# 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 L. Balcer. Location of ship rolling axis. Polish Maritime Research, pages 3–7, 2004. 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 0947-3580. Mogens Blanke and Anders C Christensen. Rudder-roll damping autopilot robustness to sway-yaw-roll couplings. In Proceedings of the 10th Ship Control Systems Symposium, Ottawa, Canada, 1993. H.K. Fathy, Dongsoo Kang, and J.L. Stein. Online vehicle mass estimation using recursive least squares and supervisory data extraction. In American Control Conference, 2008, pages 1842–1848, June 2008. T. I. Fossen. Nonlinear modelling and control of underwater vehicles. PhD thesis, Norwegian University of Science and Technology, Trondheim, Norway, 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. L. Ljung. System Identification: Theory for the User (2nd Edition). Prentice Hall, 1999. ISBN 0136566952. Lennart Ljung and Torkel Glad. On global identifiability for arbitrary model parametrizations. Automatica, 30(2):265 – 276, 1994. ISSN 0005-1098. doi: 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 London Limited, 2005. ISBN 9781846281570. Maryam Sadeghi Reineh, Martin Enqvist, and Fredrik Gustafsson. Detection of Roof Load for Automotive Safety Systems. In Proceedings of the 52nd IEEE Conference on Decision and Control, pages 2840–2845, 2013. T.S. Söderström and P.G. Stoica. System Identification. Prentice Hall International Series In Systems And Control Engineering. Prentice Hall, 1989. ISBN 9780138812362. E.A. Tannuri, J.V. Sparano, A.N. Simos, and J.J. Da Cruz. Estimating directional wave spectrum based on stationary ship motion measurements. Applied Ocean Research, 25(5):243–261, 2003. 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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