Gust Rejection Properties of VTOL Multirotor Aircraft James - PowerPoint PPT Presentation

Gust Rejection Properties of VTOL Multirotor Aircraft James Whidborne & Alastair K. Cooke Dynamics, Simulation and Control Group Centre for Aeronautics Cranfield University Friday December 8, 2017 Gust Rejection Properties of VTOL Multirotor Aircraft 1/17

Introduction Use & application of quadrotor and other multirotor unmanned aircraft proliferating Stable hovering a requirement — resistance to transient winds and gusts important particularly for operation in urban areas (U-space) Aerodynamic modelling & analysis of traditional rotor-craft is well-established; classic texts include Bramwell (1976); Seddon (1990); Leishman (2000) Most multirotor control ignores aerodynamic effects ◮ Most exceptions that include aerodynamic modelling based on these established methods (for example Pounds et al., 2004) ◮ Effects analyzed include ground effect, gust response, centre of gravity location ◮ Effect of rotor tilt has not been analyzed Gust Rejection Properties of VTOL Multirotor Aircraft 2/17

Approach Analysis of the rotor tilt is performed Usually quadrotor thrust assumed to act in parallel with body frame z -axis But rotor tilt affects stability and gust rejection properties T 2 T 1 To simplify the analysis a vertical plain birotor is modelled and analyzed: rotor aerodynamic modelling 1 steady state analysis of rotor 2 aerodynamic properties W equations of motion for planar birotor 3 including tilt angle vehicle trim conditions 4 dynamics analysis (stability & 5 non-minimum phase behaviour gust rejection properties 6 Gust Rejection Properties of VTOL Multirotor Aircraft 3/17

Rotor Aerodynamic Modelling T V cos α Consider a rotor disc in a H horizontal airflow V α Thrust in axial direction V sin α T = ρ A (Ω R ) 2 C T v i where ρ is air density From momentum theory (flapping ignored) A is rotor disc area � 1 Ω is rotor angular velocity C T = 1 1 + 3 2 µ 2 � − 1 � � 2 σ a 3 θ 2 λ R is rotor radius C T is rotor thrust coefficient where σ is solidity factor Horizontal drag force a is lift slope of blade θ is rotor blade pitch angle H = ρ A (Ω R ) 2 C H µ is advance ratio λ is inflow ratio C H is drag coefficient Blade profile drag coefficient, C H , can be similarly calculated Gust Rejection Properties of VTOL Multirotor Aircraft 4/17

Thrust and Drag Mappings Force and moment control is effected on the vehicle by adjusting the thrust of each rotor by varying Ω by means of a speed servocontroller To design controller and analyze dynamics, we need mappings from Ω to the pair ( T , H ) However, ( T , H ) is also dependent upon airspeed, V , and airspeed incidence angle, α Hence we require the mappings from triple (Ω , α, V ) , onto ( T , H ) Equations describing mapping (Ω , α, V ) �→ ( T , H ) are given (see paper for details) However, the equations for calculation of C T and hence T are implicit and an explicit solution appears intractable Hence calculation of the mapping to T requires a numerical method — MATLAB routine, ❢③❡r♦ Gust Rejection Properties of VTOL Multirotor Aircraft 5/17

Thrust Mapping — Draganflyer X-Pro quadrotor Thrust mapping for airspeeds V = { 0 , 5 , 10 } 30 25 20 15 10 5 0 -5 200 90 150 60 30 0 100 -30 -60 Gust Rejection Properties of VTOL Multirotor Aircraft 6/17 50 -90

Equations of Motion ℓ T 1 Γ y H 1 x φ T 2 Γ V w H 2 W m ¨ y = ( T 1 + T 2 ) c φ c Γ − ( T 1 − T 2 ) s φ s Γ − ( H 1 + H 2 ) s φ c Γ − ( H 1 − H 2 ) c φ s Γ − W m ¨ x = − ( T 1 + T 2 ) s φ c Γ − ( T 1 − T 2 ) c φ s Γ − ( H 1 + H 2 ) c φ c Γ + ( H 1 − H 2 ) s φ s Γ I ¨ φ = ( T 1 − T 2 ) ℓ c Γ − ( H 1 + H 2 ) ℓ s Γ Gust Rejection Properties of VTOL Multirotor Aircraft 7/17

Trim Analysis y ¨ Set ¨ x , ¨ φ = 0 to trim Implicit nature of the thrust mapping makes an explicit solution intractable Hence use MATLAB routine ❢♠✐♥s❡❛r❝❤ to minimize the residual 0 360 -10 320 -20 280 -30 240 -40 200 -50 160 -60 120 0 2 4 6 8 10 12 14 16 18 20 Gust Rejection Properties of VTOL Multirotor Aircraft 8/17

Excluding Rotor Aerodynamics Analyzing effect of rotor tilt angle Γ on linearized hover model in still air Control effected by thrusts T i Small perturbation linear system transfer function matrix model with output x � T is given by � y = y � � cos Γ / ( ms ) 0 G T = (1) − ( I sin Γ s 2 + mg ℓ cos Γ) / ( mIs 3 ) 0 ◮ Vertical and horizontal channels decoupled ◮ Pair of transmission zeros in the lateral position channel located at � s = − mg ℓ/ ( I tan Γ) ◮ If Γ is small, then zeros are high frequency — little impact on the closed loop system performance ◮ If Γ < 0, there is a nonminimum phase zero ◮ System poles independent of Γ — open-loop stability is invariant Gust Rejection Properties of VTOL Multirotor Aircraft 9/17

Including Rotor Aerodynamics Dynamics linearized at hover in 3 still air using second order 2 finite difference method giving 1 small perturbation model 0 x = Ax + B Ω u Ω + B w u w ˙ -1 -2 � T ˙ � x = ˙ ˙ y y x x φ φ -3   0 1 0 0 0 0 -4 0 Y v 0 0 0 0   -5   0 0 0 1 0 0   A = -6   0 0 0 X u − g X ω   -7   0 0 0 0 0 1 -15 -10 -5 0 5 10 15   0 0 0 N u 0 N ω Poles depend on Γ if aerodynamics are where Y v = ∂ ˙ v /∂ v , included X u = ∂ ˙ u /∂ u , N ω = ∂ ˙ ω/∂ω etc Gust Rejection Properties of VTOL Multirotor Aircraft 10/17

Eigenvalues analysis Resulting characteristic equation Q ( λ ) = λ 2 ( λ − Y v ) λ 3 − ( N w + X u ) λ 2 + ( N ω X u − N u X ω ) λ + N u g � � ◮ Note high eigenvalue sensitivity when Γ is near zero. ◮ For Γ < 0, the negative values of the N u g term causes the instability ◮ For Γ ≃ 10 . 62 ◦ stability is also lost when N u g > ( N u X ω − N ω X u )( N ω + X u ) 2 1 0 -1 -2 -3 -4 -5 -6 -7 -15 -10 -5 0 5 10 15 Gust Rejection Properties of VTOL Multirotor Aircraft 11/17

Sensitivity to Wind Disturbance 3 0.1 Recall 2 0 ˙ 1 -0.1 x = Ax + B Ω u Ω + B w u w . 0 -0.2 Inspection of B w shows the sensitivity of the pitch rotation -1 -0.3 rate to wind disturbance V w -2 -0.4 � T � B w = 0 Y V w 0 X V w 0 N V w -3 -0.5 -4 -0.6 N V w has same sign as Γ -15 -10 -5 0 5 10 15 Although Γ < 0 causes instability, it also provides moment causing tilt into wind when subjected to a lateral gust – beneficial for gust rejection Gust Rejection Properties of VTOL Multirotor Aircraft 12/17

LQR Controller LQR controller designed to investigate the gust rejection properties for various Γ LQR problem well known Given x ( t ) = Ax ( t ) + Bu ( t ) , ˙ a control input u ( t ) = − K c x ( t ) is determined such that closed loop system ˙ x = [ A − BK c ] x ( t ) is stable with a gain K c that minimizes � ∞ J = ( x ( t ) Q x ( t ) + u ( t ) R u ( t )) dt 0 where Q and R are weighting matrices Controller stabilizes the vehicle (zero rates) Nonlinear system simulations performed to evaluate response Gust Rejection Properties of VTOL Multirotor Aircraft 13/17

Lateral position response to 5 m lateral position step demand 6 5 4 3 2 1 0 -1 0 2 4 6 8 10 12 Gust Rejection Properties of VTOL Multirotor Aircraft 14/17

Lateral position response to 5 m/s horizontal wind speed step 5 0 -5 -10 0 2 4 6 8 10 12 Gust Rejection Properties of VTOL Multirotor Aircraft 15/17

Roll angle response to 5 m/s horizontal wind speed step 10 5 0 -5 -10 -15 -20 -25 0 2 4 6 8 10 12 Gust Rejection Properties of VTOL Multirotor Aircraft 16/17

Conclusions Effect of rotor tilt on stability: ◮ Positive tilt (inwards) results in increased open-loop static stability, although dynamic stability can be lost for large tilt ◮ Negative tilt (outwards) results in a loss of static stability and some non-minimum phase. For small tilt angles, the non-minimum phase behaviour is high frequency and appears to have little effect on control performance. Loss of static stability means some ‘hunting’ may result (further work) Gust rejection properties are dramatically improved with negative tilt — this introduces anhedral into the aircraft Gust disturbance rejection properties of anhedral in fixed wing aircraft are fairly well-known Body and other parasitic drag not included in model (further work) Combined effect of CoG and tilt (further work) Gust Rejection Properties of VTOL Multirotor Aircraft 17/17