sensors Article

A Damping Grid Strapdown Inertial Navigation System Based on a Kalman Filter for Ships in Polar Regions Weiquan Huang, Tao Fang *, Li Luo, Lin Zhao and Fengzhu Che College of Automation, Harbin Engineering University, Harbin 150001, China; [email protected] (W.H.); [email protected] (L.L.); [email protected] (L.Z.); [email protected] (F.C.) * Correspondence: [email protected]; Tel.: +86-451-8251-8426 Received: 26 May 2017; Accepted: 29 June 2017; Published: 3 July 2017

Abstract: The grid strapdown inertial navigation system (SINS) used in polar navigation also includes three kinds of periodic oscillation errors as common SINS are based on a geographic coordinate system. Aiming ships which have the external information to conduct a system reset regularly, suppressing the Schuler periodic oscillation is an effective way to enhance navigation accuracy. The Kalman filter based on the grid SINS error model which applies to the ship is established in this paper. The errors of grid-level attitude angles can be accurately estimated when the external velocity contains constant error, and then correcting the errors of the grid-level attitude angles through feedback correction can effectively dampen the Schuler periodic oscillation. The simulation results show that with the aid of external reference velocity, the proposed external level damping algorithm based on the Kalman filter can suppress the Schuler periodic oscillation effectively. Compared with the traditional external level damping algorithm based on the damping network, the algorithm proposed in this paper can reduce the overshoot errors when the state of grid SINS is switched from the non-damping state to the damping state, and this effectively improves the navigation accuracy of the system. Keywords: grid SINS; Schuler periodic oscillation; Kalman filter; level damping

1. Introduction Polar navigation technology is the basic condition for ships sailing in polar region safely and reliably [1,2]. Inertial navigation system has been the first choice for ships sailing in polar regions as a result of being autonomous, continuous, and comprehensive [3–7]. The working principle of the SINS is that the gyroscopes and accelerometers are strapped on the carrier. The gyroscopes and accelerometers measure the angular motion and linear motion of the carrier, respectively, then the navigation computer calculates the position, velocity, and the attitude of the carrier through the outputs of the gyroscopes and accelerometers. The lines of longitude tend to converge to a point in high latitudes, and the fictitious graticules change from rectangular in low latitudes to triangular in high latitudes, and because of this, the common SINS based on the geographic coordinate system cannot work properly [8–10]. Although the wander azimuth inertial navigation system and the free azimuth inertial navigation system can finish the calculation of the attitude and position direction cosine matrix [11,12], there are singular values when extracting the wander azimuth angle and longitude from the position direction cosine matrix [13]. Aiming at the shortcomings of the three kinds of mechanizations mentioned above, transversal SINS based on a transversal coordinate system and grid SINS based on a grid coordinate system are two main research hotspots to solve the problems of ships sailing in polar regions. The transversal SINS mechanization is built with the hypothesis that the earth Sensors 2017, 17, 1551; doi:10.3390/s17071551

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is a standard sphere, obviously, this assumption brings the principle error to the design of transversal SINS mechanization [14]. However, the principle error doesn’t exist in grid SINS mechanization, so the grid SINS mechanization can be the ideal mechanization for ships working in polar regions. Due to the presence of measurement errors of the inertial measurement units (IMU) and other error sources, the grid SINS mechanization drifts with time and includes three kinds of periodic oscillation errors, the so-called 84.4 min Schuler, 24 h earth, and Foucault periodic oscillation [15–19]. These periodic oscillation errors are detrimental to the high-precision navigation system over a long time [20–23], so it is necessary to adopt some methods to suppress these kinds of errors, such as damping technology. Aiming ships which have the external reference information to conduct a system reset regularly in a certain period of time, suppressing the Schuler periodic oscillation is an effective way to enhance navigation accuracy, the so-called level damping technology. Traditional level damping technology includes internal level damping technology and external level damping technology. In the absence of external reference velocity, the internal level damping technology can suppress the Schuler oscillation errors, however, the internal level damping technology is sensitive to the maneuvering of the ships [24,25]. In order to overcome the shortcomings of the internal level damping technology, when the external device such as the Doppler Velocity Log (DVL) can provide an accurate reference velocity, the external level damping technology can suppress the Schuler oscillation errors with the aid of the reference velocity, and this method is not sensitive to the maneuvering of the ships [26]. Traditional external level damping technology suppresses the Schuler periodic oscillation by adding the network to the level circuit of the system. Like the common SINS, when the work state of the grid SINS is switched from the non-damping state to the damping state, big overshoot errors are introduced to the system [27]. The overshoot phenomenon is extremely unfavorable to the improvement of navigation accuracy, and some methods must be adopted to solve this problem. Aiming at the shortcomings of the traditional external level damping algorithm, a new external level damping algorithm based on a Kalman filter is proposed in this paper. Connecting with the practical situation, first, the criterion to judge if the velocity measured by DVL is available is built, and when the external velocity is judged to be available, the error model of the external velocity can be simplified as a constant in a certain period of time; second, the Kalman filter, based on a grid SINS error model, is established. When the external velocity error contains constant error, only grid level attitude error angles can be accurately estimated, and then the estimated level attitude error angles are used to calibrate the system. Compared with the traditional external level damping algorithm, the proposed algorithm can not only suppress the Schuler periodic oscillation, but also restrain the overshoot errors when the work state of grid SINS is switched from the non-damping state to the damping state. In the following sections of the paper, the new algorithm proposed in this paper will be called KF-DGSINS, and the traditional external level damping algorithm will be called T-DGSINS. KF represents Kalman filter; T represents traditional; D represents damping; GSINS represents grid strapdown inertial navigation system. 2. Grid SINS Mechanization This section first introduces some common coordinate systems and the transformation relationships between each two coordinate systems, and then the grid SINS mechanization is given. 2.1. The Definitions of Common Coordinate Systems The frequently-used coordinated systems in this paper are the geocentric inertial coordinate system i, earth centered earth fixed (ECEF) coordinate system e, body coordinate system b, and geographic coordinate system T. The detailed definitions of these coordinate systems can be seen in reference [24]. The navigation coordinate system of the grid SINS is grid coordinate system G. As shown in Figure 1, when the ship moves to the point P, the grid coordinate system is defined as follows: Grid plane: the parallel plane of the Greenwich plane which passes through the point P; Meridian

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tangent plane: horizontal plane passes through the The point P. The grid north axis GN is the intersecting linelocal of the grid plane andwhich meridian tangent plane. grid up axis with GU coincides intersecting line of the grid plane and meridian tangent plane. The grid up axis GU coincides with the the up axis of the geographic coordinate system which passes through the point P. Grid east axis GE up axis of the geographic coordinate system which passes through the point P. Grid east axis GE lies in lies in thehorizontal local horizontal plane and is perpendicular to north the grid G , GN , and G U the local plane and is perpendicular to the grid axis.north GE , Gaxis. the N , and EGU constitute constitute thecoordinate right-handed coordinate system. Obviously, the grid is coordinate system coordinate is a local right-handed system. Obviously, the grid coordinate system a local horizontal horizontal coordinate system, as the reference GN is system, and GN is used as the and reference ofused the grid course angle.of the grid course angle. ze

ωie

eU ( eGU ) eG

eN

N

 P Greenwich meridian

Grid plane

eE eGE

Meridian tangent plane

φ eZ

eX

ye

eY O

λ Meridian

xe

Figure 1. Grid coordinate system G . Figure 1. Grid coordinate system G.

2.2. The Transformation Relationships of Common Coordinate Systems 2.2. The Transformation Relationships of Common Coordinate Systems As shown in Figure 1, the angle between GN and the geographic north is called the grid As shown in Figure 1, the angle between GN and the geographic north is called the grid azimuth G frameas: azimuth . So, the transformation matrix fromto T frameistodescribed is described as: angle σ. angle So, the σtransformation matrix from T frame G frame  cosσ sinσ 0   cos σ − sin σ 0 =  sinσ cosσ 0   GC  CT =  sin σ cos σ 0   0 0 1  0 0 1 G T

(1)(1)

φ and λ represent the latitude and longitude of the point P. The sine and cosine of the grid φ and λ represent the latitude and longitude of the point P. The sine and cosine of the grid azimuth σ can be as: azimuth angle described as: angle σ can be described sin φ sin λ (2) sin σ = q sin φ sin λ sin σ  2 2 φ sin (2) 1 − cos λ 2 2 1  cos φ sin λ cos λ cos σ = q (3) cos λ cos σ  1 − cos2 φ sin2 λ (3) 2 2 1  cos φ sin λ The transformation matrix from e frame to T frame is described as: The transformation matrix from  e frame to T frame is describedas: − sin λ cos λ 0 cosλ 0     sinλ T Ce = T − (4)  sin φ cos λ − sin φ sin λ cos φ  C e =  sinφcosλ sinφsinλ cosφ  (4) cos φ cos λ cos φ sin λ sin φ  cosφcosλ

cosφsinλ

sinφ 

The transformation matrix from e frame to G frame is described as:

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The transformation matrix from e frame to G frame is described as:    cos σ − sin σ 0 − sin λ cos λ 0    T CeG = CG cos σ 0  − sin φ cos λ − sin φ sin λ cos φ  T Ce =  sin σ 0 0 1 cos φ cos λ cos φ sin λ sin φ   − cos σ sin λ + sin σ sin φ cos λ cos σ cos λ + sin σ sin φ sin λ − sin σ cos φ   =  − sin σ sin λ − cos σ sin φ cos λ sin σ cos λ − cos σ sin φ sin λ cos σ cos φ  cos φ cos λ cos φ sin λ sin φ

(5)

2.3. Grid SINS Mechanization The navigation frame is the grid frame, and the differential equation of strapdown attitude matrix for grid SINS is: .G

Cb = CbG ΩbGb

(6)

where ΩbGb is a skew-symmetric matrix related to ωbGb , and ωbGb is obtained as below: b b b G ωbGb = ωib − ωiG = ωib − CbG (ωieG + ωeG )

(7)

G is described as below: ωie



cos σ  G G T ωie = CT ωie =  sin σ 0

    − sin σ 0 0 −ωie cos φ sin σ     cos σ 0  ωie cos φ  =  ωie cos φ cos σ  0 1 ωie sin φ ωie sin φ

(8)

G is described as ωG = P where ωie is the rotational angular velocity of the earth. ωeG 3×2 ∗ eG h iT G G G G . VE , VN are east and north velocity in grid frame, and P3×2 is obtained as: VE VN

  P3×2 =  

1 R Nh ) sin σ cos σ 2 sin σ cos2 σ R Mh + R Nh √ sin λ cos φ 2 ( R 1 − R1 ) sin σ cos σ Mh Nh 1−cos2 φ sin λ

( R 1Mh −

2

σ −( cos R Mh +

sin2 σ R Nh )



  −( R 1Mh − R1Nh ) sin σ cos σ  2 2 sin λ cos φ sin σ cos σ −√ + ) ( R R 2 1−cos2 φ sin λ

Mh

(9)

Nh

where R Mh = R M + h and R Nh = R N + h. R M and R N are radius of curvature in meridian and prime vertical, and h is the altitude of the ship. For ships, h is set as zero. The differential equation of velocity for grid SINS is: . G

G G V = CbG fb − (2ωie + ωeG ) × VG + gG

(10)

h iT where gG = , g represents the constant of the gravitational acceleration. VG = 0 0 −g h i VEG VNG 0 is the ship’s velocity within grid frame.  T  T The position of the ship is described as the x y z in e frame. Re = x y z , and the differential equation of position for grid SINS is: .e

R = CeG VG where CeG is the transpose of CeG .

(11)

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The KF-DGSINS Algorithm 3. The estimation theory and feedback correction technology, the schematic diagram Based on onthe theoptimal optimal estimation theory and feedback correction technology, the schematic of the KF-DGSINS is shown in Figure 2. As in shown in Figure gyroscopes and accelerometers measure diagram of the KF-DGSINS is shown Figure 2. As2, shown in Figure 2, gyroscopes and the angular motion and linear motionmotion of the and ship,linear respectively, theship, position, velocity, and and the accelerometers measure the angular motion and of the respectively, attitude of the shipand are the calculated the grid SINS mechanization. error states are position, velocity, attitudethrough of the ship are calculated through theThe gridsystem SINS mechanization. estimated byerror Kalman filter the aidby ofKalman a reference velocity measured DVL, and then the Schuler The system states arewith estimated filter with the aid of aby reference velocity measured oscillation errors be effectively suppressed bybe correcting some error states of the Kalman filter. by DVL, and then can the Schuler oscillation errors can effectively suppressed by correcting some error This section first introduces thesection error model of the marine grid SINS, the Kalman filtering and states of the Kalman filter. This first introduces the error modelthen of the marine grid SINS, then feedback correction technology are described in detail. the Kalman filtering and feedback correction technology are described in detail.

Gyros Position, Velocity, Attitude Grid SINS mechanization Accelerometers Reference velocity DVL

+ Kalman filter

Feedback correction

Figure Figure 2. 2. Schematic Schematic diagram diagram of of the the KF-DGSINS. KF-DGSINS.

3.1. 3.1. The The Error Error Model Model of of the the Marine Marine Grid Grid SINS SINS Establishing an accurate error model is the premise of Kalman filtering. Nonlinear filter Establishing an accurate error model is the premise of Kalman filtering. Nonlinear filter algorithms algorithms such as EKF, PF, and UKF has been widely studied in the field of integrated navigation such as EKF, PF, and UKF has been widely studied in the field of integrated navigation system [28], system [28], when the real-time requirement is satisfied, the accuracy of the system can be increased. when the real-time requirement is satisfied, the accuracy of the system can be increased. In order to In order to meet the needs of the engineering, the error model is established with the assumptions: meet the needs of the engineering, the error model is established with the assumptions: the attitude the attitude error angle is extremely small; measurement errors of the IMU are simplified as the sum error angle is extremely small; measurement errors of the IMU are simplified as the sum of the constant of the constant and the white noise. Based on these two assumptions, the nonlinear factor of the and the white noise. Based on these two assumptions, the nonlinear factor of the system can be system can be ignored, and the linear error model can be established. Reference [16] has given the ignored, and the linear error model can be established. Reference [16] has given the airborne polar airborne polar grid SINS linear error equations. In order to establish the error model which adapts to grid SINS linear error equations. In order to establish the error model which adapts to the shipborne the shipborne environment, the altitude channel needs to be ignored. The state and measurement environment, the altitude channel needs to be ignored. The state and measurement equation can be equation can be established as follows: established as follows: ( . AX + BW  = AX + BW XX= (12) (12)  Z = HX + η Z = HX + η   h iT G φ G δV G δV G δx δy δz εb G , φG εbby εbbzb ∇bbbx ∇b bbyT ∇bbzG , φ where X =  GφEG GφN G U G E G N b b b bx N where X = E N U δVE δVN δx δy δz εbx εby εbz bx by bz  , E , NG E and G are the grid attitude error angles; δV G , δV G are the grid east and north velocity error; δx, δy, δz and φU N G E δx along UG the δV G are the are position the grid attitude errorthe angles; north velocity error; , δy , E , in eNframe; are errors along x, y, zδV axes εbbxgrid , εbbyeast , εbbz and are the gyro constant drifts b b b b , ∇the b ,∇ xbbz, are δz x, εbx y , the arey,the position errors ∇ along in e frame; , εbiases , εbz are the z axes the z axes in b frame; accelerometer constant along thegyro x, y, constant z axes in by bx by h i T b b  bbbx ,  bby, ε,bwx, bbzεbwyare drifts along in b b frame; constant b frame; W =the εbxwx, yε,bwyz εaxes , εbwzthe areaccelerometer the gyro random driftsbiases along wz ∇wx ∇wy ∇wz T the x, y, z axes in b frame; ∇bwx , ∇bwy , ∇bwz are the random biases along in baccelerometer b b b b the x, b y, z axes εwy εwz bwx bwy bwz  , εwx b frame; W = εwx along the x , y , z axes in , εwy , εwz are b frame. x , y , zreference b frame;and bwx  bwz are the the gyro drifts along the the external axes in velocity ,  bwy , calculated The random difference between velocity byaccelerometer the system is taken as biases the observation. noise array. Coefficient matrix A, system noise driven y , measurement random along the ηx is , the z axes in b frame. matrix B, and measurement matrix H are described as follows: The difference between the external reference velocity and velocity calculated by the system is

taken as the observation. η is the measurement noise array. Coefficient matrix A , system noise driven matrix B , and measurement matrix H are described as follows:

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    A=  


G × − ωiG (fG ×)2,3 03×3 03×3 03×3

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(CωGv )3,2
G + ωG )×] VG × CωGv − [(2ωie eG 2,2
CeG 3,2 03 × 2 03 × 2



−CbG  B =  02 × 3

CωeR + CωGR
VG × (CωGR + 2CωeR ) 2,3 −CeG (VG ×)CR 03×3 03×3

03 × 3 (CbG )2,3

−CbG 02×3 03×3 03×3 03×3

03×3
CbG 2,3 03×3 03×3 03×3

     (13)  

 (14)

 

09 × 6 " H=

02×3

1 0

0 1

# (15)

02 × 9

where the Vector× means the skew-symmetric matrix related to the Vector, the (matrix)m,n means the first m rows and first n columns of the matrix. The detailed expressions of submatrices CωGv , CωeR , CωGR and CR can refer to the reference [16]. 3.2. Kalman Filtering and Feedback Correction Kalman filtering is a technology based on the minimum variance criteria, and the states are estimated according to the state and measurement equation. The equations of the discrete Kalman filtering are described as follows, and the detailed definition of each symbol can be seen in reference [29]. Xˆ k/k−1 = φk,k−1 Xˆ k−1 T T Pk/k−1 = φk,k−1 Pk−1 φk,k −1 + Γ k −1 Qk −1 Γ k −1 −1

Kk = Pk/k−1 HkT (Hk Pk/k−1 HkT + Rk ) ˆ k = Xˆ k/k−1 + Kk (Zk − Hk Xˆ k/k−1 ) X

(16)

Pk = (I − Kk Hk )Pk/k−1 φk,k−1 , Γk−1 , Hk , and Zk are determined after the discretizing the Equation (12), when the initializations of the state vector X and P matrix are finished, the Kalman filtering can be conducted. Section 3.1 has given the error model of the shipborne grid SINS, and the error model of the DVL should be considered as well. In practice, the velocity measured by DVL can easily be affected by the external environment. On one hand, the velocity measured by DVL is easily polluted by the ocean currents when the ship is maneuvering; on the other hand, in some sea areas, such as the gulfs and trenches, the DVL measurement error is sensitive to the changes of the ocean currents. For the reasons above, establishing a precise error model of the DVL becomes difficult. In order to solve this problem, when the external reference velocity is available, combining with the actual situation, the error model of the external velocity can be simplified as a constant in a certain period of time. In this way, when the external velocity is available, supposing that the external velocity error is a constant, the grid SINS works in an external level damping state; otherwise the grid SINS works in a non-damping state. Therefore, the key to the question is to find out an effective criterion to judge if the velocity measured by DVL is available. The criterion can be established according to the changes of the observation. The observation can be described as: Z = VGSI NS − VDVL , VGSI NS represents the velocity calculated by grid SINS, and VDVL represents the velocity measured by DVL. VGSI NS and VDVL can be described as follows: VGSI NS = V + δVGSI NS and VDVL = V + δVDVL , where V is the true velocity of the ship, δVGSI NS represents the velocity error of the grid SINS, and δVDVL represents the velocity measurement error of the DVL. The observation Z can be described as: Z = VGSI NS − VDVL = δVGSI NS − δVDVL

(17)

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Z  VGSINS  VDVL = δVGSINS  δVDVL

(17)

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Z1 and Z 2 can be obtained as: Z1 and Z2 represent the observation at time t1 and t2 , respectively. The difference between Z1 and 1 2 1 2 δZ = Z1  Z2 = (δVGSINS  δVGSINS )  (δVDVL  δVDVL ) (18) Z2 can be obtained as: 1 2 2 velocity errors 1 of grid2 SINS at time t and t ;  δVGSINS where δVGSINS difference 1between the δZ =isZthe (18) 1 2 1 − Z2 = ( δVGSI NS − δVGSI NS ) − ( δV DVL − δV DVL ) 1 2 δVDVL  δVDVL is the difference between the velocity measurement errors of DVL at time t1 and t2 . 1 where δVGSI NS 1− δV2GSI NS is the difference between the velocity errors of grid SINS at time t1 and t2 ; 2  δV Obviously, not change in a short errors periodof ofDVL time,atthus, 1 2δVGSINS GSINS willbetween δV − δV is the difference the dramatically velocity measurement timeast1long and as t2 . DVL DVL 1 2 which δZ changes means the external velocity measurement error has greatly changed Obviously, δVdramatically, − δV will not change dramatically in a short period of time, thus, as long as GSI NS GSI NS as achanges result ofdramatically, the ship’s maneuvering orthe changes of the sea conditions, in this case, external velocity δZ which means external velocity measurement error hasthe greatly changed as δZ changes by DVL be used. the basis ofsea above-mentioned analysis, when ameasured result of the ship’scannot maneuvering orOn changes of the conditions, in this case, the external velocity dramatically a certain period of time, SINS works in the non-damping the measured byin DVL cannot be used. Onthe thegrid basis of above-mentioned analysis,state, whenotherwise δZ changes grid SINS works the external level damping dramatically in a in certain period of time, the gridstate. SINS works in the non-damping state, otherwise the Whenworks DVL in canthe provide anlevel accurate reference grid SINS external damping state. velocity, the grid level velocity errors, position errors and grid angles can all be accurately estimated. attitude, velocity, and When DVLlevel can attitude provide error an accurate reference velocity, the grid levelThe velocity errors, position position errors grid SINS can be angles compensated byaccurately using theseestimated. three kinds ofattitude, estimatesvelocity, to calibrate errors and grid of level attitude error can all be The and the system. However, a result ofcompensated the measurement errorthese of thethree DVL,kinds the observability of calibrate the state position errors of grid as SINS can be by using of estimates to variables changes. Practical tests of show that grid levelerror velocity errors errorsofcannot be the system. However, as a result the measurement of the DVL,and theposition observability the state estimatedchanges. accurately when the reference velocity contains constant error, the grid level attitude variables Practical tests show that grid level velocity errors and but position errors cannot be error angles can still be accurately estimated. In this way, only the estimated attitude error estimated accurately when the reference velocity contains constant error, butgrid the level grid level attitude anglesangles can becan used calibrate the estimated. system. In this way, only the estimated grid level attitude error error stilltobe accurately Two angles cancorrection be used tomethods calibrateare theusually system.used to calibrate the system: one is feedback correction and the other is output correction. Obviously, if output correction is selected, only the oscillation errors Two correction methods are usually used to calibrate the system: one is feedback correction and of the grid attitude angles can be suppressed, as a resultisofselected, the system’s open-loop structure. the other is level output correction. Obviously, if output correction only the oscillation errors Feedback is theangles suitable way calibrate the can suppress not of the gridcorrection level attitude can be to suppressed, as system a resultfor of this the method, system’sand open-loop structure. only the oscillation of the grid level attitudethe angles, butfor also the oscillation of the grid Feedback correctionerrors is the suitable way to calibrate system this method, and errors can suppress not level the velocity and position. Inthe fact,grid the existence of theangles, level attitude angles is the essential reason only oscillation errors of level attitude but alsoerror the oscillation errors of the grid for the Schuler the Schuler errors can be suppressed effectively level velocity andperiodic position.oscillation, In fact, theso existence of theoscillation level attitude error angles is the essential reason through the feedback correction. Heresowe take the grid north level circuit ansuppressed example andeffectively the block for the Schuler periodic oscillation, the Schuler oscillation errors canasbe diagramthe of the KF-DGSINS algorithm is take shown Figure In Figure thethe variable through feedback correction. Here we thein grid north3.level circuit3,asSanrepresents example and block 1 diagram of the KF-DGSINS algorithm is shown in Figure 3. In Figure 3, S represents the variable after after the Laplace transformation, and means the integration process. R represents the radius of S the integration process. R represents the radius of the earth. the Laplace transformation, and S1 means ˆ G represents ˆG represents φ estimatethe of estimate the grid east attitude error angle.error angle. the of the grid east attitude E earth.  E the

Kalman filter

Figure 3. 3. Block Block diagram diagram of of KF-DGSINS KF-DGSINS algorithm. algorithm. Figure

4. The T-DGSINS Algorithm 4. The T-DGSINS Algorithm Referring to the traditional external level damping algorithm based on the damping network of Referring to the traditional external level damping algorithm based on the damping network of common SINS, the T-DGSINS based on classical control theory can be designed. Taking the grid north common SINS, the T-DGSINS based on classical control theory can be designed. Taking the grid north level circuit as an example, the block diagram of the T-DGSINS is shown in Figure 4. The external level circuit as an example, the block diagram of the T-DGSINS is shown in Figure 4. The external reference velocity is introduced to the system through the damping network 1 − H (s), from the physical sense, the introduction of the external reference velocity just adds a compensation channel

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reference velocity is introduced to the system through the damping network 1  H( s) , from the

physical sense, the introduction of the external reference velocity just adds a compensation channel to the the system, system, and and the the navigation navigation accuracy accuracy of of this this method method isis sensitive sensitive to to the the error error of of the the external external to reference velocity. velocity. reference

Figure Figure4.4.Block Blockdiagram diagramof ofT-DGSINS T-DGSINS algorithm. algorithm.

The T-DGSINS based on the damping network can effectively dampen the Schuler oscillation The T-DGSINS based on the damping network can effectively dampen the Schuler oscillation errors of the grid SINS, but several obvious disadvantages of this method are difficult to overcome: errors of the grid SINS, but several obvious disadvantages of this method are difficult to overcome: 1. The damping network added to the system breaks the applicability of the Schuler adjustment 1. condition, The damping added to the system breaks the applicability the Schuler adjustment and network an equivalent step-function signal is introduced to theof system when the state of condition, and an equivalent step-function signal is introduced to the system when the state state. of the the grid SINS is switched from the non-damping state to the external level damping grid SINS is switched from the non-damping to the damping state. Therefore, Therefore, the error overshoot phenomenonstate exists in external the statelevel switching process, and the the error overshoot phenomenon exists in the state switching process, and the navigation results navigation results cannot be used for a certain period of time. cannot be used for a certain period of time. 2. T-DGSINS is designed based on the conditions of a single-input-single-output system (SISO), so 2. the T-DGSINS designed based on the conditions of a be single-input-single-output system (SISO), couplingiseffects of each calculating circuit cannot solved. so the coupling effects of each calculating circuit cannot be solved. 5. Simulation Results and Discussions 5. Simulation Results and Discussions This section is divided into two parts: one is the simulation results’ analysis, the other is the This section is divided into two parts: one the simulation results’ analysis, the is the semi-physical simulation results’ analysis. The issimulation results demonstrated the other theoretical semi-physical simulation results’ analysis. The simulation results demonstrated the theoretical feasibility of the KF-DGSINS, and the semi-physical simulation results confirmed the validity of the feasibility ofinthe KF-DGSINS, and the semi-physical simulation results confirmed the validity of KF-DGSINS practical application. the KF-DGSINS in practical application. 5.1. Simulation Results and Discussions 5.1. Simulation Results and Discussions Aiming at the grid SINS, in order to test the damping performance of the KF-DGSINS, nonAiming at the grid SINS, in order to test the damping performance of the KF-DGSINS, damping grid SINS and KF-DGSINS are simulated on a stationary base. Gyroscopic and non-damping grid SINS and KF-DGSINS are simulated on a stationary base. Gyroscopic and accelerometer outputs are generated by the track generator. The general idea of the track generator accelerometer outputs are generated by the track generator. The general idea of the track generator is: according to the change rules of the carrier’s position, velocity, attitude, and the settings of the is: according to the change rules of the carrier’s position, velocity, attitude, and the settings of the IMU error, the outputs of the gyroscope and accelerometer can be back-stepped through the SINS IMU error, the outputs of the gyroscope and accelerometer can be back-stepped through the SINS mechanization. The simulation conditions are set as follows: simulation lasts 30 h; the initial latitude mechanization. The simulation conditions are set as follows: simulation lasts 30 h; the initial latitude and longitude are 85◦ N and ◦18 E ; the three-axis gyro constant drifts are set as 0.003 degree per and longitude are 85 N and 18 E; the three-axis gyro constant drifts are set as 0.003 degree per hour hour andthree-axis the three-axis accelerometer constant set as 0.00005 the random error of g , and and the accelerometer constant biases biases are set are as 0.00005 g, and the random error of the gyro the and accelerometer arewhite set asnoise; whitethe noise; theoferror of the external reference velocity is set as andgyro accelerometer are set as error the external reference velocity is set as white white The simulation are shown in Figures noise.noise. The simulation resultsresults are shown in Figures 5–7. 5–7. The dotted green line is the error curve of the non-damping grid SINS; solid red is line the The green line is the error curve of the non-damping grid SINS; the the solid red line theiserror error theSINS, grid SINS, adopting the KF-DGSINS. The simulation resultsthat show the periodic Schuler curvecurve of theof grid adopting the KF-DGSINS. The simulation results show the that Schuler periodic oscillation in the of grid level grid attitude, level and velocity, and The position. The oscillation exists in exists the errors of errors grid level attitude, levelgrid velocity, position. proposed proposed KF-DGSINS effectively suppressed the Schuler periodic oscillation, and at the same time KF-DGSINS effectively suppressed the Schuler periodic oscillation, and at the same time the Foucault the Foucault periodicwhich oscillation, which the Schuler periodic For griderrors, level periodic oscillation, modulates themodulates Schuler periodic oscillation. Foroscillation. grid level attitude attitude onlyerrors the constant caused by accelerometer biases remain; forvelocity grid level velocity only theerrors, constant caused errors by accelerometer biases remain; for grid level errors and errors and position errors, the 24 h earth periodic oscillation still exists in the system. The error of the position errors, the 24 h earth periodic oscillation still exists in the system. The error of the grid course grid course angleinfluenced is mainly influenced byperiodic the earthoscillation, periodic oscillation, non-damping angle is mainly by the earth therefore, therefore, non-damping grid SINSgrid and SINS and KF-DGSINS achieved almost the same effect. In conclusion, aiming at ships which have the KF-DGSINS achieved almost the same effect. In conclusion, aiming at ships which have the external

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9 of 17 information to conduct a system reset regularly, the KF-DGSINS can observably reference information to conduct atosystem regularly, theregularly, KF-DGSINS observably improve the improve the navigation accuracy of the reset grid SINS. external reference information conduct a system reset the can KF-DGSINS can observably external accuracy reference of information to conduct a system reset regularly, the KF-DGSINS can observably navigation the grid SINS. improve the navigation accuracy of the grid SINS. improve the navigation accuracy of the grid SINS.

Figure 5. Grid attitude angle errors of the simulation results. Figure 5. Grid attitude angle errors of the simulation results. Figure 5. Grid attitude angle errors ofof the simulation Figure 5. Grid attitude angle errors the simulationresults. results.

Figure 6. Grid level velocity errors of the simulation results. Figure 6. Grid level velocity errors of the simulation results. Figure 6. Grid level velocity errors of the simulation results. Figure 6. Grid level velocity errors of the simulation results.

Figure 7. Position errors in e frame of the simulation results. Figure 7. Position errors in e frame of the simulation results. Figure 7. Position errors in e frame of the simulation results. Figure 7. Position errors in e frame of the simulation results.

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When the state of grid SINS is switched from the non-damping state to the external level damping state, big overshoot errors are introduced to the system by using the T-DGSINS. The KF-DGSINS proposed in this paper can restrain the overshoot errors effectively. In order to verify the advantage of the KF-DGSINS, the T-DGSINS and the KF-DGSINS are simulated under the same conditions. The simulation conditions are set as follows: gyro and accelerometer outputs are generated by the track generator; simulation lasts 10 h; the initial latitude and longitude are 85◦ N and 18◦ E; the ship sails eastward along the latitude circle and the velocity is 10 m/s; the swing of the ship is set as the sine function:     pitch = pitchm sin(2π/TP + Ph P ) (19) roll = rollm sin(2π/TR + Ph R )    heading = 90◦ + heading sin(2π/T + Ph ) m

H

H

where pitchm , rollm , headingm are swing amplitudes, and pitchm = 3◦ , rollm = 5◦ , headingm = 4◦ ; TP , TR , TH are swing periods, and TP = 7 s, TR = 9 s, TH = 12 s; Ph P , Ph R , Ph H are initial phases and they are random values in the range from 0◦ to 90◦ . In practice, the velocity measured by DVL is unavailable when the ship maneuvers or sails in some special sea areas. When the ship sails in a steady state, according to the previous introduction, the error of the DVL can be simplified as a constant, and the velocity measured by DVL is available. Considering the practical situation, the errors of the grid east and north velocity can be set as follows: before t = 5 h, the velocity measured by DVL is set to be unavailable. In order to simulate the condition that the error of the external reference velocity changes dramatically, the errors are set as the zero mean white noise with large amplitudes; after t = 5 h, the velocity measured by DVL is set to be available, and the error model of the velocity measured by DVL is simplified as the constant. In this case, the errors are set as the sum of the white noise and the constant (1kn). Taking the error of the grid east reference velocity as an example, the error of the grid east reference velocity is shown in Figure 8. When the reference velocity error is set as zero, the results of the T-DGSINS represent the ideal damping results of the grid SINS. In this way, in order to better reflect the advantage of the proposed KF-DGSINS, the ideal damping results are given when the reference velocity error is set as zero from beginning to end. The simulation results are shown in Figures 9–11. 6

4

m/s

2

0

-2

-4

-6 0

2

4

6

8

Time (h)

Figure 8. The error of the grid east reference velocity.

10

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Figure 9. Grid attitude angle errors of the simulation results. Figure 9. Grid attitude angle errors of the simulation results. Figure 9. Grid attitude angle errors of the simulation results.

Figure 10. Grid level velocity errors of the simulation results. Figure 10. Grid level velocity errors of the simulation results. Figure 10. Grid level velocity errors of the simulation results.

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Figure 11. Position errors in e frame of the simulation results. Figure 11. Position errors in e frame of the simulation results.

The solid green line SINS; the the solid solidblack blackline lineisisthe The solid green lineisisthe theideal idealdamping dampingerror errorcurve curve of of the grid SINS; theerror errorcurve curve grid SINS, adopting T-DGSINS; the dotted red is line the error ofof thethe grid SINS, adopting the the T-DGSINS; the dotted red line theiserror curvecurve of the of grid theSINS, grid SINS, adopting the KF-DGSINS. Before theerror error of the changes t =t5=h 5, h,the adopting the KF-DGSINS. Before thereference referencevelocity velocity changes dramatically. According to to thethe criterion which is is introduced in in Section 3.2,3.2, thethe external reference dramatically. According criterion which introduced Section external reference velocity cannot bebe used. Therefore, thethe grid SINS works inin the non-damping state; 5 hthe velocity cannot used. Therefore, grid SINS works the non-damping state;after aftert =t =5 h, , the external reference velocity changes slightly. According to the criterion, the external reference velocity external reference velocity changes slightly. According to the criterion, the external reference velocity becomes available, therefore, the grid SINS works in an external level damping state. What needs to be becomes available, therefore, the grid SINS works in an external level damping state. What needs to stressed is that the feedback correction should start after the process of Kalman filtering becomes stable. be stressed is that the feedback correction should start after the process of Kalman filtering becomes Some conclusions can be made from the simulation results: (1) two kinds of algorithm can effectively stable. Some conclusions can be made from the simulation results: (1) two kinds of algorithm can suppress the Schuler oscillation errors, and in the state switching process, the overshoot errors of effectively suppress the Schuler oscillation errors, and in the state switching process, the overshoot the KF-DGSINS are obviously smaller than the overshoot errors of the T-DGSINS. For T-DGSINS, errors of the KF-DGSINS are obviously smaller than the overshoot errors of the T-DGSINS. For Twhen the state of system is switched from the non-damping state to the external level damping state, DGSINS, when the state of system is switched from the non-damping state to the external level the damping network is added to the level circuit of the grid SINS, and an equivalent step-function damping state, the damping network is added to the level circuit of the grid SINS, and an equivalent signal is introduced to the system. In this case, the equilibrium state of the system is broken, and step-function signal is introduced to the system. In this case, the equilibrium state of the system is the overshoot phenomenon occurs. For KF-DGSINS, no damping network is added to the system. broken, and the overshoot phenomenon occurs. For KF-DGSINS, no damping network is added to When the Kalman filter is stable, the feedback correction of the grid level attitude error angles can the system. When the Kalman filter is stable, the feedback correction of the grid level attitude error suppress the Schuler oscillation error without breaking the equilibrium state of the system, and this angles can suppress the Schuler oscillation error without breaking the equilibrium state of the system, is the reason why KF-DGSINS has a better performance in restraining the overshoot phenomenon and this is the reason why KF-DGSINS has a better performance in restraining the overshoot than T-DGSINS; (2) the grid course angle error is hardly influenced by the overshoot phenomenon, phenomenon than T-DGSINS; (2) the grid course angle error is hardly influenced by the overshoot therefore, T-DGSINS and KF-DGSINS achieved almost the same effect; (3) when the damping state is phenomenon, therefore, T-DGSINS and KF-DGSINS achieved almost the same effect; (3) when the stable, the damping results of the KF-DGSINS and T-DGSINS are almost the same as the ideal damping damping state is stable, the damping results of the KF-DGSINS and T-DGSINS are almost the same results, which means the constant error of the external reference velocity will not affect the damping as the ideal damping results, which means the constant error of the external reference velocity will results of thethe KF-DGSINS T-DGSINS when the damping state is stable. compared not affect damping and results of the KF-DGSINS and T-DGSINS when In thea word, damping state iswith stable. theInT-DGSINS, the KF-DGSINS has a better performance in overcoming theperformance overshoot errors and this a word, compared with the T-DGSINS, the KF-DGSINS has a better in overcoming algorithm effectively improves navigation accuracy of the grid in the state switching the overshoot errors and thisthe algorithm effectively improves theSINS navigation accuracy of the process. grid SINS

in the state switching process.

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5.2. Semi-Physical Simulation Results and Discussions 5.2. Semi-Physical Simulation Results andrestriction, Discussionsthe semi-physical simulation is conducted to verify Due to the limit of the geographic the performance of the proposed algorithm in practice. The outputs of the gyroscopes Due to the limit of the geographic restriction, the semi-physical simulation is conductedand to accelerometers can be described as follows: verify the performance of the proposed algorithm in practice. The outputs of the gyroscopes and b b b accelerometers can be described as follows:  ωib _ real = ωib _ ideal + ωib _ error ( b b (20) b + f b + ωb f_ real ==f_bideal ω ωib_ideal _ error ib_real ib_error (20) b b b f_real = f_ideal + f_error b b where ωib _ real and f_ real are the measurements of the gyroscopes and accelerometers, respectively; b b b the ideal and are and accelerometers, ωbib _ ideal ω where accelerometers, respectively; respectively; ideal f_real are the measurements of the gyroscopes ib_realf_ and b b b b ω and f are the ideal measurements of the gyroscopes and accelerometers, respectively; _ideal are the measurement errors of the gyroscopes, and f_ error are the measurement errors of the ωibib_ideal _ error b b ωib_error are the measurement errors of the gyroscopes, and f_error are the measurement errors of accelerometers. the accelerometers. In order to get more realistic simulation measurements of the gyroscopes and the accelerometers In order to get more realistic simulation measurements of the gyroscopes and the accelerometers in in polar regions, the key point is to get the measurement errors of the gyroscopes and the polar regions, the key point is to get the measurement errors of the gyroscopes and the accelerometers. accelerometers. Once the initial position, velocity, attitude, and the form of motion are determined, b Once the initial position, velocity, attitude, and the form of motion are determined, ωib_ideal and threeωb bib _ ideal and f_bideal can be generated by the track generator. With the aid of the high-precision f_ideal can be generated by the track generator. With the aid of the high-precision three-axis turntable, b b turntable, b ωib _can axis and f_berror can be obtained easily. Collecting the done IMU data was state, done in static error be obtained ωib_error and f_error easily. Collecting the IMU data was in static and the static and IMUthe data collection system is shown in Figure 12. in Figure 12. state, static IMU data collection system is shown

Turantable Turntable

SINS

Figure 12. Static Static IMU data collection system.

The SINS is installed on the three-axis turntable, and the attitude and position of the turntable The SINS is installed on the three-axis turntable, and the attitude and position of the turntable are precisely known. The experimental data is provided by the IMU. The high-precision three-axis are precisely known. The experimental data is provided by the IMU. The high-precision three-axis turntable can provide the accurate attitude reference of the carrier. With the aid of the high-precision turntable can provide the accurate attitude reference of the carrier. With the aid of the high-precision three-axis turntable, the carrier can move in preset attitude change rules. The main technical indexes three-axis turntable, the carrier can move in preset attitude change rules. The main technical indexes of the SGT-8 three-axis turntable in this experiment are shown in Table 1. of the SGT-8 three-axis turntable in this experiment are shown in Table 1. Table 1. The technical indexes of the SGT-8 three-axis turntable. Table 1. The technical indexes of the SGT-8 three-axis turntable.

Outer Axis

3 Location accuracy of angular position Outer Axis LocationMaximum accuracy ofangular angular position rate Maximum angular rate Rotation rate resolution Rotation rate resolution

±3 180 ±180 0.0010.001

Middle Axis Middle Axis

2

±2250 ±250 0.001 0.001

Inner Axis Inner Axis

2

±2400 ±400 0.001 0.001

Unit Unit arc sec  arc sec /s ◦ /s  ◦ /s/ s

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14 of 17 of the three-axis gyroscopes are only sensitive to the rotational angular velocity the earth, and the three-axis accelerometers are only sensitive to the local acceleration of gravity. Therefore, ωbib _ error and f_berror can be calculated by using (21): In static state, the three-axis gyroscopes are only sensitive to the rotational angular velocity of the earth, and the three-axis accelerometers areonly sensitive to theblocal acceleration of gravity. Therefore, b b ωib _ error = ωie _ real  ωie b b ωib_error and f_error can be calculated by using (21)  b (21): b f_ error = f_ real  f b   ( b b b ωib_error = ωie_real − ωie b b f ω where ie _ real and _ real are the measurements of the gyroscopes and accelerometers in static(21) state; b b f_error = f_real − fb b b ωie and f can be calculated by using the precise position and attitude of the turntable.

b b where ω andparameters f_real are theofmeasurements theaccelerometers gyroscopes and in staticfrom state; The related the gyroscopesof and canaccelerometers be roughly calculated the ie_real b b  8 ωstatic f can bethey calculated by using the precise position and attitude of the turntable. and are shown as follows: the three-axis gyro constant drifts are 2.573  10 rad/s , ie and test, The related parameters of 8the gyroscopes and accelerometers can be roughly calculated 8 2.607  10 rad/s , 2.387  10 rad/s , respectively; the three-axis gyro random drift variances are from the static test, and they are shown as follows: the6 three-axis gyro constant drifts are 6 (2.9938 −10 rad/s)2 , (2.4866−8106 rad/s)2 , (3.2143  10 rad/s)2 , respectively; the three-axis 8 − 8 2.573 × 10 rad/s, −2.607 × 10 rad/s, 2.387 × 10 rad/s, respectively; the three-axis gyro 4 2 2 4 2 2 , 22 accelerometer constant are10−3.214 6 rad/s  104 )m/s  10 m/s random drift variances are biases (2.9938 × ) , m/s , 3.21433.072 2.4866, × 2.847 10−6 rad/s × 10−610 rad/s ) , , 2 22 2 respectively; three-axis accelerometer random biases respectively; the the three-axis accelerometer constant biases are 3.214variances × 10−4 m/sare , 2.847 × 10−4m/s m/s (0.001594 ), , 2 − 4 2 2 the three-axis accelerometer random biases variances are 2 2 −3.072 × 10 m/s , respectively; (0.001771 m/s ) , (0.001817 m/s ) , respectively.

2

2

2

2 (0.001594 ) , (0.001771 m/s2 ) are , (0.001817 m/s2 ) , respectively. Them/s simulation conditions set as follows: simulation lasts 10 h; the initial latitude and The simulation conditions are set as follows: simulation lasts 10 h; the initial latitude and longitude are 85 N and 18 E ; the ship sails eastward along the latitude circle and the velocity is longitude are 85◦ N and 18◦ E; the ship sails eastward along the latitude circle and the velocity is 10m/s; the swing of the ship is set as (18), and each parameter is the same as before. Due to the changes 10 m/s; the swing of the ship is set as (18), and each parameter is the same as before. Due to the of the external velocity errors, before t = 5 h , the external velocity is set to be unavailable, and the changes of the external velocity errors, before t = 5 h, the external velocity is set to be unavailable, t = 5th=, the grid in theinnon-damping state;state; after after velocity is available with and theSINS grid works SINS works the non-damping 5 h, external the external velocity is available constant errors, andand the the system works in the external level damping state. The The results of the with constant errors, system works in the external level damping state. results of semithe physical simulation are shown in Figures 13–15. semi-physical simulation are shown in Figures 13–15.

Figure Grid attitude angle errors semi-physical simulation results. Figure 13.13. Grid attitude angle errors of of thethe semi-physical simulation results.

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Figure 14. Grid level velocity errors of the semi-physical simulation results. Figure 14. Grid level velocity errors of the semi-physical simulation results. Figure 14. Grid level velocity errors of the semi-physical simulation results.

Figure of the semi-physical simulation results. e frame Figure15. 15.Position Positionerrors errorsinine frame of the semi-physical simulation results. Figure 15. Position errors in e frame of the semi-physical simulation results.

The solid green line is the error curve non-damping grid SINS; solid black line is the The solid green line is the error curve of of thethe non-damping grid SINS; thethe solid black line is the The solid green line is the error curve of the non-damping grid SINS; the solid black line is the error curve thegrid gridSINS, SINS,adopting adopting the the T-DGSINS; the dotted curve of of thethe grid error curve ofofthe dottedred redline lineisisthe theerror error curve error curve of the grid SINS, adopting the T-DGSINS; the dotted red line is the error curve of the grid SINS, adopting Figures 13–15, 13–15,the thesemi-physical semi-physicalsimulation simulation results grid SINS, adoptingthe theKF-DGSINS. KF-DGSINS.As Asshown shown in Figures results SINS, adopting the KF-DGSINS. As shown in Figures 13–15, the semi-physical simulation results further verify that the proposed KF-DGSINS has a better performance than the T-DGSINS. The results further verify that the proposed KF-DGSINS has a better performance than the T-DGSINS. The results verify that the proposed coincide KF-DGSINS has a better performance than the T-DGSINS. The results of the semi-physical simulation coincidewith with the results results: on hand, the offurther the semi-physical simulation the results of the simulation simulation results: onone one hand, of the semi-physical simulation coincide with the results of the simulation results: on one hand, the the Schuler Schuleroscillation oscillationerrors errorseffectively, effectively,onon the other theKF-DGSINS KF-DGSINSand andthe theT-DGSINS T-DGSINS can can suppress the the other KF-DGSINS and the T-DGSINS can suppress the as Schuler oscillation errorsnetwork effectively, on theadded other hand, state switching process, KF-DGSINS, a result no damping network being hand, in in thethe state switching process, for for KF-DGSINS, aas result of noofdamping being added to hand, in the state switching process, for KF-DGSINS, as a result of no damping network being added the circuit, the equilibrium of the grid SINS maintains, so the overshoot errors of the KFthetocircuit, the equilibrium state ofstate the grid SINS maintains, so the overshoot errors of the KF-DGSINS to the circuit, the equilibrium state of the the gridofSINS maintains, so T-DGSINS, the errors thethe KFDGSINS aresmaller obviously than overshoot errors of the and this of effectively are obviously thansmaller the overshoot errors the T-DGSINS, and thisovershoot effectively improves DGSINS are than thegrid overshoot of switching the T-DGSINS, and thiscourse effectively improvesaccuracy the obviously navigation accuracy of SINS inerrors theprocess. state process. Grid angle navigation of the smaller grid SINS in the state switching Grid course angle error is hardly improves the navigation accuracy of the grid SINS in the state switching process. Grid course angle

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influenced by the overshoot phenomenon, therefore T-DGSINS and KF-DGSINS achieved almost the same effect. The KF-DGSINS has a better performance in overcoming the overshoot errors when the state of grid SINS is switched from the non-damping state to the external level damping state, therefore, the shipborne grid SINS based on the KF-DGSINS is feasible for ships working in polar regions. 6. Conclusions Aiming at ships which have the external reference information to conduct a system reset regularly, non-damping and external level damping working states are two kinds of frequently-used working states for ships sailing in polar regions. Therefore, the working state of the ship is frequently switched between the two working states mentioned above. In order to overcome the overshoot phenomenon of the T-DGSINS in the state switching process, the KF-DGSINS is proposed in this paper. The simulation results show that the KF-DGSINS can achieve the effective suppression of the Schuler oscillation errors, and can also reduce the overshoot errors in the state switching process. In conclusion, in view of the advantages of the KF-DGSINS, the KF-DGSINS can satisfy the needs of ships sailing in polar regions. Acknowledgments: This work is funded by the National Natural Science Foundation of China under grant No. 61633008. Author Contributions: Weiquan Huang and Tao Fang conceived and designed the experiments; Tao Fang performed the experiments; Weiquan Huang contributed experiment tools and analyzed the data; Tao Fang, Li Luo, Lin Zhao and Fengzhu Che wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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