1

Abstract—We present an overview of the state-of-the-art for

underwater vehicle autopilots. We start by reviewing reference frames, vehicle states, typical control surfaces, equations of motion, and disturbance models. We then consider different possible configurations for the autopilot, including decoupled lateral and longitudinal loops, maneuver and waypoint control. Finally, we consider possible hardware for underwater vehicle autopilots.

Index Terms—Mobile Robots, Underwater Vehicles, Underwater Vehicle Control, Underwater Vehicle Electronics.

I. INTRODUCTION UTONOMOUS Underwater Vehicles (AUV) and

Remotely Operated Vehicles (ROV) are small, unmanned submersibles. They are intended to provide researchers with simple, long-range, low-cost, rapid response capability to collect pertinent environmental data. There are numerous applications for AUV and ROV, including underwater structure inspection, oceanographic surveys, operations in hazardous environments, and military applications.

The complexity of underwater vehicles, and their control

systems and associated software, is ever increasing, due to continuous demand for more advanced features and capabilities (for example, precision maneuvering). Many parallel efforts are ongoing in industry and academia. However, no overarching, Commercial-Off-The-Shelf (COTS) autopilot solution has emerged for the market, as has happened, for example, for unmanned air vehicles (see http://www.cloudcaptech.com, for example). We seek to fill this void by providing a clear description of the functionality such a COTS autopilot for underwater vehicles would have.

The paper provides an overview of the state-of-the-art in

autopilot design for underwater vehicles. Equations of motion

Manuscript received March 30, 2007. This work was supported in part by

the University of Michigan, Ann Arbor Internal Funds. A.R. Girard is with the University of Michigan, Ann Arbor, MI 48109

USA (phone: 734-647-4692; e-mail: anouck@ umich.edu). J. Borges de Sousa is with the Electrical and Computer Engineering

Department, Faculty of Engineering, Porto University, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal (e-mail: jtasso@fe.up.pt).

J. Estrela da Silva is with the Electrical and Computer Engineering Department, Porto Polytechnic Institute, Rua Dr. António Bernardino de Almeida 431,4200-072 Porto, Portugal (e-mail: jes@isep.ipp.pt).

in 6 degrees of freedom for underwater vehicle motion are well known. The difficulties are in obtaining the proper hydrodynamic parameters for a given vehicle, in keeping the consistency between different coordinate frames and units and in the fact that different thruster configurations are used in practice. We will review these equations, the different coefficients and how they are obtained, and the effect of different thruster configurations. We will discuss simulation strategies, and the inclusion of various disturbance models (currents, wave effects) in the simulation environment. We will discuss assumptions under which the equations can be linearized and/or decoupled.

We will then discuss the standard autopilot loops for

underwater vehicles; these include speed, altitude, depth, and heading control loops, and make use of inner roll and pitch control loops. The design of the control loops will be considered and the current state of the art reviewed, from PID and classical control techniques to modern techniques such as integrator backstepping and nonlinear control. Additional functionality such as waypoint tracking, ocean current compensation, and automatic recovery systems will also be discussed.

We will use as examples the underwater vehicles of the

Underwater Systems and Technology Laboratory at Porto University in Portugal. The Underwater Systems and Technology Laboratory (USTL) from Porto University was founded in 1997 to promote research, development, deployment, and operation of advanced systems and technologies in oceanographic and environment field studies. USTL has designed and operates Remotely Operated Vehicles (ROV) for the inspection of underwater structures, and low cost Autonomous Underwater Vehicles (AUV) for coastal oceanography. Different autopilot configurations and possible hardware will be discussed.

II. UNDERWATER VEHICLE DYNAMICS

A. Coordinate Frames Several coordinate frames are typically considered when

discussing underwater vehicle dynamics. Multiple coordinate frames are used because they “make sense” in different problems, or because they are frames in which interesting phenomena are naturally expressed. For example, for navigation we are concerned with position and velocity of the

Autopilots for Underwater Vehicles: Dynamics, Configurations, and Control

Anouck R. Girard, João B. Sousa, and Jorge E. Silva

A

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vehicle with respect to the Earth, and the direction of a slewable propeller’s thrust is more naturally considered with respect to the fixed body of the underwater vehicle. We consider a few of the reference frames that are the main coordinate systems of interest.

1) The Earth-Fixed Frame, FE

This coordinate system has its origin fixed to an arbitrary point on the surface of the Earth (assumed to be a uniform sphere). xE points due North, yE points due East, and zE points toward the center of the Earth.

2) The Body-Fixed Reference Frame, FB

Body-fixed means the origin and axes of the coordinate system are fixed with respect to the (nominal) geometry of the vehicle. In underwater vehicle dynamics and control, body-fixed coordinate systems usually have their origin at the ship’s center of gravity. Determination of the orientation of the axes is as shown below (see figure 1): if the underwater vehicle has a plane of symmetry (and we will assume here that they all do) then xB and zB lie in that plane of symmetry. xB is chosen to point forward and zB is chosen to point downward. Figure 1 shows one possibility. Commonly, one chooses the origin of the body-fixed frame at the center of gravity and the body axes to coincide with the principal axes of inertia.

Figure 1. Body-Fixed Reference Frame.

3) Longitude and Latitude Position on the Earth is customarily measured by latitude

and longitude. We denote latitude by the symbol µ and longitude by the symbol λ. We will measure latitude positive north and negative south of the equator, −90 deg ≤ µ ≤ +90 deg; and longitude positive east and negative west of zero longitude, −180 deg < λ ≤ +180 deg.

4) Vehicle States and Units

The states are a minimum set of variables that completely describe the vehicle’s position, orientation, and velocity. The basic states are the inertial or body-fixed linear velocity, the inertial or body-fixed angular velocity, and a set of three transformation variables that uniquely describe the orientation of the vehicle with respect to inertial space. For the latter we usually take Euler angles. The only thing missing is the inertial position of the vehicle, which is simply the integral of the inertial velocity.

a) Euler Angles The Euler angles are used to describe the orientation of the

vehicle with respect to inertial space. This implies three different rotations will be needed (one for each axis). The

order in which these rotations are carried out is not arbitrary. The standard (SNAME convention, [2]) coordinate systems and rotations are as defined in figure 2.

The angles are termed yaw angle ψ, pitch angle θ, and

bank or roll angle φ.

Figure 2. Euler angles definition.

b) Underwater Vehicle States The twelve basic states of an underwater vehicle may

therefore be written as (this is just one possible choice, but is the standard one):

TABLE I VEHICLE STATES

Name Description Unit u Linear velocity along body-fixed x-axis m/s v Linear velocity along body-fixed y-axis m/s w Linear velocity along body-fixed z-axis m/s p Angular velocity about body-fixed x-axis rad/s q Angular velocity about body-fixed y-axis rad/s r Angular velocity about body-fixed z-axis rad/s ψ Heading angle with respect to the

reference axes rad

θ Pitch angle with respect to the reference axes

rad

φ Roll or bank angle with respect to the reference axes

rad

x Position with respect to the reference axes (North)

m

y Position with respect to the reference axes (East)

m

z Position with respect to the reference axes (Down)

m

B. Equations of Motion The Equations of Motion (EOM) are the differential

equations that describe the evolution of the twelve states of an aircraft. We will only state the equations here, as they are developed in detail in many reference texts such as Fossen [1].

1) Kinematic Equations

The kinematic equations follow from the Euler angle definitions:

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Note that there is a singularity in the equation above when the pitch angle is ±90°. Underwater vehicles typically do not operate close to this singularity. If an underwater vehicle were to operate in that zone, the Euler angles can be described by two Euler angle representations with two different singularities, or by a quaternion representation [1].

The transformations are given by:

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2) Dynamic Equations

We assume the vehicles under consideration can be approximated well as rigid bodies with gravitational, hydrodynamic and propulsive forces acting on them. Their motions may then be described by giving the position and velocity of the center of gravity and the orientation and angular velocity of a set of body-fixed axes with respect to a set of reference axes (usually Earth-fixed).

As described above, the motions in the body-fixed frame are described by 6 velocity components ν = (ν1; ν2) = [u; v; w; p; q; r] (respectively, surge, sway, heave, roll, pitch, and yaw), relative to a constant velocity coordinate frame moving with the ocean current. The six components of position and attitude in the earth-fixed frame are η = (η1; η2) = [x; y; z; φ; θ; ψ]. The Earth-fixed reference frame can be considered inertial for the AUV. The velocities in both reference frames are related through the Euler angle transformation described in the kinematics section above, that is:

!"" )( 2J=& The equations of motion are composed of the standard

terms for the motion of an ideal rigid body and, additionally, the terms due to hydrodynamic forces and moments. The usual approach to model the hydrodynamic terms is to consider three main effects: restoring forces, the simplest one, which depends only on the vehicle weight, buoyancy and relative positions of the centers of gravity and buoyancy; added mass, which describes pressure induced forces/moments due to forced harmonic motion of the body; and damping, caused by skin friction (either laminar or turbulent) and vortex shedding. The later two effects are very hard to describe accurately. They can be estimated by, usually expensive, hydrodynamic tests or by heuristic formulas, a frequent alternative which trades-off accuracy with simplicity.

In the body-fixed frame the nonlinear equations of motion

are: envactgDCM !!"##### +=+++ )()()(&

!"" )( 2J=& where M is the constant inertia and added mass matrix of

the vehicle, C(ν) is the Coriolis and centripetal matrix, D(ν) is the Damping matrix, g(η) is the vector of restoring forces and moments and τact and τenv are the vector of body-fixed forces from the actuators and the environment, respectively.

Simplications: If the vehicle’s weight equals its buoyancy and the center of

gravity is coincident with the center of buoyancy, g(η) is null. Additionally, for an AUV with port/starboard, top/bottom and fore/aft symmetries, M and D(ν) are diagonal. In that case, the damping matrix has the following form:

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For low velocities, the quadratic terms (such as Yv|v||v|) may be considered negligible.

3) Control surfaces

Figure 3. University of Porto Autonomous Underwater Vehicles.

Figure 3 depicts torpedo-like AUVs. Those AUVs are not fully actuated. There is a propeller for actuation in the longitudinal direction and fins for lateral and vertical actuation. This mechanical configuration leads to a simpler dynamic model. τact depends only on 3 parameters: propeller velocity n (0 < n ≤ nmax), horizontal fin inclination δs (

maxmax sss!!! ""# ). and vertical fin inclination δr

(maxmax rrr

!!! ""# ).The dynamics of the thruster motor and

fin servos are generally much faster than the remaining dynamics therefore, for the purposes of this work, they can be excluded from the model.

4

4) Effect of Disturbances For most marine control applications it is valid to assume

the principle of superposition can be applied when considering environmental disturbances. For underwater vehicles operating far from the surface, the main environmental disturbance to be considered is due to marine currents. Ocean and river currents arise due to a multitude of factors but, in general, may be considered low-varying (with respect to the vehicle dynamics). Therefore we can incorporate the effect of currents in the vehicle model by considering that v is relative to a frame FC moving with the water current velocity νc, where νc = [uc; vc; wc; 0; 0; 0] is the earth-fixed current velocity vector. The only change to the previously defined model will be in the kinematic equations which will come as follows:

cvJ +!"=" )( 2

&

5) Obtaining Hydrodynamic Coefficients The coefficients for the M, C and D matrices can be

obtained in a variety of ways. In general, for vehicles that move only at low speeds and have three planes of symmetry, the M and the D matrices are diagonal and the C matrix can always be parameterized as to be skew-symmetrical. The effects that need to be estimated are of three types: added mass effects (added on M and C matrices) due to the inertia of the surrounding fluid, damping effects, and restoring forces and moments.

If the vehicle has “regular” shape, coefficients for the added mass matrix can be estimated using heuristic formulas (or strip theory). Heuristic formulas are also used to estimate hydrodynamic damping. In general, they are not as accurate as the formulas for added mass effects. Software packages (such as WAMIT) and experimental validation can be used to obtain more accurate parameters.

III. UNDERWATER VEHICLE AUTOPILOTS To operate an underwater vehicle autonomously, the most

standard capability is to be able to navigate waypoints, that is, one needs to control the heading, depth and (water) speed of the underwater vehicle. For lower level control, it is desirable to be able to accept pitch and heading angle commands. To accomplish this, the most common structure is that of nested PID loops. Control surface commands can be controlled via inner PID loops. The depth (or altitude) is controlled with an outer loop, which produces commanded values for the inner loops. In addition, it may be necessary to give special consideration to how the autopilot is engaged and disengaged, so that dangerous transient motions are not produced.

1) Decoupling of the Equations

The angle between the velocity vector and the plane of symmetry, measured in the relative-to-the-current plane xC − yC, is called sideslip and is denoted by the symbol β.

The equations of motion can be simplified for level, non-side-slipping operation (φ=β=0), level skidding operation (φ=0, β≠0), or coordinated turning operation (φ≠0, β=0). The case where (φ=β=0) is the most important because in general,

it is the normal operating condition, and because it leads to decoupling of the equations of motion.

Decoupling means that the equations of motion separate

into two independent sets. One set describes the longitudinal motion (pitching, and translation in the x – z plane), and the other set describes the lateral – directional motion (rolling, side-slipping and yawing) of the underwater vehicle. In addition, in some cases vehicle speed can be decoupled as well. The decoupled equations are very much easier to handle in analytical studies. Due to lack of space we do not present them here, but the equations can be found in [8].

Longitudinal motions – x, z, u, w, q, θ, δs, n Lateral motions – y, v, p, r, φ, ψ, δr

B. Lateral Control The lateral control loop controls heading (yaw) rate and/or

heading by using the vertical fins (rudder).

Figure 4. Lateral (heading rate) control loop.

If ailerons are available (horizontal fins, used in opposition,

that is, one goes “up” as the other goes “down”, used to control roll), an inner (faster) loop is usually set up to control roll using the ailerons, and an outer (slower) loop to control yaw and/or yaw rate using the rudder.

C. Longitudinal Control There are two fast longitudinal loops, one that commands a

desired pitch angle (using horizontal fins, or elevators), and one that commands forward speed (using the propeller). In addition, a slower loop usually controls altitude (or depth).

A possible schematic for speed control is shown in figure 5.

Figure 5. Speed control loop.

The depth (or altitude) is usually controlled by a nested

structure as shown in figure 6.

5

Figure 6. Pitch/depth control loops.

In practice, consideration should be given to placing proper

saturation blocks in the above block diagrams. For ROVs, the system may have limited authority over the control surfaces and will warn the operator if the control limits have been reached.

D. Maneuver Control and Waypoint Tracking

1) Waypoint Tracking Waypoint navigation scripts are the most common means of

controlling underwater vehicles. They usually execute a set of navigation commands uploaded to the autopilot by the user [3] (waypoints are usually visited sequentially in the order that they are uploaded). The purpose of the waypoint navigation script is to navigate the underwater vehicle based on the specific command being executed, using the autopilot control algorithms. This is done by commanding heading, depth/altitude and velocity, or by commanding control surfaces, depending on the type of waypoint command under consideration. When the last command has been executed, the underwater vehicle usually either loiters about some pre-specified point, docks to an underwater dock, or surfaces at a pre-specified “safe” waypoint.

The main type of waypoint command is of the “Goto XYZ”

type, and is used to navigate to a position specified either in meters or in degrees (or seconds) of latitude and longitude. A heading is commanded based on the bearing to the waypoint. The command usually terminates when the distance to the waypoint is less than a specified parameter. At a minimum, the following parameters are required: meters East, meters North (or degrees long, degrees lat), altitude/depth (m).

2) Other Maneuvers

In addition to waypoint tracking, many underwater vehicles will have the ability to control complex maneuvers such as docking, tracking of underwater structures, and failure handling. One possibility for failure handling is that, after a failure is detected, the vehicles release drop weights, surface, and broadcast their location. For safety reasons, the vehicles usually are slightly buoyant, with the center of gravity being slightly below the center of buoyancy, which provides a restoring moment in pitch.

E. Modern Control Strategies We consider the problem of nonlinear tracking of a single

underwater vehicle. The control problem is to choose the

actuator forces to be applied such that the underwater vehicle reaches desired inertial coordinates, T

dddd yx ],,[ !=" . We utilize the “Multiple Sliding Surface” (MSS) [4] and

“Dynamic Surface Control” (DSC) [5] methods as well as the “Slotine and Li” algorithm [6], which was adapted to the control of underwater vehicles by [1].

The MSS controller involves two “sliding surfaces” [7]. The first surface defines the desired vessel position and orientation. The second surface defines a desired velocity, which, if maintained, will drive the vessel to its desired position. Thruster forces are chosen such that the second surface approaches zero.

The first surface is defined as:

ds !! "=1

.

Differentiating s1 yields: d

Js !"! && #= )(1 .

At this point a desired velocity, or “synthetic control”, νd is defined as: 11)( sJ

dd!"=#$# & , where Λ1 is a positive

definite matrix. With this definition, if ν = νd, then:

111ss !"=& , and s1→0 with a convergence rate determined

by the choice of Λ1. Because of the definition of s1, this will also guarantee that η→ηd.

The second sliding surface could be defined as s2=ν-νd, but computing the derivative of s2 could lead to a very complex control law. A method called Dynamic Surface Control (DSC) [5] eliminates the need for model differentiation. First, we pass νd through a bank of first order filters:

dzzT !=+& ,

where T is a diagonal matrix whose elements, Tii, are the filter time constants. These are chosen to be as small as possible, consistent with numerical conditioning problems. “z” now serves as an estimate of νd, with a derivative that is easily

computed as: )(1 zTzd!=

! "& .

Using z in place of νd, we now define the second sliding surface as: zs !="

2. At this point, a Lyapunov approach

[6] can be drawn upon. We select a Lyapunov function

candidate to be: 22

2

1MssV

T

= .

Differentiating V and using equations presented above

yields: ])([][ 22 zMCszMMsVT

TT&&&& !!=!= ""#"

where τT, the thruster force, is considered the only force acting on the vessel for the purpose of controller design. If τT is selected as: 2)( sKzMC

DT!+= &""# , where KD is a

positive definite, symmetric gain matrix, then:

22sKsV

D

T

!=& which guarantees that s2→0. This in turn

6

implies that ν→νd, s1→0 and η→ηd. The control law can be written in terms of z:

2

1 )()( sKzMTCDdT

!!+=! """# .

Figure 7. Experimental Results for Tracking using the University of Porto Autonomous Underwater Vehicles.

IV. AUTOPILOT HARDWARE AND CONFIGURATIONS A simple approach to understanding a typical underwater

vehicle system is to split it into self-contained subsystems.

Figure 8. High-level representation of a typical underwater vehicle’s subsystems.

The purpose of the vehicle platform is to carry the

electronics and payload. It consists of the vehicle’s frame, control surface actuators, batteries, and the propulsion system. The vehicle electronics consist of the autopilot, sensors, and communications equipment. The payload system consists of hardware not related to the vehicle platform or electronics, and the payload should help accomplish user objectives. The ground station includes the hardware and software necessary to support the vehicle electronics and payload during operations. In a standard configuration, the ground station will include a laptop computer, a communications system, a system for pilot inputs if needed (ROV). A graphical interface is sometimes provided with the autopilot and runs on the ground station laptop.

As an example, we describe the systems and autopilot

implementation on the University of Porto Isurus vehicles. Isurus is a modified version of a Remus class (Remote

Environment Measuring UnitS) class AUV, built by the Woods Hole Oceanographic Institution, MA, USA, for low cost and lightweight operations in coastal waters. Isurus has a torpedo shaped hull about 1.7 meters long, with a diameter of 20 cm and weighting about 35 kg in air. The maximum forward speed is 4 knots, with the best energy efficiency achieved at about 2 knots. At this velocity, the energy provided by a set of rechargeable Lithium-Ion batteries may last for over 20 hours. The maximum operating depth is 200m. In the standard configuration Isurus is equipped with the following devices: Ocean Sensors 200 conductivity, temperature, and depth sensor; Wet Labs optical backscatter sensor; Marine Sonics side scan sonar; Imagenex altimeter; PNI TCM2 digital compass; long baseline acoustic beacons 20-30 Khz; Benthos acoustic modem, and 802.11 compliant radio. The computer system consists of a PC-104 stack running the QNX real-time operating system. External devices, such as sensors and actuators are interfaced through the I/O cards mounted on the PC-104 stack. The PC-104 computer system runs the modular onboard software designed and implemented for command and control of the vehicle systems (power, acoustic navigation, navigation sensor, communications, actuators and payload sensors) and for interfacing with the operator.

V. CONCLUSION We have reviewed dynamics, configurations and control

strategies for underwater vehicle autopilots. We are preparing a set of Matlab files to accompany the paper, and additional material relating to sensing, state estimation, and disturbance modeling, which will be posted on the authors’ web pages.

REFERENCES [1] T.I. Fossen, Guidance and Control of Ocean Vehicles. John Wiley and

Sons, Inc., New York, 1994. [2] E. Lewis (ed.). Principles of Naval Architecture. 2nd revision. Jersey

City, N.J.: Society of Naval Architects and Marine Engineers, 1989. [3] A. Girard. A Convenient State Machine Formalism for High-Level

Control of Autonomous Underwater Vehicles. Master’s Thesis, Florida Atlantic University, Boca Raton, FL 33431, May 1998

[4] M. Won and J.K. Hedrick, “Multiple surface sliding control of a class of uncertain nonlinear systems”. International Journal of Control, 64 (4), 693-706, 1996.

[5] D. Swaroop, J.K. Hedrick, P.P. Yip and J.C. Gerdes, “Dynamic surface control for a class of nonlinear systems”. IEEE Transactions on Automatic Control, 45, (10), 1893-99, 2001.

[6] J.J.E. Slotine and W. Li, “Applied Nonlinear Control”, Prentice Hall, 1991.

[7] V.I. Utkin, “Variable structure systems with sliding modes”. IEEE Transactions on Automatic Control, AC-22, 212-222, 1977.

[8] A.J. Healey and D.B. Marco, “Slow Speed Flight Control of Autonomous Underwater Vehicles: Experimental results with the NPS AUV II”, Proceedings of ISOPE (International Symposium on Offshore and Polar Engineering), San Francisco, CA, 1992, pp. 523-532.