## 1 Introduction  
As an essential equipment in a fully-mechanized coal mining working face [1], a hydraulic powered roof support connects with the scraper chain conveyor through the pushing link. Since the cutting machine rides on the scraper chain conveyor, it is evident that the straightness of the scraper chain conveyor directly affects the working efficiency of cutting machine [2], while the straightness is determined by the pushing distance, which is produced by the hydraulic powered roof support. The pushing distance is defined as the distance that the pushing link moves forward or backward. At present, the pushing distance is measured by a displacement sensor which is embedded in the hydraulic pushing cylinder to avoid being exposed in extreme conditions. However, there are some fatal drawbacks of this approach. The displacement sensor, embedded into the hydraulic pushing cylinder, is non-replaceable due to the cumbersome mechanical structure and its unmaintainable installation method. As a consequence, it will be totally disabled once the sensor is broken, and the sensor cannot be replaced until all of the mining tasks are finished. Moreover, the method has poor accuracy, because the pushing distance is obtained either by using the pushing length of the hydraulic pushing cylinder directly, or calculated via trigonometry transformation. Both of the above methods produce large system errors. Thus, a novel pushing distance estimation approach, with comprehensive consideration of both estimation accuracy and extreme condition constraints, is much needed. Inertial sensors are widely applied in navigation [3], robotics [4], and virtual reality [5], etc., due to their positive characteristics, such as being lightweight, their low power consumption, portability, and self-contained nature [6]. They were originally proposed for aircraft navigation systems in the 1930s [7]. Later, inertial sensors are growing popular both in scientific research and entertainment technologies. The scientific studies are not limited to applications [8], but also to their performance [9]. Research on their performance improvement [10] and calibration analysis [11] provide more accurate estimation technologies for these interesting applications, e.g., equine lameness examinations [12], human arm motion tracking [13], and quantifying the evaluation of warfighter performance [14], etc. Although inertial sensors are successfully used in many fields, the application on displacement estimation is still a great challenge due to the cumulative errors [15]. As improvements in inertial sensors and rapid development of signal processing methods continue, inertial sensors also provide a possibility for operation in some extreme conditions without accuracy requirements, e.g., a fully-mechanized coal mining working face. The Kalman filter was proposed for linear filtering and prediction problems by [PERSON] in 1960 [16]. It is a state estimator based on a state-space model [17]. The Kalman filter is frequently used for inertial information processing due to uncertainty problems [18], including position estimation [19], as well as rotation estimation [20]. For non-linear problems, researchers also proposed some mutation of the Kalman filter, e.g., the extended Kalman filter [21], unscented Kalman filter [22], etc. In recent years, the Kalman filter has also been applied in intelligent computation [23], as well as information fusion through distributed [24] or parallel [25] approaches. In addition to these applications, the Kalman filter contributes to magnetometer calibration [26], moving object tracking [27], and satellite attitude determination [28], as well. Unfortunately, the estimation performance of the Kalman filter is greatly influenced by the selection of the processing covariance matrix \(Q\) and the observation (measurement) covariance matrix \(R\)[29], and the research on tuning \(Q\) and \(R\) is very valuable [30]. Thus, optimization algorithms should be applied to obtain better estimation performance through tuning the covariance.