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Processing Algorithm for a Strapdown Gyrocompass

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Processing Algorithm for a Strapdown Gyrocompass View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Scientia Iranica D (2012) 19 (3), 774–781 Sharif University of Technology Scientia Iranica Transactions D: Computer Science & Engineering and Electrical Engineering www.sciencedirect.com Processing algorithm for a strapdown gyrocompass M. Hemmati a,∗, M.A. Massoumnia b a School of Electrical Engineering, Sharif University of Technology, Tehran, Iran b Tahlile Sazegan Co., No. 18, East Second St., North Kargar Ave., Tehran, P.O. Box: 14139-83311, Iran Received 16 October 2010; accepted 26 September 2011 KEYWORDS Abstract The problem of gyrocompassing using inertial sensors, i.e., gyros and accelerometers, is Inertial navigation; addressed. North finding, with an order of accuracy of one arc-min, is not only required for the initial Gyrocompassing; alignment of inertial navigation systems, but also has a critical role to play in the guidance and navigation Kalman filter; of ships that navigate for long periods of time. In this work, after extracting the error model of an inertial Data fusion; navigation system and augmenting it with the error model of inertial sensors, a processing algorithm Heading error. based on the Kalman filter is designed and simulated to process the navigation system velocity error, and to estimate and correct tilt and heading errors along with gyro drifts and accelerometer biases. It is verified that using gyros with drift stability of 0.01 deg=hr, and accelerometers with bias stability of 100 µg, the true north direction can be determined with an accuracy of about five arc-mins. ' 2012 Sharif University of Technology. Production and hosting by Elsevier B.V. Open access under CC BY-NC-ND license. 1. Introduction provide the necessary signal for finding the north direction, and inertial north-finding is often called gyrocompassing. In contrast Direction-finding is one of the oldest problems in navigation. with inertial methods that work based on sensing and measur- There are four major directions on earth: north, south, east and ing the earth's gravity and rotation, noninertial methods are west. Precise information about these four major directions is based on some other physical characteristics. For example, a the most important prerequisite for navigation, but as the angle magnetic compass works based on the earth's magnetic field, between these major direction equals a right angle, knowledge and so finds magnetic instead of true north. Hybrid methods of one of them (usually the north) should be adequate. True use both inertial and noninertial methods together to make use north is the intersection point of the earth's rotation axis with of the benefits of both techniques simultaneously. its surface. But, magnetic north is a point near true north, at One of the most important applications of north-finding is which the lines of the earth's magnetic field will end. By north- the initial alignment of inertial navigation systems [1,2]. The finding, we mean finding true north, or the direction angle with process of the initial alignment of inertial navigation systems respect to true north (true heading) in different locations on consists of finding the angular relation between the body frame earth. North-finding techniques can be divided into three major and the computational frame. In most terrestrial navigation categories: inertial, non-inertial and hybrid methods. systems, navigation computations are done in a geographic Inertial methods use typical inertial sensors, like accelerom- (NED) frame. Therefore, the alignment process in these systems eters and gyros, for north-finding. The earth's rotation will is actually finding the horizon and the north direction, and consists of two stages: leveling and gyrocompassing. ∗ Precise north-finding for military and civil ships that Corresponding author. Tel.: +98 66165956; fax: +98 22568464. E-mail addresses: [email protected] (M. Hemmati), navigate for long periods of time is significantly important and [email protected] (M.A. Massoumnia). plays an essential role [3]. Mostly, ships travel in the same Peer review under responsibility of Sharif University of Technology. direction with a constant speed for some days. So, if the ship has a heading error of a fraction of a degree, it will be many kilometers off the planned destination after some hours of sailing. Another important application of north-finding comes from the fact that all equipment installed on the ship needs 1026-3098 ' 2012 Sharif University of Technology. Production and hosting by Elsevier B.V. Open access under CC BY-NC-ND license. doi:10.1016/j.scient.2012.01.002 M. Hemmati, M.A. Massoumnia / Scientia Iranica, Transactions D: Computer Science & Engineering and Electrical Engineering 19 (2012) 774–781 775 precise attitude information, including heading, to function factor asymmetry is negligible, the gyro error model will be as properly. For example, the pointing and aiming of weapons and follows [4]: radar antennas need precise heading information. B D C B In this work, we are going to use inertial sensors, including δ!IB Df MRSE δ!IB; (4) three gyros and two or three precise accelerometers, strapped in which the gyro constant drift vector will be as follows: to the ship's body, and an algorithm implemented in the T computer of the systems, to find true north, or in another words, Df D Dx Dy Dz ; (5) the heading angle with respect to true north with an accuracy and the Matrix of Rate-Sensitive Error is as follows: of a few arc-mins. It is supposed that integrating attitude, velocity and position equations is done by the computer of the " GSFx XGz −XGy# ship's Inertial Navigation System (INS). The north-finding or MRSE D −YGz GSFy YGx ; (6) gyrocompassing algorithm will attempt to estimate and correct ZGy −ZGx GSFz INS attitude errors like tilt and heading error. The Kalman in which the diagonal entries are scale factor errors of the gyros filter will be used as a powerful mathematical technique in and the other entries are the input axis misalignment of the estimation for this purpose. At the first step, a linearized model gyros. of the INS error will be extracted into which the inertial sensors error model will be augmented. Then, using the information 2.3. INS error model of an external speed reference, the designed Kalman filter will estimate the attitude error and the inertial sensors error. Having In this section, INS equations of position, velocity and these estimated errors in hand, it is possible to correct the attitude will be linearized using the Perturbation Method. The errors of the navigation system, like heading errors, to have error propagation model in navigation systems is extracted more precise information on heading. The main goal is for the from the inertial navigation system in the geographic frame aforementioned Strapdown Gyrocompass system to be able to and is represented in the state space equations form. Consider maintain the precision of the heading angle to a few arc-mins in position equations in the geographic frame [5]: the presence of environmental oscillations, shocks and sudden and severe movement of the ship. P VN P VE P L D I l D I h D −VD: (7) rL C h .rl C h/ cos L 2. Error modeling After differentiating the above equations and using simpli- fying assumptions: rL C h D rl C h D rL D rl D re D r, we will 2.1. Accelerometer error model have: 8 P 1 VN The main sources of error in accelerometers are input axis >δL D δVN − δh > r r2 misalignment, bias, scale factor error including nonlinearity, < P 1 VE tan L VE (8) and finally higher order errors. Assuming that accelerometers δl D δVE C δL − δh > r cos L r cos L r2 cos L are nominally configured in an orthogonal manner, and the :> P second order effects of scale factor error and asymmetry are, δh D −δVD: respectively, negligible, the accelerometer error model can be Take into consideration the velocity equations in the geographic described as follows [4]: frame [5]: f B D AB C MASE f B (1) P N D N C N − N C N × N δ ; V f g 2!IE !EN V : (9) in which the Accelerometer Bias vector is: After differentiating the above equations, we will have: T P N N N N N N N AB D Bx By Bz ; (2) δV D δf C δg − δ × V − × δV ; (10) whose elements are the constant or slow-varying bias of the in which Ω D 2!IE C !EN . three accelerometers. Also, the Matrix of Acceleration-Sensitive Expanding Eq. (10) and neglecting variations of g, with Error is as follows: respect to the ellipsoid radius, and assuming that rL C h D r C h D r D r D r D r, we will have: " # l L l e ASFx XAz −XAy D − P VD VE MASE YAz ASFy YAx ; (3) δVN D BN C g'E C δVN − 2 C !e sin LδVE ZAy −ZAx ASFz r r cos L V V 2 in which the diagonal entries are scale factor errors of the ac- N E C δVD − C 2VE !e cos L δL celerometers and the other entries are input axis misalignments r r cos2 L of the accelerometers. 2 VN VD − V tan L − E δh (11) 2 2.2. Gyro error model r P VE δVE D BE − g'N C C 2!e sin LδVN The main sources of error in the gyro angular velocity r cos L measurement are: gyro input axis misalignments, gyro drift C 2 VN tan L VD VE (including g-independent, g-sensitive and g -sensitive terms) C δVE C C 2!e and gyro scale factor error.
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