implementation, proposed by [PERSON] in 2012 [53]. FOA simulates the intelligent foraging behavior of a fruit fly group in a food-finding process [54], which combines olfactory and vision searching methods [55]. It has been widely applied in financial parameter optimization [56], forecasting [57], scheduling [58], etc. However, like other optimization algorithms, the basic FOA has the possibility of falling into local extremes, and it is difficult to jump out [55]. Bearing the above observations in mind, inertial sensors are utilized to measure the inertial information of a pushing link, and an improved fruit fly algorithm optimized Kalman filter (FOA-KF) is proposed to process the inertial information. To make it easy for jumping out of the local extreme, the basic FOA is modified by adding a perturbation intensity through a predict-update mechanism when falling into a local extreme. A simulation and application for the pushing distance estimation proves the effectiveness and superiority of the proposed method. The rest of this paper is organized as follows: In Section 2, preliminaries of Kalman filter and basic FOA are presented. In Section 3, the framework of proposed method is described and an improved FOA is applied to optimize the Kalman filter covariance. In Section 4, an artificial inertial signal is estimated to verify the effectiveness and superiority of the proposed method, and an application of pushing distance estimation is performed. Some conclusions and outlooks are summarized in Section 5. Bearing the above observations in mind, an inertial sensor is utilized to measure the inertial information of the pushing link, and an improved fruit fly algorithm optimized Kalman filter (FOA-KF) is proposed to process the inertial information. To make it easier for jumping out of the local extreme, the basic FOA is modified by adding a perturbation intensity through a predict-update mechanism when falling into a local extreme. A simulation and application for the pushing distance estimation proves the effectiveness and superiority of the proposed method.