Adaptive PD-controller for positioning of a remotely operated vehicle close to an underwater structure: Theory and experiments

ARTICLE IN PRESS

Control Engineering Practice 15 (2007) 411–419 www.elsevier.com/locate/conengprac

Adaptive PD-controller for positioning of a remotely operated vehicle close to an underwater structure: Theory and experiments Nguyen Quang Hoang, Edwin Kreuzer Hamburg University of Technology, Mechanics and Ocean Engineering, Eissendorfer Strasse 42, 21073 Hamburg, Germany Received 24 May 2005; accepted 4 August 2006 Available online 7 November 2006

Abstract The requirement for high accuracy in dynamic positioning of remotely operated vehicles (ROV), especially when tasks close to underwater structures have to be performed, demands high precision of sensor systems. Taut-wire and passive arm systems can satisfy this demand in measuring ROVs positions and orientations relative to a structure. However, the main drawback of these sensor systems is that additional forces act on ROVs due to the mechanical connection. In order to solve this problem, an adaptive PD controller is proposed and designed for dynamic positioning of ROVs working in close proximity of structures. Invoking the adaptation law, these additional forces caused by the passive arm and umbilical, and even by the uncertainties in gravity and buoyancy can be identified and compensated. By choosing an adequate Lyapunov candidate function, the system’s stability is proven. The effectiveness of this design control method is demonstrated by means of numerical simulations and experiments. r 2006 Elsevier Ltd. All rights reserved. Keywords: Remotely operated vehicles (ROVs); Underwater vehicles; Adaptive PD control; Dynamic positioning

1. Introduction The application of a ROV is mainly divided into three phases. Firstly, the vehicle is launched into water from the mother-ship and is then controlled to reach a structure. Secondly, the vehicle conducts assigned tasks, e.g. inspection or maintenance around the structure. In the last phase, the vehicle is navigated back to the mother-ship (Pinto, 1996). The Petri net-based approach introduced in Caccia, Coletta, Bruzzone, and Veruggio (2005) can be used to switch the operation of the ROV from one phase to another phase. The technical features of sensors may vary according to the working phases. Consequently, the vehicle should be equipped with different sensor systems for the different performance phases. For the first phase, traveling from mother-ship to the underwater structure, the required precision of the sensor system may be low. Therefore, a navigation system based on wave-propagation or the Corresponding author. Tel.: +49 40 4 2878 2355; fax: +49 40 4 2878 2028. E-mail address: [email protected] (N.Q. Hoang).

0967-0661/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2006.08.002

inertial navigation system is usually used to determine the position of the vehicle. In the second phase, working close to the structure, the relative position of the ROV with respect to the structure is important in positioning the ROV precisely. Accordingly, the requirement for the accuracy of the sensor system is significantly higher than that in the first phase. For the purpose of this study only the second phase is considered. The navigation sensor system used in the first phase can still be applied to this phase. However, many operations in the second stage require to maintain a ROV close to the structure. Moreover, inspecting a structure is usually the most time-consuming task throughout the ROVs working duration. Thus, a sensor system with a mechanical connection to the structure is suggested because it can measure the ROVs positions more accurately than those done by the navigation systems (Cunha, Scieszko, Costa, Hsu, & Anna, 1993). So far, taut-wire and passive arm are two of such a mechanically connected sensor system. Taut-wire is a cable stretching between a ROV and its underwater structure. The relative position of the ROV is calculated by measuring the cable length and two angles.

ARTICLE IN PRESS N.Q. Hoang, E. Kreuzer / Control Engineering Practice 15 (2007) 411–419

412

The passive arm measurement system is installed on a mechanical arm with joint angle transducers, no actuators and at least six degrees of freedom (Fig. 1). The arm is mounted on board the ROV and the end effector is attached (e.g. magnetically) to the underwater structure at some convenient points during operation (Hsu, Costa, Lizarralde, & Cunha, 1999, 2000; Pinto, 1996). However, a common challenge in employing these two systems is the existence of additional forces caused by mechanical connection which acts on ROVs as unknown uncertainties. Sliding mode is one of the control methods applied to reduce the influence of uncertainties in controlling ROVs (see, Khalil, 1996; Slotine, 1991; Utkin, 1978 for further details on sliding mode control). Sliding mode control has, however, several limitations. It requires a good dynamic model of the system and knowledge of the inaccuracies or uncertainties in the model (Hills & Yoerger, 1994). Other notable problems include chattering phenomena and the excitation of unmodeled high-frequency dynamics. In parallel with sliding mode, a traditional PD controller is still applied widely in controlling the robot manipulator because of its simplicity and capability to guarantee the stability. As is well known, a simple PD controller suffices to stabilize any kind of rigid manipulator around a reference position, provided that the gravitational forces are accurately compensated (Arimoto & Miyazaki, 1984). A number of studies have shown that PID controllers can compensate any unknown constant uncertainties, but the closed system stability is still questionable. It was shown that the PID controller is locally asymptotically stable provided the gain matrices satisfy a complex relationship (Arimoto & Miyazaki, 1984; Wen & Murphy,

Camera

ROV

Underwater structure

Umbilical

1990). Based on the result proved in Takegaki and Arimoto (1981), Fossen introduced PD plus restoring forces (gravity and buoyancy) for the set-point regulation of a ROV (Fossen, 1994). To apply this algorithm the restoring forces must be identified exactly. Unfortunately, it is difficult to meet this condition in practice. Moreover, based on the study of Arimoto and Miyazaki (1984) a PID controller was introduced in Fossen (1994), so that the positioning accuracy was improved. However, by applying a PID controller only the local stability is proven. Cheah and Sun (2004) and Sun and Cheah (2003) proposed a nonlinear adaptive PD controller, that is valid globally in the presence of uncertainties in the gravity and buoyancy forces. Although the controller’s structure in this study is similar to the controller introduced in Sun and Cheah (2003), it does not require a saturation function to ensure the global stability of the system. In the present study an adaptive PD controller for the dynamic positioning of ROVs that work in close proximity of structures is introduced, with this controller the global stability of the closed loop is proven. The approach of this study was inspired by the idea of composite control developed by Kelly (1993) and Tomei (1991) for the set-point regulation of a robotic manipulator. In the study introduced by Tomei (1991) the controller was a composition of a PD and an adaptive controller. The adaptation law is used to estimate the unknown gravity parameter online. In this study, this method for the dynamic positioning of ROVs is applied. The additional forces caused by the passive arm, umbilical and even by the uncertainties in gravity and buoyancy can be identified and compensated. The results in this study prove that the design can guarantee the global asymptotic stability of the entire system even in cases where these additional forces are only vaguely known. Throughout this paper, the notation am and aM indicate the smallest and largest eigenvalues of a symmetric positive-definite matrixpA, respectively. The norm of vector ffiffiffiffiffiffiffiffi x is defined as kxk ¼ xT x and ðÞT means the transpose of ðÞ. This paper is organized as follows. The nonlinear dynamics of ROVs with six degrees of freedom (DOF) and some useful properties of dynamic system are introduced in Section 2. In Section 3, an adaptive PD controller is designed for the dynamic positioning of ROVs working close to an underwater structure. The numerical and experimental verification of the controller is presented in Section 4 and conclusions in Section 5. 2. Dynamic model of vehicles and some properties

Passive arm

Fig. 1. ROV with its passive arm.

Dynamical behavior of a ROV are commonly described through six degrees of freedom (DOF) non-linear differential equations in two coordinate frames (bodyfixed frame and earth-fixed frame shown in Fig. 2). In general, underwater vehicles can be represented by the following vector equation (Fossen, 1994) with respect to

ARTICLE IN PRESS N.Q. Hoang, E. Kreuzer / Control Engineering Practice 15 (2007) 411–419

413

development and stability analysis. Here, it is assumed that the pitch angle is smaller than p=2, jyjop=2.

P1. The inertial matrix is symmetric and positive definite, MðgÞ ¼ M T ðgÞ 8 g 2 R6 . There are positive constants mm and mM such that mm pkMðgÞkpmM . _ P2. MðgÞ  2Cðg; g_ Þ is a skew-symmetric matrix, e.g. _ sT ½MðgÞ  2Cðg; g_ Þs ¼ 0; 8s 2 R6 . _ P3. MðgÞ ¼ Cðg; g_ Þ þ C T ðg; g_ Þ. P4. Damping matrix Dðg; g_ Þ is strictly positive and bounded, e.g. Dðg; g_ Þ40; 8g; g_ 2 R6 and there are positive constants such that d m pkDðg; g_ Þkpd M . P5. The Coriolis matrix Cðg; g_ Þ is bounded in g and linear in g_ , that is, for all x 2 R6 holds Cðg; g_ Þx ¼ Cðg; xÞ_g. There exists a positive constant kc such that kCðg; g_ Þkpkc k_gk.

Fig. 2. ROV model with two reference frames.

body-fixed frame M V m_ þ C V ðmÞm þ DV ðmÞm þ gV ðgÞ þ d V ¼ Bu,

(1)

where m ¼ ½u; v; w; p; q; rT is the vector of linear and angular velocity of the vehicle. The notation M V , C V ðmÞm, DV ðmÞm, gV ðgÞ are used to describe the mass matrix, the centripetal and Coriolis terms, the damping terms, and the gravitational and buoyant forces in the body-fixed frame, respectively. The uncertainties in buoyant forces and the forces due to the sensor arm and the umbilical are summarized in the vector d V ¼ d V ðg; hÞ, with an unknown constant h. With respect to the earth-fixed frame the equation of motion MðgÞ€g þ Cðg; g_ Þ_g þ Dðg; g_ Þg_ þ gðgÞ þ d Z ¼ J T ðgÞBu

ð2Þ

is obtained, where g ¼ ½x; y; z; f; y; cT is the position and attitude vector, MðgÞ 2 R66 is the inertia matrix including the added inertia, Cðg; g_ Þ_g 2 R6 describes the centripetal and Coriolis terms, Dðg; g_ Þ_g 2 R6 collects the damping terms, and gðgÞ 2 R6 describes the gravitational and buoyant forces the so-called restoring forces. The matrix JðgÞ 2 R66 determines the transformation between the vehicle and an inertial frame and d Z 2 R6 summarizes the uncertainties in buoyancy and uncertainties due to a tether cable and sensor arm, with d Z ¼ J T ðgÞd V ¼ d Z ðg; hÞ . The matrix B 2 R6p is a priori known constant matrix describing the thruster configuration. This matrix satisfies the condition that BB T is non-singular, and u 2 Rp is the vector whose components are thruster forces, here p is the number of thrusters. The wave-induced forces/moments are assumed negligible since the vehicle operates below the wave-affected zone (the operating depth is normally significantly greater than 20 m). The variations of water density are neglected. More details about transformation from (1) to (2) can be found in Fossen (1994). The dynamic system given by (2) exhibits the following properties that are utilized in the subsequent control

3. Adaptive PD-controller design In order to design an adaptive PD controller some assumptions are given. Firstly, the center of buoyancy rB is known but its magnitude is not exactly defined. The buoyancy B ¼ G þ DB, B is the buoyancy magnitude, G ¼ mg is the weight of the ROV, where m is the mass of the ROV and g is the gravitational acceleration. Secondly, the vectors rC and rA which describe the location of the points where the umbilical and sensor arm are attached on the ROV are known. The components of forces due to the umbilical and sensor arm are unknown constants. These assumptions are acceptable, because when the ROV holds its position close to a structure that forces are constants. With these assumptions, the uncertainties d Z in Eq. (2) can be expressed as d Z ¼ UðgÞh, where h is an unknown constant parameter vector. In this study h ¼ ½DB; F Cx ; F Cy ; F Cz ; F Ax ; F Ay ; F Az T , UðgÞ 2 R67 is the regression matrix. Eq. (2) is rewritten in the following form: MðgÞ€g þ Cðg; g_ Þ_g þ Dðg; g_ Þ_g þ gðgÞ þ UðgÞh ¼ J T ðgÞBu; Consider the following control law 8 T > < s ¼ J ðgÞ½K P g~  K D g_ ^ gðgÞ þ UðgÞh; > : u ¼ B T ðBB T Þ1 s9B # s;

h_ ¼ 0.

ð3Þ

(4)

where h^ is the estimate of h with the adaptation dynamics   e0 _^ T ~ _ g , (5) h ¼ CU ðgÞ g þ 1 þ ak~gk in which g~ ¼ g  gd ; gd ¼ const, is the position error, K P and K D 2 R66 are symmetric positive definite matrices, C 2 R77 is a positive definite matrix, e0 o1 and a are positive constants. The control schema of the system is shown in Fig. 3.

ARTICLE IN PRESS N.Q. Hoang, E. Kreuzer / Control Engineering Practice 15 (2007) 411–419

414

d

. ^ =JT[-KP-KD+g()+Φ] # u=B u

=-T



0 1+a 



ROV

precisely. In addition, as the number of adaptation variables will be increasing, therefore the amount of computation for the control law will also increase. For designing the controller, only the low and upper bounds of matrices MðgÞ, and Dðg; g_ Þ are required.

. , 

1 s

4. Case study 4.1. Numerical simulation

Fig. 3. Adaptive PD controller for ROV.

Theorem. Consider the system (3) with the control law (4) and the adaptation law (5). If the following conditions: e0 e0 kDm þ d m þ mM  kc 4 a 1 1  e0 mM  e0 kDM  e0 d M 40, ð6Þ 2 2 kPm  12kDM  12d M 40

(7)

are satisfied then the equilibrium ð~g; g_ Þ ¼ ð0; 0Þ is globally asymptotically stable except for the singular point y ¼ p=2. The proof of the theorem is given in Appendix A. 3.1. Discussion 1. The adaptive PD controller developed in this study can also be applied for ROVs which do not use the sensor systems with mechanical connection to underwater structures. If so, the number of variables for adaptation can be reduced from 7 to 4, and then the dimension of the regression matrix UðgÞ becomes 6  4 and h ¼ ½DB; F Cx ; F Cy ; F Cz T . 2. With the controller designed in the present paper, the effect of constant water current is also estimated for compensation in case its influence on the vehicle is lumped together into an unknown constant vector (Fossen & Balchen, 1991). However, it is impossible to distinguish the influence of water current and uncertainties in gravity, buoyancy and the forces due to the umbilical. According to the Barbalat’s lemma, only the convergence to zero of position errors and the boundedness of the estimated parameter are guaranteed. The experiments of this study were carried out in the still water basin, therefore the water current is not taken account in this paper. 3. Some control laws designed in Fossen, Loria, and Teel (2001), Godhavn, Fossen, and Berge (1998), Smallwood and Whitcomb (2004), for tracking control of marine vehicles, such as adaptive control, nonlinear adaptive control, and adaptive backstepping control, can also be applied for the problem of this study. However, if these control laws are used, more information about the dynamic structure of the vehicle will be needed and the matrices MðgÞ; Cðg; g_ Þ; Dðg; g_ Þ must be determined more

In order to illustrate the performance of the adaptive PD controller, simulation results are given by means of MATLAB. A comparison between the PD plus the compensation of the restoring forces is introduced in Fossen (1994) and the adaptive PD controller derived in this study is carried out. The experimental ROV (Fig. 8) developed at the Institute of Mechanics and Ocean Engineering of the Hamburg University of Technology is used as the model for these simulations. The ROV has a buoyant chamber on top and a ballast on the bottom. This structure stabilizes the roll and pitch motions. Therefore, this study considers the ROV with four DOFs as surge, sway, heave and yaw. The ROV is designed in Autodesk Inventor, this program provides the parameters such as: the mass, inertial matrix, center of mass. The added mass and the damping matrix are determined based on Hapel (1990). The damping matrix can also be determined by identification (Pepijn, Johansen, Serensen, Flanagan, & Toal, 2006). Below are the parameters of the model: M V ¼ diagðmx ; my ; mz ; I zz Þ ¼ diagð120:0; 128:0; 136:0; 14:7Þ, DV ðmÞ ¼ diagð130juj; 195jvj; 286jwj; 10:18jrjÞ, gV ðgÞ ¼ ½0; 0; 0; 0T . In case of the position control, the vehicle moves with low velocity. So one can take d M ¼ d m ¼ 0, the constant kc is determined from the mass matrix. With the considered parameters, one gets mm ¼ 14:7; mM ¼ 136, and kc ¼ 175:5. For simulation, an arbitrary constant vector h ¼ ½5; 4; 4; 0; 8; 8; 0ðNÞT is chosen. Now, setting K P ¼ diagð100; 100; 120; 55Þ and K D ¼ diagð90; 90; 90; 80Þ, by choosing e0 ¼ 0:3; a ¼ 2:0 the conditions (6) and (7) are satisfied. In order to show the advantage of the control design proposed, two simulations are conducted—one with PD plus gravity compensation, and the other with the adaptive PD control. In both simulations, the ROV is moving from the position ½x; y; z; cT ¼ ½0:52; 0:73; 0:62; 0:83T to the position ½x; y; z; cT ¼ ½0; 0; 0; 0T . The results of the simulations are shown in Figs. 4–7. These figures indicate that the system reaches the desired position by a PD controller much faster than by an adaptive PD controller. These figures show that the steady state was reached after about 15 s with the PD controller, whereas it took about 30 s with the adaptive PD controller.

ARTICLE IN PRESS N.Q. Hoang, E. Kreuzer / Control Engineering Practice 15 (2007) 411–419

0.6

1.2 PD adaptive PD

0.5

PD adaptive PD

1

0.4 yaw [rad]

0.8

0.3 x [m]

415

0.2

0.6 0.4

0.1

0.2

0

0

-0.1

-0.2 0

20

40

60

0

20

time [s]

40

60

time [s]

Fig. 4. Time history of coordinate xðtÞ.

Fig. 7. Time history of yaw motion cðtÞ.

0.8 PD adaptive PD 0.6

y [m]

0.4

0.2

0

-0.2 0

20

40

60

time [s] Fig. 5. Time history of coordinate yðtÞ.

Fig. 8. The experimental vehicle.

However, the constant uncertainties can not be compensated by the PD controller. Thus the system states obtain a hypersphere with a radius depending on the magnitude of uncertainties. This radius can be smaller by increasing matrix K P . Yet, too large K P may cause the system unstable. In the case of adaptive PD controller, it guarantees zero steady-state error. Therefore, the proposed adaptive PD controller ensures a better dynamic performance when the system is subjected to unknown constant uncertainties.

0.7 PD adaptive PD

0.6 0.5

z [m]

0.4 0.3 0.2

4.2. Experiments 0.1 0 0

20

40 time [s]

Fig. 6. Time history of coordinate zðtÞ.

60

In order to verify the theoretical results, an experimental model with the overall dimensions ð800  500  550 mmÞ was developed (Fig. 8). The model was designed to serve as a test-bed for different control, sensor, and propulsion systems. Its dry mass is about 80 kg, the ROV is designed

ARTICLE IN PRESS N.Q. Hoang, E. Kreuzer / Control Engineering Practice 15 (2007) 411–419 0.02 PD Adaptive PD 0.01

0 x [m]

so that its buoyancy is slightly greater than its weight. The propulsion system is made up of eight marine propellers, which are driven by DC motors. One passive arm with six DOFs is used as a position sensor for the region close to submerged structure. The absolute encoder at each joint has the resolution of 0:0880 . The two largest links are 0.7 m in length yielding about 1.4 m workspace radius. Through direct kinematics the position and attitude of the ROV are measured. The positioning error of this sensor system is of the order 5 mm if the flexibility of the arm is neglected. The adaptive PD controller requires ROV speed, which is not measured directly by the passive arm. Velocity could be measured by additional transducers, but this will increase the cost. Therefore, the estimation from a position measurement with the first-order lead filter is used in this study. The tests are carried out in a still water basin at the Institute of Mechanics and Ocean Engineering of the Hamburg University of Technology. The dimensions of the basin are 4000  2000  1500 mm. The proposed control law has been implemented on the vehicle in order to test its performance. In the experiments, the vehicle is forced to move to the coordinate origin from an arbitrary position in the reachable space of the sensor arm. The controller is written in C program language in Linux operation system. The sampling time of the closed loop is DT ¼ 30 ms, sampling frequency is 33 Hz. For the experiments, the following parameter have been used:

-0.01

-0.02

-0.03

-0.04 0

10

20

30 40 time [s]

50

60

70

Fig. 9. Time history of coordinate xðtÞ.

0.6 PD Adaptive PD

0.5 0.4

y [m]

416

0.3 0.2

K P ¼ diagð80; 80; 100; 30Þ; 0.1

K D ¼ diagð90; 90; 120; 45Þ; C ¼ 10I;

0

e0 ¼ 0:3; a ¼ 2:

-0.1 0

^ the Euler In order to update the adaptive variable h, method is used

20

30 40 time [s]

50

60

70

Fig. 10. Time history of coordinate yðtÞ.

^ hð0Þ ¼ 0.

Figs. 9–12 show the time history of the x; y; z, and yaw motion of the ROV. The ROV is driven by the control system to a desired position ½0; 0; 0; 0T , where it stays at rest. The set-point is reached after about 25 s. In these figures, dashed lines and solid lines represent the performances with a PD controller and with an adaptive PD controller, respectively. The experimental results prove that, the position errors are clearly improved when the adaptive PD controller is applied instead of a regular PD controller. Fig. 11 clearly shows the effect of uncertainties in buoyancy and reaction between the ROV and the sensor arm in the vertical direction. In this figure, coordinate z increased fast in the first 10 s because of vehicle’s buoyancy. After that, the adaptation force increased and compensated the difference between the buoyancy and the gravity. With an adaptive PD controller these uncertainties can be estimated online and compensated.

0.4 PD Adaptive PD 0.3

0.2 z [m]

_^ ^ þ DTÞ ¼ hðtÞ ^ þ hðtÞDT; hðt

10

0.1

0

-0.1 0

10

20

30

40

50

time [s] Fig. 11. Time history of coordinate zðtÞ.

60

70

ARTICLE IN PRESS N.Q. Hoang, E. Kreuzer / Control Engineering Practice 15 (2007) 411–419

417

University of Technology for their valuable support in implementing the experiments. This work was partially supported by DAAD, German Academics Exchange Service.

2.5 PD Adaptive PD 2

yaw [rad]

1.5

Appendix A 1 0.5 0 -0.5 0

10

20

30

40 time [s]

50

60

70

Fig. 12. Time history of yaw motion cðtÞ.

5. Conclusion

Proof. A tool commonly used in the proof of asymptotic stability are Lyapunov functions with cross terms of the form eg~ T MðgÞg_ , e40. This type of functions has been used widely in a number of studies (Berghuis & Nijmeijer, 1993; Chen, Chen, Wang, & Woo, 2001; Kelly, Santibanez, & Reyes, 1998; Loria, Lefeber, & Nijmeijer, 2000; Ortega, Loria, & Kelly, 1995; Pan & Woo, 2000). The motivation here is to construct a Lyapunov function with a derivative containing more negative terms than _gT K D g_ , for example the quadratic negative term of the position error. This allows to prove the globally asymptotic stability of the origin by invoking standard Lyapunov techniques. Considering the Lyapunov function candidate T

This paper describes the design and implementation of an adaptive PD controller for the dynamic positioning of a ROV. The main feature of this design is that it combines the PD control and the adaptive algorithm. This adaptive algorithm is used to estimate the unknown constant forces due to uncertainties in buoyancy and forces that are caused by mechanical connection between the ROV and an underwater structure. At the same time, the cable forces are estimated and then compensated. The numerical results in this study show that the controller is able to compensate the unknown constant uncertainties and provide good performance. The experimental results prove that an adaptive PD controller provides significantly higher accuracy in position control than those provided by a PD one. Moreover, an adaptive PD controller can guarantee the globally asymptotic stability of the entire system, which therefore shows the advantage in comparison with the local stability obtained through applying the PID controller, whose global stability has not been proven to date. The proposed controller requires only the regression UðgÞ, that depends only on the vehicle orientation. Therefore, the effect of noise of velocity measurements is reduced. The control technique has been finally tested in the position regulation of a ROV developed at the Institute of Mechanics and Ocean Engineering of the Hamburg University of Technology. The passive arm was used as a position sensor in the region close to an underwater structure.

V ¼ V 1 þ 12h~ C1 h~ with h~ ¼ h^  h, and V 1 is defined as follows: V 1 ¼ 12g_ T MðgÞg_ þ 12g~ T K P g~ þ eg~ T MðgÞg_ , where e¼

e0 . 1 þ ak~gk

(A.1)

In Whitcomb, Rizzi, and Kodischek (1993) it has been shown that, with a positive constant e and kpm =mM 4e2 the matrix P defined as follows is symmetric positive definite " # 1 MðgÞ eMðgÞ P¼ . (A.2) KP 2 eMðgÞ Therefore, V 1 40 and consequently V 40. Differentiating V with respect to time one gets _ V_ ¼ g_ T MðgÞ€g þ 12g_ T MðgÞ_ g þ g~ T K P g_~ T

_ þ e~gT MðgÞ€g þ e~gT MðgÞ_ g þ eg_~ MðgÞ_g T ~_ þ e_g~ T MðgÞ_g þ h~ C1 h.

Substituting dynamics (3) of ROV and control law (4) into (A.3) yields V_ ¼ g_ T ½K P g~  K D g_  Cðg; g_ Þg_  Dðg; g_ Þg_  _ þ 1g_ T MðgÞ_ g þ g~ T K P g_~ 2

 eg~ T ½K P g~ þ K D g_ þ Cðg; g_ Þg_ þ Dðg; g_ Þg_  T _ þ eg~ T MðgÞ g_ þ eg_~ MðgÞg_

Acknowledgements The first author would like to thank Mr. Borngr’’aberSander, Mr. Brennecke and Mr. Demir at the Institute of Mechanics and Ocean engineering of the Hamburg

ðA:3Þ

T _~ þ e_g~ T MðgÞ_g þ ð_g þ e~gÞT UðgÞh~ þ h~ C1 h.

With a symmetric matrix K P one gets _gT K P g~ þ g~ T K P g_~ ¼ 0.

ARTICLE IN PRESS N.Q. Hoang, E. Kreuzer / Control Engineering Practice 15 (2007) 411–419

418

From Property P2 one obtains

With this e_ the last term in (A.4) can be rewritten

1_T _ 2g ½MðgÞ

e_ g~ T MðgÞg_ ¼ 

e0 a~gT g_ g~ T MðgÞg_ ð1 þ akg~ kÞ2 kg~ k   e0 a T T g~ g~ MðgÞ g_ . ¼  g_ ð1 þ ak~gkÞ2 k~gk

 2Cðg; g_ Þ_g ¼ 0.

With adaptation law (5) one obtains ðg_ þ eg~ ÞT UðgÞg~ þ g~ T C1 g_~ ¼ g~ T ½UT ðgÞð_g þ e~gÞ þ C1 g_~  ¼ 0,

Therefore, using property P1 one obtains

and one gets



V_ ¼  g_ T K D g_  g_ T Dðg; g_ Þg_ þ eg_ T MðgÞg_

Finally, one gets  e0 e0 V_ p  kDm þ d m þ mm  kc 4 a  1 1 e0 mM  e0 kDM  e0 d M k_gk2 2 2   1 1  e0 kPm  kDM  d M kg~ k2 , 2 2

 e~gT K P g~  e~gT K D g_  e~gT Cðg; g_ Þ_g  e~gT Dðg; g_ Þ_g _ þ eg~ T MðgÞ g_ þ e_g~ T MðgÞg_ , with property P3, one obtains V_ ¼  g_ T K D g_  g_ T Dðg; g_ Þ_g þ eg_ T MðgÞg_  eg~ T K P g~ T

 eg~ K D g_  eg~ Dðg; g_ Þg_

eg~

K D g_ pe12ðg~ T K D g~ þ g_ T K D g_ Þ pe12kDM ðk~gk2 þ k_gk2 Þ pe0 12kDM ðk~gk2 þ k_gk2 Þ,

eg~ T Dðg; g_ Þg_ pe12ðg~ T Dðg; g_ Þg~ þ g_ T Dðg; g_ Þg_ Þ pe12d M ðkg~ k2 þ kg_ k2 Þ pe0 12d M ðkg~ k2 þ kg_ k2 Þ, here the property P4 has been used. With property P5 one can evaluate e0 g~ T C T ðg; g_ Þg_ 1 þ ak~gk k~gk kCðg; g_ Þk:kg_ k pe0 1 þ ak~gk k~gk kc kg_ k2 pe0 1 þ ak~gk e0 p kc k_gk2 . a

eg~ T C T ðg; g_ Þg_ ¼

Note that pffiffiffiffiffiffiffi d d=dtðeT eÞ eT e ) ðkekÞ ¼ pffiffiffiffiffiffiffi dt 2 eT e T T T e e_ eT e_ e_ e þ e e_ pffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffi ¼ . ¼ eT e kek 2 eT e

kek ¼

Differentiating (A.1) with respect to time gives e_ ¼ 

e0 ag~ T g_ . ð1 þ ak~gkÞ2 :k~gk

ðA:6Þ

ðA:4Þ

Now some terms in V_ can be estimated. At this point, note that the following inequalities hold: T

(A.5)

with conditions (6) and (7), the function V_ is negative and vanishes if and only if ðg_ ; g~ Þ ¼ ð0; 0Þ.

T

þ e~gT C T ðg; g_ Þ_g þ e_g~ T MðgÞ_g.

e0 e0 mM kg_ k2 p_eg~ T MðgÞg_ p  mm kg_ k2 . 4 4

References Arimoto, S., & Miyazaki, F. (1984). Stability and robustness of PID feedback control for robot manipulators of sensor capability. In M. Brady, & R. Paul (Eds.), Robot research (pp. 783–799). Cambridge: MIT Press. Berghuis, H., & Nijmeijer, H. (1993). Global regulation of robots using only position measurements. Systems and Control Letters, 21, 289–293. Caccia, M., Coletta, P., Bruzzone, G., & Veruggio, G. (2005). Execution control of robotic tasks: A petri net-based approach. Control Engineering Practice, 13(8), 959–971. Cheah, C. C., & Sun, Y. C. (2004). Adaptive region control for autonomous underwater vehicles. In Proceedings of the IEEE/MTS OCEANS, Japan (pp. 288–295). Chen, Q., Chen, H., Wang, Y., & Woo, P. (2001). Robust adaptive trajectory tracking independent of models for robotic manipulators. Journal of Robotic Systems, 18(9), 545–551. Cunha, J. P. V., Scieszko, J. L., Costa, R. R., Hsu, l., & Meyer Sant’ Anna, A. C. C. (1993). Short range position measurement systems for underwater vehicle dynamic positioning. In Proceedings of IEEE Oceans, (pp. 484–489) Canada: Victoria. Fossen, T. I. (1994). Guidance and control of ocean vehicles. University of Trondheim Norway: Wiley. Fossen, T. I., & Balchen, J. G. (1991). The NEROV autonomous underwater vehicle. In OCEANS ’91’. Hawaii: Honolulu. Fossen, T. I., Loria, A., & Teel, A. (2001). A theorem for UGAS and ULES of passive nonautonomous systems: Robust control of mechanical systems and ships. International Journal of Robust Nonlinear Control, 11, 95–108. Godhavn, J. M., Fossen, T. I., & Berge, S. P. (1998). Nonlinear and adaptive backstepping designs for tracking control of ships. International Journal of Adaptive Control and Signal Processing, 12, 649–679. Hapel, K. H. (1990). Festigkeitsanalyse dynamisch beanspruchter OffshoreKonstruktionen, (Strength analysis of offshore structures under dynamic loading), Braunschweig, Vieweg. Hills, S. J., & Yoerger, D. R. (1994). A nonlinear sliding mode autopilot for unmanned undersea vehicles. In OCEANS’94. Oceans engineering for today’s technology and tomorrow’s preservation. (Vol. 3) (pp. III/ 93–III/98), Brest, France. Hsu, L., Costa, R. R., Lizarralde, F., & Cunha, J. P. (1999). Passive arm based dynamic positioning system for remotely operated underwater

ARTICLE IN PRESS N.Q. Hoang, E. Kreuzer / Control Engineering Practice 15 (2007) 411–419 vehicles. 1999 IEEE International conference on robotics and automation, Detroit, MI. Hsu, L., Costa, R. R., Lizarralde, F., & Cunha, J. P. (2000). Dynamic positioning of remotely operated underwater vehicles. IEEE Robotics and Automation Magazine, 7(3), 21–31. Kelly, R. (1993). Comments on ‘‘Adaptive PD controller for robot manipulators’’. IEEE Transactions on Robotics and Automation, 9(1), 117–119. Kelly, R., Santibanez, V., & Reyes, F. (1998). A class of adaptive regulators for robot manipulators. International Journal of Adaptive Control and Signal Processing, 12, 41–62. Khalil, H. K. (1996). Nonlinear systems (2nd ed.). Upper Saddle River, NJ: Prentice-Hall. Loria, A., Lefeber, E., & Nijmeijer, H. (2000). Global asymptotic stability of robot manipulators with linear PID and PI2D control. SACTA, 3(2), 138–149. Ortega, R., Loria, A., & Kelly, R. (1995). A semiglobally stable feedback PI2D regulator for robot manipulators. IEEE Transactions on Automatic Control, 40(8), 1432–1436. Pan, L., & Woo, P. Y. (2000). PD manipulator controller with fuzzy adaptive gravity compensation. Journal of Robotic Systems, 17(2), 93–106. Pepijn, W.J. van de Ven, Johansen, T.A., Serensen, A.J., Flanagan, C., & Toal, D. (2006). Neural network augmented identification of underwater vehicle models. Control Engineering Practice. In press, corrected proof (doi: 10.1016/jconengprac.2005.11.004).

419

Pinto, F. (1996). Theoretische und experimentelle Untersuchung zur Sensorik und Regelung von Unterwasserfahrzeugen, (Theoretical and experimental study of sensors and control of remotely operated underwater vehicles), Fortschritt-Berichte VDI, Reihe 12, Nr. 292, VDI-Verlag, D’’usseldorf. Slotine, J. J. E. (1991). Applied nonlinear control. Englewood Cliffs, NJ: Prentice-Hall. Smallwood, D. A., & Whitcomb, L. L. (2004). Model-based dynamic positioning of underwater robotic vehicles: Theory and experiment. IEEE Journal of Oceanic Engineering, 29(1), 169–186. Sun, Y. C., & Cheah, C. C. (2003). Adaptive setpoint control for autonomous underwater vehicles. In Proceedings of the 42nd IEEE conference on decision and control. (pp. 1262–1267), Hawaii, USA. Takegaki, M., & Arimoto, S. (1981). A new feedback method for dynamic control of manipulators. Transactions of ASME, Journal of Dynamic Systems, Measurement and Control, 102, 119–125. Tomei, P. (1991). Adaptive PD Controller for robot manipulators. IEEE Transactions on Robotics and Automation, 7(4), 565–570. Utkin, V. (1978). Sliding modes and their application in variable structure control. Moskow: Mir. Wen, J.T., & Murphy, S. (1990). PID control for robot manipulators. Technical report 54, CIRSSE. Whitcomb, L. L., Rizzi, A. A., & Kodischek, D. E. (1993). Comparative experiments with a new adaptive controller for robot arms. IEEE Transactions on Robotics and Automation, 9(1), 59–70.