Modeling for IMU-based Online Estimation of a Ship’s Mass and Center

Modeling for IMU-based Online Estimation of a Ship’s Mass and Center
Technical report from Automatic Control at Linköpings universitet
Modeling for IMU-based Online
Estimation of a Ship’s Mass and Center
of Mass
Jonas Linder? , Martin Enqvist? , Thor I. Fossen† , Tor Arne
Johansen† , Fredrik Gustafsson?
Division of Automatic Control
E-mail: [email protected], [email protected],
[email protected], [email protected],
[email protected]
13th March 2015
Report no.: LiTH-ISY-R-3082
Submitted to the 10th IFAC Conference on Manoeuvring and Control
of Marine Craft
Address:
?
Department of Electrical Engineering
Linköpings universitet
SE-581 83 Linköping, Sweden
†
Centre for Autonomous Marine Operations and Systems (AMOS),
Department of Engineering Cybernetics
Norwegian University of Science and Technology
NO-7491 Trondheim, Norway
WWW: http://www.control.isy.liu.se
AUTOMATIC CONTROL
REGLERTEKNIK
LINKÖPINGS UNIVERSITET
Technical reports from the Automatic Control group in Linköping are available from
http://www.control.isy.liu.se/publications.
Abstract
A ship’s roll dynamics is very sensitive to changes in the loading conditions
and a worst-case scenario is the loss of stability. This paper proposes an
approach for online estimation of a ship’s mass and center of mass. Instead
of focusing on a sensor-rich environment where all possible signals on a ship
can be measured and a complete model of the ship can be estimated, a minimal approach is adopted. A model of the roll dynamics is derived from a
well-established model in literature and it is assumed that only motion measurements from an inertial measurement unit together with measurements
of the rudder angle are available. Furthermore, identifiability properties
and disturbance characteristics of the model are presented. Due to the
properties of the model, the parameters are estimated with an iterative instrumental variable approach to mitigate the influence of the disturbances
and it uses multiple datasets simultaneously to overcome identifiability issues. Finally, a simulation study is presented to investigate the sensitivity
to the initial conditions and it is shown that there is a low sensitivity for
the desired parameters.
Keywords: modelling, identification, operational safety, inertial measurement unit, identifiability, centre of mass, physical models, accelerometers,
gyroscopes, marine systems
1
Introduction
There are several factors that influence the dynamic behavior of a ship and the
mass and the center of mass (CM) are properties that have a particularly large
impact on the ship’s dynamical behavior and especially on the roll dynamics
(Tannuri et al., 2003; Fossen, 2011). Mathematical models are typically used to
enhance performance or safety, for instance, to simulate the ship’s response in an
advisory system in order to aid the crew in the operation of the ship. However,
loading conditions may change over time and large variations can be critical for
the stability of the ship. Consider, for instance, the change in mass of a fishing
vessel or a bulk tanker that changes due to the loads in the cargo holds. Online
estimation is one way to improve the model accuracy if the variations have a
large impact on the dynamic behavior.
The main challenge in both online and offline ship modeling is the complex
interaction with water. This complexity makes it difficult to compute which
forces that are acting on the ship during normal operation unless special sensors
are introduced. Without knowledge about these forces, estimation of the inertial
properties becomes challenging, for example, since large forces acting on a large
mass gives the same behavior as small forces acting on a small mass.
Online mass and CM estimation for vehicles is a hot topic, especially in
automotive applications where roll over accidents is a common type of accident,
see for instance, Fathy et al. (2008) or Sadeghi Reineh et al. (2013). However,
due to the complex interaction with the water, these methods cannot be used
directly for estimation of a ship’s mass and CM. The two major differences are
that the environmental disturbances have a larger impact on the motion in a
ship application and that the couplings between the degrees of freedom (DOF)
typically are stronger for a ship.
An approach to address the disturbance issue was proposed in Linder et al.
(2014b). A model of the decoupled roll dynamics was used and it was assumed
that only motion data from an inertial measurement unit (IMU) together with
the rudder angle were available. An instrumental variable (IV) method was
used to mitigate the influences of the environmental disturbances where the
rudder angle measurements were used to create the instruments. To overcome
identifiability issues, multiple datasets were used sequentially. In Linder et al.
(2014a), it was shown that the estimation problem is similar to closed-loop estimation and that the variance properties of the IV estimator could be improved
by considering this.
This paper presents an approach for online estimation of a ship’s mass and
CM. The approach can be seen as an extension of the method presented in
Linder et al. (2014a). The contribution of the extension is threefold. 1) A
model of a ship’s roll dynamics is derived from a well-established maneuvering
model. This model can be seen as a generalization of the model in Linder et al.
(2014b) that considers the strong coupling to the other DOF. 2) The identifiability properties and disturbance characteristics are presented together with
the implications for the parameter estimation. 3) Due to the disturbance characteristics, an iterative IV approach is formulated to estimate the parameters.
Multiple datasets are used simultaneously to overcome the identifiability issues.
In addition to the formulation of the approach, a Monte Carlo simulation is
performed to investigate the estimator’s sensitivity to initial conditions. This
paper presents theoretical aspects, the proposed method has also been validated
1
on data from scale model, see Linder et al. (2015).
The remainder of this paper is organized as follows: In Section2 the model
is derived. Section3 presents an analysis of the model’s key properties. Section4
describes an iterative IV approach for estimating the parameters. The estimator’s sensitivity to the initial conditions is investigated in Section5. Finally, in
Section 6, the paper is summarized with conclusions and suggestions for future
work.
2
Ship modeling
The basis for the approach in this paper is the maneuvering model developed
and discussed in Blanke and Christensen (1993) and Perez (2005). The model
describes the planar motion, i.e. position and heading, and the roll dynamics.
This model is particularly useful for describing the rudder induced roll motion
and is as such a good foundation for the simplified model developed in this
paper. Here, the model will be expressed in a vectorial setting described by
Fossen (1991). The model can be written as
η̇ = J (η)ν
M ν̇ + C RB (ν)ν − N (ν, η) + g(η) = R(δ) + τ
(1)
where η T = [X, Y, φ, ψ] is the generalized position descri-bed in a Earth-fixed
coordinate system (assumed to be inertial), ν T = [u, v, p, r] is the generalized
velocity described in a body-fixed coordinate system, δ is the rudder angle, τ
are the environmental forces acting on the ship and the variables of η and ν are
defined in Table1 (Fossen, 2011). A sketch relating the variables can be seen
in Figure1. The generalized position η and velocity ν are related through the
, rudder angle
ay
X, Y
yn
,
yb
t
the fron
n from
Ship see
⌧t
m
xb
as
mg
xn
M
IMU
as (tangential acceleration)
Mg
d ˙, k
r (yaw rate)
zs
zm
zg
ay
˙ = p (roll rate)
Figure 1: A sketch of a ship performing a turning maneuver which results in
an unknown acceleration ay is acting on the CR. The shift in displaced water
(green/red area) forces the ship back towards its equilibrium.
2
Table 1: The positions and velocities used to describe the ship’s position and
orientation.
Position Velocity
Generalized
η
ν
Surge
X
u
Linear
Sway
Y
v
Roll
φ
p
Angular
Yaw
ψ
r
Euler angle velocity transformation matrix which

cψ −cφ sψ 0
sψ cφ cψ 0

J (η) = 
0
0
1
0
0
0
in this case is given by

0
0

0
cφ
(2)
where ci = cos(i) and si = sin(i). The system inertia matrix (including added
mass and moment of inertia) is given by


M̃ − Xu̇
0
0
0

0
M̃ − Yv̇
−M̃ z̃g − Yṗ
M̃ x̃g − Yṙ 
 (3)
M =
2


˜
−Kṙ
0
−M̃ z̃g − Kv̇ Ix − Kṗ + M̃ z̃g
2
˜
0
M̃ x̃g − Nv̇
−Nṗ
Iz − Nṙ + M̃ x̃g
where it assumed that the ship is port–starboard symmetric (ỹg = 0) and M̃ is
the ship’s total mass that has its center of mass (CM) located at r̃ Tg = [x̃g , 0, z̃g ].
Furthermore, I˜x and I˜z are the moments of inertia about the CM and the other
parameters are added mass and moment of inertia. The Coriolis-centripetal
matrix is given by


0
−M̃ r M̃ z̃g r −M̃ x̃g r
 M̃ r
0
0
0 
,
C RB (ν) = 
(4)
−M̃ z̃g r
0
0
0 
M̃ x̃g r
0
0
0
where the (3, 4) and (4, 3) elements of C RB (ν) are 0 due to the to the chosen representation (Fossen, 2011). The non-linear hydrodynamic damping and
Coriolis effects due to added mass are assumed to be described by


X|u|u |u|u


 Y|u|v |u|v + Yur ur + Yv|v| v|v| + Yv|r| v|r| + Yr|v| r|v|




 +Yφ|uv| φ|uv| + Yφ|ur| φ|ur| + Yφuu φu2




 K|u|v |u|v + Kur ur + Kv|v| v|v| + Kv|r| v|r| + Kr|v| r|v| 


2
N (ν, η) =  +K
 (5)
φ|uv| φ|uv| + Kφ|ur| φ|ur| + Kφuu φu + K|u|p |u|p



 +Kp|p| p|p| + Kp p + Kφφφ φ3




 N |u|v + N |u|r + N r|r| + N r|v|

|u|v
|u|r
r|r|
r|v|



 +N
φ|uv| φ|uv| + Nφu|r| φu|r| + Nφ|uv| φ|uv| + Np p
+N|p|p |p|p + N|u|p |u|p + Nφu|u| φu|u|
3
where the coefficients are assumed to be constant (Perez, 2005). The hydrostatic
forces and moments are, assuming that the roll angle φ is small, given by
g(η) = 0
0
ρg∇GM T sφ
T 0 ≈ 0
0
ρg∇GM T φ 0
T
(6)
where ρ is the density of water, g is acceleration due to gravity, ∇ is the displaced
water and GM T is the transversal metacentric height (Journée and Massie,
2001). The only actuators acting on the ship are assumed to be a rudder that
has a force proportional to the rudder angle and a constant force acting as the
forward propulsion. The actuator contribution is
T
R(δ) = [C3 − C1 δ, C2 δ, −zr C2 δ, xr C2 δ]
(7)
where the coefficients Ci ≥ 0, i = 1, 2, 3 and r Tr = [xr , 0, zr ] is the position of
the rudder in the body-fixed frame.
2.1
Sensors
The motion of the ship is assumed to be measured by an IMU, and in the next
section it is shown that the key to the model developed in this paper is to
eliminate the unknown signal v̇ using the tangential acceleration measurement.
Assuming that the roll angle φ is small, the (tangential) acceleration sensed by
the IMU is
as = zs φ̈ + gφ − ay ,
(8)
where −zs is the distance from the center of rotation (CR) to the origin of
the IMU coordinate system, see Figure1. The tangential acceleration has three
contributions, the first term from the angular acceleration, the second term due
to gravity and the third term due to acceleration of the CR in the xy–plane
in the Earth-fixed frame. Note that the identity p = φ̇ only holds due to the
assumption of the model and that it is not valid in a general model. The IMU
measurements are assumed to be
y1,t = pt + b1,t + e1,t = φ̇t + b1,t + e1,t
y2,t = as,t + b2,t + e2,t
y3,t = −rt + b3,t + e3,t
(9a)
(9b)
(9c)
where pt = φ̇t is the sampled system’s angular velocity about the roll axis, as,t
is the tangential acceleration after sampling, rt is the sampled system’s angular
velocity about the yaw axis, bi,t , i = 1, 2, 3, are sensor biases and ei,t , i = 1, 2, 3,
are measurement noises.
2.2
A Limited Sensor Approach – Indirect Model
Assuming that the surge velocity u is constant and equal to U , and by introducing the states xT = [φ, ψ, v, p, r], the surge–sway–roll–yaw model (1) can be
written in the nonlinear state-space form
M̃ ẋ = F (x, δ) + τ
(10)
where M̃ is the inertia matrix and F (x, δ) is the nonlinear state transition
function. Furthermore, it is assumed that the ship is fore-aft symmetric (x̃g =
4
0), that the total mass M̃ can be split into a nominal mass M and a load mass
T
T
m with centers of gravity given by [0, 0, zg ] and [0, 0, zm ] , respectively, and
the inertia of the load mass is neglected. Linearization of the nonlinear model
(10) about x̄ = 0 and δ̄ = 0 gives
∂F ∂F x+
δ+τ
(11)
M̃ ẋ =
∂x x̄,δ̄=0
∂δ x̄,δ̄=0
where

1

0

M̃ = 0

0
0
0
1
0
0
0



∂F =
∂x x̄,δ̄=0 

and
0
0
M + m − Yv̇
−M zg − mzm − Kv̇
−Nv̇
0
0
Yφuu U 2
Kφuu U 2 − ρg∇GM T
Nφu|u| U |U |
0
0
0
0
0
∂F = 0
∂δ x̄,δ̄=0
0
0
−M zg − mzm − Yṗ
2
Ix − Kṗ + M zg2 + mzm
−Nṗ
0
0
Y|u|v |U |
K|u|v |U |
N|u|v |U |
0
C2
1
0
0
Kp + K|u|p |U |
Np + N|u|p |U |
−zr C2

0

0 

−Yṙ  , (12)

−Kṙ 
Iz − Nṙ
0
1
(Yur − M − m)U
(Kur + M zg + mzm )U
N|u|r |U |
xr C 2
T






(13)
(14)
The fourth row of (11) is
A1 ṗ − (Kv̇ + M zg + mzm )v̇ − Kṙ ṙ
= (Kφuu U 2 − ρg∇GM T )φ + (Kp + K|u|p |U |)p
(15)
+ K|u|v |U |v + (Kur + M zg + mzm )U r − zr C2 δ + τ
which is a model of the roll dynamics in component form and
2
2
A1 = Ax + M zg2 + mzm
= Ix − Kṗ + M zg2 + mzm
(16)
To get an identifiable model structure, the parameters
k = −Kφuu U 2 + ρg∇GM T − M gzg − mgzm ,
d = −Kp − K|u|p |U |
(17a)
(17b)
and
Kδ = −zr C2 ,
(17c)
are introduced together with the parameter Ax . Furthermore, Kṙ ṙ and K|u|v |U |v
are neglected since the influences from these terms are assumed to be small. This
gives
A1 φ̈ = − (k + M gzg + mgzm )φ − dφ̇
(18)
+ (Kv̇ + M zg + mzm )v̇
+ (Kur + M zg + mzm )U r + Kδ δ + τ
Here, the parameter k can be interpreted as representing the physical restoring
properties of the ship, for instance, being dependent on the hull shape. The other
two terms M gzg and mgzm are in this model representing the influence by the
5
mass and its location on the restoring properties and a big change in loading
condition is assumed to be captured by mgzm . Note that the parameters k and
d are dependent on the speed and given a speed U , assumed to be fixed and
independent of the loading condition.
The model (18) can be seen as a mass-spring-damper model with three inputs. With this point of view, the largest issue is the unknown signal v̇. Since
neither v̇ nor a model of it is known, an alternative model can be formed by
eliminating v̇. The key to this elimination is the measured tangential acceleration as defined in (8) and its relation to the signal v̇ (Linder, 2014). The sway
acceleration v̇ is related to the tangential acceleration as through the third term
in (8). This third term, i.e. the acceleration ay of the ship in the Earth-fixed
xy–plane, has two parts. The contributions emanate from the sway motion and
the angular velocity about the yaw axis. The total acceleration is given by
ay = v̇ + U r
(19)
as = zs φ̈ + gφ − ay = zs φ̈ + gφ − v̇ − U r
(20)
and combing (8) with (19) gives
Solving (20) for v̇ and substituting it into (18) give
A2 φ̈ = − (k − Kv̇ g)φ − dφ̇
− (Kv̇ + M zg + mzm )as + (Kur − Kv̇ )U r
(21)
A2 = Ax + M zg (zg − zs ) + mzm (zm − zs ) − Kv̇ zs
(22)
+ Kδ δ + τ
where
Further simplifications can be obtained since the surge velocity U is assumed
to be constant and by introducing the lumped parameter Kr = (Kur − Kv̇ )U ,
giving the model
A2 φ̈ = − (k − Kv̇ g)φ − dφ̇
(23)
− (Kv̇ + M zg + mzm )as + Kr r + Kδ δ + τ
In most cases, the true center of rotation is not known due to the complex
interaction with the water (Balcer, 2004). Instead, a known body-fixed coordinate system can be introduced and the CR zf can be estimated relative to this
body-fixed coordinate system by introducing
zg = z̄g + zf ,
zm = z̄m + zf
and zs = z̄s + zf
(24)
where it is assumed that the xb –axis of the body-fixed frame is parallel to the
rotation axis but shifted in the zb –direction. Finally, the model (23) with the
output y = φ̇ can be written on the transfer function form
y = G(p)(as + Fr r + Fδ δ + τ )
(25)
where
γ1
κ1
β1 p
, Fδ =
, Fr =
,
p2 + α1 p + α2
β1
β1
d
k − Kv̇ g
Kv̇ + M zg + mzm
(26)
α1 =
, α2 =
, β1 = −
,
A2
A2
A2
Kδ
Kr
γ1 =
and κ1 =
A2
A2
where p is the differentiation operator, A2 is defined in (22) and zg , zm and zs
are defined in (24).
G(p) =
6
3
Analysis of Model Properties
From a system identification perspective it is important to understand the
model’s properties. Firstly, an identifiability analysis is performed to investigate if the parameters in the chosen model structure can be uniquely determined. Secondly, to make an appropriate choice of estimation method, the
signals’ dependency on the process disturbance τ is analyzed.
3.1
Identifiability Issues – Using Multiple Datasets
The question whether the parameters in the model can be uniquely estimated
has two aspects, the informativity of the data and the parameterization for the
model (Ljung and Glad, 1994; Bazanella et al., 2010).
Firstly, let us assume that the data is informative enough. The goal is to
estimate the change in mass m and change in CM z̄m but there are additional
nuisance parameters that have to be estimated along with the desired parameters. It would thus be preferable if the parameters
T
ϑ̄p = M, z̄g , k, Āx , d, Kr , Kv̇ , Kδ , zf , m, z̄m
(27)
of the model (26) could be estimated using a single dataset, i.e. all parameters
except for g and z̄s that are assumed to be known. However, the model (26) is
not identifiable with respect to the parameters in (27) (Linder, 2014). To gain
identifiability, either the model structure must be changed or more information
must be introduced. Here, more information is introduced through a priori
knowledge of parameters and by using multiple datasets.
In a first calibration phase, the two datasets
N +t
N +t
c
c
n
n
and Zc = (yt , ut , δt )t=1+t
Zn = (yt , ut , δt )t=1+t
c
n
(28)
called the nominal and calibration datasets, respectively, are collected. The
nominal dataset has a known mass M and CM z̄g . The calibration dataset has
a different known mass and CM expressed in terms of the load mass m = mc
and its CM zm = zc . The parameters for the two cases are thus given by
ϑ̄np = ϑ̄p |m=z̄m =0
and ϑ̄cp = ϑ̄p |m=mc ,z̄m =z̄c
(29)
The vector of parameters (27) is extended to
ϑ̃p,2 (unknown)
ϑ̃p,1 (known)
}|
{T
}|
{ z
z
ϑ̃p = [M, z̄g , mc , z̄c , k, Ax , d, Kr , Kv̇ , Kδ , zf , m, z̄m ]
(30)
and the datasets (28) together with the known parameters ϑ̃p,1 are then used
simultaneously with the loaded dataset
N +t
l
l
Zl = (yt , ut , δt )t=1+t
,
l
(31)
collected during normal operational conditions, to estimate the unknown parameters ϑ̃p,2 .
Secondly, assuming that the model structure is identifiable using the three
datasets, the data have to be informative enough. Note that the rudder is the
only true input to the system except for the disturbances. This implies that the
7
δ
Gv̇
Gr
v̇
r
−
+
+
+
U
Fδ
as
τ
+
G
y = φ̇
Fy
Fr
Gφ
Figure 2: The model (11) expressed in terms of its subsystems. Note that due
to coupling in the system, all signals depends on each other which means that
all signals are dependent on τ .
motion induced by the rudder and observed in as and r is uniquely determined
through the dynamics. Here we assume that the complexity of the system is
sufficient and thus, that as and r will supply more information than δ does by
itself. This means that the inputs to (23) are informative if the roll dynamics
is sufficiently excited by the rudder (Bazanella et al., 2010; Linder, 2014).
3.2
Identification Issues – Correlation with τ
The ship’s motion is assumed to be affected by two inputs, the rudder and the
disturbances acting on the ship. The model (18) can be expressed in terms of
(23) and the subsystems of the linearized system (11) which results in the structure seen in Figure2. Analyzing this model reveals the dependencies between
the measured signals and shows that even though the proposed method avoids
building a model of the entire ship, it introduces some new challenges.
As mentioned in the previous section, the rudder is the only true (actuator)
input acting on the system, which implies that both as and r are dependent
on δ, i.e. all measurements (9) are correlated with δ. Due to coupling in the
system, the measurements (9) are also correlated with the process disturbance
τ . This means that there are similarities with identification in closed loop and
it is important to understand these dependencies to make the correct choices
in the identification procedure. On top of the correlation, there is also a direct
term in the loop gain from τ to as which might introduce a bias for certain
closed-loop identification methods if this is not considered (Linder, 2014).
Finally, in addition to the process disturbance τ , also the measurement noises
in (9) have to be considered. Hence, the inputs will be noisy and the identification problem will be of errors-in-variables type.
8
4
Estimation of a Ship’s Roll Dynamics
There are a lot of details to consider when estimating the unknown parameters
in (23). Firstly, a discrete-time model is introduced. Secondly, the properties
of the signals are considered to avoid pitfalls when choosing and tuning the
estimation approach.
4.1
Discretization Using Physical Parameters
The transfer function (26) is discretized using the bilinear transform p = 2/T (q−
1)/(q + 1) where T is the sample period and q is the shift operator. Applying
the transform gives
Gd (q) =
γ̄0
β̄0 (1 − q−2 )
κ̄0
, Fδ,d =
, Fr,d =
1 + ᾱ1 q−1 + ᾱ2 q−2
β̄0
β̄0
(32)
with
−2dT + (k − Kv̇ g)T 2 + 4A2
2(k − Kv̇ g)T 2 − 8A2
, ᾱ2 =
A2 (ϑ̄p )
A2 (ϑ̄p )
2T (Kv̇ + M zg + mzm )
2T Kr
2T Kδ
β̄0 = −
, κ̄0 =
, γ̄0 =
,
A2 (ϑ̄p )
A2 (ϑ̄p )
A2 (ϑ̄p )
ᾱ1 =
(33)
and A2 (ϑ̄p ) = 2dT + (k − Kv̇ g)T 2 + 4A2 . By introducing
µTt = [as,t − as,t−2 , rt − rt−2 , δt − δt−2 ] ,
(34)
the discrete-time model can be written as
where
yt = ϕTt gϑ (ϑ̄p ) + τ̃t
(35)
T
gϑ (ϑ̄p ) = ᾱ1 (ϑ̄p ), ᾱ2 (ϑ̄p ), β̄0 (ϑ̄p ), κ̄0 (ϑ̄p ), γ̄0 (ϑ̄p )
(36)
and
ϕTt = [−yt−1 , −yt−2 , µ1,t , µ2,t , µ3,t ] ,
(37)
In Linder et al. (2014a) it was possible to solve several linear problems sequentially and obey the original physical parameterization by linear constraints to
overcome the identifiability issues. In this paper, this is unfortunately not possible due to extra complexity. Instead, the model is extended and all datasets
defined in Section3.1 are used simultaneously. The joint model is
y t = ϕTt θ(ϑ̃p ) + τ̃ t
(38)
where y t = [ytn , ytc , ytl ]T ,
T
θ(ϑ̃p ) = gϑT (ϑ̄np ), gϑT (ϑ̄cp ), gϑT (ϑ̄lp ) ,

ϕnt

ϕt = 0
0
0
ϕct
0

0
0 ,
ϕlt
(39)
and the subscripts i = n, c, l correspond to the nominal, calibration and loaded
datasets, respectively.
9
4.2
The Iterative Instrumental Variable Method
An instrumental variable method uses instruments to extract the interesting information from the data. In principle, the information is estimated by requiring
that
N
1 X
ζ (y − ϕTt θ(ϑ)) = 0
(40)
N t=1 t t
i.e. that the sample covariance between ζ t and the prediction error should be
zero. There are two terms contributing to the output of (38), one containing
information about the interesting input-output relation and the second containing a contribution from disturbances. A good instrument should in this case be
correlated with the motion induced by the rudder but be uncorrelated with the
process disturbance, the sensor biases and the measurement noises. This idea
is generalized in the extended IV method, where the parameters are found by
computing
2
ϑ̂ = argmin kYN − ΦN θ(ϑ)kQ
(41)
ϑ
2
where kxkQ = xT Qx, Q 0 is a weighting matrix,
 n

ζt
" ȳ #
ϕ̄T1
.. 
 Nn
..1

= [ζ 1 . . . ζ N ] . , YN = [ζ 1 . . . ζ N ] . , ζ t =  0
ȳ N
ϕ̄TN
0

ΦN
0c
ζt
Nc
0

0

0
l
(42)
ζt
Nl
ȳ t = L(q)y t , ϕ̄Tt = L(q)ϕTt and L(q) is a diagonal matrix
of stable prefilters. See, for instance, Söderström and Stoica (1989) or Ljung
(1999) for more details.
An iterative method based on Gilson et al. (2006) is used in this paper.
Here, z i indicates that z belongs to the dataset i. The superscript i is either
n, c or l and corresponds to the datasets defined in Section3.1. For brevity,
i = {n, c, l} is not explicitly written at all places. In the j th iteration, the
parameters are estimated using the instruments and prefilters obtained from
the j − 1th iteration. The ARMA noise models H̄di (q, η̂ i,j ) are estimated from
the residuals
i
i T
ˆi,j
εi,j
(43)
t = yt − (ϕt ) gϑ (ϑ̄p )
and the prefilters are calculated as
i
i,j −1
Li,j (q, ϑ̄i,j
p ) = H̄d (q, η̂ )
(44)
The transfer functions (47) and (48) are then simulated with δti as input, which
gives the signals
i,j
i,j
i,0
i,0
i
i
i
ŷti,j = Ĝi,j
δy,d (q)δt , âs,t = Ĝδas ,d (q)δt and r̂t = Ĝδr,d (q)δt
Finally, the instrument vectors are created according to
i,j
i,j
i,j
i,j
ζti,j = Li,j (q, ϑ̄i,j
p ) × ŷt . . . ŷt−ny +1 , µ̂1,t . . . µ̂1,t−nas +1 ,
T
i,j
i,0
i,0
µ̂i,j
2,t . . . µ̂2,t−nr +1 , µ3,t . . . µ3,t−nδ +1
10
(45)
(46)
Algorithm 1 The iterative joint IV method
(A) Initialize:
i,0
i,0
(a) Set initial value of ϑ̄i,0
p and set prefilters L (q, ϑ̄p ) = 1
(b) Create initial instruments
i. Estimate blackbox models of the transfer functions
i,0
i,0
Ĝi,0
δy,d (q), Ĝδas ,d (q) and Ĝδr,d (q) in (45)
i,0
ii. Simulate ŷti,0 , âi,0
s,t and r̂t according to (45)
iii. Create the instruments ζti,0 according to (46)
(c) Set j = 1
(B) Estimate parameters:
ˆ
) and ζti,j−1
(a) Compute ϑ̃jp,2 using (41) with Li,j−1 (q, ϑ̄ˆi,j−1
p
(b) Estimate η i,j of the models H̄di (q, η i,j ) from (43)
(c) Create the prefilters Li,j (q, ϑ̄ˆi,j ) according to (44)
p
(d) Simulate the signals
ŷti,j
and âi,j
s,t according to (45)
(e) Create the instruments ζti,j according to (46)
(C) Terminate: Increase j and go to Step B while
ˆj
ˆj−1
n,j
n,j−1
c,j
c,j−1
l,j
l,j−1
kϑ̃p,2 − ϑ̃p,2 k2 + kη̂
− η̂
k2 + kη̂
− η̂
k2 + kη̂
− η̂
k2
is above a threshold or as long as a certain number of iterations is not
ˆ
ˆ
reached. Otherwise, return ϑ̃p,2 = ϑ̃jp,2 and terminate.
Note that, i = {n, c, l} is not explicitly written at all places and if the superscript
i is used, it should be understood as for i = {n, c, l}.
where the constants ni , i = y, as , r, δ, are the number of time lags (including the
non-delay signal) included in ζti,j , for instance, nδ = 0 means that µ3,t is not
included in ζti,j .
In the initializing (0th ) iteration, the transfer functions in (45) are estimated
blackbox models and in the refining iterations, the first two transfer functions
of (45) are given by
Ĝi,j
δy,d =
Ĝi,j
δas ,d
=
Gi,j
d
i,j
1 − Gi,j
d Fy,d
Gi,j
d
i,j
1 − Gi,j
d Fy,d
h
i
i,j
i,j
(Fr,d
− U )Ĝi,0
+
F
δr,d
δ,d ,
h
i
i,j
i,j i,0
Fδ,d
+ Fr,d
Ĝδr,d +
U Ĝi,0
δr,d
i,j i,j
1 − Fy,d
Gd
(47)
,
(48)
th
while Ĝi,0
iteration. Here,
δr,d are given by the blackbox models from the 0
the dependencies on q and ϑ̄ˆi,j have been dropped for brevity. The method is
p
summarized in Algorithm 1 and more details can be found in Linder (2014).
11
4.3
Summary of Approach
The method presented in this paper is an approach based on two stages. In the
first calibration phase, the nominal and calibration datasets (28) are collected.
The mass and CM relative the body-fixed coordinate system have to be known
while collecting both datasets, i.e. ϑ̃p,1 . Note that the masses also have to be
different. In the second phase, the loaded dataset (31) is collected during normal
operation and Algorithm 1 is used together with all three datasets to estimate
the unknown parameters ϑ̃p,2 online.
In addition to the datasets (28), only the position z̄s of the IMU in relation
to the body-fixed coordinate system and the acceleration of gravity g have to
be known.
Finally, we emphasize that the rudder angle δ not only is important to
extract the important information using Algorithm 1 but also to excite the ship
sufficiently to get informative data.
5
Sensitivity to Initial Conditions
A challenging aspect is that the proposed estimator is non-convex in the parameters and in this section, a brief simulation study will be presented to evaluate
the estimated parameter’s sensitivity to the initial conditions. To simplify the
analysis, all disturbances were set equal to zero. For a discussion on the estimator’s disturbance rejection, see Linder et al. (2014b), Linder et al. (2014a)
and Linder et al. (2015). The system was assumed to be given by (11) and
the parameters, given in Table2, were chosen to resemble the parameters of the
scale model in Linder et al. (2015). Note that (11) has 29 parameters while (38)
has 15 parameters where only the six parameters listed in Section4.3, i.e. M ,
z̄g , mc , z̄c , g and z̄s , are assumed to be known.
The datasets described in Section3.1 were synthesized by simulating (11)
with the masses and CMs given by Table2. The length of the datasets were
chosen to be 60 seconds and were sampled at 50 Hz. Figure3 shows the nominal
dataset as an example.
A Monte Carlo (MC) simulation with 10 000 runs was performed to test the
Table 2: The parameters used in the simulation.
Description
Parameters and Values
General
U = 1, M = 22.04, z̄g = 0, mc = 0.2, z̄c = −0.172,
m = 0.4, z̄m = −0.182, zf = −0.028, z̄s = −0.2,
g = 9.82
Sway Dynamics
Yv̇ = −100, Yṗ = Kv̇ , Yṙ = −1, Yφuu = 0, Y|u|v =
35, Yur = 0
Roll Dynamics
Kv̇ = 0.1, Ax = 0.1385, Kṙ = 0, k = 10.38, d =
0.2067, Kur = −0.4 (⇔ Kr = −0.5)
Yaw Dynamics
Nv̇ = Yṙ , Nṗ = Kṙ , Az = 3Ax , N|u|r = 1.4,
Nφu|u| = Np = N|u|p = 0
12
?_
solution’s sensitivity to the initial condition. In each run, the initial condition
was sampled uniformly between the upper and lower bound given in Table3. In
all iterations, the instruments (46) were created using the constants ny = nas =
nr = 16 and nδ = 2.
The means and standard deviations of the solutions were calculated and can
be seen on the two last rows of Table3. Five of the runs resulted in unrealistic
solutions (that could be easily identified) and were not used in the calculations.
Note that the approach was surprisingly robust for finding the mass m and the
CM z̄m despite the large variations in the other parameters. This was fortunate
since these were the desired parameters. The other parameters were in some
sense only estimated out of necessity for the estimator and should be treated
with care due to the variations.
Figure4 shows the cost function of the estimation problem as a function
of the mass m and the CM z̄m for one solution of the MC simulation (the
other parameters were fixed). Note that the gradient is largest orthogonal to
the dotted line and that there is a unique minimum (red cross). However, the
cost0.2function is quite flat close to the minimum and due to the gradient of the
cost function, it is easier to detect that there has been a change in the loading
0
condition than to separate the effect between the mass and the CM.
-0.2
a?_s
2
0.2
1
0
0
-0.2
2
0
ars
1
-0.2
-0.4
0
-0.6
0
0.6
/r
-0.2
0.4
-0.4
0.2
v/_
-0.6
0
0.1
0.6
0.05
0.4
0
0.2
-0.05
0
0.1 0
10
20
v_
0.05
0
30
40
50
60
Time [s]
Figure 3: Nominal data used in the simulation study.
-0.05
0
10
20
30
Time [s]
13
40
50
60
Table 3: Results for Monte Carlo simulation. T: true, U/L: upper/lower bounds,
M: mean and S: standard deviation.
Ax
d
k
Kδ
Kr
Kv̇
zf
m
z̄m
T
U
L
0.1385
100
0
0.2067
100
0
10.380
100
0
−0.405
0
1
−0.5
0
10
0.1
10
0
0.4
1
0
−0.182
0
0.5
M
S
0.1261
0.0296
0.2194
0.0233
13.297
5.3579
−0.4302
0.0454
−0.5318
0.0584
0.3396
0.4367
−0.028
1
−1
0.4007
0.0018
−0.1821
0.0004
6
−0.0403
0.0225
Conclusions and future work
In this paper, an extension of a previously proposed online estimation approach
for mass and CM estimation has been presented. The method relies on measurement of rudder angle and motion measurement from an IMU. The model
of the roll dynamics, parameterized with physical parameters, was derived from
a well-established maneuvering model. Due to identifiability issues, a priori
information was introduced in the form of known parameters and calibration
datasets. To mitigate environmental disturbances, sensor biases and measurement errors, an iterative closed-loop instrumental variable approach, using all
datasets simultaneously, was proposed to estimate the parameters.
A limited MC simulation was performed to investigate the estimator’s sensitivity to initial conditions. It was shown that the estimator is surprisingly
robust in the desired parameters corresponding to mass and center of mass but
that the other parameters had large variations. Although it has not been discussed in this paper, the proposed method has also been validated on data from
scale model with good results, see Linder (2014) or Linder et al. (2015).
Future work includes an investigation of the underlying cause of the sensitivity to the initial condition and to perform a more exhaustive simulation study
including a full nonlinear model and realistic process disturbances.
0
0.45
-0.05
0.4
-0.1
0.35
z7m [m]
-0.15
0.3
-0.2
0.25
-0.25
0.2
-0.3
Contour Plot
Trench
m^
^ zm = mzm
Solution
-0.35
-0.4
-0.45
-0.5
0
0.1
0.2
0.3
0.4
0.5
m [kg]
0.6
0.7
0.8
0.9
0.15
0.1
0.05
1
Figure 4: A contour plot of the estimator’s cost function as a function of m and
z̄m with the other parameters fixed.
14
Acknowledgments
This work has been supported by the Vinnova Industry Excellence Center LINKSIC and by the Research Council of Norway through the Centers of Excellence
funding scheme, number 223254 - Centre for Autonomous Marine Operations
and Systems (AMOS).
References
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Alexandre Sanfelice Bazanella, Michel Gevers, and Ljubiša Miškovic. Closedloop identification of MIMO systems: A new look at identifiability and experiment design. European Journal of Control, 16(3):228–239, 2010. ISSN
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H.K. Fathy, Dongsoo Kang, and J.L. Stein. Online vehicle mass estimation
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T. I. Fossen. Nonlinear modelling and control of underwater vehicles. PhD
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1991.
T. I. Fossen. Handbook of Marine Craft Hydrodynamics and Motion Control.
Wiley, 2011. ISBN 9781119991496.
Marion Gilson, Hugues Garnier, Peter JW Young, Paul Van den Hof, et al. A
refined IV method for closed-loop system identification. In Proceedings of the
14th IFAC Symposium on System Identification, pages 903–908, Newcastle,
Australia, 2006.
J. M. J. Journée and W. W. Massie. Offshore Hydromechanics, 2001. Lecture notes on offshore hydromechanics for Offshore Technology students, code
OT4620.
Jonas Linder. Graybox Modelling of Ships Using Indirect Input Measurements.
Linköping Studies in Science and Technology. Thesis 1681. 2014.
Jonas Linder, Martin Enqvist, and Fredrik Gustafsson. A closed-loop instrumental variable approach to mass and center of mass estimation using IMU
data. In Proceedings of the 53rd IEEE Conference on Decision & Control,
Los Angeles, CA, USA, December 2014a.
Jonas Linder, Martin Enqvist, Fredrik Gustafsson, and Johan Sjöberg. Identifiability of physical parameters in systems with limited sensors. In Proceedings
of the 19th IFAC World Congress, Cape Town, South Africa, August 2014b.
15
Jonas Linder, Martin Enqvist, Thor I. Fossen, and Tor Arne Johansen. Online
Estimation of Ship’s Mass and Center of Mass Using Inertial Measurements.
Technical Report 3081, Linköping University, The Institute of Technology,
2015.
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Hall, 1999. ISBN 0136566952.
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10.1016/0005-1098(94)90029-9.
T. Perez. Ship Motion Control: Course Keeping and Roll Stabilisation Using Rudder and Fins. Advances in Industrial Control Series. Springer-Verlag
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16
Avdelning, Institution
Division, Department
Datum
Date
Division of Automatic Control
Department of Electrical Engineering
2015-03-13
Språk
Language
Rapporttyp
Report category
ISBN
Svenska/Swedish
Licentiatavhandling
ISRN
Engelska/English
Examensarbete
C-uppsats
D-uppsats
—
—
Serietitel och serienummer
Title of series, numbering
Övrig rapport
ISSN
1400-3902
URL för elektronisk version
LiTH-ISY-R-3082
http://www.control.isy.liu.se
Titel
Title
Modeling for IMU-based Online Estimation of a Ship’s Mass and Center of Mass
Författare
Author
Jonas Linder, Martin Enqvist, Thor I. Fossen, Tor Arne Johansen, Fredrik Gustafsson
Sammanfattning
Abstract
A ship’s roll dynamics is very sensitive to changes in the loading conditions and a worst-case
scenario is the loss of stability. This paper proposes an approach for online estimation of
a ship’s mass and center of mass. Instead of focusing on a sensor-rich environment where
all possible signals on a ship can be measured and a complete model of the ship can be
estimated, a minimal approach is adopted. A model of the roll dynamics is derived from a
well-established model in literature and it is assumed that only motion measurements from
an inertial measurement unit together with measurements of the rudder angle are available.
Furthermore, identifiability properties and disturbance characteristics of the model are presented. Due to the properties of the model, the parameters are estimated with an iterative
instrumental variable approach to mitigate the influence of the disturbances and it uses multiple datasets simultaneously to overcome identifiability issues. Finally, a simulation study
is presented to investigate the sensitivity to the initial conditions and it is shown that there
is a low sensitivity for the desired parameters.
Nyckelord
Keywords
modelling, identification, operational safety, inertial measurement unit, identifiability, centre
of mass, physical models, accelerometers, gyroscopes, marine systems
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