Vessel Model and Description

2.1.1 Notation and Reference Frames

The notation in Table 2.1, defined by SNAME (1950) will be used. The geographi-cal reference frame North-East-Down (NED) and the body-fixed reference frame are used for analysis. NED is chosen as a tangent plane to the surface of the earth, and positions within the frame are denoted{n}= (x_n, y_n, z_n), where thex_n-axis points towards true north, theyn-axis points east, and thezn-axis points downwards. The body-fixed reference frame is denoted{b}= (x_b, y_b, z_b), where thex_b-axis points in the longitudinal direction of the vessel, they_b-axis in the transverse direction of the vessel, and thez_b-axis points in the direction normal to thex_b-y_b plane. See Fossen (2011d) for a closer explanation on kinematics and notation.

Table 2.1:Notation of SNAME (1950) for marine vessels.

Degree of freedom

Position in Euler coordinates

Linear and angular velocities

Forces and moments

Surge x u X

Sway y v Y

Yaw ψ r N

2.1.2 Vessel Model

A simplified control design model, based on the ReVolt model scale ship, is used as a case when considering dynamic constraints and physical properties. A fully actuated three degrees-of-freedom (DOF) model operating in the horizontal plane will be used, thus neglecting motion in roll, pitch, and heave. According to Fossen (2011e), a 3 DOF equation of motion of a marine vessel can be represented as:

˙

η=R(ψ)ν

Mν˙ =−C(ν)ν −D(ν)ν+τ+R(ψ)^>b, (2.1) where ν = [u, v, r]^> is the generalized velocity of the vessel in {b} and η = [p, ψ]^> = [x, y, ψ]^> is the generalized position in {n}. Further, M is the inertia matrix,C(ν)is the Coriolis and centripetal matrix,D(ν)is the damping matrix and τ is the forces acting on the vessel. The vectorb= [b1, b1, b3]^>, is an unknown con-stant (or slowly varying) bias expressed in{n}, accounting for model uncertainties.

R(ψ) ∈ SO(3) is the 3 DOF rotation matrix with the property that R(ψ) =˙ R(ψ)S(r)whereS(r)∈SS(3)is skew-symmetric. Thus:

R(ψ),

The force vectorτ contains the thrust forces acting on the vessel:

τ =

whereF_X andF_Y are the forces in the respective indexed directions andlxandly are the arms from whichτ_N are acting on.

Further, we assume that the ship is symmetric and has homogeneous mass distri-bution about thex_b-z_bplane. This results in a decoupling of the surge motion from sway and yaw and implies that the products of inertiaI_xy = I_yz = 0. By further assuming that the common origin (CO) of {b} coincides with the ships’ center of gravity (CG), results inxg =yg = 0, wherexg andygare the distance from CO to CG in respective directions. The system inertia matrixM =M^> >0includes both rigid-body and added mass terms, and can then be expressed as:

M =M_RB+M_A=

2.1 Vessel Model and Description wherem is the mass of the vessel, I_z is the moment of inertia around the z_b-axis, and the rest of the terms comes from the added mass contributions caused by the acceleration indicated by the subscripts. The skew-symmetric Coriolis-centripetal matrix also includes both rigid-body and added mass terms, and can then be expressed as: The damping matrixD(ν)is constructed by one linear and one nonlinear part. For a low-speed vessel such as ReVolt, the linear damping terms is dominating, such that:

D(ν) =D_L=−

where each term represents hydrodynamic damping forces caused by the velocity indicated by the subscripts.

2.1.3 Maneuverability and Vehicle Characteristics

Maneuverabilityis defined as the capability of the craft to carry out specific maneu-vers (Fossen, 2011f). The maneuverability of the ship depends on several factors, such as water depth, environmental forces, hydrodynamic derivatives, and, most im-portantly, the dynamical constraints of the vessel. For maneuvering of a marine craft, the relationship betweenheading,course, andsideslip, depicted in Figure 2.1, is im-portant. The velocity vector is given as:

U =p

u²+v². (2.7)

For a marine craft moving around, the velocity vector is not necessarily pointing in the same direction as the ship’s heading. The course angleχis the angle from the x_n-axis to the velocity vector of the craft. The headingψis defined to be the angle between thex_n-axis in{n}to thex_b-axis in{b}. The difference between them is the sideslip angleβ. Thus, we have the relation:

χ=ψ+β. (2.8)

xn

Figure 2.1: The relationship be-tweenψ,χ, andβ.

When designing a path for the vessel to fol-low, one needs to take into account that the mag-nitude and orientation of the velocity vector can-not change arbitrarily fast. For a certain velocityU, there is a limitation on the maximum angular veloc-ity the vessel can perform. The lower the speed, the sharper turns the vessel can perform. The maximum angular speedωmaxcan be estimated as a function of the vehicle speedU:ω_max(U)≥0. The angular speed of the velocity vector is given asχ˙ =ω. We know that the inequality:

|χ˙| ≤ω_max(U), (2.9) must hold to not violate the angular speed con-straint. Having the relationU = ωR, where Ris thesteady-state turning radius, and the inequality in Equation (2.9), we obtain the relation:

U ≤ω_max(U)R. (2.10)

This is an implicit relation betweenU andω_max. By further using the relation from Equation (2.20), we get:

U ≤ ω_max(U) κmax

. (2.11)

The steady-state turning radius represents the smallest turn the vessel can perform at a constant surge speed. A maneuvering test such as theTurning Circle Trialcan be used to obtain the steady-state turning radius. See Gertler and Hagen (1960) for a detailed explanation of the test. The turning radius then gives the curvatureκ, by Equation (2.20).