u (surge) X o p (roll) Body-fixed r o v (sway) w (heave) Z o Earth-fixed X Y Z r (yaw) (pitch)

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1 Nonlinear Modelling of Marine Vehicles in Degrees of Freedom Thor I. Fossen and Ola-Erik Fjellstad The Norwegian Institute of Technology Department of Engineering Cybernetics N-0 Trondheim, NORWAY Journal of Mathematical Modelling of Systems, Vol. 1, No. 1, 199. Abstract In this paper a unied framework for vectorial parameterisation of inertia, Coriolis and centrifugal, and hydrodynamic added mass forces for marine vehicles in degrees of freedom (DOF) is presented. Emphasises is placed on representing the DOF nonlinear marine vehicle equations of motion in vector form satisfying certain matrix properties like symmetry, skew-symmetry and positive deniteness. For marine vehicles, this problem was rst addressed by Fossen [1]. The proposed representations in this paper are based on extensions of this work. The main results are presented as two theorems. The rst theorem is derived by applying Newton's equations whereas the second theorem reviews the main results of Sagatun and Fossen [] where the Lagrangian formalism is applied. Key words: Modelling, marine vehicles, equations of motion, Lagrangian and Newtonian mechanics. 1 Introduction In order to describe the motion of a marine vehicle in degrees of freedom (DOF), independent coordinates representing position and attitude are necessary. The dierent motion components are dened as: surge, sway, heave, roll, pitch and yaw (see Table 1). Hence the general motion of a marine vehicle in DOF can be described by the following vectors: = [ T 1 ; T ] T ; 1 = [x; y; z] T ; = [; ; ] T = [ T 1 ; T ] T ; 1 = [u; v; w] T ; = [p; q; r] T = [ T 1 ; T ] T ; 1 = [X; Y; Z] T ; = [K; M; N] T 1

2 Table 1: DOF motion components for a marine vehicle. Forces and Body-Fixed Inertial Position DOF Moments Velocities and Euler Angles 1 motions in the x-direction (surge) X u x motions in the y-direction (sway) Y v y motions in the z-direction (heave) Z w z rotation about the x-axis (roll) K p rotation about the y-axis (pitch) M q rotation about the z-axis (yaw) N r where denotes the earth-xed position and attitude vector, denotes the body-xed linear and angular velocity vector and is used to describe the forces and moments acting on the vehicle in the body-xed frame. The body-xed and inertial reference frames are shown in Figure 1. O CG Body-fixed p (roll) u (surge) X o Y o v (sway) q (pitch) Z o r (yaw) w (heave) r o Earth-fixed X Y Z Figure 1: Body-xed and earth-xed (inertial) reference frames. Newtonian Approach The motion of a rigid body with respect to a body-xed rotating reference frame X 0 Y 0 Z 0 with origin O is given by Newton's laws: m[ _ _ r G + ( r G )] = 1 (1)

3 I 0 _ + (I 0 ) + mr G ( _ ) = () where r G = [x G ; y G ; z G ] T is the centre of gravity and I 0 is the inertia tensor with respect to the origin O of the body-xed reference frame, that is: I 0 = I x?i xy?i xz?i yx I y?i yz?i zx?i zy I z ; I 0 = I T 0 > 0; _ I0 = 0 () Here I x ; I y and I z are the moments of inertia about the X 0 ; Y 0 and Z 0 axes and I xy = I yx ; I xz = I zx and I yz = I zy are the products of inertia. These equations can be expressed in a more compact form as: M RB _ + C RB () = RB () where = [u; v; w; p; q; r] T is the body-xed linear and angular velocity vector, RB = [X; Y; Z; K; M; N] T is a generalised vector of external forces and moments and M RB and C RB () will be referred to as the rigid-body inertia and, Coriolis and centrifugal matrices, respectively. Before the parameterisation of these matrices are discussed it is convenient to dene a skew-symmetric matrix operator. Denition 1 (Skew-Symmetric Matrix Operator) Let S() be a skew-symmetric matrix operator dened such that the vector cross product a b = S(a)b, that is: S(a) = 0?a a a 0?a 1?a a 1 0 ; S(a) =?S T (a) ().1 Rigid-Body Inertia Matrix The rigid-body inertia matrix M RB can be uniquely determined from (1) and (), that is: M RB _ = m _ 1 + m _ r G I 0 _ + mr G _ 1 From this expression the positive denite inertia matrix M RB = M T RB > 0 can be dened according to: () M RB = mi?ms(r G ) ms(r G ) I 0

4 = m mz G?my G 0 m 0?mz G 0 mx G 0 0 m my G?mx G 0 0?mz G my G I x?i xy?i xz mz G 0?mx G?I yx I y?i yz?my G mx G 0?I zx?i zy I z () where I is the identity matrix and I 0 is the inertia tensor with respect to O.. Rigid-Body Coriolis and Centripetal Matrix The Coriolis and centripetal matrix can be represented in numerous ways. Two skew-symmetrical representations motivated from Newton's laws are given by Theorem 1. Theorem 1 (Parameterisation of Coriolis and Centripetal Forces) i) The Coriolis and centripetal matrix can be parameterised according to: 0?mS( 1 )? ms( )S(r G )?ms( 1 ) + ms(r G )S( )?S(I 0 ) (8) such that?c T RB() is skew-symmetrical. Notice that the matrix S( )S(r G ) = [S(r G )S( )] T. Also notice that S( 1 ) 1 = 0. Equation (8) can be written in component form according to: ?m(y G q + z G r) m(y G p + w) m(z G p? v) m(x G q? w)?m(z G r + x G p) m(z G q + u) m(x G r + v) m(y G r? u)?m(x G p + y G q) m(y G q + z G r)?m(x G q? w)?m(x G r + v)?m(y G p + w) m(z G r + x G p)?m(y G r? u)?m(z G p? v)?m(z G q + u) m(x G p + y G q) 0?I yzq? I xzp + I zr I yzr + I xyp? I yq I yzq + I xzp? I zr 0?I xzr? I xyq + I xp?i yzr? I xyp + I yq I xzr + I xyq? I xp 0 (9) ii) An alternative skew-symmetric representation is: 0?mS( 1 )? ms(s( )r G )?ms( 1 )? ms(s( )r G ) ms(s( 1 )r G )? S(I 0 ) which can be written in component form according to: where 0 C 1 ()?C T 1() C () (10) (11)

5 C 1 () = C () = 0 m(w? qx G + py G ) m(?v? rx G + pz G ) m(?w + qx G? py G ) 0 m(u? ry G + qz G ) m(v + rx G? pz G ) m(?u + ry G? qz G ) 0 0 m(vx G? uy G? I xzp? I yzq + I zr) m(?vx G + uy G + I xzp + I yzq? I zr) 0 m(?wx G + uz G? I xyp + I yq? I yzr) m(?wy G + vz G? I xp + I xyq + I xzr) m(wx G? uz G + I xyp? I yq + I yzr) m(wy G? vz G + I xp? I xyq? I xzr) 0 (1) (1) Proof: The remaining terms in (1) and () after terms associated with _ 1 and _ are removed can be written: C RB () = By using the formulas: m[ 1 + ( r G )] (I 0 ) + mr G ( 1 ) (1) a (b c) = S(a)S(b)c (1) (a b) c = S(S(a)b)c (1) where S(a)S(b) = S(S(a)b) the rst line in (1) can be rewritten to obtain the results under i) and ii) in Theorem 1 by noticing that: i) ii) m[ 1 + ( r G )] =?m 1? m (r G ) =?ms( 1 )? ms( )S(r G ) (1) m[ 1 + ( r G )] =?m 1? m( r G ) =?ms( 1 )? ms(s( )r G ) (18) The second line in (1) can be rewritten in a similar manner. Moreover: i) mr G ( 1 ) = ms(r G )S( ) 1 (19) ii) Application of the Jacobi identity: a (b c) + b (c a) + c (a b) = 0 (0) to the vector term r G ( 1 ) yields:

6 Consequently: r G ( 1 ) =? ( 1 r G )? 1 (r G ) (1) mr G ( 1 ) =?m( r G ) 1 + m( 1 r G ) =?ms(s( )r G ) 1 + ms(s( 1 )r G ) () In addition to this?ms( 1 ) 1 = 0 and (I 0 ) =?S(I 0 ). Remark to Theorem 1 Two other skew-symmetric representations of C RB () are obtained by omitting the zero term S( 1 ) 1 = 0 and by noticing that S( 1 ) =?S( ) 1. Hence, the results from Theorem 1 can be reformulated according to: i) ii) Lagrangian Approach ms( )?ms( )S(r G ) ms(r G )S( )?S(I 0 ) ms( )?ms(s( )r G )?ms(s( )r G ) ms(s( 1 )r G )? S(I 0 ) () () In order to include the eects due to hydrodynamic added mass it is advantageous to use a body-xed energy formulation of Lagrange's equations since the vector R t 0 d with = [u; v; w; p; q; r]t has no immediate physical interpretation. Moreover, Kirchho's equations which are derived from the ordinary Lagrange equations [] will be applied for this purpose. Kirchho's equations in vector form are written []: d @ + = 1 = () where T is the kinetic energy of the ambient water or the vehicle. Theorem (Coriolis and Centripetal Matrix from Inertia Matrix) Let M be an inertia matrix dened as:

7 M = M 11 M 1 M 1 M () Hence, the Coriolis and centripetal matrix can always be parameterised according to: C() = 0?S(M M 1 )?S(M M 1 )?S(M M ) () where C() =?C T (). The results of Theorem 1 can be derived from this expression by letting M = M RB. Proof: The proof follows Sagatun and Fossen [] where the kinetic energy T is written as a quadratic form: Expanding this expression yields: Hence: T = 1 T M (8) T = 1 (T 1 M T 1 M 1 + T M T M ) = M M = M M (1) From Kirchho's equations () it is recognised that: @ + = @ ) 1 which after substitution of (0) and (1) proves ()..1 Application to the Ambient Water and Vehicle Systems We will now use Theorem to derive the rigid-body and ambient water Coriolis and centripetal matrices. In order to do this we will dene an added inertia matrix []:

8 M A =? X _u X _v X _w X _p X _q X _r Y _u Y _v Y _w Y _p Y _q Y _r Z _u Z _v Z _w Z _p Z _q Z _r K _u K _v K _w K _p K _q K _r M _u M _v M _w M _p M _q M _r N _u N _v N _w N _p N _q N _r () The notation of SNAME [] is used in this expression; for instance the hydrodynamic added mass force Y A along the y-axis due to an acceleration _u in the x-direction is written as: Y A = Y _u _u where Y _u _u For control applications we can assume that M A > 0 is constant and strictly positive. This is based on the assumption that M A is independent of the wave frequency which is a good approximation for low-frequency control applications. For certain control applications like operation of underwater vehicles at great depth and positioning of surface ships we can also assume that M A is symmetrical. However, this is not true for a surface ship moving at some speed []. For notational simplicity we will write: M A = A 11 A 1 A 1 A M RB = m I?m S(r G ) m S(r G ) I 0 () for hydrodynamic added inertia and rigid-body inertia, respectively. Notice that the hydrodynamic added inertia and rigid-body inertia matrices are the same both in the Newtonian and Lagrangian approaches. Rigid-Body Coriolis and Centripetal Matrix Theorem with M RB dened in () can now be used to derive the following expression for the rigid-body Coriolis and centripetal matrix: 0?mS( 1 )? ms(s( )r G )?ms( 1 )? ms(s( )r G ) ms(s( 1 )r G )? S(I 0 ) () This is equivalent to case ii) of Theorem 1. Case i) can also be derived from this expression. Hydrodynamic Added Coriolis and Centripetal Matrix In addition to the rigid-body Coriolis and centripetal matrix, an expression for the hydrodynamic added Coriolis and centripetal matrix can be derived. Moreover, Theorem with M A dened in () yields: 8

9 C A () = 0?S(A A 1 )?S(A A 1 )?S(A A ) which can be written in component form according to: where C A () = ?a a a 0?a ?a a 1 0 0?a a 0?b b a 0?a 1 b 0?b 1?a a 1 0?b b 1 0 () () a 1 = X _u u + X _v v + X _w w + X _p p + X _q q + X _r r a = X _v u + Y _v v + Y _w w + Y _p p + Y _q q + Y _r r a = X _w u + Y _w v + Z _w w + Z _p p + Z _q q + Z _r r b 1 = X _p u + Y _p v + Z _p w + K _p p + K _q q + K _r r b = X _q u + Y _q v + Z _q w + K _q p + M _q q + M _r r b = X _r u + Y _r v + Z _r w + K _r p + M _r q + N _r r An attractive simplication of this expression is to assume that: (8) M A =?diagf X _u ; Y _v ; Z _w ; K _p ; M _q ; N _r g (9) C A () = ?Z _ww Y _vv Z _ww 0?X _uu 0 0 0?Y _vv X _uu 0 0?Z _ww Y _vv 0?N _rr M _qq Z _ww 0?X _uu N _rr 0?K _pp?y _vv X _uu 0?M _qq K _pp 0 (0) Equations of Motion Assume that the external forces and moments RB in () is written: RB =? M A _? C A ()? D()? g() + {z } {z } {z } {z} added mass damping restoring forces control input (1) Hence, the marine vehicle equations of motion can be expressed according to: where M _ + C() + D() + g() = () M = M RB + M A ; C() = C RB () + C A () () 9

10 The matrix D() can be interpreted as a non-symmetrical hydrodynamic damping matrix while g() is included to describe buoyant and gravitational forces. Notice that these equations satisfy: _M = 0; M = M T > 0; C() =?C T (); D() > 0 () For robot manipulators it is common to transform the desired state trajectory to joint space coordinates by applying the inverse kinematics. Hence the kinematic transformation matrix can be avoided in the control scheme. This is not possible for a marine vehicle since R t 0 d has no physical interpretation. Hence, the kinematic equations of motion must be included in the control design if position and attitude control are of interest. For this purpose it is common to apply an Euler angle representation (xyz-convention): where J() = diagfj 1 (); J ()g and: J 1 () = J () = _ = J() () c c?s c + c ss s s + c cs s c c c + sss?c s + ss c?s cs cc 1 st ct 0 c?s 0 s=c c=c () () This representation is undened for a pitch angle of 90 o (deg). The above system satises: where T C() = 0; _ T [ _ M? C (; )] _ = 0 (8) M () = J?T M J?1 ; C (; ) = J?T (C? M J?1 _ J)J?1 (9) which implies that the theory of passive adaptive control can be applied to design a stable control system. Conclusions In this paper methods for parameterisation of nonlinear Coriolis, centripetal and inertia forces for marine vehicles in terms of matrix properties like symmetry, skew-symmetry and positiveness have been discussed. Emphasises have been placed on deriving a model of the ambient water of the vehicle by considering the eects of hydrodynamic added inertia. The resulting model is written in vector form. Marine vehicle control design based on Lyapunov stability analyses for the proposed model structure is discussed in [] and references therein. 10

11 References 1. Fossen, T. I.: Nonlinear Modelling and Control of Underwater Vehicles, Dr. Ing. Dissertation, Dept. of Engineering Cybernetics, The Norwegian Institute of Technology, Trondheim, Norway, June Fossen, T. I.: Guidance and Control of Ocean Vehicles, John Wiley and Sons Ltd., Kirchho, G.: Uber die Bewegung eines Rotationskorpers in einer Flussigkeit, Crelle's Journal, No. 1, 189, pp. { (in German).. Lamb, H.: Hydrodynamics, Cambridge University Press, London, 19.. Meirovitch, L. and Kwak M. K.: State Equations for a Spacecraft With Flexible Appendages in Terms of Quasi-Coordinates, Applied Mechanics Reviews, 1989, pp. {8.. Sagatun, S. I. and Fossen, T.: Lagrangian Formulation of Underwater Vehicles' Dynamics, Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, Charlottesville, VA, pp. 109{10, SNAME: The Society of Naval Architects and Marine Engineers. Nomenclature for Treating the Motion of a Submerged Body Through a Fluid, Technical and Research Bulletin No. 1{,