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ON THE IMPLEMENTATION OF A SHIP DYNAMIC MODEL IN AN INTEGRATED
NAVIGATION SYSTEM
J. H. Wul der J. A. Spaan s
H.G.Stassen
Abstract
Deift University of Technology,
Royal Netherlands NavaJ Academy, and Deift University of Technology.
TECHNISCHE UNIVER$gg
Laboratorium 'icor Scheepehydromecharfica Nekeiweg Z 2628 CD Deift
Teij O15..78e873. 015.781838
This paper presents a low cost alternative for G PS/INS integration. By means of a ship manoeuvring model, the accelerations, velocities, and position of a ship are predicted. The comparison between the predicted and measured position has to result in a more reliable and more accurate position. The paper elucidates such ship manoeuvring model, and gives a method to estimate the hydrodynamical coefficients for a particular ship. A Kalman filter is used to integrate the manoeuvring model and the GPS measurements.
i. INTRODUCTION
To increase the reliability and accuracy of the GPS performance, integration of GPS with high
cost INS in a Kalman filter (KF) is often proposed [Napier, 1990; Diesel,1987]. The reason to
integrate both systems is the difference in noise characteristics of GPS and INS. The INS error
results in a low frequency position error, up to i nmi/hour. When Selective Availability (SA) is fully operational, the position noise of GPS for civilian users is expected to be an offset
(<30 m) and a high frequency error of loo m.
in this paper an alternative for INS integration for marine craft is proposed. The accelerations of the vessel are not measured, as with INS, but calculated by means of a ship manoeuvring
model. The ship manoduvring model represents the dynamics of the ship, the way a ship
responds to engine, rudder and environmental inputs. With a manoeuvring model it is possible to make a prediction of the ship's track during a longer period, even during manoeuvres. The integration of the ship dynamics and GPS avoids the cost of an INS system. It requires,
however, the knowledge of the ship behaviour. Hereafter, the usefulness ofa particular
manoeuvring model for implementation in a navigation system is investigated, with the aimto reduce the SA noise, and to increase the system reliability.
The paper is divided in four parts: first a general manoeuvring model is given. The second part describes the identification of the parameters of the manoeuvring model fora specific
ship, and elucidates model behaviour in relation to sea trials. The third part is the design of
the KF based on the ship manoeuvring model. Finally conclusions are given.
2. SHIP MANOEUVRING MODEL
The ship manoeuvring model describes the motions of a ship in the horizontal plane: the longitudinal velocity u(t), transverse velocity v(t), and the rate of turn r(t), Fig.1.
x0,y0 : earth fixed coordinate system heading
u(t) : longitudinal velocity v(t) : transverse velocity r(t) : rate of turn, 1'(t)
Figure i The definition of the coordinates
The motions of the ship are caused by forces working on the ship. The forces have different
origins: the hull, rudder, propeller and disturbances, like wind and waves. The ship
manoeuvring model is based on the Newtonian force equation. In [Inoue,1981; Abkowitz,1969]
it has been derived that the forces on a ship are functions of the ship velocities, the rudder
angle and the propeller, in formula:
20:2
i
(t)=f11{u (t),v (t)r(t)8(t)Q (t)P(t)};
(1 .a)s (t)=f,{u(t),v (t),r(t),8(t)}; (1.b)
t (t,)=frfu(t),v(t),r(t),b(t),7(t)}; (1 .c)
(t)=u(t)cos{L'(t)} - v(t)sin{(t)}; (1 .d)
0(t)=u(t)sin{b(t)} + v(t)cos{(t)}; (1.e)
with
Q(t)
: the torque of the propeller shaft;ç(t) : the roll angle;
P(t)
: the propeller pitch;the functions which represent the forces and moments acting on the ship (the mass of the ship is included); and
b(t) : the rudder angle.
Herewith the basis is found for the ship manoeuvring model. Two problems are left: first the description of the functions
1u'v'1r' and second the implementation in the model of the
disturbances acting on the ship. Next, both problems are discussed.First the descriptions of the functions are given. The functions are partly based on the physical laws of fluid mechanics, and partly based on empirically found relations
{Abkowitz:1980; Inoue:1981J. The function is given as example:
= civ(tr(t) + cu2u(t) + c3ut,flu(t.)j
+ Cu4U(t) + c5Q (t + c
pub p
.P (t)Q (t) + c f{u(t),v(t),r(t),5t)},
p ue (2)with f{u(t),v(t),r(t),h(t)}: a known function which represent the rudder force; and the hydrodynamic coefficients.
In Eq.(2) the hydrodynamic coefficients cul..cu7 are unknown. The functions and
are described in the same way [Wulder,1990:2]. The function Ç(.) has six unknown coefficients, and the function has nine unknown coefficients crl..cr9. The
hydrodynamic coefficients represent the behaviour of a particular ship: every ship has her own set of coefficients. In order to determine the set of hydrodynamic coefficients four methods are available: the theory of fluid mechanics, scale model tests, sets of coefficients of ship families,
and system identification by means of sea trials
Up to now, the theory of fluid mechanics does not provide a complete understanding of the
way the coefficients which are involved in ship manoeuvring. can be calculated
[Abkowitz,1980J. Scale model tests are often used, but suffer from scale effects, and are
expensive. The costs are comparable with a cheap INS system, eliminating the advantage of a
low cost system. The third method, the use of sets of coefficients of ship families, gives reasona.ble results [Inoue,1981; Boer,1983J. The known hydrodynamic coefficients of ship
families are used to determine the coefficients of a particular ship, by means of the dimensions of the ship. The disadvantage of this method is the unpredictable accuracy, hence, to check the accuracy, sea trials have to be executed. The last method, sea trials, means that some
special manoeuvring trials have to be performed with the ship like, turning circle, stop
manoeuvre, and so on. By means of these manoeuvres it is possible to estimate the
hydrodynamic coefficients [Abkowitz,1980; Wulder,1990:2]. The estimation of the hydrodynamic coefficients by means of sea trials is described in the next section.
The second unknown part of the manoeuvring model is the disturbances working on a ship:
the wind, the waves and the current. The wind results in a longitudinal force, a transverse force, and a moment. The magnitudes of the forces and the moment depend on the wind velocity and the angle of incidence of the wind. In [Isherwood,1972] a method is described to
estimate the wind forces, when wind velocity and directionare known. The method is based
on a set of empirical coefficients; it estimates the wind force with an accuracy of about 20%.
The second disturbance, the waves, is not measured, and with that it is not possible to
calculate the wave forces. In [Wulder,1990:2] it is shown that only theconstant components of the wave forces and moment are of importance for a navigationsystem. So, the waves cause
two unknown forces and one unknown moment on the vessel, which magnitudes depend on the
angle of attack of the waves and the wave height. The third, and last, disturbance is the current. The current is supposed to be a constant or slowly varying, unknown, speed vector which can be added to the ship velocity through the water.
In the following, the manoeuvring model, Eq.(1), will be extended with the disturbances, it
yields:
+ f
{v(t),p(t)} + zu('t);
(3.a) wu wr {u(t),v(t),r(t),ô(t)} + f {v(t)pwr(t)} + Ajt);
(3.b) V WV wrtJt)=f{u(t),v(t),r(t),8(t),ç(t,)} + f
wr{v (t),j.(t)} + zf(t);
(3.c) wr wr 0(t)=u(t)cos{L'(t)}- v(t)sin{1'(t)} + v(t);
(3.d) '0(t)=u(t)sin{'(t)} + v(t)cos{b(t)}+ v(t);
(3.e)j«t) = r(t),
(3.f)with
Çu'Çv'Çr(
: the wind forces and wind moment;Vcx(t.)vcy(t) : the current velocity in x0and y0 direction;
(t)
wr the relative wind velocity;
the relative wind direction; and
zÌ(t)4r(t)4i-(t)
: the accelerations caused by wave forces and moment.In the next section a method is proposed to estimate the hydrodynamic coefficients. The current vector and the accelerations caused by the wave forces
u(t)4(i(t)4t(t}
are unknown. In section 4 it is shown that it possible to estimate these bymeans of a KF.
3. IDENTIFICATION AND VALIDATION OF THE MODEL
The hydrodynamic coefficients are estimated by means of sea trials. The kind of sea trials which have to be performed, are dictated by the identification method. So, first the identification method is discussed, then the sea trials are discussed, and finally the identification results are validated.
The hydrodynamic coefficients are estimated step by step, which means that the input signals are not manipulated simultaneously, but sequentially: first there is an equilibrium, then only engine orders are given, and finally both the rudder and the engine are used. The advantage of
this method is the suppression of interaction effects between the terms in Eq.(2), for example:
it is not possible that the term cu3u(t) influences the term c6P(t)Q(t) and thus the value
of cu6. The disadvantage is that the system is considered to be linear, which is certainly not true for a ship.
The step by step method exists out of five steps. Step one until four are used for identification purposes. the fifth step is for the validation of the estimated hydrodynamic coefficients. Before
the five steps are discussed an overview of the required manoeuvring trials and the coefficients
to be estimated with every step is given in Table 1.
Table i The manoeuvres to be executed for the identification procedure step manoeuvres estimation of:
i 2 3 4 square natural stop (2x) gradual stop/acceleration manoeuvre (2x) turning circle (2x) zigzag manoeuvre (2x) Vcx(t) vcy(t) 6o
c2, c113, c4
c5, c6
cul c7,
CriCr9During the first step of the identification procedure, the constant rudder angle ô and the current velocity are detected. A ship is never exactly symmetric, which results in a small
rudder angle, to keep the ship on a straight line. So, to detect the ship sails with constant heading. The average of the rudder angle during the trial gives the mean rudder
angle
. If not a straight line is sailed, but a square then the wind influence is reduced. The
square exist Out of four lines with the headings perpendicular to each other (for example: 0, 90e, 18O, 27O). If there is any current, the ground track will have another shape then the sailed square; the deformation of the ground track gives information about the current
velocity and direction {Wulder,1990:1]
The second step of the identification is the determination of the coefficients
cu2, c3 and c4,
which is done with a natural stop. A natural stop starts with the ship sailing at maximum speed on a straight line. The natural stop is initiated by stopping the engines, which results in a decrease in speed of the ship. The natural stop causes that v(t)=r(t)=f(.)=O (straight line), and Q(t)=0 (engines stopped). Herewith Eq.(2) can be rewritten to:û(t)
= c9u(t) + c3u(t»u(t)J +
cu4u(t)3. (4)During the natural stop the velocity u(t) is recorded. Every ¿t second the signal u(t) is sampled, which results into the time series: u(1), u(2) ,u(n). Eq.(4) holds for every sample,
which results in a set of n equations:
I
cU3i
Ic4I,
j
u(n) u(n)lu(n»
with n is the number of samples; and ü(k)= {u(k+1)u(k-1)}/{2At}.
Eq.(5) is a set of n equations with three unknown coefficients, and can be solved for thiscase
with a least square method {Spaans,1987].
The third step in the identification is the activation of the engine,
or Q(t)O. For this step,
Eq.(2) can be rewritten as:ú(t) = c2u(t) + c3u(t)Ju(t» + c4u(t)3 + c5Q(t} +
c6P(t)Q(t).
(6)20:6
(5)
û (1)
u/i)
'..-J ui/i )I u/i II-JI-'JI
ui/i'-'J
The coefficients c5 and c6 can be estimated in the same way as the coefficients c2,
c3 and
c4 in Eq.(5). The manoeuvre required to estimate the coefficients c5 and c6 is a gradual stop/acceleration manoeuvre. The manoeuvre starts when the ship sails at maximum speed, then the power is reduced in three steps until zero, between every power reduction the ship must reach a constant speed. Thereafter the power is increased again gradually until the ship sails at full speed again. As soon as the ship starts to turn, caused by the propeller or
environmental influences, the trial is stopped. Otherwise the assumptions that r(t)=0 is not
fulfilled.
During the fourth step the rudder is activated, herewith the complete manoeuvring model,
Eq.(1), is needed to describe the ship motions. Hence, the remaining coefficients Cule Cu7
c . .c and c . .c can be estimated. The used trials are two standard manoeuvres: a
vi v6 rl r9
turning circle at half speed with a rudder angle of 300 , and a zigzag manoeuvre at full speed
with a rudder angle of 10°.
The above described identification method has been used on simulated as well on measured data, only the latter is discussed here. The sea trials have been performed with the "nv
Zeefakkel", a former hydrographic survey vessel which is used for training purposes by the
Royal Netherlands Naval Academy. The ship has a length of 45 m, a beam of 7.5 m, and a displacement of 310 ton. The ship has two propellers with controllable pitch. The complete
trials are described in {Wulder,90:1], here only a summary of the sea trials is given.
The trials were performed near Enkhuizen on the lJsselmeer, an inland sea in The
Netherlands, so current was almost absent. The wind speed was 4 Beaufort, the wind direction was South. During the trials the position was recorded with a GPS receiver, both calculated position and raw GPS data were recorded. In Deift a reference GPS station was installed by Rijkswaterstaat ( The department of the ministry of transport and public works), which gave
the possibility to process the GPS data with the Sercel track reconstruction program Trajecto [Sercel,1989] in order to obtain a reduction of disturbances. Furthermore, twelve other signals were recorded: the torque of the propeller shafts (2x), the number of propeller revolutions (2x), the rudder angle, the roll, the pitch, the heave, the heading, the horizontal (2x) and vertical accelerations. During the trials compass bearings relative to a point onshore were
taken to check the compass for errors afterwards. The wind was recorded manually.
Unfortunately, it was not possible to record the propeller pitch. To eliminate environmental
influences as much as possible the trials were performed twice with opposite headings. The
influence of the wind on the identification is reduced further by taking into account the wind
force in Eq.(2). The wind force is estimated according to Isherwood [1972].
A problem during the sea trials is the measurement of the actual variables. Instead of measuring the velocities u(t), v(t) and r(t), the position x0(t),y0(t) and the heading (t) are
measured. The derivatives of these quantities have to be calculated, and then transformed to u(t) and v(t) according to Eqs (3d) and (3e). Another problem is the assumption that r(t)=0 and t5(t)=0 during the steps 2 and 3 of the identification procedure. During the sea trials parts
of the stop manoeuvres were disturbed by rudder orders, caused by environmental influences
and the propeller. These data could not be used for identification purposes. The last practical problem is that the propeller pitch was not measurable. To estimate the coefficient c6 it is assumed that a linear relation exists between the propeller pitch P(t) and the ship velocity u(t), when the ship sails a straight line. With this assumption it is possible to replace the propeller pitch by the initial velocity of the manoeuvre.
The fifth and last step in the identification procedure is the verification of the estimated values of the hydrodynamic coefficients. Hereto a Williamson turn was sailed with the 'nv
Zeefakkel", and the manoeuvring model, with the estimated hydrodynamic coefficients, was
used to simulate the Williamson turn. A comparison of the sea trial with the simulated data
gives information on the accuracy of the marioeuvring model. In Fig.2 the performance of the manoeuvring model is shown.
In Fig.2 both the measured and the predicted track are plotted. Furthermore, the velocities u(t), v(t), the rate of turn r(t) and the heading «t) are given. It can be seen that the
manoeuvring model predicts the ship behaviour on an accurate way: the position error is less then 20 meter after a prediction period of 7 minutes of intensive manoeuvres. During the prediction of the track it is assumed that the wind had a constant velocity and direction. So, gusts of wind are neglected, which can explain the deviation in the rate of turn after 6.5
minutes.
4. IMPLEMENTATION
In section 3 it is shown that it is possible to predict the ship motion during a manoeuvre. In this section the integration of the manoeuvring model and the position measurement is made. Hereto a KF is used, which makes an optimal (with a quadratic criteria) estimation of the state of a linear system. A nonlinear system has to be linearized. [Gelb,1974; Anderson,1979].
Examples of the use of KF in navigation are given in [Cross,1986;Dove,1989;Spaans,1988].
The KF is explained briefly (Fig.3), the ship is represented as a system with the input vector i(t}, which is the rudder angle and the propeller shaft torques, and the output vector y(t), which gives the position, velocities, heading and so on. On the ship the disturbances s(t), the
environmental forces, are acting . The output y(t) is disturbed by the measurement noise
300 200 100
o
Ø -100 -200 -300-200 -100
0 100 200 300yo meter
01234567
time in minutes
measured track "nv Zeefakkel"
predicted track with manoeuvring model
Figure 2 Performance of the manoeuvring model
15 o 10 >< 20 o o O
01234567
time in minutes
1234567
time in minutes
01234567
time in minutes
1234567
time in minutes
123456
time in minutes
m(t). The theory of KF supposes that the disturbances s(t) and m(t) are white noises. If this assumption is not fulfilled then the disturbances s(t) and m(t) are assumed to be a result of two shaping filters, SF1 and SF2, and the white noise vectors s0(t) and m0('t). The
manoeuvring model, with input i(t), gives a prediction of the output vector,(t). The
estimated output (t) has the advantage that it is not disturbed with measurement noise. On the other hand, the environmental forces are not acting on the manoeuvring model. So y(t)
s(t)
s (t)
Figure 8 Kalman filter
ship manoeuvring model
_Lo
(t) SF2 m(t) t y(t)p(t)
K(t)and (t) differ. The KF makes a comparison between the predicted and the measured output vectors and feeds this error signal e(t) into the model, as a compensation for the disturbances
s(t) acting on the ship. The signal e(t)=y(t)(t) is multiplied by the Kalman gain K(t),
which depends on the statistics of s(t), m(t) and the system. So for relative large m(t) in relation to s(t) the K(t) is small: the KF gives less attention to the measurements, and more attention to the manoeuvring model. With a relative small m(t) in relation to s(t) the KF gives more attention to the measurements, and less attention to the manoeuvring model. If the system is linear it leads to a optimal reconstruction, *(t), of the state of the ship x(t). For the design of an integrated navigation system, the basic elements are given in the previous
sections: the manoeuvring model Eq.(2), with the state x(t)= [u(t) v(t) r(t) x0(t) y0(t)
L'(t) Jt; the disturbances s(t): waves, wind and current; the measurements (for example): GPS
position, GPS velocity and heading given as by the gyro compass, which gives y(t)= [x0(t) y0(t) 0(t)
i(t)
(t) ] Moreover, the measurement vector can be extended with other measurements like, speed log, radio navigation system and so on. The vector m(t) representsthe measurement noise of GPS and the gyro compass.
However disturbances 5(1) leads to a problem. In the KF it is assumed that the disturbances s(t) and m(t) are white noises. In Eq(3) it was shown that the current and the wave forces were constant or lowfrequency varying quantities. So the disturbances have to be modeled with a shaping filter. Hereto the system model is extended with five equations, which expresses that the current and wave forces are constant:
d d d . d d
-v=O;
vcy=O; Lu =0;-jEL=O; and -r=0.
The state vector of the ship is now extended with five new state variables:
x(t)=[ u(t) v(t) r(t) x(t) y0(t)
(t) Vcx(t) v(i)
(t) (t)t(t) J.
The measurement noise m(t) is also not a white noise. In [Spaans,1988} examples are given to
model the measurement noise, which results in a further extension of the state vector. Herewith the manoeuvring model is implemented in a navigation filter. The advantage of a manoeuvring model based filter above the normally used filters is threefold. First it is expected that the accuracy and reliability of the system increases, which still has to be validated by further research. The second advantage is the availability of the physical states of the ship: current velocity and ship velocity are estimated simultaneous, which can be useful (for example) for the positioning of a seismic streamer. The third advantage is the operator friendly tuning of the filter as the filter setting is be based on understandable quantities. The Kalman gain, K(t,), depends on the statistics of s(t), m(1) and the dynamics of the ship. The dynamics of the ship are defined by the inanoeuvring model. The vector s(t) contains the errors in the environmental influences: during further research the filter settings will be
modeled as function of the maximum current velocity, the m&ximum wind velocity, and the
maximum wave height. The vector m(t) contains the uncertainties of the navigation
equipment.
5. CONCLUSIONS
In this study on the implementation of a ship manoeuvring model in a navigation system, the
following conclusions can be drawn:
- The hydrodynamical coefficients of the manoeuvring model can be estimated by means of sea trials. The step by step method avoids the interaction effects of the hydrodynamical coefficient s;
- The manoeuvring model predicts the ship's track on an accurate way, the position drift is less then 20 meter during 7 minutes of intensive manoeuvring, notwithstanding the
uncertainties in the wind direction and velocity; The position drift caused by the current is
excluded;
- The environmental influences can be modeled, which results in an extension the state vector
with five states.
Further research will to be done to investigate the system performance of the integrated navigation system. Reliability, accuracy and operator friendly tuning are the major points of interest.
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