Aerial Robotics - Lecture Notes PDF

Aerial Robotics Prof. Stjepan Bogdan Prof. Matko Orsag Antun Ivanović, Marko Car, Lovro Marković Grading system  2 homeworks (as preparation for lab exercises) – 2 x 5 points  at least 2 points for each homework required to pass  2 lab exercises – 2 x 8 points...

Aerial Robotics Prof. Stjepan Bogdan Prof. Matko Orsag Antun Ivanović, Marko Car, Lovro Marković Grading system  2 homeworks (as preparation for lab exercises) – 2 x 5 points  at least 2 points for each homework required to pass  2 lab exercises – 2 x 8 points (report submission)  at least 4 points for each lab report required to pass  midterm exam (written) – 30 points  final exam (written) – 30 points  oral exam – 14 points  for students that fail midterm or final exam (or want better grade) Flying arena –  written exam - 60 points Campus  oral exam – 14 points Grades 2 – [51 - 61] 3 – [62 - 77] 4 – [78 - 90] 5 – [91 - 100] 2 Contents M. Orsag, C. Korpela, S. Bogdan, P. Oh Introduction Aerial Manipulation Mathematical models of UAVs Multirotor Aerodynamics and Actuation Sensors and Control Aerial Manipulator Kinematics Aerial Manipulator Dynamics Mission Planning 3 Introduction History of aerial systems (rotorcrafts) 425 BC – Archytas => mechanical bird (steam) 1483 – da Vinci => aerial screw 4  1843 – George Cayley => aerial coach 5 Introduction  1907 – Paul Cornu => helicopter “Like all novices we began with the helicopter but soon saw that it had no future and dropped it. The helicopter does with great labor only what the balloon does without labor….The helicopter is much easier to design than the airplane, but it is worthless when done.” Wilbur Wright, 1906 6 Introduction  1909 – Igor Sikorsky => unmanned helicopter (vibrations) 1912 – Boris Yurev => tail rotor 1916 – Elmer Sperry => gyroscope 7 Introduction Post-WWI design Cierva (1923) Bothezad (1922) Sikorsky (1939) Fa-61 (1936) 8 Introduction Modern era (unmanned helicopters) RQ-8A Fire Scout (2002) Schiebel CAMCOPTER (2005) 9 Introduction The physical interactions of aerial vehicles with the environment (manned operations) 10 Introduction The physical interactions of aerial vehicles with the environment (unmanned operations) 11 Introduction The physical interactions of aerial vehicles with the environment (unmanned operations) LARICS experiments https://www.youtube.com/watch?v=jnLDu9qTUho&t=28s https://www.youtube.com/watch?v=bXHTZHhiFVM https://www.youtube.com/watch?v=X8FP2Xlu0HI&t=14s https://www.youtube.com/watch?v=q1uAODQw9eU https://www.youtube.com/watch?v=jv089lESQ4I 12 Introduction Predictions on autonomy of UAVs (1985) ? actual status 13 Introduction The goal – fully autonomous behaviour Autorotation Rotor 5 failure 4 Turbulances Target 1 3 Tree 2 Static obstacle Birds Dynamic obstacle 14 Introduction Aerial robotics - applications Military Civilian Target and decoy Traffic and weather monitoring Surveillance and tracking Firefighting Battle activities Agriculture Logistics Search & rescue... Inspection & maintenance Research and development... 15 Introduction Mathematical models of UAVs 6 degrees of freedom (6 DOF) – x, y, z, roll, pitch, yaw UAV as a system: Nonlinear Multivariable Coupled Underactuated => Complex for control 16 Coordinate Systems and Transformations Position and orientation of the body frame with respect to the world frame Body frame {B} Origin of point B on the world frame World frame {W} Usual point where helicopter takes off 17 Mathematical models of UAVs  ? C = cos S = sin ? different frames! 18 Mathematical models of UAVs Mathematical model of a blimp xB yB zB gondola Won't move on it's own Neutral buoyancy xcg , ycg, zcg Science festival Zagreb, 28.05.2009. blimp’s center of Lift = Weight with no propulsion gravity in {B} frame 19 Mathematical models of UAVs  Equation of motion (in body frame) M ⋅ v b + D( vb ) ⋅ vb + g( ηb ) = τb external forces (stuff that you put on the blimp) mass times acceleration drag (resistance) gravity / rotational field T v b = vx vy vz ω x ω y ω z  fluid where the blimp would be ωx M M RB + M A - mass matrix (blimp’s body and added mass), = vx ωy D( vb ) - air friction matrix, ωz depends on the speed of motion vy vz g( ηb ) - gravitational and lift vector, τb - vector of forces and torques. Coriolis and centripetal forces neglected 20 Mathematical models of UAVs  - mass matrix of blimp’s body. v = rω M ⋅ v b + D( vb ) ⋅ vb + g( ηb ) = τb  m [ I ]3×3 −mS  xcg , ycg, zcg M RB =   mS Ib  a – major sami-axis  0 − zcg ycg  b – minor sami-axis   =S  zcg 0 − xcg   − ycg xcg 0  a  b for ellipsoid body one has 4  I xx − I xy − I xz   I xx 0 0 I= yy I= zz πρ ab 2 ( a 2 + b 2 )   0 15 =I b  I yx I yy =− I yz  I yy 0   8  − I zx − I zy I zz   0 0 I zz  I xx = πρ ab 4  15 21 Mathematical models of UAVs  Good to know but not by heart Body that accelerates/decelerates inside a fluid has to ‘remove’ M ⋅ v b + D( vb ) ⋅ vb + g( ηb ) = τb a volume of fluid – “added mass”. F =m ⋅ a ⇒ F =(m + madd ) ⋅ a a m M A = diag {a11 , a22 , a33 , a44 , a55 , a66 } - added mass. α0 a11 = ⋅m 2 − α0 2 (1 − e 2 )  1  1 + e   =α0  2 ln  1 − e  − e  β0 e3     a= 22 a= 33 ⋅m 2 − β0 1 1 − e2  1 + e  For ellipsoid body: β= − ln  a44 = 0 0  e2 2e3  1 − e  ( b 2 − a 2 ) (α 0 − β 0 ) 2 2 1 b 4 a55 = a66 = ⋅ ⋅m 1−   e= m= πρ ab 2 5 2 ( b2 − a 2 ) + ( b2 + a 2 ) ( β0 − α 0 ) a 3 22 Mathematical models of UAVs Matrix D – air friction M ⋅ v b + D( vb ) ⋅ vb + g( ηb ) = τb Due to symmetrical form of a blimp, D is diagonal matrix: { D= diag Dv v ⋅ vx , Dv v ⋅ v y , Dv v ⋅ vz , Dw w ⋅ wx , Dw w ⋅ wy , Dw w ⋅ wz x x y y z z x x y y z z } gap in the air tunnel for particular shapes ρ D = Ctr ⋅ S ⋅ density 2 S – intersection area of a body (perpendicular to the direction of movement) Sv v= b 2 ⋅ π x x Ctr = 0.1 - 0.4 for spherical forms Sv v = S w w = a ⋅ b ⋅ π z z z z 23 Mathematical models of UAVs Gravitational and lift vector M ⋅ v b + D( vb ) ⋅ vb + g( ηb ) = τb assumption: Flift Neutral buoyancy Lift = Weight with no propulsion xB cg = 0 0 zcg  three forces yB zcg zB  forces [ 0]3×1    xcg , ycg, zcg three torques mg cos θ sin φ ⋅ z g ( ηb ) =  cg   mg sin θ ⋅ zcg  blimp’s center of   gravity in {B} frame  − mg cos θ sin φ ⋅ z cg   Fg 24 Mathematical models of UAVs Configuration of propulsors – arbitrary number and positions M ⋅ v b + D( vb ) ⋅ vb + g( ηb ) = τb - example: 4 props attached to the blimp’s body circle out x in 𝛕𝛕𝑏𝑏 = 𝐹𝐹𝑥𝑥 𝐹𝐹𝑦𝑦 𝐹𝐹𝑧𝑧 𝑁𝑁𝑥𝑥 𝑁𝑁𝑦𝑦 𝑁𝑁𝑧𝑧 𝑇𝑇 F1 F2 F1 F1 F2 x y cg y z z cg x F3 cg F3 F4 𝛕𝛕𝑏𝑏 = 𝑓𝑓(𝐹𝐹1, 𝐹𝐹2, 𝐹𝐹3, 𝐹𝐹4) F3 F4 mapping forces into torques side view configuration mapping Γ 25 Mathematical models of UAVs top view Configuration of propulsors – an example with 3 propulsors M ⋅ v b + D( vb ) ⋅ vb + g( ηb ) = τb right 𝛕𝛕𝑏𝑏 = 𝐹𝐹𝑥𝑥 𝐹𝐹𝑦𝑦 𝐹𝐹𝑧𝑧 𝑁𝑁𝑥𝑥 𝑁𝑁𝑦𝑦 𝑁𝑁𝑧𝑧 𝑇𝑇 left 𝐹𝐹𝑥𝑥 𝐹𝐹𝐷𝐷 + 𝐹𝐹𝐿𝐿 cos 𝜑𝜑 𝐹𝐹𝑦𝑦 𝐹𝐹𝑟𝑟 𝐹𝐹 𝐹𝐹𝐷𝐷 + 𝐹𝐹𝐿𝐿 sin 𝜑𝜑 𝛕𝛕𝑏𝑏 = 𝑧𝑧 = 𝑁𝑁𝑥𝑥 −𝐹𝐹𝑟𝑟 h − 𝐹𝐹𝐷𝐷 − 𝐹𝐹𝐿𝐿 sin 𝜑𝜑 ⋅ 𝑑𝑑 𝑁𝑁𝑦𝑦 𝐹𝐹𝐷𝐷 + 𝐹𝐹𝐿𝐿 cos 𝜑𝜑 ⋅ 𝑑𝑑 𝑁𝑁𝑧𝑧 − 𝐹𝐹𝑟𝑟 ⋅ 𝑧𝑧 − 𝐹𝐹𝐷𝐷 − 𝐹𝐹𝐿𝐿 cos 𝜑𝜑 ⋅ 𝑑𝑑 tail p = rotator (rotation plane) h d Changeable rotational plane => φ fourth actuator 26 Mathematical models of UAVs  Equation of motion (in body frame) M ⋅ v b + D( vb ) ⋅ vb + g( ηb ) = τb vb T v b = vx vy vz ω x ω y ω z  vw = vb ωw = T ω b motion in world frame 27 Mathematical models of UAVs Simulation example - ellipsoidal blimp (2a = 1.74 [m], 2b = 0.76 [m]) with helium (ρ = 0.1786 [kg / m3]) - 3 propulsors (as in configuration example) Izz = Iyy = 0.5421 [kg m2 ], Ixx = 0.1737 [kg m2 ], Dvxx = 0.0162 [kgm / s2 ], Dvzz = Dwzz = 0.0371 [kgm / s2 ], z = 0.87 [m], d = 0.1 [m] 28 Mathematical models of UAVs Simulink model configuration mapping Γ blimp model 29 Mathematical models of UAVs Simulink model configuration mapping Γ 30 Mathematical models of UAVs  Simulink model Fx 1 Fy vb Fz 1 em Nx Tau Ny vbx Nz vby vbx x vbz vby 1 y dinamika_balona emu wbx s z vbz wby 1 wbx kinematika_balona emu roll 3 s wbz poz wby pitch em wbz roll yaw pitch roll emu yaw pitch 2 vw yaw blimp model 31 Mathematical models of UAVs Simulink model - FL = FD = 0.327 [N], Fr = 0 [N], φ = 0 [o] 32 Mathematical models of UAVs Simulink model - FL = FD = 0.0125 [N], Fr = 0.0125 [N], φ = 0 [o] [rad] [rad] - [rad] 33 Mathematical models of UAVs Configuration of propulsors – additional example with 3 propulsors exercise 𝛕𝛕𝑏𝑏 =? 34 Mathematical models of UAVs Mathematical model of a helicopter Yamaha R-50 Fusion 360 Smart BNF 35 Mathematical models of UAVs  swash plate 36 Mathematical models of UAVs Equations of a helicopter motion (rotations) y-z plane (roll) y x ϕ Swash plate torque (main rotor) z ϕ R  I ⋅ϕ  = τ uz β − τ µ x − τ Dx Fuz β angular acceleration (roll)  Friction torque y C.M. Fuz β Gyroscopic torque of the main  L rotor ϕ z 37 Mathematical models of UAVs  Tail rotor drag torque x-z plane (pitch) x y I ⋅ θ = τ uzα − τ µ y − τ tr − τ Dy + τ G θ z Swash plate torque θ R (main rotor) Friction Gravity  torque torque Fuz α  Ftr Gyroscopic torque of the main x rG C.M. r   rotor L G L2 θ z 38 Mathematical models of UAVs  x x ψ Tail rotor torque ψ y R ωR I ⋅ψ = τ tR − τ µ z − τ zr − τ Dz z y ωR Main rotor torque ψ x-y plane (yaw) Friction torque Gyroscopic torque of the tail L rotor 39 Mathematical models of UAVs Equations of a helicopter motion (translations)  Fzx  cos ϕ sin θ cosψ + sin ϕ sinψ  The main rotor thrust  F = R R R F=  cos ϕ sin θ sinψ − sin ϕ cosψ  ⋅ F  zy  z ,ψ y ,θ x ,ϕ    Fzz   cos ϕ cosθ  mx  Fzx − Fµ x = Friction in the direction of motion my  Fzy − Fµ y = mz = Fzz − Fµ z − G 40 Mathematical models of UAVs  x x Main rotor input U_R UR y y Swash plate input (x) Alpha alpha z z 4 inputs 6 outputs fi fi Swash plate input (y) Betha betha theta theta Tail rotor input U_r Ur psi psi Proces 41 Mathematical models of UAVs Coupling (+linearization) - roll I ⋅ϕ  = τ uz β − τ µ x − τ Dx ⇒ GRϕ , Gβϕ - pitch I ⋅ θ = τ uzα − τ µ y − τ tr − τ Dy + τ G ⇒ GRΘ , GrΘ , GαΘ - yaw I ⋅ψ = τ tR − τ µ z − τ zr − τ Dz ⇒ GRψ , Grψ mx = Fzx − Fµ x ⇒ GRx , Gβ x my =Fzy − Fµ y ⇒ GRy , Gα y mz = Fzz − Fµ z − mg ⇒ GRz 42 Mathematical models of UAVs  GRϕ ϕ + Small-scale helicopter UR Gβϕ - Most of the couplings can be neglected GRΘ θ GrΘ + α GαΘ G Rψ ψ + Grψ β GRx x + Gβ x GRy y + Ur Gαy z 43 GRz Mathematical models of UAVs  Main rotor Napon motora noseceg rotora UR [V] 15 10 X: 35.75 Y: 18.91 5 150 Z: 102.8 X: -0.3385 0 Y: 20.26 0 100 200 300 400 500 600 700 800 900 1000 Z: 103.2 100 TailNapon rotormotora repnog rotora Ur [V] X: 36.33 10 Y: 0.274 50 X: 11.18 Z: 102.7 X: 0.0764 Y: 1.515 5 Z: 101.8 Y: 0.09823 0 Z: 103.4 0 X: 12.58 0 100 200 300 400 500 600 700 800 900 1000 Y: 1.448 -50 α swash Otklon pom. lopatica - kretanje naprijed [stup] 30 Z: 29.61 10 20 30 10 20 0 0 10 0 -10 y [m] -10 -10 x [m] 0 100 200 300 400 500 600 700 800 900 1000 β swash Otklon pom. lopatica - kretanje bocno [stup] 10 Set of commands 0 vert. up=>left=>forward=>right=> backward=>vert. down -10 0 100 200 300 400 500 600 700 800 900 1000 vrijeme [s] 44 Mathematical models of UAVs Simulation model in Simulink 800 600 400 z [m] 200 0 -200 100 1 50 Operator control (students) 0 -10 0 -20 -30 y [m] -50 -40 x [m] 45 Mathematical models of UAVs  10 30 0 20 -10 10 x [m] α [st] -20 0 -30 -10 -40 -20 0 20 40 60 80 100 120 140 160 180 0 50 100 150 200 vrijeme [s] vrijeme [s] 100 30 20 50 10 y [m] β [st] 0 0 -10 -50 -20 0 20 40 60 80 100 120 140 160 180 0 50 100 150 200 vrijeme [s] vrijeme [s] 46 Mathematical models of UAVs Mathematical model of a multirotor The first quadcopter at FER (2006) (mechanical gyro) 47 Mathematical models of UAVs Reminder Position and orientation of the body frame with respect to the world frame Body frame {B} World frame {W} 48 Mathematical models of UAVs General case origin of the propulsor’s frame in CW and CCW rotation the body frame 49 Mathematical models of UAVs Translational motion Thrust produced by i-th propulsor Torque produced by propulsors Rotational motion Drag produced by i-th propulsor 50 Mathematical models of UAVs Configuration mapping Γ How to attach propulsors on a multirotor body so that all DOFs are independently controlled? - that translates into: For arbitrary value of there is no motion. This means that propulsors should be able to generate force (equal to mg) in any direction in the UAV body frame while, in the same time, keeping overall torque at 0 value. 51 Mathematical models of UAVs Quadrotor with planar configuration z3 z2 y3 y2 For Φi=0 x3 x2 =I z4 z1 y4 y1 x4 x1 } =I 52 Mathematical models of UAVs Quadrotor with planar configuration Should be satisfied for any value of ψ, θ and Φ. 0 0 = 𝑚𝑚𝑚𝑚 Satisfied only for θ = 0 and Φ = 0. => x and y cannot be controled independently! 53 Mathematical models of UAVs  Quadrotor with planar configuration 𝑙𝑙1 0 0 0 −𝑙𝑙3 0 0 0 0 0 0 0 0 𝑐𝑐𝐷𝐷𝐷 𝑐𝑐𝐷𝐷2 𝑐𝑐𝐷𝐷3 𝑐𝑐𝐷𝐷4 0 × 0 + 𝑙𝑙2 × 0 + 0 × 0 + −𝑙𝑙4 × 0 − 𝑐𝑐 0 + 𝑐𝑐 0 − 𝑐𝑐 0 + 𝑐𝑐 0 = 0 𝑇𝑇𝑇 𝑇𝑇2 𝑇𝑇3 𝑇𝑇4 0 𝑢𝑢1 0 𝑢𝑢2 0 𝑢𝑢3 0 𝑢𝑢4 𝑢𝑢1 𝑢𝑢2 𝑢𝑢3 𝑢𝑢4 0 𝑙𝑙 (𝑢𝑢2 − 𝑢𝑢4 ) 0 For li = l , cDi = cD and cTi = cT => 1 𝑙𝑙 (𝑢𝑢 − 𝑢𝑢 )3 = 0 Satisfied for any value of of 𝑐𝑐 1 3 𝑐𝑐𝐷𝐷 2 4 − 𝐷𝐷 𝑢𝑢 + 𝑢𝑢 + 𝑢𝑢 + 𝑢𝑢 0 𝑐𝑐𝑇𝑇 𝑐𝑐𝑇𝑇 All three orientations, ψ, θ and Φ can be controlled independently. 54 Mathematical models of UAVs  y Quadrotor with planar configuration 𝑓𝑓 1 x Rotational motion y y 𝑑𝑑𝜔𝜔𝑥𝑥 𝐼𝐼𝑥𝑥𝑥𝑥 = τ𝑥𝑥 + 𝐼𝐼𝑦𝑦𝑦𝑦 − 𝐼𝐼𝑧𝑧𝑧𝑧 𝜔𝜔𝑦𝑦 𝜔𝜔𝑧𝑧 𝑓𝑓 4 x 𝑑𝑑𝑑𝑑 x 𝑓𝑓 2 𝑑𝑑𝜔𝜔𝑦𝑦 friction 𝐼𝐼𝑦𝑦𝑦𝑦 = τ𝑦𝑦 + 𝐼𝐼𝑧𝑧𝑧𝑧 − 𝐼𝐼𝑥𝑥𝑥𝑥 𝜔𝜔𝑥𝑥 𝜔𝜔𝑧𝑧 assumed to be y 𝑑𝑑𝑑𝑑 negligible 𝑑𝑑𝜔𝜔𝑧𝑧 For Φi=0 𝐼𝐼𝑧𝑧𝑧𝑧 = τ𝑧𝑧 + 𝐼𝐼𝑥𝑥𝑥𝑥 − 𝐼𝐼𝑦𝑦𝑦𝑦 𝜔𝜔𝑥𝑥 𝜔𝜔𝑦𝑦 x 𝑑𝑑𝑑𝑑 𝑓𝑓 3 =I Translational motion 𝑚𝑚 𝑥𝑥̈ = 𝐅𝐅𝑅𝑅 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 𝐹𝐹𝑓𝑓𝑥𝑥 𝐅𝐅𝑅𝑅 = 𝑓𝑓 1 + 𝑓𝑓 2 + 𝑓𝑓 3 + 𝑓𝑓 4 m 𝑦𝑦̈ = 𝐅𝐅𝑅𝑅 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝐹𝐹𝑓𝑓𝑦𝑦 friction taken m 𝑧𝑧̈ = 𝐅𝐅𝑅𝑅 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑚𝑚𝑔𝑔 − 𝐹𝐹𝑓𝑓𝑧𝑧 into account 55 Mathematical models of UAVs General case rank(Γ) ≥ 6 => controllability over all 6 DOF is guaranteed 56 Mathematical models of UAVs  0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 Γ= 0 𝑙𝑙 0 − 𝑙𝑙 𝑐𝑐𝐷𝐷 𝑐𝑐𝐷𝐷 − 0 0 𝑙𝑙 − 𝑙𝑙 𝑙𝑙 0 − 𝑙𝑙 0 Γ = 𝑐𝑐𝑇𝑇 𝑐𝑐𝑇𝑇 𝑐𝑐𝐷𝐷 𝑐𝑐𝐷𝐷 𝑐𝑐𝐷𝐷 𝑐𝑐𝐷𝐷 In case of the propulsors 𝑐𝑐𝐷𝐷 𝑐𝑐𝐷𝐷 − − with a variable rotational plane 0 0 𝑙𝑙 − 𝑙𝑙 − 𝑐𝑐𝑇𝑇 𝑐𝑐𝑇𝑇 𝑐𝑐𝑇𝑇 𝑐𝑐𝑇𝑇 𝑐𝑐𝑇𝑇 𝑐𝑐𝑇𝑇 => elements of Γ are not constant 𝑐𝑐𝐷𝐷 𝑐𝑐𝐷𝐷 values 𝑙𝑙 − 𝑙𝑙 − 0 0 𝑐𝑐𝑇𝑇 𝑐𝑐𝑇𝑇 57 Mathematical models of UAVs Propulsor – generation of thrust and drag => L lift => D 1.4 1.2 2 F ≈ K ⋅Ω angle of 1 attack F[N] 0.8 drag 0.6 0.4 F=f(ω) 0.2 interpolacija 200 250 300 350 400 450 chord ω[rad/s] 58 Propulsion of UAVs Conservation of mass – momentum theory (basics) - the rotor is in vertical flight (speed Vc), - the rotor forces the air through a disk enclosed by its spinning blade, - the air vortex creates a funnel (control volume), - the thrust exerted on the air is evenly distributed across the entire disk, - an increase in air pressure Δp being ‘felt’ across the entire disk area, - the rotation of the air mass in the vortex is neglected, - the control volume is treated as a separate part of the surrounding air (all the air mass outside the volume is at rest) 59 Propulsion of UAVs  2 Qu = πR1 Vc Inflow (top of the funnel) Outflow Qi =π ( R1 − R2 ) Vc + π R2 (Vc + v2 ) 2 2 2 (bottom of the funnel) the conservation of mass principle Qi π R2 2v2 ≠ 0 ? Qu − = Not in accordance with the conservation of mass principle! 60 Propulsion of UAVs  Qi π R2 2v2 ≠ 0 Qu − = Additional air (side inflow) 𝐿𝐿𝑢𝑢 = 𝑉𝑉𝑐𝑐 𝜌𝜌 𝜋𝜋𝑅𝑅1 2 𝑉𝑉𝑐𝑐 + 𝜋𝜋𝑅𝑅2 2 𝑣𝑣2 ∆𝑡𝑡 Linear momentum – L=m·v mu 𝐿𝐿𝑖𝑖 = 𝜌𝜌 𝜋𝜋𝑅𝑅1 2 − 𝜋𝜋𝑅𝑅2 2 𝑉𝑉𝑐𝑐 2 ∆𝑡𝑡 + 𝜌𝜌𝜌𝜌𝑅𝑅2 2 𝑉𝑉𝑐𝑐 + 𝑣𝑣2 2 ∆𝑡𝑡 ∆L = Lu − Li Change in momentum There must be some force. Δ𝐿𝐿 Rotor thrust 𝐹𝐹 = = 𝜌𝜌𝜌𝜌𝑅𝑅2 2 𝑉𝑉𝑐𝑐 + 𝑣𝑣2 𝑣𝑣2 Δ𝑡𝑡 61 Propulsion of UAVs  Δ𝐿𝐿 𝐹𝐹 = = 𝜌𝜌𝜌𝜌𝑅𝑅2 2 𝑉𝑉𝑐𝑐 + 𝑣𝑣2 𝑣𝑣2 Δ𝑡𝑡 ? Assumption: airflow is incompressible throughout the control volume (funnel) 𝜌𝜌 𝑉𝑉𝑐𝑐 + 𝑣𝑣𝑖𝑖 𝑅𝑅2 𝜋𝜋 = 𝜌𝜌 𝑉𝑉𝑐𝑐 + 𝑣𝑣2 𝑅𝑅2 2 𝜋𝜋 ⇒ 𝑉𝑉𝑐𝑐 + 𝑣𝑣𝑖𝑖 𝑅𝑅2 = 𝑉𝑉𝑐𝑐 + 𝑣𝑣2 𝑅𝑅2 2 1 Bernoulli equation gives: vi = v 2 2 F 2 ρπ R (Vc + vi ) vi 2 (check the textbook) = Problems: a) measurement of vi b) how to design propeller 62 Propulsion of UAVs Blade element theory (basics) We consider: - infinitesimal rotor blade element Δr, displaced from the center of blade rotation for r, with the chord c, taken at a time instance t, - horizontal motion of the UAV by speed Vxy attacks the blade at angle β(t), - two distinct airflows affect the blade element: - horizontal, produced from rotor c spinning, Ω·r, - vertical, produced from within climb and induced speeds, Vc+vi. 63 Propulsion of UAVs  𝛼𝛼𝑒𝑒𝑒𝑒 = 𝛼𝛼𝑚𝑚𝑚𝑚𝑚 − Φ 𝑉𝑉𝑐𝑐 + 𝑣𝑣𝑖𝑖 sin Φ = Ω 𝑟𝑟 + 𝑉𝑉𝑥𝑥𝑥𝑥 sin(𝛽𝛽) VS Ω 𝑟𝑟 >> 𝑉𝑉𝑐𝑐 + 𝑣𝑣𝑖𝑖 ; Ω 𝑟𝑟 >> 𝑉𝑉𝑥𝑥𝑥𝑥 ⇒ sin Φ ≈ Φ, 𝑉𝑉𝑠𝑠 ≈ Ω 𝑟𝑟 𝑉𝑉𝑐𝑐 + 𝑣𝑣𝑖𝑖 ⇒Φ= Ω𝑟𝑟 64 Propulsion of UAVs  1 lift induced by 𝑑𝑑𝐿𝐿 = 𝜌𝜌 ⋅ 𝑉𝑉𝑠𝑠 2 𝐶𝐶𝐿𝐿 𝑑𝑑𝑑𝑑 area dS 2 𝑑𝑑𝑑𝑑 = 𝑐𝑐 𝑑𝑑𝑑𝑑 Φ lift coefficient 𝑉𝑉𝑐𝑐 + 𝑣𝑣𝑖𝑖 𝐶𝐶𝐿𝐿 = 2𝜋𝜋𝛼𝛼𝑒𝑒𝑒𝑒 = 2𝜋𝜋 𝛼𝛼𝑚𝑚𝑚𝑚𝑚 − , Ω𝑟𝑟 c 1 2 𝑉𝑉𝑐𝑐 + 𝑣𝑣𝑖𝑖 𝑑𝑑𝑑𝑑 = 𝜌𝜌𝜌𝜌𝜌 Ω𝑟𝑟 𝑐𝑐 𝛼𝛼𝑚𝑚𝑚𝑚𝑚 − 𝑑𝑑𝑑𝑑 2 Ω𝑟𝑟 65 Propulsion of UAVs  1 2 𝑉𝑉𝑐𝑐 + 𝑣𝑣𝑖𝑖 𝑑𝑑𝑑𝑑 = 𝜌𝜌𝜌𝜌𝜌 Ω𝑟𝑟 𝑐𝑐 𝛼𝛼𝑚𝑚𝑚𝑚𝑚 − 𝑑𝑑𝑑𝑑 2 Ω𝑟𝑟 100 How to design a propeller ? 90 αmeh and csirine Kompenzacija compensation i napadnog kuta 80 αmeh compensation Kompenzacija napadnog kuta Consider dL for r = 0.1 R and r = R. “flat” Bez propeller kompenzacije 70 Ω·0.1·R Ω·R 60 Omjer potiska dL 50 Ω rotor blade 40 30 20 10 1 2 𝑉𝑉𝑐𝑐 + 𝑣𝑣𝑖𝑖 𝑑𝑑𝑑𝑑 = 𝜌𝜌𝜌𝜌𝜌 Ω𝑟𝑟 𝑐𝑐(𝑟𝑟) 𝛼𝛼𝑚𝑚𝑚𝑚𝑚 (𝑟𝑟) − 𝑑𝑑𝑑𝑑 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 Ω𝑟𝑟 Udaljenost od sredista rotora x=r/R Distance from propeller center 66 Propulsion of UAVs Integration along the propeller gives 1 3 2 2 𝐿𝐿 = 𝜌𝜌𝜌𝜌𝜌𝑅𝑅 Ω 𝐶𝐶 Θ0 − 𝜆𝜆𝑖𝑖 − 𝜆𝜆𝑐𝑐 4 3 0.1 Θ0 - mechanical angle at ¾ R 0.08 an example for 2” propeller C – width of rotor at ¾ R 0.06 viR λi = ΩR 0.04 λi 0.02 Vc λc = ΩR 0 -0.02 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 λc 67 Propulsion of UAVs  Effects of vertical and horizontal motion Vxy = 0, Vc = 0 vertical motion 𝑉𝑉𝑐𝑐 + 𝑣𝑣𝑖𝑖 sin Φ = Ω 𝑟𝑟 + 𝑉𝑉𝑥𝑥𝑥𝑥 sin(𝛽𝛽) horizontal motion Vxy 𝛼𝛼𝑒𝑒𝑒𝑒 = 𝛼𝛼𝑚𝑚𝑚𝑚𝑚 − Φ = ΩR 𝐶𝐶𝐿𝐿 = 2𝜋𝜋𝛼𝛼𝑒𝑒𝑒𝑒 1 𝑑𝑑𝐿𝐿 = 𝜌𝜌 ⋅ 𝑉𝑉𝑠𝑠 2 𝐶𝐶𝐿𝐿 𝑑𝑑𝑑𝑑 68 2 Propulsion of UAVs Measurement results – 2” propeller with DC motor [V] Rotational speed Thrust i 5 vi Inducirana brzina v [m/s] λi = = 0.0802 for Vc=0 4 ΩR i 3 Interpolirana funkcija 2 Izmjerene vrijednosti 1 150 200 250 300 350 400 450 500 550 Brzina vrtnje [rad/s] 69 Propulsion of UAVs Aerial Manipulation Actuation DC motor back electromagnetic force voltage drop coils on brushes neglected Torque constant propeller drag Voltage constant neglected in 70 analysis friction J = JL + JR Actuation of UAVs DC motor static characteristics (steady state) electromagnetic static increase of load, constant ua No-load speed decrease of ua, constant load Stall torque 71 Actuation of UAVs increase of ua with propeller load reminder => Drag ui ~ ω2 72 Actuation of UAVs relation between DC motor power and torque 73 Actuation of UAVs What is the most efficient working point for a DC motor? we substitute ua = iaS·Ra and b·ω = KT· ia0 74 Actuation of UAVs An example: DC motor data sheet Pn = 80 [W], uan = 12 [V] = 8130 / (60/2π) = 851.37 [rad/s] Ke = 1/Ks · 1/(60/2π) = 0.0137 [V/(rad/s)] Find maximal efficiency, friction torque, nominal torque, nominal rotational speed, nominal power, maximal power. 75 Actuation of UAVs An example: EXAM Static characteristics determined from DC motor data sheet 76 Actuation of UAVs DC motor dynamics (transitions between steady states) 77 Actuation of UAVs Assumptions: - frictionless rotation (b = 0), - no load (TL = 0), - Laplace transformation ( ∫ => 1/s) 78 mechanical time constant Actuation of UAVs 0.63 (ω2 - ω1) 79 Actuation of UAVs Assumptions: - frictionless rotation (b = 0), - propeller as load ( TL ~ ω2 ), - Laplace transformation ( ∫ => 1/s) Nonlinear differential equation => linearization in working point 80 Actuation of UAVs Assumptions: Faster response and - frictionless rotation (b = 0), decreased sensitivity - propeller as load ( TL ~ ω2 ), as ω increases. - Laplace transformation ( ∫ => 1/s) 81 Actuation of UAVs How propeller as load impacts η? 82 Actuation of UAVs  BLDC motor i) higher efficiency, a three-phase winding ii) higher speed range, topology with star iii) longer operational lifetime, connection iv) higher torque to size ratio Electronic speed controller (ESC) 83 Actuation of UAVs Servo drive - programmable serial communication - pulse position modulation (ppm) 84 Actuation of UAVs 2-Stroke Internal Combustion Engine 85 Actuation of UAVs - highly nonlinear - difficult to control static characteristic dynamic characteristic 86 Actuation of UAVs UAV Control Situation Awareness Mission Planning Obstacle Avoidance Mode Selection Fault Tolerant / Target Tracking Control Mode Transitioning Yes Obstacle / Target Continue Flight Control System Emergency ? Identification No Mission UAV Obstacle / Target Diagnostics Detection Sensors Sensor Fusion 87 UAV control Hardware configuration On Board System Sensors Computer Airframe D-GPS Flight Control Processor M Motion Pak U 3-Axis R-1 Integrated Avionics System X Magnetometer  Control Actuators  Altitude Digital Sensor Wireless Modem Data Link  Ground Station   Hand Held Radio  Control Transmitter Ethernet Parallel Wireless Modem D-GPS Ground Unit Human Safety Pilot Mission Control Station 88 UAV control The main sensors speed IMU – inertial measurement unit - 3 accelerometers Signal processing - 3 gyros - Kalman filter position - 3 magnetomeetrs GPS – Geostationary Positioning System (USA) Galileo (EU) IMU position ‘drift’ ! (sensor movement was ±60 cm) 89 UAV control Linear and decoupled control (quadcopter) - small changes of roll and pitch - sin(x) ≈ x ; cos(x) ≈ 1 𝚪𝚪 ∗ 𝚪𝚪 𝐹𝐹𝑥𝑥 - small changes of rpms – Fthr ≈ K·Ω 𝑢𝑢1 𝐹𝐹𝑦𝑦 𝑢𝑢 2 𝐹𝐹 - cascade controller 𝐮𝐮 = 3 𝛕𝛕𝑏𝑏 = 𝑧𝑧 𝑢𝑢 𝑁𝑁𝑥𝑥 𝑁𝑁𝑦𝑦 𝑢𝑢4 𝑁𝑁𝑧𝑧 controller propulsors forces and output torques 90 UAV control Linear and decoupled control 𝑚𝑚 𝑥𝑥̈ = 𝐅𝐅𝑅𝑅 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 𝐹𝐹𝑓𝑓𝑥𝑥 Angle transfer function 𝑑𝑑𝜔𝜔𝑥𝑥 𝐼𝐼𝑥𝑥𝑥𝑥 = τ𝑥𝑥 + 𝐼𝐼𝑦𝑦𝑦𝑦 − 𝐼𝐼𝑧𝑧𝑧𝑧 𝜔𝜔𝑦𝑦 𝜔𝜔𝑧𝑧 𝑑𝑑𝑑𝑑 91 UAV control Attitude control loop 92 UAV control Attitude control – influence of arm motion 93 UAV control Position control loop 94 UAV control Simulation in Simulink angle position 95 UAV control