Ch3

1

Chapter 3 Rigid-Body Kinetics

In order to derive the marine craft equations of motion, it is necessary to study of the motion of rigid bodies, hydrodynamics and hydrostatics. The overall goal of Chapter 3 is to show that the rigid-body kinetics can be expressed in a vectorial setting according to: MRB Rigid-body mass matrix CRB Rigid-body Coriolis and centripetal matrix due to the rotation of {b} about {n} = [u,v,w,p,q,r]T generalized velocity expressed in {b} RB = [X,Y,Z,K,M,N]

T generalized forces expressed in {b}

3.1 NewtonEuler Equations of Motion about CG 3.2 NewtonEuler Equations of Motion about CO 3.3 Rigid-Body Equations of Motion

Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

MRB CRB RB #

2

The equations of motion will be represented in two body-fixed reference points: 1) Center of gravity (CG), subscript g 2) Origin ob of {b}, subscript b These points coincides if the vector Time differentiation of a vector in a moving reference frame {b} satisfies Time differentiation in {b} is denoted as

Chapter 3 Rigid-Body Kinetics

rg 0

CO CG

inertial frame

rb/i

rg/i

rg{b}

{i}={n}

id

dta

bd

dta b/i a

a :bd

dta

3

Coordinate-free vector: A vector , velocity of {b} with respect to {n}, is defined by its magnitude and direction but without reference to a coordinate frame. Coordinate vector: A vector decomposed in the inertial reference frame is denoted by

Newton-Euler Formulation Newton's Second Law relates mass m, acceleration and force according to:

where the subscript g denotes the center of gravity (CG). Euler's First and Second Axioms Euler suggested to express Newton's Second Law in terms of conservation of both linear momentum and angular momentum according to:

and are forces/moments about CG is the angular velocity of frame b relative frame i Ig is the inertia dyadic about the body's CG

3.1 Newton-Euler Equations of Motion

about CG

fg

pg hg

fg m g

vb/ni

v g/i

mv g/i fg

vb/n

vb/n

iddt

pg fg pg mvg/iiddt

hg m g hg Ig b/i

#

# b/i

Isaac Newton (1642-1726)

Leonhard Euler (1707-1783)


4

When deriving the equations of motion it will be assumed that:

(1) The vessel is rigid (2) The NED frame is inertialthat is, {n} {i}

The first assumption eliminates the consideration of forces acting between individual elements of mass while the second eliminates forces due to the Earth's motion relative to a star-fixed inertial reference system such that:

For guidance and navigation applications in space it is usual to use a star-fixed reference

frame or a reference frame rotating with the Earth. Marine crafts are, on the other hand, usually related to the NED reference frame. This is a good assumption since forces on a marine craft due to the Earth's rotation:

are quite small compared to the hydrodynamic forces.

3.1.1 Translational Motion about CG

e/i 7.2921 105 rad/s

vg/i vg/n

b/i b/n


5

CO CG

inertial frame

rb/i

rg/i

rg{b}

{i}={n}

3.1.1 Translational Motion about CG

Body-fixed reference frame {b} is fixed in the point CO and rotating with respect to the inertial frame {i}

Time differentiation of in a moving reference frame {b} gives For a rigid body, CG satisfies

bddt

rg 0 #

Translational Motion about CG Expressed in {b}

rg/i rb/i rg # rg/n rb/n rg # {n} is inertial

rg/n

vg/n vb/n bddt

rg b/n rg #

vg/n vb/n b/n rg #

fg iddt

mvg/i

iddt

mvg/n

bddt

mvg/n m b/n mvg/n

mv g/n b/n vg/n # mv g/nb Sb/n

b vg/nb fg

b #


6

The derivation starts with the Eulers 2nd axiom:

where Ig is the inertia matrix

where Ix, Iy, and Iz are the moments of inertia about {b} and Ixy=Iyx, Ixz=Izx and Iyz=Izy are the products of inertia defined as:

3.1.2 Rotational Motion about CG

Ix Vy2 z2 mdV; Ixy

Vxy mdV

Vyx mdV Iyx

Iy Vx2 z2 mdV; Ixz

Vxz mdV

Vzx mdV Izx

Iz Vx2 y2 mdV; Iyz

Vyz mdV

Vzy mdV Izy

Ig :

Ix Ixy IxzIyx Iy IyzIzx Izy Iz

, Ig Ig 0 #

Rotational Motion about CG Expressed in {b} m g iddt

Ig b/i

iddt

Ig b/n

bddt

Ig b/n b/n Ig b/n

Ig b/n Ig b/n b/n #

Ig b/nb SIgb/n

b b/nb mgb


7

The Newton-Euler equations can be represented in matrix form according to:

3.1.3 Equations of Motion about CG

MRBCG

v g/nb

b/nb

CRBCG

vg/nb

b/nb

fg

b

mgb

#

mI33 033

033 Ig

MRBCG

v g/nb

b/nb

mSb/n

b 033

033 SIgb/nb

CRBCG

vg/nb

b/nb

fg

b

mgb

#

Expanding the matrices give:

Isaac Newton (1642-1726) Leonhard Euler (1707-1783)


8

For marine craft it is desirable to derive the equations of motion for an arbitrary origin CO to take advantage of the craft's geometric properties. Since the hydrodynamic forces and moments often are computed in CO, Newton's laws will be formulated in CO as well. Transform the equations of motion from CG to CO using a coordinate transformation based on: Transformation matrix:


about CO

vg/nb vb/n

b b/nb rgb

vb/nb rgb b/n

b

vb/nb Srgbb/n

b #

vg/nb

b/nb

Hrgbvb/n

b

b/nb

#

Hrgb :I33 S

rgb

033 I33, Hrgb

I33 033

Srgb I33 #


9

Expanding the matrices


about CO

Newton-Euler equations in matrix form (about CG)

Newton-Euler equations in matrix form (about CO)

MRBCG

v g/nb

b/nb

CRBCG

vg/nb

b/nb

fg

b

mgb

#

mI33 033

033 Ig

MRBCG

v g/nb

b/nb

mSb/n

b 033

033 SIgb/nb

CRBCG

vg/nb

b/nb

fg

b

mgb

#

HrgbMRBCGHrgbv b/n

b

b/nb

HrgbCRBCGHrgbvb/n

b

b/nb

Hrgbfg

b

mgb

#


vg/nb

b/nb

Hrgbvb/n

b

b/nb

#

10

Expanding the matrices


about CO

HrgbMRBCGHrgbv b/n

b

b/nb

HrgbCRBCGHrgbvb/n

b

b/nb

Hrgbfg

b

mgb

#

MRBCO : HrgbMRBCGHrgb

CRBCO : HrgbCRBCGHrgb

#

#


MRBCO

mI33 mSrgb

mSrgb Ig mS2rgb

CRBCO

mSb/nb mSb/n

b Srgb

mSrgbSb/nb SIg mS2rgbb/n

b

#

#

The mass and Coriolis-centripetal matrices in CO are defined as:

11

3.2.1 Translational Motion about CO

Translational Motion about CO Expressed in {b}

mv b/nb S b/n

b rgb Sb/nb vb/n

b S2b/nb rgb fb

b #


An alternative representation using vector cross products is:

mv b/nb b/n

b rgb b/nb vb/n

b b/nb b/n

b rgb fbb #

Copyright Bjarne Stenberg/NTNU

12

Theorem 3.1 (Parallel Axes or Huygens-Steiner Theorem)

The inertia matrix Ib about an arbitrary origin ob is given by:

where Ig is the inertia matrix about the body's center of gravity.

3.2.2 Rotational Motion about CO

Rotational Motion about CO Expressed in {b}

Ig mSrgbSrgb Ig mS2rgb

Ib #

Ib b/nb Sb/n

b Ibb/nb mSrgbv b/n

b mSrgbSb/nb vb/n

b mbb #


Ib b/nb b/n

b Ibb/nb mrgb v b/n

b b/nb vb/n

b mbb #

An alternative representation using vector cross products is:

Christian Huygens (1629-1695) Jakob Steiner (1796-1863)

13

3.3 Rigid-Body Equations of Motion

mu vr wq xgq2 r2 ygpq r zgpr q X

mv wp ur ygr2 p2 zgqr p xgqp r Y

mw uq vp zgp2 q2 xgrp q ygrq p Z

Ixp Iz Iyqr r pqIxz r2 q2Iyz pr q Ixy

mygw uq vp zgv wp ur K

Iyq Ix Izrp p qrIxy p2 r2Izx qp rIyz

mzgu vr wq xgw uq vp M

Izr Iy Ixpq q rpIyz q2 p2Ixy rq p Izx

mxgv wp ur ygu vr wq N

Component form (SNAME 1950)

fbb X,Y,Z - force through ob expressed in b

mbb K,M,N - moment about ob expressed in b

vb/nb u,v,w - linear velocity of ob relative on expressed in b

b/nb p,q,r - angular velocity of b relative to n expressed in b

rgb xg,yg,zg - vector from ob to CG expressed in b


14

3.3.1 Nonlinear 6-DOF Rigid-Body

Equations of Motion Matrix-Vector Form (Fossen 1991)

Property 3.1 (Rigid-Body System Inertia Matrix)

MRB CRB RB u, v, w, p, q, r

MRB mI33 mSrgb

mSrgb Io

m 0 0 0 mzg myg

0 m 0 mzg 0 mxg

0 0 m myg mxg 0

0 mzg myg Ix Ixy Ixz

mzg 0 mxg Iyx Iy Iyz

myg mxg 0 Izx Izy Iz

#

MRB MRB 0, MRB 066

I33 is the identity matrix is the inertia matrix about CO is the matrix cross product operator

Srgb

generalized velocity

Ib Ib 0


15

Theorem 3.2 (Coriolis-Centripetal Matrix from System Inertia Matrix)

Let M be a 66 system inertia matrix defined as:

where M21= M12T. Then the Coriolis-centripetal matrix can always be parameterized such that

where

Proof: Sagatun and Fossen (1991).


Equations of Motion

M M M11 M12

M21 M22 0

C C

C 033 SM111 M122

SM111 M122 SM211 M222

1 u, v, w, 2 p, q, r


16

Property 3.2 (Rigid-Body Coriolis and Centripetal Matrix)

The rigid-body Coriolis and centripetal matrix can always be represented such

that is skew-symmetric, that is


Equations of Motion

CRB

CRB

CRB CRB , 6


The skew-symmetric property is very useful when designing nonlinear motion control system since the quadratic form TCRB() 0. This is exploited in energy-based designs where Lyapunov functions play a key role. The same property is also used in nonlinear observer design. There exists several parameterizations that satisfies Property 3.2. Two of them are presented on the forthcoming pages:

17

Lagrangian Parameterization:

Application of the Theorem 3.2 with M = MRB yields the following expression:

which can be rewritten according to (Fossen and Fjellstad 1995):

In order to ensure that CRB() = -CRB()T, it is necessary to use S() = 0

and include S() in CRB{21}.


Equations of Motion

CRB 033 mS1 mSS2rgb

mS1 mSS2rgb mSS1rgb SIo2


CRB 033 mS1 mS2Srgb

mS1 mSrgbS2 SIb2 #

Joseph-Louis Lagrange (1736-1813)

18

Linear Velocity Independent Parameterization: By using the cross-product property S() = -S(), it is possible to move S() from CRB

{12} to CRB{11}. This gives an expression for CRB() that is independent of linear velocity

(Fossen and Fjellstad 1995): Remark 1: This expression is useful when ocean currents enter the equations of motion. The main reason for this is that CRB() does not depend on linear velocity . This can be further exploited when considering a marine craft exposed to irrotational ocean currents. According to Property 8.1 in Section 8.3: if the relative velocity vector r = -c is defined such that only linear ocean current velocities are used:


Equations of Motion


CRB mS2 mS2Srgb

mSrgbS2 SIb2 #

MRB CRB MRB r CRBrr #

: uc,vc,wc,0,0,0 #

19

3.3.2 Linearized 6-DOF Rigid-Body

Equations of Motion


The nonlinear rigid-body equations of motion can be linearized about = [U,0,0,0,0,0]T for a marine craft moving at forward speed U. The linearized Coriolis and centripetal forces are recognized as:

MRB CRB RB #

CRB MRBLU #

L

0 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 1 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

#

fc CRB

0

mUr

mUq

mygUq mzgUr

mxgUq

mxgUr

#

MRB CRB RB #

20

(1) Origin CO coincides with the CG

This implies that such that

A further simplification is obtained when the body axes (xb,yb,zb) coincide with the principal axes of inertia. This implies that

Simplified 6-DOF Rigid-Body Equations of Motion

3.3.2 Linearized 6 DOF Rigid-Body

Equations of Motion

rgb 0,0,0, Ib Ig

MRB mI33 033

033 Ig #

Ig diagIxcg,Iycg,Izcg


21

(2) Translation of the origin CO such that Ib becomes diagonal: The body-fixed frame (xb,yb,zb) can be chosen such that:

If Ig is a full matrix, the parallel-axes theorem:

gives

where xg, yg and zg must be chosen such that:

3.3.2 Linearized 6-DOF Rigid-Body

Equations of Motion

Ib diagIx,Iy,Iz

Ib Ig mS2rgb Ig mrgbrgb rgbrgb I33 #

Ix Ixcg myg2 zg2

Iy Iycg mxg2 zg2

Iz Izcg mxg2 yg2

#

mIyzcg

xg2 IxycgIxzcg

mIxzcg

yg2 IxycgIyzcg

mIxycg

zg2 IxzcgIyzcg

#


Ch3

Documents

Transcript of Ch3