Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 56, 3, pp. 793-802, Warsaw 2018 DOI: 10.15632/jtam-pl.56.3.793 MATHEMATICAL MODEL OF LEVITATING CART OF MAGNETIC UAV CATAPULT Anna Sibilska-Mroziewicz, Edyta Ładyżyńska-Kozdraś Warsaw University of Technology, Faculty of Mechatronics, Warsaw, Poland e-mail: a.sibilska@mchtr.pw.edu.pl; e.ladyzynska@mchtr.pw.edu.pl The article presents the steps ofmodeling of the dynamics of a levitating cart of an unman- ned aerial vehicle (UAV)magnetic catapult. The presented in the article innovative catapult is based on theMeissner effect occurring between high-temperature superconductors (HTS) and a magnetic field source. The catapult suspension system consists of two elements: fi- xed to the ground base with magnetic rails and a moving cart. Generating magnetic field rails are made of neodymium magnets. Levitation of the launcher cart is caused by sixteen superconductors YBCO, placed in the cart frame supports. Described in the article model contains the system of Cartesian reference frames, kinematic constrains, equations of mo- tion and description of forces acting on the cart as well as exemplary numerical simulation results. Keywords: magnetic catapult, equations of motion, Meissner effect 1. Introduction The growing demand for commercial unmanned aerial vehicles (UAV) requires exploration of innovative technical solutions associated with critical aspects of the use of such facilities. Safe procedures forUAVtake-off and landing are one of such issues.Most of unmannedaerial vehicles have neither sufficient power supplynor a structure for self-start, especially in the field of uneven ground and insufficient runway. Only a few of UAVs such as Predator have a chassis system allowing for self-start and landing. The chassis system, however, increasesmass of the vehicle, makes its constructionmore complicated and requires implementation of advanced algorithms of control during take-off and landing. Moreover, the take-off procedure itself requires significant reserves of power. Take-off of an unmanned aerial vehicle can also be done by throwing it from a human hand. However, this way is only possible in the case of small and lightweight UAVs such as Raven used by the US Army. In other cases it is necessary to use a separate device called a launcher or airplane catapult. Nowadays, it is common to use rocket systems (RATO-Rocket Assisted TakeOff), bungee cord, hydraulic and pneumatic launchers (Fahlstrom andGleason, 2012). An attractive alternative to current systems are magnetic catapults. Magnetic catapults compared with classical solutions enable safe, non-impact service of theUAV launch process and allow achievingmuch bigger final UAV speed. NASA plans using electrodynamics catapults to launch spacecraft (Polzin et al., 2013) and hypersonic planes. Magnetic catapults are also planned to replace steam catapults used on aircraft carriers (Bertoncelli et al., 2002). Nowadays, magnetic suspension systems are used in high-speed trains (Mag-Lev). Currently, two types ofMag-Lev technology are developed commercially (Liu et al., 2015). Electromagnetic suspension (EMS) developed by the German Transrapid system. The suspension is based on the strength attraction force between metal rails and mounted on the underside of the train 794 A. Sibilska-Mroziewicz, E. Ładyżyńska-Kozdraś electromagnets. Built in 2004 Shanghai Maglev train uses the EMS technology. The second solution is an electrodynamic suspension system (EDS) proposed by Central Japan Railway Company. The EDS system uses strong repulsive forces generated by strong superconducting electromagnets built into the train path and trains chassis. Built in Japan Chuo Shinkansen line, which is the fastest train in the world (Coates, 2007), is based on the EDS technology. An alternative to EMS and EDS systems are passive suspensions using diamagnets. This paper presents a mathematical model of a newUAV catapult prototype based on “out of the box” idea, usingHTS in the launcher suspension system.According to theMeissner effect, superconductors cooled down below the critical temperature shield the external magnetic field generated by permanent magnets or electromagnets. The Meissner effect ensures spectacular levitation of the superconductor above the source of the magnetic field (Fig. 1). Fig. 1. Meissner effect The initial part of the article presents a prototype of a magnetic UAV catapult using the Meissner effect. The following considerations include itemized assumptions regarding physical model and description of the system of Cartesian coordinate frames. Subsequent discussion contains equations of motion of the levitating cart and description of kinematic constrains and loads acting on the cart. The considerations are closed by results of numerical simulations. 2. Catapult prototype Figure 2 shows photographs of the prototype of a magnetic launcher designed within FP7, EU GABRIEL (Integrated Ground and on-board system for Support of the Aircraft Safe Take-off and Landing) (Rohacs, 2015), (Falkowski, 2016), (Falkowski, Sibilski, 2013). During the take-off procedure, the UAV is attached to the levitating cart which is a movable part of the launcher. The cart is driven by a linearmotor. The construction of the cart consists of a rigid framemade of duralumin and four containers. In each container, there are four high-temperature supercon- ductors YBCO with a critical temperature of 92K. These superconductors have a cylindrical shape with diameter of 21mm and height of 8mm. After filling the containers with liquid ni- trogen, superconductors transit into superconducting state and start hovering over the launcher tracks due to the Meissner phenomena. Levitation phenomenon ends when the YBCO tempe- rature exceeds the critical temperature of 92K. Therefore, the material of the container should provide themaximumthermal isolation. The starting path,mounted to the launcher base, consi- sts of two parallel rails, eachmade of three rows of permanentmagnets. To build the prototype, rectangular neodymiummagnets polarized top-down have been used. Potential use of passive suspensions, based on HTS in transportation systems, would bring many benefits. The levitation phenomenon enables frictionless longitudinal movement of the cart. The levitation gap remains stable without any feedback loop, and low complexity of the system would improve it is reliability. The biggest defect of the solution is the need of cooling Mathematical model of levitating cart of magnetic UAV catapult 795 downandmaintaining superconductors in low temperatures.Systemswhichusepassivemagnetic suspension, despite growing popularity, have not been used so far in professional and commercial technical solutions. However, there is a small number of academic research projects on the use of HTS in transportation systems, includingSupraTrans (Schultz et al., 2005), CobraTram (Sotelo et al., 2011) and SuperMaglev (Wang et al., 2005). Fig. 2. Prototype of a magnetic UAV catapult using theMeissner effect 3. Physical model Anextremely important step in themodelingprocedure(Ładyżyńska-Kozdraś, 2011) isdefinition of the physical model being the basis of formulation of the mathematical model. The proposed physical model of the levitating cart assumes the following simplifications: • the levitating cart is an axisymmetric solid body with six degrees of freedom; • mass and center ofmass of the cart do not change duringmovement, however, the position of the taking off airplane may change; • the systemmotion is considered only in a no wind environment; • motion of the cart is controlled in only one direction by a linear motor; • the cart is levitating above magnetic rails due to theMeissner effect; • cart movement results from gravitational, magnetic and propulsion forces acting on the cart itself and gravitational, aerodynamical and propulsion forces acting on the taking off airplane; • mass of evaporating nitrogen is not considered. 4. Reference frames In order to describe the dynamics of the levitating cart, the following clockwise Cartesian coor- dinates frames are attached to the following parts of the UAVmagnetic launcher: • Themotionless base system Oxfyfzf is a rectangular Cartesian coordinate system rigidly connected with the ground. The Ozf axis is directed vertically downward, in the direction of gravitational acceleration; the Oxf axis coincides with the horizontal projection of the 796 A. Sibilska-Mroziewicz, E. Ładyżyńska-Kozdraś aircraft taking-off path; the Oyf axis completes the right-handed coordinate system. The base frame origin coincides with the starting point of the catapult cart. The position of the levitating cart, along the Oxf frame, could bemeasured by a laser sensor. • Themagnetic reference frame Oxmymzm is a coordinate system in which magnetic inter- actions and cart propulsion forces are modeled. The Oxm axis covers the catapult axis of symmetry; the Ozm axis points down perpendicularly to the catapult base; the Oym axis connects the left and right catapult rails. The system origin is moving along the catapult axis of symmetry Oxm and covers the projection of the levitating cart center of mass into the Oxm axis. Orientation of the magnetic reference frame, due to the fixed system Oxfyfzf, is described by two angles θm/f and φm/f. The angle θm/f corresponds to deli- berate inclination of the catapult base, relative to the horizontal plane. A non zero value of the φm/f angle is caused by imbalances of the substrate onwhich the launcher is placed. • The Oxsyszs coordinate system describes the position and orientation of the levitating cart. The origin of the cart coordinate system is fixedwith the cart center ofmass, and its axes are rigidly connected with the cart frame. The Oxs axis points along the longitudinal cart dimension; the Ozs axis is perpendicular to the cart surface andpoints downward; the Oys axis is parallel to the lateral cart dimensionand completes the right-handed coordinate system. Orientation of the launcher cart with respect to the magnetic coordinate system is described by three qusi-Euler angles θs/m, ψs/m and φs/m. Fig. 3. Coordinate systems fixed with the magnetic launcher: inertial Oxfyfzf andmagnetic Oxmymzm Fig. 4. The coordinate system fixed with the levitating cart Oxsyszs Mathematical model of levitating cart of magnetic UAV catapult 797 • The axes of the gravitational reference frame Oxgygzg are parallel to the inertial frame Oxfyfzf. The system origin is fixed with the cart center of mass. In that system, the gravitational forces and torques are described. • The reaction forces and torques acting between the cart and taking-off airplane are descri- bed in the Oxcyczc coordinate system. The system origin and its orientation is dictated by the way of attaching the UAV into the cart frame. 5. Kinematic constrains Motion of the cart is describedby timeand space coordinates located in the event space.The cart position, at theparticularmoment, is unambiguouslydescribedby linear andangular coordinates and velocities. Those coordinates change with time and are coupled by kinematic constrains. It is important tomaintain amutual coordinate systemwhile describing the function of particular coordinates. Delivered below equations describing cart kinematic constrains are described in the Oxsyszs system fixed with the cart center of mass. The vector of the current cart position in the fixed to the ground inertial frame Oxfyfzf is described by r= xfif +yfjf +zfkf (5.1) The vector of instantaneous linear velocity V described in the frame fixed with the cart frame Oxsyszs has three components: longitudinal U, lateral V and climb speed W V= Uis+V js+Wks (5.2) The vector of instantaneous angular velocity Ω described in the fixed to the cart frame system Oxsyszs has the following components: roll rate P, pitch rate Q and yaw rate R Ω= Pis+Qjs+Rks (5.3) Kinematic constraints between linear velocity components measured relative to the inertial co- ordinate system Oxfyfzf, and linear velocities U, V , W measured relative to the coordinate system Oxsyszs, fixed to the cart, have the following form    U V W)    = Rs/mRm/f    ẋf ẏf żf    (5.4) The rotation matrices Rs/m and Rm/f are described by equations (5.5) with notations: cα =cosα and sα =sinα Rs/m =    cθs/mcψs/m cθs/msψs/m −sθs/m sφs/msθs/mcψs/m− cφs/msψs/m sφs/msθs/msψs/m+cφs/mcψs/m sφs/mcθs/m cφs/msθs/mcψs/m+sφs/msψs/m cφs/msθs/msψs/m−sφs/mcψs/m cφs/mcθs/m    (5.5) Rm/f =    cθs/m 0 −sθs/m sφs/msθs/m cφs/m sφs/mcθs/m cφs/msθs/m −sφs/m cφs/mcθs/m    The components of instantaneous angular velocities P, Q, R are combinations of generalized velocities θ̇s/m, ψ̇s/m, φ̇s/m and trigonometric functions of angles: θs/m,ψs/m andφs/m, according to the relation    P Q R    =    1 0 −sinθs/m 0 cosφs/m sinφs/mcosθs/m 0 −sinφs/m cosφs/mcosθs/m       φ̇s/m θ̇s/m ψ̇s/m    (5.6) 798 A. Sibilska-Mroziewicz, E. Ładyżyńska-Kozdraś 6. Equations of motion To derive equations ofmotion for objects treated as rigid bodies, mostly Newtonian approach is used, i.e. forces,momentumandangularmomentumconservation laws.Examples could be found in (Baranowski, 2016). Sometimes, more involved theoretical mechanics is used, e.g. Lagrangian formulation, see (Koruba et al., 2010), Boltzmann-Hamel equations like in (Ładyżyńska-Kozdraś and Koruba, 2012) orMaggi equations (Ładyżyńska-Kozdraś, 2012). In the presented consideration, the levitating cart is treated as a rigid bodywith six degrees of freedom. The proposed equations of motions are described in the coordinate frame fixed to the cart center of mass Oxsyszs. The presented mathematical model is developed according to principal mechanical laws: the momentum and angular momentum conservation principles. —The derivative of the momentumΠwith respect to time ∂Π ∂t +Ω×Π=F (6.1) —The derivative of the angular momentumKwith respect to time ∂K ∂t +Ω×K+V×Π=M (6.2) The principle of conservation of the momentum and angular momentum can be applied to the problem in two ways. Firstly, the levitating cart and taking-off airplane are treated as a single undivided rigid body. The momentum and angular momentum of that body is defined for the entire object, relative to the one arbitrarily chosen polewhich does not necessarily coincidewith the center of mass. The second way is to designate moments and angular moments separately for the taking off plane and the levitating cart. The presented equations consider bothmovable parts of the catapult as a one rigid body. Thegeneral formof equations ofmotionof the cart in the threedimensional space is expressed by the relationship (Ładyżyńska-Kozdraś, 2011) MV̇+KMV=Q (6.3) where: —matrix of inertia M=          m 0 0 0 Sz −Sy 0 m 0 −Sz 0 Sz 0 0 m Sy −Sx 0 0 −Sz Sy Ix −Ixy −Ixz Sz 0 −Sx −Iyx Iy −Iyz −Sy Sx 0 −Izx −Izy Iz          (6.4) —matrix of kinematic constrains K=          0 −R Q 0 0 0 R 0 −P 0 0 0 −Q P 0 0 0 0 0 −W V 0 −R Q W 0 −U R 0 −P −V U 0 −Q P 0          (6.5) — vector of linear and angular accelerations V̇= [U̇, V̇ ,Ẇ,Ṗ,Q̇,Ṙ]T (6.6) Mathematical model of levitating cart of magnetic UAV catapult 799 —vector of linear and angular velocities V= [U,V,W,P,Q,R]T (6.7) — vector of external forces and torques Q= [ F M ] (6.8) After determiningkinematic constrains aswell as forcesFand torquesMacting on the levitating cart, a general model of the system dynamics is obtained. 7. Forces and torques acting on levitating cart Forces and torques acting on the considered systemresults from interactions between the taking- -off UAV, levitating cart and the generated by the catapult rails magnetic field. The vector of external forces and torques Q can be described by superposition of the vectors of forces and torques: acting only on the cart frame QS, acting on the UAV alone QB, and describing the method of attachment of the taking-off UAV into the cart frame Q=QS +QB +QSB (7.1) The cart frame undergoes the gravitational pull QSg , propulsion forces Q S T , aerodynamic inte- ractions QSA and the load resulting from the Meissner effect, called levitation forces Q S Li . The levitation forces depend on the position of the box with superconductors relative to the rails generatingmagnetic field.This distance is called the levitation gap.Taking into account changes in the orientation of the cart, the levitation forces QSLi are determined separately for each of four containers with superconductors Q S =QSg +Q S T ++Q S A 4 ∑ i=1 Q S Li (7.2) The taking-off UAV moves under the influence of the gravitational pull QBg , propulsion for- cesQBT , aerodynamic interactions Q B A and the load resulting from the UAV control systemQ B δ Q B =QBg +Q B T +Q B A +Q B δ (7.3) The loads QSg and Q B g are described in the gravitational system Oxgygzg. The linear drive propulsion forcesQST and levitation forcesQ S Li are considered in relation to themagnetic system Oxmymzm.Therefore, it is necessary to transformthe individualvectors to the coordinate system attached to the levitating cart, bymultiplying themby the corresponding transformationmatrix. The value of the nonlinear levitation force depends on the gap between the cart support and magnetic rails. The smaller the gap, the greater the force. The levitation force is modeled as a concentrated force acting on the center of the box with superconductors. 8. Preliminary numerical simulation The use of themomentum and angularmomentum laws of conservation formechanical systems makes it possible to develop the dynamical model of the take-off of an unmanned aircraft. 800 A. Sibilska-Mroziewicz, E. Ładyżyńska-Kozdraś During the take-off procedure, theUAV is attached to the levitating cart, which is amovable part of the launcher. In the analyzis, the UAV class micro Bell 540 has been taken into acco- unt. In the research, it is assumed that the linear-driven cart moves with constant horizontal acceleration. The presented in the article mathematical model of the levitating cart of themagnetic UAV catapult is the theoretical basis for preliminary numerical simulations. The obtained numerical results show correctness of the developed mathematical model. As shown in Figs. 5 and 6, the unmanned aircraft take-off takes place in a proper manner. It maintains the preset parameters resulting from the adopted guidance parameters of the levitating cart of the magnetic UAV catapult. Fig. 5. The course of changes in height of the levitating cart and UAV at the moment of take-off Fig. 6. Velocity of the levitating cart and UAV at the moment of take-off 9. Conclusions The paper is addressing an interesting subject related to dynamics andmodellingmethodology of the levitating cart of amagnetic UAV catapult. The presentedmathematical model has been developed according to principal mechanical laws – the momentum and angular momentum of conservation. The considerations are closed by exemplary numerical results. Mathematical model of levitating cart of magnetic UAV catapult 801 Described work is a prelude to wider research on the mechanical properties of the launcher levitating cart, beingunder the influenceof forces, generatedby levitation ofHTS in themagnetic field. The presented in this article model does not take into consideration the non-uniformness of magnetic field and the flux pinning effect occurring in II type superconductors. Description of levitation forces and torques, based on both theoretical investigations and laboratory tests, will be the next step of research as well as development of a more general mathematical model including UAV dynamics and non-uniformness of the magnetic field distribution. References 1. Baranowski L., 2016, Explicit “ballistic M-model”: a refinement of the implicit “modified point mass trajectorymodel”,Bulletin of the Polish Academy of Sciences Technical Sciences, 64, 1 2. 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