MEMS-based Inertial Navigation onDynamically Positioned Ships: Dead

Reckoning

Robert H. Rogne ∗ Torleiv H. Bryne ∗ Thor I. Fossen ∗

Tor A. Johansen ∗

∗ Department of Engineering Cybernetics, Norwegian University ofScience and Technology, 7491 Trondheim, Norway (e-mail:

robert.rogne@itk.ntnu.no, torleiv.h.bryne@itk.ntnu.no,thor.fossen@ntnu.no, tor.arne.johansen@itk.ntnu.no).

Abstract: Dead reckoning capabilities are vital in ship navigation if position and headingreferences are unrealizable or lost. In safety critical marine operations such as dynamicpositioning, the International Maritime Organization and classification societies require that thevessel possesses dead reckoning capabilities and position reference redundancy. In this paper, weconduct a full-scale experimental validation and comparison of the dead reckoning capabilitiesusing two different high-rate and low-cost micro-electro-mechanical inertial measurement units.The full-scale experimental validation is achieved with two nonlinear observers, aided bygyrocompasses and position reference systems, in a dynamic positioning operation carried outby an offshore vessel in the North Sea. The dead reckoning performance is evaluated after tenminutes without aiding from position reference system measurements.

Keywords: Inertial navigation; Navigation systems, Position estimation, Dead Reckoning;Nonlinear observers; Inertial Measurement Unit; Marine systems

1. INTRODUCTION

The dead reckoning (DR) term is used in the navigationcommunity to describe the process of estimating positionrelative some departure point by keeping track of thedistance covered and direction of travel. The origin of DRas an expression is not known, but one possibility is thatit stems from ded reckoning, short for deduced reckoning,Misra and Enge (2011, Ch. 1). Inertial navigation systems(INS) are DR based, by deducing position, velocity and at-titude (PVA) from previously known states by integrationof angular rate measurements and accelerometer readings,once and twice, respectively. Due to errors in the inertialsensor, such as noise and biases, DR is insufficient tomaintain accurate estimates over time. To counter INSdrift over time, aiding is introduced. Aided INS are oftenreferred to as integrated navigation systems.

INS can be aided by a number of sensors and position refer-ence (PosRef) systems such as radio, laser, hydroacousticand satellite-based systems. The latter types have the ben-efit of world-wide coverage, known as Global NavigationSatellite System (GNSS). However, these systems are ex-posed to both natural degradation and deliberate outages.Natural degradation can be caused by signal distortionfrom reflection of nearby objects, known as multipath,loss of signal due to sun storms or loss of line of sight tothe satellite. Deliberately outages can be because of signaljamming. During loss of reference, good DR capabilities isvital for the INS to provide accurate PVA estimates to the

user. Recent works on aided INS using nonlinear observers(NLO), such as Hua (2010), Grip et al. (2013), Bryne et al.(2014; 2015), Grip et al. (2015), and Rogne et al. (2016),have not discussed the DR capabilities using such observerdesigns. An exception is Fusini et al. (2016), however thisresult uses visually aided INS to improve the headingestimates and provide velocity aiding, which again is usedto improve the DR capabilities. For marine surfaces vessel,particular in dynamic positioning (DP), multiple headingreferences are available as a result of class requirements,such as DNV GL (2011), leaving the DR capabilities usingIMUs and NLOs unanswered for ship navigation.

INS applied in DP is far from novel, as proposed in anindustrial context almost 20 years ago (Vickery, 1999).Other similar products were introduced to the marketin the years that followed, Faugstadmo and Jacobsen(2003) and Paturel (2004). All these products have beendeveloped with high-end IMUs, based on ring laser gyro(RLG) or fiber optic gyro (FOG) technology. Furthermore,these products have been developed to improve or filterexisting PosRef signals before applying them as PosRefmeasurements in the DP estimator, where the integratedINS solution of Faugstadmo and Jacobsen (2003) is onlyinterfaced with hydroacoustic position reference (HPR)systems. Moreover, some of these INS products are alsosubjected to export restrictions, limiting the market po-tential and increasing the cost of installation due to apossibly lengthy approval process before installation.

Even though the cost of installation is quite high, INSintegration in DP has received considerable attention inthe industry the last years, Stephens et al. (2008), Carter(2011; 2014), Russell (2012) and Willumsen and Hals(2013). From the latter, it is stated that export restrictionsare rarely a problem since “acoustic systems are coveredby the same rules”. However, this statement does notapply to GNSS technology, which is not subjected to suchrestrictions. Furthermore, currently there exist numerousMEMS-based IMUs on the market not subjected to exportlicenses. Such units have a great potential to be used in themaritime and offshore markeds. Therefore, studies on DRcapabilities of GNSS aided INS, using high-rate MEMSIMUs, utilized in ship navigation are of great interest.

1.1 Main Contributions

This paper presents an initial study of the DR capabilitiesin DP obtained using GNSS-aided INS based on high-rateMEMS IMUs. The study is carried out:

• Based on two NLO designs, Mahony et al. (2008) andRogne et al. (2016), respectively, interconnected withtranslational motion observers (TMO).• Employing two types of MEMS IMUs.• Evaluating the DR capabilities in both position and

heading.

For a study on attitude determination using NLOs andhigh-rate MEMS IMU, Bryne et al. (2016) can be advised.

2. PRELIMINARIES

2.1 Notation

The Euclidean vector norm is denoted ‖ · ‖2. The n × nidentity matrix is denoted In, while the transpose of avector or a matrix is denoted with (·)ᵀ. Coordinate framesare denoted with {·}. S(·) ∈ SS(3) represents the skewsymmetric matrix such that S(z1)z2 = z1 × z2 for twovectors z1, z2 ∈ R3. In addition, zabc ∈ R3 denotes avector z, to frame {c}, relative {b}, decomposed in {a}.Moreover, ⊗ denotes the Hamiltonian quaternion product.Saturation is represented by sat?, where the subscriptindicates the saturation limit.

The rotation matrix describes the rotation between twogiven frames {a} and {b} and is denoted Rb

a ∈ SO(3).Similar to the rotation matrix, the rotation between {a}and {b} may be represented using the unit quaternionqba = (s, rᵀ)ᵀ where s ∈ R1 is the real part of thequaternion and r ∈ R3 is the vector part. Roll, pitch andyaw are denoted φ, θ and ψ, respectively.

2.2 Coordinate Reference Frames

This paper uses four reference frames; The Earth CenteredInertial (ECI) frame, the Earth Centered Earth Fixed(ECEF) frame, a tangent frame equivalent of a Earth-fixedNorth-East-Down (NED) frame, and the BODY referenceframe, denoted {i}, {e}, {n} and {b}, respectively (seeFig. 1). ECI is an assumed inertial frame following theEarth as it rotates around the sun, where the x-axispoints towards vernal equinox, the z-axis is pointing alongthe Earth’s rotational axis and the y-axis completes the

xi

xeyi

ye

zi, zeωie

xnyn

zn zb

yb

xb

oi, oe

obon

λ

µ

Fig. 1. Definitions of the BODY, NED (tangent), ECEFand ECI reference frames.

right hand frame. Regarding the ECEF, the x-axis pointstowards the zero meridian, the z-axis points along theEarth’s rotational axis, while the y-axis completes theright hand frame. The Earth’s rotation rate ωie = 7292115·10−11 rad/s is given by the WGS-84 datum. It is furtherdecomposed in the ECEF and NED frame as

ωeie =

(001

)ωie, ωnie =

(cos(µ)

0− sin(µ)

)ωie, (1)

where µ is the latitude on the Earth and ω??? representsangular velocity. The longitude is denoted λ. Furthermore,the navigation frame is a local Earth-fixed tangent frame,{n}, where the x-axis points towards north, the y-axispoints towards east, and the z-axis points downwards. TheBODY frame is fixed to the vessel, and the origin of {b}is located at the vessel’s nominal center of gravity. Thex-axis is directed from aft to fore, the y-axis is directed tostarboard and the z-axis points downwards.

2.3 Kinematic Strapdown Equations

Estimating position when using the tangent frame asthe navigation frame, results in implementation of thestrapdown equations,

pnnb = vnnb, (2)

vnnb = −2S(ωnie)vnnb +Rn

b fbib + gnb , (3)

where pnnb ∈ R3 is the position, relative a defined originof the tangent frame, pnnb(0) := 03×1 based on µ(0) andλ(0). Furthermore, vnnb ∈ R3 is the linear velocity. Itfollows that gnb (µ, λ) ∈ R3 is the local gravity vectorwhich may be obtained using a gravity model based onthe vessel’s latitude and longitude. Rn

b ∈ SO3 is therotation matrix. See Bryne et al. (2016) for the rotational

kinematic equations used to obtainedRnb . Moreover, f bib =

(Rnb )ᵀ(anib − gnb ) ∈ R3 is the specific force decomposed in

{b} and where anib is the accelerations decomposed in thetangent frame.

Bryne et al. (2014; 2015) extended the translational mo-tion kinematics further, by including a state of integratedvertical(down) position/heave, i.e,

pnI = pnnb,z, (4)

and is motivated by the fact that the mean verticalposition of the vessel is zero over time, since the wave-induced motion of the craft in heave oscillates about themean sea surface. This augmentation of (2)–(3) can beexploited in the INS by incorporating the virtual verticalreference (VVR) of Bryne et al. (2014; 2015; 2016).

3. IMU AND SHIP SENSOR CONFIGURATION

3.1 IMU and Error Sources

A strapdown IMU is a sensor unit measuring tri-axisangular velocity and tri-axis specific force of the unit inBODY frame relative the inertial frame,

f bIMU =(f bx, f

by , f

bz

)ᵀ, ωbIMU =

(ωbx, ω

by, ω

bz

)ᵀ,

where the subscripts x, y and z, denote the forward,starboard and downwards axes in the BODY frame. Inaddition to the specific forces and angular velocity, eachmeasurement is contaminated with sensor biases, errorsand noise. Sensor errors may consist of nonlinearity, scalefactors, cross-coupling and g-sensitivity errors, where thelatter errors only influence the angular rate sensors read-ing. In addition to internal noise sources, external noisemay be due to e.g. electrical and magnetic interference orstem from mechanical sources in the form of vibrations. Inthis paper, we assume that error sources related to sensornonlinearity, scale factors, cross-coupling and g-sensitivityare compensated for in calibration by the manufacturer,or otherwise are neglectable. Sensor biases may also becalibrated for by the manufacturer, however some time-varying bias instability and run-to-run instability is oftenpresent with MEMS IMUs. Therefore, we model the an-gular rate and accelerometer measurements as

ωbIMU = ωbib + bbg +wbg, (5)

f bIMU = f bib + bba +wba, (6)

where ωbib and f bib are the true angular rates and specificforces, respectively. Moreover, the corresponding sensorsbiases are denoted bbg and bba, while wb

g and wba represent

the sensor noise and vibration induced noise contained inthe respective measurements. Both the angular rate/rategyro and accelerometer biases are assumed constant,

bbg = 03×1, bba = 03×1. (7)

3.2 Ship Sensor Configuration

Several IMUs were installed on an offshore vessel, op-erating in the North sea, equipped with a Rolls-RoyceMarine DP system. The ship in question is owned andoperated by Farstad Shipping. In this paper we will presentresults obtained using one STIM300 and one ADIS16485MEMS IMU. The sensor configuration utilized in the aidedstrapdown INS, based on the kinematic formulation ofBryne et al. (2016, eq. (3)), (2)–(3), (4) for fusing IMU,compass, GNSS and VVR measurements, was:

• 1x dGNSS PosRef measurement,pnGNSS = (pnnb,x, p

nnb,y)ᵀ at 1 Hz.

• VVR: pnI = 0, for all t ≥ 0 at 1000 Hz. By using theVVR, other vertical references based on ranging withreduced precision due to the vertical ranging geom-etry, such as with GNSS-based and hydroacoustic-based PosRefs are avoided. For more details on theVVR measurement principle, see Bryne et al. (2015).• 2x IMUs (ADIS16485 and STIM300) providing

· Tri-axis angular rate measurements, ωbIMU· Tri-axis accelerometer-based specific force mea-

surements, f bIMUboth interfaced on 1000 Hz.

Table 1. IMU specifications

ADIS16485 STIM300 1

In-run Gyro Rate Bias Stability 6.25 degh

0.5 degh

Angular Random Walk 0.3 deg√h

0.15 deg√h

In-run Accelerometer Bias Stability 0.032 mg 0.05 mg

Velocity Random Walk 0.023m/sh

0.06m/sh

Table 2. NLO reference vectors configuration

Vector cn Vector fn

NLO A Unit vector North −gnb /‖ − gn

b ‖2NLO C Unit vector North fn

ib/‖fnib‖2 through feedback

from VVR and virtual velocityinjection

• 3x yaw measurements from gyrocompasses, ψc, at 5Hz.

The specifications of the IMUs installed on the offshorevessel are presented in Tab. 1.

4. NONLINEAR OBSERVERS

In this papers we use two NLOs, each in cascade witha TMO to evaluate the DR capabilities using the twoMEMS IMUs. The NLOs in question are the observerof Mahony et al. (2008) and Rogne et al. (2016), wherethe latter is based upon the results of Mahony et al.(2008), Grip et al.(2012; 2013) and Bryne et al. (2014;2015). In this paper we will refer to Mahony et al. (2008)as NLO A and Rogne et al. (2016) as NLO C. Theattitude determination performance using both A and Cis evaluated in Bryne et al. (2016), therefore a similarnaming convention is chosen here. An overview of themodular nonlinear observer structure used to estimatePVA is shown in Fig. 2, applicable for both NLO A andC, but where A has no aiding TMO. The main differencebetween NLO A and C is outlined in Tab. 2, see Bryneet al. (2016) for details. Both NLOs are based on the sameobserver equations,

Σ1 :

˙qnb =

1

2qnb ⊗

(0

ωbib

)− 1

2

(0ωnin

)⊗ qnb , (8a)

ωbib = ωbIMU − bbg + σi, (8b)

˙bbg = Proj

(bbg,−kI σi

), (8c)

where Proj denotes the angular rate bias projection algo-rithm of Grip et al. (2012) and the reference therein, and kIis the gain assorted with the angular rate bias estimation.The difference between observers A and C, as indicated inTab. 2 lie in the injection terms, σi, given as

σi = k1vb1 ×R(qnb )ᵀvn1 + k2v

b2 ×R(qnb )ᵀvn2 , (9)

depending on the observer setup in question, where i ∈[A,C], and where v1

b and v2b are the measurement vectors

and v1n and v2

n are the reference vectors, calculated using

v1b = f b, v2

b = f b × cb,v1n = fn, v2

n = fn × cn.For both NLO A and C, cb = (cos(ψc), − sin(ψc), 0)

ᵀand

cn = (1, 0, 0)ᵀ

as stated in Bryne et al. (2014).

1 Engineering sample used, not necessarily in compliance with spec.

ωbIMU

σ

IMU

VVRpnI PosRef

pnnb

Σ1

ψc

Attitude

Observer

Aiding

Translational

Motion

Observer

Compass

Σ2

pnnb

vnnb

Virtual

VelocityΣ3

Translational

Motion

Observer

R (qnb )

fbIMU

vnvir

AHRS

fnnb

qnb

fnnb

Accelero-

meter bias

estimate

bba

fbIMU − bba

fbIMU − bba

Fig. 2. Complete observer structure. The attitude observer estimates the quaternion qnb and angular rate/gyro bias bbg . The TMO estimate

position, pnnb, and linear velocity, vn

nb. Estimated accelerometer bias is denoted bba. An aiding translational motion observer may beutilized to aid the attitude observer. Dashed lines indicate optional feedback.

4.1 Attitude observer A

For NLO A, σA is implemented with f b and fn is basedon the injection term of Mahony et al. (2008) choosing

f b =f bIMU − b

b

a

‖f bIMU − bb

a‖2, fn =

−gnb‖ − gnb ‖ 2

,

where the local gravity vector is utilized as referencevector based on the assumption that the specific force

in the navigation frame is dominated by −gnb and bba isthe estimated bias, either statically from calibration oronline estimation. In this part of the work we apply staticaccelerometer bias compensation.

4.2 Attitude Observer C

Regarding NLO C, the reference vector fn, in the calcu-lation of σC , is chosen as

fn =satMf

(fnib)

‖satMf(fnib)‖2

,

where fnib is estimated in Σ2. k1, k2 and kI are gains. See,Bryne et al. (2016) for details.

The TMO used to aid NLO C is obtained from Rogne

et al. (2016), and is applicable in DP to estimate fnib inorder to improve the quaternion estimates of NLO C sincefnib 6= −gnb due to the wave-induced motions of the vessel.The observer takes the form of,

Σ2 :

˙pnI = pnnb,z + θhgKpIpI pI , (10a)

˙pnz = vnnb,z + θ2hgKppI pI , (10b)

˙vnnb = −2S(ωnie)vnnb + fnib + gnb

+ θ3hg

(02×1

KvpI

)pI ,+θhg

(Kvv

01×2

)v,

(10c)

ξ = −R(qnb )S(σ)(f bIMU − b

b

a

)+ θ4

hg

(02×1

KξpI

)pI + θ2

hg

(Kξv

01×2

)v,

(10d)

fnib = R(qnb )(f bIMU − b

b

a

)+ ξ, (10e)

where pI = pnI − pnI and v = vnvir −(vnnb,x, v

nnb,y

)ᵀ. K[·]pI

and K [·]p are gains, while θhg ≥ 1 is a high-gain tuning

parameter used to guarantee stability. In DP, the velocityof the ship is approximately zero, hence in that case,

vnvir = (0, 0)ᵀ. (11)

Consider the observer structure in Fig. 2, by using thevirtual velocity measurements in Σ2, instead of measure-ments obtained from PosRefs, any errors in the PosRefis prevented from entering Σ1, making the attitude es-timation more fault tolerant, Rogne et al.(2015; 2016).Any errors in the lever arm from PosRef to IMU is alsoprevented from affecting the attitude estimates using (11)as aiding measurement. The origin of the error dynamicsof observer NLO A and C, (see Tab. 2 for details on A andC) is almost globally exponentially stable and uniformlysemiglobal exponentially stable (USGES) in Mahony et al.(2008) and Rogne et al. (2016), respectively.

4.3 Accelerometer Bias Compensation for DR

Even though the angular rate and accelerometer biasesnot are mutually uniformly observable, Farrell (2008, Ch.11.9), without the vessel accelerating and rotating, someaccelerometer bias compensation have to be done in orderto obtain an INS with reasonable DR capabilities. In Bryneet al. (2016), a fixed pre-compensated accelerometer biaswas successfully applied for attitude estimation. However,some acceleration errors may be present owing to somein-run bias instability, w.r.t. Tab. 1. To atone for this,we extend NLO A with an additional observer, based onthe observer of Fossen (2011, Ch. 11.5.1) with globallyexponentially stable (GES) error dynamics. In this paperhowever, only the horizontal plane is considered sincevertical DR is not relevant for ships. Another difference isthat the attitude is not known, but estimated. This resultsin the observer,

Σ3 :

˙pnnb,xy = vnnb,xy +Kpppxy, (12a)

˙vnnb,xy = −2Sxyvnnb,xy + Rxy

(f bIMU,xy − ˆbba

)+Kvppxy,

(12b)

˙bba,r = KapR

ᵀxypxy, (12c)

ˆbba = bb

a + bb

a,r, (12d)

where bba,r is the estimate of residual uncompensatedaccelerometer bias and

Sxy =

(0 sin(ωie)

− sin(ωie) 0

), Rxy =

(R(qn

b )11 R(qnb )12

R(qnb )21 R(qn

b )22

),

and where K?p are gains chosen such that A − KC isHurwitz when writing (12) in matrix from,

˙x = Ax+Bu+K (y −Cx) (13)

with x =(pnx , p

ny , v

nx , v

ny , (b

ba,r)

ᵀ)ᵀ

, y =(pnGNSS,x, p

nGNSS,y

)ᵀand u = f bIMU,xy where x and y represent the horizon-tal components of the GNSS and accelerometer measure-ments, and

A =

(02×2 I2 02×2

02×2 −2Sxy −Rxy

02×2 02×2 02×2

), B =

(02×2

Rxy

02×2

),

C =(I2 02×2 02×2

), K =

(Kᵀ

pp Kᵀvp Kᵀ

ap

)ᵀ.

Equations (12b)–(12c) are used in an attempt to estimate

bba,r in order to compensate for any acceleration errors

in Rxy(f bIMU,xy − bba), such that velocity error growth ofv = vnnb − v

nnb in (12b) is limited, which in turn improves

the DR performance of (12a). Since the accelerometer biasis decomposed in BODY, the error dynamics of Σ1−Σ2−Σ3 cannot be proven globally stable due to the nonlinearstate dependent term, R(qnb ), from the NLO enters theerror dynamics of (13) both through the A matrix andthe injection term and not only through the input w.r.t.to the USGES result of Rogne et al. (2016, Theorem. 1)applying a cascades system argument. However, a localstability result is obtainable through linearization.

For observer C we base the estimation of the NED accel-eration error on the signal ξ, in fnib from Σ2, (10e) toperform the same task as (12b)–(12c) by replacing the

term Rxy

(f bIMU,xy − ˆbba

)with fnib,xy from Σ2.

5. FULL-SCALE VERIFICATION OF DRCAPABILITY IN DP APPLYING MEMS IMUs

In this section, the evaluation of the DR properties usingan ADIS16485 and a STIM300 MEMS IMU is presented.The DR performance evaluation is carried out with datacollected during a DP operation performed in the NorthSea during the fall of 2015. The GNSS track of the vesselin DP, obtained from the ship’s dGNSS receiver, is shownin Fig. 3.

0 2 4 6 8

East [m]

-5

-4

-3

-2

-1

0

North[m

]

Fig. 3. The path track of the DP operation used to evaluatethe DR performance of the respective IMUs. The pathtrack is obtained from the on board dGNSS solution.

First, the heading DR performance when using the IMUsavailable is discussed, and illustrated with an example.

Then, the position DR performance during the particularDP operation is evaluated, applying both IMUs and NLOA and C. The resulting DR statistics are based on amultitude of estimation runs, where NLO A and C weretuned with similar gains.

5.1 Heading Angle DR Capabilities

The heading angle DR capabilities using the IMUs avail-able were found to be in compliance with the IMUs’ theangular rate specifications, presented in Tab. 1. 12 headingDR evolutions of the absolute yaw angle error

|ψ| = |ψc − ψ|, (14)

relative the averaged ship gyrocompass measurements forboth sensors, for one hour in heading DR after disablingthe observer injection from the compass by setting k2 = 0,are shown in Fig. 4. The observer initialization time was15 minutes. In addition, the average DR error, of the 12runs, is highlighted in Fig. 4. Examples of typical angularrate bias estimates are shown in Fig. 5, where the yaw ratebias is frozen from t = 15 minutes.

Fig. 4. DR performance in yaw obtained using theADIS16485 and STIM300 IMUs. Highlighted evolu-tion indicates average DR error.

5.2 Position DR Capabilities

Evaluation of the DR capabilities in position is moreelaborate than for heading since the theoretical growth oferrors are a combination of higher order terms, (Groves,2013, Ch. 5.7), as opposed to linear growth for heading. Inorder to obtain statistically significant results related tothe position drift while performing DR, each combinationof IMU and NLO was evaluated 50 times by comparing theDR errors accumulated when disabling GNSS feedback atarbitrary times. The DR evaluation is done by taking thenorm of the difference between the horizontal componentsof pnGNSS and pnGNSS, defined pGNSS := pnGNSS − p

nGNSS

where,pnGNSS = pn +R(qnb )rb, (15)

and where rb being the lever arm from the IMU to theGNSS such that

‖pGNSS‖2 = ‖pn +Rnb r

b − pn −R(qnb )rb‖2,= ‖p+ (Rn

b −R(qnb )) rb‖2. (16)

NLO A: Fig. 6 shows the aggregated drift errors over 10minutes, after PosRef injection is disabled, applying NLOA for both the ADIS16485 and the STIM300 IMU. The

0 10 20 30 40 50 60 70

Time [min]

-0.15

-0.1

-0.05

0

0.05bb g

[deg/s]

roll rate

pitch rate

yaw rate

(a) Typical angular rate bias estimates of theADIS16485

0 10 20 30 40 50 60 70

Time [min]

-0.15

-0.1

-0.05

0

0.05

bb g[deg/s]

roll rate

pitch rate

yaw rate

(b) Typical angular rate bias estimates of theSTIM300

Fig. 5. Angular rate bias estimates after heading DR test.

Table 3. Position DR error statistics using NLO A asattitude observer in DP.

ADIS16485 STIM300

Mean error [m] 1 min 4.1631 3.9816Mean error [m] 5 min 35.2529 31.5438Mean error [m] 10 min 116.5540 102.3643Min error [m] after 10 min 12.1275 14.6419Max error [m] after 10 min 314.9137 283.8679RMSE [m] after 10 min 132.6765 119.2748

Table 4. Position DR error statistics using NLO C asattitude observer in DP.

ADIS16485 STIM300

Mean error [m] 1 min 7.2513 3.9651Mean error [m] 5 min 67.0432 28.7058Mean error [m] 10 min 167.4041 93.9744Min error [m] after 10 min 14.5668 36.3566Max error [m] after 10 min 402.2314 172.0885RMSE [m] after 10 min 190.3469 98.8203

statistical results based on the 50 DR runs are presentedin Tab. 3 using A as attitude observer.

NLO C: Fig. 7 show the aggregated drift errors over 10minutes, after PosRef injection is disabled, applying NLOC for both the ADIS16485 and the STIM300 IMU. Thestatistical results based on the 50 DR runs are presentedin Tab. 4 using C as attitude observer.

5.3 Discussions

Heading DR: It is evident, w.r.t. to Fig. 5, that the gyrobias estimates using the STIM300 is smoother and morein-run stable than those obtained using the ADIS16485,

(a) DR errors obtained with ADIS16485

(b) DR errors obtained with STIM300

Fig. 6. Aggregated DR error over 50 runs using NLO A.Red indicates the mean DR error.

resulting in the performance difference seen in Fig. 4. Thisis in compliance with the sensor specifications presentedin Tab. 1. The asymptotic angular rate bias estimationperformances seen in Fig. 5, is representative of what isseen from run to run.

Position DR: As seen from Tabs. 3–4 and Figs. 6–7,the two main conclusions from the 4 times 50 DR runsperformed over the data set collected during DP is that:

• using the STIM300 results in better DR performance,than using the ADIS16485,

• using observer structure A often resulted in betterDR performance than using observer structure C forADIS16485.

The first result is as expected considering the angularrate in-run bias stability presented in Tab. 1 and theresults from Sec. 5.1 where the STIM300 provides thebest attitude estimates. However, why observer structureA gave better DR performance than using structure C forADIS16485, is not evident since in Bryne et al. (2016)C outperformed A when it came to attitude estimation.

(a) DR errors obtained with ADIS16485

(b) DR errors obtained with STIM300

Fig. 7. Aggregated DR error over 50 runs using NLO C.Red indicates the mean DR error.

One possible answer could lie in that the mean attitudeerror of NLO A, obtained in Bryne et al. (2016), was smallsuch that the DR position estimates based on (12) wereobtained with a reasonably accurate averaged attitude.

The results indicate a large spread of the DR error over10 minutes. Especially when using the ADIS16485 andNLO structure C. This might be due to noise, mechanicaldisturbance such as vibration, or insufficient tuning ofthe observers, or that the virtual velocity measurementemployed in NLO structure C is not accurate enough when

estimating fnib for DR purposes. More probable however isthat the assumption of constant bias done in Sec. 3.1 is notsufficient over time for the NLO to provide the same DRperformance as NLO structure A, even though the sameassumption yielded satisfactory results related to attitudeestimation, (Bryne et al., 2016).

Considering the quality of the results obtained comparedto the results in Paturel (2004), using either of the twoMEMS-based IMUs available in this work, gave worseresults than in the cited works where an FOG-based IMUwas applied. In the results presented in Paturel (2004),

a position accuracy during GNSS outage stayed withinGNSS accuracy for a period exceeding two and a halfminutes. The mean position drift after a 50 seconds GNSSoutage was less than half a meter. These results areconsiderably better than the approximately 4 meters errorobtained using the STIM300 IMU after one minute DRfor both NLOs. However, in Paturel (2004) only 10 runsare presented, making a definite statistical comparisondifficult due to the few DR trajectories presented. TheFOG product in question is currently advertised to have a20 m error with a 50 % circular error probability afterfive minutes of unaided navigation, whereas we obtainapproximately 30 meters averaged error in the same timeframe, being 50 % worse.

As depicted in Figs. 6–7, a MEMS-based INS may providerelatively stable position estimates (less than four meterserror) for half a minute, without PosRef injection. From afault-tolerance perspective, such as Rogne et al. (2016), theresults obtained here indicate what kind of PosRef errorsone might detect based on MEMS IMUs. For instancea PosRef drift of 10 centimeters per second results in aPosRef error of 3 meters after half a minute, which mightbe possible to detect with the results obtained, consideringthe average DR error is two meters with either of thetwo observers and the STIM300 unit. Moreover, in thesituation of PosRef failure during DP, if four meters isan acceptable error margin, 30 seconds is available to theDP operator to decide whether the operation should beaborted or not. This might be sufficient time for PosRefrecovery e.g. if line-of-sight is established with one moresatellite, resulting in a complete GNSS solution.

Since the ship was in DP with small roll and pitch motion(between ±3 degrees), the x- and y-axis accelerometerbiases were probably not entirely observable, which will

affect the DR performance even though fnib is estimateddirectly and indirectly using NLO A and C, respectively.

The DR performance properties are not only dependenton the sensor biases, but also on the velocity-random walkand the sensed vibrations on the ship. Integrating theseover time, results in a large error even when averagingthem out using high-rate integration (1000 Hz). Regardingtuning, approximately the same tuning as in Rogne et al.(2015; 2016) was used. With more emphasis on tuning for aDR application, better results may be accomplished. Also,time-synchronization errors between our IMUs and the on-board dGNSS system may result in small errors in velocityand specific force at the time of disabling dGNSS injection,resulting in a steeper error slope than otherwise obtained ifthe position and inertial measurements were synchronized.It is also difficult to conclude with certainty that theresults found using the STIM300 are representative, sincewe used an engineering sample provided by Sensonor.This is a test unit, not necessarily in compliance with thestandard specification.

6. CONCLUDING REMARKS

Full-scale assessment of dead reckoning capabilities, inconjunction with the comparison and validation of twononlinear observers, has been carried out. The observerswere driven by two MEMS based IMUs, aided by GNSSand gyrocompasses, and evaluated in a DP operation

executed by an offshore vessel in the North Sea. Forheading, 12 cases were run for the purpose of showcasingtypical observer and IMU behavior were presented. Forposition, a more elaborate study was conducted, runningeach combination of observer and IMU 50 times eachfor amassing statistics regarding their performance. Theresults showed that STIM300 was better suited for deadreckoning than the ADIS16485 unit, and that NLO Awas the preferred observer. An in-depth study on theeffects of ship vibrations on the DR performance has tobe undertaken. Further works on optimal tuning shouldalso be carried out.

ACKNOWLEDGEMENTS

This work has been carried out at the Centre forAutonomous Marine Operations and Systems (NTNU–AMOS) and supported by the Research Council of Norwayand Rolls–Royce Marine through the Centres of Excel-lence funding scheme and the MAROFF programme, grantnumbers, 223254 and 225259 respectively. The ResearchCouncil of Norway is acknowledged as the main sponsorof NTNU–AMOS.

The authors wish to thank colleagues at the mechanicaland electronics workshop at the Department of Engineer-ing Cybernetics for help during the development of thesensor payload and thank Rolls-Royce Marine and FarstadShipping for assistance in the process of installing the sen-sor payload on board the offshore vessel. The authors wishto thank Farstad Shipping for allowing us to install theequipment on board. The authors wish to thank Sensonorfor providing an engineering sample of the STIM300 IMU.

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