HOW TO INCORPORATE WIND, WAVES AND OCEAN CURRENTS IN THE MARINE CRAFT EQUATIONS OF MOTION. Thor I. Fossen

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1 HOW TO INCORPORATE WIND, WAVES AND OCEAN CURRENTS IN THE MARINE CRAFT EQUATIONS OF MOTION Thor I. Fossen Department of Engineering Cybernetics, Noregian University of Science and Technology, NO-7491 Trondheim, NORWAY Abstract: This paper demystifies ho ocean currents together ith ind and ave loads influence the marine craft equations of motion. In the literature there exists great confusion of the use of absolute and relative velocity terms hen modeling rigid-body and hydrodynamic forces. The article is useful for engineers ho ant to simulate and predict the motions of marine craft exposed to ind, ave and ocean currents as ell as control engineers evaluating the performance of marine craft control systems. The results are also very useful for testing and tuning of integral action time constants for compensation of ocean current and 2nd-order ave-induced drift forces. Keyords: Marine systems, ind, aves, ocean currents, equations of motion INTRODUCTION The equations of motion for underater vehicles, ships, ocean structures and high-speed craft are usually derived using Netonian and Lagrangian mechanics (Fossen, 1994, 211). The resulting models are nonlinear mass-damper-springs hich include rigid-body, hydrostatic and hydrodynamic generalized forces. The motions are coupled in six degrees of freedom (DOF). Marine craft are exposed to environmental forces due to ind, aves and ocean currents, hich act like forcing on the mass-damper-spring system. In hydrodynamics it is common to assume linear superposition such that forcing due to ind and aves can be treated as generalized forces that can be directly added to the nonlinear equations of motion. Hoever, generalized forces due to ocean currents do not follo the la of linear superposition and there exists much diversity and misunderstandings in the existing literature on ho to include the effects of ocean currents in the equations of motion. This paper addresses the effect of ocean currents on marine craft in a tutorial perspective by using the concept of relative velocity, that is the velocity of the craft ith respect to the ocean current, to effectively describe current-induced forces. Different properties and representations of the marine craft equations of motion are discussed and it is shon ho the current velocity enters the equations. Nonlinear models for time-domain simulations and control systems design are presented using the compact vectorial notation of Fossen (1994, 211). 1. MARINE CRAFT EQUATIONS OF MOTION In Fossen (1991) it as shon that the coupled 6-DOF equations of a marine craft could be expressed as: η = J(η)ν (1) M ν + C(ν)ν + D(ν)ν + g(η) = τ (2) here η = N, E, D, φ, θ, ψ] and ν = u, v,, p, q, r] are the generalized position and velocity vectors,

2 respectively. The rotation and angular velocity transformation matrices beteen body coordinates (BODY) and the North-East-Don (NED) geographical reference frames are denoted R n b and T Θ, respectively. The other quantities follo the notation of (Fossen, 1994, 211): ] R n J(η) = b 6 6 Euler angle velocity 6 6 T Θ transformation matrix System inertia matrix M = M RB + M A (including added mass) Coriolis-centripetal matrix C(ν)=C RB (ν)+c A (ν) (including added mass) D(ν) Damping matrix Vector of gravitational/ g(η) buoyancy forces τ Vector of control inputs The subscripts RB and A are used for the rigidbody and added mass terms, respectively. The rigid-body system inertia matrix M RB satisfies: here M RB = M RB >, Ṁ RB = 6 6 mi3 3 ms(r b ] M RB = g) ms(r b g) S(λ) = S (λ) = I b λ 3 λ 2 λ 3 λ 1 λ 2 λ 1, λ = λ 1 λ 2 λ 3 (3) is the cross-product operator defined such that λ a := S(λ)a. The matrix C RB in (2) represents the Coriolis vector term ω b b/n vb b/n and the centripetal vector term ω b b/n (ωb b/n rb g). Contrary to the representation of M RB, it is possible to find a large number of representations for the matrix C RB. We ill present some useful representations belo. Theorem 1. (Coriolis-Centripetal Matrix from M). Consider the 6 6 constant system inertia matrix: ] M = M M11 M = 12 > (4) M 21 M 22 here M 21 = M 12. Then the Coriolis-centripetal matrix can alays be parameterized such that C(ν) = C (ν) by choosing: C(ν) = 3 3 S(M 11 ν 1 + M 12 ν 2 ) S(M 11 ν 1 + M 12 ν 2 ) S(M 21 ν 1 + M 22 ν 2 ) ith ν 1 := u, v, ] and ν 2 := p, q, r]. PROOF. Sagatun and Fossen (1991). ] (5) The rigid-body Coriolis and centripetal matrix C RB (ν) can alays be parametrized such that it is ske-symmetrical: C RB (ν) = C RB(ν), ν R 6 (6) The ske-symmetric property is very useful hen designing nonlinear motion control systems since the quadratic form ν C RB (ν)ν. This is exploited in energy-based designs here Lyapunov functions play a key role. There exists several parametrizations (Fossen and Fjellstad, 1995) that satisfy (6). Lagrangian parametrization: Application of Theorem 1 ith M = M RB yields the folloing expression: 3 3 C RB (ν) = ms(ν 1 ) ms(s(ν 2 )r b g) ms(ν 1 ) ms(s(ν 2 )r b g) ms(s(ν 1 )r b g) S(I b ν 2 ) ] (7) Velocity-independent parametrization: By using the cross-product property S(ν 1 )ν 2 = S(ν 2 )ν 1, it is possible to move S(ν 1 )ν 2 from C {12} RB to C{11} RB in (7). This gives an expression for C RB (ν) that is independent of linear velocity ν 1 : ms(ν C RB (ν) = 2 ) ms(ν 2 )S(r b ] g) ms(r b (8) g)s(ν 2 ) S(I b ν 2 ) Remark 1. Formula (8) is useful hen irrotational ocean currents enter the equations of motion since C RB (ν) does not depend on the linear velocity ν 1 (uses only angular velocity ν 2 and lever arm r b g). This is discussed in Section 3.2. Gravitational and buoyancy forces for surface vessels sho a linear behavior g(η) = Gη here G = Z z Z θ K φ (9) M z M θ For underater vehicles (W B) sin (θ) (W B) cos (θ) sin (φ) g(η) = (W B) cos (θ) cos (φ) (y g W y b B) cos (θ) cos (φ) (z g W z b B) sin (θ) (x g W x b B) cos (θ) sin (φ) + (z g W z b B) cos (θ) sin (φ) + (x g W x b B) cos (θ) cos (φ) (1) (y g W y b B) sin (θ)

3 Wind Coefficients Angle of ind γ (deg) relative bo CK CX Angle of ind γ (deg) relative bo Angle of ind γ (deg) relative bo CY CN Angle of ind γ (deg) relative bo Fig. 1. Wind coeffi cients C X, C Y, C K and C N for a research vessel. here the gravitational and buoyancy forces act through the centers of gravity (CG) and buoyancy (CB) defined by the vectors r b g := x g, y g, z g ] and r b b := x b, y b, z b ], respectively. 2. SUPERPOSITION OF WIND AND WAVE-INDUCED FORCES For control systems design it is common to assume the principle of superposition hen considering ind and ave-induced forces such that (2) takes the folloing form: M ν + C(ν)ν + D(ν)ν + g(η) = τ ind + τ ave + τ (11) here τ ind R 6 and τ ave R 6 represent the generalized forces due to ind and aves. 2.1 Wind forces and moments Wind can be defined as the movement of air relative to the surface of the Earth. For a marine craft moving at a forard speed, the ind forces and moments: τ ind = 1 2 ρ av 2 r C X (γ r )A F C Y (γ r )A L C Z (γ r )A F C K (γ r )A L H L C M (γ r )A F H F C N (γ r )A L L oa (12) are functions of relative ind speed V r and angle of attack γ r according to: V r = u 2 r + vr 2 (13) γ r = atan2(v r, u r ) (14) hich both are functions of the relative velocities: u r = u u (15) v r = v v (16) The nondimensional ind coeffi cients C X, C Y, C Z, C K, C M and C N are usually computed using h = 1 m as reference height hile ρ a is the air density. The frontal and lateral projected ind areas are denoted by A F and A L hile L oa is the length over all. The mean heights of the areas A F and A L are denoted by H L and H F, respectively. Figure 1 shos four ind coeffi cients for a typical research vessel (Blendermann, 1994). Wind coeffi cients for other vessels are found in Fossen (211) and references therein. 2.2 Wave-induced forces and moments The first- and second-order ave forces for varying ave directions β i and ave frequencies ω k are denoted τ {dof} ave1 (ω k, β i ) and τ {dof} ave2 (ω k, β i ) here dof {1, 2, 3, 4, 5, 6}. The normalized force response amplitude operators (RAOs) are complex variables given by (WAMIT Inc., 21): F {dof} ave1 (ω τ {dof} ave1 k, β i ) = (ω k, β i ) ej τ {dof} a v e 1 (ω k,β i ) ρga k F {dof} ave2 (ω τ {dof} ave2 k, β i ) = (ω k, β i ) ρga 2 ej τ {dof} a v e 2 (ω k,β i ) k The output from the hydrodynamic code is usually an ASCII file containing RAOs in table format. Let us denote the imaginary and real parts of the force RAOs by; Im ave1 {dof}(k, i) and Re ave1 {dof}(k, i). The amplitudes and phases for different frequencies ω k and ave directions β i for the first-order ave-induced forces can be computed according to the formulae: F {dof} ave1 (ω k, β i ) = ( Im ave1 {dof}(k, i) 2 + Re ave1 {dof}(k, i) 2) 1/2 (17) F {dof} ave1 (ω k, β i ) = atan2 (Im ave1 {dof}(k, i), Re ave1 {dof}(k, i)) (18) The amplitudes and phases for the second-order mean forces are: F {dof} ave2 (ω k, β i ) = Re ave2 {dof}(k, i) (19) F {dof} ave2 (ω k, β i ) = (2) Since the first- and second-order ave forces are represented in terms of the complex variables F {dof} ave1 (ω k, β i ) and F {dof} ave2 (ω k, β i ), the responses for sinusoidal excitations can be computed using different ave spectra. A frequently used family of ave spectra is: S(ω) = Aω 5 exp( Bω 4 ) (21) here different values for A and B are used. These values depend on geographical location and ind speed (see Chapter 8.2, Fossen 211).

4 Fig. 2. Computation of first- and second-order ave-induced forces from force RAOs. When computing the ave-induced forces, linear superposition is employed as illustrated in Fig. 2. The relationship beteen the a ave spectrum S(ω k ) and the ave amplitude A k for a ave component k is (Faltinsen, 199): 1 2 A2 k = S(ω k ) ω (22) here ω is a constant difference beteen the frequencies. Let the ave-induced forces in 6 DOF be denoted by the vectors: τ ave1 = τ ave2 = τ {1} ave1, τ {2} ave1 ] {6},..., τ ave1 τ {1} ave2, τ {2} {6} ave2,..., τ ave2 ] For the no spreading case, the ave direction β = constant such that: N τ {dof} ave1 = ρg F {dof} ave1 (ω k, β) τ {dof} k=1 A k cos k=1 ( ω e (U, ω k, β)t + F {dof} ave1 (ω k, β) + ɛ k ) N ave2 = ρg F {dof} ave2 (ω k, β) here (23) A 2 k cos (ω e (U, ω k, β)t + ɛ k ) (24) ω e (U, ω k, β) = ω k ω2 k U cos(β) (25) g is the encounter frequency. The assumption that β = constant can be relaxed to model spreading of the main ave propagation direction (see Chapter 8.3, Fossen 211). to turn the major currents to the East in the northern hemisphere and West in the southern hemisphere. Finally, the major ocean circulations ill also have a tidal component arising from planetary interactions and gravity. In coastal regions and fjords, tidal components can reach very high speeds, in fact speeds of 2 3 m/s or more have been measured. The forces on a marine craft due to ocean currents can be accounted for by replacing the generalized velocity vector in the hydrodynamic terms ith relative velocities: ν r = ν ν c (26) here ν c R 6 is the velocity of the ocean current expressed in BODY. Definition 1. (Irrotational fluid). The generalized ocean current velocity of an irrotational fluid is: ν c = u c, v c, }{{} c,,, ] (27) vc b here v b c = u c, v c, c ] is the linear velocity. The ocean current linear velocity vector satisfies: v n c = R n b (Θ nb )v b c (28) here Θ nb = φ, θ, ψ] are the Euler angles beteen BODY and NED, and R n b (Θ nb) SO(3) is the corresponding rotation matrix. Definition 2. (Irrotational constant ocean current). An irrotational constant ocean current in NED satisfies: here v n c = Ṙ n b (Θ nb )v b c + R n b (Θ nb ) v b c := (29) Ṙ n b (Θ nb ) = R n b (Θ nb )S(ω b b/n ) (3) This implies that the ocean current linear velocity vector in BODY coordinates is given by: v b c = S(ω b b/n )vb c (31) 3. EQUATIONS OF MOTION INCLUDING OCEAN CURRENTS Ocean currents are horizontal and vertical circulation systems of ocean aters produced by gravity, ind friction and ater density variation in different parts of the ocean. Besides ind-generated currents, the heat exchange at the sea surface together ith salinity changes, develop an additional sea current component, usually referred to as thermohaline currents. The oceans are conveniently divided into to ater spheres, the cold and arm ater sphere. Since the Earth is rotating, the Coriolis force ill try 3.1 Equations of Motion including Ocean Currents In order to simulate irrotational ocean currents and their effect on marine craft motion, the folloing model can be applied: M RB ν + C RB (ν)ν + g(η) }{{} rigid-body and hydrostatic terms + M A ν r + C A (ν r )ν r + D(ν r )ν r }{{} hydrodynamic terms = τ ind + τ ave + τ (32) here v b v b ] c ν r = ω b b/n (33)

5 is the relative velocity vector. Notice that the rigid-body kinetics is independent of the ocean current. 3.2 Equations of Relative Velocity It is possible to simplify (32) by exploiting the structure of C RB (ν r ). Theorem 2. If the Coriolis and centripetal matrix C RB (ν r ) is parametrized independent of linear velocity ν 1 = u, v, ], for instance by using (8), and the ocean current is irrotational and constant (Definition 2), the rigid-body kinetics satisfies (Hegrenæs, 21): M RB ν+c RB (ν)ν M RB ν r +C RB (ν r )ν r (34) PROOF. Since the Coriolis and centripetal matrix represented by (8) is independent of linear velocity ν 1 = u, v, ], it follos that C RB (ν r ) = C RB (ν). The property: M RB ν c + C RB (ν r )ν c = (35) is proven by expanding the matrices M RB and C RB (ν r ), and corresponding acceleration and velocity vectors according to: mi3 3 ms(r b ] g) S(ω b b/n )vc b ] + ms(r b g) I b 3 1 ms(ω b b/n ) ms(ωb b/n )S(rb g) ms(r b g)s(ω b b/n ) S(I bω b b/n ) Finally, it follos that: ] ] v b c = 3 1 M RB ν + C RB (ν)ν = M RB ν r + ν c ] + C RB (ν r )ν r + ν c ] = M RB ν r + C RB (ν r )ν r (36) Theorem 2 hen applied to (32) gives the differential equations: ] v n η = J(η)ν r + c (37) M ν r +C(ν r )ν r +D(ν r )ν r +g(η) = τ ind +τ ave +τ (38) here M = M RB + M A and C(ν r ) = C RB (ν r ) + C A (ν r ). Notice that only ν r and not ν is used in (38) if compared to (32). The model (37) (38) includes the bias v c n = at the kinematic level hile (32) models drift due to ocean currents at the kinetic level using ν r = ν ν c. 3.3 Equations of Motion for Zero Speed For lo-speed applications such as DP, ocean currents and damping can be modeled by three current coeffi cients C X, C Y and C N. These can be experimentally obtained using scale models in Fig. 3. Current speed V c, current direction β c and current angle of attack γ c relative bo. ind tunnels. The resulting forces are measured on the model, hich is restrained from moving. In many textbooks and papers, for instance Blendermann (1994), ind and current coeffi cients are defined relative to the bo using a counter clockise rotation γ c (see Figure 3). The current forces on a marine craft at rest can be expressed in terms of the area-based current coeffi cients C X, C Y and C N as: X current = 1 2 ρa F cc X (γ c )V 2 c (39) Y current = 1 2 ρa LcC Y (γ c )V 2 c (4) N current = 1 2 ρa LcL oa C N (γ c )V 2 c (41) here V c is the speed of the ocean current. The frontal and lateral projected currents areas are denoted A F c and A Lc, respectively hile L oa is the length over all and ρ is the density of ater. For vehicles at rest and motions limited to surge, say and ya, ocean currents are linearly superimposed according to: M ν + C(ν)ν + D(ν)ν + g(η) = τ current + τ ind + τ ave + τ here τ current = X current, Y current, N current ]. The current coeffi cients can also be used at forard speed U > and related to the surge resistance, cross-flo drag and the Munk moment used in maneuvering theory by using the concept of relative velocity (see Chapter 7.3, Fossen 211). 4. OCEAN CURRENT SIMULATION MODELS Let the ocean current speed be denoted by V c hile its direction relative to the moving craft is expressed in terms of to angles: angle of attack

6 α c and sideslip angle β c as shon in Figure 4. For computer simulations the ocean current speed and direction can be generated by using first-order Gauss Markov processes: V c + µ 1 V c = 1 (42) α c + µ 2 α c = 2 (43) β c + µ 3 β c = 3 (44) here i (i = 1, 2, 3) are zero-mean Gaussian hite noise processes and µ i (i = 1, 2, 3) are constants. If µ 1 = µ 2 = µ 3 =, the models reduce to a random alks, corresponding to time integration of hite noise. A saturating element is usually used in the integration process to limit the current speed to: V min V c (t) V max (45) The direction of the current can also be fixed by specifying constant values for α c and β c. 3-D Irrotational Ocean Current Model: A 3-D irrotational ocean current model is obtained by transforming the ocean current speed V c and directions (α c, β c ) from FLOW axes to NED velocities: v n c = R y,α c R z, β c = V c V c cos(α c ) cos(β c ) V c sin(β c ) V c sin(α c ) cos(β c ) (46) here the principal rotations R y,αc and R z, βc are recognized as: cos(α) sin(α) R y,α = 1 (47) sin(α) cos(α) cos(β) sin(β) R z, β = sin(β) cos(β) (48) 1 This expression can be transformed from NED to BODY using the Euler angle rotation matrix R n b. Consequently, u c v c c = R n b (Θ nb ) V c cos(α c ) cos(β c ) V c sin(β c ) V c sin(α c ) cos(β c ) (49) 2-D Irrotational Ocean Current Model: For motions in the horizontal plane, the 3-D equations (49) reduce to: u c = V c cos(β c ψ) (5) v c = V c sin(β c ψ) (51) for α c = and φ = θ =. Consequently, V c = u 2 c + v 2 c (52) Fig. 4. Angle of attack and sideslip angle for a marine craft. 5. CONCLUSIONS A tutorial on ho to include models for ind, aves and ocean currents for marine craft has been presented. The concept of equations of relative motions is used to model the effect of ocean currents hile ind and ave-induced forces are added under the assumption of linear superposition. The article is intended for control engineers ho ant to simulate and predict the motions of marine craft exposed to ind, ave and ocean currents and use time-series to evaluate the performance of control systems. REFERENCES W. Blendermann. Parameter Identification of Wind Loads on Ships. J. Wind Eng. Ind. Aerodyn, JWEIA-51: , O. M. Faltinsen. Sea Loads on Ships and Offshore Structures. Cambridge University Press, 199. T. I. Fossen. Nonlinear Modeling and Control of Underater Vehicles. PhD thesis, Department of Engineering Cybernetics, Noregian University of Science and Technology, Trondheim, Noray, June T. I. Fossen. Guidance and Control of Ocean Vehicles. John Wiley and Sons Ltd., ISBN T. I. Fossen. Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley and Sons Ltd., 211. ISBN T. I. Fossen and O. E. Fjellstad. Nonlinear Modelling of Marine Vehicles in 6 Degrees of Freedom. International Journal of Mathematical Modelling of Systems, JMMS-1(1):17 28, Øyvind Hegrenæs. Autonomous Navigation for Underater Vehicles. PhD thesis, Dept. of Engineering Cybernetics, Noregian University of Science and Technology, Trondheim, Noray, 21. S. I. Sagatun and T.I. Fossen. Lagrangian Formulation of Underater Vehicles Dynamics. In Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, pages , Charlottesville, VA, October WAMIT Inc. WAMIT User Manual. <.amit.com>, 21.