Kinematics of Ship Motion

3 Kinematics of Ship Motion by T. Perez and T.I. Fossen Within the discipline of mechanics, dynamics refers to the branch that studies the motion of particles and bodies under the action of forces. This study can be divided into two parts [203]: • • Kinematics, Kinetics. Kinematics describes geometrical aspects of motion without considering mass and forces: reference frames, variables and transformations. Kinetics describes the effects of forces on the motion. This chapter introduces the kinematics of ship motion while kinetics are discussed in Chapter 4. 3.1 Reference Frames A ship in a seaway moves in six degrees of freedom (6DOF). Thus, to describe its motion, we need to consider three coordinates to define translations and three coordinates to define the orientation. These coordinates are defined using two type of reference frames: inertial frames and body-fixed frames. The following right-hand reference frames are usually considered for marine vehicles—see Figures 3.1 and 3.2: • North-east-down frame (n-frame). The n-frame (on , xn , yn , zn ) is fixed to the Earth. The positive xn -axis points towards the North, the positive yn -axis towards the East, and the positive zn -axis towards the centre of the Earth. The origin, on , is located on mean water free-surface at an appropriate location. This frame is considered inertial. This is a reasonable assumption because the velocity of marine vehicles is small enough 46 3 Kinematics of Ship Motion p xn North Surge xb , u on yn East ob xh zn Down r xg αr q oh zg yh og zh yg Sway yb , v Heave zb , w Fig. 3.1. Notation and sign conventions for ship motion description. FP AP Lpp CG oh og xg rbCG ob rōhb xb xh T zg V CG LCG zb DW L BL zh Fig. 3.2. Main particulars and reference frames: geometric (origin og ); hydrodynamic (origin oh ); and body-fixed (origin ob ); CG—centre of gravity; LCG—lateral centre of gravity (distance); V CG—vertical centre of gravity (distance); AP —aft perpendicular; F P —front perpendicular; Lpp —length between perpendiculars; T — draught; DWL—design waterline and BL–baseline. for the forces due to the rotation of the Earth being negligible compared to the hydrodynamic forces acting on the vehicle [67]. 3.1 Reference Frames 47 • Geometric frame (g -frame; forward-starboard-up). The g-frame (og , xg , yg , zg ) is fixed to the hull. The positive xg -axis points towards the bow, the positive yg -axis points towards starboard and the positive zg axis points upwards. The origin of this frame is located along the centre line and at the intersection of the baseline (BL) and the aft perpendicular (AP), which is taken at the rudder stock – see Figure 3.2. • Body-fixed frame (b-frame; forward-starboard-down). The bframe (ob , xb , yb , zb ) is fixed to the hull. The positive xb -axis points towards the bow, the positive yb -axis points towards starboard and the positive zb axis points downwards. For marine vehicles, the axes of this frame are chosen to coincide with the principal axes of inertia; this determines the position of the origin of the frame, ob , [67]. We will further discuss the location of ob in Chapter 4. • Hydrodynamic frame (h-frame; forward-starboard-down). The hframe (oh , xh , yh , zh ) is not fixed to the hull; it moves at the average speed of the vessel following its path. The xh -yh plane coincides with the meanwater free surface. The positive xh -axis points forward and it is aligned with the low-frequency yaw angle ψ 1 . The positive yh -axis points towards starboard, and the positive zh -axis points downwards. The origin oh is determined such that the zh -axis passes through the time-average position of the centre of gravity. This frame is usually considered when the vessel travels at a constant average speed (which also includes the case of zero speed); and therefore, the wave-induced motion makes the vessel oscillate with respect to the h-frame. This frame is considered inertial. Each of these frames has a specific use. For example, the g-frame is commonly used by naval architects to define the geometry of hull, main particulars, location of the centre of gravity, location of ob etc. The n-frame is used to define the position of the vessel, and together with the b-frame it also defines the orientation of the vessel. All the measurements taken on board (velocities, accelerations, etc.) are referred to the b-frame, which is also used to formulate the equations of motion. The h-frame is used in hydrodynamics to compute the forces and motion due to the interaction between the hull and the waves—these data are important for preliminary ship and ship motion control system design. This frame is also used to define local wave-induced accelerations, which are used to calculate indices related to performance of the crew or comfort of passengers—see Chapter 7. 1 The angle ψ̄ is obtained by filtering out the 1st-order wave-induced motion (oscillatory motion), and keeping the low frequency motion, which can be either equilibrium or slowly-varying. Hence, ψ̄ is constant for a ship sailing in a straightline path. 48 3 Kinematics of Ship Motion 3.2 Vector Notation Because of the use of different reference frames, it is necessary establish a mathematical notation that allows us to identify position, velocity, and acceleration of different points of interest on the ship and to express them in the different frames considered. Thus, for a generic point of interest x on the ship, • rfx denotes the position of x with respect to a frame f , i.e. ⎡ f⎤ xx rfx = xfx fx + yxf fy + zxf fz ≡ ⎣yxf ⎦ = [xfx , yxf , zxf ]t . zxf (3.1) • vxf denotes the velocity of x with respect to a frame f . • v̇xf denotes the acceleration of x with respect to a frame f . • Θab Euler angles that take the a-frame into the orientation of the b-frame. Rotations are considered positive when made counter clockwise. • ω cab denotes the relative angular velocity of the frame b with respect to the frame a, decomposed in the frame c. The cross product of the vectors will be written as [62, 67]: a × b
S(a)b, where the skew-symmetric matrix S is defined as: ⎡ ⎤ ⎤ ⎡ λ1 0 −λ3 λ2 S(λ) = −St (λ)
⎣ λ3 0 −λ1 ⎦ , λ
⎣ λ2 ⎦ . λ3 −λ2 λ1 0 (3.2) (3.3) 3.3 Coordinates Used to Describe Ship Motion 3.3.1 Manoeuvring and Seakeeping Surface vessel operations are performed under different environmental conditions, and different assumptions are made during the study of hydrodynamics in each case. As a consequence of this, the study of ship dynamics has traditionally been separated into two main areas: • Manoeuvring, • Seakeeping. Manoeuvring deals with the the motion of a ship in the absence of waveexcitation (calm water) [2]. The motion results from the action of control devices: control surfaces and propulsion units. Manoeuvring is associated with course changes, stopping, etc. Seakeeping, on the other hand, is associated with motion sue to wave excitation, while the vessel keeps its course and its speed constant. 3.3 Coordinates Used to Describe Ship Motion 49 These two areas of study of ship motion are well established with accurate models to describe the motion characteristics according to the assumptions made in each of them. Due to the independent development of manoeuvring and seakeeping, different reference frames and coordinates are used to describe the motion of the ship. 3.3.2 Manoeuvring Coordinates and Reference Frames The north-east-down position of a ship is defined by the coordinates of the origin of the b-frame, ob , relative to the n-frame: ⎡ ⎤ n rnob
⎣ e ⎦ . d The attitude of a ship is defined by the orientation of the b-frame relative to the n-frame. This is given by the three consecutive rotations that take the n-frame into the b-frame. The rotations are performed in the following order: 1. Rotation about the zn axis; the rotation angle is called yaw ψ, 2. Rotation about the yn axis; the rotation angle is called pitch θ, 3. Rotation about the xn axis; the rotation angle is called roll φ. With the rotations performed in this particular order, the rotation angles are called Euler angles. The vector of Euler angles is defined as ⎡ ⎤ φ Θnb
⎣ θ ⎦ . (3.4) ψ Following the notation of Fossen [66, 67], the position-orientation vector (or generalised position vector ) is defined as:  rnob η
= [n, e, d, φ, θ, ψ]t . Θnb
(3.5) The linear and angular velocities of the ship are more conveniently expressed in the b-frame. The linear-angular velocity vector (or simply generalised velocity vector ) given in the b-frame is defined as: ν
 vobb = [u, v, w, p, q, r]t , ω bnb
(3.6) where • vobb = [u, v, w]t is the linear velocity of the point ob expressed in the bframe. 50 3 Kinematics of Ship Motion • ω bnb = [p, q, r]t is the angular velocity of the b-frame with respect to the n-frame expressed in the frame b. Table 3.1 summarises the adopted notation. The reader should be aware of the differences in nomenclature and reference frames used to describe ship motion in different areas of study (manoeuvring and seakeeping), and be careful when combining data and results. A further discussion on these topics can be found in [67]. Table 3.1. Adopted nomenclature for the description of ship motion, and reference frames in which the components are defined. Component n e d φ θ ψ u v w p q r Name North position East position Down position Roll angle Pitch angle Heading or yaw angle Surge velocity Sway velocity Heave velocity Roll rate Pitch rate Yaw rate Definition frame n-frame n-frame n-frame Euler angle Euler angle Euler angle b-frame b-frame b-frame b-frame b-frame b-frame 3.3.3 Seakeeping Coordinates and Reference Frames In seakeeping, the study ship motion is performed under the assumption that the ship is moving on a steady course and at a constant-average forward speed (which includes the case of zero speed). This defines a state of equilibrium of motion, and the action of the waves makes the ship oscillate with respect to this equilibrium (1st-order wave-induced motion). This fundamental assumption is the basis of the seakeeping theory of ship motion—see [28, 63, 135, 159]. In seakeeping theory, the motion of the ship is commonly described using the h-frame, which is fixed with respect to the equilibrium of motion. Here, however, we will also allow the ship to manoeuvre, but under the assumption that the manoeuvring is much slower than the motion induced by the waves. This way the h-frame can still be used to describe the motion. Once the origin oh is chosen, it coincides with the slowly-varying location of a point s in the ship, i.e. oh ≡ s̄, where the notation s̄ indicates either equilibrium or slowly varying component. Thus, oh ≡ s̄ = s when there are no waves (see Figure 3.3). The set of generalised perturbation coordinates or 3.3 Coordinates Used to Describe Ship Motion 51 ψ xh ξ6 γ =ψ+β ū xb β ū ob xn oh s̄ ξ yn v̄ s yh yb Fig. 3.3. Angles for the horizontal plane. seakeeping coordinates defined in the h-frame will be denoted by: ξ = [ξ1 , ξ2 , ξ3 , ξ4 , ξ5 , ξ6 ]t . (3.7) When these coordinates2 describe the position of s (see Figure 3.3), the linear coordinates are normally referred to as • ξ1 —surge displacement, • ξ2 —sway displacement, • ξ3 —heave displacement, whereas the angular coordinates ξ4 , ξ5 , ξ6 are the Euler angles that take the h-frame into the orientation of the b-frame: 2 Note that in some of the hydrodynamic and seakeeping literature (e.g. [63, 133]), the above variables are usually denoted by ηi . Unfortunately, this is in conflict with the variables (3.5), which are already well accepted in the literature of guidance, navigation and control of marine systems [66, 67]. For this reason we have adopted the notation ξi . 52 3 Kinematics of Ship Motion Θhb ⎤ ⎡ ⎤ ⎡ φ ξ4
⎣ξ5 ⎦ = ⎣ θ ⎦ . ξ6 ψ−ψ (3.8) These angles are referred to as • ξ4 —roll perturbation angle, • ξ5 —pitch perturbation angle, • ξ6 —yaw perturbation angle. The perturbation coordinates can be used to describe the oscillatory position of any point of interest with respect to h-frame. Indeed, for the generic point of interest x with equilibrium position x̄ with respect to oh , the following relationships hold: rhx = [ξ1 , ξ2 , ξ3 ]t + [ξ4 , ξ5 , ξ6 ]t × rhx̄ vh = [ξ˙1 , ξ˙2 , ξ˙3 ]t + [ξ˙4 , ξ˙5 , ξ˙6 ]t × rh x v̇xh = [ξ¨1 , ξ¨2 , ξ¨3 ] + [ξ¨4 , ξ¨5 , ξ¨6 ] × t t x̄ rhx̄ . (3.9) This, for example, allows one to evaluate vertical accelerations at different locations on the ship and calculate the motion sickness incidence (MSI) index— see Chapter 7. 3.3.4 Angles About the z-axis Let us define the total ship velocity vector (in the b-frame) as ū = [ū, v̄, w̄]t , (3.10) such that u = ū + δu v = v̄ + δv w = w̄ + δw. For surface vessels w̄ =0, and for slow manoeuvring the surge and sway velocities ū and v̄ are the approximately the same in both the b- and the h-frame. In this case, the surge velocity is denoted U ≡ ūh ≈ ū, which is the notation commonly used in seakeeping and hydrodynamics. Then, for the angles about the z-axis of surface ships, it is convenient to distinguish between the following (see Figure 3.3): 3.4 Velocity Transformations 53 • Heading or yaw angle ψ. This is the first rotation of the sequence of rotations (Euler angles) that take the n- into the b-frame—see Section 3.3.2. • Seakeeping or yaw perturbation angle ξ6 . This is the first rotation of the sequence of rotations (Euler angles) that take the h- into the b-frame— see Section 3.3.3. • Drift angle β. This is the angle between the positive x-axis of the b-frame and the average ship velocity vector ū , i.e.  v̄  , (3.11) β = arctan ū provided ū is not zero. • Course angle γ. This is the angle between the positive x-axis of the n-frame the ship velocity vector ū. 3.4 Velocity Transformations 3.4.1 Rotation Matrices The transformation of vector coordinates between different frames is performed via appropriate transformation matrices. Following [62], the generic vector r, can be expressed in either the frame a or the frame b as r= 3  ria ai and r = 3  rib bi , (3.12) i=1 i=1 where the vectors ai and bi are the unit vectors along the axis of the reference frames a and b respectively, and ria = r · ai and rib = r · bi . Then,   3 3   a b ri = r · ai = rib (ai · bi ). (3.13) ri bi · ai = i=1 i=1 54 3 Kinematics of Ship Motion This leads to the notation Rab for transformation matrix with entries {ai · bi }, which takes vectors expressed in the frame b to the frame a: ra = Rab rb . (3.14) This matrix is called the rotation matrix from b to a. Rotation matrices are elements in SO(3), the special orthogonal group of order 3:   SO(3) = R|R ∈ R3×3 , RRt = I3×3 , and det(R)=1 . (3.15) Thus, (Rab )−1 = (Rab )t = Rba . 3.4.2 Kinematic Transformation Between the b- and the n-frame The transformation between the body-fixed linear velocities and the time derivative of the positions in the n-frame can be expressed as ⎡ ⎤ ⎡ ⎤ ṅ u ⎣ ė ⎦ = Rnb (Θnb ) ⎣ v ⎦ , w d˙ (3.16) where linear-velocity transformation matrix Rnb (Θnb ) is given by [66, 67] ⎡ ⎤ cψcθ −sψcφ + cψsθsφ sψsφ + cψcφsθ Rnb (Θnb ) = ⎣sψcθ cψcφ + sφsθsψ −cψsφ + sψcφsθ⎦ , (3.17) −sθ cθsφ cθcφ where s ≡ sin(·) and c ≡ cos(·), and Rbn (Θnb ) = Rnb (Θnb )−1 = Rnb (Θnb )t . (3.18) This transformation is the result of three consecutive rotations about the principal axes [67]: (3.19) Rnb (Θnb )
Rz,ψ Ry,θ Rx,φ , with, Rx,φ ⎡ ⎤ ⎡ ⎤ ⎤ cψ −sψ 0 cθ 0 sθ 1 0 0
⎣ 0 cφ −sφ ⎦ , Ry,θ
⎣ 0 1 0 ⎦ , Rz,ψ
⎣ sψ cψ 0 .⎦ (3.20) 0 0 1 −sθ 0 cθ 0 sφ cφ ⎡ The transformation between the body-fixed angular velocity ω bnb and the time derivative of the Euler angles Θ̇nb can be expressed as 3.4 Velocity Transformations Θ̇nb = TΘ (Θnb ) ω bnb , or ⎡ ⎤ ⎡ ⎤ φ̇ p ⎣ θ̇ ⎦ = TΘ (Θnb ) ⎣q ⎦ , r ψ̇ 55 (3.21) where angular-velocity transformation matrix TΘ (Θnb ) and its inverse are given by ⎤ ⎡ ⎤ ⎡ 1 0 −sθ 1 sφtθ cφtθ TΘ (Θnb )
⎣0 cφ −sφ ⎦ , TΘ (Θnb )−1
⎣0 cφ cθsφ⎦ (3.22) 0 −sφ cφcθ 0 sφ/cθ cφ/cθ with t ≡ tan(·) and cθ = 0. This transformation can be obtained from [67]: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 p φ̇ ω bnb = ⎣q ⎦ = ⎣ 0 ⎦ + Rtx,φ ⎣θ̇⎦ + Rtx,φ Rty,θ ⎣ 0 ⎦
TΘ (Θnb )−1 Θ̇. r 0 ψ̇ 0 (3.23) Notice that TΘ (Θnb )−1 = TΘ (Θnb )t , and also that, TΘ (Θnb ) is not defined for θ = ±π/2. This is never a problem for surface vessels, but could be of concern for underwater vehicles in some cases. In such cases, the singularity can be avoided using unit quaternions [67]. Using the above, we can define the following kinematic transformation for the manoeuvring coordinates: η̇ = Jnb (Θnb ) ν =
n Rb (Θnb ) 03×3  03×3 ν. TΘ (Θnb ) (3.24) To transform the accelerations, we need to consider the time derivatives of (3.16) and (3.21). For example, ¨ t + Ṙb (Θnb ) [ṅ, ė, d] ˙t [u̇, v̇, ẇ]t = Rbn (Θnb ) [n̈, ë, d] n [ṗ, q̇, ṙ]t = TΘ (Θ) [φ̈, θ̈, ψ̈]t + ṪΘ (Θ) [φ̇, θ̇, ψ̇]t . (3.25) 3.4.3 Kinematic Transformation Between the b- and the h-frame The transformation for the generalized velocity vector ν (given in the b-frame) into the generalised velocity vector ξ̇ = [(vohh )t , (ω hhb )t ]t in the h-frame is performed in two steps: a translation and a rotation. Let us consider the case of zero forward speed first. The linear velocity of ōh in the b-frame is given by vōbh = vobb + ω bnb × rbōh , (3.26) 56 3 Kinematics of Ship Motion where ōh is the equilibrium position of oh with respect to to the b-frame (remember that the h-frame is not fixed to the ship and therefore the vector rboh is time varying, so we consider rbōh ). Since the h-frame is considered inertial and its relative angular velocity with respect to the n-frame is approximately zero (i.e. ω bnh ≈ 0), the following relationship holds: (3.27) ω bhb = ω bnb − ω bnh ≈ ω bnb . By expressing the cross product in (3.26) in terms of a skew-symmetric matrix ⎤ ⎡ 0 −zōbh yōbh b b t 0 −xbōh ⎦ , S(rōh ) = −S(rōh ) = ⎣ zōbh (3.28) b −yōh xbōh 0 Expressions (3.26) and (3.27) can be combined as
b 
b  vob vōh b = H(rōh ) , ω bhb ω bnb (3.29) where is the screw transformation:
 I3×3 S(λ)t H(λ)
, 03×3 I3×3 λ ∈ R3 . (3.30) The linear velocity transformation between the b- and the h-frame is given by the following rotation vōhh = Rhb (Θhb )vōbh , (3.31) where Rhb (Θhb ) is of the form of (3.17). In a similar way, the angular velocity transformation is given by ω hhb = Rhb (Θhb )ω bhb , where TΘ (Θhb ) is of the form of (3.22). By combining (3.31), (3.32) and (3.29), we obtain
b  
h 
h vob Rb (Θhb ) 03×3 vōh b ) H(r . = ōh 03×3 Rhb (Θhb ) ω hhb ω bnb (3.32) (3.33) Thus, the sought transformation is obtained by combining the above expression with Θ̇hb = TΘ (Θhb )ω bnb : ξ̇ = Jhb (Θhb , rbōh )ν, with Jhb (Θhb , rbōh )

h  Rb (Θhb ) Rhb (Θhb )S(rbōh )t . 03×3 TΘ (Θhb ) (3.34) (3.35) 3.4 Velocity Transformations 57 For the case of forward velocity, let us separate the body-fixed velocity into a slowly-varying component and an oscillatory component induced by the wave motion: ν = ν̄ + δν, (3.36) with ν̄ = [U, 0, 0, 0, 0, 0]t δν = [δu, δv, δw, δp, δq, δr]t (3.37) t = [δu, v, w, p, q, r] . Then the kinematic transformation between the b- and the h-frame for the case of forward speed is given by ξ̇ = Jhb (Θhb , rbōh )(ν − ν̄), (3.38) with Jhb (δΘ, rbōh ) given in (3.35). Note that for small angles, i.e. Θhb ≈ 0 small, the following approximations can be used: Rhb (Θhb ) ≈ I3×3 TΘ (Θhb ) ≈ I3×3 Jhb (Θhb , rbōh ) ≈ (3.39) H(rbōh ). If we expand the this kinematic transformation, consider small angles, assume a slender ship (so the DOF 1,3,5 can be decoupled from 2,4,6) and keep only the linear terms, we obtain b ξ˙1 ≈ δu + zoh δq b b ξ˙2 ≈ δv + xoh δr − zoh δp + U δψ ξ˙3 ≈ δw − xb δq − U δθ oh ξ˙4 = δp ξ˙5 = δq ξ˙6 = δr. (3.40) (3.41) (3.42) (3.43) (3.44) (3.45) The time derivative of (3.40)-(3.45) gives ˙ + z b δq ˙ ξ¨1 = δu oh ˙ + xb δr ˙ − z b δp ˙ + U δr ξ¨2 = δv oh oh b ˙ − x δq ˙ − U δq ξ¨3 = δw oh ˙ ξ¨4 = δp ˙ ξ¨5 = δq ˙ ξ¨6 = δr. (3.46) (3.47) (3.48) (3.49) (3.50) (3.51) 58 3 Kinematics of Ship Motion If we assume sinusoidal motions, the following relationships hold for yaw (and similarly for pitch): δψ = sin ωe t r = ωe cos ωe t ṙ = −ωe2 sin ωe t = −ωe2 δψ, (3.52) which can be used to express δψ = − 1 ṙ, ωe2 δθ = − 1 q̇. ωe2 (3.53) Substituting these in (3.40)-(3.45), we obtain b ξ˙1 ≈ δu + zoh δq (3.54) 1 ˙ b δp − U 2 δr ξ˙2 ≈ δv + xboh δr − zoh ωe 1 b ˙ ξ˙3 ≈ δw − xoh δq + U 2 δq ωe ξ˙4 = δp ξ˙5 = δq ξ˙6 = δr. (3.55) (3.56) (3.57) (3.58) (3.59) Expressions (3.54)-(3.59) and (3.46)-(3.51) represent a linear approximation of (3.38), and can be written in a compact form as follows: U ˙ Lδν ωe2 (3.60) ˙ + U Lδν, ξ̈ = Jhb δν (3.61) ξ̇ = Jhb δν − where Jhb
Jhb (0, rbōh ) = H(rbōh ), with H(rbōh ) given in (3.30), and ⎡ 0 ⎢0 ⎢ ⎢0 L
⎢ ⎢0 ⎢ ⎣0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎤ 0 0 0 1⎥ ⎥ −1 0 ⎥ ⎥. 0 0⎥ ⎥ 0 0⎦ 0 0 (3.62) (3.63)