Trajectory Tracking Control of Underactuated AUV in the Presence

International Journal of Control and Automation

Vol.9, No.1 (2016), pp.81-92

http://dx.doi.org/10.14257/ijca.2016.9.1.08

ISSN: 2005-4297 IJCA

Copyright ⓒ 2016 SERSC

Trajectory Tracking Control of Underactuated AUV in the Presence of Ocean Currents

Xue Yang1 and Juan Li

2, *

1 College of Science and Information, Qingdao Agricultural University, Qingdao 266109, China [email protected]

2 College of Mechanical and Electrical Engineering, Qingdao Agricultural

University, Qingdao 266109, China [email protected]

Abstract

This paper addresses the trajectory tracking problem of underactuated autonomous

underwater vehicles (AUVs) in the presence of ocean currents on the horizontal plane.

The tracking control problem is reduced the problem of stabilizing the nonlinear tracking

error system to two separate problems. The backstepping technique is applied to design

tracking controller, and the conditions of control gains are derived such that the AUV can

track a reference AUV. Simulations results show the effectiveness of the proposed

trajectory tracking controller.

Keywords: AUV, trajectory tracking control, backstepping, Lyapunov method

1. Introduction

Autonomous Underwater Vehicles (AUVs) depicted in Figure 1, are playing a crucial

role in exploration and exploitation of resources located at deep oceanic environments.

They are employed in risky missions such as oceanographic observations, military

applications, recovery of lost man-made objects, etc [1]

. AUVs present a challenging

control problem since most of them are underactuated they have less actuator than state

variables to be tracked, imposing nonintegrable acceleration constraints. However, AUVs‟

models are highly nonlinear, making control problem a hard task.

{ }I X

Y

Z

{ }B

r uv

xy

z

uF

rF

Figure1. The Underactuated AUV Model

Recently, trajectory tracking control problem of underactuated AUVs has received a lot

of attention for the control community. Trajectory tracking issue refers to the case where

the vehicle tracks a reference trajectory generated by a suitable virtual vehicle a geometric

path on which a time law is specified. Note that when moving on a horizontal plane,

Online

Vers

ion O

nly.

Book m

ade b

y this

file i

s ILLEGAL.


Vol.9, No.1 (2016)

82 Copyright ⓒ 2016 SERSC

AUVs present similar dynamic behavior to underactuated surface vessels [2]

. A high gain

based, local exponential tracking result was obtained by applying the recursive technique

for the standard chain form systems [3]

. Based on Lyapunov‟s direct method and passivity

approach, two constructive tracking solutions were proposed for an underactuated surface

ship [4]

. The trajectory tracking problems of underactuated systems were solved based on

cascaded systems by many scholars. A full state feedback control law that make the

underactuated ship follow a straight line is developed using a cascaded approach [5]

. A

new cascade approach for global -exponential tracking of underactuacted ships was

derived [6]

. Based on sliding mode with eigenvalue decomposition and cascaded theory, a

control scheme for a line of sight guidance law of an underactuated AUV was proposed [7]

.

The control problem of tracking a desired continuous trajectory for an AUV in the

presence of gravity, buoyancy and fluid dynamic forces and moments was studied [8]

. The

hybrid control law is developed by combining sliding mode (SMC) and classical

proportional-integral-derivative (PID) control methods for the tracking control of an

underactuated AUV [9]

. A adaptive switching control combined with a nonlinear

Lyapunov-based tracking control are applied to solve the problem of position trajectory

tracking control for an underactuated AUV in the presence of possibly large modeling

parametric uncertainty [10]

. Based on the variable structure systems (VSS) theory and, in

particular, on the second-order sliding-mode (2-SM) methodology, a tracking controller is

designed for the AUV which includes the unmodelled actuator dynamics and the presence

of external uncertain disturbances [11]

.

In practice, an AUV must often operate in the presence of ocean currents. Motivated by

the considerations, this paper presents a solution to the trajectory tracking problem of the

underactuated AUV in the presence of ocean currents. We divide the tracking error system

into two simple subsystems which we can stabilize independently of each other. The

control algorithm proposed builds on Lyapunov stability theory and backstepping design

techniques.

The organization of the paper is as follows. Section 2 presents the AUV model and

problem formulation. Section 3 is devoted to the control design. Some simulations are

given in Section 4 to demonstrate the effectiveness of the proposed controller. Section 5

concludes the paper.

2. AUV Model and Problem Formulation

This section describes the kinematic and dynamic equations of motion of the AUV of

Figure 2 on the horizontal plane and formulates the problem of controlling it to track a

desired trajectory in the presence of ocean currents. Following standard practice, the

general kinematic and dynamic equations of motion of the AUV can be developed using a

global coordinate frame { }I and a body-fixed coordinate frame { }B that are depicted in

Figure 1.

u

vtv

x

y

{ }I

{ }B

rCV

CuCv

u

C

Figure 2. The Model of AUV on the Horizontal Plane in the Presence of Ocean Currents

Online

Vers

ion O

nly.

Book m

ade b

y this

file i

s ILLEGAL.


Vol.9, No.1 (2016)

Copyright ⓒ 2016 SERSC 83

Furthermore, we assume that the inertia, added mass and damping matrices are

diagonal. In the presence of ocean currents, the model of the AUV on the horizontal plane

can be described by

22 11

11 11 11

11 22

22 22

3311 22

33 33 33

cos sin cos

sin cos sin

1

1

r r C C

r r C C

r r r u

r r r

r r r

x u v V

y u v V

r

m du v r u F

m m m

m dv u r v

m m

dm mr u v r F

m m m

(1)

where ( , )x y denotes the coordinates of the AUV in the earth-fixed frame, is the

heading angle of the AUV, and ,u v and r denote the velocity in surge, sway and yaw

respectively, the surge force uF and the yaw torque

rF are consider as the control

inputs. In the presence of a constant and un-rotational ocean currents, , 0c cu v , u

and v are given by r cu u u and r cv v v , where ,r ru v is the relative

body-current linear velocity vector. The positive constant terms , 1 3iim i denote the

AUV inertia including added mass. The positive constant terms , 1 3jjd j represent

the hydrodynamic damping in surge, sway and yaw [12]

. For simplicity, we ignore the

higher nonlinear damping terms. Since the only two propellers are the force in surge and

the control torque in yaw, AUV model (1) is underactuated. In the equations, and for

clarity of presentation, it is assumed that the AUV is neutrally buoyant and that the centre

of buoyancy coincides with the centre of gravity. For system (1), we assume that a feasible reference trajectory

, , , , , , ,d d d d d d d dx y u v r X N is given, a trajectory satisfying

22 11

11 11 11

11 22

22 22

3311 22

33 33 33

cos sin

sin cos

1

1

d d d d d

d d d d d

d d

d d d d ud

d d d d

d d d d r d

x u v

y u v

r

m du v r u F

m m m

m dv u r v

m m

dm mr u v r F

m m m

Where , ,d d dx y denotes the desired position and orientation of the virtual AUV.

, ,d d du v r denotes the desired velocities. ,ud rdF F is the reference inputs in surge

and yaw.

Define the tracking errors

Online

Vers

ion O

nly.

Book m

ade b

y this

file i

s ILLEGAL.


Vol.9, No.1 (2016)


cos ( ) sin ( )

sin ( ) cos ( )

e d d

e d d

e d

e d

e d

e d

x x x y y

y x x y y

u u u

v v v

r r r

In this way, the closed-loop tracking error system can be expressed as shown in the

following equation

22 11

11 11 11

11 22

22 22

3311 22

33 33 33

(1 cos ) sin cos

sin (1 cos ) sin

1( ) ( )

1( ) (

e e e d e e e d e d C C

e e e d e e e d e d C C

e e

e d d e u ud

e e d e

e d d e r

x r y r y u u v V

y r x r x v u v V

r

m du vr v r u F F

m m m

m dv u r v

m m

dm mr uv u v r F

m m m

)r dF

(2)

The tracking control problem of the AUV is transformed to a stabilization one of (2).

Then, the problem considered in this paper can be formulated as follows:

Consider the underactuated AUV with the kinematic and dynamic equation given by

(1). Derive a feedback control law for uF and

rF such that closed-loop system (2) is

input-to-state stable in the presence of known ocean currents.

3 Controller Design

As mentioned above, the control objective now is to stabilize error system (2). Our

main goal is to subdivide the tracking control problem into two simpler and „independent‟

problems: position and orientation tracking, then we design the controllers for the two

subsystems independently. Specially, the velocity of ocean current is considered with

bounded input for the position tracking system. For simplicity, the reference velocity dr

is considered with constant value.

Theroem 1. Consider underactuated AUV system (2) with the control law

22 11 22 11 2 11 3 11( )e e eu ud e x d d e x uF F m v m z m vr v r d u k m z k m z (3)

11 22 33 33 4 33r rd d d e e eF F m m uv u r d r m k m r (4)

where ik ( 1,2,3,4i ） satisfies

Online

Vers

ion O

nly.

Book m

ade b

y this

file i

s ILLEGAL.


Vol.9, No.1 (2016)


11

3 2 2

1 22 22 1 22 22 11 22 11 222 4 ( ) ( ) 0d dk m d k m d m m r m m m r

112 1

22

mk k

m

3 4, 0k k

If udF ,

r dF are bounded and reference velocity dr satisfies

11

22

0 d

mr

m

and the velocity of ocean currents CV is bounded, then trajectory tracking error

system (2) is input-to-state stable.

Proof. The specific implementation process is divided into three steps:

Step 1: error system (2) can be written into the subsystem

22 111

11 11 11

11 22

22 22

cos (1 cos ) sin

sin sin (1 cos )

1( ) ( )

e d e e C C e d e d e e

e d e e C C e d e d e e

e d d e u ud

e e d e

x r y u V u v r y

y r x v V u v r x

m du vr v r u F F

m m m

m dv u r v

m m

： (5)

and the subsystem

2 3311 22

33 33 33

1( ) ( )

e e

e d d e r r d

r

dm mr uv u v r F F

m m m

： (6)

Based on Lyapunov theory, the stability of 1 is identified with the following system

22 113

11 11 11

11 22

22 22

cos

sin

1( ) ( )

e d e e C C

e d e e C C

e d d e u ud

e e d e

x r y u V

y r x v V

m du vr v r u F F

m m m

m dv u r v

m m

： (7)

Then, the next aim is to design control input uF and rF that stabilize the system

2 and 3 .

Online

Vers

ion O

nly.

Book m

ade b

y this

file i

s ILLEGAL.


Vol.9, No.1 (2016)


Step 2: Let

1

sine

C Cx e

d

Vk y

r

where 1k is a positive constant.

Introduce error variable

1

sine e

C Cx e x e e

d

Vz x x k y

r

Choosing the candidate Lyapunov function

22 2 222

1 2

11

1 1

2 2 2ee x e

d

mV y z v

m r

and computing its derivative, one gives

2

221 2

11

1 1 1

2

22 11 22

2

11 22 22

2 2 2 22 221 1 1 1

11

( sin ) ( cos sin )

( )

( cos )

e e

e

e e

e e x x e e

d

e d e e C C x d e e C C d e e C C

e e d e

d

d e e e d x x d e e e C C e e

mV y y z z v v

m r

y r x v V z r y u V k r x k v k V

m m dv u r v

m r m m

m m dk r y y v k r z z k r y u k v V u v

m

222

2

11

e

d

vm r

Let

2e eu xk z

where 2k is a positive constant. Introduce error variablee eu e uz u , considering

candidate Lyapunov function2

2 1

1

2 euV V z , its time derivative becomes

2 1

2 2 222 22 2 221 2 1 12

11 11

2 22 22 111 2

11 11 11 11

cos

1( ) ( )

e e

e e e

e e e e

u u

d e e e d x e x C C e x

d

d e x u x e d d e u ud x

V V z z

m d k mk r y y v k k r z v z V k v z

m r m

m m dk r y z z z v vr v r u F F k z

m m m m

(8)

Let (3) holds, (8) becomes

Online

Vers

ion O

nly.

Book m

ade b

y this

file i

s ILLEGAL.


Vol.9, No.1 (2016)


2 2 222 222 1 2 1 2

11

2 22 221 1 3

11

)

cos ( )

e

e e e ue

d e e e d x e

d

x C C e x d e x

m dV k r y y v k k r z v

m r

k mz V k v z k r y z k z

m

（&

(9)

For simplicity, we choose the parameters as

2 221

11

0k m

km

and 2 1 0dk k r (10)

Then (9) becomes

2 2 2 2 222 222 1 2 1 1 32

11

222

2 1 122 22 11

11 22 22 2 1

2 42 11 1

3 1

22 22 2 1

) cos

1

2 2 2( )

4 2( )

e e e ue

e

ue

d e e e d x e x C C d e x

d

d d de e x e

d d

d dd

d

m dV k r y y v k k r z v z V k r y z k z

m r

r k k r k rm d mv y z y

m r m d k k r

m r k rk z k r

m d k k r

（

2 22 1 cos2 e e

de x x C C

k k ry z z V

Obviously

2 2 22 1 2 1 2 1cos (1 ) cos2 2 2e e e e e

d d dx x C C x x x C C

k k r k k r k k rz z V z z z V

where 0 1 . If 2 12 ( )ex C dz V k k r , then

22 1 cos 02 e e

dx x C C

k k rz z V

(11)

So far, in order to ensure 2 0V ，the following inequality hold

2 4

11 11

22 22 2 1

04 2( )

d dd

d

m r k rk r

m d k k r

Then, taking into account equation (10), we have

11

3 2 2

1 22 22 1 22 22 11 22 11 222 4 ( ) ( ) 0d dk m d k m d m m r m m m r (12)

At this result, if 2 12 ( )ex C dz V k k r ，

2 0V ，the system 3 is input-to-state

stable.

Step 3: Considering the subsystem 2 and choosing Lyapunov

Online

Vers

ion O

nly.

Book m

ade b

y this

file i

s ILLEGAL.


Vol.9, No.1 (2016)


function2 2

3 3

1 1

2 2e eV r , the time derivative of

3V becomes

3

3311 22

33 33 33

1( )

e e e e

e d d e r r d e

V r r

dm mr uv u r r F F

m m m

(13)

Let (4) holds, then the previous derivative becomes2

3 4 0eV k r which means the

subsystem 2 is globally asymptotically stable.

For the overall system, we select the Lyapunov function2 3V V V , when controller

gains satisfy

11

3 2 2

1 22 22 1 22 22 11 22 11 222 4 ( ) ( ) 0d dk m d k m d m m r m m m r ,

112 1

22

mk k

m

3 4, 0k k

It is clear that the derivative of V satisfies

222

22 1 122 22 113

11 22 22 2 1

2 42 211 1

1 4

22 22 2 1

1

2 2 2( )

04 2( )

e ue

d d de e x e

d d

d dd e e

d

r k k r k rm d mV v y z y k z

m r m d k k r

m r k rk r y k r

m d k k r

where 2 12 ( )ex C dz V k k r . So system (2) is input-to-state stable.

4. Simulation Results

In order to illustrate the performance of the proposed control scheme in the presence of

a constant ocean current disturbance, simulations are carried out with a model of an AUV.

The AUV parameters are:

11 50m kg, 22 31.43m kg, 33 30m kg m2,

11 30d kg/s, 22 262d kg/s, 33 80d kg m2/s.

The initial conditions for reference AUV are chosen as:

(0) 3dx m, (0) 5dy m, 0 6d （） rad,

(0) 5du m/s, (0) 0dv m/s, (0) 0.4dr rad/s

Online

Vers

ion O

nly.

Book m

ade b

y this

file i

s ILLEGAL.


Vol.9, No.1 (2016)


and the requirement

( ) 0.4dr t rad/s, ( ) 5du t m/s.

The initial states for the controlled AUV are chosen as:

(0) 1x m, (0) 2y m, 0 0 （） rad,

(0) 1ru m/s, (0) 0rv m/s, (0) 1r rad/s.

The velocity of ocean currents is 0.01CV m/s, orientation 3C rad. The

following results are obtained with controller gains chosen as

1 1k , 2 1 11 22k k m m ,

3 3k , 4 2k .

In Figure 3, the reference and the resulting trajectory of the AUV in the inertial X-Y

plane are displayed. From Figure 4 it can be seen that the errors of positions and

orientation converge to a very small neighborhood of zero, the AUV is not out of

control. The errors in velocities are depicted in Figure 5. After a short period of time, the

errors of velocities converge smoothly to zero. Figure 6 shows the control force uF and

the control torque rF needed for tracking.

-15 -10 -5 0 5 10 150

5

10

15

20

25

30

x,[m]

y,[

m]

Reference

Actual

Figure 3. AUV Reference and Actual Trajectory

0 5 10 15 20 25 30-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

t,[s]

xe,[

m],

ye,[

m],

e,[

rad]

xe

ye

ψe

Figure 4. Positions and Orientation Errors

Online

Vers

ion O

nly.

Book m

ade b

y this

file i

s ILLEGAL.


Vol.9, No.1 (2016)


0 5 10 15 20 25 30-4

-3

-2

-1

0

1

2

3

t,[s]

ue,[

m/s

],v e

,[m

/s],

r e,[

rad/s

]

ue

ve

re

Figure 5. Velocity Tracking Errors

0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600

700

800

900

t,[s]

Fu,[

N],

Fr,[

Nm

]

Fu

Fr

Figure 6. Control Surge Force and Yaw Torque

5 Conclusions

In this paper, the trajectory tracking control problem for an underactuated AUV has been

addressed in the presence of a constant ocean current disturbance. The resulting control laws

have guaranteed input-to-state stability of the tracking error dynamics. The proposed approach

has reduced the problem of stabilizing the nonlinear tracking error system to two separate

problems of stabilizing simpler systems. The conditions of control gains that ensure the AUV

track a reference AUV have been given. Simulations have demonstrated the validity of the

designed trajectory tracking control scheme.

Acknowledgments

This work was supported by the National Natural Science Foundation of China

(61374126, 61379029), the Natural Science Foundation of Shandong Province

(ZR2013FM021), and the Doctor Research Startup Foundation of Qingdao Agricultural

University (631114341).

Online

Vers

ion O

nly.

Book m

ade b

y this

file i

s ILLEGAL.


Vol.9, No.1 (2016)


References

[1] R. Filoktimon and P. Evangelos, “Planar trajectory planning and tracking control design for under actuated AUVs”, Ocean Engineering, vol. 34, no. 11, (2007), pp. 1650-1667.

[2] A.P. Aguiar and A.M. Pascoal, “Dynamic positioning and way-point tracking of under actuated AUVs in the presence of ocean currents”, International Journal of Control, vol. 80, no. 7, (2007), pp. 1092-1108.

[3] K.Y. Pettersen and H. Nijmeijer, “Under actuated ship tracking control: theory and experiments”, International Journal of Control, vol. 74, no. 14, (2001), pp. 1435-1446.

[4] Z.P. Jiang, “Global tracking control of under actuated ships by Lyapunov‟s direct method”, Automatica, vol. 38, no. 1, (2002), pp. 301-309.

[5] K.Y. Pettersen and E. Lefeber “Way-point tracking control of ships”, Proceedings of the IEEE Conference on Decision and Control, (2001), December, pp. 940-945; Orlando, USA.

[6] T.C. Lee and Z.P. Jiang, “New cascade approach for global -exponential tracking of underact acted ships”, IEEE Trans on Automatic Control, vol. 49, no. 12, (2004), pp. 2297-2303.

[7] E. Borhaug and K.Y. Pettersen, “Cross-track control for under actuated autonomous vehicles”, Proceedings of IEEE Conference on Decision and Control, and the European Control Conference, (2005), December, pp. 12-15; Seville, Spain.

[8] A. Sanyal, N. Nordkviot and M. Chyba, “An almost global tracking control scheme for maneuverable autonomous vehicles and its discretization”, IEEE Transaction on Automatic Control, vol. 6, no. 2, (2010), pp. 457-462.

[9] S. Mohan and A. Thondiyath, “Investigations on the hybrid tracking control of an under actuated autonomous underwater robot”, Advanced Robotics, vol. 24, no. 11, (2010), pp. 1529-1556.

[10] A.P. Aguiar and J.P. Hespanha, “A trajectory-tracking and path-following of under actuated autonomous vehicles with parametric modeling uncertainty”, IEEE Transaction on Automatic Control, vol. 52, no. 8, (2007), pp. 1362-1379.

[11] G. Bartolini and A. Pisano, “Black-box position and attitude tracking for underwater vehicles by second-order sliding-mode technique”, International Journal of Robust and Nonlinear Control, vol. 20, no. 4, (2010), pp. 1594-1609.

[12] T.I. Fossen, “Guidance and control of ocean vehicles”, Wiley: New York, USA (1994)

Authors

Xue Yang, She received her doctor degree form Ocean

University of China in 2014. Her research interest covers

autonomous robots, nonlinear system control.

Juan Li, She received her doctor degree form Ocean University

of China in 2008. Her research interest covers fault diagnosis,

fault-tolerant control and nonlinear system control.

Online

Vers

ion O

nly.

Book m

ade b

y this

file i

s ILLEGAL.


Vol.9, No.1 (2016)


Online

Vers

ion O

nly.

Book m

ade b

y this

file i

s ILLEGAL.

Documents

Trajectory Tracking Control of Underactuated AUV in the Presence