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

SLIDE 1

Gust Rejection Properties

f 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

SLIDE 2

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

SLIDE 3

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 To simplify the analysis a vertical plain birotor is modelled and analyzed:

1

rotor aerodynamic modelling

2

steady state analysis of rotor aerodynamic properties

3

equations of motion for planar birotor including tilt angle

4

vehicle trim conditions

5

dynamics analysis (stability & non-minimum phase behaviour

6

gust rejection properties T1 T2 W

Gust Rejection Properties of VTOL Multirotor Aircraft 3/17

SLIDE 4

Rotor Aerodynamic Modelling

Consider a rotor disc in a horizontal airflow Thrust in axial direction T = ρA(ΩR)2CT

where ρ is air density A is rotor disc area Ω is rotor angular velocity R is rotor radius CT is rotor thrust coefficient

Horizontal drag force H = ρA(ΩR)2CH CH is drag coefficient

T H α V V sin α vi V cos α

From momentum theory (flapping ignored) CT = 1

2σa

1

3θ

1 + 3

2µ2

− 1

2λ

where σ is solidity factor

a is lift slope of blade θ is rotor blade pitch angle µ is advance ratio λ is inflow ratio

Blade profile drag coefficient, CH, can be similarly calculated

Gust Rejection Properties of VTOL Multirotor Aircraft 4/17

SLIDE 5

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 CT 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

SLIDE 6

Thrust Mapping — Draganflyer X-Pro quadrotor

Thrust mapping for airspeeds V = {0, 5, 10}

5

200 5 10 90 150 15 60 20 30 25 30 100

30
60

50

90

Gust Rejection Properties of VTOL Multirotor Aircraft 6/17

SLIDE 7

Equations of Motion

x y ℓ T1 H1 T2 H2 W φ Γ Γ Vw

m¨ y = (T1+T2) c φ c Γ − (T1−T2) s φ s Γ − (H1+H2) s φ c Γ − (H1−H2) c φ s Γ − W m¨ x = −(T1+T2) s φ c Γ − (T1−T2) c φ s Γ − (H1+H2) c φ c Γ + (H1−H2) s φ s Γ I ¨ φ = (T1−T2)ℓ c Γ − (H1+H2)ℓ s Γ

Gust Rejection Properties of VTOL Multirotor Aircraft 7/17

SLIDE 8

Trim Analysis

Set ¨ x, ¨ y ¨ 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

2 4 6 8 10 12 14 16 18 20

60
50
40
30
20
10

120 160 200 240 280 320 360

Gust Rejection Properties of VTOL Multirotor Aircraft 8/17

SLIDE 9

Excluding Rotor Aerodynamics

Analyzing effect of rotor tilt angle Γ on linearized hover model in still air Control effected by thrusts Ti Small perturbation linear system transfer function matrix model with output y =

y

xT is given by GT =

cos Γ/(ms)

−(I sin Γs2 + mgℓ cos Γ)/(mIs3)

(1)

◮ Vertical and horizontal channels decoupled ◮ Pair of transmission zeros in the lateral position channel located at

s =

−mgℓ/(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

SLIDE 10

Including Rotor Aerodynamics

Dynamics linearized at hover in still air using second order finite difference method giving small perturbation model ˙ x=Ax + BΩuΩ + Bwuw x=

y

˙ y x ˙ x φ ˙ φ T A=         1 0 Yv 1 0 Xu −g Xω 1 0 Nu Nω         where Yv = ∂ ˙ v/∂v, Xu = ∂ ˙ u/∂u, Nω = ∂ ˙ ω/∂ω etc

15
10
5

5 10 15

7
6
5
4
3
2
1

1 2 3

Poles depend on Γ if aerodynamics are included

Gust Rejection Properties of VTOL Multirotor Aircraft 10/17

SLIDE 11

Eigenvalues analysis

Resulting characteristic equation Q(λ) = λ2 (λ − Yv)

λ3 − (Nw + Xu)λ2 + (NωXu − NuXω)λ + Nug
◮ Note high eigenvalue sensitivity when Γ is near zero.

◮ For Γ < 0, the negative values of the Nug term causes the instability ◮ For Γ ≃ 10.62◦ stability is also lost when Nug > (NuXω − NωXu)(Nω + Xu)

15
10
5

5 10 15

7
6
5
4
3
2
1

1 2 Gust Rejection Properties of VTOL Multirotor Aircraft 11/17

SLIDE 12

Sensitivity to Wind Disturbance

Recall ˙ x = Ax + BΩuΩ + Bwuw. Inspection of Bw shows the sensitivity of the pitch rotation rate to wind disturbance Vw Bw =

0 YVw

0 XVw 0 NVw T NVw has same sign as Γ

15
10
5

5 10 15

4
3
2
1

1 2 3

0.6
0.5
0.4
0.3
0.2
0.1

0.1

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

SLIDE 13

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) = −Kcx(t) is determined such that closed loop system ˙ x = [A − BKc]x(t) is stable with a gain Kc that minimizes J = ∞ (x(t)Qx(t) + u(t)Ru(t)) dt 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

SLIDE 14

Lateral position response to 5 m lateral position step demand

2 4 6 8 10 12

1

1 2 3 4 5 6

Gust Rejection Properties of VTOL Multirotor Aircraft 14/17

SLIDE 15

Lateral position response to 5 m/s horizontal wind speed step

2 4 6 8 10 12

10
5

5

Gust Rejection Properties of VTOL Multirotor Aircraft 15/17

SLIDE 16

Roll angle response to 5 m/s horizontal wind speed step

2 4 6 8 10 12

25
20
15
10
5

5 10

Gust Rejection Properties of VTOL Multirotor Aircraft 16/17

SLIDE 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