Gravity Compensation Using EGM2008 for High-Precision - MDPI

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Gravity Compensation Using EGM2008 for High-Precision Long-Term Inertial Navigation Systems Ruonan Wu, Qiuping Wu *, Fengtian Han *, Tianyi Liu, Peida Hu and Haixia Li Department of Precision Instrument, Tsinghua University, Beijing 100084, China; [email protected] (R.W.); [email protected] (T.L.); [email protected] (P.H.); [email protected] (H.L.) * Correspondence: [email protected] (Q.W.); [email protected] (F.H.); Tel.: +86-10-6279-8065 (Q.W.); +86-10-6279-8645 (F.H.) Academic Editor: Jörg F. Wagner Received: 31 October 2016; Accepted: 14 December 2016; Published: 18 December 2016

Abstract: The gravity disturbance vector is one of the major error sources in high-precision and long-term inertial navigation applications. Specific to the inertial navigation systems (INSs) with high-order horizontal damping networks, analyses of the error propagation show that the gravity-induced errors exist almost exclusively in the horizontal channels and are mostly caused by deflections of the vertical (DOV). Low-frequency components of the DOV propagate into the latitude and longitude errors at a ratio of 1:1 and time-varying fluctuations in the DOV excite Schuler oscillation. This paper presents two gravity compensation methods using the Earth Gravitational Model 2008 (EGM2008), namely, interpolation from the off-line database and computing gravity vectors directly using the spherical harmonic model. Particular attention is given to the error contribution of the gravity update interval and computing time delay. It is recommended for the marine navigation that a gravity vector should be calculated within 1 s and updated every 100 s at most. To meet this demand, the time duration of calculating the current gravity vector using EGM2008 has been reduced to less than 1 s by optimizing the calculation procedure. A few off-line experiments were conducted using the data of a shipborne INS collected during an actual sea test. With the aid of EGM2008, most of the low-frequency components of the position errors caused by the gravity disturbance vector have been removed and the Schuler oscillation has been attenuated effectively. In the rugged terrain, the horizontal position error could be reduced at best 48.85% of its regional maximum. The experimental results match with the theoretical analysis and indicate that EGM2008 is suitable for gravity compensation of the high-precision and long-term INSs. Keywords: gravity disturbance vector compensation; high-precision INS; EGM2008; deflections of the vertical (DOV); update interval

1. Introduction In inertial navigation systems (INSs), the accelerometer-sensed specific force consists of the kinematic acceleration and the gravitational acceleration. Thus gravitational information along the route plays an important role to extract the kinematic acceleration of the vehicle first. Generally, the normal gravity is employed in order to achieve a balance between the accuracy and computation efficiency. The difference between the actual and the normal gravity vector, namely the so-called gravity disturbance vector, is one of the error sources in INSs. The influence of the gravity disturbance vector is mostly negligible compared to inertial measurement unit (IMU) errors and some other factors. However, for the long-term INSs with precise inertial sensors and efficient algorithms, the gravitational errors should be taken into account to achieve high navigation precision. A number of researchers have used the covariance propagation analysis to theoretically investigate the gravity Sensors 2016, 16, 2177; doi:10.3390/s16122177

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induced navigation errors as early as the second half of the 20th century. Levine and Gelb evaluated the gravity-induced velocity, position, azimuth and platform tilt errors for a wide range of vehicle speeds, and concluded that they were significantly affected by mission speed, operating latitude and system damping ratio [1]. Schwarz’s results showed that the gravity-induced position errors mainly resulted from the poorly modeled deflections of the vertical (DOV), while the effect of the gravity anomaly and geoidal undulations could be in general neglected [2]. Harriman and Harrison reported that errors in a damped INS were predominately cross-track at all times, and that the position and velocity errors resulting from gravity disturbance errors would decrease if the flight speed or altitude was increased [3]. Soon it was observed that on each of several westbound transpacific flights the Schuler oscillation within the velocity errors would grow rather significantly along the Kuril trench [4], which provided some evidences for the theoretical works. Besides the pure-inertia systems, even though most errors can be corrected by the Global Positioning System (GPS) updates in a GPS/INS integrated system, it should be noted that the gravity-induced attitude errors still exist [5,6] and that position and velocity errors would arise during the time of GPS signal loss [7]. Therefore, the compensation of the gravity disturbance vector is imperative for those applications demanding high-accuracy inertial navigation solutions, such as submarines. Statistic models can be used not only to analyze the error propagation of unknown gravity, but also for optimal prediction and filtering of the gravity disturbance in INSs [5], airborne gravimetry [8] and precise orbit determination for Earth-orbiting satellites [9]. The third-order Markov undulation model is reported to be both convenient and appropriate for the analysis of the gravity uncertainties induced errors in INSs, and suitable for the Kalman filter technique [10]. However, the statistic models of the gravity disturbance vector have two inherent disadvantages. First, a single low-order model is not sufficient to describe a large-scale gravity field due to the diversity of topography; and secondly, a large amount of a priori information must be collected to precisely estimate the parameters of the statistic models. Nowadays, benefiting from advanced gravimeters and gradiometers and easier access to accurate high-resolution gravity data archives, it has become a better option using the attainable gravitational information directly. Several theoretical works and simulations indicated that the gradiometer would help improve the navigation performance with in situ measurements of the gravity field [11,12], yet the high cost still limits its application in comparison to accurate high-resolution data archives. According to Kwon and Jekeli’s research, with ground data gridded with 2 arc-minutes resolution and accurate to better than 3 mGal, the error in gravity compensation contributes less than 5 m to the position error after one hour of free-inertial navigation for a typical flight trajectory at 5 km altitude and 300 km/h speed [13]. A few global or near-global maps of DOV and gravity disturbances have already been released, which are adequate to meet such requirement. For instance, the DOV data set released under the model GGMplus has a much higher resolution of 7.2 arc-seconds, covering 80% of Earth’s land masses [14]. To make the best of such ground data, the methods of interpolation and upward continuation must be carefully chosen [13,15]. Another alternative to obtain the gravitational information, which is chosen in this work, is to make use of spherical harmonic models. The Earth Gravitational Model 2008 (EGM2008) is such a model developed by a least squares combination of the satellite-only ITG-GTACE03S gravitational model with a global set of area-mean free-air gravity anomalies [16]. Assessments in various regions around the world indicate that it performs comparably with contemporary detailed regional geoid models [17–21]. For example, the EGM2008 DOVs over USA, Europe and Australia are within 1.1 to 1.3 arc-seconds of independent astrogeodetic values [16,17]. This model represents significant improvements by a factor of six in resolution, and by factors of three to six in accuracy over its predecessor EGM96 that is inadequate for very precise navigation [16,22]. Thus it is reasonable to take advantage of the EGM2008 for gravity compensation in INSs. A few studies based on the theoretical analysis and simulation on such methods indicate that it is effective to compensate gravity induced INS errors with the aid of EGM2008 [5,23–25].

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As the errors of free-inertial navigation diverge over time, most INSs utilize a kind of feedback As the errors of free-inertial navigation diverge over time, most INSs utilize a kind of feedback loop with external altitude and velocity references to restrain the divergence in the vertical channel loop with external altitude and velocity references to restrain the divergence in the vertical channel and the Schuler oscillation in the horizontal channels, namely the damping network. Existing studies and the Schuler oscillation in the horizontal channels, namely the damping network. Existing studies basically aim at short-term free-inertial navigation or INSs with simple damping networks, while this basically aim at short-term free-inertial navigation or INSs with simple damping networks, while paper focuses on the high-precision and long-term INSs with high-order damping networks. First, this paper focuses on the high-precision and long-term INSs with high-order damping networks. the error propagation of the gravity disturbance vector in such systems is analyzed and two methods First, the error propagation of the gravity disturbance vector in such systems is analyzed and two for gravity compensation using the EGM2008 are provided. Then a formula is developed to methods for gravity compensation using the EGM2008 are provided. Then a formula is developed to characterize the compensation error resulting from the gravity update interval and the computing characterize the compensation error resulting from the gravity update interval and the computing time time delay, which can provide some references for the implementation of real-time gravity delay, which can provide some references for the implementation of real-time gravity compensation. compensation. Correspondingly, the computation burden of gravity vectors using the high degree Correspondingly, the computation burden of gravity vectors using the high degree and order spherical and order spherical harmonic model is reduced by investigating and optimizing the calculation harmonic model is reduced by investigating and optimizing the calculation procedure. Finally, a few procedure. Finally, a few off-line experiments using the data from an actual sea test are presented to off-line experiments using the data from an actual sea test are presented to validate the theoretical validate the theoretical analysis and simulation results. analysis and simulation results. 2. Induced Position Position Errors Errors 2. Gravity Gravity Disturbance Disturbance Vector Vector Induced 2.1. Error Propagation To maintain long-term precise navigation, INSs must introduce external altitude and velocity references to provide suitable damping for the vertical vertical channel channel and the the Schuler Schuler loops. loops. Under this circumstance, errors errorsin inthe thehorizontal horizontalchannels channelsare are more concerned than the vertical Ignoring more concerned than the vertical one.one. Ignoring the the cross-coupling the vertical channel and the cross-coupling with the rotation Earth’s rotation rate,1 cross-coupling withwith the vertical channel and the cross-coupling with the Earth’s rate, Figure Figure 1 illustrates the propagation of sources, typical including error sources, including the gyro bias εbias , the illustrates the propagation of typical error the gyro bias ε, the accelerometer ∇ accelerometer bias ∇ and the reference velocity error δV , in the generalized damped Schuler loop [25]. and the reference velocity error δV r , in the generalized rdamped Schuler loop [25].

Figure 1. Error propagation of the generalized damped Schuler loop. Figure 1. Error propagation of the generalized damped Schuler loop.

Here, g and R are the local gravity and the average radius of the Earth, respectively. The Here, g and are the local gravity and theby average radius of the Earth, respectively. The velocity velocity error andRplatform tilt are represented δV and δθ. The INSs discussed in this paper adopt error and platform tilt are represented by δV and δθ. The INSs discussed in this paper adopt a kind a kind of high-order damping network, which is designed based on the complementary filtering to of high-order damping network, which is designed based on the complementary filtering to obtain obtain 40 dB/10 dec or higher attenuation rate to both low-frequency and high-frequency reference 40 dB/10errors dec or[26]. higher rate toofboth low-frequency and high-frequency reference velocity velocity Theattenuation transfer function such a damping network, Q(s), is given by: errors [26]. The transfer function of such a damping network, Q(s), is given by: 2 2 2  ) ( 2μs s  2μωs s  ωs
  2 2 2  s + 2µω  2µs s + ω + s 2
s 2 2 ωs s  2μω2s  1  4ζμ  s2  2  ζ  μ  ωs s  ωs  2    Q s  ωs s + 2µωs (1 + 4ζµ)s + 2(ζ + µ)ωs s + ωs ) (1) Q(s) = ( 2μs (1)  2 s2  2  ζ  μ  ω s   1  4ζμ  ω2 2   2 s2+  2µs 2(ζ + µ)ωs ss + (1 + 4ζµ)ωs s +
  3 3   s2 4ζµω +s2 ω s  ωωs sss+  22µω μωs2s2  ss22 +22(ζζ+ μ µω)ω s s ω 4ζμω s s s+ s s  p is the Schuler angular frequency, and ζ ζand where ωωss ==√g/Rg/R is the Schuler angular frequency, and andμµ are are two two coefficients coefficients determining the attenuation response. response. The Thevalues values forandμ ζ have and been ζ have been and optimized and are for µ optimized are assigned µ =assigned 0.5 and μ andinζ our = 1.296 in our INS shipborne INS obtain the required rate attenuation rateabove. mentioned above. ζ ==0.51.296 shipborne to obtain thetorequired attenuation mentioned The analysis The analysis of the gravity disturbance vector induced position errors will be based on these values. of the gravity disturbance vector induced position errors will be based on these values.













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2.2. Gravity Disturbance Vector and Its Induced Position Errors The accelerometer sensed specific force vector f is the combination of the kinematic acceleration vector a and the gravitational acceleration vector G, as: f = a−G

(2)

Considering the centrifugal effect of the Earth rotation, Equation (2) can be rewritten in the following form: a = f + g + ωie × (ωie × r) (3) where g is the gravity vector and ωie is the Earth’s rotation vector. The radius vector r defines the position to the Earth’s center of mass. Obviously it can be seen from Equations (2) and (3) that INSs need gravitational information to extract the kinematic acceleration of the vehicle. The normal gravity model is frequently employed because it can meet the accuracy requirement in most cases and is both simple and convenient to be calculated. This model is based on an ellipsoid of revolution having the same mass and rotation rate with the Earth, namely the so-called reference ellipsoid. As the normal gravity vector γ is perpendicular to the surface of the reference ellipsoid, its vertical component equals its magnitude γ, as:  T γn = 0 0 γ (4) where the superscript n indicates the vector in the navigation coordinate system (n-frame). Since both the shape and mass distribution of the Earth are not ideal, there exists difference between the actual and the normal gravity vector at the same position. This difference is called the gravity disturbance vector, expressed in n-frame as: δgn =



−ξg −ηg δg

T

(5)

where ξ and η are the north and the west component of DOV respectively, which represent the difference between the orientations of the actual and the normal gravity vector. δg is the magnitude of the gravity disturbance vector, called the gravity disturbance. The gravity disturbance vector barely affects the vertical channel damped by the external altitude reference input, thus we can focus on the latitude and longitude errors only. According to Figure 1, the accelerometer error induced horizontal position error δr is given in: δr (s) = Rδθ (s) = where: I (s) =

s2

1 I (s)∇(s) ωs2

ωs2 Q(s) + ωs2 Q(s)

(6)

(7)

It can be concluded from Equation (3) that the gravity disturbance vector has the same propagation with the accelerometer error. Replacing ∇ in Equation (6) with the horizontal components of Equation (5), namely ξg and ηg, yields their induced latitude and longitude errors: δL(s) =

ωs2 Q(s) s2 +ωs2 Q(s)

δl (s) cos( L) =

ξ (s)

ωs2 Q(s) s2 +ωs2 Q(s)

  η (s) 

(8)

where L and l represent the latitude and the longitude, respectively. In other words, δL(s) and δl (s) cos( L) are the responses of a linear system to the corresponding components of the DOV, whose transfer function is I (s). Using Equations (1) and (7), we can draw the pole plot and the Bode plot of I (s), as shown in Figures 2 and 3, respectively.

the pole plot and the Bode plot of I(s), as shown in Figures 2 and 3, respectively. Figure 2 shows that all of the poles have negative real parts and that four of them are complex Figure 2 shows that all of the poles have negative real parts and that four of them are complex with imaginary components near the Schuler angular frequency. This means that the system is stable with imaginary components near the Schuler angular frequency. This means that the system is stable but has underdamped transient responses similar to the Schuler oscillation. The stable Schuler loop but has underdamped transient responses similar to the Schuler oscillation. The stable Schuler loop acts as a low-pass filter, whose detailed frequency response has been illustrated in Figure 3. From the acts as a low-pass filter, whose detailed frequency response has been illustrated in Figure 3. From the Bode 2016, plot 16, it can Sensors 2177 be concluded that the gravity disturbance vector induced latitude and longitude 5 of 17 Bode plot it can be concluded that the gravity disturbance vector induced latitude and longitude errors consist of two parts. errors consist of two parts.

Figure 2. Pole plot of I(s) (μ = 0.5, ζ = 1.296). Figure 2. 2. Pole plot of of I (s)I(s) 0.5, ζζ == 1.296). (µ(μ== 0.5, Figure Pole plot 1.296).

Figure 3. Bode plot of I(s) (μ = 0.5, ζ = 1.296). Figure 3. Bode plot of I(s) (μ = 0.5, ζ = 1.296). Figure 3. Bode plot of I (s) (µ = 0.5, ζ = 1.296).

First, at low frequencies there are no amplitude or phase distortions, hence ξ and η, and their First, at low frequencies there are no amplitude or phase distortions, hence ξ and η, and their induced δL(s) and δl(s)cos(L) share thehave samenegative low-frequency components, respectively. Figure 2 shows that all of the poles real parts and that four of them areSecondly, complex induced δL(s) and δl(s)cos(L) share the same low-frequency components, respectively. Secondly, the peak around the Schuler angular frequency indicates that This δL(s)means and that δl(s)cos(L) also isinclude with imaginary components near the Schuler angular frequency. the system stable the peak around the Schuler angular frequency indicates that δL(s) and δl(s)cos(L) also include underdamped Schuler transient oscillations with amplitude to theoscillation. fluctuations of stable ξ andSchuler η. Sinceloop the but has underdamped responses similar torelated the Schuler The underdamped Schuler oscillations with amplitude related to the fluctuations of ξ and η. Since the global of thewhose DOV is more than 100 arc-seconds, forbeen high-precision long-term acts as amaximum low-pass filter, detailed frequency response has illustrated inand Figure 3. FromINSs the global maximum of the DOV is more than 100 arc-seconds, for high-precision and long-term INSs the resulting errors cannot bethat neglected, and disturbance must be carefully Besides, Figure 3 shows Bode plot it can be concluded the gravity vectorcompensated. induced latitude and longitude errors the resulting errors cannot be neglected, and must be carefully compensated. Besides, Figure 3 shows consist of two parts. First, at low frequencies there are no amplitude or phase distortions, hence ξ and η, and their induced δL(s) and δl (s) cos( L) share the same low-frequency components, respectively. Secondly, the peak around the Schuler angular frequency indicates that δL(s) and δl (s) cos( L) also include underdamped Schuler oscillations with amplitude related to the fluctuations of ξ and η. Since the global maximum of the DOV is more than 100 arc-seconds, for high-precision and long-term INSs the resulting errors cannot be neglected, and must be carefully compensated. Besides, Figure 3 shows that there is a significant attenuation, e.g., higher than 30 dB at angular frequencies above 0.01 rad/s,

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which corresponds a spatial wavelength of 6.28 km for a speed of 10 m/s. As the spatial frequency of the DOV is fixed, higher speed means faster change with the time. Thus, the gravitational data used in such a system do not require extremely high spatial resolution because of the low-pass characteristic. 3. Gravity Compensation Using a Spherical Harmonic Model Assuming that the density outside the Earth is zero, the gravitational potential V satisfies the Laplace equation and can be expressed by a harmonic function. At a position defined by its geocentric distance r, geocentric co-latitude ϕ (defined as 90◦ -latitude) and longitude l, V is given by [27]: V (r, ϕ, l ) =

KM ∞ n a n ∑ ( ) (Cnm cos ml + Snm sin ml )· Pnm (cos ϕ)] r n∑ =0 m =0 r

(9)

where KM is the geocentric gravitational constant, a is the semi-major axis of the reference ellipsoid, C nm and Snm are fully-normalized, unit-less, spherical harmonic coefficients, and Pnm (cos ϕ) is the fully normalized associated Legendre function (ALF) of the first kind, of degree n and order m. Gravitational acceleration is the gradient vector of the gravitational potential, of which each component deriving from Equation (9) is given by [27]: Gr

Gϕ

Gl

=

∂V (r,ϕ,l ) ∂r

=

KM r2

     Nmax n  KM a n  = − r2 [1 + ∑ (n + 1)( r ) ∑ (C nm cos ml + Snm sin ml )· Pnm (cos ϕ)]     n =2 m =0   ∂V (r,ϕ,l )   = r∂ϕ 

= =

Nmax

n

n ∑ ( ar ) ∑ (C nm cos ml + Snm sin ml )·

m =0 n =2 ∂V (r,ϕ,l ) r sin ϕ∂l Nmax n KM a n ( ) m(−C nm sin ml ∑ ∑ 2 r r sin ϕ n =2 m =0

∂Pnm (cos ϕ) ∂ϕ

+ Snm cos ml )· Pnm (cos ϕ)

Then the transformation to the n-frame can be written in the form:     − sin L 0 cos L cos ϕ 0 sin ϕ Gϕ     Gn =  0 1 0 0 1 0  Gl   Gr − cos L 0 − sin L − sin ϕ 0 cos ϕ

(10)

              

(11)

Finally, the Earth’s gravity consisting of the gravitational acceleration and the centrifugal acceleration of the Earth’s rotation can be expressed as: 

 −ωie2 ( R N + h) cos L sin L   gn = Gn +  0  2 2 −ωie ( R N + h) cos L

(12)

where ωie is the Earth’s rotation rate and h is the altitude. R N is the normal radius of curvature taken in the direction of the prime vertical, given in: RN =

a

(1 − e2 sin2 L)

1/2

(13)

where e is the first eccentricity of the reference ellipsoid. EGM2008 provides a set of estimated spherical harmonic coefficients, up to degree 2190 and order 2159 [28]. Then we can use Equations (10)–(13) to calculate the gravity vector at any given position on or outside the Earth. Since the ultra-high degree and order ALFs in Equation (10) could range over thousands of orders of magnitude, we have to use some special techniques to avoid underflow

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position on or outside the Earth. Since the ultra-high degree and order ALFs in Equation (10) could range over thousands of orders of magnitude, we have to use some special techniques to avoid underflow andproblems overflowwhen problems whenthem. computing them. Existingshows algorithms shows different and overflow computing Existing algorithms different performance performance in numerical stability and and in it geodesy is common in geodesy to use Clenshaw’s in numerical stability and accuracy, and accuracy, it is common to use Clenshaw’s method. Here method. Here we will utilize the modified forward column method because it is equivalent to we will utilize the modified forward column method because it is equivalent to Clenshaw’s method Clenshaw’s method in both efficiency and precision, while the mechanisms within the computation in both efficiency and precision, while the mechanisms within the computation process are highly process are highly intuitiveand and also transparent, also because it can output individual of ALFs intuitive and transparent, because and it can output individual values of ALFsvalues and their first and their first derivatives [29]. derivatives [29]. The only implement real-time gravity compensation. There are two The only problem problemleft lefthere hereisishow howtoto implement real-time gravity compensation. There are options: (1) compute the EGM2008 and record an offline database for real-time interpolation; and (2) two options: (1) compute the EGM2008 and record an offline database for real-time interpolation; compute the EGM2008 directly in situ. Generally, the first one is preferred because the calculation of and (2) compute the EGM2008 directly in situ. Generally, the first one is preferred because the the sphericalofharmonic model, to ultra-high degree and order, is thought to be is complicated calculation the spherical harmonic model, to ultra-high degree and order, thought toand be time-consuming and thus a huge burden for INSs. However, with analysis and optimization, we have complicated and time-consuming and thus a huge burden for INSs. However, with analysis and found that the we second also thechoice requirement of satisfy real-time This will be optimization, havechoice foundcan that thesatisfy second can also thecompensation. requirement of real-time discussed in detail thebe next section.in detail in the next section. compensation. Thisinwill discussed 4. Real-Time Gravity Compensation 4.1. Time Requirements 4.1. Time Requirements for for Real-Time Real-Time Compensation Compensation A A test test result result on on aa digital digital signal signal processor processor (DSP) (DSP) showed showed that that spherical spherical harmonic harmonic models models of of degree 12 are applicable to lowand middle-precision INSs with update frequencies less than degree 12 are applicable to low- and middle-precision INSs with update frequencies less than 400 To further resolution and and reduce reduce the the computational computational complexity, complexity, 400 Hz Hz [25,30]. [25,30]. To further improve improve the the spatial spatial resolution aa low-order low-order polynomials polynomialswas wasused usedtotoapproximate approximate the spherical harmonic model in a small the spherical harmonic model in a small areaarea and and showed good performance for real-time free-inertial solutions [24]. However, it is actually showed good performance for real-time free-inertial solutions [24]. However, it is actually not not necessary to update the gravity data that frequently because they change much slower than the typical And aa spherical spherical harmonic harmonic model model of of degree degree 12 12 is typical IMU IMU outputs. outputs. And is obviously obviously not not suitable suitable for for the the high-precision and long-term inertial navigation. Therefore, a new time requirement is developed in this section. It has been been concluded concludedabove abovethat thatititisismainly mainly low and medium frequency components of thethe low and medium frequency components of the the DOV propagate position errors, whichsuggest suggestthat thatwe wecan cansafely safely lengthen lengthen the the time DOV thatthat propagate intointo thethe position errors, which interval of gravity updating. In addition, the time spent on interpolation from database and calculation interval of gravity updating. In addition, the time spent on interpolation from database and using the EGM2008 the delays values’ the update, although first one too one fast istotoo be fast observed. calculation using thedelays EGM2008 values’ update,the although theisfirst to be The compensation error resulting the gravity update interval and computing time delaytime tc is observed. The compensation errorfrom resulting from the gravity updatetminterval tm and computing illustrated in Figure 4. delay tc is illustrated in Figure 4.

Figure of gravity gravity update update interval interval and and computing computing time time delay delay on on compensation compensation accuracy. accuracy. Figure 4. 4. Effect Effect of

Although in practice t might change within a small range, it can Although in practice tcc might change within a small range, it can simplicity. During the navigation process, the computation processes of simplicity. During the navigation process, the computation processes of

be assumed constant for be assumed constant for navigation solutions and navigation solutions and

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gravity data are concurrent. Gravity calculation is triggered at a constant interval tm , and the gravity vector will be maintained at its current value until the gravity calculation process outputs a new one after a time delay tc . Such arrangement guarantees that the gravity calculation does not interrupt the navigation process. In Figure 4, xr (t) and xc (t) denote the truth and actually used values of the gravitational information. The root mean square (RMS) of their discrepancy (denoted by the hatched areas in Figure 4) is given by: Z T 1 2 W 2 = lim (14) [ xc (t) − xr (t)]2 dt T →∞ T − T2 As xc (t) has a staircase shape, Equation (14) can be written in the form of piecewise integrations: N 1 1 ∑ t N →∞ 2N + 1 k =− N m

W 2 = lim

Z tm 0

[ xr (ktm − tc ) − xr (ktm + t)]2 dt

(15)

Expanding Equation (15) and interchanging the order of summation and integration, we have: W2

1 N →∞ 2N +1

N

= lim

k =− N

R 1 tm 1 t 0 2N +1 m N →∞

T

1 2 T T →∞ T − 2

∑ xr2 (ktm − tc ) + lim

R

xr2 (t)dt

N

−2 lim

(16)

∑ xr (ktm + t) xr (ktm − tc )dt

k =− N

If the sample frequency criterion is satisfied, that is, tm < 1/(2 f max ), the discrete sequence of samples xr (ktm − tc ) are able to capture all the information from the continuous-time signal xr (t). Then the first term in the right-hand side of Equation (16) equals the second one. Moreover, as the element to be integrated in the last term is bounded for every N, according to the dominated convergence theorem we can also interchange the order of limit and integration and get the integral of the autocorrelation function. Thus Equation (16) can be finally written in the following form: W 2 = 2Φr (0) − 2

1 tm

Z tm 0

Φr (τ + tc )dτ

(17)

where Φr (τ ) is the autocorrelation of xr (t). Local gravity field can be characterized by exponential correlation function [1], such as: Φr (τ ) = σr2 e−d|τ |

(18)

where σr2 is the variance of xr (t), and d, defined by v/D (where v is the speed of the vehicle and D is the correlation distance of the gravitational information), represents the reciprocal of the correlation time. Substituting Equation (18) into Equation (17) yields: s W=

  e−tm d − 1 −t d 2σr2 1 + e c tm d

(19)

Both Equations (17) and (19) show that the increase of tm or tc decreases the accuracy of compensation. When both of them become zero, W is also reduced to zero, which matches with the fact that in this case there is no discrepancy between the truth and actually used value. p As tm and tc approach √ infinity, W is reaching its maximum 2Φr (0), which in the case of Equation (18) becomes 2σr . According to the DOV data set released under the model EGM2008, the global arithmetic RMSs of the DOV are 5.417 (ξ) and 5.503 (η) arc-seconds [16,28]. Over the area of our sea test (whose scope will be described in Section 5), the arithmetic RMSs are 4.724 (ξ) and 7.404 (η) arc-seconds. In addition, the horizontal components of gravity disturbance vectors can be assumed to behave like

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the first-order Gauss-Markov stochastic process, whose autocorrelation is given as in Equation9 (18). Sensors 2016, 16, 2177 of 17 The values chosen to fit the gravity field of the Texas-Oklahoma region is that σr equals 21.8 mGal (around both thethe along-track and and cross-track directions, and that D that equals km (around 4.59 4.59arc-seconds) arc-seconds)in in both along-track cross-track directions, and D 181 equals for component and 838and km838 forkm thefor cross-track component. Thus,Thus, to produce a timea 181the kmalong-track for the along-track component the cross-track component. to produce requirement suitable for most a situation is assumed which the hasthe a relatively time requirement suitable foroccasions, most occasions, a situation is inassumed inDOV which DOV hasbiga amplitude andamplitude changes quite and drastically, the values for σr the andvalues D are assigned as 10 relatively big anddrastically, changes quite and for σr and D arc-seconds are assignedand as 181 km, respectively. Considering the common experimental flight condition, the speed is assumedthe to 10 arc-seconds and 181 km, respectively. Considering the common experimental flight condition, be 80 m/s. Using these a setthese of simulated gravity was generated as xr (t) with speed is assumed to beparameters, 80 m/s. Using parameters, a setdisturbances of simulated gravity disturbances was −4 A series of t covering the range σ and dσr= × 10−4 to verify Equation (19). generated as xr (t) with = 104.4199 arc-seconds and d = 4.4199 × 10 to verify Equation (19). A series r = 10 arc-seconds m from s to 200 sthe was usedfrom to sample xc (ts) was fromused xr (t)to , with the xtime delay x1r (t), s and 20 the s respectively. of tm 0covering range 0 s to 200 sample with time delay c (t) from Then the20RMS difference between xr (t) was compared predication of 1 s and s respectively. Then the xRMS difference between xc (t) with and the xr (t)theoretical was compared with the c ( t ) and Equation as shownofinEquation Figure 5.(19), as shown in Figure 5. theoretical(19), predication

Figure 5. 5. Simulation Simulation and under the the Figure and theoretical theoretical values values of of the the RMS RMS difference difference between between xxc(t) and xxrr((t) t) under (t) and condition. common experimental flight condition.

In Figure 5, simulation results show good agreements with theoretical values. The RMS In Figure 5, simulation results show good agreements with theoretical values. The RMS difference difference grows with tm , fast at the beginning and later approaching its steady state. On the other grows with tm , fast at the beginning and later approaching its steady state. On the other hand, hand, the increase in tc shifts the entire RMS curve upward and makes it faster to approach the the increase in tc shifts the entire RMS curve upward and makes it faster to approach the maximum √ maximum √2σr . This effect is more notable when tm is smaller. The parameter σr determines the 2σr . This effect is more notable when tm is smaller. The parameter σr determines the amplitude amplitude and the upper limit of W. Moreover, the requirement of tm and tc for gravity and the upper limit of W. Moreover, the requirement of tm and tc for gravity compensation of the compensation of the airborne INSs can also be concluded from Figure 5. If it takes more than 1 s to airborne INSs can also be concluded from Figure 5. If it takes more than 1 s to calculate the single-point calculate the single-point DOV, there will be a very strict limit for the update interval. And if the DOV, there will be a very strict limit for the update interval. And if the calculation time is under 1 s, calculation time is under 1 s, an update interval of 20 s can ensure the compensation error less than 1 an update interval of 20 s can ensure the compensation error less than 1 arc-second, as what the marker arc-second, as what the marker shows in Figure 5. shows in Figure 5. The requirement of tm and tc in marine navigation applications is also analyzed using The requirement of tm and tc in marine navigation applications is also analyzed using Equation (19). The values of σ and D remain unchanged, while the speed is chosen as 15 m/s, Equation (19). The values-5 of σrr and D remain unchanged, while the speed is chosen as 15 m/s, resulting in × 10 6 shows howhow RMSRMS changes as a function of tm of and tc under this −5 resulting in dd ==8.2873 8.2873 × 10. Figure . Figure 6 shows changes as a function tm and tc under circumstance. this circumstance. Comparing Figure 6 with Figure 5, it can be seen that the growth of RMS compensation errors Comparing Figure 6 with Figure 5, it can be seen that the growth of RMS compensation errors becomes slower as a result of a smaller d. As the error resulting from tc contributes a lot to the whole becomes slower as a result of a smaller d. As the error resulting from tc contributes a lot to the whole compensation error, it is still recommended that the calculation time should be no more than 1 s. compensation error, it is still recommended that the calculation time should be no more than 1 s. Under this condition, using an update interval under 100 s can obtain a compensation accuracy better Under this condition, using an update interval under 100 s can obtain a compensation accuracy better than 1 arc-second by a margin, as what the markers show in Figure 6. than 1 arc-second by a margin, as what the markers show in Figure 6.

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6. RMS compensation errors vs. update interval and computing time delay under the Figure Figure 6. RMS compensation errors vs. update interval and computing time delay under the shipborne condition. shipborne condition.

In a word, higher speed, bigger amplitude and more drastic change of the DOV result in higher on the update interval and computing time. In general, the actual gravity field changes Inrequirements a word, higher speed, bigger amplitude and more drastic change of the DOV result in higher more gently, and for the DOV the damped Schuler loop acts as a low-pass filter, both of which lead requirements on the update interval and computing time. In general, the actual gravity field changes to longer correlation time and allow a longer update interval, but smaller computing time delay is more gently, and for the DOV the damped Schuler loop acts as a low-pass filter, both of which lead to still better. When the compensation error is required to be no more than 1 arc-second, it is longer correlation time and allow a longer update interval, but smaller computing time delay is still recommended that the single-point DOV should be computed within 1 s and updated at an interval better. less When error is required toairborne be no more thanthe 100compensation s for marine navigation, and 20 s for INSs. than 1 arc-second, it is recommended that the single-point DOV should be computed within 1 s and updated at an interval less than 100 s 4.2. Improvement Computation for marine navigation,of and 20 s forEfficiency airborne INSs. Originally it took over 30 s to compute a gravity vector using the EGM2008 to degree 2190 and

4.2. Improvement of Computation Efficiency order 2159 with a desktop computer (Intel dual core processor i3-3240, 3.40 GHz, 3.40 GHz; physical RAM 3.41GB available; 32-bit C language compiled by Microsoft Visual Originally it took over 30 s toWindows compute7 aProfessional; gravity vector using the EGM2008 to degree 2190 and Studio 2010 Ultimate). Such a long time delay does not meet the time requirement and will result in order 2159 with a desktop computer (Intel dual core processor i3-3240, 3.40 GHz, 3.40 GHz; physical unacceptable compensation errors. RAM 3.41 Program GB available; 32-bit Windows 7 Professional; C language compiled by Microsoft Visual profiling shows that most of the computation time are spent on locating and reading Studio the 2010 Ultimate). Such a long timeThe delay does requirement and willthe result in spherical harmonic coefficients. reason is not that meet in thethe filetime provided by the EGM2008, unacceptable errors. stored in ASCII format records are first arranged by their sphericalcompensation harmonic coefficients corresponding degree n and then sub-arranged by their ordertime m. Therefore, needless Program profiling shows that most of the computation are spentwe onremoved locatingthe and reading the information and rearranged the coefficients a binary filefile firstprovided by m and then by n. This modification spherical harmonic coefficients. The reason isinthat in the by the EGM2008, the spherical matches with the modified forward column method to calculate the ALFs, and thus allows the harmonic coefficients stored in ASCII format records are first arranged by their corresponding degree program to read every coefficient sequentially just along with the recursion of ALFs and their first n and then sub-arranged by their order m. Therefore, we removed the needless information and derivatives without the process of locating and transforming. In this way the average computing time rearranged the coefficients binary file first by m and n. This modification matches of a gravity vector under in theasame computing environment hasthen been by shortened to less than 1 s, which with the modified forward column method to calculate the ALFs, and thus allows the program to makes it possible to calculate gravity vectors from the EGM2008 directly in situ. The size of necessary read every sequentially just alongthan with the recursion ALFs and their first derivatives data iscoefficient under 40 MB, which is much smaller that of the original of coefficient file (239.29 MB) or the a large area (aboutIn 1 GB global gridded at 1 arc-minute). withouthigh-resolution the process ofdatabase locatingforand transforming. thisfor way the data average computing time of aIngravity addition, as the gravity vector can be calculated anywhere on and outside the Earth, both the vector under the same computing environment has been shortened to less than 1 s, which makes it interpolation and upward continuation, which bring errors when using ground databases, are no possible to calculate gravity vectors from the EGM2008 directly in situ. The size of necessary data is longer needed. under 40 MB, which is much smaller than that of the original coefficient file (239.29 MB) or the high-resolution database a large (about 1 GB for global data gridded at 1 arc-minute). 4.3. Compromise betweenfor Accuracy and area Computing Efficiency In addition, as the gravity vector can be calculated anywhere on and outside the Earth, both the If the maximum degree of the spherical harmonic model used to calculate the gravity vector is interpolation continuation, which errors using ground databases, reduced,and bothupward the computing time and size of thebring coefficient filewhen will decrease, but accompanied by aare no longer loss needed. of detailed gravitational information and the non-gravitational artefacts. To find a compromise between the accuracy and computing efficiency, 4000 points on the route of the sea test are chosen to

4.3. Compromise between Accuracy and Computing Efficiency If the maximum degree of the spherical harmonic model used to calculate the gravity vector is reduced, both the computing time and size of the coefficient file will decrease, but accompanied by a loss of detailed gravitational information and the non-gravitational artefacts. To find a compromise between the accuracy and computing efficiency, 4000 points on the route of the sea test are chosen

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to calculate their DOVs from EGM2008 to degree 12, 180, 360, 600, 800, 1000, 1200, 1400, 1600, 1800 Sensors 2016, 16, 2177 11 of 17 and 2190 with coefficient files. Using the set of DOVs corresponding to degree 2190 as a Sensors 2016,matched 16, 2177 11 of 17 reference, standard deviations of calculation errors of ξ360, and η are in Figure 7, and of calculate their DOVs from EGM2008 to degree 12, 180, 600, 800,plotted 1000, 1200, 1400, 1600, 1800changes and calculate their DOVscomputing from EGM2008 to degree 180, 360, 600,corresponding 800, 1000, 1400, 1600, 1800as and 2190 with matched coefficient files. Using the12, set ofof DOVs to degree a It can the average single-point time and the sizes coefficient files 1200, are plotted in 2190 Figure 8. 2190 withstandard matcheddeviations coefficientoffiles. Using errors the setofofξ DOVs to degree as a calculation η corresponding are plotted in Figure 7, and2190 changes be seenreference, that the maximum degree has to be bigger thanand 1000 to guarantee the calculation accuracy reference, standard deviations of calculation errors of ξ and η are plotted in Figure 7, and changes of the average single-point computing time and the sizes of coefficient files are plotted in Figure 8. It better than 1 arc-second. Taking into account thethe compensation errors resulting from the computing of the computing time and coefficient files are in Figure 8. It can be average seen thatsingle-point the maximum degree has to be biggersizes thanof1000 to guarantee theplotted calculation accuracy time and update interval, thedegree maximum degree should no less than 1400, with a minimum can the bethan seen the maximum has to be bigger than 1000be to guarantee the from calculation accuracy better 1that arc-second. Taking into account the compensation errors resulting the computing average computing time within 0.4 s maximum and a minimum file size less than 20 MB. better than 1 arc-second. Taking account the compensation errors from theacomputing time and the update interval, theinto degree should be no lessresulting than 1400, with minimum time andcomputing the update interval, the maximum degree should be no less than 1400, with athe minimum It average should be noted that the results about truncation in this section are gained over mid-latitude time within 0.4 s and a minimum file size less than 20 MB. ◦ , 80 ◦ size ◦than average computing time within 0.4 s and a minimum file less 20 MB. areas. Repeat tests were conducted in latitudes of 75 and 85 , over the longitude scope of 0◦ ~180◦ It should be noted that the results about truncation in this section are gained over the midIt should be noted results about truncation in this section are gained over thescope mid◦the areas. Repeat were conducted latitudes of 75°, and over thebecomes longitude with alatitude discretization step tests ofthat 0.1 .The resultsinshowed that the80° loss of85°, accuracy bigger near latitude areas. Repeat tests were step conducted in latitudes ofshowed 75°, 80°that andthe 85°,loss overofthe longitude scope of 0°~180° with a discretization of 0.1°.The results accuracy becomes the pole and increases with the latitude. For example, when truncating the model at degree 1800, of 0°~180° a discretization stepwith of 0.1°.The results showed that the loss of accuracy becomes bigger nearwith the pole and increases the latitude. For example, the model at the differences become 2.07(ξ) and 1.84(η), 2.50(ξ) and 2.40(η), when 2.78 truncating (ξ) and 2.3(η) arc-seconds, bigger near the pole and increases with the latitude. For example, when truncating the model at degree 1800, the differences become 2.07(ξ) and 1.84(η), 2.50(ξ) and 2.40(η), 2.78 (ξ) and 2.3(η) arcrespectively. Thistheindicates that the 2.07(ξ) gravityand compensation around the2.78 polar areas needs more degree 1800, differences become 1.84(η), 2.50(ξ) and 2.40(η), (ξ) and 2.3(η) arcseconds, respectively. This indicates that the gravity compensation around the polar areas needs investigation, which could be one of our future research works. seconds, respectively. This indicates that the gravity compensation around the polar areas needs more investigation, which could be one of our future research works. more investigation, which could be one of our future research works.

Figure 7. Calculation errors vs. maximum degree.

Figure 7. Calculation vs.maximum maximum degree. Figure 7. Calculationerrors errors vs. degree.

Figure 8. (a) Sizes of spherical harmonic coefficient files; and (b) Average computing time vs. Figure 8. (a) Sizes of spherical harmonic coefficient files; and (b) Average computing time vs.

degree of the used sphericalcoefficient harmonic model. Figuremaximum 8. (a) Sizes of spherical harmonic files; and (b) Average computing time vs. maximum maximum degree of the used spherical harmonic model. degree of the used spherical harmonic model.

Besides, although discussion in this part is aimed to provide some reference for systems with limited hardware resources, we suggest that the truncation of the model should be taken only as a last resort.

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Besides, although discussion in this part is aimed to provide some reference for systems with limited hardware as17a Sensors 2016, 16, 2177 resources, we suggest that the truncation of the model should be taken only 12 of last resort. 5. The Sea Test Test of of aa Shipborne Shipborne INS INS The shipborne INS used in the sea test is the same as in [6], where the specifications of the instruments are described in detail. Two dual-axis gyros with ultra-low drift and three orthogonal orthogonal pendulous accelerometers are mounted on a gyro-stabilized gyro-stabilized gimbaled gimbaled platform. An altimeter and a velocity log are used to provide the altitude and velocity reference for the damping damping of the vertical and horizontal channels. The high-order horizontal damping network network used in the system has been introduced in Section 1. The position solutions of the INS are compared with the outputs of a GPS to obtain the real-time real-time position position errors. errors. Some quite quite drastic drasticchanges changesininthe theoriginal original position errors were observed during the test, sea position errors were observed during the sea test, which doesmatch not match with the typical error propagation of thevarying slowly INS varying INS errors. which does not with the typical error propagation of the slowly errors. But they But showcorrelation a strong correlation withdepths the ocean along the route, from the showthey a strong with the ocean alongdepths the route, acquired fromacquired the 2 arc-minutes 2global arc-minutes globalETOPO2v2 relief model ETOPO2v2 the NationalData Geophysical Data Center relief model released by thereleased NationalbyGeophysical Center (NGDC). As a (NGDC). Asspeculated a result, it that is speculated that thisphenomenon anomalous phenomenon wasthe caused by the gravitational result, it is this anomalous was caused by gravitational errors, and errors, a static and experiment and a few dynamic experiments aretoconducted to such compensate a static and experiment a few dynamic experiments are conducted compensate gravity such gravity induced position errors. The route of a round-trip experiment is shown inthis Figure 9. induced position errors. The route of a round-trip experiment is shown in Figure 9. Along route Along route several symmetrically distributed peaks canoriginal be observed in errors, the original severalthis symmetrically distributed peaks can be observed in the position whichposition implies errors, which implies some relevance with topography. the local underwater topography. some kind of relevance withkind the of local underwater

Figure 9. The round-trip experiment experiment during during the the sea sea test. test. Figure 9. The route route of of aa round-trip

According to the coverage area of the sea test, we used the EGM2008 and Equations (10)–(12) to According to the coverage area of the sea test, we used the EGM2008 and Equations (10)–(12) generate a 5 arc-minutes gridded local database of gravity vectors, covering latitude 5~25°◦ N and to generate a 5 arc-minutes gridded local database of gravity vectors, covering latitude 5~25 N and longitude 105~120° E. The values of the gravity vectors were interpolated from this off-line database longitude 105~120◦ E. The values of the gravity vectors were interpolated from this off-line database using the bilinear interpolation and were updated every 10 s. There will be a detailed discussion of using the bilinear interpolation and were updated every 10 s. There will be a detailed discussion of the the compensation results in the next section. compensation results in the next section. 6. Results and Discussion 6. Results and Discussion 6.1. The and the the Round-Trip Round-Trip Experiment Experiment 6.1. The Static Static and The static static experiment experiment was was conducted conducted during during aa period period of of anchoring, anchoring, of of which which the the compensation compensation The results is shown in Figure 10. All of the curves illustrated in Section 6 are normalized using the the results is shown in Figure 10. All of the curves illustrated in Section 6 are normalized using maximum absolute absolute value value of of the the uncompensated uncompensated latitude latitude or or longitude longitude errors errors in in the the corresponding corresponding maximum segment. It can be observed that a constant offset exists between the position errors before and after segment. It can be observed that a constant offset exists between the position errors before and compensation, which is almost the same as the corresponding components of the DOV that after compensation, which is almost the same as the corresponding components of the DOV at at that position. The standard deviations of the difference between the compensated position errors and the

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position. The standard deviations of the difference between the compensated position errors and the

corresponding components of the DOV are and0.95% 0.95% ofη ηinin the corresponding components of the DOV are2.38% 2.38%of ofξ ξininthe thelatitudinal latitudinal direction direction and of13 Sensors 2016, 16, 2177 of 17 longitudinal direction. the longitudinal direction. position. The standard deviations of the difference between the compensated position errors and the corresponding components of the DOV are 2.38% of ξ in the latitudinal direction and 0.95% of η in the longitudinal direction.

Figure 10. Results of the static experiment: (a) Latitude errors before (line 1) and after (line 2)

Figure 10. Results of the static experiment: (a) Latitude errors before (line 1) and after (line 2) compensation; (b) Longitude errors before (line 1) and after (line 2) compensation. compensation; (b) Longitude errors before (line 1) and after (line 2) compensation.

The results of the round-trip experiment is shown in Figures 11 (the latitudinal direction) and 12 Figure 10. Results of the static experiment: (a) Latitude errors before (line 1) and after (line 2)

(the longitudinal direction), whose route has(line been Figure 9. It locates at the very beginning The results of the round-trip experiment is shown ininFigure 11 (the latitudinal direction) and 12 compensation; (b) Longitude errors before 1) shown and after (line 2) compensation. of the sea test where the accumulation of INS errors has not become prominent. (the longitudinal direction), whose route has been shown in Figure 9. It locates at the very beginning 11a,bofshows that the symmetrically peaks in δLprominent. appears where ξ reaches The theaccumulation round-trip experiment isdistributed shown Figures 11 (the latitudinal direction) and its 12 of the seaFigure testresults where the of INS errors hasinnot become peaks. Most of the peaks are removed after compensation, yielding a relatively steady (the longitudinal direction), whose route has been shown in Figure 9. It locates at the very beginning Figure 11a,b shows that the symmetrically distributed peaks in δL appears where ξ reaches its 24-h whichaccumulation is typical for of theINS long-term INSnot errors. Figure 11c illustrates the difference of theperiodic sea test form errors has become prominent. peaks. Most of thewhere peaksthe are removed after compensation, yielding a relatively steady 24-h periodic between the latitude errors before and after compensation, which is almost the same as ξ. The Figure 11a,b shows that the symmetrically distributed peaks in δL appears where ξ reaches its formdifference which is between typical for the long-termerrors INS and errors. Figure 11c illustrates the difference between the compensated ξ represents the Schuler oscillation excited ξ, as the peaks. Most of the peaks are removed after compensation, yielding a relatively bysteady latitude errors and after compensation, which is almost the same as ξ. Thebecomes difference between shown in before Figure More intensely fluctuates, theerrors. oscillation bigger. 24-h periodic form 11d. which is typical for theξ long-term INS Figure amplitude 11c illustrates the difference the compensated errors and ξ represents the Schuler oscillation excited by ξ, as shown in Figure Figure 12 illustrates similar results, except for using δlcos(L) instead of δl in Figure 12c,d according between the latitude errors before and after compensation, which is almost the same as ξ. The 11d. Equation Thethe maximum error errors compensated is 48.85%, in which the Schuler12 oscillation Moreto intensely ξ(8). fluctuates, the oscillation amplitude becomes bigger. Figure illustrates difference between compensated and ξ represents the Schuler oscillation excited by takes ξ, similar as 18.83%, of the maximum absolute value of the uncompensated horizontal position errors. results, except using11d. δl cos in Figure 12c,d accordingamplitude to Equation (8). Thebigger. maximum ( L) instead shown in for Figure More intenselyof ξδl fluctuates, the oscillation becomes These results indicates that: (1)except the component the gravity vector hardly 12 illustrates similarin results, for using δlcos(L)ofinstead of δl disturbance in of Figure 12c,d according error Figure compensated is 48.85%, which thevertical Schuler oscillation takes 18.83%, the maximum absolute affects the accuracy of INS solutions; (2) low-frequency components of the DOV will propagate into to Equation (8). The maximum error compensated is 48.85%, in which the Schuler oscillation takes value of the uncompensated horizontal position errors. the latitude and longitude errors atvalue a ratioofofthe 1:1;uncompensated and (3) fluctuations in the DOV excite a time-varying 18.83%, of the maximum position errors. These results indicates absolute that: (1) the vertical component ofhorizontal the gravity disturbance vector hardly error response in the form of Schuler All of them the disturbance theoretical predication in results that: (1) theoscillation. vertical component of verifies the gravity vector hardly affects theThese accuracy ofindicates INS solutions; (2) low-frequency components of the DOV will propagate into Section 1 and the successful realization the gravity compensation affects the accuracy of INS solutions; (2)oflow-frequency componentsusing of theEGM2008. DOV will propagate into the latitude and longitude errors at a ratio of 1:1; and (3) fluctuations in the DOV excite a time-varying the latitude and longitude errors at a ratio of 1:1; and (3) fluctuations in the DOV excite a time-varying errorerror response in the form of Schuler oscillation. All of them verifies the theoretical predication in response in the form of Schuler oscillation. All of them verifies the theoretical predication in Section 1 and the realization compensationusing using EGM2008. Section 1 and successful the successful realizationofofthe thegravity gravity compensation EGM2008.

Figure 11. Latitudinal results of the round-trip experiment: (a) ξ; (b) Latitude errors before (line 1) and after (line 2) compensation; (c) Compensated latitude error; (d) Difference between ξ and the compensated latitude error. Figure 11. Latitudinal results of the round-trip experiment: (a) ξ; (b) Latitude errors before (line 1)

Figure 11. Latitudinal results of the round-trip experiment: (a) ξ; (b) Latitude errors before (line 1) and after (line 2) compensation; (c) Compensated latitude error; (d) Difference between ξ and the and after (line 2) latitude compensation; (c) Compensated latitude error; (d) Difference between ξ and the compensated error. compensated latitude error.

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Figure 12. Longitudinal results of the round-trip experiment: (a) η; (b) Longitude errors before Figure 12. Longitudinal results of the round-trip experiment: (a) η; (b) Longitude errors before (line 1) (line 1) and after (line 2) compensation; (c) Compensated longitude error; (d) Difference between η and after (line 2) compensation; (c) Compensated longitude error; (d) Difference between η and the and the compensated longitude error. compensated longitude error.

6.2. Dynamic Experiments after a Long-Time Navigation 6.2. Dynamic Experiments after a Long-Time Navigation As the inertial navigation has already lasted for quite a long time, the accumulation of INS As the inertial navigation has already lasted for quite a long time, the accumulation of INS position errors has become prominent enough to conceal the low-frequency components of the position errors has become prominent enough to conceal the low-frequency components of the gravity gravity induced errors. Over the areas where the topography changes drastically, we choose seven induced errors. Over the areas where the topography changes drastically, we choose seven segments segments with time span around 10 h to observe the Schuler oscillation. In order to remove the 24-h with time span around 10 h to observe the Schuler oscillation. In order to remove the 24-h periodical periodical components, both the normalized latitude and longitude errors are fitted to quadratic components, both the normalized latitude and longitude errors are fitted to quadratic polynomials. polynomials. Time durations, scopes of corresponding latitude and longitude, and sum squared Time durations, scopes of corresponding latitude and longitude, and sum squared errors (SSEs) of the errors (SSEs) of the fittings, which evaluate the intensity of Schuler oscillation, are listed in Table 1. fittings, which evaluate the intensity of Schuler oscillation, are listed in Table 1. Table 1. SSEs of fittings of the normalized horizontal position errors before and after compensation. Table 1. SSEs of fittings of the normalized horizontal position errors before and after compensation. No. Time No. span (h)

II-1 7.22 II-1

II-2 10.28 II-2

II-3 10.55 II-3

II-4 6.66 II-4

II-5 13.88 II-5

Latitude scope Time span (h)(°)

13.15~14.65 7.22

10.07~10.44 10.28

9.80~10.21 10.55

9.60~9.85 6.66

13.41~15.51 13.88

Latitude scope (◦ )

13.15~14.65

10.07~10.44

9.80~10.21

9.60~9.85

13.41~15.51

◦) Longitude Longitude scope scope ((°)

115.52~115.61 115.52~115.61

114.21~115.34 114.21~115.34

114.23~115.54 114.23~115.54

114.64~115.53 114.64~115.53

112.33~112.52 112.33~112.52

Uncompensated δL Uncompensated δL SSE

62.01

65.95

92.88

33.06

78.33

62.01

65.95

92.88

33.06

12.41

26.57

49.09

13.49

SSE Compensated δL SSE Compensated δL SSE Uncompensated δl SSE Uncompensated δl Compensated δl SSE SSE Compensated δl SSE

78.33 34.27

II-6

II-7

7.22 II-6

7.50 II-7

13.96~15.10 7.22 13.96~15.10

112.35~112.44 112.35~112.44 36.52

36.52 9.66

12.41

26.57

49.09

13.49

34.27

9.66

34.89

75.69

367.66

384.84

187.18

236.06

34.89

75.69

367.66

384.84

187.18

236.06

20.65

33.21

157.40

99.80

66.22

81.99

20.65

33.21

157.40

99.80

66.22

81.99

14.05~15.16 7.50 14.05~15.16

112.30~112.43 112.30~112.43 39.25

39.25 19.19

19.19 58.58

58.58 15.88

15.88

Here, the error curves of Segment II-5 are illustrated in Figure 13 as an example. All SSE values in the error curves of Segmentwhich II-5 are illustrated 13induced as an example. SSE values TableHere, 1 decreased after compensation, indicates that in theFigure gravity Schuler All oscillation has in Table 1 decreased after compensation, which indicates that the gravity induced Schuler oscillation been attenuated. Besides, the more direct illustration in Figure 13 shows that the peaks of the DOV do has only beencause attenuated. Besides, the more direct illustration Figure the 13 nearby shows that the peaks of the not larger errors at corresponding points, but alsoinincrease oscillation amplitude. DOV do not only cause larger errors at corresponding points, but also increase the nearby oscillation amplitude.

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Figure 13. Compensation results of Segment II-5: (a) ξ; (b) Latitude errors before (line 1) and after Figure 13. Compensation results of Segment II-5: (a) ξ; (b) Latitude errors before (line 1) and after (line 2) compensation; (c) η; (d) Longitude errors before (line 1) and after (line 2) compensation. (line 2) compensation; (c) η; (d) Longitude errors before (line 1) and after (line 2) compensation.

7. Conclusions 7. Conclusions In the high-precision and long-term INSs with both the altitude damping and the horizontal In the high-precision and long-term INSs with both the altitude damping and the horizontal velocity damping networks, the gravity disturbance vector induced errors exist almost exclusively in velocity damping networks, the gravity disturbance vector induced errors exist almost exclusively the horizontal channels and are mostly caused by the DOV. Low-frequency components of the DOV in the horizontal channels and are mostly caused by the DOV. Low-frequency components of the propagate into the latitude and longitude errors at a ratio of 1:1. Moreover, the fluctuations in the DOV propagate into the latitude and longitude errors at a ratio of 1:1. Moreover, the fluctuations DOV excite Schuler oscillation since the system is underdamped. To compensate these errors, two in the DOV excite Schuler oscillation since the system is underdamped. To compensate these errors, methods based on the EGM2008 are provided in this paper, namely, interpolation from an off-line two methods based on the EGM2008 are provided in this paper, namely, interpolation from an off-line database generated beforehand using the spherical harmonic model and computing the values of database generated beforehand using the spherical harmonic model and computing the values of gravity vectors from the model directly in situ. gravity vectors from the model directly in situ. A formula is developed to characterize the relationship of the update time interval, the A formula is developed to characterize the relationship of the update time interval, the computing computing time delay and their resulting compensation errors, which produces a time requirement time delay and their resulting compensation errors, which produces a time requirement for the for the real-time gravity compensation in INSs. Typically, it is recommended that the gravity vector real-time gravity compensation in INSs. Typically, it is recommended that the gravity vector should be should be calculated within 1 s and update at an interval less than 100 s for the marine navigation, calculated within 1 s and update at an interval less than 100 s for the marine navigation, and 20 s for and 20 s for the airborne INSs, to ensure the compensation accuracy better than 1 arc-second. After the airborne INSs, to ensure the compensation accuracy better than 1 arc-second. After optimizing the optimizing the layout of spherical harmonic coefficients, the average single-point computing time layout of spherical harmonic coefficients, the average single-point computing time has been reduced has been reduced greatly to less than 1 s, which makes it possible to implement the second method greatly to less than 1 s, which makes it possible to implement the second method for real-time gravity for real-time gravity compensation applications. compensation applications. Several off-line compensation experiments were conducted using the data of a high-precision Several off-line compensation experiments were conducted using the data of a high-precision shipborne INS and auxiliary test instruments collected during an actual sea test. With the aid of shipborne INS and auxiliary test instruments collected during an actual sea test. With the aid of EGM2008, both low-frequency components and Schuler oscillation of the gravity induced position EGM2008, both low-frequency components and Schuler oscillation of the gravity induced position errors are attenuated, up to 48.84% in total of the regional maximum in the rugged terrain. The errors are attenuated, up to 48.84% in total of the regional maximum in the rugged terrain. experimental results agree well with the theoretical predication, and indicate that the EGM2008 has The experimental results agree well with the theoretical predication, and indicate that the EGM2008 has enough accuracy and resolution for the gravity compensation in such high-precision long-term INSs. enough accuracy and resolution for the gravity compensation in such high-precision long-term INSs. It should be noted that the sea test is conducted in regions with unrestricted gravity anomaly It should be noted that the sea test is conducted in regions with unrestricted gravity anomaly data during the development of EGM2008. As for areas where gravity anomaly data are unavailable, data during the development of EGM2008. As for areas where gravity anomaly data are unavailable, such as Antarctica, more tests will be needed to further investigate the EGM2008’s performance in such as Antarctica, more tests will be needed to further investigate the EGM2008’s performance in gravity compensation. Our future work will focus on the further improvement of gravity gravity compensation. Our future work will focus on the further improvement of gravity compensation compensation accuracy over such regions. For example, the model GGMplus with ultra-high accuracy over such regions. For example, the model GGMplus with ultra-high resolution will be taken resolution will be taken into consideration in the possible future flight test over land areas. into consideration in the possible future flight test over land areas. Acknowledgments: The authors would like to acknowledge the support offered by the following programs: National Natural Science Foundation of China (61603208) and Tsinghua University Initiative Scientific Research Program (2014Z09106, 20151080461).

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Acknowledgments: The authors would like to acknowledge the support offered by the following programs: National Natural Science Foundation of China (61603208) and Tsinghua University Initiative Scientific Research Program (2014Z09106, 20151080461). Author Contributions: Ruonan Wu analyzed the time requirement for the real-time gravity compensation, and optimized the calculation procedure of the spherical harmonic model; Qiuping Wu provided the original code of the inertial navigation algorithm; Qiuping Wu and Fengtian Han designed the entire experimental scheme and R.W. performed the experiments; Ruonan Wu, Qiuping Wu and Fengtian Han jointly contributed in writing this paper; Tianyi Liu helped analyzing the program profile of the code to calculate the spherical harmonic model; Q.W., Fengtian Han, Peida Hu and Haixia Li participated in the sea test and provided the original data of the INS. Conflicts of Interest: The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

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