Trajectory Optimization Strategies for Supercavitating Underwater Vehicles

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DOI: 10.1177/1077546307076899

2008 14: 611Journal of Vibration and ControlM. Ruzzene, R. Kamada, C.L. Bottasso and F. Scorcelletti

Trajectory Optimization Strategies for Supercavitating Underwater Vehicles

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Trajectory Optimization Strategies forSupercavitating Underwater Vehicles

M. RUZZENER. KAMADASchool of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Drive, Atlanta, GA30332-0150, USA ([email protected])

C.L. BOTTASSOF. SCORCELLETTIDipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa 34, Milano 20156,Italy

(Received 14 March 2005� accepted 6 December 2006)

Abstract: Supercavitating vehicles are characterized by substantially reduced hydrodynamic drag, in com-parison with fully wetted underwater vehicles. Drag is localized at the nose of the vehicle, where a cavitatorgenerates a cavity that completely envelopes the body, at the fins, and on the vehicle after-body. This uniqueloading configuration, the complex and non-linear nature of the interaction forces between vehicle and cavity,the unsteady behavior of the cavity itself and memory effects associated with its formation process make thecontrol and maneuvering of supercavitating vehicles particularly challenging. This study presents an initialeffort towards the evaluation of optimal trajectories for this class of underwater vehicles. Flight trajectoriesand maneuvering strategies for supercavitating vehicles are obtained through the solution of an optimal con-trol problem. Given a cost function, and general constraints and bounds on states and controls, the solution ofthe optimal control problem yields control time histories that maneuver the vehicle according to the desiredstrategy, together with the associated flight path. The optimal control problem is solved using the direct tran-scription method, which does not require the derivation of the equations of optimal control and leads to thesolution of a discrete parameter optimization problem. Examples of maneuvers and resulting trajectories aregiven to demonstrate the effectiveness of the proposed methodology and the generality of the formulation.

Keywords: Supercavitating vehicles, trajectory optimization, optimal control, flight mechanics.

1. INTRODUCTION

When a body moves through water at sufficient speed, the fluid pressure drops, locally, be-low the level that sustains the liquid phase, and a low-density gaseous cavity forms. Flowsexhibiting cavities entirely enveloping the moving body are called supercavitating. In super-cavitating flows, the liquid phase does not contact the moving body over most of its length,and the skin friction drag is thus almost negligible. Several new and projected supercavitatingunderwater vehicles exploit supercavitation as a means to achieve extremely high submerged

Journal of Vibration and Control, 14(5): 611–644, 2008 DOI: 10.1177/1077546307076899

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612 M. RUZZENE ET AL.

speeds and low drag (Miller, 1995). Existing and notional supercavitating high-speed bodiesrange in size from small projectiles to full-scale heavy-weight torpedoes. Recent researchefforts have lead to important advances in the comprehension, modeling and simulation ofsupercavitating vehicles. The development of reliable and effective theoretical and compu-tational procedures is particularly important for the future design of such vehicles. Recentstudies have examined a variety of computational methods for investigating supercavitatingflows, including boundary element methods (BEM) and first-principle simulation proceduresbased on state-of-the-art computational fluid dynamics (Kirschner et al., 2001� Stinebring etal., 2001). Using these methods, important parameters, such as the cavity shape, drag andbody-cavity and fin-water-cavity interactional forces, can be computed.

In parallel with these efforts, flight mechanics simulators have been developed to allowfor the initial assessment of the flight dynamics characteristics of this class of vehicles. Thecontrol and maneuvering of supercavitating vehicles presents unique challenges associatedwith their distinctive operating conditions. Such challenges are related to the location ofthe hydrodynamic force generated by the cavitator with respect to the center of gravity, thecomplex and non-linear nature of the interactional forces between vehicle and cavity, thedynamic nature of the cavity itself, the need for roll control during curved flight (Kirschneret al., 2002), and the presence of hull vibrations induced by after-body planing, surfing andtail-slapping.

This paper describes an ongoing effort dedicated to the problem of guidance, maneu-vering control and design of supercavitating vehicles. Achieving aggressive maneuveringcapability and maximum steady flight performance is a primary goal in the design of such ve-hicles. In this work we address the problem of trajectory optimization as a first step towardsthese goals. The functionalities provided by the methodology described in the followingpages will later on be used for:

1. Investigating vehicle performance in the maneuvering flight regime,2. optimizing vehicle configurational parameters in the design phase, and3. the actual guidance of the vehicle, wherein a real-time controller tracks the optimal tra-

jectory and regulates and stabilizes the vehicle against perturbations.

With regard to this last point, we note that trajectories generated a priori, for exampleby interpolation of way-points, are typically not compatible with the vehicle dynamics. Forhigh performance vehicles such as the ones considered here, this can imply that the real-time controller might be unable to effectively realize the desired trajectory, or that the flightmay result in excessive loads or high frequency oscillations and repeated impact interactionsbetween the vehicle and the cavity, which are all clearly undesirable effects.

In the following, we first describe a flight mechanics model of the vehicle. Next, weformulate a maneuver as an optimal control problem, where a suitable cost function is min-imized while satisfying the vehicle equations of motion and other conditions on the vehiclecontrols and states, as required by the problem at hand. The optimal control problem is thensolved using a direct transcription approach, by discretizing the governing equations in thetime domain. Finally, we test the proposed methodology using some representative exampleproblems.

The results presented here are based on simplified models of the cavity shape, hydro-dynamic coefficients and vehicle/cavity interactions. These models were chosen based ontheir availability in the open literature and on their computational convenience within the op-



SUPERCAVITATING UNDERWATER VEHICLES 613

Figure 1. Configuration of supercavitating torpedo, with reference frames and interactional forces andmoments with corresponding reference points.

timization framework. The proposed approach is however a general one, which can accom-modate newer and more sophisticated models as needed. The example problems discussedin the present paper are therefore intended to demonstrate the capabilities of the proposedtrajectory optimization approach, and are not intended to provide absolute information re-garding specific maneuvers.

2. FLIGHT MECHANIC MODEL OF A SUPERCAVITATING VEHICLE

A 6 degrees-of-freedom rigid body model is used to describe the dynamic behavior of thevehicle. A schematic of the vehicle configuration and of the applied forces is shown inFigure 1. The body is acted upon by a system of forces corresponding to the interaction ofthe vehicle control surfaces with the cavity boundaries. The control surfaces include the finsat the back of the vehicle and the cavitator, whose primary function is the generation of thesupercavity. The control surfaces support the vehicle in the vertical direction by providinglift, and allow for roll, pitch and yaw control. Finally, the vehicle motion is sustained bya propulsion force directed along the body axis. The equations of motion are convenientlyformulated in a body-fixed reference frame �P��, with origin P and triad � � �b1� b2� b3�.A reference inertial frame �O�� is centered at point O and has a triad of unit vectors � ��e1� e2� e3�, as shown in Figure 1.

2.1. Equations of Motion

The equations of balance of linear and angular momentum (Euler’s equations) of the vehicle,using components expressed in terms of the body-attached frame, can be written as




�l� � �� l� � s� (1a)

�h�P � v�P � l� � �� h�P � m�P (1b)

where the linear momentum is l � mvP �STP�� and the angular momentum is given by hP �

SPvP � JP��. �V is the vehicle’s density, m � �V �V dV is the mass of the vehicle, SP ��

V �V r�dV is the first moment of inertia, JP � ��

V �V r�r�dV is the inertia dyadic, vP and�� denote the linear velocity of point P and the angular velocity of the body, respectively,and s and mP are the resultants of the applied forces and moments, respectively. Here, andin what follows, the notation �� denotes components in the generic � triad. If R is therotation tensor that brings triad � into triad �, then the components of a generic vector ain the two triads are related as a� � Ra�. Furthermore, a� is the skew-symmetric tensorassociated with a. Finally, the symbol �� d ��dt indicates a derivative with respect to time.

Equations (1a) and (1b) can be conveniently rewritten in the compact form

M� �w� � w�� M�w� � f� (2)

where the generalized inertia tensor is defined as

M ��

mI STP

SP JP

�(3)

and the generalized velocity w and generalized force f are defined (respectively) as

w � �vTP� ��

T �T � f � �sT �mTP�

T � (4)

In equation (2), �� is the South-West cross product operator (Bottasso et al., 2001):

w�� 0

vP� ��

�� (5)

The position and orientation of the vehicle with respect to the inertial frame can beexpressed through the position vector rP � �P � O� and a set of rotation parameters (morespecifically quaternions, for this work). The vehicle kinematic equations can be written as

�d ��

R 0

0 E

�w� (6)

where d is the vector of the generalized coordinates defined as

d � �rTP� q

T4 �

T (7)




with q4 � �q0� qT �T denoting the quaternion vector. With reference to equation (7), Erelates the time rates of the rotation parameters to the body-frame components of the angularvelocity and is defined as

E � 1

2

� �qT

q0I� q�

�� (8)

Finally, the unit quaternion condition is expressed as

qT4 q4 � 1 (9)

The forces s acting on the vehicle can be written as

s � sT � sN �nF�i�1

sFi � sI � sG (10)

where sT � �T b1 is the propulsive thrust, sN is the hydrodynamic force at the vehicle nosegenerated by the cavitator, sFi is the hydrodynamic force generated by the i th of the nF fins,sI are the contact forces due to the interaction of the vehicle with the cavity, and sG � �mge3

is the gravitational force. Similarly, the moments mP can be written as

mP � rPT � sT � rP N � sN �nF�i�1

�mFi � rP Fi � sFi ��mI � rP I � sI � rPG � sG (11)

where rAB indicates a distance vector from point A to point B, T is the point of applicationof the thrust, N is the cavitator location, Fi is a reference point on the i th fin, I is the tail-cavity contact point, mFi is the hydrodynamic moment generated by the i th fin, mI is theinteractional moment, and G is the center of gravity.

2.2. Cavity Shape and Dimensions

The behavior of the cavity affects the forces at the nose of the vehicle, the immersion of thefins in the fluid, and the contact forces between the vehicle and the cavity boundary. In thiswork, we consider an approximate, simplified (Munzer-Reichardt) model for the cavity assummarized by Kirshner et al. (2002) and May (1975). The model estimates cavity length,maximum diameter and shape for an assigned diameter of a circular, flat cavitator disk. Thenominal axisymmetric shape of the cavity is computed as

dc�� dmax

�1�

� � lc�2

lc�2

�2�1�2�4

(12)

where dc�� is the cavity diameter at distance from the cavitator along the cavity axis, whiledmax and lc are the maximum diameter and the length of the cavity, respectively, which can be




expressed (according to Garabedian (1956), as summarized by Kirschner et al., 2002� May,1975) as

dmax � dN

CD� � 0�

(13a)

lc � dN

CD� � 0�

2ln

�1

�� (13b)

In equations (13a), and (13b), dN is the cavitator diameter, CD� � 0� is the cavitator dragcoefficient at zero angle of attack and is the cavitation number, defined as

� p � pc

1�2��2N

(14)

where p� pc are respectively the ambient and cavity pressures, �N is the cavitator speed,and � is the water density.

The expression of the cavity shape defined by equation (12) ignores distortions due toturning, gravity and buoyancy. It is also assumed that the cavity axis t is aligned with thevelocity vector of the cavitator vN . Furthermore, it is worth observing that the model doesnot include any of the cavity dynamics and related memory effects mentioned by Kirschneret al. (2002). A more complex model, similar to the one presented by Dzielski and Kurdila(2003), which originally dates back to 1972, is described in Bottasso et al. (2008). Cavitymodels with memory effects lead to delay differential equations (DDEs), and a frameworkfor the solution of optimal control problems for systems governed by DDEs detailed in thesame bibliographical reference.

2.3. Cavitator Force Model

The force generated on the cavitator as a result of its interaction with water can be usedto assist in controlling the vehicle by orienting the cavitator at an appropriate angle. Theachievement of an optimal orientation of the cavity with respect to the vehicle during turningmaneuvers requires this angle to be controllable.

The hydrodynamic forces acting on a circular cavitator can be conveniently expressedin terms of a reference frame �N�� located at the cavitator center N and with triad of unitvectors � � �n1�n2�n3�, as shown in Figure 2. Unit vector n1 is perpendicular to the disksurface. Its orientation with respect to the vehicle axis b1 is defined by the control angle �N ,so that the components of n1 in the body-fixed triad �, labelled n�1 , are

n�1 � �cos �N � 0�� sin �N �T � (15)

Unit vector n2 is orthogonal to the plane formed by the pair of vectors vN and n1, i.e.,

n2 � vN � n1

vN � n1 (16)




Figure 2. Detail of cavitator with relevant quantities and reference frames.

where vN � vP��rP N is the cavitator velocity, rP N being the vector distance between thereference point P on the vehicle and the cavitator center N . Finally, unit vector n3 completesa right handed triad: n3 � n1 � n2.

The components of the three unit vectors n1, n2 and n3, measured in the body-attachedtriad �, readily give the components in � of the rotation tensor R�� that rotates the � triadinto the� triad:

R�� n�1 �n�2 �n�3

�� (17)

Hence, if v�N denotes the components of the cavitator velocity in the � triad, the componentsof the same vector in the cavitator triad� are

v�N � R�T�� v�N � �u�N � 0� ��N �T � (18)

The cavitator angle of attack N is measured in the vN , n1 plane (see Figure 2), and iscomputed as

tan N � ��Nu�N

� (19)

In the vN , n1 plane, the hydrodynamic force acting on the cavitator can be decomposedinto lift and drag components, which can be computed (Kirschner et al., 2002) as

L N � 1

2��2

N AN CD� � 0� sin N cos N (20a)

DN � 1

2��2

N AN CD� � 0� cos2 N (20b)

where AN is the cavitator area. The hydrodynamic force can hence be expressed in the �triad as




Figure 3. Detail of fin with relevant quantities.

s�N � �L N sin N � DN cos N � 0� L N cos N � DN sin N

�T(21)

and transformed to the � triad as s�N � R�� s�N . This formulation of the cavitator force ne-glects the effects of hydrodynamic added mass and damping, which are discussed by Uhlmanet al., (2001).

2.4. Fin Force Model

The fins are controlled to provide lift in the after-body section and to maneuver the vehicle.We consider a generic nF -fin configuration. Each fin interacts with the surrounding fluidwith forces that depend on the immersion depth in the fluid, the velocity at the fin locationwith respect to the fluid, the fin geometry and the angle of attack.

For convenience, the forces are first expressed in a reference frame�Fi ��i , with origin Fi

and triad �i � �f 1� f 2� f 3� fixed to the i th fin, as shown in Figure 3(a). Triad �i is obtainedby a rotation that first brings � into the undeflected fin configuration �i � � f 1� f 2� f 3�,




f k � Ri bk , k � 1� 2� 3, followed by a rotation with rotation vector �Fi f that accounts for

the fin deflection �Fi , f k � R��Fi f� f k , k � 1� 2� 3. The total rotation from � to �i is hence

f k � R��i bk , R��i � R��Fi f � Ri .

In the fin-fixed reference system, forces are determined in terms of the angle of at-tack and the immersion depth, using previously published results for wedge-shaped fins(Kirschner et al., 2002). The i th fin force and moment components in �i (i � 1� � � � � nF )are given by

s�iFi

� 1

2��2

F�i Sfin

Cx�� F�i �

�dFi ��Cy�� F�i ��dFi ��Cz�� F�i �

�dFi ��T� (22)

m�iFi

� 1

2��2

F�iSfinL fin

Cmx�� F�i �

�dFi ��Cmy�� F�i ��dFi ��Cmz�� F�i �

�dFi ��T� (23)

The fin force components are transformed from the �i to the body-fixed triad � as s�Fi�

R��is�i

Fi, as are the moment components. The latter are, however, typically negligible,

and hence were not considered in the present study. The magnitude of the velocity vectorcomputed at point F�

i on the fin is denoted by �F�i , Lfin is the fin reference length (span),Sfin � L2

fin is the fin reference surface, and Cx �Cy�Cz are force coefficients defined in termsof the fin angle of attack � F�i and of the non-dimensional penetration distance �dFi � dFi �Lfin,dFi being the fin immersion depth.

The fin angle of attack in the local �i triad is obtained from the components of thevelocity v�i

F�i � ��x � �y� �z�T . Accordingly, we have

� F�i � tan�1

��z

�x

�� (24)

This formulation for the fin forces neglects the effects of the hydrodynamic added mass anddamping. Furthermore, in maneuvering flight characterized by non-zero roll and yaw rates,the velocity of the fluid with respect to the fin, and hence the angle of attack, will vary alongthe fin span. Hence, the use of the reference point F�

i , here chosen to be the mid pointalong the wetted fin span, implies another approximation, although this approximation isacceptable for the typical small spans of the torpedo fins.

The fin immersion depth dFi is computed as follows, based on the configuration shownin Figure 3(b). Q denotes the point on the fin span where the fin pierces the cavity boundary,the position of which is computed as

rQ � rFi � � f 2 (25)

where � defines the unknown non-submerged length of the fin. The projection of point Qonto the cavity centerline is point R, whose position vector can written as

rR � rC � � t (26)




Figure 4. Fin force coefficients versus angle of attack, for varying penetration depth.




Figure 4. Fin force coefficients versus angle of attack, for varying penetration depth. (Continued)

where t � �vN�vN is the axis vector of the Munzer-Reichardt cavity, pointing from thecavitator to the tail of the torpedo. The unknown location � of point R along the cavity axiscan be obtained by imposing the condition

rRQ � t � 0 (27)

where

rRQ � rC Fi � � f 2 � �t (28)

which gives � � �� f 2�rC Fi �� t. Consider now the cavity radius corresponding to the locationof the i th fin, which is denoted by rc � dc� Fi

��2 (see equation (12)), Fibeing the projected

distance between each fin root and the cavitator along the cavity axis, i.e., Fi� rC Fi � t.

Substituting � into equation (28) and imposing the condition that rRQ2 � r2c , one obtains

a quadratic scalar equation which can be solved for � . Finally, the non-dimensional finimmersion can then be calculated as �dFi � 1� ��L�n.

Plots of the force coefficients are shown in Figure 5. Their approximately bilinear behav-ior for assigned penetration depth is associated with two different flow regimes developingon the fin. The first flow regime occurs for low angles of attack, when two separate cavitiesare formed, at the base and leading edge of the fin. For larger angles of attack, the two cav-ities merge to form a supercavity that envelopes all the surfaces except for the pressure face(Kirschner et al., 2002).




Figure 5. Detail of planing cross section.

2.5. Vehicle/Cavity Interactions

Supercavitating vehicles operate over a wide speed range (50–900 m/s), and experienceinteractions of the cavity with the after-body. Such interactions can be divided into twobasic types: Tail-slapping and planing. Planing usually occurs at speeds of the order of50–200 m/s, while tail-slap motion is observed at 300–900 m/s. During tail-slap motion,the vehicle undergoes an oscillatory motion with periodic impacts against the cavity, whileduring planing the vehicle is in steady contact with the internal surface of the cavity. Thecorresponding planing interaction force may provide sufficient lift to counteract the vehicleweight and may thus stabilize the motion.

Hassan (2006) presents a theory which describes forces and moments experienced bya cylindrical body steadily planing on flat and cylindrical free surfaces. The model is ingood agreement with experimental data, and is intended for application to supercavitatingvehicles. The formulation extends the theory based on the work of Logvinovich (1972,1980) for inviscid flows by adding the skin friction force induced by fluid viscosity. Themodel considers a planing slender body which is animated by steady forward motion on anundisturbed free surface, under the assumption of large Froude numbers and a low ratio ofimmersion depth to body radius. The vertical force on a cylindrical foil planing along anundisturbed planar free surface can be expressed (Logvinovic, 1972, 1980) as

P � m� �h � �M �h� (29)

The impact mass m� and apparent added mass M associated with the non-holonomic dynam-ics of the spray sheet are related (Logvinovic, 1980� Hassan, 2006) as

�m� � 2�1� k�

2k � 1�M (30)




where k is defined as k � Vs��2 �h sin��, where Vs denotes the average spray sheet velocity,while � is the angle of Vs with respect to the horizontal surface, as shown in Figure 5.Hassan’s model considers the steady planing force associated with the apparent added massin equation (29) assuming a constant immersion rate ( �h � 0). Accordingly, the specific forceon the planar section is expressed as

P � �M �h � �h2 �M

�h� (31)

The planing force on the immersed portion of the body can be obtained by integratingthe force per unit length P (equation (31)) over the planar section of the wetted surfacemeasured along the longitudinal axis. Similarly, the planing moment about the center of thecross section at the trailing edge of the immersed body is readily obtained.

To this end, consider the reference frame �I�� , which has its origin at the center of thebody-cavity interaction section, I , and the triad of unit vectors � � �i1� i2� i3�. The unitvectors are defined as i1 � b1, i2 � b1 � t, and i3 � i1 � i2. Clearly, i2 and hence i3 areundefined when b1 is parallel to t, but in this case the two bodies are certainly not in contact.From this basis, the planing force and moment can be expressed as

sp � �� l p

0

�V��

�2�M

�hdx i3 (32a)

mp �� l p

0x

�V��

�2�M

�hdx i2 (32b)

where V� is the component of the velocity of the generic section along i3, and l p is the lengthof the wetted region along the body centerline.

The transport parameter � is defined (Hassan, 2006) as

� � 1� tan p

2�c

�m�

�x(33)

where p is the angle of attack between the longitudinal axis of the body and the free surface,and c is the lateral width of the spray sheet as shown in Figure 5.

Assuming a small wetted portion of a cylindrical after-body planing on a cylindrical freesurface, the planing forces and moment are (Hassan, 2006� Kirschner et al., 2002)

sp � ��r2c �

2e sin e cos e

r � h p

r � 2h p

�1�

��

�� h p

�2�

i3 (34a)

mp � ��r2c �

2e cos2 e

r � h p

r � 2h p

h2p

�� h pi2 (34b)

where � � rc � R is the difference between the cavity radius at the tail rc � dc� I ��2(equation (12)) and the body radius R, with I � rC I � t. Furthermore, h p � R � �rc �




b�� cos p is the planing immersion depth at the impact section, with b ��rC I2 � 2

I .The effective fluid speed in the i1, i3 plane at the impact section is defined as

�e ��

u2I � �2

I (35)

where the components of the impact section velocity vI � vG � �� rG I in the � triad aredenoted by v�I � �uI � � I � � I �

T . Finally, the effective angle of attack is defined as

e � tan�1

�� I

u I

�� (36)

The drag and moment due to viscous effects can be written as

sp f � �1

2��2

e cos2 eCdp S� i1 (37a)

mp f � �1

2��2

e cos2 eCdp S�m i2� (37b)

where Cdp is the drag coefficient over a flat surface, which for a smooth plate can be esti-mated as

Cdp � 0�031

�Re�1�7(38)

where Re is the Reynolds number based on the wetted longitudinal length, Re � �el p��k , �k

denoting the kinematic viscosity.The wetted area S� and wetted area including moment arm, S�m , are defined as

S� �� l p

02r��x� dx (39a)

S�m �� l p

0

� ��x�

02r 2 cos � d� dx �

� L p

02r2 sin��x� dx (39b)

where � and ��x� are the circumferential and maximum angle, respectively, along the wettedplanar surface section shown in Figure 5, and x and r denote (respectively) the longitudinalcoordinate and the cylindrical body radius.

Finally, the total cavity vehicle interaction force sI and moment mII with respect to pointI can be computed as

sI � sp � sp f (40a)

mII � mp �mp f � (40b)

The latter is readily transported to reference point P and added to equation (11) to yield thetotal applied moment on the vehicle, mP .




3. TRAJECTORY OPTIMIZATION

3.1. Overview

In this work we are interested in computing maneuvers of supercavitating vehicles. In thecontext of this paper, computing a maneuver means determining the time histories of thevehicle controls and the associated time histories of vehicle states. Any computed maneuvermust always satisfy a certain number of requirements, as detailed below.

First, maneuvers must be compatible with the vehicle dynamics, i.e., they must satisfythe equations of motion within the admissible limits imposed by the vehicle flight envelopeand the necessarily limited control authority of the vehicle actuators. Clearly, this require-ment is also relevant to the guidance and navigation problem. In fact, guiding a supercav-itating vehicle along a compatible maneuver is evidently more easily accomplished thanwhen trajectories are specified upfront, for example through spline interpolation of givenway-points. Such strategies may in fact easily result in unfeasible trajectories, especially foraggressive and high performance maneuvering.

Second, maneuvers should if possible be optimal in some sense, i.e., they should min-imize some cost function, such as the time necessary to accomplish a given goal, or thecontrol effort necessary to steer the vehicle, or maximize the final vehicle velocity. In fact,optimality provides a way to select one meaningful solution among the typically infinitepossible different ways of achieving a same goal.

Finally, maneuvers must satisfy possible operational constraints imposed by the vehicleuser, resulting from a requirement to satisfy safety, cost, effectiveness and other needs.

3.2. The Maneuver Optimal Control Problem

All the above-mentioned requirements can be met by expressing each maneuver as the solu-tion of an appropriate optimal control problem (Bryson and Ho, 1975). In order to formulatethe problem, and also describe the solution technique used for solving it, we will now in-troduce additional notation. The problem domain is here denoted by � � �0� T � � �, andits boundary is � � �0� T �, t � �, where the final time T may be unknown. The dynamicequations of a rigid supercavitating vehicle introduced in the previous section (equations (2)and (7)) are for convenience rewritten in compact form as

�y� z �y�u� � 0 (41)

where y � �ny denotes the vehicles states, y � �vP� �� rP� q4�T , ny � 13, while the controls

u � �nu are u � ��T � �N � � � � � �Fi � � � ��T , nu � 2� nF , and include the propulsion force �T ,

the cavitator angle �N , and the fin deflections �Fi , (i � 1� � � � � nF ).The optimal vehicle state time histories yopt�t� and associated control policy uopt�t�

define an optimal maneuver and minimize the cost function

J � l�y�u� t��

T��

L�y�u� t� dt (42)




where the first term is the terminal cost, while the second is the integral term of the costfunction.

As previously stated, the optimal solution must satisfy the vehicle equations of motion(equation (41)), which can therefore be interpreted as constraints of the optimization prob-lem. Constraints on the states and the controls further characterize and define the maneuver,for example by providing initial and final conditions, or by providing operational and flightenvelope limits. For purposes of generality, all these conditions can be expressed as inequal-ity constraints of the form x � [xmin� xmax], i.e. xmin � x � xmax. Equality constraints areenforced simply by selecting xmin � xmax. The initial and terminal state conditions can bewritten

��y�0�� [��0min� ��0max

] (43a)

��y�T �� [��Tmin� ��Tmax

] (43b)

while non-linear constraints on states and controls can be expressed in general as

g�y�u� t� � [gmin� gmax] (44)

similarly, constraints at a (possibly unknown) internal event Ti are

g�y�u� Ti� � [gTimin� gTimax

] (45)

integral conditions on states and controls can be given as��

h�y�u� t� dt � [hmin�hmax] (46)

and, finally, the upper and lower bounds are

y � [ymin� ymax] (47a)

u � [umin�umax]� (47b)

According to optimal control theory, an optimal solution to this problem is determinedby the following process. First, an augmented performance index, obtained by adjoiningthe system governing equations (41) and constraints (43a–46) to the performance index (42)through the use of Lagrange multipliers (co-states), must be defined. Next, the stationarity ofthe augmented index is imposed, resulting in the definition of a set of differential equationsin the states, co-states and controls, together with a set of associated boundary conditions(Bryson and Ho, 1975).

3.3. Numerical Solution

This approach is however not always either necessary or convenient. In fact, one can avoidthe derivation of the optimal control equations altogether, using a very simple and effective




method (Betts, 2001). First, one discretizes the system equations (41) on a grid �h of thecomputational domain using some numerical discretization method. This defines a set ofunknown parameters, which are represented by the discrete values of the states and controlson the computational grid, here denoted by x � �nx . At this point, the problem cost func-tion (42) and the boundary conditions and constraints (43a–46) are expressed in terms ofthe discrete parameters x. This process defines a finite-dimensional non-linear programming(NLP) problem, which can be written as

minx

K �x�

s.t.: : c�x� � [cmin� cmax](48)

where K : �nx � � is the discrete counterpart of the cost J of equation (42), whilec : �nx � �nc are the optimization constraints, which include the discretized system dy-namic equations, the discretized constraints and the boundary conditions. Here, again, nec-essary conditions for a constrained optimum are obtained, similarly to the optimal controlcase, by combining the objective K with the constraints through the use of Lagrange mul-tipliers, and imposing the stationarity of the augmented cost function. The resulting largebut sparse problem can be solved efficiently by sequential quadratic programming (SQP)methods (Barclay et al., 1997) or interior point (IP) methods (Renegar, 2001).

The discretization of the equations of motion can, in principle, be based on any validnumerical method. For example, we have used both a finite element method in the temporaldomain (Bottasso et al., 2003), and a non-linearly unconditionally stable energy preservingmethod (Bottasso and Croce, 2003) in previous work. Many other valid choices are clearlypossible. In this work we use the mid-point rule, which yields a second order solution, forreasons of simplicity. This method can also be interpreted as the lowest order member of thediscontinuous Petrov-Galerkin finite elements of our previous work (Bottasso et al., 2002),which provide a method for possible inclusion of higher order solvers in future versions ofthe numerical procedures.

To introduce the discretization of the equations, we consider a grid �h of�. In particular,we let the partition 0 � t0 � t1 � � � � � tn�1 � tn � T be composed of n � 1 intervals T i �[ti � ti�1] of size hi , i � 0� � � � � n�1. Since T is in general unknown, we introduce a mappingof time onto a fixed domain parameter s, i.e. s : �0� T � �� 0� 1�, s � t�T , s � [0� 1]. Thisyields the generic time step length as hi � T �si�1 � si�, i � 0� � � � � n � 1, which is nowexpressed in terms of the step length in the s space and of the unknown maneuver duration.

The discretized system dynamics equations can be written on the generic interval T i as

yi�1 � yi � hi z�

yi � yi�1

2�ui

�� 0� i � 1� � � � � n � 1 (49)

where yi and yi�1 are the values of the states at times ti and ti�1, respectively, and ui is theconstant value of the controls within T i . Note that, coherently with their algebraic nature,controls are treated as internal unknowns, which reflects the fact that no boundary conditionscan be associated with these variables.

Given the discretization of the equations expressed by (49), the NLP variables x aredefined as




Figure 6. Fin configuration and fin numbering system (view from behind the vehicle looking forward).

x ��

yTi �i � 0� � � � � n��ui T

�i � 0� � � � � n � 1�� T�T

(50)

i.e., they include the state values at the grid vertices, the control values on each grid elementand, possibly, the final time. The cost function and all problem constraints and bounds,including equations (49), are expressed in terms of the NLP variables x to yield the finitedimensional optimization problem (48).

4. NUMERICAL EXAMPLES AND APPLICATIONS

In this section, the proposed procedures are tested on some representative maneuvers.The vehicle configuration used for these simulations reflects projected designs for a

supercavitating torpedo. The vehicle has a length of 4.0 m and a diameter of 0.2 m, whilethe cavitator is a circular disc with a diameter of 0.07 m. We consider the cruciform finarrangement shown in Figure 6� fins are numbered as shown in the figure, which gives aview from behind the vehicle looking forward. Fins 1 and 3 are oriented parallel to the pitchaxis of the vehicle (b2)� being used only as elevators, they always deflect by the same amountas each other. Fins 2 and 4, on the other hand, are independent, and hence act both as ruddersand for the roll control of the vehicle. The fins, located 3.5 m aft of the cavitator, have a spanof 0.25 m and zero sweep angle, and feature a symmetric wedge shape which provides goodstrength characteristics as well as a transition between partial cavitation and supercavitationconfined to a very limited range of deflection angles (Kirschner et al., 2002).

The controls, u, include the three independent fin angles �F1 � ��F3 , �F2 and �F4 , thecavitator angle �N , and the thrust �T . Additional controls can be added in future studies, andin particular the possibility of thrust vectoring and additional actuation degrees of freedom




Table 1. Torpedo configuration.

Total vehicle length 4.0 mNominal vehicle diameter 0.2 mCavitator diameter 0.07 mTotal vehicle mass 150 kgFin span 0.25 mFin location (aft of nose) 3.5 m

on the cavitator can be considered as viable possibilities for improving the maneuverability ofthe vehicle. The total mass of the torpedo is 150 kg and is considered constant during flight.Extension to reflect mass reduction associated with fuel consumption and expulsion of ven-tilation gases can be considered as part of future developments. The torpedo configuration issummarized in Table 1.

All the maneuvers presented below have been computed assuming a fixed cavitationnumber, which has been properly selected to ensure that the cavity fully envelops the bodyat all times. This assumption relies on the hypothesis that the cavity pressure can be adjustedto compensate for variations in the vehicle velocity and so maintain the desired value of .This could, in principle, be achieved through proper cavity ventilation and control of thecavity pressure. In general, however, it is still not clear whether fine control of the cavitypressure can in fact be achieved, and what supporting technology might be needed. An alter-native perspective may envision that the maneuvers considered here occur with only smallvariations of the cavitation number, and hence that the selected constant values representthe corresponding average value of . A more refined formulation should include cavitydynamics equations and additional information regarding the level of control available overthe cavitation pressure. Such models can be accommodated in the present formulation, withrelatively minor modifications.

The optimization cost function used in the following studies is

J � T 2 � �� T

0�u � �u dt (51)

where T denotes the total unknown time required to perform a desired maneuver, whilethe second term of the cost includes the control velocities �u. This second contribution isintroduced here to limit the control rates and the tendency to obtain a “bang-bang” typesolution. � is a user adjustable weighting parameter used for scaling the contribution of thetwo terms of the cost.

4.1. Dive Maneuvers

We consider a vehicle initially flying at trim conditions at a horizontal velocity of 85 m/s,and we wish to find an optimal trajectory that involves diving to an assigned depth where thevehicle returns to operation in the initial trimmed state. The cavitation number is set to 0�02for all the maneuvers presented in this and the following examples.




Figure 7. Family of dives for increasing final depth.

Optimal trajectories for a final depth ranging from 5 m to 50 m are shown in Figure 7.These results were obtained by using as a starting guess for the first dive the steady state andcontrol values corresponding to the trimmed initial condition. The trajectory resulting fromthe first optimization is then used as a starting guess for the next dive, thus applying a contin-uation technique. These results are obtained for � � 1�100, which essentially describes theobjective as performing the prescribed dives in minimum time. The temporal domain cor-responding to the maneuver is discretized using a uniform grid consisting of 80 time steps.The optimization is performed by imposing constraints on the controls and on their rates ofchange. The results of Figure 7 are obtained with the thrust constrained to vary between 0and �Tmax � 33� 000 N, which is an arbitrary upper limit imposed by the generic propulsionsystem considered in this study.

Minimum time maneuvers have also been evaluated with the thrust fixed at �T � 0�5�Tmax .A comparison of the maneuver durations obtained with free and fixed thrust is shown in Fig-ure 8, which, as expected, shows how controlling the thrust allows for faster dives.

Figure 9 presents the time histories of the controls corresponding to the free thrust, min-imum time maneuvers. The plots show how the thrust and cavitator angle vary in an approx-imately linear fashion between their minimum and maximum values. Such linear variationsare the result of the control rates reaching the corresponding imposed bounds. The maximumlevel of thrust appears to be required during part of the 50 m dive, as shown in Figure 9(a).

It is worth noting that all the controls return to their initial value as the maneuver iscompleted and the vehicle reaches the designated trim state. The time histories of the ruddercontrols are omitted in this case, as they both remain equal to zero throughout.

Figure 10 shows the time histories of the horizontal and vertical velocity components inthe body-fixed frame, together with the pitch rate.




Figure 8. Diving times for fixed and free thrust.

Finally, Figure 11 compares optimal 50 m dives obtained for increasing values of thetunable weight parameter �. As expected, these results indicate how an increase in the weightgiven to the control rates in the cost function corresponds to longer times needed to completethe maneuver, as indicated by the greater longitudinal distances required for the achievementof the prescribed depth.

4.2. Turn Maneuvers

Figure 12 shows a family of turns for heading changes varying from 10 to 180 degrees.The vehicle is initially at trim, flying with a horizontal velocity of 85 m/s, and is requiredto achieve a trimmed state at the same velocity after the specified change in heading. Themaneuvers in Figure 12 are minimum time turns obtained by considering the cost functionof equation (51) with � � 1�100.

The temporal domain was discretized with a uniform grid of 100 elements. Here, again,the solution for each value of heading change was used as the initial guess for the nextproblem, at a larger turn angle. It is interesting to observe how all of the resulting trajectoriesare three dimensional, and are somewhat different from the simple level turn in the xy plane




Figure 9. Control time histories for the diving problem.




Figure 9. Control time histories for the diving problem. (Continued)

Figure 10. State time histories for the diving problem.




Figure 10. State time histories for the diving problem. (Continued)




Figure 11. Optimal 50 m dives for varying optimization weight �.

Figure 12. Family of turns for heading change of between 10 and 180 degrees.

which one might expect. The vehicle Euler angles (in the 3-2-1 sequence) shown in Figure 13for some representative turns confirm that significant roll and pitch occur during these ma-neuvers. The Euler angles of the figure were computed from the quaternions used for theparametrization of rotations in the formulation.

It is also important to note that the vehicle does not come into contact with the cavityin any of these maneuvers. Figure 14 shows the time history of the angle p between thevehicle and cavity axes during three representative turns. In all cases, the angle remainsmuch lower than the critical angle corresponding to the contact of the vehicle after-bodywith the cavity, indicated in the figure by a horizontal dotted line. This result is dependenton the cavity model considered, which is rather crude and does not include delay effects.However, it is fully expected that similar results, where contacts between vehicle and cavityare avoided, may also be found with a more sophisticated cavity model. The lack of contactmay in fact be dictated by the requirements imposed by the selected cost function, since




Figure 13. Time histories of Euler angles during representative turns.




Figure 13. Time histories of Euler angles during representative turns. (Continued)

Figure 14. Time history of angle of attack p between vehicle and cavity axes during representativeturns.




Figure 15. Schematic of trajectory tracking maneuver.

cavity/vehicle contacts cause an increase in drag and a resulting need for a thrust increaseas well as an increase in the control rates in response to the contact itself. As an alternative,specific constraints can be introduced in the optimization in order to enforce the non-contactcondition as an operational requirement.

The development of an improved cavity model based on Logvinovich’s independenceprinciple (Dzielski and Kurdila, 2003) including time delay cavity advection effects, andthe evaluation of the advantages and disadvantages of maneuvering with the vehicle planingon the cavity surface are underway, and will be documented in future developments of thisstudy.

4.3. Target Tracking

Our final set of examples considers the problem of tracking a target moving along an assignedtrajectory. Specifically, optimal maneuvers are found for a target travelling at a constantlongitudinal velocity of 12 m/s, and for the torpedo initially travelling at trim in straight flightat 85 m/s, with a heading angle with respect to the target trajectory varying between 10 and180 degrees. A schematic of the maneuver is shown in Figure 15. The target is modeled as asphere with a diameter of 4 m, at an initial distance from the torpedo of 1000 m. The vehicleis required to enter the sphere at the end of the maneuver, this condition being imposed as anappropriate inequality constraint on the vehicle states. In this case, we used a non uniformdistribution of the grid points� the 100 elements are more refined in the proximity of thebeginning of the maneuver, where a trimmed condition is imposed and where the vehiclestarts accelerating and turning, becoming coarser towards the final instants.

The minimum time trajectories obtained for this specific problem are presented in Fig-ure 16. It appears that, as in the case of the turns analyzed in the previous section, theresulting optimal trajectory is characterized by a complex, three dimensional behavior. Thecorresponding maneuver durations are displayed in Figure 17. The time histories presentedin Figures 18 and 19 show a rapid variation of the controls in the initial part of the maneuverwhen one can assume that the torpedo becomes aware of the target and must rapidly modifyits state of motion in order to reach the target in minimum time. This is particularly evident inthe time history of the thrust (Figure 18(a)), whose rapid increase and subsequent saturationat the maximum value are driven by the minimum time requirements of the maneuver.

Finally, Figure 20 presents the time history of the angle p, to demonstrate that the targetis reached without contacts with the cavity. As mentioned above, this might be considered




Figure 16. Minimum time trajectories for initial target relative movement angle ranging between 10 and180 degrees.

Figure 17. Maneuver duration versus initial relative movement angle.




Figure 18. Time histories of thrust and cavitator angle for selected tracking maneuvers.




Figure 19. Time history of fin control angles for selected tracking maneuvers.




Figure 19. Time history of fin control angles for selected tracking maneuvers. (Continued)

Figure 20. Time history of angle of attack p between vehicle and cavity axes during representativetarget tracking maneuvers.




to be due in part to the simplified cavity model used herein, and may not hold true for moresophisticated models of the cavity dynamics.

5. CONCLUSIONS

We have presented a framework for the optimization of trajectories of supercavitating vehi-cles. The procedures are based on a flight mechanics model of the vehicle, and maneuversare defined as optimal control problems. A numerical solution is computed using the directtranscription approach, which leads to a standard parameter optimization problem. Finally,we have tested the new proposed procedures on some preliminary but representative exam-ples, involving dives, turns and target-tracking problems.

This general framework can now be applied to a variety of other, more complex, scenar-ios. Optimal trajectories can be evaluated for different cost functions, for example to maxi-mize performance. Configuration parameters can be cast as design variables to be optimizedtogether with the maneuver, leading, in this case, to an integrated configuration-maneuveroptimization problem. In addition to these studies, further efforts could also be aimed atcomparing the behavior of maneuver tracking controllers. In fact, we speculate that the tra-jectories obtained through the proposed optimization framework, being compatible with thevehicle dynamic equations which appear explicitly as constraints of the optimization prob-lem, will be more easily trackable than trajectories obtained as interpolations of way pointsor other ad-hoc procedures. We think that this could possibly reduce cavity-body interactionsand vibrations during maneuvering flight, with obvious advantages.

Acknowledgments. This work is supported through a grant from the Off ice of Naval Research. The authors gratefullyacknowledge the guidance and support of Dr. Kam Ng, Technical Monitor and Program Manager. The authors alsoexpress their gratitude to the reviewers, whose accurate comments have contributed to a signif icant improvement inthe quality of the paper.

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