Advanced Borehole Attitude Determination without Measuring Axial Angular Rate Component

The problem of borehole azimuth determination by means of gyro inclinometer which operates in the gyrocompassing mode but don't measure the axial angular rate component is considered. Basing on the existence of two solutions for azimuth which differ in accuracy the optimal algorithm is derived, algorithm performance is examined and the efficiency of new algorithm versus conventional one is demonstrated. The problem is also solved based on the general approach for the attitude determination from 2 vector observations. In the bounds of this approach the existence of two solutions for the unknown DCM element is proven and the equivalence of these solutions to two ones for the azimuth is shown. The universal optimal DCM attitude algorithm is obtained which can be used in the vertical boreholes too. In conclusion of the paper gyro inclinometer mockup test results which verify the advantage of new algorithm are presented. I. INTRODUCTION In many cases the underground navigation tasks while borehole drilling and surveying can not be adequately solved without the use of gyro inclinometers (GI). The state-of-art GI intended for small diameter borehole navigation use strapdown measurement unit design with incomplete composition of gyro sensors (as compared to a strapdown INS). So, the scheme with three accelerometers and one two- axis rate gyro (RG) or 2 single-axis RG which measure 2 cross-section angular rate components is of considerable current use. Such a scheme is called "axial" throughout this paper. The GI is conventionally used in one of two possible modes: the directional gyro (DG) mode which is based on the RG outputs integration and the point-to-point gyrocompassing (GC) mode when the borehole azimuth is determined from the Earth angular rate measurements. In both GI modes leveling is provided using the accelerometer outputs. It is well known that in the GC mode the azimuth decreases in accuracy with increasing the borehole inclination angle and gyro inclinometer loses operation under horizontal drilling (when the inclination angle approaches 90˚). In practice it leads to further limitations on the value of inclination angle θ up to 60˚-70˚. In the paper the problem of accuracy improvement of GC mode as applied to the "axial" scheme of GI is considered. The results obtained by the authors on investigation of solution uniqueness when defining the borehole azimuth are presented. Such statement of the problem was not considered earlier, probably, due to seeming evidence of the conventional algorithm, in which arctg function is used. It was shown in the authors' paper (1) that there exists another azimuth calculation algorithm with the use of arcsin function. In this case the properties of the desired angle are changing: in particular, the limitation area for the use of "axial" scheme is substantially narrowed down, i.e. in horizon borehole it is impossible to accomplish navigation procedure only in East- West direction. This approach is further developed in the present paper. The list of problems discussed below was determined by the availability of two possible azimuth calculation algorithms and includes: (1) transition from combined algorithm to an optimal one, in which both solutions are used with weights inversely proportional to their variances; and (2) analysis of the optimal algorithm properties in the cases of inadequate RG error models. In a special section of the paper the problem under consideration is treated from the viewpoint of the general approach for the attitude determination from two vector observations. Duality in azimuth calculation is justified in the bounds of this approach reasoning from availability of two solutions for unknown DCM element. To calculate this value, it is required to know the value of the axial angular rate component, which cannot be measured in the "axial" scheme. A DCM calculation algorithm was derived basing on TRIAD algorithm implementing a well-known algebraic approach (2). This algorithm uses optimal weighting of the sought-for DCM element and is fully equivalent to the optimal azimuth determination algorithm derived above. In the final section of the paper the GI mockup test results are presented which confirm the validity of conclusions made basing on the analytical study.