Flight Dynamics and Stability of a Tethered Inflatable Kiteplane E. J. Terink,��� J. Breukels,��� R. Schmehl,��� and W. J. Ockels�� Delft University of Technology, 2629 HS Delft, The Netherlands DOI: 10.2514/1.C031108 The combination of lightweight flexible-membrane design and favorable control characteristics renders tethered inflatable airplanes an attractive option for high-altitude wind power systems. This paper presents an analysis of the flight dynamics and stability of such a kiteplane operated on a single-line tether with a two-line bridle. The equations of motion of the rigid-body model are derived by Lagrange���s equation, which implicitly accounts for the kinematic constraints due to the bridle. The tether and bridle are approximated by straight line elements. The aerodynamic force distribution is represented by four discrete force vectors according to the major structural elements of the kiteplane. A case study comprising analytical analysis and numerical simulation reveals that the amount and distribution of lateral aerodynamic surface area is decisive for flight dynamic stability for the specific kite design investigated. Depending on the combination of wing dihedral angle and vertical tail plane size, the pendulum motion shows either diverging oscillation, stable oscillation, converging oscillation, aperiodic convergence, or aperiodic divergence. It is concluded that dynamical stability requires a small vertical tail plane and a large dihedral angle to allow for sufficient sideslip and a strong sideslip response. Nomenclature A = aspect ratio CD = aerodynamic drag coefficient CL = aerodynamic lift coefficient Cmac = pitch moment coefficient c = mean aerodynamic chord, m D = aerodynamic drag, N d = tether damping constant, Ns=m Fa = aerodynamic force vector, N FB = bridle force, N FGS = ground-station force vector, N FZ = gravitational force, N g = gravitational acceleration vector, m=s2 I = inertia matrix, kgm2 k = tether spring constant, N=m L = aerodynamic lift, N l = length, m Ma = aerodynamic moment vector, Nm Mwac = wing pitching moment, Nm mg = kite mass excluding confined air, kg mk = kite mass, kg Q = generalized force, N q = generalized coordinate r = position vector, m S = surface area, m2 T = kinetic energy, J TBA = transformation matrix for frame A to B V = potential energy, J v = velocity, m=s v = velocity vector, m=s X = aerodynamic force in X direction, N X,Y,Z = Cartesian axis system x,y,z = Cartesian coordinates, m Y = aerodynamic force in Y direction, N y = spanwise location of mean aerodynamic chord, m = angle of attack, deg = sideslip angle, deg = dihedral angle, deg = bridle geometry angle, deg = pitch angle, deg T = pendulum mode angle, deg = sweep angle, deg = wing taper ratio = air density, kg=m3 = zenith angle, deg = bridle angle, deg = azimuth angle, deg ! = angular velocity vector, rad=s Subscripts app = apparent c.g. = center of gravity f = vertical tail fin HT = horizontal tail LE = leading edge lw = left wing rw = right wing T = tether t = tail VT = vertical tail W = wind w = wing Superscripts a = aerodynamic reference frame B = body reference frame E = Earth reference frame T = tether reference frame I. Introduction K ITES are among the earliest man-made flying objects in history and have been used for a wide variety of purposes [1]. Especially from 1860 to 1910, kites emerged as an important Received 20 May 2010 revision received 9 October 2010 accepted for publication 7 November 2010. Copyright �� 2010 by E. J. Terink, J. Breukels, R. Schmehl, and W. J. Ockels. Published by the American Institute of Aeronautics andAstronautics, Inc., withpermission. Copies ofthispaper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923 include the code 0021-8669/11 and $10.00 in correspondence with the CCC. ���Researcher, Institute for Applied Sustainable Science Engineering and Technology, Kluyverweg 1. ���Ph.D. Candidate, Applied Sustainable Science Engineering and Technology Institute, Kluyverweg 1. ���Associate Professor, Institute for Applied Sustainable Science Engineer- ing and Technology, Kluyverweg 1. ��Professor, Chair Holder, and Director, Institute for Applied Sustainable Science Engineering and Technology, Kluyverweg 1. JOURNAL OF AIRCRAFT Vol. 48, No. 2, March���April 2011 503

technology for scientific and technical applications such as in mete- orology, aeronautics, wireless communication, and aerial photog- raphy. Although the airplane has subsequently taken over these application areas, kites have made a comeback as major recreational devices. The increasing shift toward sustainable energy generation and propulsion has triggered a renewed interest in kites for industrial applications, and a major driver is the potential of the technology to efficiently exploit the abundant wind at higher altitudes [2,3]. Using kites for power generation was first proposed and systematically analyzed by Loyd in 1980 [4] however, subsequent research and development activities were rare before the presentation of the laddermill concept by Ockels in 1996 [5,6]. Since then, the number of institutions actively involved in kite power has increased rapidly, including several multimillion-dollar projects [7,8]. Various concepts and ideas have been proposed to exploit thewind currents at higher altitudes [8,9]. One of the concepts is the pumping- kite concept [10,11], where the tether, pulled by lifting bodies, drives a drum that is connected to a generator to produce electricity. By alternating between a high-power-producing upstroke and a low- power-consuming downstroke, net energy is generated. The main advantages of such a system over conventional wind turbines are the higher operational flexibility and the ability to exploit the stronger and steadier wind at higher altitudes. However, the high degree of freedom in the design and operation of kite power systems also leads to control challenges. Compared with an airplane, the flight dynamics of a kite are constrained by the tether and bridle system. However, this does not mean that kites are more stable and easier to control. Research indicates that the presence of a tether may raise stability issues [12���14]. A successful pumping-kite power system requires a kite that is not only agile and aerodynamically efficient to maximize the power output, but that is also stable, to minimize the control effort. In addition, a low-lift mode (in kite terminology, called depower) is necessary to implement a swift low-power-consuming downstroke. Some kite types are naturally stable on a single line (such as box kites, sled kites, delta kites, and some ram-air kites), but none meets the full set of requirements. The kiteplane depicted in Fig. 1, has been developed at the Institute for Applied Sustainable Science Engineering and Technology of Delft University of Technology to operate in a pumping-kite concept [15���17]. The airplane-shaped kite is constructed from inflatable beams and canopy surfaces. It features a bridled wing, efficient aerodynamics for a kite, and easy angle-of- attack control. However, flight tests have indicated that the prototype of 2009 suffers from a pendulum instability, which is an unstable oscillation in the crosswind plane. In an early scientific contribution, Bryant et al. [12] proposed equations of motion for kites and towed gliders. However, the basic empirical cable model becomes singular when approaching a straight line. Glauert [18] investigated the stability of towed bodies behind an aircraft and provided insight into tether interaction, and Jackson [19] analyzed kites in a large 24 ft wind tunnel. Sanchez [20] started from Lagrangian equations of motion to formulate a model for longi- tudinal dynamics of a kite and used this to develop a controller for the symmetric flight mode. Alexander and Stevenson [21] dealt with the existence of possible static equilibrium points, and Jackson [22] investigated the shape of the kite in relation to its optimal loading using a static analysis based on Prandtl���s lifting-line theory. This paper presents a rigid-body model of a single-line bridled kiteplane to investigate the influence of geometry on flight dynamics and stability. The bridle is used not only as a structural element to alleviate the wing bending moment, but it also constrains the rotational freedom of the kite. The roll and yaw motions of the kite are coupled, because the bridle acts as a revolute joint between the tether and the kiteplane. At a high elevation angle, the roll motion is constrained, and at a low elevation angle, the yaw motion is constrained. At intermediate angles, the motion is a combination of roll and yaw. The continuous aerodynamic load distribution is represented by discrete forces and moments acting on the aero- dynamically active structural elements of the kiteplane to make the model largely parametric in geometry. The kite system model is described in the Methodology section and applies to kites that are built out of planar aerodynamic elements and for which global geometry and inertia can be assumed constant. The developed model is subsequently employed to analyze the stability of the kiteplane in a case study. Both analytical and numerical results are obtained and compared qualitatively with each other and with a flight test. Stable and unstable domains are identified and tested on robustness by varying geometry and operational conditions in the simulations. II. Methodology This paper focuses on the influence of geometrical design parameters on the global dynamics of a single-line kite system. Several modeling methods are found in Williams et al. [23]. Optimal control problems for kite systems are often solved using point-mass models [24���26]. Kite performance is investigated using lumped- mass and massless models [27]. For stability, however, attitude dynamics are essential and a point-mass model cannot be used. Also, a flexible-body model would not provide pure geometry-stability relations. Given this scope and the philosophy that the best model is the smallest model that describes the behavior of interest, a rigid- body approach is selected to model the kite. Moreover, to investigate the impact of geometry changes on stability, a parametric approach is required for the aerodynamic forces. To this end, strip theory is used for the aerodynamic element discretization, similar to the approach used by Meijaard et al. [28]. Strip theory is especially popular in flapping-wing modeling [29,30]. The equations of motion are derived using Lagrange���s equation of the second kind [31], a method used frequently for the constrained kite systems [32���34]. A. Kite System Definition The kite system consists of a ground station, tether, bridle, and kite. The ground station is represented as a point acting as a forced sink and source of tether length. Tether and bridle are modeled as a single massless rigid body that is free to rotate about the longitudinal axis of the tether. In this idealized system, the tether is connected to the ground station by a spherical and a translational joint. The kite is represented as a rigid body and connected to the bridle by a revolute joint. The fivedegreesoffreedom (DOF) of this system are illustrated in Figs. 2 and 3. The rotation around the spherical joint is described by the azimuth angle , the zenith angle , and the bridle rotation angle . The translation along thetether longitudinal axis is described bythe tether length lT, and the rotation of the kite around the axis connecting the two bridle attachment points (the dashed line in Fig. 3) is described by the pitch rotation angle . The bridle constrains the freedom of the kite with respect to the tether and couples the roll and yaw motions, as explained in the Introduction. The orientation of the bridle with respect to the tether is assumed to be invariant, although, in theory, this specific geometry will allow for some rolling motion. This motion is neglected, because 1) such asymmetric flight conditions are not sustained and the inertia for this motion is low and 2) this motion has not been observed in flight tests. Fig. 1 Photograph of the kiteplane in flight. 504 TERINK ET AL.