Control Theory and Informatics                                                                           www.iiste.orgISSN...
Control Theory and Informatics                                                                                            ...
Control Theory and Informatics                                                                                            ...
Control Theory and Informatics                                                                                            ...
Control Theory and Informatics                                                                             www.iiste.orgIS...
Control Theory and Informatics                                                                                            ...
Control Theory and Informatics                                                                                            ...
Control Theory and Informatics                                                                      www.iiste.orgISSN 2224...
Control Theory and Informatics                                                                                www.iiste.or...
Control Theory and Informatics                                                                  www.iiste.orgISSN 2224-577...
Control Theory and Informatics                                                                                            ...
Control Theory and Informatics                                                        www.iiste.orgISSN 2224-5774 (print) ...
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Structural design and non linear modeling of a highly stable

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Structural design and non linear modeling of a highly stable

  1. 1. Control Theory and Informatics www.iiste.orgISSN 2224-5774 (print) ISSN 2225-0492 (online)Vol 2, No.4, 2012 Structural Design and Non-linear Modeling of a Highly Stable Multi-Rotor Hovercraft Ali Shahbaz Haider1* Muhammad Sajjad2 1. Department of Electrical Engineering, COMSATS Institute of Information Technology, PO Box 47040, Wah Cantt, Pakistan 2. Department of Electrical Engineering, University of Engineering and Technology, Taxila, Pakistan * E-mail of the corresponding author: engralishahbaz@gmail.comAbstractThis paper presents a new design for a Multi-rotor Unmanned Air Vehicle (UAV). The design is based on therequirement of highly stable hover capability other than typical requirements of vertical takeoff and landing (VTOL),forward and sidewise motions etc. Initially a typical Tri-rotor hovercraft is selected for modeling and analysis. Thendesign modifications are done to improve the hover stability, with special emphasis on compensating air dragmoments that exist at steady state hover. The modified structure is modeled and dynamic equations are derived for it.These equations are analyzed to verify that our structural modifications have the intended stability improvementeffect during steady state hover.Keywords: Multi-rotor Crafts, T-Copter, Rotational Matrix, Pseudo Inertial Matrix, Coriolis Acceleration, Air dragmoment, Swash Plate1. IntroductionThe term UAV is used interchangeably with terms Auto Pilots or Self Piloted air craft’s. UAVs are Unmanned AirVehicles that can perform autonomous maneuvers hence are auto pilot based. They have got applications in militaryperspective as well as in civilian purposes. They are used for various functions such as surveillance, helicopter pathinspection, disaster monitoring and land sliding observation, data and image acquisition of targets etc. (Salazar 2005,Jaimes 2008, Dong etc 2010 and Penghui 2010).This is the era of research and development of mini flying robot thatmay also act as a prototype for the controller design of bigger rotor crafts. Multi Rotor crafts have a lot of advantagesover air planes and helicopter; they do not have swash plate and do not need any runway. Changing the speeds ofmotors with respect to each other, results in changing the flight direction of Multi rotors. They employ fixed pitchpropellers, so are much cheaper and easy to control (Soumelidis 2008, Oosedo 2010 and Lee 2011). This researchpaper comprises of our work corresponding to tri rotor UAV Structure selection. There are various configurationspossible as shown in figure 1. We have chosen the Tee structure, known as T-Copter. It has its own merit ofsomewhat decoupled dynamics in six degree of freedom as compared to other possible structures; moreover it is achallenging system to be modeled and controlled (Salazar 2005). we would show that replacing the tail rotor servo,which is used in most of the existing T-copter designs (Salazar 2005, Jaimes 2008, Dong 2010 and Penghui 2010),with air drag counter moment assembly has resulted in a much more simple model and precise control on the hovermaking the yaw control decoupled and independent.2. T-Copter Dynamic ModelingStructure of a typical tri-rotor hovercraft is shown in figure 2. It has two main rotors namely A and B and one tailrotor C. Roll  , pitch  and yaw are defined in figure 3. In this typical configuration, tail rotor is at a tilt angle which is adjustable by some sort of servo mechanism (Salazar 2005). This is usually done to have a componentof force F3 in E1E2-plane to compensate for net air drag moment in hovercraft body as explained in sections to follow.But this configuration has a severe drawback that it results in an acceleration of the body even in steady state. So firstof all we would model structure of figure 2, then we will modify the structure so that we eliminate this effect,resulting in stable hover. 24
  2. 2. Control Theory and Informatics www.iiste.orgISSN 2224-5774 (print) ISSN 2225-0492 (online)Vol 2, No.4, 20122.1 Symbols and TermsVarious symbols and terms used in the structure T-Copter in figure 2 and in its modeling equations, as describedahead, are enlisted below. =Real Space,  =roll,  =Pitch,  =yaw, M=mass of each rotor,  =angular velocity 33 33m=mass of entire Hovercraft Structure,   =Pseudo inertial matrix, R  =Rotational matrix  E1 , E2 , E3  =Body Frame of Reference,   Ex , E y , Ez  =Inertial Frame of ReferenceC.G. =center of gravity of Hovercraft body ( x, y , z )  3 =position of C.G in  ( ,  , )  3 =Angular displacement vector of hovercraft body j =lengths of respective sides of hovercraft bodyF =force vector in  ,   =torque vector, Fi =Force vector of the ith rotor 3 3  3 =parasitic moment term in system dynamics  3 =parasitic acceleration term in system dynamics2.2 Newton-Euler’s equationsNewton-Euler’s equations govern the dynamics of general multi rotor hovercrafts (Atul 1996, Francis 1998 andBarrientos 2009). These equations describe the dynamics of body in inertial frame of reference. These equations areas follows, d 2 m  R33 F31  mg Ez  31 (b , i , rj ) dt 2 31 31 (1) d 2  33   31  31 (b , i , rj ) dt 2 31Where:    x y z  ,       and Ez   0 1 T T T 0In the following sub sections we explain various terms within these equations, develop these equations for typicalT-Copter. Then we would modify this structure so that resulting dynamic equations of modified structure are simpleand well controllable.2.2.1. Force Vector FForces on the structure in  frame are shown in figure 4 in E 2 E 3 -plane. So the force vector comes out to be,F  0 F3 sin  F1  F2  F3 cos   T2.2.2 Rotational MatrixForce vector F in Newton-Euler’s equations is described with respect to body frame of reference  . In order totranslate this vector in inertial frame of reference, a transformation matrix is needed, which is known as rotationalmatrix. Since body can perform three rotations with respect to inertial frame of reference, we have a possible of sixrotational matrices R , R , R , R , R , R .Out of the six of these possible transformation matrices; weselect the standard one namely R and denote it by R. This rotational matrix is given by, 25
  3. 3. Control Theory and Informatics www.iiste.orgISSN 2224-5774 (print) ISSN 2225-0492 (online)Vol 2, No.4, 2012  cos cos sin cos  sin    R  cos sin  sin   sin cos  sin sin  sin   cos cos  cos  sin    cos sin  sin   sin cos  sin sin  sin   cos cos   cos  cos   2.2.3 Parasitic forces vectorThis vector includes the effects of centripetal and centrifugal acceleration of the structure and propellers. It alsoincludes Coriolis acceleration terms.2.2.4 Pseudo Inertial MatrixPseudo inertial matrix is a diagonal matrix of form,  I roll 0 0   0  I pitch  0   0  0 I yaw   Iroll Is moment of inertia of T-Copter for rotation around E1 axis. Ipitch is moment of inertia of T-Copter for rotationaround E 2 axis. Iyaw is moment of inertia of T-Copter for rotation around E 3 axis. Our structure is made up ofcylindrical shells of radius “R”. For a cylindrical shell of figure 5 with M =mass of the shell, the principle moment ofinertia are given by, I xx  M ((3r 2  2h2 ) / 6)  M d 2 , I yy  M ((3r 2  2h2 ) / 6)  M d 2 , I zz  3r 22.2.3.1. Moment of Inertia of T StructureNow we derive the moment of inertia of the T structure of figure 6 during roll, pitch and yaw. The moment of inertia ofthe T structure during Roll i.e. rotation across E1 axis, as shown in figure 7.A, is denoted by IE1, T . Using the resultsof multi body dynamics, it is sum of moment of inertia of shell 1 around E1 axis IE1, S 1 and shell 2 around E1axis IE1, S 2 . I E1,T  I E1, S 1   I E1, S 2 If ms1 and ms2 are masses of shell1 and shell2 then we get the following expression using parallel axis theorem,  I E1,T  I E1, S 1  I E1, S 2   mS 1 R 2   m S2 R / 2  mS 2l2 / 3 2 2 The moment of inertia of the T structure during pitch i.e. rotation across E2 axis, as shown in figure 7.B, is denotedby I E 2,T .  3 3   I E 2,T  I E 2, S 1  I E 2, S 2   mS 1 R / 2  mS 1 (l3  l1 ) / (3(l1  l4 ))  mS 2 R  mS 2l4  2  2 2 The moment of inertia of the T structure during yaw i.e. rotation across E3 axis, as shown in figure 7.C, is denotedby I E 3,T .  3 3   I E 3,T  I E 3, S 1  I E 3, S 2   I E 3,T  mS 1 R / 2  mS 1 (l1  l3 ) / (3(l1  l3 ))  mS 2 R / 2  mS 2 l2 / 3  [mS 2 l4 ] 2 2 2 2 2.2.3.2. Moment of Inertia of MotorsMotors are assumed to be solid cylinder of length Imot , radius Rmot and masses mmot . During roll the situation isshown in figure 8.A. Moment of inertia of three motors during roll are given below, I Mot1, roll  mMot RMot 2 / 4  mMot lMot / 3, I Mot 2,roll  mMot RMot 2 / 4  mMot lMot / 3  [mMot l22 ] 2 2 I Mot 3,roll  mMot RMot 2 / 4  mMot lMot / 3  [mMot l23 ] 2 2 26
  4. 4. Control Theory and Informatics www.iiste.orgISSN 2224-5774 (print) ISSN 2225-0492 (online)Vol 2, No.4, 2012During pitch the situation is shown in figure 8.B. Moment of inertia of three motors during pitch are given below, I Mot 1, pitch  mMot RMot / 4  mMot lMot / 3  [mMot l3 ], I Mot 2, pitch  mMot RMot / 4  mMot lMot / 3  [mMot l4 ] 2 2 2 2 2 2 I Mot 3, pitch  mMot RMot / 4  mMot lMot / 3  [mMot l4 ] 2 2 2During yaw the situation is shown in figure 8.C. Moment of inertia of three motors during yaw are given below, I Mot 1, yaw  mMot RMot / 2  [mMot l3 ], I Mot 2, yaw  mMot RMot / 2  [mMot ], I Mot 2, yaw  mMot RMot / 2  [mMot 2 2 2 2 2 2   ]2.2.3.3. Moment of Inertia of the Control Unit RackControl unit rack holds the batteries, sensors and electronics boards etc. It is assumed to be in the form of aparallelepiped as shown in the figure 9. Its mass, including all parts placed in it, is mrack . During roll, pitch and yaw,moment of inertia of rack are given to be.I Rack , roll  m ( a  4c ) / 12, I Rack , pitch  m (b  4c ) / 12  m l Rack 2 2 Rack 2 2  2 Rack rack , I Rack , yaw  m ( a  b ) / 12  m l Rack 2 2  2 Rack rack 2.2.3.4. Diagonal Elements of Pseudo Inertial MatrixResults of multi body dynamics show that moment of inertia T-Copter during roll i.e. Iroll is sum of moment ofinertia of all parts of T-Copter during roll. Same is true for pitch and yaw moment of inertia, hence, 3 3 I roll  I E1,T   I Moti , roll  I Rack ,roll , I pitch  I E 2,T   I Moti , pitch  I Rack , pitch i 1 i 1 3 I yaw  I E 3,T   I Moti , yaw  I Rack , yaw i 12.2.4. Nominal Torque Vector Figure 10 shows forces produced by propellers P 1, P2 and P3 on T-Copter body. Since torque is product of force andmoment arm, so from figure 10, nominal torque vector comes out to be,    2 ( F2  F1 )  1 ( F1  F2 )  F3 cos   3 F3 sin   T 32.2.5. Parasitic Moment VectorParasitic moments’ vector contains all those moments that creep into the structure as a by-product of desirablemoments. They are inevitable. Brief introduction of these moments is given in following sub sections. Expressionfor given by,   g   d   f   ic , where, g =Gyroscopic moment,  d =Air Drag moment,  f =Frictional moment,  ic =Inertial Counter moment2.2.5.1. Gyroscopic MomentConsider figure 11 in which a rotating wheel is subjected to an angular acceleration d / dt . It results in a gyroscopicmoment orthogonal to both angular acceleration vector and angular velocity vector of rotating wheel. Expression forgyroscopic moment is given by, g  I   (d / dt )Gyroscopic moment is produced due to rotating propellers and rotors. If Ipc is moment is inertia of rotor and propellerthen gyroscopic moment vector, as denoted by M g , is given by, M g   M gp 0  I pc (1  2  3 )d / dt (1  2  3 )d / dt 0 T T  M gr  27
  5. 5. Control Theory and Informatics www.iiste.orgISSN 2224-5774 (print) ISSN 2225-0492 (online)Vol 2, No.4, 2012Here Mgr is gyroscopic moment resulting from roll rate and Mgr is gyroscopic moment resulting from pitch rate.2.2.5.2. Air Drag MomentThese moments are produces as propellers rotate and drag through air. They are typically taken to be proportional topropellers’ angular velocity squared, directed counter to direction of rotation of propellers. Denoting air drag momentsby Q , we get the net air drag moment of the T-Copter, denoted by M d , to be, M d  0 0 Q1  Q2  Q3  T2.2.5.3. Frictional MomentsFrictional moments are proportional to angular velocities. Frictional moment vector ( M f ), is given by, M f  k f  d / dt d / dt d / dt  TWhere k f =frictional constant2.2.5.4. Inertial Counter momentsInertial counter moment is proportional to rate of change of angular velocity of propellers. It acts only in yaw direction.Net inertial counter moment vector is given by, M ic  I pc 0 0 d1 / dt  d2 / dt  d3 / dt  TNow the final expression for is given by,   g   d   f   ic  (     )       1 2 3   0     0         k    I pc  (1  2  3 )   (2) 0    I pc  0    f        Q  Q  Q    1 2 3 1  2  3    0            2.2.6. Developing Newton-Euler’s EquationsFirst of all we develop Newton’s Equation, d  2 m 2  RF  mg E z  dt x  0  0  x   y  R    mg 0    2 dm 2 F3 sin  dt         y z    F1  F2  F3 cos     1       z Letting u z = F1  F2  F3 cos  and putting value of R vector, we get the results as, 28
  6. 6. Control Theory and Informatics www.iiste.orgISSN 2224-5774 (print) ISSN 2225-0492 (online)Vol 2, No.4, 2012 2 d xm  u z sin    sin  cos   F sin   dt 2 3 x 2 d ym  u z cos  sin    sin  sin  sin   cos  cos   F3 sin   (3) dt 2 y 2 d zm  u z cos  cos   mg   sin  sin  sin   cos  cos   F sin   dt 2 3 zNow we develop Euler’s Equation, d  2 d 2  d  2 d  2 I 2 I    2  I 1  I 1  2  I 1  I 1  2  I 1  I 1 dt dt dt dtPutting the values we get, 1     I 0    I roll  1 0 2 ( F2  F1 )   1  2d        0 I 0 ( F1  F2 )  3 F3 cos    I pitchdt 2      1   1      0    0 I      3 F3 sin    I yaw     Or,d 2 1   2 ( F2  F1 )  I roll 1 dt 2 Id 2 1 (4)    1 ( F1  F2 )  F cos   +I pitch 3 3 1 dt 2 Id 2 1    3 F3 sin    I yaw 1 dt 2 INear to stability i.e.   0;   0; d / dt  0; d / dt  0; d / dt  0 and neglecting transient disturbance terms,Equation 2 becomes,  0 0 Q1  Q2  Q3   0 0 Qnet  T TEquation set 3 becomes, d 2x d2y d 2zm  sin  F3 sin  , m  cos F3 sin  , m  uz  mg  cos F3 sin  (5) dt 2 dt 2 dt 2Equation set 4 becomes,d 2 1 d 2 1 d 2 1   (F  F ) ,   1( F1  F2 )  3 F3 cos   ,   F sin    I yawQnet 1dt 2 I  2 2 1  dt 2 I   dt 2 I  3 3  (6)Now it is evident from Equation set (5) and (6) that tail rotor tilt angle  ,that was introduced to compensate for Qnetin equation set (6) has made accelerations in x, y and z directions dependent upon  in a non-linear fashion and has 29
  7. 7. Control Theory and Informatics www.iiste.orgISSN 2224-5774 (print) ISSN 2225-0492 (online)Vol 2, No.4, 2012made the model much complicated to control even near to the steady state condition. Even at steady state, T-copterbody would accelerate in xy-plane, a fact evident from equation set (5). Hence results in failure to achieve stable hover.2.2.7. Design Modification for Stable HoverSince Qnet has comparatively very small magnitude, we can counter balance it by a small counter air drag couple bytwo small motors’ set Mt , 1 and Mt , 2 as shown in the figure 12, and turn  =0. Two propellers of counter air dragcouple motor assembly are chosen to be of opposite pitch so they rotate in opposite direction with equal speed. Thiswould balance out their gyroscopic moments and air drag moments all the time. Moment of inertia for this counter airdrag couple motor assembly can easily be calculated during roll, pitch and yaw by the procedure described above.Moment of inertia of this assembly can then be added in respective diagonal elements of Pseudo inertial matrix.2.2.8. Developing Newton-Euler’s Equations for Modified T-Copter StructureIf the distance between motors Mt , 1 and Mt , 2 is t in figure 12 then moment produced by them has themagnitude t Ft , which is counter to air drag moment. So Equation set (3) becomes, d2x d2y d 2zm  uz sin   x , m  u z cos  sin   y , m  u z cos  cos   mg  z (7) dt 2 dt 2 dt 2And equation set (4) becomesd 2 1 d 2 1 d 2 1   (F  F )     ( F  F )  3 F3  +   F   (8) I  2 2 1  I  1 1 2  I  t t  2  , 2 , 2 dt dt dtAn immense simplification in the structure dynamics is evident from equation sets (7) and (8) as compared to equationsets (3) and (4). Most of the nonlinearities of previous model are eliminated. Linear accelerations have been madedecoupled from yaw angle as clear from equation set (7).Now we observe the behaviour near to steady state i.e.  0;   0; d / dt  0; d / dt  0; d / dt  0Neglecting transient disturbance terms, equation set (7) becomes, 2 2 2 d x d y d zm  0, m  0, m  u z  mg (9) dt 2 dt 2 dt 2Equation set (8) becomes,d 2 1 d 2 1 d 2 1   2 ( F2  F1 ) ,    1 ( F1  F2 )  3 F3  ,    t Ft   Qnet (10)dt 2 I dt 2 I dt 2 ISo we see that our structural modification has resulted in an extremely stable model near to steady state whichis evident from equation sets (9), (10) as compared to equation set (5) and (6).5. ConclusionIn typical Tri-rotor Hovercrafts, air drag moments of propellers are compensated by tilting the tail rotor through anangle, by using some sort of servo mechanism. Such arrangement results in an unstable steady state hover and muchmore complicated system dynamics. If tail rotor servo mechanism is replace by a cheap counter air drag couple motors’assembly, we get immensely simple system dynamics and completely stable steady state hover. 30
  8. 8. Control Theory and Informatics www.iiste.orgISSN 2224-5774 (print) ISSN 2225-0492 (online)Vol 2, No.4, 2012ReferencesAtul Kelkar & Suresh Joshi (1996) Control of Nonlinear Multibody Flexible Space Structures, Lecture Notes inControl and Information Sciences, Springer-Verlag London Publications.Barrientos, A.; Colorado, J. (22-26 June 2009). Miniature quad-rotor dynamics modeling & guidance forvision-based target tracking control tasks. Advanced Robotics, 2009. ICAR 2009. International Conference on , vol.,no., pp.1-6Dong-Wan Yoo; Hyon-Dong Oh; Dae-Yeon Won; Min-Jea Tahk. (8-10 June 2010). Dynamic modeling and controlsystem design for Tri-Rotor UAV. Systems and Control in Aeronautics and Astronautics (ISSCAA), 2010 3rdInternational Symposium on , vol., no., pp.762-767Francis C.Moon (1998) Applied Dynamics With Application to Multibody Systems, Network Books, AWiley-Interscience publications.Jaimes, A.; Kota, S.; Gomez, J. (2-4 June 2008). An approach to surveillance an area using swarm of fixed wing andquad-rotor unmanned aerial vehicles UAV(s). System of Systems Engineering, 2008. SoSE 08. IEEE InternationalConference on , vol., no., pp.1-6Lee, Keun Uk; Yun, Young Hun; Chang, Wook; Park, Jin Bae; Choi, Yoon Ho. (26-29 Oct. 2011). Modeling andaltitude control of quad-rotor UAV. Control, Automation and Systems (ICCAS), 2011 11th International Conferenceon , vol., no., pp.1897-1902Oosedo, A.; Konno, A.; Matumoto, T.; Go, K.; Masuko, K.; Abiko, S.; Uchiyama, M. (21-22 Dec. 2010). Designand simulation of a quad rotor tail-sitter unmanned aerial vehicle. System Integration (SII), 2010 IEEE/SICEInternational Symposium on , vol., no., pp.254-259Penghui Fan; Xinhua Wang; Kai-Yuan Cai. (9-11 June 2010). Design and control of a tri-rotor aircraft. Control andAutomation (ICCA), 2010 8th IEEE International Conference on , vol., no., pp.1972-1977Salazar-Cruz, S.; Lozano, R. (18-22 April 2005). Stabilization and nonlinear control for a novel Tri-rotormini-aircraft," Robotics and Automation, 2005. ICRA 2005. Proceedings of the 2005 IEEE International Conferenceon , vol., no., pp. 2612- 2617Soumelidis, A.; Gaspar, P.; Regula, G.; Lantos, B. (25-27 June 2008). Control of an experimental mini quad-rotorUAV. Control and Automation, 2008 16th Mediterranean Conference on , vol., no., pp.1252-1257 31
  9. 9. Control Theory and Informatics www.iiste.orgISSN 2224-5774 (print) ISSN 2225-0492 (online)Vol 2, No.4, 2012 Figure 1. Possible Tri-rotor Craft Structures (structures’ topological names are indicated) E3 3 F3 C 2 E3 Tail Rotor F2 M E1 E2 A l3 l2 O C.G Main Rotor1 1 mg F1 M l2 B Ez l1 Main Rotor2  M Ey Ex Figure 2. T-Copter Structure and Parameters ve  ve   ve  E1 E2 E3 Figure 3. Roll, Pitch and yaw definitions 32
  10. 10. Control Theory and Informatics www.iiste.orgISSN 2224-5774 (print) ISSN 2225-0492 (online)Vol 2, No.4, 2012 Figure 4. Forces on structure in body frame Figure 5. A cylindrical shell E1 3 +ve Roll 3 shell1 +ve Pitch +ve Yaw E2 O(C.G) 4   1 2 shell 2 1 2 2 Figure 6. T structure made up of cylindrical Shells 33
  11. 11. Control Theory and Informatics www.iiste.orgISSN 2224-5774 (print) ISSN 2225-0492 (online)Vol 2, No.4, 2012 shell1   shell1 shell1  O(C.G)  O(C.G)  O(C.G)  shell 2 shell 2 shell 2 A. T structure during roll B. T structure during pitch C.T structure during yaw Figure 7. T structure during three angular rotations in space M1 Mot Mot RMot M1 Mot RMot RMot shell1 M1   M2 shell1 shell1 l  l2  l4 2 2  O(C.G)  O(C.G)  O(C.G) M2 M2 shell 2  shell 2 shell 2 M3 M3 M3 A. Motors during roll B. Motors during pitch C. Motors during yaw Figure 8. Motors during three angular rotations in space E3 E1 Yaw Roll O(C.G) mrack  Pitch E2 c b lrack a Figure 9. Control unit rack 34
  12. 12. Control Theory and Informatics www.iiste.orgISSN 2224-5774 (print) ISSN 2225-0492 (online)Vol 2, No.4, 2012 Figure 10. Force Diagram to Torque vector on T-Copter. Z,   F Gyroscopic   moment   Y, g  I    X F Figure 11. Gyroscopic moments Figure 12. Modified T-copter Structure 35
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