Control of an underactuated remotely operated underwater vehicle
Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering http://pii.sagepub.com/
Control of an underactuated remotely operated underwater vehicle M W S Lau, S S M Swei, G Seet, E Low and P L Cheng Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 2003 217: 343 DOI: 10.1177/095965180321700501 The online version of this article can be found at: http://pii.sagepub.com/content/217/5/343
Published by: http://www.sagepublications.com
On behalf of:
Institution of Mechanical Engineers
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343
Control of an underactuated remotely operated underwater vehicle M W S Lau, S S M Swei, G Seet*, E Low and P L Cheng Robotics Research Centre, School of Mechanical and Production Engineering, Nanyang Technological University, Singapore
Abstract: In this paper, a steady state model of a thruster and a general equation of rigid-body motion for an underwater robotic vehicle ( URV ) is presented. By means of modelling, simulation and experiments, the model parameters have been identied. These are used in the analysis and design of closed-loop stabilizing controllers for two control modes: manual cruise and station keeping. Since the URV under study has fewer actuators than possible degrees of freedom, it is necessary to limit the controllable degrees of freedom. These variables are eventually selected based on the inherent vehicle dynamics. Using the Lyapunov direct method, which has been shown to be appropriate for such non-linear systems, appropriate stabilizing controllers have been designed. The manual cruise mode controller is non-linear and would result in chattering in the thruster outputs, but simulations show that the desired results can be achieved. The station-keeping mode controller has a proportionalintegral-derivative (PID) structure and its gain values are designed using a non-linear optimizing approach. Simulation and swimming pool tests for the heave and yaw directions have shown that such a controller is possible. Keywords: underwater robotic vehicle ( URV ), thruster, closed-loop stabilizing controllers, remote control
1
INTRODUCTION
In the last two decades, underwater robotic vehicles ( URVs) have experienced tremendous growth in industrial, scientic and naval applications. It has in turn set o much work in the study of URV dynamics, which are highly non-linear, and the control of such vehicles in environments with a degree of uncertainty. Most control research has been focused on one subclass of the URV, the autonomous underwater vehicle (AUV ), where automatic control is of prime importance. Nonlinear control methods that take into account model uncertainty have been studied. These include the use of sliding mode control (SMC ) [1], e.g. by Rodrigues et al. [2], adaptive SMC [3, 4] and adaptive control dynamic compensation with a proportional-derivative (P+D) controller by Antonelli et al. [5]. Another commonly used non-linear controller is that of feedback linearization ( FBLN ). A combined P+D and optimal error compensation is proposed by Chellabi and Nahon [6 ]. The MS was received on 31 January 2002 and was accepted after revision for publication on 19 May 2003. * Corresponding author: Robotics Research Centre, School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798. I00202 © IMechE 2003
Linear controllers have been proposed by Kaminer et al. [7 ] using H synthesis, which can handle uncertainty 2 but the resulting controller is potentially of high order. Comparisons of some of these various schemes are given in reference [8], where the proportional-integralderivative (PID) controller with a linearized AUV model fares poorly against SMC and FBLN. These techniques and the approach can be adopted for use with the older subclass, the remotely operated vehicle (ROV ), which is the subject of this project. In spite of the main focus on the AUV, the ROV is still the industry current ‘work-horse’ and it dominates much of the subsea tasks below 200–300 m, especially in the oil and gas industry. This is likely to be the case until the Hybrid AUV [9]—a combination of the ROV and AUV—becomes a reality. Some of the control concerns are dierent; in particular, the ROV has a pilot at a remote station operating the vehicle and the equipment, and its tether and shape generate larger hydrodynamic drag. Department of Energy UK sponsored tests of ROVs, cited in reference [10], reported that one of the diculties associated with station-keeping or stationholding is the demand on the pilot to both hold the vehicle and to operate on-board equipment. With auto-heading and auto-depth systems, the overall
Downloaded from pii.sagepub.com by guest on June 3, 2011
Proc. Instn Mech. Engrs Vol. 217 Part I: J. Systems and Control Engineering
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M W S LAU, S S M SWEI, G SEET, E LOW AND P L CHENG
performance is improved. The more desirable autoposition system would require an advanced form of control. Other improvements in ROV control include the use of a tether management system ( TMS ), locking arms or devices to position the ROV to structures and various sensing systems to provide feedback to the pilot. The strategy adopted for the project is a fully integrated ROV system that assists the pilot to ‘y’ the ROV in a true dynamic and kinematic correct environment [11]. It has an immersive three-dimensional graphical display [12] and control assistance subsystems. The three-dimensional display system gives a simulated environment for the pilot to rehearse his approach and evaluate the suitability of his chosen tools. The control assistant modules, comprising the manual cruise and the station-keeping modes, help to de-skill some control operations as much as possible. This combined ‘man-inthe-loop’ and automatic control of certain prescribed modes allows a far more robust system to exist than can be ensured with a fully autonomous computer-controlled system. Furthermore, the control system module is an integral modular unit that can be implemented (a) as a software component that communicates with the overall system and (b ) on the embedded controller inside the Super Sar. With this conguration, it is possible to switch from software for training to hardware-in-theloop for actual operation. This paper focuses on the development of the control assistant modules and discusses its analysis and design. The vehicle, the Super Sar (Fig. 1), used in the project has four thrusters, fewer than the permissible degrees of freedom (DOF ), i.e. it is underactuated. Some of the control strategies (e.g. see references [5] and [13]) deal with systems or assume systems having either more actuators or at least an equal number of actuators for the available DOF. Others [2, 8] restrict motion to a certain plane and hence avoid discussing the problem of underactuation. The general control for a class of underactated mechanical systems is discussed in reference [14] and specically for underactuated AUVs in reference
Fig. 1
The Super Sar ROV manufactured by Hytec Hydrotechnologie, France
[15]. Unlike the general underactuated system where no control inputs are possible in certain directions, the Super Sar has directions in which the inputs cannot be independently controlled. To deal with this problem, the proposed design approach separates out the permissible DOF into two groups and uses a property of the vehicle dynamics discussed in reference [15]. The Super Sar non-linear model is used in the design of two dierent controllers, one for each of the control assistant modules. Models for the thrusters and the Super Sar are given rst, followed by the design and analysis of controllers. Swimming pool tests have been conducted using the Super Sar and the results obtained show reasonable agreement with the simulation model. 2
SYSTEM MODEL
The Super Sar uses unducted direct-drive d.c. thrusters without feedback compensation. At low operating speeds, the thruster’s non-linear dynamics can dominate the response of the vehicle; e.g. during station-keeping, such subsea vehicles exhibit oscillations about the desired position if no compensation is made for the thrusters’ dynamics [16, 17]. In most cases, the thruster dynamics are often neglected and the oscillations are usually tolerated. Local closed-loop controls of the thrusters, e.g. Seaeye Marine type thrusters, have been used for some ROVs. However, such thrusters can be expensive as additional electronics in the thruster packaging are needed. Where direct thruster velocity feedback is not either possible or practical, a mathematical model predicting the thruster performance would be very useful. Such a model can be derived from the propeller’s hydrodynamics and the motor’s electromechanical properties. The hydrodynamic propeller model can be derived from basic Newtonian uid mechanics theory [18], given by b +K V 2 T =K V (1) T0 T where T is the thruster output thrust, V is the propeller rotational speed and K and K are lump parameters T T of various constants. The0 electromechanical model [19] of a motor that relates the applied voltage V to the thruster output T is given by V =R I+L Ib +K V (2) m m E T =K I (3) M In the above equations, I is the current owing in the motor armature, R is the motor resistance, L is the m m motor inductance while K is the motor back-EMF (elecE tromotive force) constant and K is the modied motor M constant. The three equations above constitute the thruster model. All the thruster model’s parameters were measured experimentally using the test rig discussed in
Downloaded from pii.sagepub.com by guest on June 3, 2011 Proc. Instn Mech. Engrs Vol. 217 Part I: J. Systems and Control Engineering
I00202 © IMechE 2003
CONTROL OF AN UNDERACTUATED REMOTELY OPERATED UNDERWATER VEHICLE
reference [20]. The responses of the thruster model and that obtained experimentally are shown in Fig. 2. For clarity only the thrust in response to ±20 and ±40 V is shown, although voltages in the 10 V interval from 0 to 40 V are used in both the reverse and forward directions. The model steady state responses are almost in good agreement with those observed in the experiments. The simulated transient response, however, does not match very well with the observed oscillatory response, in part contributed by the experimental set-up [20]. It is observed that most oscillations diminish by about 0.5 s. This is relatively shorter than the ROV dynamics response time. In this paper the thruster transience is ignored and only the more accurate steady state component of the thruster model is considered. It can be derived from equations ( 1) and (2) as follows: T=K V 2 (4) T V =R I +K V (5) m E Substituting equation (3) in equations (4) and (5), a simplied steady state model is obtained:
S
R T V= m T+K (6) E K K M T Two dierent models are subsequently used: one for the forward thrust and the other for the reverse thrust. The open-loop control of the thrusters use equation (6) to compute the desired input voltages for various desired thruster levels computed by the ROV controller. The outputs of this model are compared with the steady state thrust measurement in the forward and reverse directions and are shown in Fig. 3. The model ts the experimental measurement for reverse downscale quite well over the useful range and
Fig. 2 I00202 © IMechE 2003
345
in the forward direction it is quite representative of the actual responses between 10 and 40 N. Below 10 N, the model will give a higher thrust value. On the whole the model given by equation (6) is quite acceptable and is used in the subsequent design of the system controller. The model of the ROV consists of two parts, namely the dynamics and kinematics. The derivation of a 6 DOF ROV dynamics can be found in references [13] and [21] and is described by Mvb +C(v)v+D(v)v+g( ) =F +F (7) I E where v represents the body-xed velocity vector comprising the Cartesian linear {u, v, w}T and angular velocities {p, q, r}T, denotes the earth-xed vector of ={x, y, z}T, and Euler angles ( R–P–Y ) of position, 1 orientation, ={w , h, y)T. The term Mvb represents the 2 inertial force, C(v)v the centrifugal and Coriolis force, D(v)v the hydrodynamic damping force and g( ) the gravitational and restoring forces. F and F give the I E input and external (or environmental ) force vectors respectively. The mass matrix M and the centrifugal and Coriolis matrix C(v) are governed by inertial properties of the ROV. They are obtained from the CAD/CAM (computer aided design/manufacture) data le of assembly and part drawings of the Super Sar. The buoyancy and gravitational vectors in g( ) are similarly obtained. A Jacobian coordinate transformation matrix J relates v and as follows: b=
C
J ( ) 1 2 0
0
D
=J( )v (8) J ( ) 2 2 where J and J are functions of the Euler angles only. 2 1 The elements of the hydrodynamic drag matrix D(v) are found using the method of similitude [18] or
Time responses of the thruster model and the measured thrust Downloaded from pii.sagepub.com by guest on June 3, 2011
Proc. Instn Mech. Engrs Vol. 217 Part I: J. Systems and Control Engineering
346
M W S LAU, S S M SWEI, G SEET, E LOW AND P L CHENG
Fig. 3
Model and actual thrust steady state responses
dimensional analysis using the drag parameters of a similarly shaped ROV [22]. Only the drag coecient in the surge direction so obtained is compared with that determined from a simple tow tank test. The Super Sar is propelled forward using thrusters 1 and 2 under a xed applied voltage and the terminal velocity is measured. At steady state, with the measured maximum velocity u and known thrusts T +T , the drag coecient in the 1 2 surge direction C can be estimated from: x T +T C u|u |=0 (9) 1 2 x Similar to other ROV models [13, 22], the Super Sar is made to be neutrally buoyant by the addition of a oat board, so that the buoyancy force is equal and opposite to the weight of the ROV in water when it is close to the surface. It is also stable, with its centre of buoyancy located above its centre of gravity.
3
OPEN-LOOP ANALYSIS OF THE ROV
3.1 Open-loop stability of the ROV It is essential to ascertain the vehicle’s open-loop stability with regards to body-xed velocities. To show this, consider a candidate Lyapunov function for the open-loop system described in equation (7) with F =0 and F =0 I E for a controls-xed stability analysis [13]: a 1 V(v, t )= vTMv+ eÕ bt b 2
(10)
(11)
Substituting for Mvb from equation ( 7) into equation (11) gives b =vTCvvTDvvTg( )a eÕbt V
b = vTDv vTg( ) a eÕbt V b V D
b V g
b V exp
(13)
b >0 is always positive denite, The damping term V D reecting the fact that the system’s energy is being dissipated through uid friction. Although the restoring force b is not guaranteed to be positive for all time, it term V g converges to zero at some rate, say larger than b¯ >0. b >0 is introduced The exponentially decaying term V exp to represent an envelope of convergence for the restoring b . Hence, if a is chosen to be suciently force term V g b +V b large and b
Control of an underactuated remotely operated underwater vehicle M W S Lau, S S M Swei, G Seet, E Low and P L Cheng Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 2003 217: 343 DOI: 10.1177/095965180321700501 The online version of this article can be found at: http://pii.sagepub.com/content/217/5/343
Published by: http://www.sagepublications.com
On behalf of:
Institution of Mechanical Engineers
Additional services and information for Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering can be found at: Email Alerts: http://pii.sagepub.com/cgi/alerts Subscriptions: http://pii.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav Citations: http://pii.sagepub.com/content/217/5/343.refs.html
Downloaded from pii.sagepub.com by guest on June 3, 2011
343
Control of an underactuated remotely operated underwater vehicle M W S Lau, S S M Swei, G Seet*, E Low and P L Cheng Robotics Research Centre, School of Mechanical and Production Engineering, Nanyang Technological University, Singapore
Abstract: In this paper, a steady state model of a thruster and a general equation of rigid-body motion for an underwater robotic vehicle ( URV ) is presented. By means of modelling, simulation and experiments, the model parameters have been identied. These are used in the analysis and design of closed-loop stabilizing controllers for two control modes: manual cruise and station keeping. Since the URV under study has fewer actuators than possible degrees of freedom, it is necessary to limit the controllable degrees of freedom. These variables are eventually selected based on the inherent vehicle dynamics. Using the Lyapunov direct method, which has been shown to be appropriate for such non-linear systems, appropriate stabilizing controllers have been designed. The manual cruise mode controller is non-linear and would result in chattering in the thruster outputs, but simulations show that the desired results can be achieved. The station-keeping mode controller has a proportionalintegral-derivative (PID) structure and its gain values are designed using a non-linear optimizing approach. Simulation and swimming pool tests for the heave and yaw directions have shown that such a controller is possible. Keywords: underwater robotic vehicle ( URV ), thruster, closed-loop stabilizing controllers, remote control
1
INTRODUCTION
In the last two decades, underwater robotic vehicles ( URVs) have experienced tremendous growth in industrial, scientic and naval applications. It has in turn set o much work in the study of URV dynamics, which are highly non-linear, and the control of such vehicles in environments with a degree of uncertainty. Most control research has been focused on one subclass of the URV, the autonomous underwater vehicle (AUV ), where automatic control is of prime importance. Nonlinear control methods that take into account model uncertainty have been studied. These include the use of sliding mode control (SMC ) [1], e.g. by Rodrigues et al. [2], adaptive SMC [3, 4] and adaptive control dynamic compensation with a proportional-derivative (P+D) controller by Antonelli et al. [5]. Another commonly used non-linear controller is that of feedback linearization ( FBLN ). A combined P+D and optimal error compensation is proposed by Chellabi and Nahon [6 ]. The MS was received on 31 January 2002 and was accepted after revision for publication on 19 May 2003. * Corresponding author: Robotics Research Centre, School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798. I00202 © IMechE 2003
Linear controllers have been proposed by Kaminer et al. [7 ] using H synthesis, which can handle uncertainty 2 but the resulting controller is potentially of high order. Comparisons of some of these various schemes are given in reference [8], where the proportional-integralderivative (PID) controller with a linearized AUV model fares poorly against SMC and FBLN. These techniques and the approach can be adopted for use with the older subclass, the remotely operated vehicle (ROV ), which is the subject of this project. In spite of the main focus on the AUV, the ROV is still the industry current ‘work-horse’ and it dominates much of the subsea tasks below 200–300 m, especially in the oil and gas industry. This is likely to be the case until the Hybrid AUV [9]—a combination of the ROV and AUV—becomes a reality. Some of the control concerns are dierent; in particular, the ROV has a pilot at a remote station operating the vehicle and the equipment, and its tether and shape generate larger hydrodynamic drag. Department of Energy UK sponsored tests of ROVs, cited in reference [10], reported that one of the diculties associated with station-keeping or stationholding is the demand on the pilot to both hold the vehicle and to operate on-board equipment. With auto-heading and auto-depth systems, the overall
Downloaded from pii.sagepub.com by guest on June 3, 2011
Proc. Instn Mech. Engrs Vol. 217 Part I: J. Systems and Control Engineering
344
M W S LAU, S S M SWEI, G SEET, E LOW AND P L CHENG
performance is improved. The more desirable autoposition system would require an advanced form of control. Other improvements in ROV control include the use of a tether management system ( TMS ), locking arms or devices to position the ROV to structures and various sensing systems to provide feedback to the pilot. The strategy adopted for the project is a fully integrated ROV system that assists the pilot to ‘y’ the ROV in a true dynamic and kinematic correct environment [11]. It has an immersive three-dimensional graphical display [12] and control assistance subsystems. The three-dimensional display system gives a simulated environment for the pilot to rehearse his approach and evaluate the suitability of his chosen tools. The control assistant modules, comprising the manual cruise and the station-keeping modes, help to de-skill some control operations as much as possible. This combined ‘man-inthe-loop’ and automatic control of certain prescribed modes allows a far more robust system to exist than can be ensured with a fully autonomous computer-controlled system. Furthermore, the control system module is an integral modular unit that can be implemented (a) as a software component that communicates with the overall system and (b ) on the embedded controller inside the Super Sar. With this conguration, it is possible to switch from software for training to hardware-in-theloop for actual operation. This paper focuses on the development of the control assistant modules and discusses its analysis and design. The vehicle, the Super Sar (Fig. 1), used in the project has four thrusters, fewer than the permissible degrees of freedom (DOF ), i.e. it is underactuated. Some of the control strategies (e.g. see references [5] and [13]) deal with systems or assume systems having either more actuators or at least an equal number of actuators for the available DOF. Others [2, 8] restrict motion to a certain plane and hence avoid discussing the problem of underactuation. The general control for a class of underactated mechanical systems is discussed in reference [14] and specically for underactuated AUVs in reference
Fig. 1
The Super Sar ROV manufactured by Hytec Hydrotechnologie, France
[15]. Unlike the general underactuated system where no control inputs are possible in certain directions, the Super Sar has directions in which the inputs cannot be independently controlled. To deal with this problem, the proposed design approach separates out the permissible DOF into two groups and uses a property of the vehicle dynamics discussed in reference [15]. The Super Sar non-linear model is used in the design of two dierent controllers, one for each of the control assistant modules. Models for the thrusters and the Super Sar are given rst, followed by the design and analysis of controllers. Swimming pool tests have been conducted using the Super Sar and the results obtained show reasonable agreement with the simulation model. 2
SYSTEM MODEL
The Super Sar uses unducted direct-drive d.c. thrusters without feedback compensation. At low operating speeds, the thruster’s non-linear dynamics can dominate the response of the vehicle; e.g. during station-keeping, such subsea vehicles exhibit oscillations about the desired position if no compensation is made for the thrusters’ dynamics [16, 17]. In most cases, the thruster dynamics are often neglected and the oscillations are usually tolerated. Local closed-loop controls of the thrusters, e.g. Seaeye Marine type thrusters, have been used for some ROVs. However, such thrusters can be expensive as additional electronics in the thruster packaging are needed. Where direct thruster velocity feedback is not either possible or practical, a mathematical model predicting the thruster performance would be very useful. Such a model can be derived from the propeller’s hydrodynamics and the motor’s electromechanical properties. The hydrodynamic propeller model can be derived from basic Newtonian uid mechanics theory [18], given by b +K V 2 T =K V (1) T0 T where T is the thruster output thrust, V is the propeller rotational speed and K and K are lump parameters T T of various constants. The0 electromechanical model [19] of a motor that relates the applied voltage V to the thruster output T is given by V =R I+L Ib +K V (2) m m E T =K I (3) M In the above equations, I is the current owing in the motor armature, R is the motor resistance, L is the m m motor inductance while K is the motor back-EMF (elecE tromotive force) constant and K is the modied motor M constant. The three equations above constitute the thruster model. All the thruster model’s parameters were measured experimentally using the test rig discussed in
Downloaded from pii.sagepub.com by guest on June 3, 2011 Proc. Instn Mech. Engrs Vol. 217 Part I: J. Systems and Control Engineering
I00202 © IMechE 2003
CONTROL OF AN UNDERACTUATED REMOTELY OPERATED UNDERWATER VEHICLE
reference [20]. The responses of the thruster model and that obtained experimentally are shown in Fig. 2. For clarity only the thrust in response to ±20 and ±40 V is shown, although voltages in the 10 V interval from 0 to 40 V are used in both the reverse and forward directions. The model steady state responses are almost in good agreement with those observed in the experiments. The simulated transient response, however, does not match very well with the observed oscillatory response, in part contributed by the experimental set-up [20]. It is observed that most oscillations diminish by about 0.5 s. This is relatively shorter than the ROV dynamics response time. In this paper the thruster transience is ignored and only the more accurate steady state component of the thruster model is considered. It can be derived from equations ( 1) and (2) as follows: T=K V 2 (4) T V =R I +K V (5) m E Substituting equation (3) in equations (4) and (5), a simplied steady state model is obtained:
S
R T V= m T+K (6) E K K M T Two dierent models are subsequently used: one for the forward thrust and the other for the reverse thrust. The open-loop control of the thrusters use equation (6) to compute the desired input voltages for various desired thruster levels computed by the ROV controller. The outputs of this model are compared with the steady state thrust measurement in the forward and reverse directions and are shown in Fig. 3. The model ts the experimental measurement for reverse downscale quite well over the useful range and
Fig. 2 I00202 © IMechE 2003
345
in the forward direction it is quite representative of the actual responses between 10 and 40 N. Below 10 N, the model will give a higher thrust value. On the whole the model given by equation (6) is quite acceptable and is used in the subsequent design of the system controller. The model of the ROV consists of two parts, namely the dynamics and kinematics. The derivation of a 6 DOF ROV dynamics can be found in references [13] and [21] and is described by Mvb +C(v)v+D(v)v+g( ) =F +F (7) I E where v represents the body-xed velocity vector comprising the Cartesian linear {u, v, w}T and angular velocities {p, q, r}T, denotes the earth-xed vector of ={x, y, z}T, and Euler angles ( R–P–Y ) of position, 1 orientation, ={w , h, y)T. The term Mvb represents the 2 inertial force, C(v)v the centrifugal and Coriolis force, D(v)v the hydrodynamic damping force and g( ) the gravitational and restoring forces. F and F give the I E input and external (or environmental ) force vectors respectively. The mass matrix M and the centrifugal and Coriolis matrix C(v) are governed by inertial properties of the ROV. They are obtained from the CAD/CAM (computer aided design/manufacture) data le of assembly and part drawings of the Super Sar. The buoyancy and gravitational vectors in g( ) are similarly obtained. A Jacobian coordinate transformation matrix J relates v and as follows: b=
C
J ( ) 1 2 0
0
D
=J( )v (8) J ( ) 2 2 where J and J are functions of the Euler angles only. 2 1 The elements of the hydrodynamic drag matrix D(v) are found using the method of similitude [18] or
Time responses of the thruster model and the measured thrust Downloaded from pii.sagepub.com by guest on June 3, 2011
Proc. Instn Mech. Engrs Vol. 217 Part I: J. Systems and Control Engineering
346
M W S LAU, S S M SWEI, G SEET, E LOW AND P L CHENG
Fig. 3
Model and actual thrust steady state responses
dimensional analysis using the drag parameters of a similarly shaped ROV [22]. Only the drag coecient in the surge direction so obtained is compared with that determined from a simple tow tank test. The Super Sar is propelled forward using thrusters 1 and 2 under a xed applied voltage and the terminal velocity is measured. At steady state, with the measured maximum velocity u and known thrusts T +T , the drag coecient in the 1 2 surge direction C can be estimated from: x T +T C u|u |=0 (9) 1 2 x Similar to other ROV models [13, 22], the Super Sar is made to be neutrally buoyant by the addition of a oat board, so that the buoyancy force is equal and opposite to the weight of the ROV in water when it is close to the surface. It is also stable, with its centre of buoyancy located above its centre of gravity.
3
OPEN-LOOP ANALYSIS OF THE ROV
3.1 Open-loop stability of the ROV It is essential to ascertain the vehicle’s open-loop stability with regards to body-xed velocities. To show this, consider a candidate Lyapunov function for the open-loop system described in equation (7) with F =0 and F =0 I E for a controls-xed stability analysis [13]: a 1 V(v, t )= vTMv+ eÕ bt b 2
(10)
(11)
Substituting for Mvb from equation ( 7) into equation (11) gives b =vTCvvTDvvTg( )a eÕbt V
b = vTDv vTg( ) a eÕbt V b V D
b V g
b V exp
(13)
b >0 is always positive denite, The damping term V D reecting the fact that the system’s energy is being dissipated through uid friction. Although the restoring force b is not guaranteed to be positive for all time, it term V g converges to zero at some rate, say larger than b¯ >0. b >0 is introduced The exponentially decaying term V exp to represent an envelope of convergence for the restoring b . Hence, if a is chosen to be suciently force term V g b +V b large and b
