Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 2950376, 10 pages

http://dx.doi.org/10.1155/2016/2950376

## A Temperature Compensation Model for Low Cost Quartz Accelerometers and Its Application in Tilt Sensing

^{1}College of Automation, Chongqing University, Chongqing 400044, China^{2}College of Computer Science, Chongqing University, Chongqing 400044, China^{3}Department of Computer and Information Science, University of Macau, Macau^{4}Zhengzhou Horizon Electronic Science and Technology Co. Ltd., Zhengzhou 450000, China

Received 23 May 2016; Revised 14 July 2016; Accepted 1 August 2016

Academic Editor: Michael Vynnycky

Copyright © 2016 Weibin Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Although the quartz accelerometer has made great advances, the performance, in some specific applications such as tilt sensing, needs to be well compensated in high temperature environment. Based on the high temperature testing of low cost quartz accelerometers, we found that the normalized positive and negative parts are asymmetrical at high temperature and the temperature curve of zero sensor output is related to the roll angle of the sensor. Traditional temperature compensation method only considers the temperature factor and ignores the roll sensitivity, which leads to deteriorated accuracy. To solve this problem, this paper proposes a novel and simple mathematical model to obtain a more accurate expression of zero sensor output, which makes the sensor output more robust at high temperature. Experimental results on two low cost quartz accelerometers demonstrate that the proposed model is feasible and effective, which could reduce the temperature drift error of the sensor output typically from 0.01 g to 0.001 g. Furthermore, we introduce the compensated sensors in the three-axis inclinometer system for tilt sensing, and the evaluation results show that the temperature drift error of the inclination in the range (, ) is reduced typically from to compared to the traditional method.

#### 1. Introduction

Quartz accelerometers have become popular in many applications like motion control [1] and inertial measurement units [2] due to their simple construction and high accuracy. However, along with the increasing requirement for excellent performance of sensors, especially in harsh environment, for example, high temperature, high shock, and high vibration, some recalibration or recompensation methods should be adopted to obtain a satisfying accuracy depending on the specific usage [3, 4]. Generally, the accuracy of the accelerometer can be influenced by manufacturing technique, internal structure, and surrounding environment in which the environmental temperature makes the performance much worse. A directed compensation method is to add a temperature control system [5], which could make the sensor work under the controlled temperature. However, the main drawbacks of the temperature control system include long stabilization time, large power, and increasing size, which cannot meet the actual demand in many applications. A commonly used alternative solution is to construct a mathematical model [6], which first models how the performance changes with temperature and then, according to the measured temperature of the sensor, corrects the sensor output. Apparently, such mathematical models are more suitable for most applications due to their simplicity and practicality.

Estimation methods need to be settled for the accelerometer in the thermal calibration in which iterative methods are mostly utilized to solve the calibration parameters (e.g., scale factors, misalignments, and biases of the accelerometer triad) and to achieve high estimation accuracy. Qian et al. [7] used least squares method to estimate the parameters of the proposed linear model. Yang et al. [8] presented an improved iterative nonlinear calibration method by using least squares method and sequence quadric program method. Ang et al. [9] proposed a nonlinear regression model to reduce the deterministic errors associated with scale factor, bias, and misalignment of the dual axis accelerometer, and the performance was verified by tilt and motion sensing. Won and Golnaraghi [10] proposed a mathematical model of six calibration parameters and use an iterative method to estimate the parameters of the nonlinear model. The objective of the proposed iterative method is to make the calculated gain factors and biases of each axis match the corresponding true values. Because the need of an initial rough estimate makes this algorithm inconvenient, Zhang et al. [11] proposed an improved multiposition calibration for solving the unknown parameters without any initial guess. However, the accelerometer temperature drift is still left as an unresolved problem. As a consequence, many trials have been made to establish thermal models of calibration parameters to attack this problem. Aggarwal et al. [12] explored the effects of thermal variations on biases and scale factors at different temperature values through the thermal chamber and then proposed three-order polynomial thermal models for ADI microelectromechanical system sensors. To investigate the thermal property in varying temperature conditions, Aggarwal et al. [13] considered the thermal ramp experiment from which a simple polynomial temperature model is developed for the inertial sensor biases and scale factors. After compensating the thermal errors, the inertial navigation solution was significantly improved. These two trials have to utilize a turntable with a temperature-controlled incubator that requires the precise orientation information. To improve the robustness to the turntable error, in [14], an indirect calibration technique for estimating the body-frame drift induced by the variation of the accelerometer sensitivity axes due to temperature changes is proposed. However, this estimation method depends on and is limited to the accuracy of horizontal accelerometer measurements. Moreover, Zhang et al. [15] utilized a nonvertical rotation axis observation method to attack the problem of the relationship between the gyroscope triad and the accelerometer triad in constant thermal conditions.

In this paper, we mainly focus on the usage of the quartz accelerometer in tilt sensing systems [16], such as Measurement While Drilling (MWD) and Logging While Drilling (LWD), which are used to monitor and guide the down borehole directional drilling in oil and gas exploration. Two main involved angles, inclination and roll angle, are illustrated in Figure 1. Suppose that an accelerometer is mounted and its sensitive axes are arranged along the -axis in the coordinate system, inclination is the angle that -axis makes with the down direction . It is 0° when -axis is down and 90° when -axis is horizontal. Roll angle is defined as the angle of counterclockwise rotation about the -axis (looking in the positive -axis direction) in the gravitational field. Particularly, the inclination plays a very important role in tilt sensing systems, and the precision of inclination directly determines the performance of the whole system, so the accuracy of the sensor output is the key for accurate measurement at high temperature. In this paper, we pay our attention to the temperature compensation of low cost quartz accelerometers, and the main contributions include proposing a mathematical temperature compensation model for zero sensor output, especially, when the temperature curve changes with the roll angle, presenting a feasible and effective solution for the whole compensation of quartz accelerometer, and applying the proposed temperature compensation method in the tilt sensing system.