Journal of Ship Research, Vol. 38, No. 2, June 1994, pp. 123-132

Cross Track Error and Proportional Turning Rate Guidance ofMarine Vehicles

Fotis A. Papoulias

The problem of turning rate guidance and control of marine vehicles is considered. Feedback with feed-forward rudder control is used to deliver a specified turning rate for the vehicle, while a guidance lawis employed to create the necessary sequence of turning rate commands which would allow conver-gence to a desired geographical path. Two different guidance schemes are presented and analyzed,namely, cross track error and proportional turning rate guidance. Stability conditions are computed ex-plicitly, while nonlinear analysis techniques illustrate the significance of design parameters on the finalsystem response that cannot be inferred from linearized stability results.

Introduction

SMALL UNMANNED marine vehicles suitable for use in bothnaval and commercial operations have unique mission re-quirements and dynamic response characteristics. In partic-ular, they are required to be highly maneuverable and veryresponsive as they operate in obstacle-avoidance and object-recognition scenarios. The need, therefore, arises to main-tain accurate path-keeping in confined spaces and shallowwaters under the influence of steady- and time-varying ex-ternal forces. The primary vehicle guidance system is basedon heading or turning rate commands that are generatedbased on a specified geographical sequence of desired waypoints. Speed commands can be generated by incorporatingtemporal attributes to the way points. These guidance com-mands are then passed to the vehicle controller which at-tempts to deliver the commanded heading and/or headingrate of change by an appropriate use of the vehicle controlsurfaces (Healey et al 1990). Unlike open sea operations, forvehicle missions in coastal areas and confined waters, theway point sequence must be very dense so that satisfactorypath accuracy is maintained. One efficient way of maneu-vering through a given way point sequence is by using aline-of-sight guidance law which commands a heading anglethat is directly related to the line-of-sight angle between thevehicle position and a desired destination point. The vehiclecontroller is then an orientation control law which deliversthe commanded heading. Previous studies (Papoulias 1991,1992), have demonstrated that this scheme is guaranteedstable only if the way point separation is above some criticalvalue. This conclusion is true regardless of the particularform of the line-of-sight guidance or the heading control lawused. Similar results hold for vertical plane guidance (Pa-poulias 19921, although additional instabilities are possiblehere due to the existence of the metacentric height. In thiswork we analyze the turning rate guidance and control prob-lem in the horizontal plane, where the guidance law de-mands a specific yaw rate response from the controller. Alinear state feedback with a feedforward term (Friedland1986) control law is used, while two different guidanceschemes are considered. The first, a cross track error guid-ance, is very popular in land-based robotic applications

Assistant professor, Department of Mechanical Engineering, NavalPostgraduate School, Monterey, California.

Manuscript received at SNAME headquarters September 23, 1992; re-vised manuscript received February 24, 1993.

(Kanayama et al 1990), and the second, a proportional guid-ance law, is predominantly used in aerospace applicationsand interception/evasion problems (Brainin & McGhee 1968).Stability analysis is performed and bifurcation theory tech-niques (Hassard & Wan 1978, Guckenheimer & Holmes 19831are utilized in order to assess the dynamics of the systemupon initial loss of stability. All computations are performedfor the Naval Postgraduate School autonomous test-bed ve-hicle for which a complete set of geometric properties andhydrodynamic characteristics is available (Bahrke 19921.Unless otherwise mentioned, all results are presented instandard dimensionless form with respect to the vehicle lengthP = 2.3 m and nominal forward speed u = 0.6 m/s, whichcorresponds to a Froude number of 0.13. At these conditions,the vehicle is very maneuverable due to a total of four rud-der surfaces with a maximum turning rate of about 9 degper second and a turning radius of less than two vehiclelengths.

1. Problem formulation

In this section we present the vehicle equations of motionin the horizontal plane. The control law is based on the dy-namic equations in sway and yaw, whereas guidance isachieved through the use of the kinematic relations.

Equations of motionRestricting our attention to the horizontal plane, the

mathematical model consists of the nonlinear sway and yawequations of motion. In a moving coordinate frame fixed atthe vehicles geometrical center (see Fig. 11, the maneuver-ing equations of motion are

mid + ur + xCf) = Y,r + Y,d + Y,ur + Y,uv + Y&3

-0.5pCDJh(S)(u + Sr)lu + SrldS (1)

I,? + mxG(ti t ur) = N,f + N,ti t N,ur t N,uu + N,&%-0.5pColh(S)(u + Sr)lu + 5r15 d5 (2)

where u is the vehicle forward speed, v and rare the relativesway and yaw velocities of the moving vehicle with respectto the water, and the rest of the symbols are explained inthe Nomenclature. Equations (1) and (2) can be written astwo first-order decoupled equations in the form

d = aiiuu t q2ur + b,uS + d,(u, r-1 (3)

I: = azluu t aY2ur + b2u26 t d,(u, r-1 (4)

JUNE 1994 0022-4502/94/3802-0123$00.45/O JOURNALOFSHlPRESEARCH 123

Y7v u

+

~

Y

*

dX

Fig. 1 Vehicle geometry and definitions of symbols

where the coefficients ad, bj are functions of the hydrody-namic derivatives, geometric properties, and rudder coeff-cients. The terms d&u, r) and d,(u, r) represent the contri-butions from the quadratic drag terms in (1) and (2). In theabove form, the equations of motion are valid for both smalland large drift angles. Drag related terms are relatively smallfor regular cruising operations, u + (u + xr), and the vehicleresponse is, therefore, predominantly linear. For a vehicleoperating near hover, u 4 (v + xr), the quadratic drag forcesdominate the response. The surge velocity u is clearly af-fected during the turn due to the added drag in turning. Forthe purposes of this study it is assumed to be constant. Thisis a valid approximation since experimental experience hasshown that the propulsion control law is, in general, capableof keeping the forward speed relatively constant at the com-manded value (Bahrke 19921.

Feedback controlA linear rudder feedback control law based on the linear-

ized set of equations (3) and (41,

has the form

d = alluu + a12ur + blu6

r = azluu + az2ur + b2u2S

(5)

(6)

6 = k,,u + k,r (7)

where k,, k, are the feedback gains. By substituting (7) into(5) and (6) we can find the closed loop characteristic equa-tion

where

X2 + Aih + A, = 0 (8)

Ai = -[all + uz2 + (blk, + b2k,)ulu, and

AZ = [a11a22 - u12az1 + (b,a22 - b2a&k,+ (bzall - bla21)uk,lu2.

If the desired characteristic equation is

x2 + ci,h + IX.2 = 0 (9)

we can equate the coefficients of (8) and (9) and get the fol-lowing system of linear equations

k,.b1u2 + k,b& = -aI ~ (aI1 + a22)u

k,(a22b1 - a12b2h3 + k,hbz - a21bl)u3= a2 ~ (a11a22 - a12a21V

to be solved for the gains k, and k,.The coefficients ol, o2 of the desired characteristic equa-

tion (9) can be specified according to standard second-ordersystem transient response specifications (Friedland 19921. In

Nomenclature

a, = open loop state coefficientsin U, r model

A = linearized system matrixb, = open loop rudder coeffr-

cients in u, r modelCo = drag coefficient

d = proportional guidance lawpreview distance

I, = vehicle mass moment of in-ertia

T = matrix of eigenvectors of AT,, = zeroth-order approximation

of limit cycle periodTc = control law time constantT, = guidance law time constant

u = vehicle forward speedu = sway velocity

uO = ratio of steady-state swayvelocity to steady-stateturning rate

X = cubic stability coefficient x = state variables vectork,, k, = control law feedback gains xc = body-fixed coordinate of ve-

k, or k, = control law feedforward gain hicle center of gravitykJr, k, = cross track error guidance y = deviation off commanded

gains pathk,,, ku = proportional guidance gains

m = vehicle massN = yaw moment

N, = derivative of N with respectto a

Y = sway forceY, = derivative of Y with respect

to az = state variables vector in ca-

nonical formPAH = Poincare-Andronov-Hopf bi-

furcationzl, .zp = critical variables of zz3, z4 = stable coordinates of z

r = yaw rater, = commanded yaw rateR = polar coordinate of trans-

formed reduced systemt = time

Greek symbols

(Y, = coefftcients of desired con-trol characteristic equa-tion

124 JUNE 1994

01 = derivative of o with respectto d evaluated at d,,,t

pL = coefficients of desired guid-ance characteristic equa-tion

6 = rudder angle6,., = saturation level of rudder

angleE = difference between a bifur-

cation parameter and itscritical value

8 = polar coordinate of trans-formed reduced system

I) = vehicle heading angleCT = line-of-sight angle

0, = positive imaginary part ofcritical pair of eigenval-ues evaluated at criticalpoint

w = derivative of w with respectto bifurcation parameterevaluated at critical value

JOURNALOFSHIPRESEARCH

this work we use the controller time constant, T,, as theparameter. Then the desired characteristic equation is

( !A+;2

C

=0 orX+$X+$=OC C

and comparing with (9) we see that

(10)

Specification of a controller time constant Tc then deter-mines the feedback gains k,, k, uniquely.

Feedforward control

The control law (7) guarantees stability of u = r = 0 of (5)and (6), in other words, straight line motion at an arbitraryheading. When the commanded angular velocity r, is non-zero the control law is slightly modified to

6 = K,u + Iz,(r - rJ + k,r, (11)

where k, is the feedforward gain. The feedback gains k,, k,remain the same since the drag terms d&u, r), d,(u, r) aresmall and, therefore, the linearized dynamics of (3) and (4)around r, do not differ significantly from (5) and (6). Thefeedforward gain k, is computed based on steady-state ac-curacy requirements. At steady state, equations (5) and (6)yield

ha22 - ha12U=

ball - ha21r,, 6 = a2lal2 - mh2

(b2aIl - bla2du(12)

Substituting (12) into (11) and requiring that r = r, at steadystate we can solve for k, and finally write the control law(11) in the form

Analogously to the control law design, if the time constantof the guidance law is selected to be Tc, then (20) results in

where

6 = k,u + k,r - kOaZrc (13)k,= -$ k,= -A

G G

1k,, =

(bpall - b,azlh3(14)

With the above feedforward gain the control law is complete.It should be mentioned that all gains k,, k,, k, depend ex-plicitly on the forward speed u and are, therefore, continu-ously updated every time a different forward speed is com-manded.

Selection of TG then determines k, and k, uniquely.Although this development followed the small angle ap-

nroximation sin $ = I), it is not difficult to see that negativevalues of k, and k, guarantee stability of the nonlinear SYS-tern (15) and (16). The associated total energy of the systemis

E(,j,, ,J,) = f $ - k,u(l - cos $1

The feedforward gain k. computed from (14) ensures thatthe steady-state turning rate r equals the commanded valuer, for the linear system (5) and (6). In general, we can seefrom (3) and (4) that at steady state r # r, unless d, = d, =0. As the controller time constant Tc is decreased, the con-trol law becomes tighter and the steady-state error Ir - r,lwill be smaller. In practice, the above steady-state error couldnot be made zero due to uncertainties in the vehicle hydro-dynamic description and other unmodeled dynamics. One wayto ensure steady-state accuracy in r would be to abandon theuse of the feedforward gain k, and to introduce integral con-trol. This approach is not favored since it results, in general,in oscillatory transient response (Friedland 1992). The otheralternative is to use a time varying r, such that convergenceto a specified geographical path is achieved. This is accom-plished through the introduction of the guidance law pre-sented in the following sections.

which can be viewed as the sum of kinetic and potential en-ergy. Using (15) and (17) this is written as

~(9, y) = f (k,+ + k,yY - k,u(l - ~0s 4~)

We note that E(+, y) provides a Lyapunov function candidatefor (15) and (16) since E(0, 0) = 0 at the unique equilibrium(+, yl = (0, 01 and E(I), y) > 0 for (9, y) f (0, 01, because k,< 0. Moreover, we have

&=!$+$.t$

= l(k,+ + k&k, - k,u sin 91(k,$ + Iz,y)

Cross track error guidance

+ (k,$ + k,y) k,u sin IJI

= k6(kb$ + kg,

In order to achieve path control to a commanded route inthe horizontal plane, the commanded turning rate r, must

which, since k, < 0, is negative semi-definite. Therefore,Lyapunovs theorem guarantees stability of the nonlinear

be appropriately selected. This constitutes the guidance law system (15) and (16) (Guckenheimer & Holmes 19831.

design. Without loss in generality we can assume that thecommanded path is a straight line. This is not a very re-strictive assumption since every smooth path can be discre-tized into a series of straight-line segments as accurately asdesired.

The guidance law is based solely on kinematics, whereasvehicle dynamics are handled by the rudder control law.Guidance law development is therefore based on

$ = r, (15)

3 = u sin * (161

where r, is the commanded turning rate and the lateral ve-locity v is assumed to be zero in (16). Cross track error guid-ance is achieved by

r, = k+J, + k,y (17)

The closed loop characteristic equation of (15), (161, and (17)is

x2 - k,h - k,u = 0 (18)

If the desired characteristic equation is

A2 + p,x + pz = 0 (191

the guidance law gains k,, k, are obtained by equating thecoefficients of (18) and (19)

JUNE 1994 JOURNAL OF SHIP RESEARCH 125

Proportional guidance

Proportional guidance with an integral term (Brainin &McGhee 1968) is achieved by

r, = ki,ti + k,(o - *) (22)

In this fashion, the commanded turning rate attempts to closein on the difference between the vehicle heading + and theline-of-sight angle cr. The additional term which is propor-tional to the line-of-sight rate of change ir adds damping incases where o changes rapidly in time such as obstacle-avoidance, object-recognition, or terrain-following tasks. Theline-of-sight angle is defined as the angle between the ve-hicle longitudinal axis and a target point located ahead ofthe vehicle on the nominal path at a constant preview dis-tance d, as shown in Fig. 1. For the straight-line nominalpath case we have

The proportional guidance characteristic equation is ob-tained from (151, (161, (221, and (23) as

(24)

and by comparing coefficients of (19) and (24) we get

k = !@ 0, and (33)

D>O (34)

Explicit evaluation of conditions (33) and (34) results in

Tc