Robot Dynamics Rotorcrafts: Dynamic Modeling of Rotorcraft & Control

Transcription

1 Robot Dynamics Rotorcrafts: Dynamic Modeling of Rotorcraft & Control V Marco Hutter, Roland Siegwart and Thomas Stastny Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 7..5

2 Contents Rotorcrafts (Rotary Wing UAV). Introduction Rotorcrafts. Dynamics Modeling & Control 3. Thrust Force of Propeller / Rotor 4. Rotor Craft Case Study 7..5 Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

3 Introduction Rotorcrafts Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

4 Rotorcraft: Definition Rotorcraft: Aircraft which produces lift from a rotary wing turning in a plane close to horizontal A helicopter is a collection of vibrations held together by differential equations John Watkinson Advantage Ability to hover Power efficiency during hover Disadvantage High maintenance costs Poor efficiency in forward flight If you are in trouble anywhere, an airplane can fly over and drop flowers, but a helicopter can land and save your life Igor Sikorsky Robot Dynamics: Rotary Wing UAS

5 Rotorcraft Overview on Types of Rotorcraft Helicopter Autogyro Gyrodyne Power driven main rotor The air flows from TOP to OTTOM Tilts its main rotor to fly forward Un-driven main rotor, tilted away The air flows from OTTOM to TOP Forward propeller for propulsion No tail rotor required Not capable of hovering Power driven main propeller The air flows from TOP to OTTOM Main propeller cannot tilt Additional propeller for propulsion Robot Dynamics: Rotary Wing UAS

6 Rotorcraft Rotor Configuration Single rotor Multi rotor Most efficient Mass constraint Need to balance counter-torque Reduced efficiency due to multiple rotors and downwash interference Able to lift more payload Even numbered rotors can balance countertorque 7..8 Robot Dynamics: Rotary Wing UAS 6

7 Rotorcraft at UAS-MAV Size Quadrotor / Multicopter Std. helicopter Four or more propellers in cross configuration Direct drive (no gearbox) Very good torque compensation High maneuverability Very agile Most efficient design Complex to control 7..8 Robot Dynamics: Rotary Wing UAS 7

8 Rotorcraft at UAS-MAV Size Ducted fan Coaxial Fix propeller Torques produced by control surfaces Heavy Compact Complex mechanics Passively stable Compact Suitable for miniaturization 7..8 Robot Dynamics: Rotary Wing UAS 8

9 Dynamics Modeling & Control Rotorcrafts Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

10 Modeling Introduction Two main reasons for dynamic modeling System analysis: the model allows evaluating the characteristics of the future aircraft in flight or its behavior in various conditions Stability Controllability Power consumption Control laws design and simulation: the model allows comparing various control techniques and tune their parameters Gain of time and money No risk of damage compared to real tests Modeling and simulation are important, but they must be validated in reality Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 7..5

11 Modeling a Quadrotor Assumptions Main modeling assumptions The CoG and the body frame origin are assumed to coincide Interaction with ground or other surfaces is neglected The structure is supposed rigid and symmetric Propeller is supposed to be rigid Fuselage drag is neglected Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 7..5

12 Modeling of Rotorcraft Overview The model of the rotorcraft consist of three main parts.. ω R,cmd ω R F, M v, ω Input to the system are commanded rotor speeds or the input voltage into the motor Motor dynamics block: Current rotor speeds Due to fast dynamics w.r.t. body dynamics. Not necessary for control design if bandwidth is taken into account Propeller aerodynamics: Aerodynamic forces and moments ody dynamics: Speed and rotational speed Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 7..5

13 Multi-body Dynamics General Formulation T T M b, g J F S Generalized coordinates M Mass matrix b, Centrifugal and Coriolis forces g Gravity forces act Actuation torque F External forces (end-effector, ground contact, propeller thrust,...) ex J ex S Jacobian of external forces Selection matrix of actuated joints ex ex act 7..5 Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 3

14 The Quadrotor Example F 4 4 F F 3 3 F M T T b, g J F S ex ex act 7..5 Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 4

15 The Quadrotor Example F 4 4 F F 3 x R 3 F y R z R x I y I T T M b, g J F S z I ex ex act 7..5 Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 5

16 Modeling a Quadrotor Properties Arm length l Planar distance between CoG and rotor plane Rotor height h Height between CoG and rotor plane Mass m Total lift of mass Inertia I Assume symmetry in xz- and yz-plane I I xx I yy I zz Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

17 Modeling a Quadrotor Simplifications Quadrotor shall only be considered in near hover condition Main forces and moments come from propellers Depend on flight regime (hover or translational flight) In hover: Hub forces and rolling moment small compared to thrust and drag moment Since hover is considered, thus thrust and drag are proportional to propellers squared rotational speed: T i =bω p,i b: thrust constant Q i =dω p,i d: drag constant Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

18 Modeling of Rotorcraft Coordinate System Coordinate systems E Earth fixed frame ody fixed frame The information required is the position and orientation of (Robot) relative to E for all time Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

19 Modeling of Rotorcraft Representing the Attitude Yaw Pitch 3 Roll cos ( z, ) sin sin cos C E cos C ( y, ) sin sin cos ( x, ) C cos sin sin cos Roll (-<<) Pitch (-/<</) Yaw (-<<) C E : Rotation matrix from to E 7..5 Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 9

20 Modeling of Rotorcraft Rotational Velocity Split angular velocity into the three basic rotations ω = ω roll + ω pitch + ω yaw Angular velocity due to change in roll angle ω roll = (,,) T Angular velocity due to change in pitch angle ω pitch = C T (x,ϕ) (,,) T Angular velocity due to change in yaw angle ω yaw = [C (y,θ) C (x, ϕ)] T (,,) T Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 7..5

21 Relation between Tait-ryan angles and angular velocities Linearized relation at hover 7..8 Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor Modeling of Rotorcraft Rotational Velocity r r E ω r r E ; cos cos sin cos sin cos sin ω r E, Singularity for = 9 o

22 Modeling of Rotorcraft ody Dynamics T T M b, g J F S Change of momentum and spin in the body frame me x v 3 3 ω m v F I ω ωi ω M Position in the inertial frame and the attitude E x C E v Forces and moments Aerodynamics and gravity F M E r F G M ex ex act Aero ω F Aero E 3x3 : Identity matrix Torque!! T FG C E mg Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 7..5 E

23 Modeling a Quadrotor Aerodynamic Forces Hover forces Thrust forces in the shaft direction Additional Forces: Can be neglected Hub forces along the horizontal speed T i = ρ/ac H (Ω i R prop ) v h = [u v ] T 4 F Aero i T H T 4 i H b i i p, i i v v i h h Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

24 Hover moments Thrust induced moment Drag torques Additional moments Inertial counter torques Propeller gyro effect Rolling moments Hub moments 7..5 Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 4 Modeling a Quadrotor Aerodynamic Moments 4 4 Pr Pr ) ( i h h P i H h h i i i P op G P op CT i H R I I v v p M v v R ω M M,, i i p i i i p i T b Q d Aero T i i i l T T M M Q l T T Q

25 Control of a Quadrotor Analysis of the Dynamics me x v 3 3 ω m v I ω ωi ω Rotational dynamics near hover I p qr( I I ) lb( ) xx yy zz p,4 p, I q rp( I I ) lb( ) yy zz xx p, p,3 Izzr d( p, p, p,3 p,4 ) p E rr q ; r r pi, Full control over all rotational speeds Not dependent on position states F M p Ixx p qr I zz I yy I q I q rpi I r Izz r pqiyy Ixx yy xx zz b : thrust constant d : drag constant l : distance of propeller from the CoG : rotational speed of propellor i Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

26 Control of a Quadrotor Analysis of the Dynamics me 3x3 v ω m v I ω ωi ω Translational dynamics in hover mu m( rv qw) sin mg mv m( pw ru) sincos mg mw m( qu pv) coscos mg b( ) u x C E v CE v w F p u mqw rv m v q m v mru pw M r w mpv qu Only direct control of the body z velocity Use attitude to control full position p, p, p,3 p,4 b : thrust constant : rotational speed of propellor i pi, sin T F C sincos mg mg coscos E G E Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

27 Control of a Quadrotor General Overview Goal: Move the Quadrotor in space Six degree of freedom Four motor speeds as system input Under-actuated system Not all states are independently controllable Fly maneuvers Control using full state feedback Assume all states are known Needs an additional state estimator Assumption Neglect motor dynamics Low speeds only dominate aerodynamic forces considered linearization around small attitude angles Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

28 Modeling a Quadrotor Control Overview Four propellers in cross configuration Two pairs (,3) and (,4), turning in opposite directions Vertical control by simultaneous change in rotor speed Directional control (Yaw) by rotor speed imbalance between the two rotor pairs Longitudinal control by converse change of rotor speeds and 3 Lateral control by converse change of rotor speeds and 4 F 3 F F F Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

29 Control of a Quadrotor Virtual Control Input Define virtual control inputs to simplify the equations Get decoupled and linear inputs Moments along each axis I xx p qr( Iyy Izz ) U I q rp( I I ) U yy zz xx I r zz U 4 Total thrust mu m( rv qw) sin mg mv m( pw ru) sincos mg mw m( qu pv) coscos mg U 3 U U U lb( ) p,4 p, lb( ) 3 p, p,3 d( ) 4 p, p, p,3 p,4 U b( p, p, p,3 p,4 ) Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

30 Control of a Quadrotor Hierarchical Control Rotational dynamics independent of translational dynamics Use a cascaded control structure Inner loop: Control of the attitude Input: U, U 3, U 4 Outer loop: Control of the position Input: U, ϕ, θ U U 3,, U 4 U p q r X Y Z u v w Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

31 Control of a Quadrotor Linearization of Attitude Dynamics Linearize attitude dynamic at hover Equilibrium point: p q r ; U U U ; U mg 3 4 I p qr( I I ) U xx yy zz I q pr( I I ) U yy zz xx I r E zz r U 4 p q r 3 U I xx U3 I yy U 4 I zz This model has no coupling We use 3 individual controller Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

32 Control of a Quadrotor PID Control PID control strategy Generic non model based control Consist of three terms P - Proportional I - Integral D - Derivative Different tuning methods Ziegler-Nichols method Pole placement Widely used r(t) e(t) C u(t) t ut () ket () k etdt () k Plant p i d det () dt y(t) Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

33 Control of a Quadrotor PID Attitude Control Consider roll subsystem U I xx P controller can t stabilize the system Harmonic oscillation I xx ( des ) Use a PD controller k p kp( des ) kd( des ) I xx Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

34 Control of a Quadrotor Attitude Control Design 3 separate control loops additional thrust input U Tdes U ( ) k k U ( ) k k 3 U ( ) k k 4 des proll droll des ppitch dpitch des pyaw dyaw Aspect on the implementation Get angle derivatives from transformation of the body angular rates Avoid integral element in controller d d d U U 3 U 4 real real real Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

35 Control of a Quadrotor Parameter Tuning k p is chosen to meet desired convergence time k d is chosen to get desired damping In theory any convergence time can be reached Actuator dynamics allow for a limited control signal bandwidth Closed loop system dynamics must respect the bandwidth of actuator dynamics Actuator signals shall not saturate the motors Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

36 Transformation of virtual control inputs to motor speeds Rewrite virtual control inputs in matrix form Invert matrix to get motor speeds Limit motor speeds to feasible (positive) values 7..8 Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor 36 Control of a Quadrotor Control Allocation 4 3,4,3,, U U U U d lb b d lb b d lb b d lb b p p p p,4,3,, 4 3 p p p p d d d d lb lb lb lb b b b b U U U U

37 Control of a Quadrotor Altitude Control A possible control flow for altitude control ϕ des,θ des,ψ des U, U 3,U 4 ω i,des z des T z U Control attitude and the altitude Attitude controller to control the angles Parallel control loop. Altitude controller to control the horizontal thrust and nonlinear transformation to calculate the corresponding thrust input Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

38 Control of a Quadrotor Altitude Control Rewrite translational dynamics in the inertial frame X V V C g E E m U Only look at the altitude dynamics Virtual Control PD controller for the height Transformation to the body frame z gcoscos U m z g TZ TZ cos cos U m T k ( z z) k z mg U z p des d T z cos cos Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

39 Control of a Quadrotor Position Control Possible control structure for full position control Ψ des x des y des z des T x T y T z U ϕ, θ U 3 ω i,des U 4 U Control position and yaw angle Hierarchical controller Outer loop: Position control Thrust vector control. Desired thrust vector in inertial frame Transform to thrust and roll and pitch angles Inner loop: Attitude control Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

40 Control of a Quadrotor Position Control Thrust vector control Assume that the control input is the thrust in the inertial frame Inertial position system dynamics x y z T T m T x y z g Use 3 separate PD controller for each direction Transform to desired thrust, roll and pitch Tx sin cos T T Tx Ty Tz C ( z, ) T sin E y T Tz cos cos Robot Dynamics - Rotary Wing UAS: Control of a Quadrotor

41 Thrust Force of Propeller / Rotor Rotorcrafts Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

42 Modeling of Rotorcraft Rotor/Propeller Aerodynamics Highly depends on the structure Propeller: Generation of forces and torques Thrust force T Hub force H (orthogonal to T) Drag torque Q Rolling torque R Rotor: Generation of forces and torques. Additional tilt dynamics of rotor disc Modeling of forces and moments similar to propeller Modeling the orientation of the TPP (Tip Path Plan) Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

43 Modeling a Quadrotor Propeller Aerodynamics Propeller in hover Thrust force T Aerodynamic force perpendicular to propeller plane T = ρ A P C T (ω P R P ) Drag torque Q Torque around rotor plane Q = ρ A P C Q (ω P R P ) R P C T and C Q are depending on lade pitch angle (propeller geometry) Reynolds number (propeller speed, velocity, rotational speed) A p Similar to blade element Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

44 Modeling a Quadrotor Propeller Aerodynamics Propeller in forward flight Additional forces due to force unbalance at position. and. Hub force H Opposite to horizontal flight direction V H H = ρ A P C H (ω P R P ) Rolling moment R Around flight direction R = ρ A P C R (ω P R P ) R P C H and C R are depending on the advance ratio μ μ= V ωprp V rel ω P Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

45 The Propeller Thrust Force Momentum Theory Analysis of an ideal propeller/rotor Power put into fluid to change its momentum downwards Actio-reactio: Thrust force at the propeller/rotor Assumptions Infinitely thin propeller/rotor disc area A R Thrust and velocity distribution is uniform over disc area One dimensional flow analysis Quasi-static airflow Flow properties do not change over time No viscous effects No profile drag Air is incompressible Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

46 Momentum Theory Recall of Fluid Dynamics Conservation of fluid mass () V nda Mass flow inside and outside control volume must be equal (quasistatic flow) Conservation of fluid momentum p nda VV nda F () The net force on the fluid is the change of momentum of the fluid Conservation of energy de V V nda P dt (3) : density of fluid V : flow speed at surface element n : suface normal unity vector da : patch of control surface area : surface pressure Work done on the fluid results in a gain of kinetic energy p E : energy P : power Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

47 Momentum Theory Derivation One-dimensional analysis Area of interest,, and 3 Area and 3 are on the far field with atmospheric pressure Conservation of mass V dav u da 3 V da V u dav da V u da 3 AV A V u A V u A V u R R 3 3 (4) u u (5) V, A, p V u, A, p R V u, A, p R No change of speed across rotor/propeller disc, but change in pressure V u3, A3, p Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

48 Momentum Theory Derivation Conservation of momentum Unconstrained flow: net pressure force FThrust V da V u3 da 3 is zero = AV A3V u3 = AR V uu3 (6) Conservation of energy 3 Thrust Thrust pnda P F V u V da V u da 3 3 AV A3V u3 AR V u V u3u3 (7) With eq. (4) and (5) With eq. (4) and (5) V, A, p V u, A, p R V u, A, p R V u3, A3, p Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

49 Momentum Theory Derivation 3 Comparison between momentum and energy Far wake slipstream velocity is twice the induced velocity u u (8) 3 Thrust force formula F A V u u (9) Thrust R With eq. (6) and (7) V, A, p Hover case (V = V = ) Thrust force F A u () Slipstream tube A Thrust R 3 A R ; A = () V u, A, p R V u, A, p R V u3, A3, p Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

50 Momentum Theory Power consumption in hover Ideal power used to produce thrust P FThrust V u () Hover case P F Thrust mg 3 3 Thrust F = () A A R mg (3) R With eq. () V, A, p V u, A, p R V u, A, p R Power depends on disc loading F Thrust / A R Decreasing disc loading reduces power Mechanical constraint: Tip Mach Number More profile/structural drag Longer tail V u3, A3, p Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

51 Propeller/Rotor Efficiency Defining the efficiency of a rotor Figure of Merit FM Ideal power required to hover FM Actual power required to hover Ideal power: Can be calculated using the momentum theory Actual power: Includes profile drag, blade-tip vortex, FM can be used to compare different propellers with the same disc loading Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

52 EMT (lade Elemental and Momentum Theory) Combined lade Elemental and Momentum Theory Used for axial flight analysis Divide rotor into different blade elements (dr) Use D blade element analysis Calculate forces for each element and sum them up R R ω R Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

53 EMT introduction Forces at each blade element V u ind θ R α ϕ dt dl V P V T = ω R r With the relative air flow V we can determine angle of attack and Reynolds number The corresponding lift and drag coefficient are found on polar curves for blade shape (see next slide) Problem: What is the induced velocity u ind? Calculation of angle of attack (AoA) depends on axial velocity V P and induced velocity u ind (influence of profile) Axial velocity V P =V P +u ind Can be calculated with the help of the Momentum Theory dd dq/r 7..5 Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 53

54 EMT Example of Lift and Drag Coefficient c L c L c D c D α x cord c M c L /c D α α 7..5 Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 54

55 EMT Example of Lift and Drag Coefficient c L c D 7..5 Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 55

56 EMT Example of Lift and Drag Coefficient c L c L /c D α α 7..5 Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling 56

57 dt dl V u ind θ R α ϕ EMT blade element Lift and drag at blade element Thrust and drag torque element N b : Number of blades V T : ω R r c : cord length Approximation (small angles) dl C cdrv L dd C cdrv L dt Nb( dl cos ddsin) dq Nb ( dlsin dd cos) r V V T dt N b dl V P V T = ω R r V ' P V T dd dq/r Describing the lift coefficient Assume a linear relationship with AoA (angle of attack) Thin airfoil theory (plate) Experimental results Linearization of polar Thrust at blade element C C ( ) L L CL C 5.7 L Linearize polar for Reynolds number at /3 R ' P dtbe Nb CL ( R ) cdrvt (4) VT Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling V α : zero lift angle of attack. Used for asymmetric profiles

58 EMT Momentum Theory V p is still unknown Use momentum theory at each blade annulus to estimate the induced velocity Equate thrust from blade element theory and momentum theory Solve equation for V p With solved V p calculate the thrust and drag torque with the blade element theory Integrate dt and dq over the whole blade dt V u u da V V V rdr ' ' mt ( P ind ) ind 4 P ( P P ) (5) dtmt dt be x r R R, ' VP R R R, VP V R R R, Nbc R C C ( ) ( ) x 8 8 and/or u ind L L V R T dt N ( dl cos ddsin) b Q dq Nb ( dlsin dd cos) r R With eq. (4) and (5) Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

59 Modeling of Rotorcraft Motor Dynamics Use DC motor model Consists of a mechanical and an electrical system Mechanical system Change of rotational speed depends on load Q and generated motor torque T m I m dω m /dt =T m -Q Generated torque due to electromagnetic field in coil U(t) i(t) ω m (t) Q(t) T m =k t i k t : Torque constant Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling

60 Modeling of Rotorcraft Motor Dynamics Electrical System Voltage balance Ldi/dt = U-Ri-U ind Induced voltage due to back electromagnetic field from the rotation of the static magnet i(t) ω m (t) Q(t) U ind =k t ω m Motor is a second order system Torque constant k t given by the manufacturer U(t) Electrical dynamics are usually much faster than the mechanical Approximate as first order system Robot Dynamics - Rotary Wing UAS: Propeller Analysis and Dynamic Modeling