Vol.:(0123456789) 1 3 Multiscale Science and Engineering (2020) 2:135–142 https://doi.org/10.1007/s42493-020-00045-2 ORIGINAL RESEARCH A Field Calibration Method for Low‑Cost MEMS Accelerometer Based on the Generalized Nonlinear Least Square Method Mahmood ul Hassan 1  · Qilian Bao 1 Received: 13 March 2020 / Revised: 31 July 2020 / Accepted: 1 September 2020 / Published online: 11 September 2020 © Korean Multi-Scale Mechanics (KMSM) 2020 Abstract This paper proposes a feld calibration method for an accelerometer without the need of having any external devices for calibration. In the proposed calibration method, generalized nonlinear least-square (GNLS) is used to estimate deterministic errors. A novel sensor’s data collection procedure is developed to collect data of an accelerometer along all three axes and all possible orientations where the expectation of infuence of all possible errors is very high. The proposed calibration method is verifed by applying it to two diferent accelerometers. The proposed calibration method achieved an accurate estimation of calibration parameters. The results of the proposed GNLS based calibration method are compared with two other commonly used algorithms, such as Levenberg–Marquardt (LM) and Gauss–Newton (GN). Simulation and experimental results show that the proposed GNLS-based calibration method is slightly more accurate than the LM and GN. The GNLS convergence rate for estimating the calibration parameters is also faster than the LM and GN. Keywords Calibration · Low-cost MEMS accelerometers · Sensor error model · Generalized nonlinear least square method · Gauss–newton and levenberg–marquardt Introduction The accelerometer is one of the essential components of an inertial navigation system (INS). During navigation, its measurement errors directly cause navigation errors of the same order of magnitude. Moreover, its measurement errors also afect the calibration of the gyroscope and ini- tial alignment, which indirectly cause navigation errors [1]. The accelerometer errors are divided into two categories: Deterministic errors and stochastic errors [2]. Random errors mainly contain random noise, which can be modelled sto- chastically. Systematic errors consist of scale-factor, bias ofset, misalignment errors, and non-orthogonality errors, which can be eliminated by specifc calibration procedure [3]. Calibration is the procedure of fnding unknown deter- ministic errors such as scale-factor, bias, and misalignment of a sensor [4]. The desired deterministic errors in a vector form are represented as  in Eq. (1) where b x , b y , and b z are bias along x-axis, y-axis, and z-axis. s x , s y , and s z are scale-factors along x-axis, y-axis, and z-axis. s xy, s xz, and s yz are misalignment angles. There are various techniques used for the calibration of an accelerometer. These calibration techniques can be divided into two major categories based on the requirement of any external devices for calibration. In the frst type, calibration methods require costly machinery such as an accurately con- trolled rotation table to precisely place a sensor into various known positions and orientations while the output of the sen- sor is measured. That is why these methods are limited to a laboratory environment or sensor production industry. These methods are expensive and need plenty of time, so these are economically more suitable for high-grade accelerom- eter sensors. The second type is known as feld calibration methods, which do not need any particular devices. In these methods, a sensor’s output is measured at various random orientations [5]. These calibration methods are economi- cally suitable for low-cost accelerometer sensors. These feld calibration methods can be easily implemented in a feld environment. The sensor output is afected by environmental (1)  = [ s x , s y , s z , s xy , s xz , s yz , b x , b y , b z ] * Mahmood ul Hassan mahmood-ul-hassan@sjtu.edu.cn 1 School of Electronic Information and Electrical Engineering (SEIEE), Shanghai Jiao Tong University, Shanghai, China