An Unscented Kalman Filter Based Wave Filtering Algorithm for Dynamic Ship Positioning

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1 Proceeding of the IEEE International Conference on Automation and Logistics Chongqing, China, August An Unscented Kalman Filter Based Wave Filtering Algorithm for Dynamic Ship Positioning Xiaocheng Shi, Xingyan Sun, Mingyu Fu, Wenbo Xie and Dawei Zhao Department of Automation Harbin Engineering University Harbin, Heilongjiang Province, China Abstract - A dynamic positioning (DP) system is used to maintain a ship on a specified position and at a desired heading. A ship is eposed to the wind, currents and waves in the ocean. In order to counteract these disturbances, the dynamic positioning ship uses solely active propulsion. Since the first-order wave force is too large for the ship s thrust system to deal with and it only causes the high-frequency motion, which do not induce the finial position drift. A filtering algorithm is needed to separate the high frequency motions from the low frequency ones. In this paper the Unscented Kalman Filter (UKF) method is used for the wave filtering of dynamic positioning ships. Noisy position and heading measurements are used by the UKF to estimate the lowfrequency motion and the high-frequency motion. A nonlinear platform supply vessel (PSV) model is employed for both the model simulation and the estimator design. An approimation method for the first and second order wave force calculation is used to produce the high frequency and low frequency ship motion. With considering of the wind, currents and waves, some illustrative results of the UKF based wave filtering algorithm are presented. Inde erms - dynamic positioning, wave filtering, Unscented Kalman Filter. I. INRODUCION With the development of offshore oil industry, a kind of automatic position control system, which called ship positioning or dynamic positioning (DP), has been manufactured for the offshore engineering vessels since the 96s. he objective of the system is to maintain the ship s position and heading at specified values by the use of thruster system. Fig. General structure of the DP system. Its main component includes the power system, the thruster system, position reference and sensor system and the control system. Fig. shows a general structure of the DP system. In general, the vessel motions are modeled under the assumption of the principle of superposition, which means the total motion is the sum of low-frequency (LF) and highfrequency (HF) component, as shown in Fig.. he LF motions are the results of the wind, current, the second-order wave drift forces and the control thrust force. he drift of the vessel is attributed to the former three forces. he HF motions are assumed to be caused by the first-order wave motions. hese cause the oscillatory motion of the vessel at the wave frequencies between.3-.6 rad/s. he LF motions are to be controlled and the HF motions that cause thruster modulation are to be eliminated []. Psi ( o ) otal Motion LF Motion HF Motion Fig. Ship motion components. he first generation of dynamic positioning system was designed using conventional control principle like PID control. In order to avoid the thruster to counteract the first order wave induce high frequency oscillatory motion, ordinary analogy low-pass filter and/or notch filter were/was employed []. However, this kind of filters inevitably impose a phase lag into the control loop, as the filters will reduce the control system bandwidth and lead to lower control system stiffness in order to maintain system stability [3]. Around 974, a new modelbased control concept was introduced in which mathematical modelling of vessel movement, state and parameter estimation his work is supported by Fundamental Research Funds for the Central Universities (HEUCFR8) //$6. IEEE 399

2 was employed in the form of Etend Kalman Filters (EKF) and multivariable control using linear quadratic Gaussian (LQG) theory [4]. Comparing with the conventional filter, the model-based method, EKF, is computational heavily, but allows for an improved separation of the low frequency vessel motion and the wave inducing high frequency motion. Also, it reduces the phase lag introducing by the filtering process and resulted a better control performance [5]. What is more, the EKF method allows the use of several redundant position reference systems and sensors to obtain an optimal estimation of the position and heading, as well as the velocity. his greatly improves the DP system s reliability. Also, by the function of parameter estimation, the environment forces can be predicted, which is an important information for the controller. For these reasons, the EKF methods are still widely used in today s commercial DP system. Wave filtering is one of the most important components in dynamic positioning control system. Its performance has direct impact on the finial DP system. With the development of the nonlinear controllers and estimators, the inherent nonlinear feature of the vessel dynamics can be handled even well. he UKF is a new kind of recursive linear estimator for filtering systems with nonlinear process and observation models [6]. It represents a derivate-free alternative to the EKF, and provides superior performance at an equivalent computational compleity. his paper makes use of the UKF method to deal with the wave filtering problem. Meanwhile, instead of using the sinusoidal generator or spectrum generator method to simulate the HF wave motions, an approimate method is used to calculate the first-order wave force, so the HF motions could be simulated through a force-motion relation that seems more complies with the reality. II. MODEL OF VESSEL MOION AND ENVIRONMEN FORCES A nonlinear vessel model will be used for the simulation in this paper. he model includes kinematics and kinetics two parts, which is developed by the. I. Fossen can be found in [7]-[9]. While, the environment forces model includes forces which are caused by the wind, waves and ocean currents. A. Mathematical Modeling of Vessel Dynamics o describe the motion of a vessel, two reference frames are used. One is a local geographical Earth-fied frame (NED frame), the other is a body-fied frame (B frame), which is attached to the center of gravity (CG) of the vessel. For the DP ship, only three degrees of freedom (DOF) needed to be considered. he components of the position-orientation vector η = [n, e, ψ] are the north-east positions (n, e) of the vessel relative to the local geographical frame and the heading angle ψ is relative to the north. he components of the velocity vector v = [u, v, r] are surge and sway line velocities (u, v) and the yaw angular velocity r. hese variables are depicted in Fig. 3. Where U = u + v is the vessel s forward speed in the horizontal plan. he following dynamic model represents the LF motions of the vessel: η = R( ψ ) v () ( MRB + MA ) v + CRB ( v) v+ D( v) v = τcontrol + τenv () Equation () is the kinematic transformation, which relates the body-fied velocities to the time derivative of the positions in the NED frame. Where the rotation matri is cosψ sinψ R( ψ) = sinψ cosψ, R ( ψ) = R ( ψ) (3) he terms on the right hand side of () represent the vectors of forces due to control and environment. he left hand side of M RB and C RB denote the rigid-body mass matri and the skew-symmetric Coriolis-centripetal matri. Where M A is the hydrodynamic added mass matri; D(v) is hydrodynamic damping force. For further details about these matries, see [7] and [8]. ψ North( n) b NED o Body r u v y b U East( e) Fig. 3 he reference frames and related variables. he parameters for the simulation PSV model used in this paper is obtained by scaling a scale model s parameters which is named CybershipⅡ. Detailed descriptions about the complete mathematical model of LF ship dynamics can be found in [] and the references cited in. he CybershipⅡis a test scale physical model of a supply ship developed by the department of engineering cybernetics at the Norwegian University of Science and echnology (NNU). B. Mathematical Modeling of Environmental Force ) Wind Force he static wind forces and moments can be represented by using Isherwood s empirical formula A C ( ) w X γ w rw τwind = ρavrw AL C ( ) w Y γ w rw (4) AL L ( ) w oacn γ w rw where ρ a is the air density, A,w and A L,w are the frontal and lateral projected wind areas, L oa is the vessel s overall length. he wind speed V rw and direction γ rw relative to the vessel are given by Vrw = urw + vrw (5) γ rw = atan ( vrw, urw) (6) with u = u V cosψ (7) rw w w 4

3 vrw = v Vw sinψ w (8) Here V w and γ w are the true wind speed and direction in NED frame. he relative relationship is depicted in Fig.4. North( n) NED o V rw v rw U V w North u rw ψ γ w U b y b γ rw East( e) Fig. 4 he relationship between true wind and relative wind. In the paper a PSV is used as the simulation object, by using the Blendermann s wind coefficient calculation method we could get the wind coefficients for the PSV, which are given in Fig.5. Its main particulars are given in able І. C N C X C Y ABLE I A PSV S MAIN PARICULARS ype Length (L oa / L pp) Beam (B) Draft () Mass (M) U 745 Design PSV 8.45/76. (m) 8.8 (m) 6.68 (m) 459 (ton) Frontal Area (A,w ) 9.4 (m ) Lateral Area (A L,w ) 573.(m ) Relative Wind Angle ( o ) Relative Wind Angle ( o ) Relative Wind Angle ( o ) Fig. 5 Wind Coefficients of the PSV. ) Current Force Usually, the current force can be represented in a similar way to the wind forces and moments by means of the current force coefficient. But here we do not have the empirical method to calculation the current force coefficient for the PSV. So in the simulation, we approimate the current force as a constant force, which is proportional to the wind force, although it is not entirely correct. hrough this way, the ocean currents forces effect is modeled. 3) Wave Force Wave is the main component in the environmental modeling. It is difficult to eactly model it as the wave is a compleity random process. In the literatures to deal with this problem, the HF motions are usually simply modeled by the summation of sinusoids or by passing white-noise sequences through a wave transfer function. In order to simulate the HF motion through a forcemotion method, which relates wave-elevation and waveacceleration to forces affecting the vessel, some approimate calculations are used. As the HF motion is mainly caused by the first-order wave force, an approimation method in [] is used to calculate the Froude-Krilov forces and the second order drift forces. A modified Pierson-Moskowitz (MPM) spectrum S(ω) is used, N frequencies are picked from the range Δω. he total wave elevation of a fied point is given as N ζ () t = S( ω) Δ ωsin( ωit+ εi) (9) And the wave accelerations are given as N k a () t = ωi e S( ω) Δ ω cos( ωit+ εi) () N k 3() = ωi ( ω) Δ ωcos( ωi + εi) a t e S t () where ω i is the ith frequency in the range Δω; k is the wave number; is the draft of the vessel. In this way, the Froude-Krilov forces and second-order drift forces are models as ρwva τ ρwva FK = () ρwva 3 BR ( ( ωζ ) ) τwd = ρwg Lpp( Ry( ) ) 8 ω ζ (3) BLpp ( R ψ ( ωζ ) ) where R (ω), R y (ω) and R ψ (ω) are reflection coefficients. Fig. 6 gives an illustration about the first and second order wave forces calculation method which shows that under a sea state 4 (significant wave height is.8 meter, modal wave period is 7 second) when the vessel stays in a beam sea, it will drift away slowly and with an oscillation motion which period is also about 7 second. 4

4 Psi ( o ) North Position (m) Finial Position Initial Position -4 - East Position (m) Fig. 6 Vessel s Open-Loop Response to the Wave Forces. III. UKF BASED WAVE FILERING ALGORIHM DESCRIPION Reference [8] gave a definition for the Wave filtering the reconstruction of the LF motion components from noisy measurements of position, heading and in some cases velocity and acceleration by means of a state observer or a filter. Since Norwegian scholar Balchen first employed the Kalman filter to deal with the wave filtering problem, this method has become a standard component in the DP system. Besides estimating the LF motion and velocity, it can also estimates the HF motion components, which will be useful in the case of dead rocking. he standard wave filter contains two components: LF motion and HF motion. Equation () is used to design the LF part of the wave filter in this paper. For the reason that () is a nonlinear model, the EKF or UKF method can be used. he main difference between these two algorithms is mainly concern about the manner in which Gaussian random variable (GRV) are represented for propagating through system dynamics. For the EKF, the state distribution is approimated by a GRV, which is then propagated analytically through the first-order aylor epansion of the nonlinear system. But it has three well-known drawbacks: the linearization can produce highly unstable filters if the assumption of local linearity is violated; the linearization can be applied only when the Jacobian matri is eists which means the system should be differentiable at the estimate; the derivation of the Jacobian matries is nontrivial in most applications which can lead to significant implementation difficulties. While for the UKF, it uses a deterministic sampling approach to address these problems. hese sample points completely capture the true mean and covariance of the GRV, and when propagated through the true nonlinear system, it can captures the posterior mean and covariance accurately to the second-order aylor series epansion for any nonlinearity. A method which called Unscented ransformation is used to calculate the statistics of a random variable which undergoes a nonlinear transformation [6]. Wave filtering model used in the paper: η = R( ψ ) v (4) Mv =C() v v D() v v+ τ + R ( ψ ) b+ w (5) b = w (6) ξ = A ξ + Ew (7) ω ω 3 y = η + Cωξ + v (8) where W, W, W 3 R 3 are vectors of zero-mean Gaussian white noise with covariance matrices Q L, Q b, Q H, b R 3, is a vector of bias terms, ξ R 6 is the HF state vector, A R 6 6, E R 6 6, C R 3 6 are constant matrices of appropriate dimension, v R 3 is the vector of zero-mean Gaussian white noise with covariance matri R. he complete model can be written as: = f( ) + Bu+ Ew (9) y = H+ v with parameters given by the obvious association of (4)-(8). A. Discrete UKF wave filtering algorithm Reference [] gave a simplified UKF method which uses scale unscented transformation. he UKF algorithm used for the DP wave filtering is as below: Sigma points and related weights setting: Sigma points are setting according to the covariance matri P and the weight are given as: χ k = k k ( n λ) k k ( n λ) + + P + + Pk ( m) W = λ/( n+ λ) () ( c) W = λ/( n+ λ) + ( α + β) ( m) ( c) Wi = Wi = /(( n+ λ)) i =,, Prediction: χ = f( χ, u ) Z kk k k ( m) kk = Wi χikk, ( c) kk Wi [ χikk, kk ][ χikk, kk ] P = + Q z = h( χ, u ) kk kk kk ( m) kk = Wi Zikk, Correction: zkzk z k k k ( c) Wi [ Zi, t t k k ][ Zi, t t k k] ( c) Wi [ χikk, kk ][ Zikk, kk ] z k k zz k k P = z z + R P = z K = P P () () k = k k + Kk( zk zk k) Pk = Pk k KkPzkz K k k where χ is the sigma points set; k, P k- are the mean and covariance of the points in the sigma set; λ = α (n+κ)-n is a scaling parameter and κ is a secondary scaling parameter; ( m) ( c) W, W are the weight for the initial mean and covariance; ( m) ( c) Wi, W i are the weight for the ith sigma point; the constant α determines the spread of the sigma points around k and is set to.; β is used to incorporate prior knowledge of the distribution of ; K k is the Kalman gain matri. 4

5 IV. SIMULAION RESULS AND ANALYSIS he simulation weather condition is set to sea state 4 which corresponding to beaufort number 5, meanwhile a.5 knots currents is also used to test the UKF wave filtering algorithm. he directions for the wind, waves and currents are 45, 45 and 9 in the NED frame. Vessel s initial station was set to η = [,, ]. Under the influences of above environment forces, the open-loop motion of the vessel is showed in Fig. 7. that were filtered by the UKF wave filtering algorithm was given for verify its performance. From these results, it can be seen that the vessel position and heading could be eactly estimated. For the vessel s velocity estimation, the first order wave forces influence to the LF velocities has been effectively reduced, but the HF velocities has not been fully estimated. Fig. 9 shows the approimation calculation results of the first order wave forces. From the simulation result, we can see that the vessel s HF motion had been successfully generated by the force-motion method. North Position (m) 6 4 Finial Position - -4 Initial Position Wave Force / Moment (kn / knm ) Surge direction st order wave force Sway direction st order wave force Yaw direction st order wave moment r (rad/s) East Position (m) North Position (m) Psi ( o ) u (m/s) v (m/s) East Position (m) 5 Fig. 7 Vessel state in the NED frame. Real data Filtered LF data Filtered HF data Fig. 8 Position and heading of vessel. Real data Filtered LF data Filtered HF data Fig. 9 Surge, sway and yaw rate of vessel. Fig. 8 and Fig. 9 illustrated the vessel model s real data about the motions, meanwhile, the data of LF and HF motion Fig. First order wave force and moment applied to the vessel. V. CONCLUSIONS A UKF based wave filtering algorithm are design and successfully tested against the motion response of a PSV eposed to the environmental forces which caused by the wind, waves and currents. Also, instead of using the traditional sinusoidal generator or the spectrum generator to simulate the vessel HF motion a simplified approimate force generator method is used. he simulation process has shown that the use of UKF method has avoided the derivate of Jacobian matri for the traditional EKF, and also can deal with the nonlinear of the vessel dynamic even better. But it should be noted that, the tuning of UKF and EKF are usually done in a trial and error mode, it will be much better to apply the selftuning to improve the wave filtering algorithm s practical use value. REFERENCES [] M. J. Grimble and M. A. Johnson, Optimal Control and Stochastic Estimation heory and Applications, vol. II, A Wiley-Interscience publications, 988, pp [] J. S. Sargent and P. N. Cowgill, Design Considerations for Dynamically Positioned Utility Vessels, 8th Annual Offshore echnology Conference, Dallas, U.S.A, 976, pp [3] H. R. Sørheim and F. L. Galtung, Wave Filter Performance Evaluation, 9th Annual Offshore echnology Conference, Houston, U.S.A, 977, pp [4] J. G. Balchen, N. A. Jenssen, and S. Sælid, Dynamic Positioning Using Kalman Filtering and Optimal Control heory, in Proc.IFAC/IFIP Symp. Automation in Offshore Oil Field Operation, Bergen, Norway, 976, pp [5] E. A. annuri and.. Bravin, Dynamic Positioning Systems: Comparison between Wave Filtering Algorihms and heir Influence on Performance, Proceedings of d International Conference on Offshore Mechanics and Arctic Engineering, Cancun, Meico, 3, pp

6 [6] S. J. Julier, J. K. Uhlmann, Unscented Filtering and Nonlinear Estimation, Proceedings of the IEEE, vol. 9, no. 3, pp. 4-4, March 4. [7]. I. Fossen, Guidance and Control of Ocean Vehicles. New York: Wiley, 994. [8]. I. Fossen, Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. rondheim, Norway: Marine Cybernetics,. [9]. I. Fossen and. Perez, Kalman Filtering for Positioning and Heading Control of Ships and Offshore Rigs. IEEE Control Systems Magazine, pp.3-46, December 9. [] M. omera, Model Matematczny statku Cybership II, Zeszyty Naukowe Akademii Morskiej W Gdyni, nr 6, grudzien 9, pp. 8-. (in Polski) [] A. V. Fannemel, Dynamic Positioning by Nonlinear Model Predictive Control, Master Dissertation, Norwegian University of Science and echnology, Norway, June 8. pp [] S. J. Julier, J. K. Uhlamn and H. F. Durrant-Whyte. A new Approach for Filtering Nonlinear Systems. Proceeding of the American Control Conference, 995, pp

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