# A Designed Calibration Approach for the Measurement -While-Drilling Instrument
## 1 Introduction  
With extensive application of nearly-horizontal drilling equipment in the coal mining industry, in for example the gas drainage, the water detection, and the coal bed methane exploitation, directional drilling technology has made a great development [1; 2; 3; 4; 5; 6]. In particular, the equipment used for monitoring the state of the borehole and measuring the trajectory of the borehole have been an intensive research subject in recent decades [1]. The core of underground in-seam directional drilling technology is exactly this equipment, namely the Measurement-While-Drilling (MWD) instrument. Many scholars have focused on improving the detection function of this instrument to avoid coal mine disasters, such as measuring vibrations to prevent rock bursts [2; 3; 4; 5], while ignoring the measurement of the borehole trajectory itself. Rebuilding the borehole trajectory aims to guide the bottom drilling tool along a predefined line, which is very important during the drilling process [7]. Thus, it is very necessary to accurately draw the borehole trajectory with the MWD instrument [6; 7; 8; 9]. The borehole trajectory is geometrically regarded as a 3D curve, which is calculated from the attitude information of the borehole and the length of the drillpipe. Actually, the MWD instrument, installed in the nonmagnetic drill collar behind the screw motor, can provide the attitude information [8; 9; 10], such as the azimuthal angle, the pitch angle, and the tool-face angle. This attitude information is mainly solved by the output of one tri-axial accelerometer and one tri-axial magnetometer mounted in the carrier platform of the MWD instrument [11]. Ideally, the tri-axial accelerometer and the tri-axial magnetometer are mathematically considered as two standard Cartesian coordinate frames, and the platform is considered another. These frames form a conceptual sensor model. However, many kinds of errors exist in these tri-axial sensors, such as the null-bias, the sensitivity error, and the non-orthogonal error, etc. [12]. In addition, the misalignment error also exists between the three coordinate frames. [PERSON], [PERSON], and [PERSON] et al. [13; 14; 15; 16; 17; 18] have fully analyzed the sources of these errors and unified the errors into an error model with 12 parameters. Furthermore, the random noise of the sensor should be taken into account as well during the calibration process and the measurement process [17]. The above descriptions indicate that these errors must be eliminated appropriately for the gaining of the attitude information from the MWD instrument. The counterparts have acknowledged that the error elimination is contingent upon the deployment of the calibration algorithms [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23]. However, they ignored the method of obtaining the sampled data. For example, [PERSON] [13] determined all the error parameters of a tri-axial accelerometer by matching the output vector with the gravity acceleration; however, this method has a restrictive condition of knowing the reference of the acceleration vector. Moreover, some expensive instruments were employed to aid calibration, such as the Magnetic Shielding Room [15]. Based on the same theory, [PERSON] et al. [14; 15; 16] presented and elaborated an approach, the ellipsoid fitting method, for a stand-alone sensor triplet. [PERSON] et al. [18] proposed a fast calibration method specific for a spinning projectile based on the ellipsoid fitting method. [PERSON] et al. [19] improved the ellipsoid fitting method with a truncated singular value decomposition method to enhance the success rate of calibration. It is notable that the above-mentioned studies have an underlying assumption: that is, the axes of the sensor triplet are aligned with the axes of the carrier platform. However, the phenomenon will never happen in practice. Moreover, the magnitude of sample data used in the ellipsoid fitting method is not taken into consideration.