Static and Dynamic Modeling and Simulation of the Umbilical Cable in A Tethered Unmanned Aerial System

Transcription

1 Statc and Dynamc Modelng and Smulaton of the Umblcal Cable n A Tethered Unmanned Aeral System by Sna Doroudgar B.Sc. (Mechancal Engneerng), Shraz Unversty, 2010 Thess Submtted n Partal Fulfllment of the Requrements for the Degree of Master of Appled Scence n the School of Mechatronc Systems Engneerng Faculty of Appled Scences Sna Doroudgar 2016 SIMON FRASER UNIVERSITY Summer 2016

2 Approval Name: Degree: Ttle: Sna Doroudgar Master of Appled Scence Statc and Dynamc Modelng and Smulaton of the Umblcal Cable n A Tethered Unmanned Aeral System Examnng Commttee: Char: Krshna Vjayaraghavan Assstant Professor Samak Arzanpour Senor Supervsor Assocate Professor Carolyn Sparrey Co-Supervsor Assstant Professor Kevn Oldknow Internal Examner Lecturer School of Mechatronc Systems Engneerng Date Defended/Approved: June 27, 2016

3 Abstract The research at hand has been accomplshed n collaboraton wth our ndustry partner, Rgd Robotcs Inc. and ams to predct and suggest solutons for some of the known and unknown ssues that mght appear n operaton of a tethered Unmanned Aeral System (to be referred to as UAS hereafter). In ths work, the statc and dynamc behavour of the power cable connectng a hoverng UAS to ts base staton s studed n dfferent flght scenaros. The mathematcal modellng of the cable s carred out usng catenary equatons for the statc case and mult-body dynamcs prncples are employed for the dynamc condton. In the prelmnary stages of the project, for the purpose of the cable and UAS desgn, a smple technque s used to estmate the maxmum tenson forces present n the statc state of the cable as well as the cable shape functon and other related parameters when UAS s n hoverng mode. The dervaton of the system s equatons of moton s done usng Lagrange s Equatons by consderng the cable as a dscrete multbody system. The equatons of moton are derved for a system of fnte segments and are solved numercally usng MATLAB software package n order to smulate the cable s moton. The effect of wnd on the dynamcs of the cable s also mplemented usng theoretcal methods and smulatons. The system s dynamcs s modeled n a planar moton as well as a 3D space n separate chapters. Keywords: Mathematcal Modellng; Cable Dynamcs; Tethered Systems; Mult-body Dynamcs; Lagrange s Equatons; Umblcal Cables

4 Dedcaton To my lovely wfe, Anahta. To my beloved parents, Javad and Smn. v

5 Acknowledgements I would lke to take ths opportunty to thank everyone who helped me drectly or otherwse through ths process. Frstly, I would lke to express my grattude towards my senor supervsor, Dr. Samak Arzanpour, who provded me wth ths exctng opportunty to pursue my graduate studes n ths program under hs supervson. He helped me follow through ths whole journey by hs gudance and advces. Hs comments and suggestons helped sgnfcantly towards composton of ths thess. Secondly, I would lke to thank the members of my supervsory commttee, Dr. Carolyn Sparrey, Dr. Kevn Oldknow, as well as the char of the sesson, Dr. Krshna Vjayaraghavan for ther valuable tme kndly put nto revewng ths work and makng useful comments n a relatvely short tme. In addton I would lke to thank Natural Scences and Engneerng Research Councl of Canada (NSERC) for ther scholarshp support n form of an Industral Postgraduate Scholarshp. Many thanks go to Mehran Motamed, presdent and techncal drector of Rgd Robotcs Inc. for ther fnancal and ntellectual support durng ths nternshp. I am grateful to my frends and colleagues at Rgd Robotcs, Thomas Huryn and Morteza Najafvand for ther professonal support durng two years of workng at Rgd Robotcs. Ths acknowledgement would not be complete wthout thankng my old frend Ehsan Asad for hs contnued support and frendshp, and my offce mates at SFU, Azm Keshtkar, Vahd Zaker, Sohel Sadeq, and Shahab Azm for ther professonal support. I hope I can be of some help to all of you n future. I cannot emphasze enough how mportant the role of my beloved parents s n my professonal lfe. I would lke to thank my father and mother who have always dedcated themselves to ther famly n every aspect of lfe. I cannot thank them wth words for all ther efforts, fnancal and emotonal, durng every step of my undergraduate and graduate studes. v

6 Last but defntely not the least, I would lke to thank my lovely wfe, Anahta wthout whose patence and support, I could not accomplsh ths work. I would lke to thank her for the contnued support and encouragement as well as all the challenges and dffcultes she has endured durng ths tme; even though, I wll never be able to express my grattude towards her devotons. v

7 Table of Contents Approval... Abstract... Dedcaton... v Acknowledgements... v Table of Contents... v Lst of Tables... x Lst of Fgures... x Lst of Acronyms... x Nomenclature... x Chapter 1. Introducton Research Objectve and Motvaton Lterature Revew Thess Organzaton Contrbutons... 6 Chapter 2. Prelmnary Modelng of the Cable Introducton Development of the Case-Specfc Cable Model Solvng the System of Equatons The Basc Optmzaton Problem Numercal Soluton to the System of Equatons Results and Dscussons Case I: Known UAS Coordnates Case II: UAS Thrust and Ptch Effect Concluson Chapter 3. Two-Dmensonal Mult-Body Dynamcs Modelng and Analyss of the Tether Methodology Cable Model Cable Segment Model Coordnate System and Model Set-up Dervaton of Equatons of Moton Usng Lagrange s Equatons Knetc Energy and Its Dervatves Potental Energy Dervaton of Generalzed Forces External Forces Cross-Flow Prncple UAS Dynamcs Implementaton of UAS Segment nto the Equatons Numercal Method for Solvng the System s Equaton of Moton Order Reducton of Dfferental Equatons Model Verfcaton v

8 Smple Pendulum Test Generalzaton of the Model to a Fnte-Segment Cable Model s Convergence Results and Dscussons Effect of Wnd Speed Take-Off n Hgh-Velocty Wnd Effect of UAS Thrust Fluctuatons Concluson Chapter 4. Three- Dmensonal Mult-Body Dynamcs Modelng and Analyss of the Tether Methodology Cable Model Cable Segment Model Coordnate System and Model Set-up Dervaton of Equatons of Moton Usng Lagrange s Equatons Knetc and Potental Energy and Ther Dervatves Assemblng Equatons n a Matrx Form External Forces Cross-Flow Prncple UAS Dynamcs Implementaton of UAS Segment nto the Equatons Numercal Method for Solvng the System s Equaton of Moton Order Reducton of Dfferental Equatons Model Verfcaton Smple Pendulum Test Generalzaton of the Model to a Fnte-Segment Cable Results and Dscussons Effect of Wnd on Lateral Moton Varable Wnd Model Effect of UAS Thrust Fluctuatons on Lateral Moton Conclusons Chapter 5. Conclusons and Future Work Summary Future Work Bendng Stffness Longtudnal Stran Reelng Process Contact Modellng Sophstcated UAS Modellng Base Staton Motons Recommendatons References Appendx A. MATLAB Program Codes v

9 Lst of Tables Table 2.1. Parameters Used n Smulaton Table 2.2. UAS Operaton Requrements Table 3.1. Forces appled at UAS for Valdaton Table 3.2. UAS Parameters Table 3.3. UAS Thrust wth fluctuatons Table 4.1. UAS Parameters Table 4.2. Wnd Velocty Vector Table 4.3. Wnd Parameters Table 4.4. UAS Thrust wth fluctuatons x

10 Lst of Fgures Fgure 2.1. UAS Cable Geometry Confguraton... 8 Fgure 2.2. Free-body dagram of the cable n statc equlbrum Fgure 2.3. Cable shape functon Fgure 2.4. Cable Angle Defnton Fgure 2.5. Cable Tenson and Angle vs. Horzontal Offset at Wnch for Varous Lengths, Constant Alttude Fgure 2.6. Cable Tenson and Angle vs. Horzontal Offset at UAS for Varous Lengths, Constant Alttude Fgure 2.7. Vertcal and Horzontal Components of Tether Tenson at UAS for Varous Lengths, Constant Alttude Fgure 2.8. Cable Tenson and Angle vs. Alttude at Wnch for Varous Lengths, Constant Offset Fgure 2.9. Cable Tenson and Angle vs. Alttude at UAS for Varous Lengths, Constant Offset Fgure Cable Tenson and Angle vs. Alttude at Wnch for Varous Lengths, Constant Offset of 25 m Fgure Vertcal and Horzontal Components of Tether Tenson at UAS for Varous Offsets, Versus UAS Alttude Fgure Cable Tenson and Angle vs. Cable Length at Wnch for Varous Offsets, Constant Alttude Fgure Cable Tenson and Angle vs. Cable Length at UAS for Varous Offsets, Constant Alttude Fgure Vertcal and Horzontal Components of Tether Tenson at UAS for Varous Offsets, Constant Alttude Fgure Effect of Length on UAS poston, Constant Thrust and Ptch Fgure UAS Alttude and Offset vs. Cable Length at constant UAS Thrust and Ptch Fgure 3.1. Orgnal cable n a hoverng operaton vs. Dscretzed model of the same cable Fgure 3.2. Dfferent cable segment models used n lterature Fgure 3.3. Smple Pendulum cable segment Fgure 3.4. Global and local coordnate systems of the dscretzed cable Fgure 3.5. Cartesan vs. Polar coordnates for a cable segment Fgure 3.6. Drag and Lft coeffcents of crcular cylnders, wres and cables [32] Fgure 3.7. Drag model on a cable segment x

11 Fgure 3.8. UAS Coordnate Setup Fgure 3.9. Smple Pendulum Perods, Comparson of Smulated and Theoretcal Results Fgure Comparson of statc cable shapes; catenary model vs. dscrete model Fgure Cable shapes n the Take-Off process Fgure Comparson of longtudnal tenson n cable Fgure Convergence of UAS poston for varous number of cable segments Fgure Solvng duraton for dfferent number of cable segments Fgure Fnal Cable Shapes at Dfferent Horzontal Wnd Speeds Fgure UAS poston and alttude tme hstory under dfferent wnd speeds Fgure Cable tenson and angle at UAS connecton pont Fgure Cable tenson and angle at the wnch Fgure Cable shape durng take-off under the effect of ncreasng thrust Fgure UAS poston and ptch under the effect of ncreasng thrust Fgure UAS alttude and the thrust change Fgure UAS poston and atttude under thrust fluctuatons Fgure Cable tenson at UAS under thrust fluctuatons Fgure Cable tenson at base under thrust fluctuatons Fgure 4.1. Global and local coordnate systems of the cable Fgure 4.2. UAS 3D Coordnate Setup Fgure 4.3. Smple Pendulum Perods, Comparson of Smulated and Theoretcal Results Fgure 4.4. Comparson of statc cable shapes; catenary model vs. 2D and 3D dscrete models Fgure UAS poston n dfferent planes, Lateral Wnd Effect Fgure UAS Poston and Atttude, Lateral Wnd Effect Fgure Wnd Varatons; left: speed varatons; rght: headng varatons Fgure UAS poston n dfferent planes, Lateral Wnd Effect Fgure UAS Poston and Atttude, Lateral Wnd Effect Fgure UAS poston results due to thrust fluctuatons Fgure UAS poston and atttude under thrust fluctuatons x

12 Lst of Acronyms 2D 3D AUV DOF FEA GPS NSERC ODE ROV SFU UAS UAV VTOL Two Dmensonal Three Dmensonal Autonomous Underwater Vehcle Degrees of Freedom Fnte Element Analyss Global Postonng System Natural Scences and Engneerng Research Councl of Canada Ordnary Dfferental Equaton Remotely Operated Underwater Vehcle Smon Fraser Unversty Unmanned Aeral System Unmanned Aeral Vehcle Vertcal Take Off and Ladng x

13 Nomenclature Roman A Cable s cross-sectonal area, m 2 C D Drag Coeffcent C L F F D F L F x F z f V f Lft Coeffcent Objectve/Cost functon Drag Force, N Lft Force, N Tangental wnd force n the local segment coordnate, N Normal wnd force n the local segment coordnate, N Wnd Velocty Frequency, Hz Wnd Headng Angle Frequency, Hz g Gravtatonal acceleraton, m/s 2 G Cable s Effectve Shear Modulus, Pa I CG Mass Moment of Inerta Matrx, kgm 2 J Tether Cross Sectonal Polar Moment Of Inerta, m 4 L l L M m U Cable s total length, m Cable segment length, m Lagrangan Functon, J Mass Dstrbuton Matrx, kg UAS s mass, kg Q Generalzed force, m/s 2 Q Generalzed force matrx, m/s 2 R s T 0 T Transformaton matrx Cable length, m Cable s horzontal tenson at vertex, N Longtudnal cable tenson, N x

14 T U U V Knetc energy, Watt Potental energy, Watt Horzontal Wnd Velocty, m/s Lateral Wnd Velocty, m/s V w0 Wnd Velocty Vector, m/s V w, rel Relatve wnd velocty, m/s W Vrtual Work, J W X x Vertcal Wnd Velocty, m/s Global horzontal coordnate Local (cable segment) tangent coordnate x Vrtual Dsplacement Y y Z z Global lateral coordnate Local (cable segment) lateral coordnate Global vertcal coordnate Local (cable segment) normal coordnate Greek η V Angle of Attack, rad and/or deg Cable s mass per length, Kg/m UAS Segment Rotaton Angle, rad and/or deg Vrtual Angular Dsplacement (Vrtual Rotatons) UAS State Vector Atttude Vector Ptch Angle, rad and/or deg Cable s weght per unt length, N/m Wnd Velocty Magntude Mean Value, m/s Wnd Headng Angle Mean Value, deg Cable s densty, Kg/m 3 xv

15 V Wnd Velocty Varance, m/s Wnd Headng Angle Varance, deg Roll Angle, rad and/or deg Headng Angle, rad and/or deg xv

16 Chapter 1. Introducton 1.1. Research Objectve and Motvaton Ths thess s focused on analyzng the behavor of a cable used to tether a hoverng Unmanned Aeral System (to be referred to as UAS hereafter) to ts statonary base staton n dfferent operaton scenaros. The purpose of the cable s to power the UAS n order to ncrease ts operaton duraton as well as to communcate data securely. Ths research was ntated n collaboraton wth Rgd Robotcs Inc. to analyze the tether s statc and dynamc effects on the UAS whch they ntended to desgn and manufacture. Flexble cables have been n use for several years n many structures ncludng brdges; electrcal power lnes, towng and moorng systems, hoverng tethered aerostats, etc. For example, a cable mght be used for towng an underwater vehcle usng a vessel or to connect two flyng arcrafts called an aeral tow system [1]. Tetherng dfferent mechancal systems are among applcatons whch beneft sgnfcantly from the use of cables. These systems are referred to as cable-body systems and are requred to satsfy certan physcal requrements dependng on ther applcaton and the envronment n whch they are operatng [1]. Cables are wdely used n these applcatons because of ther flexblty and versatlty. The fact that cables provde tenson whle not resstng compresson, makes them unque and useful for certan applcatons. They are also able to mtgate vbratons n comparson to smlar components of same mechancal propertes. To better understand the statc and dynamc behavor of the system ncludng the cable, ts effect on the base staton, as well as the tenson that the cable exerts on the 1

17 UAS, t s mportant that the system s modeled so dfferent operaton scenaros can be smulated and analyzed. By havng a relable dynamc model for the cable-body system, one can nvestgate the cable tensons n each specfc flyng condton and acheve desgn crtera for system components, ncludng the cable, UAS propellers and motors, cable- UAS connecton jont, the wnch and ts motor, etc. The effect of wnd on the system and mplement these consderatons nto the controller desgn procedure for both the wnch controller and the UAS controller. Although the UAS autoplot mostly consders the tether tenson as an external dsturbance rather than a real-tme estmated nput, t s crtcal to understand the operaton pont whch enables the control desgn engneers to desgn a more robust length controller and autoplot for the system Lterature Revew Usng cables n mechancal applcatons s very common; thus a wde varety of analytcal and numercal research has been carred out around analyss of cable dynamcs and statc behavour. Many methods have been employed to analyze the cable dynamcs n lterature, as noted n Choo and Cassarella s survey of dfferent cable modellng studes [1]. These methods nclude contnuous and dscrete approaches such as method of characterstcs, fnte element method, lnearzaton method, equvalent lumped-mass method, etc. Contnuous approaches are mostly used to analyze a cable s oscllaton modes or when the statc confguraton of the cable s of nterest. It s common for the contnuous models to be ncapable of yeldng a transent response for the whole system due to the complcaton or neffcency of solvng the equatons of moton. Dscrete models are used to analyze larger dsplacements n the cable when the transent response of the system s desred. Among these smulaton approaches, method of fnte-segment has been nvestgated properly to study the cable s behavor. Several works have been carred out by dscretzng the cable nto several segments and the equatons were derved by wrtng the system equatons for a multbody system of fnte cable elements. In 1972, Domnguez et al. employed a dscrete approach to analyze the statc and dynamc behavor of cable systems [2]. In 1976, Wnget and Huston employed a non-lnear 2

18 3D fnte segment approach to nvestgate the dynamc behavor of a cable or chan. Dreyer and Van Vuuren [3] have studed statc confguraton of an nflexble 2D cable anchored at ts endponts n 1999 wthout consderng external forces on the cable except gravtatonal forces. The soluton was obtaned by numercal analyss for both contnuous and dscrete models and results were valdated expermentally. As mentoned before, cables are used n towng aeral and marne vehcles. Kamman and Huston modeled towed and tethered cable systems wth a focus on reeln/pay-out and towng confguratons for marne vehcles n 1999 [4]. Same authors also numercally smulated cables for marne applcatons n 2001 [5]. Dynamc analyss of cable structures as dscrete systems have been studed n several works and are avalable wdely through lterature. Buckham et al. have carred out extensve research n ths area, specfcally for underwater vehcles ncludng both low-tenson and taut cable applcatons [6] [11]. Many research papers have also been publshed on analyss of cable dynamcs and statcs for aeral applcatons. In 1967, Genn and Cannon nvestgated equlbrum confguraton and tensons of flexble cables n a unform flow feld wth a focus on aeral towed vehcles [12]. They consdered the effect of cable weght, frcton, and drag n ther mathematcal analyses. In 1971, Huffman and Genn n a contnuaton of Genn s prevous work, studed the dynamcal behavor of the same system [13]. Non-lnear mathematcal model for extensble cables were derved and computer smulatons were run. Ths model consders large dsplacements as well as mpact forces and drag forces whle studyng the stablty of the system. Qusenberry have worked on aeral tow systems by mathematcal modellng of the cable as a dscrete cable-body system usng Lagrange s method [14] [16]. In these works, the authors have developed a model for the cable n an arcraft tow system and predcted the dynamc response of the towed arcraft. In a research by Hembree and Sleger [17], authors have modeled a tether usng recursve rgd-body dynamcs n whch they estmate the tenson n the tether based on a low stran model. The elements used n ths work are not elastc and each cable element 3

19 has only 2 DOF. Johansen et al. [18] have modeled nextensble cable dynamcs usng a dscrete cable model and valdated ther results wth a hangng cable. Montano et al. [19] studed dynamcs of a tethered buoy usng mxed fnte elements. In ths work, the model consders a quas-extensble tether attached to a buoy vessel and mplemented the buoy s dynamcs. Surendran and Goutam [20] nvestgated reducton of the dynamc ampltudes of moored cable systems. They modeled the cable as a dscrete lumped mass system and modeled the statc and dynamc responses of the system. Aerostats are smlar to the system we are analyzng n ths work n the sense that they are aeral systems that exert tenson to the upper end of the tether, wth the excepton that they usually have no vertcal thrust and have more drag due to the wnd. There are many works on tethered aerostats n lterature whch date back to as early as to the 70 s. Kang and Lee [21] have developed equatons of moton for a tethered aerostat under atmospherc turbulence usng nonlnear cable dynamcs for an extensble tether. They have obtaned results on tether forces due to dfferent wnd gusts. Rajan et al. [22] have studed dynamc stablty of a tethered aerostat n whch the cable s modeled as a dscrete system of segments modeled as a sprng-mass-dashpot system wth the segment mass concentrated at the nodes. In a smlar work, Stanney and Rahn [23] developed equatons of motons for a tethered aerostat wnd under turbulence and obtaned results on aerostat poston, atttude, and tether tenson due to wnd dsturbances. In two separate works n 1982 and 2001 by Jones and Krausman [24], and Jones and Schroder [25], respectvely, the authors smulated the response of a tethered aerostat to atmospherc turbulence. They used a Frozen-Feld turbulence model and solved for the dynamcs of the system usng a nonlnear dynamc smulaton. Few works have been accomplshed modelng a tethered aeral vehcle. Among these, Ncotra et al. [26] studed taut cable control of a tethered UAV; although the focus s mostly on controllng the UAV and less attenton s pad to the cable model. The cable s consdered taut and the cable tenson s consdered a dsturbance to the UAV. Therefore 4

20 no specfc cable model s developed n ths work. Lupashn and D Andrea [27] accomplshed a work on stablzaton of a flyng vehcle on a taut tether, however, the focus of ths research s mostly on the control of the vehcle rather than modellng the cable tself. Muttn [28], [29] has developed equatons of moton for a flyng ducted fan for ol slck survellance and detecton. In ths work, the model s approxmated usng fnteelement method and an updated Lagrangan approach to model the reelng process for the UAV as well as effect of wnd on Vertcal Take Off/Landng (VTOL) process. The research consders the effect of wnd on the ducted fan as well as smulates the cable reelng durng operaton and emergency mode Thess Organzaton Chapter 2: Prelmnary Modelng of the Cable, s dedcated to prelmnary modellng of a statc cable. Although all the catenary equatons have been derved and used extensvely n research n all applcatons, no organzed solvng algorthm for such a system has been developed. Ths chapter models the cable as a smple statc catenary shape, however t mplements the equatons n a very user-frendly code. Chapter 3: Two-Dmensonal Mult-Body Dynamcs Modelng and Analyss of the Tether, s allocated to mathematcal modellng and smulaton of the cable as a multbody dynamc system by modellng the system usng Lagrange s method. Ths method s an energy-based method and s more capable and less labor-ntensve than Newtonan Mechancs n dervng equatons of motons of dynamc systems, more partcularly mult-body systems.. The equatons are then solved and smulated usng MATAB software. Some results are presented at the end for a specfc case. Chapter 4: Three- Dmensonal Mult-Body Dynamcs Modelng and Analyss of the Tether, analyzes the same problem n a 3D space usng the same method as Chapter 3. Smulatons are run to capture the non-planar moton of the cable. Results are presented and observatons are made at the end. 5

21 Fnally, Chapter 5: Concluson, summarzes the work done n ths thess. A few recommendatons are made and some future works are proposed at the end Contrbutons The algorthm developed n Chapter 2, although based on smple catenary equatons, can help the user n analyzng a statc cable effcently wthout beng nvolved n solvng catenary equatons. Although the purpose of ths work was to analyze a captve UAS, the statc model can be used n any applcaton usng a statc cable when the only external forces appled to the cable are the ones at the cable ends as well as the gravtatonal force. The user can defne mechancal propertes of the cable as well as the forces at the top end, the cable length, as well as horzontal and vertcal dstance between both cable ends. Although varous extensve research has been carred out on dynamcs of flexble cables, there s no straghtforward organzed approach or tool avalable to analyze dynamcs of a cable n specfc applcatons. The cable model derved n ths work along wth the computer algorthm developed, allow the user to ntate analyzng the cable s behavor effcently. The cable model s very versatle and can be appled to a wde range of applcatons n several envronments. Gven a correct and accurate vehcle model, such as a flyng vehcle, or a submarne equpment, one can ntegrate these models nto the algorthm, allowng the software to generate results of a specfc smulaton. As an example, one can use the cable model to analyze the behavor of a tethered aerostat, or a towed submarne vehcle, as long as the user s famlar wth the dynamcs of the vehcle and s able to model t mathematcally. 6

22 Chapter 2. Prelmnary Modelng of the Cable 2.1. Introducton As mentoned n the frst chapter, the man objectve of ths research s to predct the behavour of the cable whle the UAS s n dfferent operaton scenaros e.g. hoverng above the base staton, or vertcal landng or take-off. As a frst attempt to predct these cable shapes for the prelmnary phases of the project, analyss of statc behavor of the cable as a catenary shape was ntated. Knowng certan constant parameters of the system, such as cable s mechancal propertes, as well as varable parameters, one can solve for the unknowns of the problem usng the algorthms offered n ths chapter. In order to determne the unknown parameters of the problem, one should solve the equatons for some parameters such as the cable shape functon, the length of the cable, and the tenson on both ends as well as the tenson functon along the tether. Although the latter s of less mportance snce the maxmum tenson does not occur along the cable. The followng sectons wll explan how calculaton of some of these parameters was performed durng the prelmnary phases of the project n order to acheve an estmate of the cable s behavour. The theory behnd ths chapter s content wll be presented followed by the applcaton to ths specfc problem as well as soluton of the equatons and results. A concluson secton wll follow at the end dscussng the obtaned data. The MATLAB codes and nstructons on how to use these codes for other smlar casespecfc applcatons are presented n Appendx A. 7

23 2.2. Development of the Case-Specfc Cable Model As dscussed n Chapter 1, ths applcaton concerns a UAS flyng at a certan alttude and s connected to a base staton usng a flexble power cable. Ths confguraton s smplfed assumng the cable s hangng only under effect of ts own weght and s llustrated n Fgure 2.1. It s assumed that the UAS s flyng n the same plane as the reel.e. xz-plane; therefore the problem s consdered two-dmensonal. In practce ths s not the case snce the UAS s flyng n a three-dmensonal space. The 3D modellng and smulaton of the cable wll be nvestgated n Chapter 4. Fgure 2.1. UAS Cable Geometry Confguraton In addton, assume that there s no wnd present so the varables are beng acqured under a statc condton. In practce, usually wnd s present and affects the physcs of problem. In the ntal phase of ths project t was necessary to estmate the relaton between parameters such as UAS heght and horzontal poston, cable length, tensons and cable slopes n order to capture some basc workng assumptons for the UAS. As a result, the analyses n ths chapter are lmted to a smplfed case where 8

24 external loads are not present. The effect of wnd on the cable shape and forces wll be nvestgated n Chapter 3. Assumng the reel poston s fxed, t wll have a certan horzontal and vertcal dstance from the hypothetcal vertex of the catenary n each dfferent settng; named x A and z A respectvely. Therefore the coordnate system s not fxed and s defned relatve to the reel s poston. That s, after solvng the problem the parameters x A and z A represent the dstance between the reel and the vertex of the vrtual curve. At ths stage, the orgn of the vrtual catenary curve s defned wth respect to the base staton A and s fxed for ths specfc combnaton of L, p, and H. From ths pont all other varables such as z A are defned n ths coordnate system. Generally, ths system has sx man varables, namely, L, p, H, x A, z A, and T 0. Once these sx varables are known, the system s defned and all other varables,.e. x B, z B, θ A, θ B, T 1 and T 2 whch are dependent on these varables and can be calculated. Snce z A s also a drect functon of x A, one can represent the system by 5 man equatons and 5 varables. As an example, f some of the parameters of the problem ncludng UAS heght H, cable length L, and UAS horzontal offset from the base staton, labeled p are known, the system can be solved. To reduce the number of unknowns and equatons, the equatons are reordered by substtutng some of these varables wth others. Lookng at catenary equatons, one can express the vertcal poston of any pont on the cable n terms of ts horzontal poston. The followng equatons hold true for both ponts A and B or generally any pont on the cable. As stated before, ponts A and B, respectvely represent the reel and the UAS and the dfference between ther vertcal coordnates s the UAS heght H. z A T 0 x A cosh 1 T0 ( 2.1 ) z B T 0 x B cosh 1 T0 ( 2.2 ) 9

25 Therefore, one can see that z A tself s a functon of x A and T 0 and can be consdered as a dependant varable and be calculated after solvng the system equatons. Snce the dfference between these two equatons represents the heght of the UAS, by geometry, one can wrte: xb xa p zb za H ( 2.3 ) Equaton ( 2.1 ) can be combned wth ( 2.2 ) to get H T x 0 A p cosh T x cosh T 0 0 A ( 2.4 ) On the other hand, takng a dervatve of the angle equaton wll yeld the slope equaton of the curve. Applyng the slope equaton at each pont A and B wll result n Equatons ( 2.5 ) and ( 2.6 ). dy xa tan A snh ( 2.5 ) dx T xxa 0 tan B dy xb snh ( 2.6 ) dx T xxb 0 Also, the tenson at any pont can be expressed n terms of ts tangent angle as: T T cos 0 ( 2.7 ) that s, for any pont along the cable, the longtudnal cable tenson s related to the cable angle by ths equaton. 10

26 Fgure 2.2 shows the free body dagram of the whole cable n equlbrum. Assumng the reel s locked and therefore prevented from rotaton, the cable can be assumed to be suspended between two fxed ponts A and B. Wrtng the equlbrum equatons n both x and z drectons wll yeld: TA cosa TB cosb T0 L TB snb TA sna T0 ( 2.8 ) Fgure 2.2. Free-body dagram of the cable n statc equlbrum By substtutng T B from the frst lne n Equaton ( 2.8 ) n the second equaton and some smple operatons one can conclude the followng. L tanatanb ( 2.9 ) T 0 expressed as: By substtutng Equatons ( 2.5 ) and ( 2.6 ) nto ( 2.9 ), Equaton ( 2.9 ) can be 11

27 xa p xa L snh snh ( 2.10 ) T T T Overall, by substtutng some of the varables and reorderng equatons, one can reduce the number of varables to fve and the number of equatons defnng the system to two. Equaton ( 2.11 ) shows the system of two equatons that should be solved n order to determne the system s soluton. The method for solvng these equatons wll be presented n Secton 2.3. At ths pont, there are fve varables present n ths system of equatons whch are L, p, H, x A, and T 0. The equatons can be solved gven any three parameters of the system. T x 0 A p x A H cosh cosh T0 T0 T 0 xa p x A L snh snh T0 T0 ( 2.11 ) 2.3. Solvng the System of Equatons Equaton ( 2.11 ) s a set of two nonlnear equatons whch express varables L and H n terms of ndependent varables x A, T 0, and p. Dependng on the confguraton of the problem, ths equaton can be changed to match the unknowns of the problem. As an example, n an applcaton cable length L, UAS heght H and horzontal poston p mght be known, therefore the only unknowns of the problem wll be x A and T 0. However ths confguraton wll become a set of mplct equatons n terms of x A and T 0, snce x A cannot be drectly expressed n terms of T 0 or vce versa. In general, Equaton ( 2.11 ) represents two mplct nonlnear equatons and cannot be solved analytcally. As a result, a numercal method should be used to solve ths equaton, whch s explaned n Sectons and

28 The Basc Optmzaton Problem Consder a system of n equatons and n unknowns as shown n Equaton ( 2.12 ). f1 x1, x2,, xn 0 f2 x1, x2,, xn 0 fn x1, x2,, xn 0 ( 2.12 ) Now, the varables are defned as a vector called x, where x1 x x n ( 2.13 ) One can then wrte Equaton ( 2.12 ) as f f 1 2 x x 0 0 ( 2.14 ) when Equaton ( 2.14 ) has a unque soluton specfcally for ths system of equaton, x * n ( ) 0 ( 2.15 ) f x 2 F x 1 where Fx s usually referred to as the cost functon or objectve functon [3]. In order to solve Equaton ( 2.15 ), One can reduce ths problem to an optmzaton problem of the followng form 13

29 mnmze F x x subject to approprate constrants ( 2.16 ) * The soluton to Equaton ( 2.16 ), x s the mnmzer of the objectve functon F x ; thereby satsfyng the requrements of the orgnal system Equaton, ( 2.12 ). Usually a numercal search method such as Steepest-Descent method, Newton Method, or Gauss- Newton Method should be employed to solve Equaton ( 2.16 ) Numercal Soluton to the System of Equatons As explaned n Secton 2.3.1, before solvng the problem, t should be formulated properly. One can wrte Equaton ( 2.11 ) n the followng form to comply wth the format of the optmzaton problem dscussed n Secton T x p x cosh cosh 0 0 A A H T0 T0 T x p x snh snh 0 T0 T0 0 A A L ( 2.17 ) If frst and second equatons are denoted by f 1(x) and f 2(x) respectvely and consder T 0 and x A as two unknown varables x 1 and x 2, one can wrte x x p x f x x H x x p x f x x L , 2 cosh cosh 0 x2 x , 2 snh snh 0 x2 x2 ( 2.18 ) where Now defne a vector varable x denotng the unknown varables of the problem, 14

30 xa x1 x T0 x2 ( 2.19 ) One can then re-wrte Equaton ( 2.18 ) as f f 1 2 x x 0 0 ( 2.20 ) whch can be reduced to Equaton ( 2.21 ) smlar to the optmzaton problem shown n Equaton ( 2.16 ). mnmze F x x subject to x1, x2 0 ( 2.21 ) In ths work, the optmzaton problem for Equaton ( 2.17 ) s solved usng MATLAB s nonlnear equaton solver functon fsolve. The programmng detals and the MATLAB codes are presented n Appendx A. Ths tool needs the nput equatons as a vector functon of the followng form where f 1 and f 2 are the same functons as descrbed n Equaton ( 2.18 ). f1 G f 2 ( 2.22 ) Ths search algorthm s able to solve Equaton ( 2.17 ) wth a convergence tolerance of less than After runnng each algorthm, the tolerance s checked to ensure convergence of the problem. Some example results for dfferent scenaros one can apply ths tool are presented n the next secton along wth dscusson on how ths tool has helped acheve some ntal workng assumptons for the UAS operaton. Note that the 15

31 mentoned equatons are set up for a specfc case when the cable length, alttude and horzontal poston of the UAS s known. However the equatons can be set up accordngly to accommodate other scenaros. As an example magne that the alttude and horzontal poston are known and a certan tenson on the cable s requred. One can set up the equatons n a way that the algorthm yelds the requred cable length to keep the desred cable tenson. Ths would be partcularly helpful f the purpose of the analyses s to desgn a tenson controller system Results and Dscussons Ths secton s dedcated to explorng some scenaros of UAS operaton. The methods and algorthms dscussed n prevous sectons are appled to some specfc cases and present results and dscussons about the observatons. The MATLAB codes whch solve specfc problems are presented n Appendx A. It should be noted that n all cases, the constant µ, cable s weght per length should be entered at the begnnng of the code. Moreover, all unts used n ths algorthm are n metrc system. More nstructons regardng runnng the codes to solve the problem are presented n Appendx A. It should be emphaszed agan that ths chapter assumes a planar moton wthout presence of wnd or dsturbance on the system s dynamcs and do not consder the dynamcs of the system and specfcally the UAS. The UAS s consdered to be able to hold ts stablty and poston autonomously and that t ntroduce no dsturbances on the cable. That s, a steady state equlbrum s assumed for the system. Ths assumpton s vald only for the cases where wnd s not present and there s not external forces appled to the system. As mentoned prevously, one can defne the problem as they desre based on the known factors of the system. For example, one can use three nputs ncludng L, H and p to determne the cable s shape functon as well as tensons along the cable as a functon of horzontal coordnate, x. Also, the codes wll help the user develop some general nsght about how the cable wll acheve equlbrum n some scenaros. Three cases are nvestgated n the followng three sectons. 16

32 Case I: Known UAS Coordnates In general, for the system developed, knowng three parameters wll make the problem defned. In the event that the UAS s coordnates are known, one can solve the problem by feedng the known parameters to the algorthm wrtten n the fle named Mtethsolve3.m. Ths code solves the catenary equatons and determnes cable varables of nterest ncludng tensons and cable angle at both ends or on any pont of nterest on the cable. Generally, parameters such as alttude (H) and the horzontal offset (p) can be avalable through the UAS s GPS system or other systems such as an nertal navgaton systems. Length of the cable (L) can also be known by trackng the reel s payout, although other methods mght be used to determne these values. In the case consdered n ths work, snce the project s n the research and development stage, the purpose s only to predct dfferent scenaros that could occur durng flght and gan a better understandng of the physcs of the system. To llustrate the results from ths case, a sample algorthm s run for a system wth parameters p=15, H=120, and L= 122. The cable parameters used n the smulaton are presented n Table 2.1 Table 2.1. Parameters Used n Smulaton Smulaton Parameter Value Unts Mass per length 0.14 [kg/m] Cable Modulus of Elastcty 80 [GPa] Cable Dameter [m] Cable Length 122 [m] UAS Heght 120 [m] UAS Offset 15 [m] The syntax for the algorthm s n the form of >>Mtethsolve3(p,H,L) 17

33 As an example, runnng a scrpt called Mtethsolve3 wth the parameters of Table 2.1 and runnng the followng command wll yeld the results. >> Mtethsolve3(15,120,122) Fgure 2.3. Cable shape functon Usng ths algorthm, once the system equatons are solved, one can plot the cable shape functon as well as evaluate all cable values. Usng another scrpt called Mtethplot3, the cable shape can be plotted n a 2D plane. Fgure 2.3 shows the cable s shape functon usng four dfferent values of horzontal offset for the UAS wth fxed alttude and cable length. The base staton s located at the orgn of the coordnate system, and the red rectangle represents the hoverng UAS. In order to see the tenson ranges of the cable both at the UAS and at the wnch, the algorthms to solve for the cable parameters are used and cable tensons are plotted at the wnch and at the UAS along wth the cable angle at both ends. Ths s useful specfcally n understandng the ranges of cable tensons occurrng n dfferent hoverng scenaros. It has helped wth understandng the power ranges for the UAS based on the vertcal thrust t needs to provde to mantan a specfc cable tenson range. 18

34 Table 2.2 shows the parameter ranges consdered for the cable length as well as UAS postons accordng to the operaton requrements. The UAS s allowed to move n a 40 meters horzontal dstance from ts base staton whle t s supposed to hold an alttude between 110 to 130 meters. Table 2.2. UAS Operaton Requrements Smulaton Parameter Value Unts Cable Length (L) [m] UAS Alttude (H) [m] UAS Horzontal Offset (p) 0-40 [m] Fgure 2.4 shows postve, zero and negatve tenson angles at the wnch for a certan UAS heght and cable length at three dfferent offsets. Note that the angles defned are wth respect to the horzontal lne. For each cable length, the angles wll be zero at a specfc offset. Ths essentally means that the base staton wll be located at the vertex of the catenary makng the tangent at the curve,.e. cable angle zero. Negatve angles techncally mean that the cable s slack and s hangng below the vrtual base staton. In practce, f the base staton s located on the ground the cable wll be lyng on the ground and the physcs of the cable wll change and ths analyss wll not be applcable to the system. However, assumng the cable s allowed to hang below the wnch, one can use ths tool to solve for the system s parameters. Fgure 2.4. Cable Angle Defnton 19

35 Fgure 2.5 shows the magntude of the cable tensons and angles at the wnch versus the horzontal offset for dfferent cable lengths for a UAS hoverng at a constant heght of 120 m. For each cable length, the fgures are plotted aganst the offset to a maxmum offset value as the geometry permts. For an nextensble cable, the lmt of the offset s determned by the UAS heght and length of the tether. One can observe that the tenson values approach small numbers as the UAS gets closer to the base staton. For hgher cable lengths, when the horzontal offset gets smaller, the cable wll gan some slackness and the tenson drecton wll be downwards. Ths can be confrmed by the cable angle curves. Fgure 2.5. Cable Tenson and Angle vs. Horzontal Offset at Wnch for Varous Lengths, Constant Alttude Fgure 2.6 shows the cable tensons and angles at dfferent cable lengths plotted aganst the UAS horzontal offset. Agan, one can observe the UAS tensons when the UAS s closer to the base staton tend to converge to a certan range of numbers. These numbers are techncally the weght of the cable hangng from the UAS. Ths could be nterpreted n another way;.e. when the UAS s closer to the base staton, changng the 20

36 cable length affects the tensons at the UAS less sgnfcantly than when t s horzontally postoned further,.e. larger p values. Fgure 2.6. Cable Tenson and Angle vs. Horzontal Offset at UAS for Varous Lengths, Constant Alttude Fgure 2.7. Vertcal and Horzontal Components of Tether Tenson at UAS for Varous Lengths, Constant Alttude 21

37 Although these two fgures are useful n understandng the cable tenson, they do not gve transparent mplcaton of the vertcal loads on the UAS. Therefore, the vertcal and horzontal components of the tenson at UAS are plotted and presented n Fgure 2.7. To observe the effects of the UAS alttude on the tensons, for a UAS hoverng at a constant offset, Fgure 2.8 and Fgure 2.9 are presented below. As can be seen n Fgure 2.8, the tenson at the reel takes a mnmum value at any certan heght. That s, for ths specfc horzontal poston, the mnmum tenson at the reel s 5 N. However by comparng Fgure 2.8 and Fgure 2.10 one can confrm that ths mnmum value ncreases as p ncreases. Fgure 2.10 shows the results for the same cable lengths and alttudes at the offset of 25 meters. Fgure 2.8. Cable Tenson and Angle vs. Alttude at Wnch for Varous Lengths, Constant Offset Fgure 2.9 presents the cable tensons and pull angle at the UAS at dfferent heghts when the UAS holds offset and constant cable length. Ths fgure helps one understand how much the tenson ncreases as the UAS gans heght holdng other parameters constant. It s useful to know the senstvty of the tensons both at the UAS and at the wnch to heght so the user can understand what to expect n dfferent scenaros. 22

38 Fgure 2.9. Cable Tenson and Angle vs. Alttude at UAS for Varous Lengths, Constant Offset Fgure Cable Tenson and Angle vs. Alttude at Wnch for Varous Lengths, Constant Offset of 25 m 23

39 To llustrate the vertcal and horzontal components of the tenson at the UAS, they are plotted aganst the UAS alttude H n Fgure Gven any desred horzontal offset from the base staton as well as cable length, one can plot the cable tensons at the UAS or the base staton aganst dfferent flyng alttudes to understand how the tenson changes as the UAS ncreases ts operaton heght. Fgure Vertcal and Horzontal Components of Tether Tenson at UAS for Varous Offsets, Versus UAS Alttude Consderng a UAS at a certan heght and dstance from base staton, t s worthwhle to observe the tenson senstvty to cable length changes at both cable ends. Fgure 2.12 and Fgure 2.13 show the tensons at wnch and UAS respectvely as a functon of cable length when the UAS heght s held constant. The results are plotted for dfferent horzontal offsets. 24

40 Fgure Cable Tenson and Angle vs. Cable Length at Wnch for Varous Offsets, Constant Alttude Fgure Cable Tenson and Angle vs. Cable Length at UAS for Varous Offsets, Constant Alttude 25

41 As mentoned before, the tenson tself at the UAS mght not defne the need for the power requrement of UAS thrust system. Thus, Fgure 2.14 has been generated showng the two horzontal and vertcal component of these forces. Fgure Vertcal and Horzontal Components of Tether Tenson at UAS for Varous Offsets, Constant Alttude The results n ths secton were essentally used n the frst stages of research and development of the UAS system n order to estmate the wnch motor s torque as well as UAS thrust producton capablty for a more confdent conceptual desgn. In the next secton, another approach s used and some new results are presented Case II: UAS Thrust and Ptch Effect In order to analyze the UAS thrust effects on the cable, another algorthm was developed to whch one can feed the UAS vertcal and horzontal thrust. In equlbrum, the thrust force from the UAS wll be equal to the tenson at the upper end of the cable. Therefore, by nputtng the vertcal and horzontal components of the UAS thrust to the algorthm along wth the cable length, the algorthm s able to solve the equatons and fnd 26

42 the parameters of the catenary ncludng horzontal and vertcal dstance from the base staton called offset and alttude, respectvely as well as the vertex locaton x A. Havng obtaned these parameters, the system s defned and all other parameters are calculated accordngly. At ths pont, the cable shape functon can be plotted as well as the UAS trajectory. It s worthwhle to emphasze agan that UAS and the cable are under effect of no external forces and the system s consdered stable and n equlbrum. The purpose of ths test s to see the effect of the length on the cable shape and UAS poston whle the UAS thrust and ptch reman constant. The UAS thrust s vertcal and horzontal components can be defned n the form of thrust and ptch meanng that the UAS propellers produce a constant vertcal thrust. Fgure Effect of Length on UAS poston, Constant Thrust and Ptch Fgure 2.15 shows the UAS poston for varous cable lengths when the UAS thrust and ptch are kept constant. It can be observed that by keepng all UAS parameters 27

43 constant, ncreasng the length ncreases both the horzontal poston and the flyng alttude. Ths helps understand how the cable length affects the poston of the UAS. In addton, usng a smlar algorthm, the UAS hoverng alttude and horzontal offset s plotted as a functon of the cable length for the same UAS thrust and ptch n Fgure 2.16.The two plots n ths fgure mply that both the alttude and offset change lnearly wth the cable length as long as the forces on the upper end of the tether reman constant. Ths s very mportant n understandng the reelng n / payng out process. Of course, ths analyses can be appled to the cases where the system s n a quas-statc condton,.e. the changes n the speed of reelng s small to the pont that at each tme step, one can assume the system s n equlbrum wthout the presence of nertal forces. Fgure UAS Alttude and Offset vs. Cable Length at constant UAS Thrust and Ptch 28

44 2.5. Concluson In ths chapter, a smple user-frendly tool has been developed to help any user model and solve equatons of a statc catenary. The focus of ths chapter, however, was applcaton of ths tool to a cable whch tethers an unmanned aeral system (UAS) to the ground base staton. Wth the correct mplementaton of varables and equatons, ths tool can be appled to any statc cable to acheve forces and cable shape functons n the absence of external forces. As an example, by applyng the correct weght per length for a moorng cable t can be appled to a cable connectng a floatng object to the seabed. As mentoned before, the model s lmted to cases where wnd or water current, or generally external forces are not present and where the statc condton of the cable s of nterest. As another example, one can solve the equatons for a certan system confguraton based on dfferent weghts per length of the cable to observe the effect of cable weght on the outputs of the problem. In ths chapter, two dfferent case studes are presented to demonstrate the capablty of the developed algorthm n applcaton to dfferent scenaros and some results are presented. The results are used to observe the cable behavor to dfferent confguratons of the system. Accordng to these results one can better understand the senstvty of the tensons both at the UAS and at the wnch to dfferent parameters, ncludng length, and UAS vertcal and horzontal poston. In addton, t helps understand how the UAS behaves by changng length and what ptch and thrust are requred to mantan poston. Another outcome of these tests s the effect of cable length on UAS poston by keepng UAS thrust and ptch constant. As mentoned n Secton 2.1, the codes are lmted n terms of applcaton, n the sense that they are unable to predct the results of a dynamc system. It s also lmted to the cases where external forces are absent. However, t s powerful n predctng a quasstatc system where the acceleraton s almost zero and as a result, no nertal force s present. Wth some n-depth mathematcal dervaton, one may extend ths model to one that can analyze external forces such as unform wnd, etc. 29

45 Chapter 3. Two-Dmensonal Mult-Body Dynamcs Modelng and Analyss of the Tether As mentoned n Chapter 1, nonlnear analyss and modelng of a flexble cable s not a straght forward task and can become qute complcated n terms of analytcal soluton [16]. In an effort to approach ths problem, a fnte segment model s developed and solved n order to nvestgate the cable s behavor and ts nterference wth the UAS. Ths chapter s dedcated to explanng how the system s modelled and what tools and methods are used to derve the equatons of the system and how they are solved. Some smulatons have been run and some observatons are presented at the end Methodology Choo and Casarella [1] have carred out a comprehensve research revew on dfferent analytcal methods for modellng and smulaton of cables. Several methods are dscussed along wth merts and lmtatons of each method. The common methods nclude method of characterstcs, fnte element method, lnearzaton method, equvalent lumpedmass method, etc. Among these, fnte element method s consdered the most versatle one snce t can be appled almost to any applcaton and physcal system [1]. To model a flexble cable, a dscretzaton method s employed. The cable s consdered as a dscrete system consstng of small segments of the same length as shown n Fgure 3.1. The system s a combnaton of multple rgd bodes to whch multbody dynamcs rules are appled n order to model and solve for the dynamcs of the system. Note that n ths work, the effect of varable length dynamcs s not consdered. 30

46 Fgure 3.1. Orgnal cable n a hoverng operaton vs. Dscretzed model of the same cable The equaton of moton could be developed usng Newtonan mechancs. However, n order to develop the model usng ths method one would need to have full nformaton of the external forces actng on each mass node as well as all ntal condtons of the whole system. In ths work, consderng the fact that the system conssts of several bodes n nteracton, t was chosen to develop the mathematcal model for the mult-body system usng prncples of Lagrangan Mechancs. Dynamcs of the system can also be studed usng Lagrange s method. Ths method has two man advantages over Newtonan mechancs n dervng the systems equatons. Frstly, Lagrange s method s energy-based whle Newtonan mechancs s vectoral n nature [16]. Therefore usng Lagrange s equatons wll elmnate the use of vectors n the fnal stages of the dervatons of equatons of moton snce t uses scalar values rather than vectors. Ths can make the dervatons less complcated and labour- 31

47 ntensve. The second advantage s that unlke Newton s laws, dervaton of the equatons usng Lagrange s method can be carred out through a certan set of procedures as long as the system and ts constrants are well-defned [19]. The derved equatons wll have to be solved. However as t wll be explaned n followng sectons, the equatons that are derved for ths system are a set of second order ordnary dfferental coeffcents of whch nclude the varable and ther rates whch makes these equatons farly complcated to solve usng common analytcal methods. Hence the best optons for solvng the equaton of motons would be numercal ntegraton technques such as Euler s method, Trapezodal method, Modfed Euler s method, Runge-Kutta methods, etc. [16]. In ths work, t s chosen to use the Runge-Kutta method due to smplcty of mplementaton Cable Model Cable Segment Model The fnte-element method that s used should mplement a sutable cable segment model n order to satsfy the physcal requrements of the problem as well as to yeld an acceptable approxmaton of the system dynamcs. Therefore, t s crtcal that one defnes the ntal and boundary condtons accordngly. As mentoned prevously, there are several FEA modelng methods for a dscrete flexble cable and dfferent methods use dfferent cable segment models. Some of the models dscussed by Choo and Casarella [1] are smple pendulum model, thn rod model, sprng-mass model, etc. Other models ncorporatng a cable segment model as several pont masses connected by vscoelastc sprngs are also used n some works such as the research carred out by Buckham et al. [11] and also Wllams et al. [31]. Fgure 3.2 shows these dfferent segment models used for FEA modelng of flexble cables. 32

48 Fgure 3.2. Dfferent cable segment models used n lterature a) Smple pendulum, b) Sprng-mass, c) Thn-rod [4], d) Curved beam e) Vscoelastc sprngs [1],[6] For the purpose of ths research, the smple pendulum model s used due to a few reasons. Frst of all, usng ths model wll help effectvely model the wnd drag forces snce straght cylnders are used. Drag forces on straght cylnders n a unform flow are well researched and avalable for mplementaton nto the equatons [32]. Moreover, snce the cable modeled here has a hgh stffness relatvely, by neglectng the effect of stran, ths model yelds an approprate physcal representaton of the cable n dfferent scenaros and elmnates the need for usng a sprng-mass model. The use of smple pendulum model for each segment s also suggested n [19] to reduce computaton costs. Model of the cable s consdered as a system of several cable segments connected together n sequence as shown n Fgure 3.1. Snce t s assumed that the cable does not undergo any bendng moment due to the large bendng rad occurred n operaton scenaros of ths work, the connectons are modeled as deal sphercal jonts;.e. there are only tenson forces present at the connectons. However the segments can tolerate bendng snce they are modeled as a smple pendulum consstng of a weghtless rgd rod and a concentrated mass. The cable segment model used n ths thess s shown n Fgure

49 Fgure 3.3. Smple Pendulum cable segment Coordnate System and Model Set-up The two dmensonal model of the cable s developed n a Cartesan coordnate system labeled XZ-plane. XZ s an nertal frame of reference n whch the poston of each node on the cable s represented. An nertal or namely Newtonan frame of reference s defned as a reference frame whch does not undergo any acceleraton ncludng rotatonal or translatonal [20]. The orgn of ths system s at the cable s connecton to the ground staton, the X axs s parallel to the ground and the Z axs s perpendcular to the ground pontng upwards to represent the heght of each node. In addton to ths nertal coordnate system, a local coordnate system called xz s used whch s defned separately for each and every segment n a way that x-axs s always parallel to the cable segment towards the mass end of the segment, called tangent coordnate from here on, and the z-axs s perpendcular to the cable segment called normal coordnate hereafter. Ths confguraton s shown n Fgure 3.4. Now, length of the th segment s labeled by l and the angle each segment makes wth the horzon s called θ. Havng ths model n mnd helps smply express the poston of each mass node on the cable model as follows: X Z j j j1 l cos l sn j j j1 ( 1,2,3,, n) ( 3.1 ) where X and Z are the coordnates of the th mass node. 34

50 Fgure 3.4. Global and local coordnate systems of the dscretzed cable Snce there are two dfferent frames, one should be able to convert any vector expressed n these frames to one another usng a transformaton matrx. Now, consder a vector represented n the nertal reference frame by (U,W) and n the local xz coordnate system by (u,w). A transformaton matrx can be used to convert the vector representaton to/from the nertal frame of reference. The equaton below shows the transformaton matrx used to convert any vector n the xz coordnate (u,w) to ts XZ representaton,.e. (U,W). U u W R ( 3.2 ) w 35

51 expressed as: where the transformaton matrx from the body frame to the nertal frame s cos sn R sn cos ( 3.3 ) It s clear that the opposte of ths transformaton could be done usng the nverse of the matrx R. Usng Equatons ( 3.1 ) - ( 3.3 ) and snce each mass s located at (l j,0) n ts own local frame, one can wrte the poston of each mass node n the nertal frame as: X x 1 j 1l j ( 1,2,3,, n) Z R j1 z R j j1 0 ( 3.4 ) Now that the model s geometry s set up and the coordnate systems properly, dfferent terms for Lagrange s equatons can be used to derve the equatons of moton as t s presented n Secton Dervaton of Equatons of Moton Usng Lagrange s Equatons As explaned n Secton 3.1, Lagrange s equatons are used for ths system n order to derve ts equatons of moton. It s mportant to be careful wth the degrees of freedom of the system n the process of dervng the Lagrangan for a mult-body system. Accordng to Greenwood, the number of degrees of freedom (DOF) of a system s equal to the number of ndependent coordnates used to specfy the confguraton mnus the number of ndependent equatons of constrant [34][33]. It must be noted that the DOF of a system s a characterstc of the system and does not depend on the coordnates chosen to represent ts moton. 36

52 As an example, the system dscussed n Secton 3.2.1, can be defned by only two coordnates; examples would nclude X and Z n a Cartesan system or r and θ n a polar system. Consder only one segment as shown n Fgure 3.5. In ths segment there are two coordnates defnng the poston of the mass; however, ths system has only one DOF snce these two coordnates are not ndependent,.e. they can be related to each other usng a constrant equaton. Fgure 3.5. Cartesan vs. Polar coordnates for a cable segment The equaton of constrant, for a cable segment of length l could be wrtten as follows whch reduces the DOF to one l X X 1 Z Z 1 ( 1,2,, n) ( 3.5 ) In the same way one can see that the mass could be located usng only the coordnate θ n polar coordnates snce the length of the pendulum s constant. Hence one can verfy that the DOF s the same regardless of the coordnate system chosen. In the mult-body model of the cable, snce n cable segments are used, the DOF of the system wll be n. Ths could also be verfed n another way. Snce there are n segments and each segment s defned by 2 coordnates (X and Z), the system needs 2n coordnates overall to be defned. On the other hand, there are n constrant equatons as shown n Equaton ( 3.5 ) whch make the DOF equal to 2n-n;.e. n degrees of freedom. Therefore the 37

53 mnmum number of generalzed coordnates needed for the system to be defned thoroughly s n. In order to smplfy the dervaton of the equatons usng Lagrange s equatons, one should use generalzed coordnates q n the dervatons [34]. In ths analyss, θ could be used as the generalzed coordnates n the development of Lagrange s equatons. The Lagrangan of a system s shown byl and s defned as [23]: L TU ( 3.6 ) where T and U are the system s knetc and potental energy, respectvely. Lagrange s equatons for a mult-body system consstng of n rgd bodes can be wrtten n the followng form [34]: L L d Q ( 1,2,, n) dt q q ( 3.7 ) where q s are the generalzed coordnates used to defne the system and Q s are appled or generalzed forces that are not dervable from the potental energy [34]. Snce n ths system, the potental energy s ndependent of velocty, one can reduce the Lagrange s equatons to the followng format: d T T U Q Q ( 1,2,, n) dt q q q ( 3.8 ) where Q s are generalzed forces whch nclude appled forces derved from the potental energy. Therefore, by makng the rght selecton of the segment angles as the generalzed coordnates, Equaton ( 3.8 ) can be expressed as: 38

54 d T T Q ( 1,2,, n) dt ( 3.9 ) Now that Lagrange s equatons are establshed for the system, t s crtcal that all the energy terms are derved correctly n order to be ncorporated nto the man equatons Knetc Energy and Its Dervatves The knetc energy for a system of n partcles can be stated as [20]: T n 1 2 mkvk ( 3.10 ) 2 k1 where V k represents the velocty vector for the k th partcle n the nertal reference frame. Therefore, accordng to the choce of reference frame, one can express the squared velocty vector as: T k k k k k V V V X Z ( k 1,2,, n) ( 3.11 ) However, ths equaton should be wrtten n terms of the generalzed coordnates. Dfferentatng and arrangng Equaton ( 3.1 )yelds: X Z l sn j j j j1 l cos j j j j1 ( 1,2,3,, n) ( 3.12 ) By substtutng ths equaton nto the velocty vector equaton, one can wrte: 39

55 2 k k 2 2 k k k j j sn sn j cos cos j 1 j1 V X Z l l ( 3.13 ) whch can then be reduced to k k 2 V l l cos( ),( k 1,2,, n) ( 3.14 ) k j j j 1 j1 where l s the length of the th segment. Now, havng derved the velocty n terms of the generalzed coordnates one can wrte the knetc energy of the system as: n k k 1 T mkll j j cos( j ) ( 3.15 ) 2 k1 1 j1 where m k s the mass of the node at the end of the th segment. The Lagrange s equatons shown n Equaton ( 3.9 ) requres fndng some of the dervatves of the knetc energy. Dfferentatng the knetc energy wth respect to the generalzed coordnate and also wth respect to ts tme dervatve respectvely yelds: n k T T m l l sn( ),( 1,2,, n) k j j j ( 3.16 ) q k j1 n k T T m l l cos( ),( 1,2,, n) k j j j ( 3.17 ) q k j1 It s very crtcal to note that n equatons ( 3.16 ) and ( 3.17 ), the summaton over k, wll start at the th component. The reason s that the nner summaton covers the 1 st to 40

56 the k th terms. Therefore, for all k s smaller than I, there are no or present n the summaton; hence, the dervatve wth respect to these terms are zero. Followng ths, one can wrte the frst term on the left hand sde of the Lagrange s equaton n Equaton ( 3.9 ) as : n k d T d T d m l l cos( ) dt q dt dt k j j j k j1 n k = mkll j j cos( j ) j ( j )sn( j ),( 1,2,, n) k j1 ( 3.18 ) And fnally substtutng Equatons ( 3.16 ) and ( 3.18 ) nto ( 3.9 ) yelds the followng equaton as the Lagrange s equaton for ths system: n k 2 mkll j j cos( j ) j sn( j ) Q,( 1,2,, n) ( 3.19 ) k j Potental Energy Wth the assumpton of no stran or elastcty n the lnks, the only potental energy present n ths mult-body system s of gravtatonal form. Gravtatonal potental energy s usually measured wth respect to a datum lne at whch U=0. Assume a datum lne on the ground, that s at Z=0. Then for the system shown n Fgure 3.4, the potental energy could be stated as: n U m Z g ( 3.20 ) j1 j j 41

57 Dervaton of Generalzed Forces Durng the dervaton of Lagrange s equatons, as seen n Equaton ( 3.19 ), one needs to evaluate the generalzed forces assocated wth the system s generalzed coordnates [33]. In hs book, Dynamcs of Multbody Systems, Shabana states that these generalzed forces can be ntroduced by applyng the prncple of Vrtual Work whch can be used both n statc and dynamc analyss of a multbody system [35]. Qusenberry has also appled ths method n ther thess for an arcraft towng system [19]. Consder a system consstng of k rgd bodes, whose centres of mass are gven by 3k Cartesan coordnates, x 1, x 2,, x 3k. Now consder force vectors F 1, F 2,, F k are appled to these bodes at these coordnates and moments M 1, M 2,, M k are appled to these rgd bodes, and the system goes through nfntesmal poston and angular change of δr 1, δr 2,, δr k and δϕ 1, δϕ 2,, δϕ k, respectvely. Then the vrtual work done by these forces on the system can be wrtten as: 3k W F x M x j j j1 1 k ( 3.21 ) Usng the prncple of vrtual work, one can obtan the generalzed forces for a dscretzed cable n terms of the external forces. Accordng to Qusenberry [19], for the system defned n ths work, the generalzed forces whch cannot be derved from the potental energy can be wrtten as: n X j Zj j Q FX F ( 1,2, ) j Z M n j j j1 ( 3.22 ) Also, snce U Q Q ( 1,2, n) ( 3.23 ) 42

58 by subtractng the dervatve of the potental energy, one can express the generalzed forces for ths dscrete cable system as: n X j Zj j Q FX F ( 1,2, ) j Z m j jg M n j j1 ( 3.24 ) After calculatng the dervatve of the X and Z coordnates wth respect to the generalzed coordnates the generalzed forces can be smplfed to: n cos sn + ( 1,2, ) j j Q F m g l F l M n ( 3.25 ) Z j X j Combnng wth the derved Lagrange s equaton ( 3.19 ), one can wrte: n k k j1 m l l n 2 k j j cos( j ) j sn( j ) FZ m cos sn +, ( 1,2,, ) j jg l FX l j M n j ( 3.26 ) Equaton ( 3.26 ) actually s a set of n second-order ordnary dfferental equatons whch altogether represent the equatons of moton of the cable n terms of n unknown varables θ and ther two dervatves θ andθ. Defnng the appled external forces as well as proper ntal condtons, one can solve ths set of equatons for the cable segment s angular acceleraton whch wll fully defne the dynamc behavor of the system. Note that these equatons are hghly nonlnear and cannot be solved usng common analytcal methods. Therefore, a sutable numercal technque should be employed for solvng the equatons of moton of the system. Lookng at the equatons of moton they can be reduced to a matrx/vector format for smpler manpulaton and representaton. The generalzed forces can be mplemented n a vector of a generalzed force n 1 vectorq where: 43

59 n F m gl cos F l sn + M ( 1,2, n) j j Q ( 3.27 ) Z j X j Also, after manpulaton and smplfcaton of the summatons on the left hand sde of Equaton ( 3.26 ), the followng matrces can be defned to represent the Lagrange s equatons smlar to Qusenberry s work n [19]: n j k j j A = m l l cos (, j 1,2, n) ( 3.28 ) kmax, j n Β j k j j = m l l sn (, j 1,2, n) kmax, j ( 3.29 ) where matrx M s the mass dstrbuton matrx for the multbody system and s defned as: M j n = m (, j 1,2, n) kmax, j k ( 3.30 ) Usng these the matrces ntroduced n Equatons ( 3.27 ) ( 3.30 ), Equaton ( 3.26 ) can be expressed as: Aθ Bdagθ θ Q ( 3.31 ) where A and B are n by n matrces and functons of s and Q s a vectors wth n rows and a functon of and. Also,θ andθ are n 1 vectors consstng of and respectvely. Consderng these notatons, calculaton of the external forces wll conclude the dervaton of Lagrange s equatons n the next secton. 44

60 3.4. External Forces In general, cables used n most applcatons n ndustry are subject to certan types of external forces. As an example, consder a moorng cable connectng a shp to a towed ROV or an arcraft towng an autonomous UAV. In all these cases as well as the case of a tethered flyng vehcle for ths work, external forces e.g. those caused by wnd or water current may be appled to the system. Ths mght sgnfcantly affect the cable s shape functon as well as ts dynamc behavour and tensons actng on the base staton and the UAS. Thus t s mportant to derve and mplement these forces nto the system s equatons for the purpose of ths analyss. As Equaton ( 3.31 ) suggests, the generalzed force vector Q should be defned so that one can solve the Lagrange s equatons. Equaton set ( 3.27 ) shows that ths generalzed force vector, conssts of the external forces and moments exerted on each cable segment namely F, F and M. Note that all components of the generalzed forces X Z are nherently moments rather than lnear forces n ths analyss snce the selected generalzed coordnates are angles. In ths secton, wth the assumpton of unform 2D wnd forces beng the only external load on the system, the equatons for the external forces appled to each cable segment are derved to form the generalzed force vector; then the results are substtuted nto the Lagrange s Equatons. The drag model used n ths research s borrowed from the work of Hoerner [32] n whch the effect of flow around crcular cylnders on lft and drag coeffcents n a unform flud flow s studed. The cross-flow prncple or cosne prncple as labeled by Hoerner s mostly vald for a straght cylnder of crcular cross secton n a subcrtcal flow [16], [32]. The prncple can also be confrmed by the expermental work done by Pouln and Larsen [36]. Therefore t deems approprate to model the smplfed drag and lft forces on the cable segments of the system usng ths prncple. 45

61 Cross-Flow Prncple For the purpose of modellng the external forces actng on the cable segments due to wnd, each cable segment s consdered as a straght cylndrcal rod n a unform flow feld wth a velocty of V. Although the model set up constrans the moton of the cable n a 2D plane, ths wnd model predcts the external forces approprately. The rod wll be subjected to lft and drag forces due to the mpact of flud on ts surface. The lft forces consst of pressure lft and the drag forces consst of pressure and frcton forces actng on the surface. Fgure 3.6. Drag and Lft coeffcents of crcular cylnders, wres and cables [32] The cable s nclned at an angle of α aganst the drecton of flow - at subcrtcal Reynolds numbers Fgure 3.6 shows the expermental data and the curve ft to these data. From here on, ths drag/lft models wll be adapted nto the equatons. Assume a cylnder n a unform wnd flow wth velocty of V. Defnng the angle of attack α as the angle between the wnd velocty vector and the longtudnal axes of the cylnder as shown n Fgure 3.6, assumng 46

62 tangental pressure drag to be neglgble, and neglectng tp vortces, Hoerner states that the lft and drag coeffcents for ths confguraton of a slender cylnder n the unform subcrtcal wnd flow can be calculated as [32]: 3 3 sn cos 2 2 sn cos cos sn C C C C C CL CD, basc CD, basc D D, basc f D, basc f ( 3.32 ) Fgure 3.7. Drag model on a cable segment A cable segment used n ths work s shown n Fgure 3.7. For a more general case, assume the wnd s actng on the cylnder wth a speed of V w0 wth horzontal and vertcal velocty components of U and W as: 47

63 V w0 U W ( 3.33 ) Assumng the cylnder s small enough so that one can neglect the dfference n translatonal speeds of ts dfferent ponts, the wnd s velocty vector relatve to the segment, can be defned as: V w, rel U X W Z ( 3.34 ) Note that angle of attack α s defned as the angle between V w, rel wth the postve drecton of axes x as shown n Fgure 3.7. The relatve ar velocty causes two forces on the segment; a drag force and a lft force, namely F and F. These forces can be calculated as: D L 1 F dlv C 2 1 F dlv C 2 2 D w, rel D 2 L w, rel L ( 3.35 ) Therefore substtutng the drag and lft coeffcents derved by Hoerner,.e. Equaton ( 3.32 ), namely C andc, and reorderng and smplfyng the equatons yelds D L the forces exerted by the wnd on the cable segment n the body frame as: dlc fvx, rel Vx, rel Vz, rel Fx 2 Fz dlvz, rel C f Vx, rel Vz, rel CP V z, rel 2 ( 3.36 ) 48

64 where C p s called the pressure coeffcent for the cylnder and the relatve wnd velocty tangental and normal components n the segment local frame, Vx, rel and V, z rel can be determned by rotatng the relatve wnd velocty n the nertal frame,.e. Vw, rel as shown n Equaton ( 3.37 ) usng the rotaton matrx R defned n Equaton ( 3.3 ). Vx, rel U X 1 R V z, rel W Z ( 3.37 ) It s also worthwhle to note that the tangental and normal forces n each segment s local frame shown n Equaton Error! Reference source not found. can be transformed back and forth to the nertal frame usng the same rotaton matrx; that s: FX F x F R Z F ( 3.38 ) z 3.5. UAS Dynamcs The UAS s moton and dynamcs can affect the dynamc of the cable sgnfcantly, snce the nterface between the cable and the UAS s assumed to be an deal sphercal jont whch cancels moments about ts axs but can exert forces and be affected by cable tenson. As a result the equatons of moton for both the UAS and the cable are coupled and should be derved and solved smultaneously. In ths secton, the UAS s dynamc equatons wll be derved to be mplemented nto the system equatons whch are prevously derved n Equaton set ( 3.26 ) Implementaton of UAS Segment nto the Equatons To accomplsh modelng the cable consderng the vehcle s dynamcs, the UAS can be consdered as an addtonal cable segment wth the mass of the UAS lumped at 49

65 the end, whch wll form the n+1 th element of the multbody system. Ths element s n fact a vrtual element extendng from the end pont of the last cable segment (n th element),.e. the cable connecton pont to the center of mass of the UAS. It wll serve as another cable segment wth a dfferent mass and length as others as shown n Fgure 3.8. The mass for ths vrtual segment wll be the mass of the UAS and the length wll be the dstance between the connecton pont and the UAS center of mass. In addton, t s mportant to model the drag forces on ths vrtual element n complance wth the approprate drag forces modeled for the UAS. Ths approach s the best and smplest way of mplementng the UAS s equatons of motons as t s also performed n a smlar work for a towed aeral vehcle by Qusenberry [19]. Wth the assumpton that the connecton nterface s a torque cancellng jont, such as a smple sphercal jont that s the two segments apply only forces on the neghbor segments and are free to rotate under any moment, ths selecton ensures that mplementaton of ths element can be completed n consstence wth the prevous modelng approach. The only dfference of ths element wth the rest of the system s n the length and mass of the segment as well as addton of the UAS mass moment of nerta. Fgure 3.8. UAS Coordnate Setup Note that the length between the connecton pont of the cable and the UAS s center of mass s consdered as an extra segment 50

66 Moreover, to be able to defne the orentaton of the vehcle properly, a local reference frame xz q q needs to be defned wth the orgn postoned at the center of mass of the UAS, x q axes pontng towards the forward face of the UAS and z q axes pontng towards the top of the UAS as llustrated n Fgure 3.8. Usng ths confguraton, vectors n the UAS segment coordnate system,.e. xn 1zn 1 can be transformed to the xzcoordnate q q system by usng a transformaton matrx R as defned below: γ cos sn R sn cos ( 3.39 ) x x n1 q R ( 3.40 ) zn1 zq In ths system, the connecton pont s exactly below the center of mass, makng 90 degrees. Ths causes the UAS coordnates to become equal to: xq zn1 zq xn1 ( 3.41 ) Addton of ths segment wll modfy the dervaton of the Lagrange s equaton n the sense that t wll add another term to the system. As ths segment has moment of nerta about the orgn of xn 1zn 1 unlke the cable segments, t causes a rotatonal knetc energy to appear n the dervatons. Snce planar moton of the system are beng studed, one can assume that the moment of nerta of the UAS conssts of a moment of nerta about the y axes only as shown n Equaton ( 3.42 ). 51

67 I CG Iyy ( 3.42 ) Ths matrx represents the nerta matrx of the UAS about ts center of gravty, however, n the calculaton of knetc energy, the nertas should be calculated about the nstantaneous center of rotaton,.e. the connecton pont whch s also the orgn of the n+1 th segment. Ths requres the nerta matrx to be calculated about ths pont usng parallel axs theorem [37], [38]. Then, nerta about the orgn of the UAS segment coordnate system wll be: I I m l ( 3.43 ) 2 yy yycg U n 1 where mu represents the UAS s mass and ln 1s the length of the n+1 th segment. Therefore the nerta matrx can be wrtten as: I 0 I yy mu ln ( 3.44 ) Then the knetc energy of the multbody system ncludng the UAS segment whch conssts of a translatonal and a rotaton term could be wrtten as: T T T ( 3.45 ) trans. rot. where the translatonal knetc energy s calculated n the same way as n Equaton Error! Reference source not found. wth the addton of a n+1 th segment and the rotatonal knetc energy could be wrtten as: 52

68 T n1 n1 T j T rot. rot. j j j j1 2 j1 1 ω I ω ( 3.46 ) Takng the dervatves of the new knetc energy of the system wth respect to the generalzed coordnate and velocty, and, respectvely, and takng a tme dervatve of the latter, one can mplement these terms nto Equaton ( 3.9 ) and establsh the Lagrange s Equatons as: d T trans. Ttrans. I yy ( 1,2,, 1) Q n dt ( 3.47 ) Therefore, the Lagrange s Equatons can be wrtten as: n1 k k j1 m l l n 2 k j j cos( j ) j sn( j ) I yy k k FZ m cos sn +, ( 1,2,, ) j jg l FX l j M n j ( 3.48 ) Ths confguraton mples that the equatons can be wrtten n a smlar form to Equaton ( 3.31 ) wth slght changes as follows: ( 1,2,, ) Aθ Bdag θ θ Q n ( 3.49 ) where A A dagi ( 3.50 ) 53

69 where I s an (n+1) 1 vector representng nertas of system segments to be referred to as the segment nerta vector hereafter. Ths notaton ensures that the term I s added to the correspondng dagonal elements of matrx A. yy Equaton ( 3.50 ) shows that addton of the UAS s segment to the system, affects the equatons only by one term, whch s I yy beng added to the terms on the dagonal of the matrx A. It should be noted that the generalzed forces for the last segment,.e. the UAS segment wll change as the forces on the UAS are calculated wth a dfferent coeffcent of drag and lft based on the UAS s geometry and materal whch wll be dscussed n Secton 3.8. For the purpose of ths chapter, however, the cable segments are modeled usng a lumped mass model allowng all moments of nertas to be zero except that of the last segment. Therefore the rotatonal component of the knetc energy of the system can be wrtten as: T rot., U 1 T ω Iω ( 3.51 ) 2 Snce the angular velocty of the system s also defned about the y axes only, wth x and z components equal to zero, Equaton ( 3.51 ) s expressed as: T 1 I ( 3.52 ) 2 2 rot., U yy n 1 Dervaton of ths rotatonal part of knetc energy, wll yeld the term I. Therefore, for the lumped mass model, one can wrte the th element of the segment nerta vector as: yy 0 =1,, n I ( I ( 3.53 ) yy ) UAS = n 1 54

70 Ths causes the nerta term to be added to the equaton of moton of the n+1 th element only. Ths wll conclude the dervaton of equatons of motons for a cable-body system consstng of a UAS and a cable consderng the followng assumptons: 1. The moton s planar. 2. The UAS dynamcs s smplfed and the UAS s consdered robust wthout a flght controller. 3. The cable s consdered nextensble. 4. The base staton s consdered statonary. 5. The cable s modeled as a dscrete lumped-mass model. In order to solve these equatons gven proper ntal condtons, t s mportant to employ a sutable numercal technque. Ths wll be dscussed n the followng sectons Numercal Method for Solvng the System s Equaton of Moton Now that the equatons of the system s dynamcs are completely derved and defned n Equaton set ( 3.49 ), one can proceed to solvng these equatons usng the approprate ntal condtons. Snce the system s equatons are hghly coupled and nonlnear as well as large n terms of number of unknowns and equatons, t s necessary to employ an approprate numercal method to solve the system s dynamc equatons. As a result, the Runge-Kutta scheme s chosen to solve the system s equatons usng MATLAB software package [39]. As for any numercal analyss method, Runge-Kutta ntegraton method needs to receve the system s equaton n a specfc arrangement. Secton explans how the equatons are manpulated to be solved usng the fourthorder Runge-Kutta solver (RK4). Note that the name fourth-order Runge-Kutta method does not concern the mathematcal order of the equatons; but rather the truncaton errors n the ntegraton routne, detals of whch are beyond the scope of ths work. More n-depth nformaton on detals of ths method s vastly avalable through lterature and partcularly n [40]. 55

71 Order Reducton of Dfferental Equatons form: The 4 th -order Runge-Kutta method can only drectly solve frst-order ODE s of the dy f ( x, y), y(0) y 0 dx ( 3.54 ) Therefore only frst order ODE s can be solved usng ths technque. The system at hand conssts of n+1 nonlnear second-order dfferental equatons n whch the unknowns are the angles of the segments,.e. as well as ther frst and second tmedervatves, and, respectvely. In order to avod the need to use the Taylor Expansons to solve these second-order equatons, system s order wll be reduced by convertng these n+1 second-order equatons to 2(n+1) frst order ODE s usng the followng procedure. The general form of the system s equaton for each segment s presented below: 2 2 d d A(, ) B(, ) f ( t,, ) (0) 2 0, (0) 0 dt dt ( 3.55 ) where, can be a functon of tme,, and. By assumng the followng vector, one can change the varables n order to acheve a frst-order ODE set wth the number of equatons twce as the orgnal number of equatons. The vector s called the angular poston vector and s represented by: 1 2 n 1 ( n1) 1 ( 3.56 ) The vector s called the angular velocty vector and s represented by: 56

72 ( 1) 1 n n ( 3.57 ) The vector s called the angular acceleraton vector and s represented by: ( 1) 1 n n ( 3.58 ) Now, defne a vector consstng of and, namely the angular state of the system as: ( 1) 1 n n n ( 3.59 ) Then, one can wrte the tme-dervatve of Equaton ( 3.59 ) as:

73 1 2 n1 1 2 n1 2( n1) 1 ( 3.60 ) By comparng these equatons to the system s equaton of moton derved n Equaton ( 3.49 ), and by multplyng the equaton by nverse of matrx ths equaton can be expressed as: A from the left, 1 A Bdag I ( n1) ( n1) 2( n1) 1 2( n1) 1 A 1 ( n1) 1 ( n1) ( n1) ( n1) ( n1) Q 2( n1) 2( n1) ( 3.61 ) where Q s the vector of generalzed forces shown n Equaton ( 3.27 ). Equaton ( 3.61 ) s a set of 2n+2 frst-order ODE s whch could be solved usng the Runge-Kutta numercal ntegraton routne. The codes wrtten to mplement these equaton nto a MATLAB program and to solve the problem can be found n Appendx A. These equatons are then solved numercally usng approprate ntal condtons to smulate dfferent scenaros. Usng the acceleratons obtaned from the numercal solutons, the forces on each cable segment are calculated n order to estmate the cable tenson at each node. Ths model s tested for correctness n Secton

74 3.7. Model Verfcaton In order to confdently use the smulatons and rely on the results, t s necessary to verfy the model aganst experment or avalable smlar results. As mentoned before, gven the correct parameters, and ntal condtons ths cable model s very versatle and can be appled to a wde range of applcatons, from subsea to aerospace and even n space applcatons as long as the cable s not experencng very low tensons and the cable s stffness s hgh enough for the cable extenson to be neglgble. To verfy the correctness of the dervatons and valdty of the model mplemented n MATLAB, several tests have been run the result of whch wll follow Smple Pendulum Test The model has been talored to smulate a smple pendulum and the oscllaton frequency of the pendulum s verfed aganst theoretcal calculatons. In theory the perod of an deal smple pendulum wth a massless rod of length L oscllatng wthout any wnd frcton s calculated by: T L 2 ( 3.62 ) g A seres of tests were run wth the same condtons to replcate varous pendulum lengths and the smulated results are compared to the calculated theoretcal values. Results are compared n Fgure

75 Fgure 3.9. Smple Pendulum Perods, Comparson of Smulated and Theoretcal Results. The plotted results show that the model s agreement wth theory n predctng the oscllaton perod for a smple pendulum. Ths approves that the equatons of moton for the system were derved correctly. However more tests are needed to prove the valdty of ths model n terms of dervng the equatons of moton for the general n-segment model Generalzaton of the Model to a Fnte-Segment Cable In prevous secton, t was confrmed that a smple pendulum usng the developed model can replcate the oscllaton of an deal pendulum. In ths set of tests, the model s taken further and the cable s modeled usng n rgd segments. Dfferent sets of UAS thrust and ptch are appled at the top end of the model and the cable shapes as well as UAS postons are compared to smlar smulated results usng the statc model developed n Chapter 2. Fgure 3.10 shows four dfferent force sets appled at the top end of the 60

76 cable, both smulated usng the catenary model used n Chapter 2 and the dscrete model developed n ths chapter. Note that the smulated models n ths case do not consder the weght and nerta of the UAS snce the purpose of these tests are merely valdaton of the cable shape regardless of the UAS weght. One can observe that the dscrete model s able to replcate the shape functon usng 31 segments. Even wth ths number of segments, there s an error of up to 8% whch s acceptable n most applcatons. In addton, the number of segments can affect the convergence and accuracy of the model, hence there s an error present due to the model approxmaton. Fgure Comparson of statc cable shapes; catenary model vs. dscrete model 61

77 A verfcaton of the forces n the tenson concludes the valdty of the model. The smulaton s run wth the parameters defned n Table 3.1 and are compared to the results of the same condtons usng the model used n Chapter 2. Fgure 3.11 shows dfferent stages of the cable when the UAS s takng off under the effect of UAS thrust and ptch shown n Table 3.1. The cable s ntally lyng on a horzontal lne when at tme 0, the forces are appled at ts top end movng the UAS upwards untl the cable reaches a steady state. The cable tenson calculated usng both models are then compared n Fgure Table 3.1. Forces appled at UAS for Valdaton Smulaton Parameter Value Unts UAS Thrust 180 [N] UAS Ptch 4 [deg] UAS Weght 0 [-] Fgure Cable shapes n the Take-Off process 62

78 The results show that the cable model can predct the tenson n the cable to a close approxmaton. However further from the base staton, the error ncreases. The reason behnd ths s manly the error ncluded n the end pont poston of the UAS. As you can see the fnal pont on Fgure 3.12 represents the UAS endpont and whle the tensons are equal, the horzontal poston of nodes are not the same. As a result, these dfference n the values occur. However n the same fgure, t can be seen that a model wth 40 segments behaves very closely to the catenary model. Ths concludes the equatons have been derved correctly. Fgure Comparson of longtudnal tenson n cable Model s Convergence Snce the developed model s a fnte-segment model, ts accuracy can be dependent on the number of segments used n the smulaton as t was observed n Fgure It s clear that modelng the cable wth 11 segments yeld less accurate results than the same model wth 40 segments. In order to acheve a better approxmaton of the system s behavor, t s necessary to understand the model s convergence. As a result, 63

79 the smulaton s run under same condtons of Table 3.1 for varous cable segment numbers and the results are presented n Fgure Fgure Convergence of UAS poston for varous number of cable segments Accordng to Fgure 3.13, one can see that the model convergence wth an error of less than 2% wth 80 cable segments. Obvously addng more cable segments wll ncrease the computatonal load for the solvng process. Fgure 3.14 shows how the duraton for solvng the system s dynamcs changes wth the number of segments for smulatng a 120 second flght. One can see that the tme needed for the algorthm to solve for the acceleratons as well as calculatng the postons along wth all other parameters changes sgnfcantly wth the number of segments; however, dscussng the computatonal complexty of the algorthm s beyond the scope of ths work. 64

80 Fgure Solvng duraton for dfferent number of cable segments Overall, based on the observatons dscussed above, one can conclude that selectng 80 segments for a cable of length 122 m could be an approprate opton n terms of accuracy as well as computatonal load. Note that as the cable length changes, the requred number of nodes change drectly as well;.e. for a longer cable, larger number of cable elements wll be needed for the same accuracy and vce versa Results and Dscussons The model developed here can be used to approxmate the cable shape and tensons n dfferent flght scenaros. Although, t was shown before that 80 segments wll provde an optmum soluton n terms of solvng tme and accuracy, n these case studes, lower number of segments are used n order to reduce the processng tme snce the 65

81 purpose of these results are llustraton of the behavor of the system rather than exact calculaton of varable values. Few case studes are nvestgated consderng the UAS s weght and moment of nerta as shown n Table 3.2 and the results are dscussed n the followng sectons. The UAS thrust s modeled as a vectored thrust always perpendcular to the UAS horzontal axs. The thrust then s a vector consstng of a horzontal and vertcal component n the nertal coordnate system whch both depend on the UAS ptch;.e. the angle wth respect to the horzontal axs X. It s assumed the UAS thrust s held exactly constant n magntude unless stated otherwse. In addton the normal and frctonal (tangental) drag coeffcents used for the cable are 1.5 and 0.02 respectvely as suggested n Hoerner s book [32]. Table 3.2. UAS Parameters Smulaton Parameter Value Unts UAS Weght [kg] UAS-Cable Connecton Lnk Length 0.1 [m] UAS Moment of Inerta (I xx) CG 1.22 [kg.m 2 ] UAS Moment of Inerta (I yy) CG 1.22 [kg.m 2 ] UAS Moment of Inerta (at connecton pont) (I yy) [kg.m 2 ] UAS Frame Rod Length [m] UAS Frame Rod Dameter [m] UAS Vertcal Thrust [N] Effect of Wnd Speed Ths secton s dedcated to llustratng effect of horzontal wnd speed on the cable shape and UAS locaton. The cable s postoned n a vertcal shape n a way that the UAS s hoverng exactly above the base staton. For each test, a certan wnd speed s appled to the system at t=1 sec and the fnal states of the system s llustrated after 150 seconds. The steady state cable shapes for dfferent wnd speeds rangng from 0 to 27 m/s are llustrated n Fgure

82 Fgure Fnal Cable Shapes at Dfferent Horzontal Wnd Speeds It could be observed from ths fgure that the UAS becomes unstable for wnd veloctes larger than 25 m/s. In order to keep the UAS stable, the UAS autoplot should be able to compensate the wnd force by algnng the thrust vector so that the horzontal component of the thrust can overcome the wnd force whle the vertcal thrust can mantan alttude of the UAS. Fgure 3.16 shows the UAS horzontal offset and heght as well as ts ptch versus tme for the same set of tests and wnd condtons. From these plots, one can conclude that gven ths amount of thrust, the UAS s not able to stablze tself as soon as the wnd speed ncreases to 26 m/s. 67

83 Fgure UAS poston and alttude tme hstory under dfferent wnd speeds Fgure Cable tenson and angle at UAS connecton pont 68

84 It could be also observed that the cable tensons at the UAS connecton pont along wth the cable angle n Fgure Also, for all the cases where the UAS s stable, that s wnd speeds lower than 25 m/s, the cable tenson reaches a constant value after stablzaton. It s observable that for hgher wnd speeds, the tenson at the UAS decreases sgnfcantly at 11 seconds. Ths means that the cable becomes a lttle slack. Ths causes a sudden change n the tenson whch can cause a phenomenon called snap loadng. Ths essentally means that can pull the UAS down suddenly whch s a hazard to the system. Another mportant parameter that should be consdered, s the tenson at the base staton. Ths helps better understand the forces and therefore torques actng on the wnch motor and the requrements for desgn. Fgure 3.18 shows how the cable tenson and the cable angle changes at the base staton when the UAS s flyng n a wndy condton. One can see that almost for all wnd cases, when the wnd starts beng appled nstantly, the cable tenson at base ncreases at once sgnfcantly almost by 600%. Fgure Cable tenson and angle at the wnch 69

85 Take-Off n Hgh-Velocty Wnd In ths case study, the UAS s behavor durng take-off s nvestgated when the UAS rses from a heght of zero n a hgh-velocty wndy condton. The effect of thrust on the maxmum acheved flyng alttude s nvestgated as a gude towards the UAS desgn. Not consderng the reelng process, and wth focus on thrust effects, the UAS s smulated to takes off from the ground n a wnd velocty of 28 m/s wth a thrust of mentoned n Table 3.2. The thrust then starts to rse at tme t=20 sec, lnearly wth tme and the UAS heght s observed. Fgure 3.19 shows the cable shapes durng the ncrease n the UAS thrust for dfferent wnd speeds. Fgure Cable shape durng take-off under the effect of ncreasng thrust The change n UAS poston and atttude at varous wnd speeds under the effect of ncreasng thrust are also llustrated n Fgure One can see that n all cases the UAS can acheve the heght of almost 122 meters as long as a large thrust s provded. Snce ths s not the case n realty, and snce generatng a larger thrust means a larger 70

86 and heaver UAS, one can conclude the lmtatons of the system and wnd operaton condton from ths analyss. Fgure 3.21 presents the UAS thrust versus tme along wth UAS alttude due to ths thrust. Based on ths graph, one can conclude that wth the assumpton the UAS can provde up to 400 N of thrust, at no wnd speed larger than 4 m/s can the UAS acheve ts target alttude of 122 meters. Fgure UAS poston and ptch under the effect of ncreasng thrust Fgure UAS alttude and the thrust change 71

87 Effect of UAS Thrust Fluctuatons Another usage of ths smulator s to learn more about the effects of thrust fluctuatons on the cable and UAS dynamcs. In ths example, the effect of thrust changes on the UAS s poston as well as cable tenson at both the UAS and the base staton s nvestgated. The thrust s modeled as a smple sne wave of the form: 2 Thrust Tm Tf sn t f ( 3.63 ) where the constant does not affect the results. The varable thrust parameters are defned n Table 3.3. Table 3.3. UAS Thrust wth fluctuatons Parameter Label Value Unts UAS Mean Thrust T m [N] Fluctuaton Ampltude T f 10 [N] Fluctuaton Frequency f 1 [Hz] Fgure Fgure 3.24 show the UAS poston and atttude as well as cable tensons at the base and at the UAS due to ths UAS thrust fluctuatons. It could be observed that the cable tensons vary wth the same frequency of 1 Hz at both the UAS and the wnch; however the thrust change does not propagate to the base wth the same value, but rather wth an ampltude of 4 N n comparson to 10 N. 72

88 Fgure UAS poston and atttude under thrust fluctuatons Fgure Cable tenson at UAS under thrust fluctuatons 73

89 Fgure Cable tenson at base under thrust fluctuatons 3.9. Concluson In ths chapter, a 2D model s developed n order to predct the planar moton of a flexble cable. The cable s consdered as a dscrete system of fnte rgd bodes connected wth frctonless jonts. The equatons of motons are derved usng Lagrange s Method for mult body systems. The equatons are then solved numercally gven proper ntal condtons. Wnd forces are also modeled usng a cable drag model adapted from lterature. The model s valdated aganst theory n terms of oscllaton perods as well as aganst results of Chapter 2 n terms of cable shape functon, longtudnal tenson, and model convergence due to number of elements. The results show close approxmaton and relablty of the dscrete cable model. The results show that the model s capable of smulatng the cable s shape functon wth a close approxmaton. Ths cable model s very versatle and can be appled to a wde range of applcatons, usng proper ntal condtons and system confguratons. 74

90 A few case studes are nvestgated and results are generated usng MATLAB software. The effect of UAS thrust on the maxmum acheved heght s studed. Moreover, effect of wnd speed on the cable shape, tenson and UAS poston and atttude are nvestgated and presented. In addton, effects of UAS thrust fluctuatons on these parameters are llustrated n another secton. In summary, the cable dynamcs can be smulated for a planar moton usng the 2D model developed and mplemented n ths chapter. Ths algorthm can be used to analyze a wde range of propertes of the system ncludng but not lmted to: effect of cable dameter, cable mass per length, wnd condtons, UAS sze and weght, etc. on the dynamc and steady state behavour of the system. It should be noted that ths model does not consder the effect of cable stran as well as bendng n the cable. A more thorough modelng could be carred out to mplement and nvestgate these parameters. In addton, a more n-depth modellng of the UAS could yeld n more accurate and relable results. 75

91 Chapter 4. Three- Dmensonal Mult-Body Dynamcs Modelng and Analyss of the Tether The results of Chapter 3, although valuable and mportant to consder n the prelmnary stages of an analyss, cannot be fully extended to every system to reflect a realstc representaton of the system s dynamc behavor. Ths s due to the fact that most real-world systems wll mostly experence a moton n all 3 drectons as opposed to the 2D planar assumpton of the prevous chapter. Therefore, those effects are nvestgated by modellng the whole system n a 3D space. Presented n followng sectons are the dervatons and applcaton of the system s equaton as well as the smulaton results for the cable system movng n the 3D space Methodology The concept behnd ths chapter s smlar to the one used n Chapter 3. The cable s dscretzed to a set of small rgd segments nstead of a contnuous non-rgd member. Each segment wll then be assgned an approprate coordnate system. Usng these ndvdual local coordnate systems, Lagrange s Equatons are employed for the whole system whch then concludes the equatons of moton for each segment. The system s equatons are then solved accordngly to capture the behavor of the cable under dfferent flght scenaros. Smlar to the prevous chapter, the effect of varable length dynamcs as well as cable extenson are not consdered Cable Model Cable Segment Model For smplcty and to avod complexty of the system of equatons, the segment model used n ths chapter s a lumped-mass model as t was used prevously n Chapter 3. However the factor complcatng equatons of moton for a 3D system s the fact that 76

92 any rgd body such as one cable segment n a 3D space must be defned by sx ndependent coordnates, meanng that each cable segment has 6 degrees of freedom as opposed to the planar case where each segment could be defned by two coordnates. Ths s due to the fact that, n order to defne the poston and orentaton of each segment, one should specfy the poston of the mass node wth three coordnates n addton to ts orentaton wth three angular coordnates. However, snce there s a length constrant on each node, only the orentaton of the segment would defne the poston of the mass node elmnatng the need for 6 degrees of freedom. Thus, the sze of the system s equatons wll be three tmes larger. Ths adds to the complexty of analytcal dervaton of the system s equatons as well as computatonal tme durng solvng the equatons of moton. In ths chapter, the effect of longtudnal stran s neglected snce the steel cable used n ths applcaton has relatvely hgh young s modulus. Should the cable n use have low stffness, such as a very elastc materal, t s defntely recommended to use the sprng mass model to model the stran effect. Ths wll complcate dervaton of equatons as well as computatonal load of the smulatons. Due to the scope of work, ths study s left for future work Coordnate System and Model Set-up An mportant key to the modelng of a rgd body n 3D space s the concept of Euler Angles. Ths s extensvely dscussed n lterature and text books, however the man reference for ths concept for the purpose of ths study s derved from the valuable Advanced Dynamc book by Greenwood [33]. Consder a rgd body n the 3D space, the center of mass of whch n the nertal reference frame XYZ s defned by (x c,y c,z c); To fully defne the locaton and orentaton of ths object 6 coordnates are needed, 3 of whch are the locaton of the center of mass. Now consder a local coordnate system xyz attached to the rgd body wth the orgn located at an arbtrary pont. The relatonshp between the orentaton of the local coordnate system xyz and the nertal coordnate XYZ s essentally what defnes the orentaton of the rgd body. Ths relatonshp, techncally a rotaton matrx, s what 77

93 determnes the Euler Angles and vce versa. Therefore, any vector n the xyz local coordnate system can be represented n the nertal coordnate system, XYZ by utlzng ths transformaton matrx. In essence, each transformaton conssts of three successve rotatons where each rotaton s done about the latest body axs. The type of Euler Angles used for ths chapter s called type I or Arcraft Euler Angles. The frst rotaton s about the Z axs an angle to acheve the new rotated coordnate system xyz. Then ths coordnate system s rotated about y axes by an angle of and coordnate system xyz wll result. At the end ths coordnate system s rotated about the x axs through an angle of and fnally the requred local coordnate system xyz s located. The rotaton matrces for each of the rotatons can be presented as [33]: cos sn 0 R sn cos 0 ( 4.1 ) cos 0 sn R ( 4.2 ) sn 0 cos R 0 cos sn ( 4.3 ) 0 sn cos Note that the whole rotaton from the nertal frame to the local body frame s actually a combnaton of these 3 successve rotatons and can be wrtten as: R RR R ( 4.4 ) 78

94 In ths work, the local frame should be rotated to the nertal coordnate system. Ths can be completed by usng the nverse of matrx R snce each of the three rotatons s ndependent of the other. Therefore, one can obtan any vector n the body frame by: X x 1 Y y R ( 4.5 ) Z z Matrx R s an orthogonal matrx, therefore ts nverse can be obtaned by transposng the matrx; and then t could be expressed as: cos cos ( sn cos cos sn sn ) sn sn cos sn cos 1 R sn cos cos cos sn sn sn cos sn sn sn cos ( 4.6 ) sn cos sn cos cos The reader s referred to [33] for more explanaton regardng Euler Angles. Error! Reference source not found. shows the nertal and local coordnate systems for the cable model n 3D space. Length of the th segment s labeled by l as before. For each segment, the angles,, and, namely roll (bank), headng, and ptch angles, are what defnes the local coordnate system wth respect to the nertal frame XYZ. One can now smply express each node s locaton n terms of the Euler Angles as: X x j 1 Y R y ( 1,2,3,, ) j j j j n ( 4.7 ) j1 Z z j where X, Y and Z are the coordnates of the th mass node. 79

95 Fgure 4.1. Global and local coordnate systems of the cable Snce each node can be located at n ts local coordnate by the vector (l,0,0),one can defne the poston of each node as: X lj 1 Y R 0 ( 1,2,3,, n) j j j ( 4.8 ) j1 Z 0 Therefore, the coordnates wll be defned as: X lj cos j cosj j1 Y l j sn j cos j ( 1,2,3,, n) j1 Z lj sn j j1 ( 4.9 ) 80

96 It can be observed that n ths coordnate confguraton, each node locaton wll be ndependent of the bank angle snce each node locaton s not dependant on torson of the segment due to the fact that a smple lumped-mass model s used and each node s consdered a partcle of mass m. Each node wll be fully defned by 4 coordnates X, Y, Z as well as the segment s bank angle, n general; however the system s constraned by the cable segment length. That s: l X X 1 Y Y 1 Z Z 1 ( 1,2,, n) ( 4.10 ) Presence of ths constrant reduces the number of system DOF by one for each segment. Therefore, n fact the system has a DOF of 3n. Havng set up the coordnate systems properly, equatons of moton for the system can now be developed usng Lagrange s Equatons Dervaton of Equatons of Moton Usng Lagrange s Equatons As dscussed before, dervaton of the equatons of moton usng Lagrange s Equatons, requres selectng, defnng and usng generalzed coordnates for the system. In ths case, the Euler Angles can be consdered as generalzed coordnates for the dscrete cable model snce Euler Angles are orthogonal and ndependent of each other and the confguraton descrbed here can fully defne the node postons for the system. Smlar to the prevous chapter, the generalzed coordnates, q are used n the dervatons for the 3D model. As used before, the Lagrangan of the system s shown by L and s defned as [23]: L TU ( 4.11 ) 81

97 Smlar to the dervaton of Lagrange s equatons for a mult-body system consstng of n rgd bodes n Chapter 3, the knetc and potental energes are derved as well as ther dervatves to substtute n the Lagrange s Equatons of the form: d T T Q ( 1,2,,3 n) dt q q ( 4.12 ) where q s are the generalzed coordnates used to defne the system and Q s are generalzed forces whch nclude appled forces and nertal forces derved from the potental energy. The knetc and potental energes of the system wll be derved n order to complete the dervaton of Lagrange s Equatons n the followng sectons Knetc and Potental Energy and Ther Dervatves expressed as: The squared velocty vector of each node on the cable wth a mass m can be T k k k k k k V V V X Y Z ( k 1,2,, n) ( 4.13 ) Also, takng dervatve of Equaton Error! Reference source not found. wth respect to tme, one can defne the velocty components for each node n terms of Euler Angles as stated n Equaton ( 4.14 ). k Xk lj j sn j cos j j cos j sn j j1 k Yk l j j cos j cos j j sn j sn j ( k 1,2,3,, n) j1 k Zk lj j cos j j1 ( 4.14 ) Therefore, by takng each component to the power of two one can wrte: 82

98 j sn sn j cos cos j k k 2 X k llj j cos cos j sn sn j 1 j1 2 j sn cos cos j sn j j cos cos j cos cos j k k 2 Yk llj j sn sn j sn sn j 1 j1 2 j cos cos sn jsn j k k 2 Zk ll j j cos cos j 1 j1,( k 1,2,3,, n) ( 4.15 ) Substtutng nto Equaton ( 4.13 ) to acheve the squared velocty vector n terms of the Euler Angles, one can wrte: 2 k k V l l sn sn cos cos cos cos cos cos k j j j j j j 1 j1 cos cos sn sn sn sn sn sn j j j j j 2 sn cos cos sn cos cos sn sn cos cos j j j j j j j ( 4.16 ) whch can further be reduced to: V 2 k k l l cos cos cos( ) k j j j j 1 j1 sn sn cos( ) cos cos j j j j 2 cos sn sn ( k 1, 2,, n) j j j ( 4.17 ) Now, usng the velocty vector derved for each mass, the knetc energy of the system can be represented as: 83

99 n k k 1 T mkll j j cos cos j cos( j ) 2 k 1 1 j1 sn sn cos( ) cos cos j j j j 2 cos sn sn j j j ( 4.18 ) In order to complete the dervaton of Lagrange s equatons, the dervatves of ths knetc energy wth respect to the Euler Angles and ther tme rates should be derved. 1 T 2 n k k k 1 1 j1 m l l k j j cos cos j cos( j ) j cos sn j sn j j cos j sn sn j j sn sn j cos( j ) cos cos j ( 4.19 ), yelds: Dfferentatng the knetc energy wth respect to the headng angle and ts rate T n k k j1 m l l k j j cos cos j sn( j ) j cos sn j cos j j sn sn j sn( j ) j cos j sn cos j,( 1,2,, n) ( 4.20 ) T n k k j1 m l l k j cos cos cos( ) cos sn sn,( 1, 2,, n) j j j j j j ( 4.21 ) 84

100 The next term needed n the Lagrange s Equatons s the tme-dervatve of Equaton ( 4.21 ). Takng dervatve of Equaton ( 4.21 ) wth respect to tme yelds: d T dt n k k j1 k j cos cos cos( ) cos sn sn( ) 2 2 cos cos sn( ) cos cos sn( ) cos cos sn( ) sn sn sn( ) j2cos sn j cos( j ) j cos sn j cos( j ) j j j j j j j j j j j j j j j j j j j m l l j sn cos j cos( j ),( 1,2,, n) ( 4.22 ) Followng the same logc as n secton for dervaton of generalzed forces, and assumng there s no stran or elastcty n the lnks, the only potental energy the system preserves s of gravtatonal form. Selectng the datum lne at whch U=0, to be on the ground,.e. Z=0, the generalzed forces for ths generalzed coordnates can be expressed as: n X j Yj Z j j Q FX F j Y F j Z m j jg M j j1 ( 4.23 ),( 1,2,, n) Based on Equaton ( 4.9 )Error! Reference source not found., dervng the rate of global coordnates wth respect to the headng angle yelds: X j l sn cos Yj l cos cos ( 1,2,3,, n), j Z j 0 ( 4.24 ) 85

101 Substtutng back nto Equaton ( 4.23 ), the generalzed forces n terms of the external forces and the segment mass could be wrtten as: n Q ( sn cos ) ( cos cos ),( 1,2,, ) FX l j FY l j M n ( 4.25 ) j Therefore Lagrange s Equatons for the headng angles as one set of the system s equatons, can be stated as follows. d T T dt n k k j1 m l l k j cos cos cos( ) cos sn sn( ) 2cos cos sn( ) 2cos cos sn( ) 2cos sn cos( ) j j j j j j j j j j j j n j j j j FX ( l sn cos ) ( cos cos j FY l j ) M,( 1,2,, n) j ( 4.26 ) Ths concludes the dervaton of the system s equaton of moton n terms of the headng angles of the segments. Followng the same path and dervng the equatons n terms of the ptch angles, dervatve of the knetc energy wth respect to and can be wrtten as: 86

102 T n k k j1 m l l k j sn cos cos( ) sn sn sn cos sn cos( ) sn cos cos cos sn j j j j j j j j j j j j j,( 1,2,, n) ( 4.27 ) T n k k j1 m l l k j sn sn cos cos cos sn cos sn( ),( 1,2,, n) j j j j j j j ( 4.28 ) Therefore, after takng the tme-dervatve of Equaton ( 4.28 ), t can be wrtten as: d T dt n k k j1 m l l k j sn cos sn( ) sn sn cos( ) cos cos 2sn cos cos( ) 2sn cos cos( ) cos sn sn cos cos( ) cos sn c 2sn sn sn( ) sn sn sn( ) cos cos sn( ),( 1,2,, ) j j j j j j j j j j j j j j j j j j j j j j j j j j os( ) sn cos j j j j j n ( 4.29 ) follows: Dervatves of the global coordnates n terms of the ptch angle can be wrtten as 87

103 X j l cos sn Yj l sn sn ( 1,2,3,, n), j Z j lcos ( 4.30 ) Then, these equatons are substtuted nto the followng equaton, to defne the generalzed forces for the ptch angle generalzed coordnates. n X j Yj Z j j Q FX F j Y F j Z m j jg Mj j1 ( 4.31 ),( 1,2,, n) Therefore, the generalzed forces are reduced to: Q n j F ( cos sn ) ( sn sn ) ( cos ) X l FY l FZ m jg l M j j j,( 1,2,, n) ( 4.32 ) Therefore Lagrange s Equatons for the ptch angles can be expressed as follows: 88

104 d T T dt sn cos sn( ) sn sn cos( ) cos cos 2 2 sn cos cos( ) sn cos cos( ) cos sn 2sn sn sn( ) X j n k k j1 m l l k j j j j j j j j j j j j j j j n j j j j j F ( lcos sn ) ( sn sn ) ( cos ) FY l FZ m jg l M j j,( 1,2,, n) ( 4.33 ) Equaton ( 4.33 ) shows the equatons of moton based on the ptch angle coordnate usng Lagrange s Equatons whch should be solved n conjuncton wth all other equaton sets derved. Please note that Equatons ( 4.26 ) and ( 4.33 ) are n fact two sets of equatons each wth n equatons n terms of the correspondng generalzed coordnates. Therefore these two equatons represent 2n equatons for the 2n-DOF system whch can be solved numercally. The last set of generalzed coordnates to consder for the Lagrange s Equatons s the bank or roll angle. Lookng at the equatons of knetc energy, t can be confrmed that both the translatonal knetc energy as well as the generalzed forces are ndependent of ths coordnate. Therefore, dervaton of Lagrange s Equatons based on the bank angles s nconclusve at ths pont. However, t wll be shown later n Secton that mplementaton of the rotatonal knetc energes wll add bank angle s equatons of moton to the system. To ths pont, all the equatons of moton for the cable consderng zero twst has been derved and should be solved numercally after substtutng the proper external forces nto the generalzed force equatons,.e. Equatons ( 4.25 ) and ( 4.32 ). 89

105 Assemblng Equatons n a Matrx Form All the 2n equatons of moton derved n the prevous secton should be solved n parallel to yeld the solutons for the system dynamcs. All the equatons should be assembled n an easy-to-use matrx form to be plugged nto a smlar algorthm developed n Chapter 3. However, n order to comply wth the equatons for the UAS dynamcs derved later, one needs to mplement these equatons nto a general 3n 3n matrx format. Based on Equatons ( 4.26 ) and ( 4.33 ), the system s equatons of moton can be assembled n the followng form: θ ψ Aη Bdagη η Cdag θ Q ψ ( 4.34 ) n whch, ψ and θ, and are vectors representng headng and ptch angles of all elements. Vector ηs also called atttude vector representng the Euler Angles n a vector of 3n 1 elements and s represented by: ψ η = θ ( 4.35 ) Therefore, we can wrte Equaton ( 3.26 ) as: A A A ψ B B B ψψ C C C θψ Q Q Q dag A21 A22 A23 θ B21 B22 B2 θ θ C21 C22 C23 θ A31 A32 A33 B31 B32 B33 C31 C32 C33 ψ ( 4.36 ) Although, snce there s no equaton n terms of the roll angle n ths case, and all terms of the Lagrange s Equatons are zero, the prevous equaton can be expressed as: 90

106 A A ψ 12 B11 B ψ ψ 12 C θψ Q dag 0 0 A21 A22 θ B21 B22 θ θ C21 θ Q ψ ( 4.37 ) where 0 represents a zero matrx of sze n n and sub-matrx components of matrx A are defned as: A11 Mjll, j cos cos j cos( ) j j A12 M, jll j cos sn j sn( ) j j A21 M, jll j sn cos j sn( j ) j A22 Mjll j sn sn j cos( j ) cos cos j, j (, j 1,2,, n) ( 4.38 ) and sub-matrces of matrx B are: B11 Mjll, j cos cos j sn( ) j j B12 M, jll j cos cos j sn( ) j j B21 M, jll j sn cos j cos( j ) j B22 Mjll j sn cos j cos( j ) cos sn j, j (, j 1,2,, n) ( 4.39 ) and fnally, matrxc conssts of two sub-matrces: C11 Mjll, j 2cos sn j cos( ) j j C21 Mjll j 2sn sn j sn( j ), j (, j 1,2,, n) ( 4.40 ) based on equatons of moton derved n ( 4.26 ) and ( 4.33 ). Matrx M s also called the mass dstrbuton matrx by Qusenberry [16] and s represented as: 91

107 n M j= mk (, j 1,2, n) ( 4.41 ) kmax, j By mplementng the generalzed forces n terms of external forces n Equaton ( 4.36 ), the equatons can be solved numercally usng approprate ntal condtons External Forces For the case of the 3D model, the cross flow prncple ntroduced by Hoerner [32] s used to calculate the external forces due to wnd. Equatons ( 4.25 ) and ( 4.32 ) suggest that generalzed force vectorsq andq requre the external forces and moments appled to the system to be derved and entered n terms of the Euler Angles n order to solve the equatons. It should be emphaszed once agan that the so-called generalzed forces are n fact moments n these equatons,.e. the dmenson s actually N.m and not N. Ths s due to the fact that the choce of generalzed coordnates are angles rather than lnear postons. Ths ensures that the equatons derved whch are energy-based are consstent n terms of dmenson. It s assumed that the flow feld s unform n each drecton and the wnd profle n a certan drecton s constant wth respect to the other 2 global coordnates; that s, wnd speed n the X drecton s constant n terms of Y and Z, and so on. However; the model can be smply extended to a varable profle wnd condton wth correct mplementaton of the wnd profle values n the solver algorthm developed here. Cross-Flow Prncple Accordng to Secton 3.4, one can extend the cross flow prncple to a 3D cable segment model. Thus havng the rod model llustrated n Secton 3.4, consder the wnd velocty s defned as the followng vector: 92

108 V w0 U V W ( 4.42 ) Therefore one can express the wnd velocty relatve to the segment s speed as: U X Vw, rel V Y W Z ( 4.43 ) Defnng the angle of attack α as the angle between the wnd velocty vector and the longtudnal axes of the cylnder as shown n Fgure 3.6, and assumng tangental pressure drag to be neglgble, and neglectng tp vortces, and employng Hoerner s cross flow prncple for a slender cylnder n the unform subcrtcal wnd flow [32], the forces exerted by the wnd on a cable segment can be expressed as: dlc fvx, rel Vx, rel Vy, rel Vz, rel 2 F x F y dlvy, rel C f Vx, rel Vy, rel Vz, rel CD Vy, rel V z, rel 2 F z dlvz, rel C f Vx, rel Vy, rel Vz, rel CD Vy, rel Vz, rel 2 ( 4.44 ) n whch the relatve wnd velocty normal components n the segment local frame, V x, rel V y, rel,and Vz, rel can be determned by transformng the relatve wnd velocty n the nertal frame to each body coordnate system usng Euler Angle transformaton matrx R defned n Equaton Error! Reference source not found.. 93

109 V x, rel U X Vy, rel R V Y ( 4.45 ) V W Z z, rel It s also worthwhle to note that the tangental and normal forces n each segment s local frame shown n Equaton Error! Reference source not found. can be transformed back and forth to the nertal frame usng the same rotaton matrx; that s: F X F x 1 FY R F y ( 4.46 ) F F Z z These forces wll be used n dervng the generalzed forces for the purpose of solvng the equatons of moton UAS Dynamcs It s mportant to consder the UAS dynamcs n the modellng of the system n order to capture the dynamc and statc behavor of the system wth a closer approxmaton. The same procedure used n Chapter 3 s followed n order to mplement the dynamcs of the UAS nto the equatons of moton Implementaton of UAS Segment nto the Equatons The UAS s consdered to be another extra segment added to the n-segments modeled before and derve the Lagrange s Equatons for ths n+1 th segment. Snce each segment s consdered to have 2 degrees of freedom, there are two equaton derved for the UAS whch are then added to the matrx equatons derved before. It s assumed ths partcular segment expands from the last mass node whch happens to be on the cable- UAS connecton pont to the center of gravty of the UAS so that t comples wth the 94

110 lumped mass model. In essence, the UAS, s reduced to a cable segment wth a mass equal to the UAS s and mass moment of nerta accordng to that of the UAS. A new axs s defned as the UAS s prncpal axs and s labeled as xq yqz q to be able to defne the orentaton of the vehcle wth respect to the nertal frame n terms of Euler Angles. The orgn of ths coordnate system s located at the center of mass of the UAS, x q axes pontng towards the forward face of the UAS and z q axes pontng towards the top of the UAS whle the y q axes pont towards port as shown n Fgure 4.2. In ths work, the cable/uas connecton s constraned not to rotate about ts longtudnal axes. That s, t can rotate about x q and y q axs but s not able to rotate freely about the z q axes. Thus, the UAS xz q q plane wll always be n the same plane as the vrtual segment plane xn 1zn 1. Usng ths confguraton, vectors n the UAS coordnate system, xq yqz q, can be transformed to the UAS vrtual segment coordnate system,.e. xn 1yn1zn1 through rotaton wth an angle by usng the followng rotaton matrx: cos 0 sn R ( 4.47 ) sn 0 cos In ths system, the connecton pont s exactly below the center of mass, whch makes the angle 90. Ths causes the UAS coordnates to become equal to: x z n1 q yn 1 yq z x n1 q ( 4.48 ) 95

111 Fgure 4.2. UAS 3D Coordnate Setup Note that the length between the connecton pont of the cable and the UAS s center of mass s consdered as an extra vrtual segment Consder that the UAS has the mass moment of nerta I CG and about ts prncpal axs x q y q z q. at ts center of gravty I I I I I I I I I I xx xy xz CG yx yy yz zx zy zz ( 4.49 ) n whch the off-dagonal elements I xy, I xz,and I zy are consdered to be very close to zero and therefore neglected due to symmetry of the UAS. Ths matrx of mass moment of 96