Model-Based Output Feedback Control Of

930 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 16, NO. 5, SEPTEMBER 2008

Model-Based Output Feedback Control ofSlender-Body Underactuated AUVs:

Theory and ExperimentsJon E. Refsnes, Asgeir J. Sørensen, and Kristin Y. Pettersen, Senior Member, IEEE

Abstract—This paper presents the design and experimentalresults of a novel output feedback controller for slender-bodyunderwater vehicles. The controller is derived using model-baseddesign techniques. Two separate control plant models are em-ployed: a 3-degree-of-freedom (DOF) current-induced vesselmodel accounting for the current loads acting on the vehicleand a 5-DOF model describing the vehicle dynamics. The maindesign objective behind this strategy is to incorporate the vehicledynamics when estimating the current influence on the vehicle.Furthermore, the transit model is based on the notion of constantpropeller revolution resulting in a partly linearized model, whichsubsequently leads to perspicuous and implementable controllerand observer structures. The controller is derived using the ob-server backstepping technique, and the closed loop is proved tobe asymptotically stable using Lyapunov and cascaded systemstheory. The control objective is to track the desired pitch andheading angle generated by the line-of-sight guidance system whilekeeping constant forward thrust. Experimental results demon-strate successful performance of the proposed output feedbackcontroller implemented on the Minesniper MkII AUV/ROV.

Index Terms—Autonomous underwater vehicle (AUV), experi-mental results, nonlinear model-based output feedback control.

I. INTRODUCTION

F OR underwater vehicles, moving with some forwardspeed, the dynamics are highly nonlinear and coupled.

This presents control challenges that have led to considerableinterest on nonlinear observer and controller design for un-derwater vehicles during the last decades. There are, however,relatively few reported results on model-based control (MBC)designs for underwater vehicles that include experimental tests.The main reason for this lies probably in the great difficulties inobtaining an accurate model of the vessels. Furthermore, unpre-dictable current loads and poor position measurements presentchallenges when employing MBC due to their potentially stronginfluence on the controller. For these reasons, in addition to

Manuscript received June 7, 2007; revised August 21, 2007. Manuscriptreceived in final form January 7, 2007. Published July 30, 2008 (projected).Recommended by Associate Editor F. Caccavale. This work was supported bythe Norwegian University of Science and Technology and Kongsberg ASA,Norway.

J. E. Refsnes and A. J. Sørensen are with the Department of Marine Tech-nology, Norwegian University of Science and Technology, N-7491 Trondheim,Norway (e-mail: [email protected]; [email protected]).

K. Y. Pettersen is with the Department of Engineering Cybernetics, Nor-wegian University of Science and Technology, N-7491 Trondheim, Norway(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2007.916347

issues related to the actual implementation, tuning, and debug-ging, nonmodel-based solutions are often preferred. There are,however, important advantages that can follow from employingMBC. Based on the model, one can predict the motion of thevehicle by using controller actuator inputs and available statemeasurements. Moreover, a model-based observer can provideestimates of unmeasured states in addition to filtering of noisysignals. This paper presents successful results of an MBCsystem of a slender-body underwater vehicle, demonstratingorientation tracking, estimation of unmeasured states, filtering,and dead reckoning. We will, in this paper, denote the modelthat is designed for the purpose of control design as the controlplant model (CPM). A CPM is, according to [31], defined asa model that captures the main characteristics of the physicalsystem. Unfortunately, poorly formulated CPMs that do notcapture the important characteristics of the dynamic systemmay cause reduced performance and also stability problems.Hence, when deriving the CPM, emphasis should be placed onstability and robustness issues related to the system in additionto simplifying the model such that analysis is feasible. We willthus, in this paper, describe the development of the dynamicmodel and explain the most important hydrodynamic featuresof slender-body autonomous underwater vehicles (AUVs). Themore complex process plant model (PPM) is a comprehensivedescription of the actual process and should be as detailed asneeded. The main purpose of this model is to simulate the realplant and to test controllers and observers that are designedbased on the corresponding CPM.

A. Background

There are some reported results on MBC of AUVs in theliterature. In [16], a state feedback controller is proposed fortracking of the NPS ARIES AUV. The model is linearized abouta constant forward velocity and decoupled into three separatesystems: surge, horizontal steering (sway and yaw), and thediving system (heave and pitch). Sliding-mode controllersand observers [9] are proposed to solve the tracking problem.Experimental results, reported in [22], demonstrate successfulcontroller performance. The NPS ARIES is an underactuatedslender-body AUV intended for orientation tracking whilemaintaining some forward speed. This kind of streamlinedAUVs should be distinguished from open box-framed vehi-cles. These are low-speed vehicles, usually fully actuated,and with hydrodynamic and stability properties that may varysignificantly. In [30], a model-based positioning system isproposed for robotic vehicles where experimental evaluation

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REFSNES et al.: MODEL-BASED OUTPUT FEEDBACK CONTROL OF SLENDER-BODY UNDERACTUATED AUVs 931

of the different controllers is performed on the JHRUROVvehicle. The proposed model is completely decoupled, andthe hydrodynamics are dominated by linear and nonlineardamping, i.e., the Coriolis forces are not explicitly includedin the model. The paper concludes that fixed model-basedcontrollers outperform the PD controller. However, accordingto [30], the performance is greatly degraded when employingincorrect model parameters. In [34], experimental study ontracking of the open frame vehicle ODIN is presented. Al-though employed on a nonlinear model of the ODIN vehicle,the reported controller is a linear PID controller. Hence, it is notmodel-based since it does not incorporate the model dynamicsin the controller. Nevertheless, the controller provides goodtracking results. In [30] and [34], velocity measurements areavailable for feedback. Successful tracking results of an MBCderived using the backstepping theory are presented in [2].The vehicle, an open-frame hovercraft, is descried by a threeDOF horizontal model without nonlinear damping. All thesementioned results have in common that the velocity is availablefor feedback, and all, except [2], assume that the destabilizingCoriolis forces are dominated by the hydrodynamic damping insome sense. Comparing with low-speed applications for ships,e.g., dynamic positioning [21], this is a common approach forcontrol plant modeling. Moreover, the hydrodynamic propertiesof a box shaped vehicle indicates that the damping is dominant,and that the hydrodynamic Coriolis forces are negligible. How-ever, for slender-body vehicles with some forward speed, thisassumption is not realistic.

The work presented in this paper is motivated by the Mines-niper MkII developed by Kongsberg ASA. The AUV/ROV isa low-cost, torpedo-shaped underwater vehicle. The relativelysmall weight compared to the nominal speed implies that thedynamics are speed dominant and that the nonlinear character-istics of the hydrodynamics become decisive. Moreover, due tocost reasons, this generation of the Minesniper does not carryany velocity or inertial measurement units (IMU). The positionis measured by using a short base line acoustical measurementsystem. The sensor suite also provides measurements of theheading, pitch, roll, and depth. This restriction in the instrumen-tation contributes to increased challenges for accurate tracking.Therefore, to improve the performance, we propose an observerproviding position and velocity estimates. For underwater vehi-cles, speed measurements can be obtained by using Doppler Ve-locity Log (DVL) [19] or by integrating accelerations measuredby the IMU. However, the DVL can only generate accurate ve-locity measurements provided that the distance to the seaflooris within a certain boundary. Furthermore, IMUs are subjectto drift in the derived velocity when integrating faulty accel-eration measurements. Consequently, the output feedback con-troller proposed in this paper may also improve the performancefor slender-body vehicles with more sophisticated sensor suitessince the proposed observer and controller may work indepen-dently of these velocity measurements, and thus contribute toincreased reliability of the control system by providing analyt-ical redundancy to the measurements. This makes the systemmore tolerant to faults.

In underwater applications, the ocean current has severe in-fluence on the vehicle performance, and the current influence

is difficult to predict even though measurements of both vehicleand water velocity are available. Therefore, a common approachis to model the disturbance as a constant, or slowly varying bias,see, e.g., [34] and [14]. A drawback of this method is that the hy-drodynamic properties of the vehicle are not properly accountedfor when modeling the current loads. Other reported methods in-volve using kinematics and filtering techniques to obtain an es-timate of the current velocity. Examples of this can be seen in,e.g., [5], [3], and [4], in which all require velocity feedback ofsome kind. In this paper, we will employ the modeling approachfirst introduced in [26] and more thoroughly described in [29].A 3-degree-of-freedom (DOF) model in surge, sway, and heaveis derived to serve as a foundation for current observer design.This is a current-induced vessel model that can be interpreted asa third-order filter with constants obtained based on the vehicleparameters. The goal is to provide an estimate of the currentvelocity and thereby estimate the influence of the current loadson the vehicle. With this approach, the key hydrodynamic prop-erties are taken into account when estimating the effect of theenvironmental disturbance, since the estimated current velocityis explicitly used in calculation of the nonlinear hydrodynamicdamping and Coriolis forces. Furthermore, since only the orien-tation, and in particular the position, which can be contaminatedby severe noise, are measured, a higher order model is preferredin order to avoid large jumps and oscillations in the current es-timate. Successful experimental results of this observer conceptcan be found in [27] which reports the design of a three DOFcurrent-induced vessel model coworking with a complete non-linear six DOF vehicle model. An output controller has not yetbeen tested with the observers in [27]. This paper adds to theresults on output feedback control presented in [29] since wein this paper consider the case where velocity measurementsare not available, and in addition, this paper presents the resultsfrom experimental tests carried out on a full-scale vehicle in theocean.

In this paper, we consider underactuated AUVs, a vehicleproperty which often complicates the overall analysis. In [10]and [3], the guidance kinematics algorithms are included in thecontroller derivation. This makes it possible to prove conver-gence to the desired path despite the lack of control actuators.We use a slightly different approach in this paper by consideringthe desired trajectories as external, time-varying, and boundedsignals. This contributes to relatively simple solutions for theobserver-controller design. We then show that the unactuatedstates are bounded due to hydrodynamic damping by analyzingthe inherent dynamics of the proposed controller. This approachwas first introduced in [13], and it is a convenient tool whichfollows from using the backstepping method [20]. The three-di-mensional guidance system is based on the line-of-sight method,which has been thoroughly described and analyzed in the liter-ature; see, e.g., [7] and [8].

B. Main Contribution and Paper Outline

The main contribution of this paper is the following. The de-sign and results of a guidance and control system for slender-body underactuated AUVs measuring only position, depth, andorientation. The control objective is to track the desired pitchand heading angles while keeping constant forward thrust. The


output feedback controller consists of a pair of coworking non-linear Luenberger observers providing filtering and estimationof the position, Euler-angles, and the vehicle and current veloc-ities. The proposed vehicle CPM is semi-linearized; the desta-bilizing Coriolis forces and moments are linearized about therelative forward speed when applying constant thruster revolu-tion. However, nonlinear damping is included. Experimental re-sults show satisfactory performance of the proposed observersand controller. Cascaded systems theory is employed to proveasymptotical stability of the closed loop system. Neither highgain nor bounded controller feedbacks are required, and the ob-server and controller gains can be tuned separately. These fea-tures are convenient for practical implementation. This paper isan extension to the work presented in [28], displaying all proofsand also elaborating on the guidance system and the actuatormodeling. An important objective of this work is to develop anobserver-controller system that is easily implementable. Due tothe nonlinear coupling between the three DOF current-inducedvessel model and the five DOF vehicle model, the stability anal-ysis becomes quite involved, but the resulting observers andcontroller are easily implementable.

This paper is organized as follows: A description of the math-ematical modeling is given in Section II. The observer and con-troller design and analyses are given in Sections III and IV, re-spectively. Furthermore, a case study on the Minesniper MkIIis presented in Section V, describing the actuator modeling,the guidance system, and finally, the experimental results areshown. Some concluding remarks are given in Section VI.

II. MATHEMATICAL MODELING

The CPMs proposed in this paper are based on the followingsix DOF dynamics presented as the PPM in ([12], chap. 4.3) and[31]

(1a)

(1b)

where and denotethe North-East-down (NED)-frame position/orientation andbody-frame velocities in six DOF, respectively. The relative ve-locity is given by , whereis the current velocity vector containing the body-fixedcurrent velocities in surge, sway, and heave. Furthermore,

and are the frame transformation, mass,rigid-body Coriolis, added mass Coriolis, and damping ma-trices, respectively. The vectors and capture the restoringand control forces and moments, respectively. The followingsubsections present the corresponding CPMs for the applicationconsidered in this paper.

A. Steady-State Analysis of Surge Motion

The motivation behind this section is to determine the con-stant about which to linearize the CPMs. We decouple the surgemotion from the rest of the model and study the steady-state be-havior. This decoupling of the dynamics is a method that hasshown to provide a successful basis for underwater vehicle con-trol design; see, e.g., [16]. Considering only steady-state, i.e.,

straight line motion, gives the following steady state surge forceequality:

(2)

where represents the linear and nonlinear hydrody-namic damping. The relative surge velocity is given by

, where and denote the body-fixed vehicle and cur-rent velocity, respectively. Furthermore, Whitcomb and Yoerger[33] have presented a quasi-steady approach for mathematicalmodeling of thrust which has shown to match the thrusters onthe Minesniper MkII well

(3)

where [rad/s] denotes the propeller revolution, and andare positive constants. More details regarding the thruster

forces will be presented in Section V-A1. Other commonthruster models for underwater vehicles omit the term propor-tional to ; see, e.g., [17]. However, full-scalebasin tests of the Minesniper MkII have shown that (3) matchesthe actual thruster forces in a satisfactory manner, although

; see Section V. In transit, it is reasonable to determinea constant propeller revolution which the forward thruster isoperating on. This is mainly because obtaining a certain speedis of less importance compared to tracking of the orientationand depth. Therefore, employing constant propeller revolution,it follows from (2) and (3) that the relative forward velocity

is constant in steady-state , i.e., , whereis a known constant velocity. The reason for this can be

verified by the following: It follows that , thismeans that the even though is dependent on the orientationof the vehicle relative to the current, which is time-varying,it follows from (2) and (3) that will vary accordingly, suchthat the relative velocity remains constant in steady state. Thisis valid independently of the vehicle orientation. Note thatis obtained analytically by solving (2) and (3) with respect to

. Inaccuracies in the parameters are equivalent to the effectof a current which will be estimated by an observer. Hence, apotential error in the calculated will be automatically com-pensated for, provided that the observer converge to the actualstate. To illustrate the scenario, Fig. 1 depicts a simulation usingthe PPM (1) of the Minesniper MkII performing way-pointtracking in a current. The right plot shows that the relativeforward velocity is the same for all headings, onlyslightly varying when changing course direction. Moreover,note that the vehicle velocity is altering with the heading.Hence, the modeling approach proposed in this paper is anextension to the results in which constant forward velocity isassumed, (i.e., ; see, e.g., [16] and [12, Ch. 13], since theapproach in this paper is also valid when there is ocean currentspresent. In this paper, we will consider AUVs in transit wherethe forward velocity is larger than the current velocity such that

.

B. Two Separate Systems

The CPM presented in this section is based on the approachpresented in [29]. The key idea is to apply two coworking


Fig. 1. Simulation with the Minesniper MkII performing way-point tracking.Left: Horizontal position. Right: Surge velocities u; u and u .

dynamic models: a five DOF vehicle model and a three DOFcurrent induced vessel model (surge, sway, heave) which cap-tures the main current loads on the vehicle. When utilizingthis method, the vehicle dynamics are taken into accountwhen modeling the current loads since the current velocity isused explicitly in the vehicle model. Furthermore, since weconsider underwater vehicles measuring only the position andorientation, and since the current loads are strongly dependenton the vehicle dynamics, we choose to employ a model-basedobserver for the current estimation.

CPM 1: Vehicle Model: Key properties of a slender-bodyAUV are taken into account when deriving the CPM: port-star-board symmetry, self-stabilizing roll, and the fact that the lengthis much larger than the width. Thus, by neglecting roll and ap-plying a constant propeller set-point, the following CPM is pro-posed:

(4a)

(4b)

(4c)

where denotes the NED-frame position andorientation vector, and the vector contains the pitchand yaw angles. The matrix transforms body-fixed vectorsinto NED-frame coordinates and is given by

(5)

and

(6)

is the vector containing the gravity and buoyancy moments,is the vehicle mass, and denotes the Earth gravity. Further-more, and are the distances between the buoyancy andgravity center in surge and heave direction, respectively. It is as-sumed that the vehicle is neutrally buoyant. The velocity vector

denotes the surge, sway, heave, pitch, andyaw velocity, respectively. In the NED-frame, the current is de-scribed by the vector . This vector isrotated into the body-frame as , where

. This gives the following expression for therelative velocity . The bias

captures the external disturbances such as drag froman optical fibre connecting the vehicle to the surface vessel.Moreover, it is included to compensate for unmodeled vehicleand thruster dynamics, [12]. It is modeled as a slowly varyingprocess, where is a diagonal matrix containing the positivetime constants, and and are tunable scalar gains. Theseare included for stability purposes and will be determined inthe upcoming section. In this paper, we consider vehicles withcontrol actuators in surge, pitch, and yaw resulting in the fol-lowing control vector , which is commonfor slender-body underwater vehicles in transit. The mass anddamping matrices yield

(7)

(8)

where denotes the mass elements included added mass. Fol-lowing slender-body theory presented in [23, Ch. 7], we let thedamping coefficients in (8), which are all positive, be defined asfollows:

(9)

where and so forth are the linear and nonlineardamping coefficients, respectively. Only the linear off-diagonalterms are included since these are dominating for slender-bodyvehicles [23, Ch. 7.4]. The total damping is thus given by a linearand a nonlinear part according to ,


where and denote the linear and non-linear damping matrices, respectively. The Coriolis matricesare defined as follows:

(10)

(11)

where , and are the added mass coefficients in surge,sway, and heave, respectively. See, e.g., [12, Ch. 3] for furtherdetails regarding the coefficients in Coriolis matrices. The mainelements of the destabilizing overall Coriolis forces and mo-ments shown in (1) are comprised in (10) and (11), includingthe Munk-moment. The Munk-moment is a hydrodynamicphenomenon which affects all geometric shapes in water exceptspheres. This CPM includes the most dominant Munk-mo-ments, i.e., and in pitch andyaw, respectively. Especially for slender-body vehicles withlength much larger than the width causing ,the Munk-moment is decisive. For box framed vehicles with

, the resulting moments are often negligible.The Munk-moment is destabilizing in the sense that it tries toturn the vehicle perpendicular to the flow. In this model, theMunk-moments in pitch and yaw are captured in andin , respectively. For more details on this, see, e.g., [32] and[11, Ch. 6)].

CPM 2: Current-Induced Vessel Model: This is a currentinduced vessel model that captures the slowly varying loadscaused by the current. The key task of this model is to functionas a basis for observer design to obtain an estimate of the currentvelocity . The following three DOF model is proposed:

(12a)

(12b)

(12c)

where represents the vehicle velocities insurge, sway, and heave, respectively, anddenotes the vehicle position in the NED-frame. The matrices

are the top left submatrices ofand in (4), respectively, and the control vector

yields . Underwater vehicles with no controlactuators in sway and heave are incapable of counteractingthe forces induced by the current in these directions. Hence,in CPM 2, it is clear that given , and since and arediagonal matrices, any nonzero velocities in sway and heave

must originate from the bias , which captures theslowly varying current forces. Hence, (12) is not a model of

the current, but a model that serves as basis for an observerintended to estimate the dominant response of the vehicle dueto the current, i.e., a current-induced vessel model. The modelmay also be interpreted as a third-order filter with gains ob-tained from the vehicle parameters. By comparing the nominalvelocity with the actual velocity , the induced currentvelocities in surge, sway, and heave may be obtained accordingto the following:

(13)

Although the current velocity also affects the motions in pitchand yaw, we assume that the main current loads are capturedin the linear motions: surge, sway, and heave. In fact, it can beshown by using the assumption on constant current velocity inthe NED-frame, i.e., ,that , where (skew-symmetric), [12]. Hence, in a Lyapunov sense, rotation of thevehicle is contributing to neither stabilizing or destabilizing thesystem. Thus, the proposed CPM 2 of the main current loads onthe vehicle is valid.

III. OBSERVER DESIGN

In the following section, we propose two separate nonlinearLuenberger observers for the CPMs presented in Section II.

A. Preliminaries

The following assumption and properties yield throughout thepaper.

A.1: The pitch angle is limited by . For mostunderwater vehicles, this is realistic given the inherent restoringmoments preventing the vehicle from large pitch angles.

P.1: Only the position and the Euler angles, i.e., the vector, is measured. Although the signals in , especially the hori-

zontal position , are updated at low frequencies, we con-sider the error generated by the zero-order-hold in discrete timeas bounded. Hence, is considered continuous.

P.2: The magnitude and direction of the current is unknownbut upper bounded, i.e., there exists a constant suchthat .

We use the following notation in this paper. For any positivedefinite and symmetric matrix , and denotethe minimum and maximum eigenvalue of , respectively. Fur-thermore, denotes the Euclidian norm of a vector.

In order to fully utilize the properties of the hydrodynamicdamping, the five DOF damping matrix in CPM 1 is separatedinto a diagonal and an off-diagonal part according to

, where . Here,and denote the diagonal and off-diagonal part of the lineardamping matrix, respectively. The reason for this partition is thematrix is not necessarily positive definite for slender-body AUVs. However, this yields the following property.

P.3: It follows from (8) that the nonlinear diagonal dampingmatrix satisfies . Moreover, thereexists a constant such that .In CPM 2, the three DOF damping matrix is diagonal.Hence, it follows that . To simplify


the notation we define the function , which byemploying the mean value theorem gives

where , and is on the line segment joiningand . Furthermore, the damping matrix (8) gives that there

exists a constant such that .P.4: The mass matrix and its submatrices are positive

definite and symmetric. Hence, and, where is

the mass matrix transformed into the NED-frame.P.5: The rotation matrix is orthogonal and satisfies

. Furthermore, under A.1, there exist constantssuch that yields

B. Current Observer

The following Luenberger observer is proposed. Note that thegray box indicates that this is implemented in the control system

(14a)

(14b)

(14c)

where , and are positivedefinite and diagonal observer gain matrices. According to (13),the estimated current velocity is derived as

(15)

Furthermore, we define the error vectors and. Subtracting (14) from (12) and letting

and , the observer errordynamics become

(16a)

(16b)

(16c)

where refers to in P.3. Let

(17)

denote the overall current estimation error state. The error dy-namics (16) will be proven uniformly globally exponentiallystable (UGES) in the upcoming section.

C. Vehicle Observer

In this section, we derive a nonlinear Luenberger observer forthe vehicle dynamics. To avoid technicalities using two frames

(NED and body-frame), CPM 1 (4) is rewritten in NED-framecoordinates as follows:

(18a)

(18b)

(18c)

where denotes the NED-frame velocities,the NED-frame relative velocity yields . The su-perscript marks the NED-frame matrices. See, e.g., [12, Ch. 3.3]for details regarding the matrix transformations. The followingproperty yields:

P.6: There exist sufficiently large constants suchthat the Coriolis matrices are upper bounded according to

We propose the following Luenberger observer by copying thedynamics in (18) and adding correction terms

(19a)

(19b)

(19c)

where , and are positive definiteand diagonal observer gain matrices. Subtracting (19) from (18)and using P.3 gives the following observer error dynamics

(20a)

(20b)

(20c)

where and . Employing cascaded systemstheory [24], we define the following perturbation vector

(21)

since is proportional to the current estimation error. This re-sults in the following nominal observer error dynamics:

(22a)

(22b)

(22c)

Notice that also evolves linearly with the estimated velocity, which growth is unknown. Hence, we apply the following

assumption:A.2: There exists a constant such that the relative

velocity is bounded according to . This as-sumption will be lifted when the overall closed loop system,including the controller error dynamics, are analyzed. This is acommon method in output feedback controller design; see, e.g.,


[6]. Under A.2, we have that , whichonly consists of error variables and thus corresponding with cas-caded systems theory methods.

In order to fully exploit the dissipative property of the hy-drodynamic damping, we will analyze the current estimationerror and the vehicle observer error dynamics using one Lya-punov function. Let the error vectors be defined as

, and . Note here that theerror vector includes the current estimation error. The nominalobserver error dynamics (16) and (22) can then be written incompact form as the following:

(23)

where

and where

The nonlinear and the diagonal linear damping matrices are col-lected in using P.3 as follows:

Moreover, the vector consists of the following functions:

Notice that system (23) is autonomous since all the error vari-ables are concatenated in the error vector . Consequently,we apply autonomous Lyapunov stability theory. The matrix

is Hurwitz, and hence, there exist positive definite and sym-metric matrices and such that . More-over, since the current velocity is upper bounded, as claimedin P.2, gives that there exists a constant such that

.Proposition 1: The origin of the nominal observer

error dynamics (23) is globally exponentially stable (GES) if thefollowing condition is satisfied:

(24)

where is the minimum eigenvalue of , and.

Proof: Consider the following positive definite and radiallyunbounded Lyapunov function candidate .Differentiating with respect to time gives

Using P.6, can be upper bounded as follows:

(25)

Recall that which gives thatfor all

. We thus arrive at the following upper boundon the Lyapunov function derivative:

(26)

Using Lyapunov theory [18], it follows that if (24) is satis-fied, there exists a constant such that

. Thus, the origin of the nominal observererror dynamics (23) is GES. .

Recall that the current estimation error dynamics (16) is astand-alone system independent on the vehicle observer errordynamics (22). Notice, however, that the right-hand side of thesystem (16) includes the time-varying vector , whereas thecurrent error dynamics only involve . To circum-vent this problem, we can consider as a general time-varying signal using forward completeness as in [21]. We as-sume that the time-varying vector exists for all . Thisis trivial since in the following proofs is part of the trackingerror which automatically lifts this assumption. Hence, it fol-lows that the origin of system (16) is UGES. We thushave that the error variable defined in (17) can be boundedby

(27)

where and are positive constants. Moreover, sinceof system (23), and of (16) are GES and UGES,

respectively, it follows that of system (22)is GES.

Remark 1: The Lyapunov analysis provides only sufficientconditions for stability which often leads to conservative results.In this case, the demand on (24) may be unnecessarily strictsince the nonlinear damping, which is dominant for most states,is not contributing to alleviate the condition on . However, theresult clearly indicates that the observer feedback gains must besufficiently high to dominate the destabilizing Coriolis forcesand the bias loads.

Remark 2: Note that the globalness is given with respect tothe chosen coordinate frame. It is not topologically possible toobtain results that are global in using any coordinateframe of like the Euler angles, Euler parameters, Euler-Rodrigues parameters, or similar. Due to the topological prop-erties of , these representations will either have one sin-gularity or two equilibrium points, something which precludes


global results on . Therefore, the state space defined inthis paper does not include , and the results in thispaper are thus only global in the chosen coordinate frame.

IV. OUTPUT FEEDBACK CONTROL

In this section, we design a nonlinear controller utilizing theobserver backstepping technique [20, Ch. 7)]. Since the vehicleis underactuated, the controller design and analysis becomemore involved. However, the fact that all states, including theunactuated states sway and heave, are subjects to linear hy-drodynamic damping, enables a separate study of the trackingerror dynamics and the unactuated states. This approach wasfirst introduced in [13], and it is a convenient feature of thebackstepping procedure. The method is shortly described byfirst designing the control vector considering it as an arbitraryvector in . Second, we analyze the inherent dynamics of thecontroller that arise since there are no controls in sway andheave. It will become clear that due to hydrodynamic dampingin all degrees of freedom, the velocities of the unactuated statesconverge to a bounded set.

The control objective is defined as tracking the desired orien-tation as follows:

(28)

as , where contains the smoothand continuously differentiable reference trajectories. The keyidea behind the approach shown in this paper is to split the totaltracking task in two.

1) Design a control system such that the control objectivestated in (28) is guaranteed.

2) By carefully designing the guidance system, globaltracking of, for instance, way-points or a path can be ob-tained if the control objective (28) is met. For more details,see, for instance, [7], in which they prove global conver-gence to the desired track/path using a line-of-sight-basedmethod in the guidance system. This concept is basedon the fact that given a nonzero forward speed and someorientation of the vehicle, any point in the global framecan be reached.

A. Controller Design

Based on the observers designed in the prior section, the es-timated position and velocity are available for feedback.Hence, we utilize the observer backstepping method. Further-more, since the control vector naturally evolves in the body-frame, it is desirable to back-step into the following observerdynamics:

(29a)

(29b)

(29c)

In this section, the Euler angle symbol is omitted when it isused in a transformation matrix for notational simplicity. More-over, the time variable is included for the reference trajec-tories since these are external time-varying signals. Thus, thecontroller error dynamics are nonautonomous.

Step 1: The nonlinear damping introduces undesirable cou-pling terms that complicate the stability analysis. Therefore,since the pitch and heading angles are measured, we define twotracking error vectors as follows:

(30)

The reason for this will become clear in the upcoming stabilityanalysis. Then, computing the corresponding error dynamics bydifferentiating and with respect to time and using (29a)and that give

(31a)

(31b)

where

Here, and are the continuous reference trajectoriesfor the angular velocities. The second error vector is defined as

(32)

where is a vector of stabilizing func-tions that we will choose, andis the velocity tracking error vector. Inserting for in (32) into(31) yields

(33a)

(33b)

where contains the stabilizing functions forthe actuated states: pitch and yaw. In order to render (33) stabledifferential equations, we choose the stabilizing function toevolve according to

(34)

where is a positive diagonal controller gain matrix.Note that we have used in the -function in order to avoidnonlinear coupling terms involving the estimation error . Thisresults in the following expression for the - and -dynamics

(35a)

(35b)

where the stabilizing function is rewritten asto fit into (35b), and where we have mul-

tiplied with the positive and diagonal controller gain matrixin order to increase the design flexibility in the

controller.


Step 2: Proceeding with the -dynamics

(36)

We choose the control vector according to

(37)

where , and is a diagonal controller gainmatrix. Notice that we have used feedback gain and not

in (37) in order to keep the level of measurement noisein the controller to a minimum. Furthermore, note that the con-troller is considered as a general vector in and thatare left undecided. This is, however, trivial since the Lyapunovanalysis is valid, provided that (37) is satisfied, independentlyof the contents of vector . The fact that the second and thirdelement are zero, i.e., will be considered inSection IV-C, where also will be determined.

Remark 3: The proposed controller (37) is not directly de-pendent on the estimation error . This is an advantage sincethe horizontal position provided by the acoustical measurementsystem can be contaminated by severe noise. If controller termsproportional to are included, scattering and thereby degrada-tion of the controller is likely to occur.

B. Controller Analysis

In [21], a solution for output feedback control of dynamicpositioning (DP) of ships is reported. The paper presents, byemploying cascaded systems theory [25], a convenient methodfor deriving observers and controllers for nonlinear systems. Itis shown that the separation principle holds for the nonlinearcase in the sense that the controller and observer can be tunedseparately. The DP model in [21] is nonlinear only because ofthe rotation matrix between the NED and the body-frame. Thispaper is thus an extension to that result since the CPM models inthis paper are coupled and involve nonlinear damping. Anotherfavorable result of the following analysis is that the controllergains and do not need to meet high gain criteria inorder to ensure stability.

We want to formulate the controller error dynamics and ob-server estimation error as a cascaded system. In order to achievethis, we need to define new controller error vectors; the errorbetween the desired state and the actual state

, and not the estimated state , which is the naturalresult of the observer backstepping technique used for the con-troller design in Section IV-A. Hence, we define the followingnew error vectors:

(38)

With the new error states, we will show that the controller errordynamics can be written in a cascade with the nominal ob-server dynamics as follows:

(39a)

(39b)

where the perturbation is to be defined. The errorsystem (39b) represents the nominal observer error dynamicsdepicted in (23), where the origin is proven GES inthe Proof of Proposition 1. The complete observer system, in-cluding the perturbation vector (21) will be analyzed in theproof of the final theorem of this paper. Recall that the pertur-bation vector is derived under A.2, which claims boundedvelocities of the vehicle. In order to avoid circularity in the fol-lowing stability analysis, we lift A.2 and replace it with the fol-lowing assumption on bounded desired velocities.

A.3: There exists a constant such that the desiredvelocity is bounded according to .

Under A.3, we have that the perturbation vector to thenominal observer error dynamics (23) can be upper boundedaccording to

(40)

by using (32) and (34). Hence, consists only on error vari-ables and can be treated as a stand-alone perturbation. Note thatthe proof of Proposition 1 still holds when replacing A.2 withA.3.

In order to rewrite the control vector (37) so that it consistsof the actual states and not the estimated states, we use that

and resulting in

(41)

Using (41), the controller (37) can be rewritten as follows:

(42)

Here, we have collected the terms involving the estimation errorinto the perturbation vector as follows:

where we have used (41), and that. Inserting the modified control vector (42)


into the actual vehicle dynamics (4) results in the following-dynamics:

(43)

where , and from (35a) and(41) the perturbation vector for the tracking error capturing theterms involving the estimation error becomes .Having established this, it follows that the rewritten trackingerror state (43) can be viewed as a cascade with the nominalobserver error dynamics (23), and where the perturbation vectoris given by , as depicted in (39).Notice here that the nominal observer error dynamics (23)includes the time-varying signals , while weonly consider the tracking error . We thus consider

as general time-varying signals using forwardcompleteness [21]. The following assumption, which will beproven to hold in the Appendix, is applied.

A.4: The time-varying signals exist for all.

Proposition 2: The origin and of the cas-caded system (39), described in (43) and (23), is uniformly glob-ally asymptotically stable (UGAS) under A.3, A.4, and if con-dition (24) is satisfied.

Proof: Following cascaded systems theory arguments eachof the functions in (39) are analyzed separately in three steps.

1) The origin of the nominal observer error dynamics(23), denoted as in (39), is proven GESin the Proof of Proposition 1.

2) At this step, we want to establish the stability propertiesof the unperturbed system shown in(43). Similarly as in Section III, we want to exploit thedissipative property of the hydrodynamic damping. Wethus include the current estimation error in the controllerstability analysis and propose the following radially un-bounded Lyapunov function candidate

(44)

where and . Differentiating(44) with respect to time and inserting for the tracking dy-namics (43) without the perturbation gives

(45)

where we have inserted for the current estimation error dy-namics (16), which are rewritten in the body-frame. Ac-cording to [1, Lemma 2], the origin is UGAS if theLyapunov function (44) satisfies

(46)

where , and is a class functionsatisfying , where , and issome constant. In this case, since is proven UGESand satisfying (27), it follows that condition (46) holds, andthus is UGAS, and therefore also the originof the unperturbed system (39a) is UGAS.

3) This step involves determining the growth of the perturba-tion vector . Using P.2, P.6, and P.5 gives

where we have used that, under A.3. The perturbation can be shown

to be upper bounded by the tracking error state and the es-timation error as follows:

(47)

where

Thus, the linear growth restriction on in the pertur-bation is satisfied. Consequently, based on the three priorsteps, it follows that the origin and of thecascaded system (39) is UGAS [24, Theorem 2.8]. Thiscompletes the proof.

Up to this point, we have only considered the nominal ob-server error dynamics (22). The following theorem establishesuniform global asymptotic stability of the overall output feed-back controller. Let the error vector denote the com-plete error state excluding the current estimation error, i.e.,

.Theorem 1: The origin and of the cascaded

system (16), (20), and (43) is UGAS under A.1, A.2, and A.4,and if (24) is satisfied.

Proof: We write the overall system including the currenterror dynamics in the following compact form:

(48a)

(48b)

. The origin of the unperturbedsystem in (48a) is UGAS given Proposition 1and 2. Furthermore, the current estimation error dynamics (48b)are proven UGES in Proposition 1. Following cascaded systems


theory, it remains to show bounded growth on the perturbationvector . It follows from (40) that it can be boundedlinearly by and hence satisfying

(49)

where

The linear growth restriction on in the perturbation term issatisfied. It thus follows from [24] that the origin

of the system (48) is UGAS. Consequently, it follows that theorigin of the error dynamics (16), (20), and (43) are UGAS.

C. Underactuated

In this section, we consider the fact that the vehicle is under-actuated. Recalling that the control vector yields

(50)

This results in a dynamic constraint in the controller for theunactuated states, i.e., sway and heave since we cannot assigncontrol force in these directions. It is clear from the analysisin Section IV that the -dynamics are UGAS provided thatthe control vector satisfies (37). For pitch and heave, the ob-vious choice is to design the stabilizing function renderingthe tracking error dynamics stable since we can assignand arbitrarily. This is, however, not feasible for and

. Instead, and must satisfy the differential equation(37) that arises since there are no controls in sway or heave

. Hence, in this sense, instead of tracking, the -dynamics track the actual velocities .

It will become clear that the -dynamics converge ex-ponentially to a bounded set due to the linear hydrodynamicdamping in sway and heave.

From (37), we have that

(51)

where the bounded and converging variables are concatenatedin the function . All the signals in

are shown to be bounded or converging to zero exceptthe bias term . Recall from Section II that this captures theunmodeled dynamics. Obviously, the proposed CPM (4) needsto resemble the real world to a certain extent in order to ob-tain a stable MBC for real systems. This is the fundamentalidea behind MBC in this paper. Therefore, it follows naturallyto declare the effect of the unmodeled dynamics as bounded andhence, is bounded. Another common ap-proach in marine applications is to define the bias as constant,

i.e., [12], which clearly manifests the statement above.This approach, however, is not employed in this paper since

only leads to negative semidefinite Lyapunov functionderivatives, which subsequently enhance the complexity of thestability analysis.

We proceed by analyzing the -dynamics by rewriting(51) into compact form and collecting all the bounded and con-verging signals into the vector function

(52)

where and

The vector captures the off-diagonal terms in the mass,damping and Coriolis matrices ( and ). Furthermore, no-tice that consists solely of bounded and converging signals

. Hence, we add to the function .Proposition 3: The —subsystem is input-to-state stable

(ISS) from to .Proof: This can be proven by applying for instance the Lya-

punov function candidate , which differ-entiating with respect to time along the solutions of gives

. This leads to

(53)

where is inserted into since is provenbounded in Section III-B. Then, there exists a constant thatsatisfies , and the following upper bound on (53) isobtained:

Clearly, due to the dissipative hydrodynamic damping, weachieve UGES of of the unforced system. This implies thatthe system is ISS from to [18, Lemma 4.6]. Hence,converges to the bounded set .Moreover, since and as , it follows that

as . Finally, as impliesthat as . Consequently, the velocities in swayand heave, i.e., are bounded and converge to the sameset. This completes the proof.

V. CASE STUDY: THE MINESNIPER MKII

The parameters of the Minesniper MkII (see Fig. 2) areshown in Table I. The added mass coefficients were obtainedby using the computer program WAMIT, whereas the dampingand thruster coefficients were calculated based on basin tests


Fig. 2. Minesniper MkII. Courtesy of KDA, Norway.

TABLE IMINESNIPER MKII CPM CONSTANTS

TABLE IISENSOR PROPERTIES

at Kongsberg Defence and Aerospace at Stjørdal, Norway.Table II presents the sensor suite of the vehicle.

A. Control Forces and Moments

This section describes the mapping from the control momentsderived in Section IV-A to the control action of the actuators onthe vehicle.

1) Thrusters: On the Minesniper MkII, the heading controlis obtained by two horizontal thrusters located on each side ofthe hull at the center of the vehicle. Recall from (3) that thethruster force is , where the constantsare defined as [33]

(54)

Fig. 3. Pitch control by mass movement.

where is the water density, and and denote the propellerdiameter and the thrust deduction coefficient, respectively. Fur-thermore, is the ambient water velocity insteady-state where (typically 0.1–0.4) is denoted asthe wake fraction number [12]. Based on the linear approxima-tion and are positive constants. The equations for forwardthruster force and yaw moment can then be described as

(55)

where and are port and starboard propeller revolutions,and is the distance from the propeller to the center of thebody. The corresponding revolutions are then obtained by

(56a)

(56b)

where is the constant revolution providing surge velocity.2) Pitch Actuator: The Minesniper is equipped with a move-

able mass for pitch control. It is a common phenomenon thatunderwater vehicles can have difficulties obtaining initial pitchangle when surfacing. This is avoided by this type of controlaction since the static pitch angle varies with the location of thepitch mass. Fig. 3 describes the system of pitch control by massmovement. Movement of the pitch mass leads to the followingexpression for the center of gravity: ,where is the distance from the center to the mass, anddenotes the weight of the actuator. Recall that the vehicle is neu-trally buoyant, hence . Notice also that the con-trol moment obtained in (37) involves , which contains theexpression for the static pitch angle given a certain ; see (6).Therefore, to obtain the expression for the distance of the pitchmass actuator , we solve the equationwith respect to . Then, to generate the pitch moment given thispitch angle, it follows from Fig. 3 that the pitch mass distance is

(57)


Fig. 4. Kinematics of the horizontal LOS guidance system.

where is saturated such that whichinherently includes the limit .

B. Line-of-Sight Guidance System

Guidance is provided by two decoupled line-of-sight systemsfor heading and pitch, respectively, [22].

1) Heading: In order to avoid potentially large cross-trackerrors, the aiming point is moved from the next way-point to apoint on the track line by setting the length of the aiming vectorto ; see Fig. 4. We start by defining the total track length be-tween way-point and , i.e., ,

where and

where the ordered pairs andare the present and the previous

way-points, respectively. The distances and are givenby

where

Note that ranges from 0% to 100% of . The anglebetween these two vectors is then defined by

(58)

where . The cross-track errormay now be defined as . Based on the

Fig. 5. Kinematics of the vertical LOS guidance system.

cross-track-error and the length of the aiming vector, the desiredheading yields , where

if

sign otherwise.

The LOS-point on the track (see Fig. 4) is given by

In order to ensure correct desired heading at all times, the guid-ance system automatically switches to way-point tracking ac-cording to the following:

if or

then

Care must be taken when determining the size of prior to arun. It is shown in [15] that for way-point maneuvering of ships,

must be larger than some constant which is dependent on thevehicle dynamics, actuators, and the forward speed. This corre-sponds well with tests carried out on the Minesniper revealingthat when was too small, excessive control action and an in-crease in the cross-track error occurred.

The next way-point is activated if one of two conditions ismet. The most usual case is if the vehicle has penetrated theway-point watch radius or if the vehicle has passed away-point by a distance larger than

if or

then

Activate next way-point.

2) Pitch: The LOS method is modified for pitch control ac-cording to Fig. 5. The pitch angle between two way-points is


Fig. 6. Left: Measured (red dots) and estimated (blue line) horizontal position of the Minesniper MkII performing way-point (green squares) tracking. A 5-mradius of acceptance around each way-point is included. Right: Measured (red) and estimated (blue) depth and orientation of the Minesniper MkII.

. The distance betweenthe next way-point and the LOS point is given by

(59)

Hence, the desired pitch angle is given by, where

according to Fig. 5.

C. Reference Trajectories

In order to obtain smooth and continuously differentiable ac-celeration, velocity, and position reference trajectories, we uti-lize a second-order filter cascaded with a low-pass filter [12].The reference trajectories are then given by

(60)

(61)

where is the relative damping ratio, is thenatural frequency, and the guidance functions are collected in

. A saturation is included

on the final reference pitch angle to ensure that the desiredpitch angle is feasible at all times, i.e.,

(62)

where are determined by setting the pitch weight inits maximum and minimum position, respectively.

D. Sea Trials—Results and Discussion

The sea trials of Minesniper MkII were performed inTrondheimsfjorden nearby Stjørdal, Norway. The observers,guidance, and controller were implemented as functions writtenin C code with 20-Hz update rate. Fig. 6 shows the measuredand the estimated position and orientation of the MinesniperMkII. The bay area is quite shallow and narrow which limitedthe location of the way-points. However, the heading shownin the bottom right plot of Fig. 6 shows large variations of theheading, which demonstrates the performance of the controller.The observer provides satisfactory estimates for the entirerun with small deviations and little noise. Toward the endof the run, we experienced increased noise in the acousticalmeasurements mainly due to the topography of the seabedblocking the view of the acoustical receivers. Nevertheless, theobserver seemed to cope with measurement drop outs and thenoise in a satisfactory manner. From Fig. 6, dead-reckoningcan be seen between the second and third way-point. Clearly,the observer provides satisfactory position estimated despitemeasurement drop outs. The way-point watch radius was setto m. This caused the vehicle to slightly miss thethird and the fourth way-point. Moreover, the reference systemgain was chosen relatively low, , in order to keep


Fig. 7. Tracking results. Left: The desired trajectories provided by the reference generator. Center: The tracking error in pitch and heading, i.e., z and [z ; z ].Right: The controller action, thruster revolution, and position of the pitch weight.

the yaw rate small. The main reason for this was that giventhe center location of the thrusters, several sea trials revealedthat erratic usage of the thrusters could cause intractable yawmotion. Therefore, emphasis was placed on keeping the yawmotion within relatively restrictive boundaries. Fig. 7 presentsthe tracking results in pitch and heading and the actuator action.The tracking performance is satisfactory with relatively smalldeflections of the actuators. In pitch, some error occurred in thebeginning of the run. This was because the forward accelerationgenerated a pitch moment. However, the pitch controller calmsthe motion relatively fast despite limited rate in the pitch massactuator. The estimation results are presented in Fig. 8 showingsatisfactory observer performance. We were unable to measurethe actual current. However, the estimates indicate a slight cur-rent from South-East. Table III shows the controller/observergains used in this run. The current observer gains were set equalto the upper left matrices of the vehicle observer gains. Thecontroller gains were found by trial and error since there exists,to the author’s best knowledge, no formal methods for tuninggains in a backstepping controller at the present time. Severalruns indicated that the controller gain should be set smalland large for optimal tracking results. This is mainly due touncertainty of the damping coefficients which, togethers with alarge , aggravated the performance.

VI. CONCLUSION AND FURTHER WORK

An output feedback controller was proposed for slender-bodyunderactuated underwater vehicles. The CPM system consisted

of two separate models: a five DOF vehicle model and a threeDOF current induced vessel model accounting for the main cur-rent loads. Part of the vehicle CPM was linearized about the rel-ative surge velocity. The nonlinear Luenberger observers andthe controller, which was designed using the observer backstep-ping technique, were proven UGAS using Lyapunov and cas-caded system theory. An advantage of the employed method isthat it does not require any high gain nor bounded feedback con-troller gains. Furthermore, the observer and controller gains canbe tuned separately. Experimental sea trials on the MinesniperMkII were presented showing satisfactory observer and trackingperformance. Further work involves obtaining more accurate ve-hicle parameters and optimal tuning of the controller feedbackgains in attempt to optimize the performance of the vehicle.

APPENDIX

In this section, we prove that A.4 holds by showing forwardcompleteness of the closed loop system. This is carried outby including the reference system algorithms presented inSection V-C in the overall stability analysis and thereby showthat the global position exists all .

Proposition 4: The time-varying vectorexists for all .

Proof: Consider the following Lyapunov function:

(63)


Fig. 8. Estimation results. The left column shows the estimation error ~�(t). The center column presents the estimated velocities �(t). The right column describesthe estimated current velocity � (t) and the ��(t)-dynamics versus [v(t); w(t)].

TABLE IIICONTROLLER, OBSERVER, AND LOS GAINS

Differentiating with respect to time gives

(64)

Before proceeding we note the following:1) Based on the guidance algorithms presented in

Section V-B, it can be shown that, where is a sufficiently large

constant. Moreover, and can be linearly upperbounded by signals captured in .

2) The overall observer-controller error dynamics given in(16), (20), and (43) can now be analyzed using the Lya-punov function (63) without applying A.4 since is in-cluded in . Based on these observations, it can beshown by inserting the error dynamics and using Schwartzinequality that satisfies , where

is a constant. Hence, the vector exists and canbe continued for all . Thus, A.4 holds.

ACKNOWLEDGMENT

The authors would like to thank their colleagues at CeSOSfor valuable discussions and contributions to the testing of theMinesniper MkII.

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Jon E. Refsnes received the M.Sc. degree in ma-rine technology from the Norwegian University ofScience and Technology (NTNU), Trondheim, in2003, where he is currently working toward thePh.D. degree in marine technology . The projectis in cooperation with Kongsberg Defence andAerospace, Stjørdal, Norway.

His research interests are underwater vehicles,modeling and hydrodynamics, observer design, andcontrol.

Asgeir J. Sørensen received the M.Sc. degree in ma-rine technology and the Ph.D. degree in engineeringcybernetics, both from the Norwegian University ofScience and Technology (NTNU), Trondeim, in 1988and 1993, respectively.

From 1993–2002, he was employed in the ABBGroup in various positions as a Research Scientist,Project Manager, Department Manager, and Tech-nical Manager in the Business Area AutomationMarine and Turbochargers. Since 1999, he has heldthe position of Professor of Marine Cybernetics at

the Department of Marine Technology, NTNU.

Kristin Ytterstad Pettersen (SM’04) received theM.Sc. and Ph.D. degrees in electrical engineeringfrom the Norwegian University of Science andTechnology (NTNU), Trondheim, Norway, in 1992and 1996, respectively.

She became a Associate Professor in 1996, andin 2002 Professor, in the Department of Engi-neering Cybernetics, NTNU. In 1999, she was aVisiting Fellow in the Department of Mechanicaland Aerospace Engineering, Princeton University,Princeton, NJ. She has published more than 70

conference and journal papers. Her research interests include nonlinear controlof mechanical systems with applications to robotics, satellites, AUVs, andships.

Dr. Pettersen received the IEEE TRANSACTIONS ON CONTROL SYSTEMS

TECHNOLOGY Outstanding Paper Award in 2006. She holds several boardpositions.

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