536 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 3, JUNE ... Collection/TRO/2006/june/10.pdf · trol literature for 3-DOF vertical takeoff and landing (VTOL) [21], [22]. More details

536 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 3, JUNE 2006

Approaches for a Tether-Guided Landingof an Autonomous Helicopter

So-Ryeok Oh, Kaustubh Pathak, Sunil K. Agrawal, Hemanshu Roy Pota, and Matt Garratt

Abstract—In this paper, we address the design of an autopilotfor autonomous landing of a helicopter on a rocking ship, due torough sea. A tether is used for landing and securing a helicopter tothe deck of the ship in rough weather. A detailed nonlinear dy-namic model for the helicopter is used. This model is underac-tuated, where the rotational motion couples into the translation.This property is used to design controllers which separate the timescales of rotation and translation. It is shown that the tether ten-sion can be used to couple the translation of the helicopter to therotation. Two controllers are proposed in this paper. In the first,the rotation time scale is chosen much shorter than the translation,and the rotation reference signals are created to achieve a desiredcontrolled behavior of the translation. In the second, due to cou-pling of the translation of the helicopter to the rotation throughthe tether, the translation reference rates are created to achieve adesired controlled behavior of the attitude and altitude. ControllerA is proposed for use when the helicopter is far away from the goal,while Controller B is for the case when the helicopter is close to theship. The proposed control schemes are proved to be robust to thetracking error of its internal loop and results in local exponentialstability. The performance of the control system is demonstratedby computer simulations. Currently, work is in progress to imple-ment the algorithm using an instrumented model of a helicopterwith a tether.

Index Terms—Attitude control, position control, robustness,tethered helicopter.

I. INTRODUCTION

I N RECENT years, considerable research has been per-formed on the design, analysis, and operation of autonomous

helicopters. Helicopters can perform low-speed tracking ma-neuvers and operate in situations where steady platforms arenot available for takeoff and landing, such as a ship deck.

A problem of importance for autonomous helicopters is thedesign of autopilots for landing on moving decks, subject to dis-turbances, such as in rough sea. The control problem for landingof an autonomous helicopter is challenging, since the vehicledynamics is highly nonlinear and coupled with unknown motion

Manuscript received May 6, 2005; revised August 16, 2005. This paper wasrecommended for publication by Associate Editor D. Sun and Editor H. Araiupon evaluation of the reviewers’ comments. This work was supported by theNational Science Foundation under Award IIS-0117733. This paper was pre-sented in part at the IEEE International Conference on Robotics and Automa-tion, Barcelona, Spain, April2005.

S.-R. Oh, K. Pathak, and S. K. Agrawal are with the Mechanical SystemLaboratory, Department of Mechanical Engineering, University of Delaware,Newark, DE 19716 USA (e-mail: [email protected]; [email protected];[email protected]).

H. R. Pota is with the School of Electrical Engineering, University of NewSouth Wales, Canberra, ACT 2600, Australia (e-mail: [email protected]).

M. Garratt is with the School of Aerospace and Mechanical Engineering, Uni-versity of New South Wales, Canberra, ACT 2600, Australia (e-mail: [email protected]).

Digital Object Identifier 10.1109/TRO.2006.870657

Fig. 1. A tethered helicopter with an autopilot to land on the deck of a ship inrough sea.

of the sea. Furthermore, helicopters are underactuated systems,i.e., have a smaller number of control inputs than the number ofgeneralized coordinates.

The control of a helicopter in hover has been dealt with fromdifferent points of view. Linear control design includes the useof adaptive controllers [1], linear quadratic Gaussian (LQG) op-timal control [2], [3], [4], -synthesis [5], [6], and dy-namic inversion methods [7]. These methods are based on lin-earized helicopter models around hover and trim conditions.Nonlinear control designs include sliding mode [8], nonlinear

[9], neural-network-based controller [10], fuzzy control[11], [12], approximate input–output linearization [13], differ-ential flatness [14], and backstepping [15].

The landing of autonomous vehicles is typically attemptedusing vision and global positioning systems. Vision-guidedlanding uses the assumption that the target’s shape is knownand the target is moving slowly [16]–[18]. A more recentapproach for landing of a manned helicopter is to use a tether,which is reeled out from the helicopter as it comes near thedeck (see Fig. 1). The end of the tether is then secured to thedeck. The tether is kept taut, and it provides a useful referencefor the pilot to make maneuvers and react to the relative motionof the helicopter and the ship. The pilots of the Canadian navyand others are using such a protocol to land a helicopter onsmall-sized ships in rough weather.

1552-3098/$20.00 © 2006 IEEE

OH et al.: APPROACHES FOR A TETHER-GUIDED LANDING OF AN AUTONOMOUS HELICOPTER 537

Fig. 2. Schematic of helicopter dynamics. At the inputs are the thrusts T and T generated by the main and tail rotors, and a, b are the flapping angles.

In this paper, we address the problem of autonomous landingof a helicopter on a ship’s deck using a tether. Although the navyadopts this protocol for landing manned helicopters on ships inrough sea, the unmanned autonomous landing problem using atether is unexplored.

The helicopter has a nonlinear underactuated dynamic model,where the rotation couples into the translation. One method tocontrol underactuated systems is to use a subset of degrees offreedom (DOFs) as control inputs. This typically forces a re-lationship between two carefully chosen sets of DOFs, a vari-able set that is used to control another independent set. The di-vision into these two sets is done such that the resulting systemis fully actuated. In practice, this scheme is realized using twotime-scale controls: fast dynamics for the controlled variablesand slow for the independent variables. The main contribution ofthe paper is to design controllers which separate the time scalesof translation and rotation. We consider two cases: Controller A,where the rotation constitutes the faster dynamics and is used tocontrol the translation of the helicopter; and Controller B, wherethe translation has faster dynamics and is used to control the atti-tude and altitude of the helicopter. The two controllers are usedduring different regimes of operation.

The rest of the paper is organized as follows. Section II intro-duces the spatial nonlinear dynamic model used in this paper. InSection III, a two time-scale position controller is designed, inwhich the rotational dynamics is faster than the translation. InSection IV, a two time-scale altitude–attitude controller is de-signed, in which two of the helicopter’s Cartesian coordinatesconstitute the faster dynamics. Simulation results for the con-trollers are provided in their respective sections. These are fol-lowed by conclusions of the paper.

II. HELICOPTER MODEL

The helicopter dynamics is based on the model presented in[14]. A similar model has been widely used in the nonlinear con-trol literature for 3-DOF vertical takeoff and landing (VTOL)[21], [22]. More details on helicopter dynamics can be foundin [23] and [24]. The model considers the fuselage of the heli-copter as a rigid body attached to the main rotor and a tail rotor.

SO(3) is a rotation matrix between the body axes relativeto an inertial coordinate frame. We parameterize byEuler angles, with , along the axes, respectively,and define . The position of the center of mass

(COM) of the helicopter is given by in the iner-tial frame. The forward, sideways, and downward velocities ofthe helicopter COM are given by along axes of the bodyframe, , respectively. The angular rates ,where , , . The angular velocity is defined as

. The kinematics is given by , where is defined as

(1)

Note that this mapping has singularities at . For thefollowing discussions, we assume that the pitch angle of thehelicopter does not reach these singularities. The other variablesare the main rotor lateral and longitudinal flapping angles and.

Fig. 2 shows a schematic of a helicopter, which includes therotary wing dynamics, force and moment generation, and rigid-body dynamics. In Fig. 2, the longitudinal and lateral cyclicpitches, and , control the longitudinal and lateral direc-tion of the tilt of the main rotor, while the collective and cyclicpedal, and , control the angle of the main rotor bladesand the tail rotor blades, respectively. However, in this paper,due to the fact that the complete dynamics of a helicopter isquite complex and somewhat unmanageable for the purpose ofcontrol, we consider a helicopter model as a rigid body, and thecontrol inputs to the helicopter are assumed to be the main rotorlateral and longitudinal flapping angles and , the main rotorthrust , and the tail rotor thrust .

A. Rigid-Body Dynamics

The equations of motion for a rigid body subject to an externalwrench applied at the COM can be described byNewton–Euler equations. For the helicopter fuselage, these canbe written as

(2)

where is the inertia matrix of the helicopter around its COM.The superscipt represents a vector in the body frame.

B. Force and Moment Generation

The force experienced by the helicopter is a sum of forcesgenerated by the main and tail rotors, aerodynamic forces from


the fuselage, and gravitational force. The torque is composedof the torques generated by the main rotor, tail rotor, and thefuselage. In hover or forward flight with slow velocity, we canignore the drag contributed from the fuselage. So, resultant force

and moment can be written as [1]

(3)

(4)

where , , , are functions to map the con-trol inputs to the forces and moment. In addition, subscript , ,and stand for terms related to the cable tension, gravitationalterm, and anti-torque, respectively, as follows:

(5)

(6)

Note that is the unit vector normal to the main rotor blades’tip path plane (TPP)

(7)

(8)

In the above, reactive anti-torques generated at the two hubsdue to aerodynamic drag are denoted by and at themain rotor and the tail rotor, respectively. As in [14], these aremodeled as for , where ’s and

’s are experimental parameters. and are the geometricparameter matrices.

The cable force and moment are given by

(9)

where is a position vector from the COM of the helicopterto the cable attachment point, expressed in the local frame. isa unit vector of the cable, shown in Fig. 3. The additional pa-rameters are listed in Table I. Note that the inputs do not ap-pear affinely. We note that the system is underactuated by two.As shown in (2), the orientation couples into the translation dy-namics through the rotation matrix and changes the directionof the force , while the translation is linked to the orientationdynamics by the tether term , where containsthe translation of the helicopter.

Remark 1: The tether dynamics should be considered be-cause it may bring perturbation to the helicopter in rough seaconditions. However, we assumed that the mass of the tether isextremely small relative to the helicopter’s weight, so that it canbe neglected. In addition, from the control design point of view,

Fig. 3. Geometric modeling of the helicopter: the helicopter frameH; the shipframe S ; and the inertial frame N .

TABLE ISYSTEM PARAMETERS IN METER–KILOGRAM–SECOND (MKS) UNITS

the distributed model of the tether makes the overall system socomplicated that it makes designing the control law almost im-possible. The tether may perturb the helicopter in rough sea con-ditions. However, we have neglected the mass of the tether, sinceit is small in comparison with the mass of the helicopter. In addi-tion, the analysis does not assume the tether as an elastic storageelement (or a spring). It is modeled as an element which appliestension to the helicopter without any dynamics. Also, from thecontrol point of view, the distributed model of the tether makesthe system complicated. We have addressed cases of such prob-lems in the context of a satellite-deployed tether system [28].

III. POSITION CONTROL

The automation problem of a helicopter landing on a movingship is divided into two control problems, position control andattitude–altitude control. The position mode is active when thehelicopter is far from the destination, while the attitude–altitudemode is switched on when the helicopter is close to the deck ofthe ship.

Remark 2: The switching between two controllers dependson the mission objectives, or the ship’s behavior. Hence, this canbe considered as a behavior-based approach, often used in thehelicopter control [29]. For example, if the ship is stationary, thesecond control mode is not required. On the other hand, in case


Fig. 4. Time-scale-separation-based outer-loop and inner-loop autopilot structure.

the ship is vigorously rolling, pitching, and yawing, we need toengineer the helicopter’s attitude to follow the ship’s deck whenthe helicopter is hovering near the deck. Also, the stability is nota concern, because the individual controllers are designed to bestable for the tracking control.

We now proceed to design a two-level controller for thissystem. Unlike other existing approaches, we do not ignorethe small body forces which arise due to the coupling be-tween translation and rotation of the system. In this paper,the separation is made between the position dynamics and theattitude dynamics. The idea is to control the position ata slow rate, and the attitude at a faster rate. The slowerhigh-level controller (HLC) runs at a sampling time period

, an order of magnitude slower than the lower level fastercontroller (LLC) . Therefore, HLC assumes that the refer-ence signals generated by it at any sampletime has been attained, due to the faster action of the LLC. Inother words, at the slow time scale, the fast states have alreadyreached their steady-state values. The controls for the slowdynamics are the reference states of the fast variables . Theother control for the slow dynamics is the main rotor thruster

, since it is the source for vertical lift and horizontal force.Hence, the position dynamics is controlled through the bodyattitude and the main thruster . Specifically, the helicopterwill be controlled by using to adjust the magnitude of thethrust vector, and by using the body attitude to orient the thrustvector in the desired direction.

Fig. 4 shows a block diagram of the proposed controller. Theslow dynamics is designed to track the reference signal of theposition coordinates through attitude com-mands and the main thruster . The fast dynamics track theattitude commands of the slow system using the longitudinal,lateral flapping angles, and the tail rotor thrust, . Notethat the fast system is constructed as an inner controller loop,while the slow dynamics constitutes the outer loop. A summaryof the steps of the control law is given in the next few paragraphsand is elaborated on in [25] and [26].

A. Design of LLC

This controller is designed to track a reference attitudegenerated by the HLC. In addition, the HLC spec-

ifies a nominal value for the main rotor thrust .

Proposition 1: Given which stays constant in the fast timescale and , the attitude of the helicopter will converge to ,if at every sample time , , and are computed so that

(10)

Remark 3: The LLC is in the form of three nonlinear alge-braic equations, since the helicopter model is not affine in theinputs. To numerically solve these for inputs , , and , aGauss–Newton or Levenberg–Marquardt method is used. Trimcondition is used to compute the initial guess for , , and .

Proof: Consider a Lyapunov function as follows:

(11)

(12)

where . On substitution of (2) and , wearrive at

(13)

Substituting the LLC control law (10), we get

(14)

Now we use LaSalle’s invariance theorem to characterize theset to which the system converges. At equilibrium, there existsone invariant manifold , and therefore, in theinvariant manifold . Substituting these into (10), we get

.

B. Design of HLC

At this level, the attitude vector is assumed to have con-verged to asymptotically. At each sample time of HLC, it isassumed that LLC has already converged to the last command

, and , , and take after desired , , and .If we compute and to satisfy the following equation:

(15)converges to .


C. Robustness of the Controller

The proposed controller is based on an assumption that theresponse of the inner controller is fast enough to converge tothe desired attitude command within the sampling period .Hence, we examine the robustness of the overall control systemby investigating the case where the inner controller has trackingerror between and due to various reasons. When trackingerror exists, (15) is transformed into

(16)

Note that , , , and have been replaced by nonsteady-state values , , , and in the above equation. In prac-tice, body forces which couple torque inputs , , to trans-lational dynamics are much smaller than those generated bythe main rotor thruster . Hence, these small body forceshave been neglected to get an approximate model [14], [15],[21]. Under this assumption, we know that

. In addition, the Taylor expansion offor the small perturbation is given by

(17)

Substituting two approximationsand into (16) gives

(18)

which can be expressed in terms of

(19)

By simple manipulation, it can be expressed in a state-spaceform

(20)

where , .The perturbation term satisfies

(21)

From (15), the bound on can be estimated as

(22)

Similarly, the upper bound on can be obtained

(23)

For the nominal system with Hurwitz matrix and , thereexists a Lyapunov function

(24)

The time differentiation of along the trajectory of the per-turbed system is given as

(25)

where .In summary, if the outer-loop control gains satisfy

the following conditions, the proposed control scheme becomesrobust to small perturbation due to the inner controller’s trackingerror:

(26)

D. Simulation

We apply our control method on a model of the system. Theinertial, geometric, and aerodynamic parameters are listed inTable I. The initial conditions are , ,

, , , . The other states are intrim conditions. The cable tension is selected as 10% of the he-licopter weight. The controller designed to hover the helicopterback to the destination from a relativelylarge initial position. As shown in Fig. 5, the error converges tozero.

For the fast convergence of the inner-loop attitude vari-ables, LLC requires high gains. These were selected as

and . Thesecond plot in Fig. 5 shows that the attitude variables convergeto the corresponding reference signal within the slow timescale, or 0.2 s. Fig. 6 shows the input histories, which are withinreasonable bound throughout the simulation.

Remark 4: The simulation does show promise of implemen-tation in real time. The numerical solution of nonlinear equa-tions within the controllers takes about 0.06 s using MATLABon a 2.4-GHz computer. Assuming that the flapping angles’ dy-namics has a time constant of about 0.2 s, the LLC should beable to compute control inputs fast enough in a real scenario.

Remark 5: The time constant of the first-order flapping an-gles’ dynamics is assumed to be approximately 0.2 s. Due to thisfast response, we think that the time response of flapping anglesshown in Fig. 6 is realistic.

Remark 6: Although the simulation considers a special casewhen the ship is stationary, the controller of (15) can be usedin tracking a time-varying target’s motion without altering theform of the controller.


Fig. 5. Cartesian states and Euler angles time trajectory. In the first figure, theouter-loop controller is designed to move the helicopter from the initial posi-tion (x; y; z) = (2;�2;10) to the destination (x; y; z) = (0;0; 1). In thesecond figure, the attitude command is computed by the outer-loop controllerevery 0.2 s, to which the attitude of the helicopter converges. This is achievedby using high control gain K .

Fig. 6. Control inputs. These values are computed by solving a set of algebraicequations, (10) and (15), at each sampling time.

IV. ATTITUDE–ALTITUDE CONTROL

A helicopter with tether is subjected to tether moments as afunction of helicopter position. This creates a coupling betweenposition and orientation variables which is normally absent ina helicopter free of a tether. In this section, position DOFs areused as controlled variables, leaving the freedom to choose theorientation variables arbitrarily. As a result of this selection ofcontrolled and independent set, the position variables constitutethe variables with the fast dynamics, and the orientation variablethe slow dynamics. This is different from what is normally donewith helicopters, where orientation is controlled to achieve achange in position, as shown in the previous section.

The slow dynamics variables are selected to be the altitudeand attitude , while constitute the fastdynamic variables. This grouping is chosen to keep the heli-copter safe from collision with the ship, while the slow dy-namic variables track the corresponding ship motion ,

. We assume that all the states are measurable ac-curately.

A. Design of LLC

This controller is designed for fast variables, e.g., , totrack their reference signals . The three control inputs for

. Since there are three inputs, we add tothe control variables of LLC, and the structure of LLC is se-lected as

(27)which makes the system linear and converge to

(28)

Note that are generated by the HLC, while is the ob-served motion of the ship along the axis.

B. Design of HLC

Due to the dynamics of fast variables achieved by the LLC,the HLC at its time scale expects the following conditions to besatisfied:

(29)

Proposition 2: Given , and , the attitude of the heli-copter will converge to , if at every sample time, , and

are computed such that

(30)

where are the gain matrix and are selected to be diag-onal and positive definite.

Proof: Formulate a Lyapunov function as follows:

(31)

(32)

where , , and . Onsubstitution of (2) and , we arrive at

(33)


Substituting the HLC control law of (30), we get

(34)

Now, we use LaSalle’s invariance theorem to characterize theset to which the system will converge. At equilibrium, there ex-ists one invariant manifold , or .Therefore, in the invariant manifold, . In summary, weget the following condition:

or(35)

Substituting these in (30), we get . As a result,as long as is nonsingular.

C. Motion Range Over the Ship’s Deck

Although (30) has three equations and three control inputs,there may not be any feasible solution to satisfy (30),since the pseudoinputs appear weakly through the unitcable vector of the tether moment , as follows:

RHS of Eq. (30) (36)

Hence, we choose the parameter as an additional controlinput. is defined as

(37)

This changes the form of algebraic equations in (30) slightly, asfollows:

RHS of Eq. (30) (38)

where is constant and a positive design parameter.Remark 7: Equation (37) can be interpreted with the tensionhaving a spring-like effect on the helicopter. In other words,

the further the helicopter is from the destination, the more ten-sion is supplied. The level of the cable tension is determined by

. This approach also ensures positive tension during maneu-vers, and increases the feasibility of the solution of the algebraicequation, since it replaces with .

Remark 8: We can identify the range , the excursion ofand during landing. The is important, since the area of theship’s deck is limited. We investigate the relation betweenand the tension coefficient . From (38), since multiplies

, we can reduce by increasing . This is equivalentto saying that the more cable tension, the less is .

D. Simulation

Fig. 7 shows the landing motion of the helicopterwhen the ship motion is assumed to be stationary, or

Fig. 7. Cartesian states and Euler angles time trajectory. The first figure showsthe translation motion of the helicopter under the attitude–altitude control mode,where x; y are used as controls of the outer-loop controller to manipulate theattitude of the helicopter. In the second plot, we observe that the orientationmotions of the helicopter are perfectly controlled to follow its command.

Fig. 8. Control inputs. The flapping angles a; b hit 40� and 30�, respectively.This implies that the attitude–altitude controller requires excessive control ef-fort, compared with the position controller.

. As shown in the figure, andrest at the corresponding ship motion . The motion ofthe helicopter on the surface of the ship deck (see in thefirst plot) varies over and , respectively,where the cable tension parameter is set to 200. HLC runsevery 0.2 s, while LLC has a sample time of 1 ms. Hence,the reference signal of the helicopter in the first two plotsare updated every 0.2 s. LLC is designed to guarantee theconvergence of and within this time period.In addition, Fig. 8 shows the input histories applied to HLC,which are kept in reasonable bound throughout the simulation.

In order to observe the effect of the tether’s tension on thelanding region, we have plotted trajectories of the heli-copter according to (see Fig. 9). For

, and . On the other hand, for, and . Hence, by


Fig. 9. Simulation plots for a x � y maneuver of the helicopter. C = 10

(circle), C = 100 (square), C = 1000 (triangle). As the cable tension in-creases from C = 10 to C = 100, the area over which the helicopter movesbecomes small. This implies that we can impose restrictions on the motion ofthe helicopter within the boundary of the ship’s deck by manipulating the stiff-ness of the cable without hurting the stability of the helicopter.

applying high cable tension, we can reduce the tracking error ofmotion relative to the ship motion .

V. CONCLUSIONS AND FUTURE WORK

This paper presented an autopilot for a tethered helicopter forautonomous landing on a moving ship platform. The tether wasintroduced for safety and ease of landing. Two control modes,position mode and attitude–altitude mode, were developed inthis paper for achieving the task. For the position mode, the po-sition variables were controlled as slow variables, and the atti-tude angles were the fast variables. On the other hand, in theattitude–altitude mode, the outer loop converts commanded at-titude and altitude to position command for the innerloop. It was shown that the magnitude of the cable tension sig-nificantly alters the landing range, which is inversely propor-tional to the magnitude of the cable tension. For both controlmodes, we showed that the time-scale separation approach re-sults in asymptotic stability of the outer loop’s reference signals,while bounding the inner loop’s states. Through simulations, theproposed approaches were demonstrated to be effective for theproblem. Based on this analytical study, our future work is to im-plement autonomous helicopter landing on a simulated movingdeck.

APPENDIX

SYSTEM DESCRIPTION

A helicopter platform shown in Fig. 10 is based on a Hi-robo Eagle radio-controlled model helicopter. The Eagle he-licopter has a mass of 7 kg when instrumented with an au-tonomy payload. The autonomy payload of about 1.5 kg consistsof an inertial measurement unit (IMU), autopilot flight com-puter, cable angle sensor, and a Bluetooth radio modem. A cableangle sensor was implemented to measure the tether angle andwas mounted on the helicopter. This sensor was based on an

Fig. 10. Complete construction of embedded hardware.

off-the-shelf industrial joystick with the centering springs re-moved to assist alignment of the joystick with the cable. In oursystem, we intend to fuse sensor information from the cableangle sensor and accelerometers to estimate the horizontal ve-locity and position of the helicopter. We assume that the lengthof tether cable can be measured in real time.

In our system, a servomotor is used to maintain constant ten-sion in a cable between a hovering helicopter and the simulateddeck of a ship. The servomotor is located under the deck plat-form and has a pulley fitted, to which the cable is connected.As the helicopter attempts to keep station over the deck landingpoint in the presence of disturbances, there will be some up anddown spooling of the pulley system. If a landing is desired, thecable will be reeled in, also under constant tension.

A hardware switch on the helicopter selects the source ofservo control inputs. The switch selects between manual controlinputs from a radio control receiver or automatic control inputsgenerated by the autopilot. The radio control receiver decodescontrol inputs transmitted from the safety pilot’s radio controltransmitter. This allows the helicopter to be manually flown intothe correct position to begin an experiment if desired. A toggleswitch on the pilot’s radio control transmitter can be used toswitch between manual and autopilot modes.

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[22] S. Sastry, “Nonlinear systems: Analysis, stability, and control,” inIAM. New York: Springer-Verlag, 1999, vol. 10.

[23] W. Johnson, Helicopter Theory. Princeton, NJ: Princeton Univ.Press, 1980.

[24] R. W. Prouty, Helicopter Performance, Stability, and Control. Mel-bourne, FL: Krieger, 1980.

[25] K. Pathak and S. K. Agrawal, “An integrated spatial path-planningand controller design approach for a hover-mode helicopter model,”in Proc. IEEE Int. Conf. Robot. Autom., Barcelona, Spain, 2005, pp.1902–1907.

[26] S.-R. Oh, K. Pathak, S. K. Agrawal, H. R. Pota, and G. Matt, “Au-tonomous helicopter landing on a moving platform using a tether,”in Proc. IEEE Int. Conf. Robot. Autom., Barcelona, Spain, 2005, pp.3960–3965.

[27] A. Isidori, Nonlinear Control Systems: An Introduction. Berlin, Ger-many: Springer-Verlag, 1995.

[28] K. K. Mankala and S. K. Agrawal, “Equilibrium-to-equilibrium ma-neuvers of flexible electrodynamic tethers in equatorial orbits,” in Proc.AIAA Guid., Navig., Control Conf. Exhibit., 2004, pp. 2004–5342.

[29] M. J. Mataric, “Behavior-based control: Examples from navigation,learning and group behavior,” J. Exp. Theoretical Artif. Intell., vol. 9,no. 2–3, pp. 67–83, 1997.

So-Ryeok Oh received the B.S. and M.S. degreesin mechanical engineering from Pusan National Uni-versity, Pusan, South Korea in 1997 and 1999. Heis currently working toward the Ph.D. degree in me-chanical engineering at the University of Delaware,Newark.

His research interests include control of cablerobots, autonomous helicopters, and passive controlof rehabilitation device.

Kaustubh Pathak received the Bachelors degree inmechanical engineering from the Indian Institute ofTechnology, Kanpur, India, in 1996. In 2001, he re-ceived the Masters degree in mechatronics from theAsian Institute of Technology, Bankok, Thailand, andthe Ph.D. degree in mechanical engineering from theUniversity of Delaware, Newark, in Fall 2005.

Sunil K. Agrawal received the Ph.D. degree in me-chanical engineering from Stanford University, Stan-ford, CA, in 1990.

Currently, he is a Professor of Mechanical En-gineering at the University of Delaware, Newark.His research has made contributions in roboticsand control, including novel design of robots andautonomous systems, computational algorithms forplanning and optimization of dynamic systems, anddevices for medical rehabilitation. He has workedwith universities, government laboratories, and

industries throughout the world. His work has yielded over 225 technicalpublications and two books.

Dr. Agrawal has received numerous awards, including the NSF PresidentialFaculty Fellowship from the White House and a Freidrich Wilheim Bessel Prizefrom the Alexander von Humboldt Foundation in Germany.

Hemanshu Roy Pota received the B.E. degree fromSardar Vallabhbhai Regional College of Engineeringand Technology, Surat, India, in 1979, the M.E. de-gree from the Indian Institute of Science, Bangalore,India, in 1981, and the Ph.D. degree from the Univer-sity of Newcastle, Newcastle, Australia, in 1985, allin electrical engineering.

He is currently an Associate Professor with theUniversity of New South Wales at the AustralianDefence Force Academy, Canberra, Australia. Hehas held visiting appointments at the University

of Delaware, Iowa State University, Kansas State University, Old DominionUniversity, the University of California, San Diego, and the Centre for AI andRobotics, Bangalore, India. He has a continuing interest in the area of powersystem dynamics control and modeling control of mechanical systems such asflexible structures, acoustical systems, and UAVs.

Matt Garratt spent ten years’ service in the RoyalAustralian Navy, working as a dual specialist inalternate aeronautical and marine engineering post-ings. On leaving the service, he spent two yearsworking as consultant for LEAP Australia in thearea of computer-aided engineering problems. Heleft the commercial world to work as the ControlSystems Engineer on a Defense Advanced ResearchProjects Agency (DARPA)-funded project to buildan autonomous helicopter using biologically inspiredvision, which led to successful visual control of a

helicopter in hover and forward flight in 2000. Since 2001, he has been withthe University of New South Wales, Canberra, Australia, as a Lecturer in theSchool of Aerospace, Civil, and Mechanical Engineering. His main researchareas are helicopter dynamics and sensing and control for autonomous systems.

Documents

536 IEEE TRANSACTIONS ON ROBOTICS, VOL. 22, NO. 3, JUNE ... Collection/TRO/2006/june/10.pdf · trol literature for 3-DOF vertical takeoff and landing (VTOL) [21], [22]. More details