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Robust and Decoupled Cascaded Control System of Underwater Robotic Vehicle for Stabilization and Pipeline Tracking CHENG SIONG CHIN*, MICHEAL WAI SHING LAU, EICHER LOW, GERALD G. L. SEET Robotic Research Centre (RRC), Mechanical and Aerospace Engineering Department, Nanyang Technological University, 50 Nanyang Ave, Singapore 639798, Abstract In this paper, we proposed a robust and decoupled cascaded control system with output feedback control for simultaneous stabilization and pipeline tracking of a remotely operated vehicle (ROV) under hydrodynamic uncertainties.

One of the ROV applications on the simultaneous stabilization and

tracking was global output feedback with backstepping method on an ODIN ROV. However, the controller design becomes complex, as partial differential equations are required in the backstepping control law and the ROVs is inherently nonlinear, highly coupled in motion, unsymmetrical in vehicle design and vulnerable to hydrodynamic uncertainties. Compared with the backstepping control and other controllers, the computer simulation shows that the proposed method is simpler and performed better in time domain response and other performance measures such as robustness and stability.

Keywords: decoupled; ROV; stabilization; tracking; cascaded control.

1. Introduction As compared to other marine vehicles such as the ship[1,2] and autonomous underwater vehicle(AUV)[3,4], the remotely-operated vehicle(ROV)[5,6,7,16] dynamics are nonlinear, highly coupled in motion, unsymmetrical in vehicle design and susceptible to hydrodynamic uncertainties. The performance can suffer significantly when the vehicle is performing a pipeline inspection that involves both station-keeping and pipeline tracking at points on the pipeline (in order to locate the actual location of pipe’s leakage), that emphasize on these nonlinearities and hydrodynamic uncertainties. This deteriorating performance can be observed when the ROV is operating in the openloop condition. *Corresponding author email: [email protected]

1

This cause the ROV to exhibit a whirling motion whereby the linear velocities (and its corresponding linear positions) are not asymptotically stable during stabilization. This whirling ROV motion (as seen in Fig. 1) is due to the added mass components in the Coriolis and centripetal term creating a cross coupling effect on the ROV’s motion – the Munk moments[8] arising from the change in the directions of the fluid as the ROV moves with a small pitch and roll angles.

Hence, we need to have a robust cascaded control system to control both the velocities and position simultaneously. One of the ROV application on simultaneous stabilization and tracking was the global output feedback with backstepping method on an ODIN ROV by Jiang et. al. [9,10]. In the paper, the restriction in this ROV’s velocity can be conservative and the controller design becomes complex, as partial differential equations are required in the control law. Moreover, the approaches described in [9,10] solve the problem in the two-dimensional space only. Besides, to perform state estimation on the velocities, the measurement noise in the output measurement and the quadratic cross terms of the unmeasured velocities appearing in the Coriolis and centripetal matrix makes the estimation complicated. However, Jiang et. al. used the coordinate transformation [9,10] to cancel the quadratic cross terms before the observer design.

Motivated by the above nonlinear dynamic and Jiang et. al. observer design on the ROV, we proposed a robust cascaded system with output feedback for both tracking and stabilization under the hydrodynamic uncertainties on the ROV operating in two and higher dimensional spaces. Throughout the paper, the ROV Design and Analysis (RDA)[11] written in MATLABTM scripting language and graphical block diagram SIMULNIKTM are used for the computer simulation.

The paper is organized as follows. The proposed cascaded ROV model is described in Section 2. This is followed by the nonlinear observer design in Section 3. A proposed nonlinear decoupled P-controllers and its cascaded system stability is described in Section 4. The proposed cascaded system was compared with the other controllers in Section 5. Conclusions are drawn in Section 6.

2

2. Cascaded ROV Model for Stabilization and Tracking A station-keeping model is used to describe the ROV dynamic during point stabilization on the pipeline while the vertical plane model is used for vertical pipeline tracking. Since the roll and pitch motions are self-stabilizable, they are neglected in most ROV’s control system design [12,13]. The following are the nonlinear model used for the station-keeping and vertical plane maneuver.

2.1 Station-Keeping Model The vehicle model used for the model-based control design is the RRC ROV[16]. This vehicle (see Fig. 2) is neutrally buoyant but is controllable in four degrees of freedom (where the roll and pitch motions are self-stabilizable or passive), containing two thrusters orientated longitudinally along the X-direction and the remaining two thrusters is orientated at an angle of 45 degrees from the X-Z plane.

The following assumptions can be made when deriving the ROV dynamic equation in order to simplify the effort in modeling. There are namely: (a) ROV is a rigid body and is fully submerged once in water; (b) ROV is slow moving for operation such as pipeline inspection; (c) The earth-fixed frame of reference is inertial;(d) Disturbance due to wave is neglected as it is fully submerged; (e) Tether dynamics attached to the ROV is not modeled (will be included in future work). With that, the stationkeeping model was developed for ROV hovering at a fixed position above the pipeline. The standard notations [14] used for the marine vehicle can be seen in Table 1.

The nominal nonlinear dynamic equation for the station-keeping model (variables labeled with subscript ‘s’) can be written as: 1 1 ν s   M s [C s ( ν s )  D s ]ν s  M s τ s

where ν s  u

(1)

T T v w r  is the body-fixed velocities, τ s  Ts FT u  [ x , y , z , ] is the thrust

input in which the voltage input

u  4 , FT  fT I 4 4   4 4 ( fT  0.92 for forward thrust and

fT  0.61 for reverse thrust) is the static thrust vs voltage relationship, Ts   4 4 is the thruster configuration matrix, Cs ( ν s )  

4 4

and moments matrix, Ds ( ν s )  

is the sum of Coriolis and centripetal and the added mass forces

4 4

is the linear damping matrix due to the surrounding fluid and 3

M s   4 4 is the sum of the rigid body inertia mass and added fluid inertia mass matrix that is defined as M A  diag{[ X u Yv Z w N r ]} . The details of the matrices are shown in Appendix 1.

The Kinematics equation that translate the body-fixed coordinate to the earth-fixed coordinate [14] can be written as: η s  J s ( ηs ) ν s

where

the

earth-fixed

cos  sin J s ( ηs )    0   0

 sin cos 0 0

position

0 0 1 0

ηs  x

y

z 

T

(2) and

the

Euler’s

transformation

0 0 . Equation (1) and (2) can be rewritten in state-space form: 0  1 x s  fs (xs , t )  gs ( u , t )

where

(3)

xs  [ ηs ν s ]T ,

J s (ηs ) ν s    ; gs ( u , t )   f s (x s , t )    1 g s1   M s [C s ( ν s )  D s ] ν s 

031 g s2

g s3

 ; g s4 Ts FT u 

and the details of the matrices can be seen in Appendix 1. The corresponding perturbed model of the station-keeping mode can be written in the form:

x s  fs (x s , t )  gs ( u , t )  Δds (xs , u , t )

(4)

where

Δd s (xs , u , t )  Δfs (x s , t )  Δg s ( u , t ) 0 41 0 41    ;     1 1 ΔM A [ΔD  ΔC A ( ν s )]ν s  ΔM A Ts FT u  are the uncertainties in the hydrodynamic added mass and damping forces terms. The upper bound of

ΔM A , ΔC A ( ν s ) and ΔD as shown in Appendix 2 are obtained from WAMITTM [15] and ANSYSTM [16].

2.2 Vertical Plane Model Similarly the vertical plane model for vertical pipeline tracking or inspection can be expressed as: 4

x v  f v (x v , t )  g v ( u , t ) where

η v  x

x v  [ ηv ν v ]T ,

T z  ,

ν v  u

(5)

w q

T

in

which

u  4 ,

τ v  Tv FT u  [ x , z ,  ]T , the G v ( ηv )  3 is the gravitational and buoyancy force, J v (ηv ) ν v    ; g v (u , t )   f v (x v , t )    1 g v1   M v (C v ( ν v )  D v ) ν v  G v ( η v ) 

 ; g v3 Tv FT u 

031 g v2

and the details of the matrices can be seen in Appendix 1. Then perturbed model for vertical plane can be written in the form:

x v  f v (x v , t )  g v (u , t )  Δd v (x v , u , t )

(6)

where

Δd v ( x v , u , t )  Δf v (x v , t )  Δg v ( u , t ) 031 031    ;     1 1 ΔΜ Α [ΔD  ΔC A ( ν v )]ν v  ΔM A Tv FT u  are the uncertainties in the hydrodynamic added mass and damping forces terms. The upper bound of

ΔM A , ΔC A ( ν v ) and ΔD can be obtained from the perturbation matrices for station-keeping mode in Appendix 2.

2.2 Cascaded Structure of ROV In the proposed cascaded structure as shown in Fig. 3, there is an inner loop for stabilization and a cascaded loop for pipeline tracking. The inner loop equation is used to represent the dynamic during the station-keeping operation whereby the ROV’s body-fixed velocity is regulated at desired velocity. The detail of the important notations used in this Section can be found in Table 2.

The velocity error states,

~ xs2e  u~e

v~e

~ w e

T ~ re  in the inner loop equation can be defined as:

~ xs2e  x s2d  xˆ s2 where x s2d

 ud

vd

wd

estimate of velocity state

(7)

T T rd  is the desired velocity states and xˆ s2  uˆ vˆ wˆ rˆ is the

x s2 . As seen in the state-space representation in (32b), the inner loop

equations for the station-keeping mode can be written as: 5

~ x

s2e

where

 fs2e (~ xs2e , t )  gs2e ( uI , ~ xs2e , t )

(8)

uI  4 is the inner-loop control law. In the Fig. 3, the full-order system is used in the

simulation with roll and pitch velocity uncontrolled as seen in the block diagram (named “6 to 4 DOF”). In ROV tracking control, it is convenient to compute the tracking error in the body-fixed frame as shown in Fig. 2 as:

~ xs1e  J s1 (xs1 )(xs1d  xˆ s1 ) where

x s1d  xd

T zd  d  is the desired position states, xˆ s1  xˆ

yd

estimate of position state 1

 J s1 [J s1 J s (x s1d  xˆ s1 )]  J s1 (x s1d  xˆ s1 )  J 1 J ~ x  J 1 (x  xˆ ) s

where

T zˆ ˆ  is the

s

s1e

x s1 is not shown in

~ x s1e becomes:

(10) for clarity). The tracking error dynamic,

s1e



xs1 and (xs1d  xˆ s1 ) is the position tracking error in the earth-fixed frame.

The J s ( xs1 ) is the inverse of the Euler’s transformation (and its dependence of

~ x

(9)

s

s1d

(10)

s1

x s1d  J s (xs1 )x s2d , xˆ s1  J s ( xs1 )xˆ s2 and ~ xs1e  J s1 ( xs1 )(xs1d  xˆ s1 ) . Note that the

components of the vector

~ xs1e correspond to the error in the ROV longitudinal direction, the cross track

error, the altitude error and heading error. Using the state-space representation in (32a), the outer loop equations for the full-order system can be written as:

~ x s1e  f s1e (~ x s1e , ~ x s2e , u O , t ) where the

(11)

uO  4 refers to the control law and similarly the roll and pitch angles are not controlled.

3. Nonlinear Observer Design for Station Keeping Mode A coordinate transformation [9,10] from body-fixed frame to earth-fixed frame is used in the observer design for the ROV. Similar method can be applied on the station-keeping and vertical plane model. The method cancels the quadratic cross term that appears in the Coriolis and centripetal matrix but it solves the problem in two-dimensional (in the horizontal plane) instead of three-dimensional space (in

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the station-keeping mode). The following steps are used in the nonlinear observer design for the station-keeping mode: 

Coordinate transformation



Observer equation



Exponential convergence proof of the observer states



Full state observer error equations

3.1 Coordinate Transformation A coordinate transformation matrix of

ν s to earth-fixed frame is used to cancel the quadratic cross

terms appearing in the Coriolis and centripetal matrix. This can be written as:

ηQ  Q o ( ηs ) ν s where Q o ( ηs )  

4 4

(12)

, is to be determined, is used to transform the body-fixed velocity

ν s to

ηQ  4 . The coordinate ( ηs , ηQ ) can be obtained by substituting ν s  J s ( ηs ) 1 η s into (12) gives:

ηQ  Qo ( ηs )J s ( ηs ) 1 η s  η s  J s ( ηs )Qo ( ηs ) 1 ηQ Differentiate (13) and substituting M ν s  C( ν s ) ν s  Dν s  τ s with

(13)

ν s  Qo ( ηs ) 1 ηQ from (12),

yields:

  Q ( η )M 1C ( ν )]Q 1η  Q ( η )M 1D Q 1η  Q M 1τ η Q  [Q o o s s s s o Q o s s s o Q o s s The goal is to find

(14)

Q o ( ηs ) such that the algebraic sum of the two terms in the first square bracket in

(14) that contains the coupling term Cs ( ν s ) is zero. By expanding:

 ( η )  Q ( η )M 1C ( ν )  0 Q o s o s s s s and substituting the solution[9,10] as

(15)

Q o ( ηs )  J s ( ηs ) back into (15), the ten partial differential

equations(PDEs) in (15) can be satisfied as seen in Appendix 3. Hence, the solution for the PDEs. With that, substitute

Q o ( ηs )  J s ( ηs ) is indeed

Q o ( ηs )  J s ( ηs ) into (12) and (15) yields:

η s  ηQ 7

(16)

η Q   D η ( ηs ) ηQ  J s ( ηs )M s1τ s where

(17)

D η ( ηs )  J s ( ηs )M s1Ds J s ( ηs ) 1 . The problem has now been transformed into the earth-

fixed frame without the coupling term due to Coriolis and Centripetal matrix,

Cs ( ν s ) . The term η Q

can be interpreted as the velocity along the earth-fixed frame.

3.2 Observer Equations The full states observer equations for the unmeasured position and velocity states in station-keeping mode are to be determined. The observer gains

K 01 and K 02 satisfying the following constraint are

obtained as follows: Equation (16)-(17) with the full-states observer becomes:

ηˆ s  ηˆ Q  K 01 ( ηs  ηˆ s )

(18)

ηˆ Q  D η ( ηs ) ηˆ Q  K 02 ( ηs  ηˆ s )  J s ( ηs )M s1τ s where

(19)

ηˆ s and ηˆ Q are the estimates of ηs and ηQ respectively. The observer error dynamic in (18)

and (19) becomes:

where gains,

~ η s  ~ ηQ  K 01~ ηs

(20)

~ η Q  D η ( ηs )~ ηQ  K 02 ~ ηs

(21)

~ ηs  ηs  ηˆ s , ~ ηQ  ηQ  ηˆ Q , K 01 and K 02 are the observer gains. To obtain these observer we

used

the

Lyapunov

first

method.

We

defined

a

Lyapunov

function

T T Vobs  ~ ηs P01 ~ ηs  ~ ηQ P02 ~ ηQ  0 for all value of ~ ηs  0 , ~ ηQ  0 and determining its

derivative

Vobs for all ~ ηs and ~ η Q as shown. T T Vobs  ~ η sT P01~ ηs  ~ ηsT P01~ η s  ~ η Q P02 ~ ηQ  ~ ηQ P02 ~ η Q T η ~ ηT K ]P ~ η ~ ηT P [~ η K ~  [~ η] Q

s

01

01

s

s

01

Q

01

s

 [ ~ ηQ D η (ηs )  ~ η K 02 ]P02 ~ ηQ  ~ ηQ P02 [D η (ηs )~ ηQ  K 02 ~ ηs ] T

T

T s

8

(22)

Then the

K 01 and K 02 are obtained by ensuring that the Vobs  0 and Vobs  0 or simply they must

satisfy the following conditions: T

T

K 01 P01  P01K 01  0, P01  P02 K 02  0, where

P01  P01  0 T

(23)

T

P02  P02  0 , D η ( ηs )P02  P02 D η ( ηs )  0

(24)

D η ( ηs ) is positive definite.

3.3 Exponential Convergence of Observer States The estimated state

ηˆ s and ηˆ Q can be shown to be exponential convergence by means of the

Lyapunov first method. For compactness in the equations, we defined

Then, the

~ ηO (t )  [~ ηs (t ) ~ ηQ (t )]T .

T P Vobs in previous Section 3.2 becomes Vobs  ~ ηO  01 0

Differentiating

0 ~ T ηO  ~ ηO Po ~ ηO .  P02 

Vobs along the solutions of (20) to (21) gives:

T T Vobs  ~ η O P0 ~ ηO  ~ η O P0 ~ η O

ηs  [ ~ where the

H0                ~ 2P K 2P  01 01 02 K 02  P01  P01 D η   η s  ~ η Q ]  ~   P02  2P 02 D η   ηQ    2P 01  K 02 P02

(25)

~ ηs  ηs  ηˆ s , ~ ηQ  ηQ  ηˆ Q and hence implying ~ ηO  ηO  ηˆ O . After some manipulation,

~ ηO (t ) becomes: 2 2 ~ ηO (t )   0 ~ ηO (to ) e  o (t  t o )

where

 o  0 , 0  0

are positive constant and

(26)

. is the Euclidean norm. Expanding (26), we can

obtain one part that contains: 2 2 ~ ηs (t )   o ~ ηs (to ) e  o (t  t o )

Since the velocity estimate error, of

(27)

~ν  ν  νˆ can be transformed to the earth-fixed frame by means s s s

~ν  J ( η ) 1 ~ ηs , the velocity estimate error is exponentially convergence as well. s s s 9

3.4 Observer Error Equations To include the observer into the cascaded system, we need to express the observer equation in (16) to (17) as function of the position estimate error

~ ηs and velocity estimate ~νs .By equating (21) to

~ η Q  J s ( ηs )~νs  J s ( ηs )~νs and substitute (15) with Q o ( ηs )  J s ( ηs ) yields:

~ν  M 1C (~ν )~ν  M 1D ~ν  K J ( η )-1 ~ ηs s s s s s s s s 02 s s where

(28)

~ ηs  ηs  ηˆ s and ~νs  ν s  νˆ s . With ~ ηQ  J s ( ηs )~νs  ~νs  J s ( ηs ) 1 ~ ηQ , equation (18)

can be written as:

~ η s  J s ( ηs )~νs  K 01 ~ ηs

(29)

3.5 Observer Error Equation for Station-Keeping Mode The station-keeping model is used to describe the ROV during the stabilization about the equilibrium whereby the roll and pitch motions are self-stabilizable. The parameters used in deriving the tracking and velocity error dynamic for the station-keeping mode has been defined in Section 2.1. Details of the parameters used in (30) and (31) can be seen in Table 2. Since there is no tracking requirement for the ROV during the station-keeping mode but the tracking errors dynamic in body-fixed dynamics still exist. The outer loop equation in (2) for station-keeping mode with control law

where

u o and the observer in (29) can be expressed as:

~ xe  ud cos  vd sin  uˆ  rˆ~ ye  u Ox   xe

(30a)

~ y e  ud sin  vd cos  rˆ~ xe  vˆ  u Oy   ye

(30b)

~z  w  wˆ  u   e d Oz ze

(30c)

~ e  rd  rˆ  u O  e

(30d)

~ xe , ~ ye , ~ze ,~e are the position tracking errors in body-fixed frame and  xe ,  ye ,  ze , e

are the first to third and last row of

K 01~ ηs in (29) respectively. The u Ox , u Oy , u Oz and u O are the

outer loop control law for the surge, sway, heave and yaw dynamic respectively.

10

The stabilization dynamic equation for station-keeping mode in (1) with the control law

u I and

observer in (28) can be expressed as:

u~e  (m  X u ) 1 X u u~e  ( m  X u ) 1[(Yv  m)v~e ~ re   x  u Ix ]   u

v~e  ( m  Yv ) 1Yv v~e  (m  Yv ) 1[(m  X u )u~e ~ re   y  u Iy ]   v

~  ( m  Z ) 1[ Z w ~ w e w w e   z  u Iz ]   w

(31a) (31b) (31c)

~ re  ( I z  N r )1 N r r~e  ( I z  N r ) 1[(Yvu~e  X u u~e )v~e    u I ]   r (31d) where

~  w  wˆ and ~ u~e  ud  uˆ , v~e  vd  vˆ , w re  rd  rˆ are the velocity errors. The e d

u , v ,  w ,  r are the first, second, third and last row of K 02 J s ( ηs )-1 ~ ηs  K 02 ~νs respectively. The

u Ix , u Iy , u Iz and u I are the inner loop control law for the surge, sway, heave and yaw dynamic

respectively. The

 x ,  y ,  z , 

are the thrust inputs to the ROV.

Now, the error dynamic for both tracking and stabilization for the cascaded system can be written in state-space form as:

where

~ xs2e  u~e

v~e

~ w e

~ x s1e  fs1 (~ xs1e , ~ xs2e , uO , t )

(32a)

~ x s2e  fs2 (~ xs2e , t )  g s2 (uI , ~ xs2e , t )

(32b)

T ~ re  is the velocity error ~ xs1e  ~ xe

~ ye

tracking errors in body-fixed frame and the remaining functions are defined as:

11

~z ~ T is the position e e

 cos  sin fs1 (~ xs1e , ~ xs2e , uO , t )    0   0  (m  X u ) 1 X u  0 f s2 (~ xs2e , t )    0  0 

sin cos 0 0

0 0 ye   uˆ  rˆ~    rˆ~  0 0 x vˆ  e x s2d    uO  K 01 ~ xs1   wˆ  1 0    0 1   rˆ 

 (m  X u ) 1 (Yv  m)~ re 1  (m  Yv ) Yv 0 1  ( I z  N r ) (Yv  X u )u~e

0 0  (m  Z w ) 1 Z w 0

 0  ~  (m  Yv ) (m  X u )ue  ~ x  s2e 0   ( I z  N r ) 1 N r  1

~

(m  X u ) 1 0 0  1  0 ( ) 0 m Y v g s2 (uI , ~ xs2e , t )    0 0 (m  Z w ) 1  0 0 0 

xs2    1  0 cos  sin 0 0    0  (u  τ )  K  sin cos 0 0 ~x I s 02 s1   0 0 1 0 0    0 0 1 ( I z  N r ) 1   0

uI , uO  4 are the inner and outer control law to be derived in Section 4 and K 01 , K 02  4 4 are the observer gains. With the observer designed on the station-keeping mode, the unmeasured states for a desired input of x1d = [5 0 3 0 0 0]T (as an example) can be estimated. The differences between the estimated and actual states can be seen in Fig. 4. As observed, the estimated states (dotted line) converged to the actual states (solid line) in less than ten seconds. For brevity, the observer design for the vertical plane subsystem is not shown as similar method can be used.

4. Robust Cascaded Controller Design After the observers are designed for the station-keeping (and vertical plane subsystems), the next step is to design robust cascaded controllers. However, the cascaded systems with the observers are of a different form as those in [17,18,19]. This affects the direct application of the GUAS proof on the cascaded system. Therefore, there is a need to identify the difference and to revise the GUAS proof in order to make it applicable for the proposed cascaded system established here. Basically, the differences in the cascaded control system are as follows:

a) However, [17,18,19], the perturbations due to hydrodynamic forces are not included in the equations. They are often required when the robustness of the control needs to be evaluated. In the case presented

12

here, the perturbations in the hydrodynamic damping and added mass are included. For example, the nonlinear dynamic for the station-keeping model with perturbation can be written as:

~ x s1e  fs1 (~ xs1e , ~ xs2e , uO , t )

(33)

~ x s2e  fs2 (~ xs2e , t )  gs2 (~ xs2e , uI , t )  Δds2 (~ xs2e , uI , t ) where

the

Δds2 (~ xs2e , uI , t )  Δfs2 (~ xs2e , t )  Δg s2 (~ xs2e , uI , t )

hydrodynamic forces. The functions

is

the

perturbation

(34) in

the

fs1 (~ xs1e , ~ xs2e , t ) , fs2 (~ xs2e , t ) and gs2 (~ xs2e , uI , t ) are

continuously differentiable in their argument and locally Lipschitz.

b) Also, the first equation is usually a function of its same states as seen in the literature [17,18,19]. This is not true as we can see in (33) that contains two states. In our case, the inner loop controller required to make

uI is

~ xs2e GUAS such that ~ xs1e is GUAS with the outer loop controller uO .

c) In most cascaded stability proof, the system is GUAS with respect to an equilibrium point. However the present situation, the roll and pitch angles are not fully actuated and thus remain unspecified at all time. As a result, there is a need to use an equilibrium sub-manifold [20] instead of equilibrium point to represent the equilibrium point where the roll and pitch orientation of the ROV in three-dimensional space are unspecified. Fortunately, the roll and pitch angles are bounded and do not destabilized the ROV.

4.1 Cascaded Systems Stability under Perturbation To ensure that the above-mentioned cascaded controller design for position tracking and stabilization is GUAS (for the pipeline tracking), a cascaded stability proof similar to [17,18] is needed. Due to the above differences in the cascaded structure, there is a need to reformulate the [Theorem 3, 19] for the GUAS of the nonlinear-cascaded system about the submanifold equilibrium.

In the Theorem 1, the conditions for the existence of the Lyapunov function called converse Lyapunov theorem [21] is used. The Theorem provides sufficient conditions to guarantee the GUAS of the

13

nonlinear-cascaded system in (33) and (34). With the P-decoupled controller used in the inner loop the sub-system involving the states

uI ,

~ x s 2e in (34) is GUAS. As the VO (~x s1e , t ) for the outer loop is

semi-negative definite with the P-decoupled controller

uO , the ~ xs1e is also GUAS. As the result, the

entire cascaded system is GUAS.

Theorem 1: GUAS of the cascaded system The nonlinear system for the station-keeping mode in the (33) and (34) are used. The assumption used is as follows: the inner and outer controllers can be designed such that the nonlinear coupled dynamic terms and hydrodynamic perturbations are cancelled. This results in some upper bound on the variables as seen in the (37) and (38).

With that, if the condition (a) and (b) below are satisfied then the cascaded system defined by the equation (33) and (34) are said to be GUAS.

a) With the P-decoupled inner loop control law

x s2e is GUAS if the Lyapunov uI , the system ~

function VI (~ xs2e , t ) ,VI : 4   0   0 satisfies the following inequality used in the converse Lyapunov theorem [21]:

1 ( ~xs2e )  VI (~xs2e , t )   2 ( ~xs2e )

(35a)

VI VI  fs2 (~ xs2e , t )   3 ( ~ xs2e ) t ~ xs2e

(35b)

VI ~ xs2e   4 ( ~ xs2e ) ~ xs2e for all

(35c)

(~ xs2e , t )  4  0   where 1 (.), 2 (.), 3 (.), 4 (.) are class  function and there exist a

positive constant b1 such that:

[d1 ( ~ xs2e )   3 ( ~ xs2e )] 4 ( ~ xs2e ) and the

gs2 (~ xs2e , t ) and Δds2 (~ xs2e , t ) are bounded as follows: 14

~ xs2e  b1VI (~ xs2e , t )

(36)

where

gs2 (~ xs2e , t )  d1 ( ~ xs2e )

(37)

Δds2 (t )  d 4

(38)

d1 (.) is continuous function and d 4 is positive constant.

b) With the

~ xs2e becomes GUAS, ~ xs1e is GUAS with the P-decoupled outer loop controller uO as seen

in Step (e) of Section 4.2.

Proof: First, we need to see whether the VI (~ xs2e , t ) is semi-negatively definite. To do this, we evaluate the time derivative of

VI (~ xs2e , t ) in (35a) along the trajectories of ~ x s2e : V V V VI (~ xs2e , t )  I  ~ I fs2 (~ xs2e , t )  ~ I gs2 (~ xs2e , t ) t xs2e xs2e

(39)

Using (35b) and (37) ,we can write:

V VI (~ xs2e , t )  [d1 ( ~ xs2e )   3 ( ~ xs2e )] ~ I xs2e

(40)

Substituting (35c) into (40) yields:

VI (~ xs2e , t )   [d1 ( ~ xs2e )   3 ( ~ xs2e )] 4 ( ~ xs2e ) Using the defined function [ d1 (

~ xs2e )   3 ( ~ xs2e )] 4 ( ~ xs2e )

~ xs2e

(41)

~ xs2e  b1VI (~ xs2e , t ) in (36),

equation (41) becomes:

VI (~ xs2e , t )  b1VI (~ xs2e , t )

(42)

VI (~ xs2e , t )  eb1 (t t0 )VI (~ xs2e (t0 ), t0 )

(43)

The solution of VI (~ xs2e , t ) becomes:

If the initial time, t0  0 , the expression can be simplified to become:

VI (~ xs2e , t )  eb1tVI (~ xs2e (0))

15

(44)

where b1 is a positive constant. Equation (44) shows that the VI (~ xs2e , t ) is exponentially bounded and hence implying that the ~ xs2e is also exponentially bounded.

In the second part of the proof, the

~ xs1e becomes GUAS as the ~ xs2e is GUAS with the P-decoupled

inner loop controller (as shown in the first part of the proof). With that the P-decoupled outer loop controller,

uO is designed such that it would not destabilized the entire state ~ xs1e as seen in the

computation of VO that is semi-negative definite for the case of station-keeping model (see Step (e) of Section 5.2). With the roll and pitch bounded, the nonlinear-cascaded system as defined in (33) and (34) is GUAS about the equilibrium submanifold.  4.2 P-Decoupled Cascaded Controller Design for Station-Keeping and Vertical Plane Mode In this Section, the robust decoupling controllers for inner and outer loop are designed for the subsystems such that the resulting closed-loop cascaded system satisfied the Theorem 1.

The

followings are the five steps used in the controller design. a) Design inner loop controller. The inner control law

uI to be designed will have three components

such as: u I  u I,p  u I,dyn  uI,pert  m  X u  X u K pv  m X     u  0 0  k pu 0  0  0 k  0 0 ~  pv   x   0 0 k pw 0  s2e  0    0 0 k pr   0  0    m  X u  X u  X u   0  -  0   0  

0

0

m  Yv  Yv Yv

0

0

0

m  Yv  Yv m  Yv

0

0

m  Z w  Z w Z w

0

0

0

m  Z w  Z w m  Z w

0

0

    0    0  I z  N r  N r   N r 

    0    0  I z  N r  N r   I z  N r  0

re   (Yv  m)v~e ~  (m  X )u~ r~  u e e     0  ~ v~  ( )   Y X u u e e  v

0

re   (Yv  m)v~e ~  (m  X )u~ r~   u e e     0   ~ ~ ( Yv  X u )u e ve 

(45)

16

where the component uI, p ensure a faster response. The component uI, dyn is to cancel some nonlinear coupled dynamic behavior, for which there is a prior knowledge and to ensure robust stabilization with least amount of coupling. The component uI, pert is to attenuate the perturbation due to hydrodynamic uncertainties. Hence, the inner loop controller is to make the state velocity error to zero and keep it small until it reaches the desired position in space.

b) Substitute u I into the

g s2 (~ x s2e , u I , t ) and Δd s2 (~ x s2e , u I , t ) , they are bounded as shown in

(37) and (38) respectively. By substituting the

uI into the g s2 (~ x s2e , u I , t ) , the uI disappeared and becomes:

pv x s2         0 0 0  k pu  cos sin  0 0  0 k  sin  cos 0 0 0 0  ~ pv ~ g s2 (~ x s2e , t )    x s2e  K 02  x s1  0 0 k pw 0   0 0 1 0     0 0 k pr  0 0 1  0  0

~

K

0 0 (m  X u )  0 (m  Yv ) 0   0 0 ( m  Z w )  0 0 0 (I z 

(46)

1

0   0  τ  s 0   N r )

Then, apply triangular inequality rule, g s2 ( ~ x s2e , t ) is bounded as:

g s2 (~ x s2e , t )  d 1 ( ~ x s2e ) where the input thrust

(47)

τ s is finite and bounded. By substituting uI into Δds2 (t ) , it becomes:  1 X  u  0  Δd s2 (t )    0    0 

0

0

1 Yv

0

0

1 Z w

0

0

 0   0   τs 0   1  N r 

(48)

Then, apply triangular inequality rule on Δds2 (t ) yields: Δd s2 (t )  d 4

17

(49)

c) Check whether the inner loop is GUAS. With

gs2 (~ xs2e , t ) and Δds2 (t ) are bounded, the

1 T xs2e , t )  ~ xs2e M s ~ xs2e , it becomes obvious that it VI (~ xs2e , t ) must be semi-negative definite. Let VI ( ~ 2 satisfies (35a) and (35c) as

V I (~ x s2e , t ) is positive definite. The remaining is to verify whether the

inequality in (35b) is satisfied. By performing the time derivative of

VI (~ xs2e , t ) , it becomes:

2 T VI  ~ xs2e Ms ~ x s2e  Ds ~ xs2e  d1 ( ~ xs2e ) ~ xs2e

As shown in (50), the GUAS with the

(50)

VI (~ xs2e , t ) is semi-negative definite. Thus, the inner closed loop system ~ xs2e is

gs2 (~ xs2e , t ) and Δd s2 (t ) bounded as shown in (47) and (49) respectively.

d) Design outer loop controller. The outer control law

uO to be designed will have two components

satisfying:

uO  uO,dyn  uO,p px   ~ 0 0   ud cos  vd sin  uˆ  rˆye   k px 0   u sin  v cos  rˆ~  0  ~ xe  vˆ   0 k py 0 d  d  x 0 k pz 0  s1e   0  wd  wˆ     0 0 k p   rd  rˆ    0

K

(51)

where the component uO, p is to ensure a faster response and uO, dyn is to cancel some nonlinear couple dynamic behavior for which there is a prior knowledge and also to ensure robust tracking to the desired trajectory with least amount of coupling.

e) Check whether the outer loop is GUAS. Applying the same method in Theorem 1 for

~ xs1e and let

1 T VO (~ xs1e , t )  ~ xs1e M s ~ xs1e , it is obvious that both condition stated in (35a) and (35c) are satisfied as 2 the

VO ( ~ x s1e , t ) is positive definite. The remaining is to check whether the inequality in (35b) is

satisfied. By performing the time derivative of

VO (~ xs1e , t ) , it becomes:

18

T VO  ~ xs1e Msx s1e

 k px 0 0  0      0 k 0 0   py T ~ xs1eMs    x  K01~ xs1  ~  0 0 k pz 0  s1e      0 0 0 k p       2 ~ VO  d5 ( xs1e )

(52)

where

~ x

s1e

Since the

k px 0   0   0

0 k py 0

0 0 k pz

0

0

  ~ x  K 01 ~ xs1  s1e  k p  0 0 0

VO (~ xs1e , t ) is semi-negative definite, the outer-loop subsystem ~ xs1e is GUAS. As shown in

the first part of the proof, the inner loop controller bounded and hence the subsystem loop controller

xs2e , t ) and Δd s2 (t ) to be uI resulted in the gs2 (~

~ xs2e becomes GUAS. Since the ~ xs2e is GUAS and with the outer-

uO , the ~ xs1e is also GUAS. In summary, the entire cascaded system is GUAS.

Repeating the Step (a) to (e), the inner controller for the vertical plane subsystem becomes:

u Ih  u I,p  u I,dyn  u I,pert  m  X u  X u  ~ k pu u e   (m  X u )  ~  0    k pv ve     k pr r~e      0   m  X u  X u  X u   0     0 

0 m  Yv  Yv ( m  Yv ) 0 0

    0  I z  N r  N r  ( I z  N r )  0

    0  I z  N r  N r  N r  0

m  Yv  Yv Yv 0

re   (Yv  m)v~e ~  ( m  X )u~ r~   u e e   ~~ ( Yv  X u )u e ve 

while the outer loop controller for the vertical plane subsystem becomes:

19

 (Yv  m)v~e r~e   (m  X )u~ ~  u e re   ( Yv  X u )u~e ~ ve 

(53)

uOh  uO,dyn  uO,p xe  ye   k px ~  ud cos  vd sin  uˆ  rˆ~  ~    ~   ud sin  vd cos  rˆxe  vˆ    k py ye    k p~e   rd  rˆ

(54)

where the selection of the controller are similar to the case of the station-keeping mode.

5 Comparisons of Control Systems Design 5.1 Time Domain Response Time domain performance of the nonlinear decoupled controller has been investigated and compared with other controllers such as backstepping [9,10], single loop PID controller [22] and other controllers such as P and PI controller. The controller parameters were obtained through some trial and error method as seen in Table 3 in Appendix 4. The simulation time was executed in 200 seconds using ODE45 solver with variable step size and a relative tolerance of 0.001. As shown in Fig. 5, the block diagrams of the nonlinear robust decoupled controller (for the inner and outer-loop) were built using the available blocks in the RDA [11] written in the MATLABTM scripting language and graphical block diagram simulation, SIMULINKTM. The desired position command is set as x1d = [5 0 3 0 0 0]T and they are chosen to cater to the limited number of sensors correctly fitted on the vehicle. The initial position of the ROV is set as zero. To cater for both the stabilization and tracking operation, the ROV will be in the station-keeping mode after the desired position in the vertical plane is reached.

As observed in Fig. 6 and 7 (or in Table 4 in Appendix 4), the nonlinear P-decoupled controller has a faster response as seen in its shorter settling time. The output approaches the reference input, x = 5m and z = 3m with smaller steady state position and velocities error under the hydrodynamic perturbation. Besides the P-decoupled controller exhibits less oscillatory response in both position and velocity as compared to other controllers. The response of the roll and pitch motions are plotted in Fig. 6 and 7 as well; both are asymptotically stable and bounded.

On the other hands, the backstepping controller has the highest steady-state position and velocity error. As observed, the single loop PID and cascaded PI-P are both oscillatory in the position and velocity responses. In addition, the settling time is longer (greater than 100 sec) as compared to the nonlinear P20

decoupled controller. As shown in Fig. 7, the PI and single loop PID exceeded the maximum velocity (=0.6m/s). From the response in Fig. 6 and 7, all controllers exhibit self-regulating in the roll and pitch motion. Alternatively, the time domain results are summarized in Table 4 in Appendix 4.

5.2 Robustness and Stability Performance A ten percent perturbation on the added mass and damping matrix, was used to test the robustness of the proposed controller design. For the worst-case simulation, the upper limits of the perturbations as shown in (4) and (6) were used. To quantify the robustness of the cascaded system due to these perturbations, the sensitive of input on the output position and velocity was used. As shown in Table 4, the proposed P-decoupled controller has a lowest value in the sensitivity to the perturbation in both the position and velocity. This implies that the output is less susceptible to model parameter changes. Besides, the input thrust for the nonlinear P-decoupled controller is lower as compared to the singleloop PID and PI controllers. Thus the control effort used to maneuver the ROV is reduced.

For the cascaded-loop stability, the

VI and VO for the inner and outer loops are all semi-negative

definite and thus satisfied the asymptotically stable criteria in Theorem 1. As observed in the Table 4 in Appendix 4, the P-decoupled controllers meet the following condition:

VI  0 , VO  0 ,



positive definite



semi-negative definite VI



radially unbounded

 0 , VO  0 ,

xs1e   and VO   , ~ xs2e   . VI   , ~

Hence, the ROV is said to be GUAS to an equilibrium submanifold at a desired position under the hydrodynamic perturbation. Besides, the P-decoupled controller is able to perform simultaneous stabilization and tracking operation as seen in the small steady state position error and velocity error.

6. Conclusions The robust cascaded controller with output feedback for stabilization and tracking of a ROV for the underwater pipeline inspection has been presented. The proposed cascaded structure of RRC ROV had been described followed by the nonlinear full states observer design. For robust cascaded controller 21

design, a nonlinear P- decoupled controller was designed for both the inner and outer loop to cancel the persistent velocities cross coupling, nonlinearity and hydrodynamic perturbation at different plane models. This provides a simultaneous stabilization and position tracking for different maneuvering. The proposed robust cascaded control design yields a GUAS in both output positions and velocity as shown in the cascaded stability proof in Theorem 1.

The time domain responses were simulated using the RDA package written in the MATLABTM scripting language and graphical block diagram simulation, SIMULINKTM with VRML visualization capability. As observed, the nonlinear P-decoupled controller exhibits a shorter settling time (less than 100 sec). The output approaches the reference input with smaller steady state position error even under the hydrodynamic perturbation. As observed in the simulation, the PI and single loop PID exceeded the maximum velocity (=0.6 m/s). Besides, the P-decoupled controller exhibits less oscillatory in both the position and velocity response as compared to the other controllers.

Hence, it deems to fair better in

the time domain response as compared to the other controllers.

To test for robustness, a ten percent perturbation on the added mass and linear damping was applied on the nominal model.

The proposed P-decoupled controller has a lowest value in the sensitivity to the

perturbation in both the position and velocity. This implies that the output is robust to model parameter changes. Besides, the input thrust for the nonlinear P-decoupled and P-P controllers were lower as compared to the single-loop PID and PI controllers. As the steady state position error and velocity errors are small, the P-decoupled controller was able to achieve simultaneous stabilization and tracking despite of the perturbation and is GUAS to an equilibrium submanifold.

Acknowledgements The technical and financial support from NTU is sincerely appreciated. The authors would like to thank the reviewers for their invaluable comments and Mr. Eng Y. H. for the ROV modeling.

References [1] Lefeber E. Pettersen K. Y., and Nijmeijer H. Tracking control of an underactuated Ship, IEEE Transactions on Control Systems Technology, 2003, 11(1), 52-61. 22

[2] Behal A., Dawson D. M., Dixon W. E. and Yang F. Tracking and regulation control of an underactuated surface vessel with nonintegrable dynamics. IEEE Transactions on Automatic Control, 2002, 47(3), 495–500. [3] Chiok, C. C. Hariharan K. and Teo C. L. A comparison of controller performance for an autonomous underwater vehicle. Proceedings of the 2nd International Conference on Recent Advances in Mechatronics, Istanbul, Turkey, 1999, 402–407. [4] Jalving B. The NDRE-AUV flight controls system. IEEE Journal of Oceanic Engineering, 1994, 19(4), 497-501. [5] Smallwood, D. A. and Whitcomb L. L. Preliminary Experiments in the Adaptive Identification of Dynamically Positioned Underwater Robotic Vehicles. Proceedings of the 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2001, 1803-1810. [6] Press W. H., Neukolsky S. A., Vetterling W. T. and Flannery B. P. Numerical Recipes in C. Cambridge, UK, Cambridge University Press, 1992. [7] Chin C. S, Lau M. W. S., Low E. and Seet G. G. A pipeline tracking control of an underactuated remotely operated vehicle. The Mediterranean Journal of Measurement and Control, 2006, 2(1), 22-34. [8] Anderson J.M. and Chhabra N.K. Maneuvering and Stability Performance of a Robotic Tuna, Integrative and Comparative Biology, 2002, 42(1), 118-126. [9] Do K. D., Jiang Z. P. and Pan J. A global output feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robot, IEEE Transactions on Robotic and Automation, 2004, 20(3), 589-594. [10] Do K. D., Jiang Z. P., Pan J. and Nijmeijer H. A global output feedback controller for stabilization and tracking of underactuated ODIN: A spherical underwater vehicle, Automatica, 2004, 40 (1), 117-124. [11] Chin C. S., Lau, M. W. S.,Low E. and Seet G. G. Software for Modelling and Simulation of a Remotely Operated Vehicle, International Journal of Simulation Modelling, 2006, 5(3), 114-125. [12] Caccia M., Bono R., Bruzzone G. and Veruggio G. Unmanned Underwater Vehicles for scientific applications and robotics research: the ROMEO Project, Marine Technology Society Journal, 2000, 34 (2), 3-17. 23

[13] Press W. H., Neukolsky S. A., Vetterling W. T. and Flannery B. P. Numerical Recipes in C. Cambridge, UK, Cambridge University Press, 1992. [14] Fossen, T. I. Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles, Marine Cybernetics AS, 2002. [15] Lee C. H. WAMIT theory manual report 95-2, Dept of Ocean Engineering, MIT, 1995. [16] Eng Y. H. Master First Year Report, Robotic Research Center, NTU, 2007. [17] Panteley E. and Loria A. Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems. Automatica, 2001, 37(3), 453-460. [18] Loria A. and Morales J. L. A separation principle for dynamic positioning of ships: theoretical and experimental results. IEEE Transactions on Control Systems Technology, 2000, 8(2), 332-343. [19] Panteley E. and Loria A. On global uniform asymptotic stability of nonlinear time-varying non-autonomous systems in cascade. System Control Letters, 1998, 33(2), 131-138. [20] Bloch A. M., McClamroch N. H. and Reyhanoglu M. Control and stabilization of nonholonomic dynamic systems, IEEE Transactions on Automatic Control, 1992, 37(11), 17461757. [21] Khalil H. Nonlinear systems. Macmillan Publishing Company, 2nd edition, New York, 1996. [22] Chin C. S, Lau M. W. S., Low E. and Seet G. G. L. Robust controller design method and stability analysis of an underactuated underwater vehicle, International Journal of Applied Mathematics and Computer Science, 2006, 16(3), 101-112.

24

Appendices Appendix 1- Station-keeping and Vertical Plane Model Station-keeping mode:

1 1 0 0 Ts   0 0    

 m  X u  0 Ms    0   0 Xu 0 Ds   0  0

0 Yv 0 0

0 0  ; 0  Nr 

0 0 Zw 0

 gs2  0 0 

0

0

m  Yv 0

0 m  Z w

0

0

  0 0 0 ; gs2  0  

1 m  Z w

 0  0   I r  N r  0

0 (Yv  m) r 0 0 Cs ( ν s )   0 0  0  Yv u  X u u T

 1 gs1    m  X u

0   sin   ; cos      sin  

0 sin  cos   sin 

T

 0 0 ; 

1 m  Yv

T

  0 ; gs4  0 0 0  

0 0  0 ( m  X u )u   0 0  0 0 

T

1  ; I r  N r 

Vertical Plane Model:

m  X u  Mv   0  0 

0 m  Z w 0

Xu  Dv   0 0 

 0 0 (m  Z w )q    0 ( X u  m)u  ; 0  ; C v ( ν v )  0 0 I y  M q  0 ( Z w  X u )u  0

0 Zw 0

0  1  0  ; Tv   0   M q 

1 0

0 cos 

   cos 

0   cos     0 Gv    ; J v (ηv )   sin  ( zG  z B )W sin    0

25

0  cos   ;   cos  

sin  cos 0

0 0 ; 1

 1 g v1    m  X u

  0 ; g v3  0  

T

T

  1 0 0 ; g v2  0 m Z w   

0

1   I y  M q 

Appendix 2- Perturbation Matrices for Station-Keeping Mode

0 0 0 0 0  21.7240  0 51 . 7013 0 0 0 0    0 0 94.3000 0 0 0  M A     0 0 4.2176 0 0   0  0 0 0 0 7.1148 0    0 0 0 0 4.7673  0

 0   0  0  CA ( ν )    0  a u 3  u  a2

 a3

0

0

0

0

0

a3

0  a3 0 u 1

a

 a2

0 u

a2

u

u

0 u

u

b3

u

 b2

0

u

a1

 b3

0

 a1

u

u

0 u

u

b1

where l

u

a1  21.0981u , a1  21.7240u l

u

l

u

a 2  51.0002v, a 2  51.7013v a3  91.8370 w, a3  94.3000 w l

u

l

u

b1  3.5760 p, b1  4.2176 p b2  2.7687 q, b2  7.1148q l

u

b3  2.4393r , b3  4.7673r 0 0 0 0 0  4.07  0 3 . 20 0 0 0 0    0 0 2.60 0 0 0  D     0 0 1.82 0 0   0  0 0 0 0 1.59 0    0 0 0 0 1.11  0

26

u a2  u  a1  0  u  b2  u  b1   0 

T

Appendix 3 - Derivation of Q O (η s ) Assuming that the elements of as function of

Q O (ηs ) are qO ij (ηs ) , i  1,2,3,4 , the equation (15) can be written

v s . This can be written as:

 v  Q M 1C ( v ) ν  0 Q o s o s s s s where

(55)

v s  Qo1 ηs and for brevity we omit the argument ηs of qo ij (ηs ) . Expanding (55) and

grouping the same terms such as

u 2 , v 2 , w2 , r 2 , uv, uw, vw, rw, vr , ur together gives the following

PDE:

 qoi1   q q q q q  cos  oi1 sin u 2    oi1 sin  oi1 cos  oi 2 cos  oi 2 sin y y x y  x   x  qoi 4

 Yv  X u I r  N r

  q   q  q q q uv    oi 2 sin  oi 2 cos v 2   oi1  oi 3 cos  oi 3 sin uw y x y  x   z  

 q  q   q q q  q   q     oi 3 sin  oi 3 cos  oi 2 vw   oi 3  w2   oi 4 r 2   oi 3  oi 4  rw y z  z   z        x  q q q Y m  vr   oi 2  oi 4 sin  oi 4 cos  qoi1 v x y m  X u     q  q q   oi1  oi 4 cos  oi 4 sin  qoi 2 ur  0 x y    (56) As observed in (56), they are ten PDEs to be solved. Now, we assumed the solution of the PDEs in (56) as:

qoi1 

m  X u [(Ci 3 x  Ci1 ) sin  (Ci 3 y  Ci 2 ) cos ] Yv  m

(57a)

qoi 2 

m  X u [(Ci 3 x  Ci1 ) cos  (Ci 3 y  Ci 2 ) sin ] Yv  m

(57b)

qoi 3  Ci 4 qoi 4  where

(57c)

I r  N r Ci 5  Yv  X u

(57d)

Ci1 , Ci 2 , Ci 3 , Ci 4 are arbitrary constant with the following values:

27

C13  C11  C14  C15  0, C12  1 ,

C23  C22  C 2 24  C25  0, C21  1

C33  C31  C32  C35  0, C34  1 ,

C43  C41  C 242  C45  0, C44  1

substituting the solution

(58)

qoi1 , qoi 2 , qoi 3 , qoi 4 into the PDEs in (56), the resulted PDEs can be

individually satisfied. The first PDEs in (56) gives:

 qoi1  q m  X u  cos  oi1 sin   0  Ci 3 sin cos  Ci 3 cos sin  0 y Yv  m  x 

(59a)

The second PDEs in (56) gives:

 Yv  X u  q q q  q oi1 0   sin  oi1 cos  oi 2 cos  oi 2 sin  q oi 4 I r  N r  y x y  x (59b) m  X u m  X u 2 2 2 2  C i 3 sin   C i 3 cos   C i 3 sin   C i 3 cos   C i 5  0 Yv  m Yv  m









The third PDEs in (56) gives:

 qoi 2  qoi 2 m  X u   sin   cos   0    Ci 3 sin cos  Ci 3 sin cos   0   x y Yv  m   (59c) The fourth PDEs in (56) gives:

 qoi1 qoi 3  qoi 3   z  x cos  y sin   0  

(59d)

The fifth PDEs in (56) gives:

 qoi 3 q q   sin  oi 3 cos  oi 2   0 y z   x

(59e)

The sixth PDEs in (56) gives:

 qoi 3   z   0  

(59f)

 qoi 4     0   

(59g)

The seventh PDEs in (56) gives:

The eighth PDEs in (56) gives:

28

 qoi 3 qoi 4   0  z   

(59h)

The ninth PDEs in (56) gives:

 q oi 2 q oi 4 q oi 4 m  X u    sin  cos  [(C i 3 x  C i1 ) sin  (C i 3 y  C i 2 ) cos ]    q  oi 1    y Yv  m x   m  X u  [(C i 3 x  C i1 ) sin  (C i 3 y  C i 2 ) cos ]  0 Yv  m (59i)

The tenth PDEs in (56) gives:

 qoi1 qoi 4  qoi 4 m  X u     x cos  y sin  qoi 2   Y  m [(Ci 3 x  Ci1 ) cos  (Ci 3 y  Ci 2 ) sin ]  v  m  X u  [(Ci 3 x  Ci1 ) cos  (Ci 3 y  Ci 2 ) sin ]  0 Yv  m (59j)

Appendix 4

29

Figure 1

2

Equilibrium

1.5

z

1 0.5 0 -0.5 10 0 -10 -20 y

-30

-20

-15

-10 x

Figure 1: 3D view of ROV in open loop

1

-5

0

5

Figure 2

Figure 2: Tracking error in body-fixed frame

1

Figure 3

outer loop 6 to 4 DOF

x1d +

Outer Controller

J (.)-1 -

inner loop +

+ ~x 1e

x 2d

uO

+ 6 to 4 DOF

+

uI

Inner Controller

u

TFT

-

τs +

M 1

+

1 s

+

-

x 1

x2 J(x1 )

x1 1 s

f 2e (.)

xˆ 2

x2 Observer

xˆ 1

x1

Figure 3. Closed-loop systems of cascaded RRC ROV (simulated as a full-order system by ignoring the variable without subscript ‘s’)

1

Figure 4a

Time response

0

10 20 Time(sec)

-0.2

30

Pitch rate(rad/s)

Roll rate(rad/s)

0.1 0.05 0 -0.05 -0.1

0

10 20 Time(sec)

30

0

10 20 30 Time(sec)

-0.5

0.2

2

0.1

1

0 -0.1 -0.2

0

10 20 30 Time(sec)

0

0

10 20 Time(sec)

2 1 0

30

Heave pos(m)

3

0

2 1

0

10 20 30 Time(sec)

0

10 20 Time(sec)

30

-3

4

0 -1 -2

actual estimate

2

0

10 20 30 Time(sec)

Sway pos(m)

0

4

3

0

2 0

-2

10 20 30 Time(sec)

x 10

0

10 20 Time(sec)

30

0.15

1.5

0.1

1

Yaw angle(rad)

-1

0

0.5

4

Pitch angle(rad)

0

0.2

1

Surge pos(m)

1

0.4

6

Roll angle(rad)

2

Time response

1.5

Yaw rate(rad/s)

actual estimate

Heave vel(m/s)

0.6 Sway vel(m/s)

Surge vel(m/s)

3

0.05 0 -0.05

0

10 20 30 Time(sec)

0.5 0 -0.5

0

10 20 30 Time(sec)

Figure 4a: Actual and estimated state for velocity (left) and position (right) of station keeping mode

1

Figure 4b

10 20 Time(sec)

0 -2

30

0

10 20 Time(sec)

0.5 0 -0.5

30

-5

x 10

-5

0

10 20 Time(sec)

30

5 Yaw rate(rad/s)

0.1

0

-10

10 20 Time(sec)

2 0

30

-4

Pitch rate(rad/s)

Roll rate(rad/s)

5

0

4

0 -0.1 -0.2 -0.3

0

10 20 Time(sec)

30

x 10

4

-5

0

10 20 Time(sec)

0

10 20 Time(sec)

0 -2 -4

30

Time response 3

0

2 1 0

50 100 Time(sec)

0

-6

0

-10

actual estimate

x 10

Heave pos(m)

2

1

2 Sway pos(m)

4

6

x 10

0.15

2 0 -2

30

0

10 20 Time(sec)

10 20 Time(sec)

30

-4

30

5

0.1

Yaw angle(rad)

0

-4

1.5

Roll angle(rad)

-1

Time response

Surge pos(m)

0

x 10

Heave vel(m/s)

actual estimate

1

Sway vel(m/s)

Surge vel(m/s)

6

Pitch angle(rad)

-5

2

0.05 0 -0.05

0

10 20 30 Time(sec)

x 10

0 -5 -10

0

10 20 Time(sec)

30

Figure 4b: Actual and estimated state for velocity (left) and position (right) of vertical plane

1

Figure 5

Figure 5: RDA used for proposed cascaded control system simulation

1

Figure 6

Figure 6: Position response of all controllers

1

Figure 7

Figure 7: Velocity response of all controllers

1

Table 1

Positions and Orientations

Linear and Angular Velocities

Motions in the xdirection(surge)

x

u

2

Motions in the ydirection(sway)

y

v

3

Motions in the zdirection(heave)

z

w

4

Rotations about the x-axis(roll)



p

5

Rotations about the y-axis(pitch)



q

6

Rotations about the z-axis(yaw)



r

DOF

Motion Descriptions

1

Table 1: Notations used in ROV

1

Table 2

Descriptions

Symbols

Actual and estimated velocity state

x s2  v s  u

v

w

Actual and estimated position state

x s1  η s  x

y

z



T ,

Body-fixed desired velocity state

x s2d  u d

vd

wd

rd

T

Body-fixed desired position state

x s1d  x d

yd

zd



Body-fixed velocity error state

~x ~ ˆ s2e  x s2d  x s2  u e

Body-fixed tracking error state

~x ~ s1e  x e

Outer and inner controllers



u O  u Ox

~y e

u Oy

~z e

u Oz

r

T

d

,

T xˆ s2  vˆ s  uˆ vˆ wˆ rˆ 

xˆ s1  ηˆ s  xˆ

T zˆ ˆ 

T

v~ e

T ~ re 

~ w e

~ e T u O

 ,u T

I



 u Ix

Table 2: Notations used in cascaded structure derivation

1



u Iy

u Iz

u I



T

Table 3

Table 3: Controller parameters

1

Table

Item

Parameters

P-Decoupled

Backstepping

P-P

SinglePID

PI-P

1

10

200

50

150

200

15

200

60

150

150

0.027

3.730

0.055

1.570

0.335

0.002

0.002

0.002

0.220

0.286

98.61

21.17

25.25

339.4

260.1

6

Settling time-position (in seconds) Settling timevelocity(in seconds) Steady state position error Steady state velocity error Thrust- Euclidean norm Stability- inner

-5675

-9837

-5302

-6487

-6229

7

Stability- outer

-1925

-3968

-1760

-2038

-1928

Input perturbation to pos Input perturbation to vel Radially unbounded Simultaneous stabilization & tracking

147.0

327.1

173.7

753.1

148.1

1797

1797

1797

1797

1797

YES YES

YES YES

YES YES

YES NO

YES YES

2 3 4 5

8 9 10 11

loop( VI  0 )

loop( VO  0 )

Table 4: Controller comparison against various parameters

1