3-Dimensional Trajectory Tracking Control of an AUV “R-One Robot” Considering Current...

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Proceedings of The Twelfth (2002) International Offshore and Polar Engineering Conference

Kitakyushu, Japan, May 26-31, 2002

Copyright 0 2002 by The International Society of Offshore and Polar Engineers

ISBN l-880653-58-3 (Set); ISSN 1098-6189 (Set)

3-Dimensional Trajectory Tracking Control of an AUV “R-One Robot” Considering Current Interaction

Kangsoo Kim and Tamaki Ura

Underwater Technology Research Center, Institute of Industrial Science, University of Tokyo

Tokyo, Japan

ABSTRACT

This paper describes the feature of newly devised 3-dimensional target

trajectory tracking control scheme for an AUV “R-One Robot”, which

was developed by joint cooperation between Institute of Industrial

Science, University of Tokyo and Mitsui Engineering & Shipbuilding,

aiming at various practical underwater missions. Dynamics model of R-

One robot was derived as a set of equations of motion, describing

coupled 6-D.O.F. motions in space. In completing this model, iterative

CFD analyses were carried out in order to evaluate hydrodynamic

forces and moments, which consist of external loads to the dynamic

system. Derived lateral and longitudinal dynamics models were applied

in designing heading and depth control systems, both of which are to

run to achieve trajectory tracking mission in underwater space.

Modification of control input to cancel out external disturbance

component normal to the reference trajectory realized the trajectory

tracking control under current interaction. Application of this advanced

control scheme is expected to enable the AUV to challenge more

difficult and practical missions, getting over adverse underwater

disturbances even in the vicinity of complicated underwater geography.

KEY WORDS: Trajectory Tracking; R-One Robot; Dynamics

Model;

CFD Analysis; Current Interaction.

INTRODUCTION

Like any other moving vehicles such as airplane, automobile or space

shuttle, AUV(Autonomous Underwater Vehicle) also requires properly

operating motion control mechanism in order to complete their

expected missions satisfactorily. Successfully designed motion control

system requires refined dynamics model, for this fundamentally affects

the performance of the integrated system. In case the vehicle dynamics

is represented by a set of differential equations, building up of

dynamics model implies the determination of coefficient terms in this

set of equations of motion.

Since these coefficients, i.e. stability

derivatives, have functional relationship with hydrodynamic forces and

moments, external hydrodynamic loads should be evaluated by some

means. In this research, hydrodynamic forces and moments were

assessed via numerical simulation based on CFD analyses instead of the

scaled model tests conducted in towing tank or wind tunnel facilities.

Due to the adoption of this computational approach, much savings in

time and expense have been attained as well as the flexibility in coping

with the frequent changes in design requirements. Based on dynamics

models completed via CFD analyses, heading and depth contro

systems were designed using classical PID controllers. As an advanced

application, trajectory tracking control scheme under current interaction

was devised and modeled into the heading controller. In this application

control command making the vehicle’s heading direct the tangent o

reference trajectory is additionally modified to cancel out current

velocity vector component normal to given trajectory within horizontal

plane. Without this kind of special consideration, AUV can not trace

the reference trajectory correctly merely with an ordinary heading

controller, for there certainly happens drift due to interaction with

current. Proposed control scheme basically assumes the steady drift o

the vehicle when it is subject to sea current. It also assumes that th

vehicle and current velocity vectors relative to inertial frame are

vectorially added to lead the resultant velocity vector. This means th

availability of linear superposition in velocity vector field. Therefore,

their exists abruptly varying current distribution over the cruising

region, or if the current vector varies or fluctuates drastically in time

domain, transient or nonlinear response characteristics due to vehicle-

current interaction becomes important so that the proposed contro

scheme may not work properly. In order to cope with these dynamic

conditions, advanced dynamic modeling or control scheme that is able

to take into account nonlinear or irregular vehicle-current interaction

should be introduced.

In case the trajectory or disturbance becomes 3

dimensional, deviation within longitudinal plane is compensated by th

depth controller, regulating vertical position of the vehicle.

MODELLING OF SYSTEM DYNAMICS

Equations of Motion for Vehicle Dynamics

Equations of motion expressing vehicle motion in 3-dimensional space

can be derived from the conservation law of linear and angular

momentum with respect to the inertial frame of reference. If the time

derivatives of vector quantities defined relative to inertial frame o

reference are transformed into the ones to moving(body fixed) frame o

reference, compacter form of the set of equations can be obtained, i

which every vector quantities are treated relative to this frame.



Fig. 1 Coordinate System

m(U+QW-RV)=Xt,+(~V-m)gsinO

m(V+RU-PW)=Yh +(m-pV)gcosOsin@

m@’ + PV - QU) = Z, + (in - pV)gcosOcos@

I,i, -I,& + (I, - IJQR - I,PQ =

Lh +

(pV - m)gz, cos 0

sin @

I,Q+(I,-I,)RP+I,(P’-R2)=

M,+(pV-m)g{z,sinO+x,cosOcos~}

- I,P + I,Ei + (Iyy - I,)PQ + I,QR =

N,

+ (m - pV)gx, cos Ocos @

where,

1)

(2)

(3)

(4)

(5)

(6)

U, V, W, P, Q, R : linear and angular veloci@ components relative to

bodyfixedframe of reference

&, Yh, Z,, Lh, Mh, Nh : hydrodynamic force and moment components

relative to bodyfixedframe of reference

@, 0, Y: roll/pitch/yaw angles

I,, I,, , etc. : mass moment(products) of inertia

V: displacement of the vehicle

(xB ,O, zB) : center of buoyancy

If we assume that instantaneous motion variables of the vehicle can be

divided into the reference value at steady equilibrium state and small

perturbation from that state like U=U&, P=P&p, 0=0,+0, etc., and

consider the center of buoyancy becomes (0, 0, zB) in case of R-One

robot, we can obtain a set of equations of motion having the unknowns

of perturbations.

m(ti+qW,) = 8(pV-m)gcos8, +AX,

(7)

m(~+rU,

- pW,) = (m - pV)g cos B. + AY,

@I

m(G-qU,) =8(pV-m)gsin8, +AZ,

(9)

I& -I,,+ =

c,bpVgz,cos8, + ALh

(10)

I, =

BpVgz, cos8, + AMh

(11)

-Ixzp+Izzk = AN,

(12)

where,

‘A ‘: perturbed amountfrom equilibrium state

u,v,w,p,q,r : perturbed linear and angular veloci@ components

In Eqs.7-12, nonlinear terms such as products or powers of unknowns

are linearized considering the relative order of magnitudes of each

terms. If we also apply perturbative expansion to hydrodynamic forces

and moments about probable smallness parameters which are believed

to be the functions of hydrodynamic loads, we can obtain a new set of

equations containing so called ‘stability derivatives’ such as X, , Yp , M4

and so forth, as coefficients of the unknowns(Eqs.l3-18). These

equations will be used as the dynamics model of the given problem.

[(m-x,)D-xu]U-xaa

+[mW,D+(pV-m)gcos8,]8 =Xnmnm

[(mUo-Y~)D-Y~lp-(mw,+Y,)P

+[(mUo -Y,)-Y,D]r =YSpr8pl

-Z,u+[(mU,-Z,)D-&]a-[(mU,+Z,)D+Z,D2

- Cm - PVk sin

8

= Z,n,, + z,a n,

-[(I, + L,)D + L,]r + pVgz, cos8, = LSp,&

-M,u - (M,D +M,)a + [(I, -M,)D’ -M,D

+pVgz, cos P =-Z,n, +Z,ann,L

where,

cx 0 : angle of attack and sideslip angle

X, =&Y/&x),, Yp =aYlap),, etc.

D : differential operator with respect to time

n,, n , n,, : rpm of the main/fore vertical/ rear vertical thruster

S,,. : deflection angle of the main thruster axis

lvf, I,, : distance between fore/ rear vertical thruster and e.g.

Evaluation of Stability Derivatives

(13

(14

(15

(16

(17

(1Q

In order to complete the equations of motion, stability derivatives are t

be evaluated by some means. In this research, these values ar

estimated by numerical simulation based on CFD analysis. As

vehicle for the analysis, one of the cruising type AUV named “R-One

Robot” belonging to authors’ group was selected, for it requires

somewhat superior dynamic characteristics so as to meet the successful

achievement of given missions being practical. Principal dimensions

of R-One robot are summarized in Table1

Table1 Principal Dimensions of R-One Robot

T’

mass moments products) of metia

Estimation of Hydrodynamic For ces and Moments via CFD Analysk

Physical properties of the fluid and flow for CFD analysis are assumed

to be viscous, incompressible and n-rotational with the density o

1025(kg/m3). Considering the flow characteristics of this problem being

fully turbulent, one of the most appropriate turbulence model named

‘RANS(Reynolds Averaged Navier-Stokes)’ was adopted, which now

has become very popular and practical tool for design oriented

engineering applications. Reynolds stress in RANS equation

evaluated on the basis of k-s hypothesis.

278



Fig.2 shows the grid system for R-One Robot, which was generated for

CFD analysis. Since the detailed body shape of R-One Robot near

aftbody is fairly complicated, entire grid system was completed via the

assemblage of several local sub-grid systems, such kind of technique in

grid generation is called Multi Block Method. Two types of boundary

conditions such as free stream on which any disturbance owing to the

existence of body is assumed to diminish and solid one are imposed at

outer and wall boundaries, considering each of corresponding

appropriate physical conditions. Fig.3 shows the drag curve for R-One

Robot over the speed range of 2.0-4.0(knots). In Fig.3, another drag

curve obtained from the towing tank test conducted by Mitsui

Engineering & Shipbuilding was represented together for comparison.

Over the entire speed range, it is shown that calculated results are more

or less larger than from experiment. But actual discrepancy is expected

to be smaller than as it is shown, since a pair of vertical tins was added

after the experiment, even though the increase of drag due to this may

be quite small. Figs.4 and 5 shows pressure and velocity vector field

obtained from this analysis.

Fig.2 Grid System for R-One Robot (overall /parts)

veloc ity kfs)

Fig.3 Drag Curve for R-One Robot

Fig.4 Pressure Distribution around R-One Robot(U0=3.0(kts))

Fig.5 Velocity Vector Field around R-One Robot(U0=3.0(kts))

Calccllation of Stability Derivatives by using estimated H ydrodynamic

For ces and Moments

Concluding that the application of CFD analysis in evaluating

hydrodynamic forces and moments is useful from the results o

abovementioned sample calculations, additional calculations were als

accomplished to derive the stability derivatives. Over the attack(a)

and sideslip angle range of -16”@: O”)-16”, hydrodynamic force

and moments are iteratively evaluated, since it is known that if th

Reynolds no. exceeds 106, only cx and 0 with body geometry uniquely

determine the nondimensional hydrodynamic forces and moments such

as C, , C, or C, etc., normalized by corresponding parameters. Lif

and drag curve relative to cx s represented in Fig.6. Due to the slightly

different body shape between upper and lower parts of the body, values

of these are not completely symmetric about cx=O”. Side lift and dra

relative to 0 are also represented in Fig.7, which are fundamentals o

lateral dynamics of the vehicle.

Fig.6 Lift and Drag Curve for R-One Robot

Fig.7 Side Lift and Drag Curve for R-One Robot

79



Figs. 14 shows the calculated stability derivatives by manipulating

obtained hydrodynamic forces and moments. Definition of the each

derivative can be interpreted as the fist subscript represents force or

moment component to be differentiated, while the second represents the

variable of differentiation.

angle of attack deg.)

Fig.8 Cx, for R-One Robot

Fig

0.01

---

‘-..P

0~o”ls-14-i2-in -8 -6 -4 -2

0 2 4 6 8 ,n 12 14 16


.9 Cm, for R-One Robot

Fig. 10 CL, for R-One Robot


Fig.12 CZ, for R-One Robot

sdesip angle deg.)

Fig. 13 Cyp for R-One Robot

3 012341

desip angle deg.)

Fig. 14 Cnp for R-One Robot

Other stability derivatives virtually very difficult to be derived from th

results of CFD analyses are estimated by the simple assessment

formulae suggested by McRuer(l973) or Kato(1982), as constant

values(Table2). Since each of coefficient terms in equations of motion

should have compatible physical dimensions from their definitions,

estimated nondimensional stability derivatives should be transformed

into corresponding dimensional ones. By means of these

transformations, dynamics model of R-One Robot has been completed.

Table2. Stability Derivatives of Constant Values

~


Fig. 11 Cx, for R-One Robot

280



RESULTS OF VEHICLE MOTION SIMULATION

In order to verify how well the derived dynamics model represents

actual vehicle motion, time domain simulations have been carried out

under the known operating conditions recorded during an actual sea

cruising. R-One Robot achieved successful autonomous exploration in

this cruising so that surveyed a dormant but suspicious undenvater

volcano named “Teisi knoll”, and took its live image via side scanning

sonar. R-One Robot made a few ascending, descending and turning

motions during this cruising by changing rpms of two vertical thrusters

and thrust direction of the main thruster, respectively. In Figs. 15 and 16,

sway and yaw velocities from simulation show quite good agreement

with the recorded ones. Stability derivatives are updated at every time

step in this real time simulation surveilling the present sideslip angle,

which is thought to give more accurate results compared to those by

constant stability derivatives at zero sideslip angle. Turning trajectory

was calculated and compared to the recorded one(Fig.l7), which also

confiis accuracy of derived lateral dynamics model.

Fig. 15 Sway Step Response of R-One Robot( s,~ = 15”

Fig. 16 Yaw Step Response of R-One Robot( s,l = 15”

Fig. 17 Turning Trajectory of R-One Robot( s,~ = 150, t=60(sec))

In activating two vertical thrusters, rpms for the fore and rear vertica

thrusters are set different deliberately in order to obtain pure heav

motion, since the each independent run leads slightly different amoun

of pitch response of opposite sign, though the exerted thrust is the sam

This is mainly due to the fact that fore and rear body shape of R-One

robot is not symmetric about the yz-plane, which implies the necessity

of adjustment of different thrust setting on the two vertical thrusters.

For this reason, rpm of rear vertical thruster is set to run 0.91 times o

the fore one in this simulation. Fig. 18 shows the response of integrated

heave velocity(depth) when the fore and rear vertical thrusters ar

commanded to run 880 and SOl(rpm) from rest.

= 16.0

E -

-: simulated

-: recorded

I

Fig.18 Heave Step Response of R-One Robot(n,+SO, n,,=SOl(rpm) )

Since both lateral and longitudinal dynamic responses from simulation

showed good agreements with the recorded ones, derived dynamics

models were certified to be usable in control system design.

MOTION CONTROL SYSYEM DESIGN

Design of Heading Control System

As a following application of the derived lateral dynamics model

design of heading control system was attempted. Classical PID

controller was considered for this purpose, because R-One Robot ha

the simple mechanism of generating necessary side force and yaw

moment for heading alternation only by the azimuthal change of main

thruster axis. Fig. 19 shows block diagram expressing constitution of th

entire system. Fig.20 represents the step response of designed system

showing acceptably good transient and steady-state performances.

Fig. 19 Block Diagram for Heading Control System

Fig.20 Step Response of Designed Heading Control System

281



Design of Depth Control System

Depth control system was also designed in order to regulate vehicle’s

vertical position within longitudinal plane. Basic configurations are

similar to that of heading control system. Satisfactory response

characteristics(Fig.22) are obtained by design procedure in which

optimum gains of

K3

and

K4

are searched and determined via root locus

technique.

Fig.21 Block Diagram for Depth Control System

Fig.22 Step Response of Designed Depth Control System

Restriction of Maximum Control Command

Since there exists limitation in obtainable control force determined by

mechanical specifications of the individual actuator, restriction of

control command by setting maximum available value seems quite

natural. For this reason, maximum control input restriction algorithm

has been installed in the designed controller, which is validated in case

currently issued control command exceeds the predefined maximum

value. In case of heading controller, symbolic expression of this scheme

is as follows.

(19)

ScP,. command of main thruster axis dejlech’on

6cmax maximum available ScP,.

TRAJECTORY TRACKING CONTROL UNDER CURRENT

INTERACTION

Basic Concept of Trajectory Tracking Control considering

Current Interaction

Target trajectory tracking mission under current interaction was

examined as a practical application of designed control systems.

Consideration of current interaction in treating vehicle dynamics and

motion control is very important since there exist more or less strong

current most of underwater environments. Current velocity can be

estimated by subtracting vehicle velocity vector relative to water from

relative to ground, each of which is able to be measured by Doppler

velocity sonar and Inertial Navigation System(INS), respectively. Once

the current velocity vector has been obtained, supplementary targe

heading correction by canceling out(if possible) the component o

current velocity vector perpendicular to intended trajectory(Eq.20)

conducted to make the vehicle trace original target trajectory. In Fig.23

modified target heading i,ul should be determined by adding AyO to th

original target heading I,Q, to satisfy the following requirements o

canceling out current velocity vector component V,,, normal to targe

trajectory at present position.

U, sin Ai,uO= -V,, = -fc nf

(20

Y, =Y,+AY,

(22

where,

yO, yI, AyO : original/modiJied/modiJication amount of target heading

U, : vehicle speed

fc : current veloci@ vector

nf : unit normal vector along the target trajectory

modified headq

Fig.23 Concept of Trajectory Tracking Control via Cancellation o

Current Vector Component normal to Target Trajectory

In addition to this modification, in order to obtain as superior trajectory

following control as possible, when the distance deviation exceeds

predetined allowable one, controller is designed to issue maximum

heading alternation command(&& instead, so as to make the vehicle

approach the target trajectory as fast as possible.

Trajectory Tracking Control under Current Interaction -

2-Dimensional Cases

Simulations have been carried out to confirm the usefulness of th

proposed control scheme under current interaction. Fig.24 shows th

effects of running trajectory tracking control in case there exists curren

of 1 O(m/s), perpendicular to the target straight trajectory(vertica1 axis)

Speed of the vehicle was set to 3.0(knots), the designed speed of R-One

Robot. As shown in this figure, by applying the trajectory tracking

control scheme, deviation from the target trajectory becomes extremely

suppressed, even under the somewhat strong current interaction.

Fig.25 shows the cruising trajectory of the vehicle in case the targe

trajectory consists of a rectangular loop, sized 500(m) of each side. I

this simulation, when the vehicle reaches a sequential target waypoint

identified as inner region of a circle having diameter of SO(m), vehicle

starts to turn to change the target heading increased by 90”. During thi

turning motion, trajectory tracking control is temporally abandoned bu

the controller commands maximum available heading alternation unti

the heading reaches the next target value. Devised control scheme

showed its effectiveness for this looped target trajectory too(Fig.25).

Usefulness of trajectory tracking control was also testified during actua

sea operation and good target trajectory following result under curren

interaction was obtained as shown in Fig.26.

282



current of V,= .o m/s)

1, I I 8, I I

-300 -250 -200 -150 -100 -50 00

50 100 150 200 250

300

transverse position m)

Fig.24 Trajectory Tracking Control along Straight Target Trajectory

B

-50

z -100 -

n

a -150 -

‘p -200 -

P

-250 r

-300

-100-50 0 50 100 150200250300350400450500550601

transverse position m)

Fig.25 Trajectory Tracking Control along Looped Target Trajectory

Fig.26 Recorded Trajectory of R-One Robot operating with Trajectory

Tracking Control under Current

Trajectory Tracking Control under Current Interaction -

3-Dimensional Case

Finally, cruising simulation tracking along the 3-dimensional S-shaped

reference trajectory,

which detours two imaginary undenvater

mountains, has been carried out. Since the reference trajectory forms 3-

dimensional spatial curve, depth controller as well as heading one

should be run in order to achieve the target trajectory tracking mission.

As shown in Fig.27, by activating the trajectory tracking control,

vehicle successfully completed the detouring cruising under 3

dimensional current interaction, while the vehicle was much deviated

from and missed the target trajectory in the end, if the controllers

simply regulate heading and depth without consideration of current

interaction. This simulated result explains the necessity of introducing

trajectory tracking control particularly well because during se

operation, it is not extraordinaq for an AUV to face with current

interaction near complicated underwater geography, which may cause

extremely serious danger.

h

current vec tor -0.5i-0.5j+0.OZk m/s))

Fig.27 Simulated Result of 3-D Trajectory Tracking Control

CONCLUSION

Dynamics model of an AUV, named R-One Robot has been

established by manipulating hydrodynamic forces and moments

obtained from CFD analyses. Simulated motion responses of R-One

Robot based on derived dynamics model showed very good agreement

with the recorded ones, which convinced the usefulness and

effectiveness of the CFD analysis in evaluating vehicle hydrodynamics.

Heading and depth control systems were designed based on the derived

lateral and longitudinal dynamics model, which aims to be applied to

newly devised control strategy of tracking trajectory under current

interaction. Successful trajectory tracking results by additional targe

heading modification have been confirmed from both simulation and

actual field operation. This means that by applying the proposed control

strategy, AUV is expected to upgrade its potential to challenge more

difficult and advanced missions, coping with severe underwater

environments.

REFERENCES

Kato et a1.(1982),

Introduction to Flight Mechanics.

University o

Tokyo Press.

Kim, Sutoh, Ura and Obara(2001), “Route Keeping Control of AUV

under Current by using Dynamics Model via CFD Analysis,” Proc o

MTS/IEEE OCEANS2001,

MTS/IEEE, pp4 17-422.

McRuer, Ashkenas and Graham(l973),

Aircraft Dynamics and

Automatic Control.

Princeton University Press.

Nise(

1995), Control System Engineering.

The Benjamin/Cummings

Publishing Company, Inc.

Ura, Obara, Takagawa and Gamo(2001), “Exploration of Teisi Knoll

by Autonomous Undenvater Vehicle R-One Robot,”

Proc o

MTSWEEE OCEANS2001, MTS/IEEE, pp456-461.

283

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