Underwater Vehicles Part 9 ppsx

Orientation control of multiple underwater vehicles with symmetry-breaking potentials, Proceedings of the 40th IEEE Conference on Decision and Control, pp.. MARES – Navigation, Control

rad rad, (0) 0.15

Fig 7 Schooling of the vehicles in a triangular movement

Fig 8 Schooling geometry for a triangular movement

5.3 Equilateral triangular schooling with obstacle avoidance

In this case, we consider an equilateral triangular schooling of the vehicles with obstacle avoidance The obstacle is modelled as a circle located at q p=(40,10) with radius as 3m For

the schooling, two virtual leaders are chosen as in Fig 3 (c) with the initial positions taken as

rad q

u2(0)= 3(0)=0.5 /

= with all other variables taking zero values Other design parameters

are taken as k ui=18,kψi=10,k ri=12,i=1,",3 and γα=0.15,γβ=0.2,γγ =0.15,γu=240,

Simulation results are depicted in Fig 9~12 Fig 9 shows the vehicles schooling in the

equilateral triangular movement with obstacle avoidance From Fig 10, we can see that

there is not any collision between vehicles Fig 11 presents the vehicles’ velocity and

heading matching in the schooling, and Fig 12 shows the histories of proposed formation

control laws for τui and δri

Fig 9 Schooling of the vehicles in an equilateral triangular movement with obstacle

avoidance

Fig 10 Schooling geometry for an equilateral triangular movement with obstacle avoidance

Fig 11 Group velocity and heading matching

Fig 12 Histories of proposed formation control laws

6 Summary

In this chapter, we have investigated an asymptotic schooling scheme for multiple

underactuated underwater vehicles For each vehicle, there are only two control inputs –

surge force and yaw moment available for its three DOF motion in the horizontal plane The

main difficulty in the tracking of this kind of vehicle is how to properly handle the vehicle’s

sway dynamics To deal with this problem, in this chapter, we have introduced a certain

polar coordinates transformation, through which the vehicle’s dynamics can be reduced to a

two-inputs strict-feedback form The vehicles schooling has been conducted by properly

selected smooth potential function, which consists of three different parts: one is for the

interaction between vehicles, another is for group navigation, and the third one is for

obstacle avoidance The proposed formation algorithm guarantees the vehicles asymptotic

schooling and velocity and heading matching while keeping obstacle avoidance

Proposed schooling scheme has been derived under the condition of u t)≥ umin>0, which

inversely can be guaranteed by proposed formation control laws being combined with some

suitable initial conditions Therefore, the proposed schooling method only can guarantee the

local stability Moreover, it is notable that the following issues should be considered in our

future works

• Finite cut-off (b<+∞ in Definition 1) of potential function, which was applied in the

previous works (Leonard and Fiorelli, 2001; Olfati-Saber, 2006; Do, 2007), also plays an

important role in the vehicles schooling in this chapter However, since b<+∞, it is

easy to verify that ∂f p(ζ,a,b)/∂ζ=0 if ζ ≥b For this reason, the proposed schooling

scheme only guarantees certain local minimum It is of interest to upgrade the present

result to the one where the global minimum can be guaranteed in our future works

• Another practical concern is for the robustness of proposed schooling scheme In

practice, there various uncertainty terms have to be faced, such as vehicle’s modelling

error, measurement noise, and disturbance, etc All of these terms should be considered

in our future practical applications

7 Acknowledgements

This work was supported by the Ministry of Land, Transport and Maritime Affairs in Korea

under Grant PMS162A and by the Korea Ocean Research & Development Institute under

Grant PES120B

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Douglas-Westwood Limited, Canterbury, UK, 2007

MARES – Navigation, Control

and On-board Software

Aníbal Matos and Nuno Cruz

of these fields is already well established, there is still a great research effort in areas related

to the design and operation of these vehicles The Ocean Systems Group at FEUP (Faculty of Engineering at the University of Porto) and ISR – Porto (Institute for Systems and Robotics – Porto) conducts research activities in marine robotics and has accumulated expertise in the utilization of AUVs and in the development of particular subsystems This chapter addresses the design of the navigation and control systems of the MARES AUV (Cruz & Matos, 2008), a state-of-the-art small size AUV developed by the authors and already demonstrated at sea operations in 2007 The implementation of these systems in the vehicle on-board software is also discussed

Fig 1 MARES AUV ready for an open sea mission

2 MARES AUV

MARES, or Modular Autonomous Robot for Environment Sampling (Fig 1), is a 1.5m long AUV,

designed and built by the Ocean Systems Group The vehicle can be programmed to follow

predefined trajectories, while collecting relevant data with the onboard sensors MARES can

dive up to 100m deep, and unlike similar-sized systems, has vertical thrusters to allow for

purely vertical motion in the water column Forward velocity can be independently defined,

from 0 to 2 m/s Major application areas include pollution monitoring, scientific data

collection, sonar mapping, underwater video or mine countermeasures

MARES configuration can change significantly according to the application scenario, so that

it is difficult to define what is a standard configuration In table 1 we summarize the main

characteristics of the AUV version that was demonstrated at sea in November 2007

Diameter 20 cm

Weight in air 32 kg

Depth rating 100 m

Propulsion 2 horizontal + 2 vertical thrusters

Horizontal velocity 0-2 m/s, variable

Energy Li-Ion batteries, 600 Wh

Autonomy/Range about 10 hrs / 40 km

Table 1 MARES main characteristics

2.1 Mechanical

All mechanical parts were designed using Solidworks® CAD software (Fig 2) and machined

from polyacetal in a local machine shop, with small parts in aluminium and stainless steel

Polyacetal is a high performance polymer, with a high degree of rigidity and mechanical

strength that makes it an excellent weight-saving metal replacement It is completely

corrosion proof and it is readily available in a wide range of sizes of tubes and rods, at

reasonable prices

Fig 2 MARES CAD model

The vehicle hull evolves around a central watertight cylinder, where all electronic boards

are installed, with the battery packs located at the bottom to lower the center of mass To

simplify the design, this is the only watertight enclosure and therefore all other equipment has to be waterproof The other polyacetal sections are designed to carry wet sensors and thrusters and they are fully interchangeable This allows for very easy sensor swapping and/or repositioning, or even to test different configurations of thrusters The main cylinder has 9 holes in each end cap, to accommodate standard bulkhead connectors and at the moment there are still several unused, sealed with dummy plugs

The overall vehicle shape resembles that of a torpedo, with ellipsoids both at the nose cone and at the tail This configuration is very simple to construct and allows for the vehicle length to be easily extended, as compared to other hull shapes without constant cross-sections The central cylinder provides most of the vehicle flotation and it is also possible to increase its length, for example if more batteries are needed

Typical small-size AUVs use vertical and horizontal fins to adjust heading and pitch, but this requires a minimum forward velocity for the control surfaces to be effective (von Alt et al., 1994; Crowell, 2006) On MARES, four independent COTS thrusters provide attitude control both in the horizontal and in the vertical plane Two horizontal thrusters located at the tail control both forward velocity and rotation in the horizontal plane, while another set

of thrusters, in the vertical direction, control vertical velocity and pitch angle This arrangement permits operations in very confined areas, with virtually independent horizontal and vertical motion at velocities starting at 0 m/s This is one of MARES innovations, as it cannot be seen in any AUV of similar size and weight Furthermore, the modularity of the system allows the integration of other thrusters, for example to provide full control of the lateral motion

It should be stressed that fins are usually more efficient for diving than thrusters, but with simple fins it is not possible to control pitch angle independently of depth In mission scenarios where bottom tracking is important, such as sonar or video acquisition, a fin controlled AUV will pitch up and down to follow the terrain, affecting data quality On the contrary, MARES AUV can control both pitch angle and depth independently, being able to maintain data quality even if the terrain has significant slopes

Another advantage of using thrusters is that all moving parts can be fully shrouded and there are no fins protruding from the hull, which minimizes the risk of mechanical failure

In the end, we deliberately traded some of the efficiency with increased maneuverability and robustness

2.2 Power and energy

Most of the power required by an AUV is spent in propulsion, with only a small amount permanently needed for onboard electronics In MARES, all energy is stored in rechargeable Li-Ion battery packs, currently with a total amount of 600 Wh, at 14.4 V Battery power is directly available to the motor controllers and, through a set of voltage converters, to the rest of the onboard electronics

Battery endurance greatly depends on vehicle velocity, both in the horizontal and in the

vertical plane For typical horizontal missions, with relatively slow changes in depth, there is

sufficient energy for about 8-10 hours of continuous operation (around 20-25 miles or 40 km) These are relatively modest numbers, but they seem to be sufficient for the great majority of envisaged missions In any case, there is still some available volume for a few more battery packs It should be stressed that these numbers refer to standard horizontal motion and it is also necessary to account for any significant vertical motion For example,

the vehicle can hover almost motionless in the water column, at a specific depth, but still

requiring some small amount of power to provide depth corrections In this case, the total

endurance will be longer in time but relative to a shorter horizontal range

2.3 Computational system

The onboard computational system is based on a PC104 stack (Fig 3), with a power supply

board, a main processor board, and additional boards to interface with peripherals, such as

health monitoring systems, actuation devices, and navigation and payload sensors A flash

disk is used to store both the onboard software and also the data collected during

operations

Fig 3 MARES on-board computer

2.4 Payload

The modularity of the vehicle allows for a simple integration of different payload sensors,

involving three sub-tasks: mechanical installation, electronics interfacing and software

Mechanically, a new sensor may be installed in a dedicated section of the hull, if it is

relatively small Alternatively, it can be externally attached to the vehicle body, since there

are many fixing points available In any case, it is important to verify the weight of the

sensor (and adapter) in the water, to compensate with extra flotation if necessary Naturally,

the overall vehicle trim has also to be adjusted, particularly in the case of bulky or heavy

payloads

Most of the payload sensors transported by the AUV need energy and a communications

link with the onboard computer MARES has several spare connectors on both end caps of

the main electronics compartment, that can be wired to provide power and receive data

from these sensors At the same time, the computational system has spare communication

ports, easily configurable according to the payload specs

As far as software is concerned, the integration of a new payload sensor requires the

development of a dedicated software module, known as a device driver Device drivers

establish a communication link between the sensor and the onboard software core, allowing

for the configuration of the sensor as well as data logging

Naturally, these tasks are greatly reduced after the first time the sensor is tested Since then,

it becomes very simple to swap payload, just by integrating the proper set: sensor, electronics and software

3 Navigation

The design of the navigation system of an AUV has to take into account several issues such

as desired positioning and attitude accuracy, size, weight, and power consumption of available sensors and systems, and also overall cost Available technologies include inertial navigation systems, digital compasses, tilt sensors, pressure cells, acoustic positioning systems, and Doppler based velocity meters The MARES navigation package was selected taking into account the above mentioned issues but also the envisaged missions and application scenarios

3.1 Sensors and systems

To estimate its position in real time, the vehicle carries a pressure sensor, a digital compass with a set of tilt sensors, and an acoustic system for long baseline (LBL) positioning (Vaganay et al., 1996) Vehicle depth is directly given by the pressure cell, while roll and pitch angles are obtained from the set of tilt sensors The estimation of the horizontal position and velocity also employs dead-reckoning data

The pressure cell has a centimetre level accuracy and outputs new data at a rate exceeding

100 Hz This allows for a quite accurate vertical positioning and also for a software based estimation of heave velocity The digital compass and tilt sensors unit provides data at 20

Hz with accuracies better than 0.5º

The acoustic system is a second generation of multi frequency boards, developed by the Ocean Systems Group, following some excellent results with previous versions (Cruz et al., 2001) These boards are installed on-board the vehicle but also on the acoustic beacons that are deployed in the operation area This system is completely reconfigurable (pre mission or

on line programming of detection/reply frequency pairs, channel sensitivity, etc.), allows complete control of signal transmission times, and provides access to low level signal detection data

3.2 Horizontal position estimation

The real time estimation of the horizontal position of MARES is computed by a Kalman filter based algorithm (Matos et al., 1999) that combines dead-reckoning data (vehicle surge velocity and heading) with range measurements based on times of flight of acoustic signals For a typical operation, two acoustic beacons are deployed in the operation area in a way such that the AUV, during the execution of its mission, does not cross the line connecting them In this way, the range measurements between the AUV and both beacons unambiguously determine its horizontal position Usually, these beacons are attached to surface buoys and signal detection and transmission time are transmitted to a shore station

by a wireless link This information allows for the real time external tracking of the AUV according to the algorithm proposed in (Cruz et al., 2001)

Since velocity data is obtained with respect to water, and no direct measurement of water currents is available, the horizontal components of such current are also estimated in real time The navigation algorithm periodically updates the real time estimates of the horizontal

Fig 4 Surface buoy attached to an acoustic beacon

position of the vehicle (x, y) and of the water current (c x , c y) at a 20 Hz rate, according to the

following dynamic system,

00sincos

=

=

+ψ

=+ψ

=

y x

y x

c c

c v y

c v x

At the same time, the associated error covariance matrix is updated Whenever a new range

measurement is obtained by the vehicle, the state variables and the error covariance matrix

are corrected accordingly This is accomplished by an iterative procedure based on the

extended Kalman filter algorithm (Gelb, 1994), since range measurements are related to the

state variables of the filter by the nonlinear relationship

2 2

where (x, y, z) is the 3D position of the AUV, (x i , y i , z i ) is the 3D position of beacon i, and r i is

the predicted range measurement Fig 5 presents the range measurements between the

AUV and two navigation beacons during an autonomous operation It shows erroneous

measurements caused multipath propagation of acoustic waves, as well as intervals of time

when there are no measurements from at least one of the beacons, typically due to adverse

propagation conditions In order to increase the accuracy of the horizontal positioning, a

validation mechanism is used to maximize the probability of rejecting those erroneous

measurements

Even in ideal propagation conditions, and neglecting error sources associated with the

electronics of the transmission and reception acoustic boards, acoustic range measurements

are directly affected by the sound speed in water This, is turn, depends on the

characteristics of the medium (temperature, salinity, pressure), and variations as high as 2%

are not unusual It is therefore mandatory to determine the sound speed in the operation area and adjust range measurements accordingly This can be done either by comparing range measurements close to the surface with DPGS based distance measurements, or by measuring the most relevant water characteristics with a CTD (conductivity, temperature, and depth) sensor These calibration procedures allow horizontal position accuracies about 1

to 2 meters with respect to the positions of the beacons

0 50 100

Fig 5 Acoustic range measurements

All MARES navigation data is stored in the flash disk carried by the vehicle, and real time GPS positioning data from the surface buoys attached to the acoustic beacons is transmitted

to a shore station where it is also stored This allows the post processing of all navigation data by a smoothing algorithm, therefore improving the accuracy of the vehicle positioning (Matos et al., 2003) and the space location of collected payload data

4 Control

The general dynamic model of an underwater vehicle follows the 6 degrees of freedom template presented in (Fossen, 1994) It is a complete model that takes into account all the forces and moments acting on a submerged body, but for which it is not easy to design adequate controllers Typically, AUVs evolve according to two dimensional motions, either

in the vertical or in the horizontal plane Therefore, the traditional approach for the control

of theses vehicles is based on mode decoupling (Healey & Lienard, 1993)

Following such approach, the MARES control system is organized into four basic controllers: surge, heading, pitch and depth The first two determine the horizontal motion

of the vehicle and their outputs are combined to obtain the actuation for the horizontal thrusters The outputs of the pitch and depth controllers are combined to provide the actuation for the vertical thrusters Each one of the four basic controllers can operate in either closed or open loop mode

4.1 Controllers

At the lowest level, all four controllers are assumed independent Surge is controlled by the

common mode of horizontal thrusters, heading is controlled by the differential mode of

these thrusters, while depth is controlled by the common mode of vertical thrusters, and

pitch by their differential mode In this approach, the couplings between modes are treated

as external disturbances, which must be taken into account when designing decoupled

feedback controllers This typically results in a small reduction of performance, which is

largely balanced by the simplicity of the design and by the modularity of the approach

Each one of these controllers can operate in different modes, ranging from a pure open loop

operation to more complex structures with more than one feedback loop, as follows:

• Velocity loop — input is a heading rate reference

• Position loop — input is a heading reference

• Line tracking loop — input is a directed horizontal line reference

Depth:

• Open loop — input is a direct common mode command for the vertical

thrusters

• Velocity loop — input is a heave velocity reference

• Position loop — input is a depth reference

Pitch:

• Open loop — input is a direct differential mode command for the vertical

thrusters

• Velocity loop — input is a pitch rate reference

• Position loop — input is a pitch reference

4.2 Mission plan and elemental maneuvers

The autonomous operation of MARES is defined by a mission plan Besides configuring a

large set of variables that affect the vehicle behaviour (such as controller gains, maximum

operating depth, operating frequencies of the acoustic systems, etc.), the mission plan also

includes a set of elemental maneuvers that the vehicle should execute in sequence

Each maneuver prescribes the behaviour all the four basic controllers (therefore defining the

vehicle motion) It also defines its end condition, and a timeout for safety reasons Besides

some maneuvers that are mainly used for debugging purposes, the basic MARES

• hovering – a maneuver that stops the vehicle at the current position;

• gotoxy – a horizontal plane maneuver that drives the vehicle along a straight line

The possibility of independently defining each basic controller allows for very different vehicle behaviours, making the operation of MARES very flexible For example, a pure vertical motion can be easily obtained by a dive maneuver with a closed loop pitch with zero reference, and a zero surge command; a combined vertical and horizontal motion can

be achieved with a dive maneuver with a closed loop pitch with a negative (downward looking) reference and an appropriate surge command

Furthermore, each basic controller is already prepared to accept inputs defined by external processes This allows for the implementation of unconventional guidance strategies which can be based on payload data collected in real time

5 On-board software

The onboard software was developed in C++, runs on a Linux kernel, and is composed by a set of independent processes In this way, not only the system modularity and robustness are increased but also its debugging and recovery from unexpected events are much simpler Communications between the modules rely on a message passing mechanism, using the User Datagram Protocol This allows connectionless data transmissions, with reduced processing overhead, as required in this kind of applications The data path between different processes can be easily reconfigured, increasing therefore the flexibility of the system and simplifying the integration of new modules Moreover, it is also possible to

modify data paths when the software is running by issuing special commands to the relevant

Fig 6 Basic on-board software structure

The basic on-board software follows the structured depicted in Fig 6 The interface with the hardware is managed by dedicated processes that provide an abstraction layer Processes that deal with navigation sensors and systems transmit their data to the navigation module This one implements all necessary algorithms to estimate the AUV state in real time It also sends such estimate to the control module

The control module is responsible for the execution of the mission In each control cycle it verifies the completion of the current maneuver and schedules the next one whenever the current ending conditions are met This module also executes the procedures required by the different control loops

A supervision module continuously monitors the behaviour of the vehicle and aborts the

autonomous operation if safety margins are exceeded or unexpected events occur This

module can also be used to configure the whole software of the vehicle For that purpose, it

establishes a communication link with a shore control station whenever the vehicle is at the

surface

A black box data logging system registers all information related to the vehicle operation on

the flash disk The information includes raw data from the navigation sensors and systems,

health monitoring data, processed navigation data and control data This black box system

is also prepared to register payload data

Payload sensors and systems are typically controlled by additional dedicated modules In

general, these modules interact with the supervision module, for configuration and

communication with the control station, and also with the data logger

Additional modules implementing sensor based control strategies or other advanced

features are easily integrated with the basic software structure Besides programming and

installing the new module, it is only necessary to redefine data paths, assuring that required

sensor data is also sent to the new module, and that the new module sends its output data to

the control module The operation of such new module is then configured in the mission

plan

6 Conclusions

The first MARES water tests were conducted in a local pool in late 2006 Those tests served

to validate the integrity of the system, adjust buoyancy and trim, and test simple

maneuvers During the first semester of 2007 a set of tests took place in a reservoir in the

Douro river, with a maximum depth of 20 meters and about 200 meters wide These tests

allowed the fine tuning of motion control parameters as well as the final adjustments on the

acoustic navigation system

Fig 7 Salinity map close to the sewage outfall diffuser

MARES was first demonstrated at sea in November of 2007 This demonstration mission took place in the neighbourhood of a sewage outfall located 2 km off the Portuguese coast

at Foz do Arelho MARES was equipped with a Seabird Fastcat 49 CTD and collected 16 samples/second of CTD data for about one hour Upon vehicle recovery, CTD data was analyzed to infer the location of the sewage plume in the vicinity of the diffuser Fig 7 shows a salinity map produced from CTD data, and although the salinity signature was very weak, it was also very consistent This demonstrated the potential for detecting minute anomalies with the onboard CTD, which was the main objective for this mission at sea

The success of the demonstration mission at sea proved that the initial requirements and the design decisions contributed to the development of an operational vehicle that can be effectively used in real application scenarios One of the major advantages of the MARES AUV, when compared with other AUVs of similar size, is the ability to independently control the motion in the vertical and in the horizontal planes This allows for some new primitives of motion, such as commanding the vehicle to be completely motionless in the water column (for example, waiting for some triggering event), or diving and emerging vertically, which greatly simplifies its launching and recovery

7 References

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Boston, USA, Sept 2006

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Navigation and Remote Tracking of Multiple Underwater Vehicles, Proc MTS/IEEE Oceans’01, Honolulu, HI, USA, Nov 2001

Cruz, N.; Matos, A (2008), The MARES AUV – A Modular Autonomous Robot for

Environment Sampling, Proc MTS/IEEE Oceans’08 Quebec, Quebec, Canada, Sept

2008

Fossen, T (1994) Guidance and Control of Ocean Vehicles, John Wiley & Sons Ltd., ISBN

0471941131

Gelb, A (1974) Applied Optimal Estimation, MIT Press, ISBN 0471941131

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Healey, A.; Lienard, D (1993), Multivariable sliding mode control for autonomous diving

and steering of unmanned underwater vehicles, IEEE Jornal of Oceanic Engineering,

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AUV, Proc MTS/IEEE Oceans’03, San Diego, CA, USA, Sept 2003

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Navigation System for an AUV, Proc MTS/IEEE Oceans’99, Seattle, WA, USA, Sept

1999

Vaganay, J.; Leonard, J.; Bellingham, J (2006), Outlier Rejection for Autonomous Acoustic

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Proc IEEE Symp AUV Techn AUV’94, Cambridge, MA, USA, July 1994

Identification of Underwater Vehicles for the

Purpose of Autopilot Tuning

Nikola Mišković, Zoran Vukić & Matko Barišić

University of Zagreb, Faculty of Electrical Engineering and Computing

Croatia

1 Introduction

Underwater vehicles (UVs) lately found their use in many activities such as underwater mapping, habitat exploration, different types of inspections (underwater cables, dams, ship hulls), rescue missions and many others All of these applications speeded up the research related to modeling and control of UVs Modeling of UVs is important because many applications demand that the mission is carefully planned in a simulated environment Control, on the other hand, is essential if higher level tasks are to be performed effectively Finding a mathematical model describing an underwater vehicle dynamics can be a tedious task The greatest problem is the complexity of the rigid body dynamics which is augmented

by additional forces and effects which appear in fluid The hydrodynamics observe this problem with great care, paying extra attention to dependencies between different variables Another problem are the couplings that appear due to motion in different degrees of freedom simultaneously For the control purposes, most of these dependencies are often neglected in order to obtain a simple model which can later be used for designing autopilots

This chapter deals with methods for obtaining a precise enough mathematical model, which can be used for control purposes, using cheap and commercially available sensors such as cameras or compasses First section deals with description of a full mathematical model of

an underwater vehicle, starting with actuators (thrusters) and methods of determining their static characteristics The section is followed by actuator allocation where some common configurations and allocation matrices are mentioned Kinematic model is briefly addressed and dynamic model is presented in its full form The coupling effects are observed in the horizontal plane and a short methodology of determining dominant parameters in the coupled model is given For the uncoupled case, two model equations are taken into account: linear and nonlinear In section two, three vision-based data acquisition methods are presented Here it is explained which equipment is needed for making a laboratory apparatus for data acquisition, and the process of data acquisition via image analysis is presented The following section deals with the identification algorithms which use the data obtained via the vision-based methods Here we present in short the least-squares method (used for determining the coupled model), open loop, zig-zag methods and identification based on the self-oscillations All of these methods are followed with results obtained either from real vehicles or simulations

2 Mathematical models of underwater vehicles

In order to define the full mathematical model of an underwater vehicle (UV) we will use the terminology adopted from Fossen (1994) Vector of positions and angles of an underwater vehicle is defined in the Earth-fixed coordinate frame and vector of linear and angular velocities (surge, sway, heave, roll, pitch and yaw velocity, respectively) is defined in a body-fixed coordinate frame, see Fig 1 Vector represent the external forces that act on the vehicle, vector are commanded thrusts for each actuator and are commanded inputs for the actuators themselves Here we make an assumption that the vehicle is actuated by thruster force, even though other actuator types possible and appear in practice Using this notation, the complete mathematical model can be represented with Fig 2 In the following sections, all parts of the model will be described

Fig 1 Body-fixed and Earth-fixed coordinate frames (taken from Omerdic (2004))

Fig 2 Scheme of a complete mathematical model

of the model can also be neglected, i.e 0

However, the force exerted by thrusters is rarely the same when the propulsor is rotating in

both directions This is why a more complex model (1) should be used where sub indices f and b denote ‘forward’ and ‘backward’, and super index i stands for a specific thruster