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A Novel Online Model-Based Wind Estimation Approach for Quadrotor

Micro Air Vehicles Using Low Cost MEMS IMUs

L.N.C. Sikkel, G.C.H.E. de Croon, C. De Wagter and Q.P. Chu

Abstract—This work extends the drag-force enhanced

quadrotor model by denoting the free stream air velocity as

the difference between the ground speed and the wind speed.

It is demonstrated that a relatively simple nonlinear observer

is capable of estimating the local wind components, provided

accelerometer and GPS-velocity measurements are available.

We perform a wind tunnel experiment at various wind speeds

using a quadrotor vehicle with a low-cost Inertial Measurement

Unit (IMU) and a motion tracking system to provide accurate

ground speed measurements. It is shown that the onboard

Extended Kalman Filter (EKF) accurately estimates the wind

components.

I. INTRODUCTION

Autonomous ﬂight of Micro Air Vehicles (MAVs) has

gained much attention in recent years. Increasing effort is

taken in designing novel control approaches for trajectory

tracking and path-following in unknown and increasingly

complex environments [1]. Trajectory-tracking is similar to

the path-following as both control strategies require the

vehicle to converge to a predeﬁned reference trajectory. How-

ever, the trajectory tracking problem adds a time reference

component to the trajectory itself, for which a time-optimal

solution should be found.

While ﬂying in strong winds the trajectory reference

may need to slow down or speed up accordingly not to

overshoot some reference position [1]. Wind is a predomi-

nantly (slowly) varying disturbance, make the time reference

tracking task potentially very difﬁcult if no wind information

is available to the system. For example, a predeﬁned velocity

reference may, in a wind ﬁeld, require a steadily increasing

bank angle. This might violate both the time and space

reference trajectory convergence requirements due to a loss

of altitude. Also, limited power capacity may require onboard

path generation to take advantage of the wind itself [2].

Typically, wind is estimated on a ﬁxed-wing aircraft by

measuring the difference between the airspeed coming from

a pitot-static system and the GPS-velocity. If a GPS-velocity

estimate is unavailable an Extended Kalman Filter (EKF) is

used to estimate the wind ﬁeld with respect to the aircraft

states and measured position [3]. The static port of an

airspeed measuring device is to be located in an area of

Micro Air Vehicle laboratory, Control and Simulation Division,

Faculty of Aerospace Engineering, Delft University of Technology

g.c.h.e.decroon@tudelft.nl. (c)2016 IEEE. Personal

use of this material is permitted. Permission from IEEE must be

obtained for all other users, including reprinting/ republishing this

material for advertising or promotional purposes, creating new collective

works for resale or redistribution to servers or lists, or reuse of any

copyrighted components of this work in other works. Original publication:

http://ieeexplore.ieee.org/document/7759336/

undisturbed air. The aerodynamic effects caused by the rotors

of a quadrotor and the interaction between the different

ﬂow ﬁelds around each motor will disrupt the local airﬂow.

Therefore, it is assumed that a pitot-static tube cannot be

used to determine the airspeed of such a vehicle.

Fig. 1: The quadrotor MAV is shown to operate in front

of the cross section of the open-jet windtunnel. The vehicle

uses its onboard accelerometers and position measurements

to estimate the wind velocity.

Even though atmospheric disturbance rejection with re-

spect to quadrotor MAVs is widely studied, there does not

exist much literature on truly estimating the wind itself.

Often simulation studies are performed showing the effec-

tiveness of novel control approaches ignoring the actual

estimation of the wind [1], [4], [5], [6]. Researchers in the

past have created aerodynamic models of the vehicle. This

allows them to analytically compute the total force acting

on the vehicle as a result of the free stream air velocity The

difference between the accelerometer measurements and the

total predicted speciﬁc force can directly be attributed to the

wind components [7], [3].

More recently, interest in improving the traditional quadro-

tor dynamic model to allow for accurate estimates of at-

titude and velocity provided useful insight. Traditionally,

accelerometers were used to predict the attitude of a MAV,

assuming that they would measure some static component

of thrust [8]. Paradoxically, the traditional model states

that these planar accelerometers shouldnt measure anything

while in ﬂight. Recently it was shown that those planar

accelerometers did indeed not measure a component of the

thrust force, but instead measured the so-called rotor drag.

This drag force appeared to be directly proportional to the

airspeed of the vehicle and a drag-force enhanced quadrotor

model was presented [9], [10], [11], [12], [8], [13]. The

airspeed is typically described as the difference between the

ground speed and the wind speed. Because the accelerometer

measurements are directly proportional to this difference an

analytical solution should exist from which one could derive

the wind components [7], [3], [14]. The unknown IMU bias

and GPS observation noise of real-time systems however

limit the use of such analytical deﬁnitions.

In [15] vehicle pose and wind component estimation

using a model-aided visual-inertial EKF is discussed. A

Visual Simultaneous Localisation and Mapping (VSLAM)

algorithm is shown to produce both unbiased position and

orientation measurements, which are then used to correct the

process model. Experimental results, generated by processing

IMU and motion tracking position measurements off-board,

infer that the estimator approximates the wind speed and

sensor bias quite accurately. The motion tracking system

served as a substitute for the vision system as these type

of sensors would add a large degree of complexity to the

system. Despite the accomplishments made in estimating the

wind, the applications of this method are limited as it requires

a complex vision system and is not applicable to unknown

highly-dynamic environments.

In this work the accelerometer measurements are inte-

grated with an exteroceptive GPS-velocity sensor, which

does not provide external pose information. This allows us

to reduce the order of the process model and consequentially

reduce the computational load. An EKF is employed to

estimate the wind speed components and sensor bias on an

embedded system. An observability analysis is added for

completeness providing a weak guarantee of the convergence

of the estimator. It is shown that the nonlinear observer is

capable of estimating the wind quite accurately. However, the

poor performance of the low-cost IMU in a highlydynamic

environment adds a signiﬁcant noise component to the esti-

mated values. To the best of our knowledge this is the ﬁrst

work in which the implementation of a GPS-velocity sensor

to determine the wind components on a real-time embedded

platform is investigated.

The paper is organised as follows: the augmentation of the

drag-enhanced quadrotor model with the wind components is

shown in Section II while the nonlinear observer is presented

in Section III. The observability of the observer is discussed

in Section IV. Finally, the results of conducted wind tunnel

test at various airspeeds are shown in Section V and some

concluding remarks are given in Section VI.

II. MODEL DESCRIPTION

The traditional quadrotor model relied on the assumption

that the most signiﬁcant forces acting upon the vehicle were

gravity (Fg) and the thrust (T) generated by the actuators,

see Fig. 2. It may be described by a system of nonlinear

equations:

˙

φ

˙

θ

˙

ψ

=

1 sin

φ

tan

θ

cos

φ

tan

θ

0 cos

φ

sin

φ

0sin

φ

cos

θ

cos

φ

cos

θ

p

q

r

(1)

˙u

˙v

˙w

=Rb

E

0

0

g

+

vr−wq

wp −ur

uq−vp

−

0

0

T/m

(2)

where Rb

Eis the rotation matrix from the inertial reference

frame Eto the body-ﬁxed reference frame B,[u v w]T∈

{B}and [pqr]T∈ {B}are the linear velocity and angular

velocity along the principal axes of the body-ﬁxed reference

frame. The gravitational acceleration is given by g, which is

assumed to point along the kEaxis.

Fig. 2: The quadrotor is assumed to be controlled by varying

the rotational velocity

ω

i,i=1,...,4 of each of its motors. A

body-ﬁxed reference frame is attached to the vehicles center-

of-gravity. The ibaxis is pointing forward while jbpoints

towards the right. The kbaxis complements the system.

The thrust force is considered to be the input to the

system and, given the choice of reference frame, is aligned

with the kbaxis (Fig. 2). This paradoxically implies that

accelerometers aligned with the iband jbaxes will always

measure zero [8]. However, many researchers have used

the measurements of the accelerometer triad to estimate the

vehicles attitude successfully, implying that other external

forces must be acting on the body. It was recently shown that

the so-called rotor drag had been neglected in the traditional

model. The rotor drag is a force proportional to the linear

velocity of the quadrotor and it prevents the vehicle from

accelerating indeﬁnitely [8].

An improved model was required and the drag-force

enhanced model was introduced [8], [13]. The vehicle is

assumed to consist of multiple independently rotating rotors

and the drag force is considered to be directly dependent

on the rotational velocity of the motors. During normal

operations the rotational velocity of the motors is quasi

constant, so a constant positive lumped rotor drag coefﬁcient

(

µ

) is introduced such that

˙ub

˙vb

˙wb

=Rb

E

0

0

g

+

vr−wq

wp −ur

uq−vp

−

0

0

T/m

−

(

µ

/m)u

(

µ

/m)v

0

(3)

in which the latter term describes the rotor drag force. The

accelerometers measure speciﬁc acceleration ab=1

m(FT−

Fg), i.e. the difference between the gravitational acceleration

and the linear acceleration of the vehicle. It is assumed that

FTis the total force acting on the vehicle. The Coriolis terms

may be neglected, such that the accelerometer measurements

are assumed equal to the components of the drag force along

the principal axes of the body-ﬁxed reference frame [8]

ab

x

ab

y=˙u+gsin

θ

+

β

x

˙v−gsin

φ

cos

θ

+

β

y≈−(

µ

/m)u+

β

x

−(

µ

/m)v+

β

y(4)

in which

β

idescribes an unknown but constant bias. The

lumped drag coefﬁcient is typically estimated [8], [13] or

found using test ﬂight data. The latter approach requires an

accurate estimate of both the speciﬁc acceleration measured

by the accelerometers and the linear velocity of the vehicle.

Most researchers typically use a motion tracking system to

this end.

Extending the model to also include the effects of wind

requires looking at the free stream velocity of the ambient

air with respect to the quadrotor [15]. Wind is described

as the motion of the surrounding air. When a vehicle is

submerged in this ﬂow ﬁeld it is being accelerated by the

air particles up to the point it has an equal velocity as the

air itself. The dragforce- enhanced model presented in (3)

may be augmented

˙vb

x

˙vb

y

˙vb

z

=Rb

E

0

0

g

+

vb

yr−vb

zq

vb

zp−vb

xr

vb

xq−vb

yp

−

0

0

T/m

−

µ

m

vb

x−Wb

x

vb

y−Wb

y

0

(5)

where Wb= [Wb

xWb

yWb

z]Tare the wind velocity compo-

nents along the axes of the body-ﬁxed reference frame. The

linear velocity components V= [vb

xvb

yvb

z]Tconsequently

describe the ground speed of the vehicle in the body-ﬁxed

reference frame. The accelerometer measurement model is

necessarily adopted to include the wind velocity components

as well

ab

x

ab

y=˙vb

x+gsin

θ

+

β

x

˙vb

y−gsin

φ

cos

θ

+

β

y≈ −

µ

mvb

x−Wx+

β

x

vb

y−Wy+

β

y(6)

for which the same conditions as for the drag-enhanced

quadrotor model hold.

III. NOVEL EKF DESIGN FOR ESTIMATING WIND

It is assumed that the drag force coefﬁcient is estimated

a-priori. The ground speed is most-likely available from

external sensors, e.g. GPS, but with considerable latency

and at a very low frequency. Despite the notion that (6)

is analytically solvable a nonlinear model-based observer is

proposed capable of providing a smooth estimate of the wind

components along the axes of the body-ﬁxed reference frame

and the vehicles attitude.

We deﬁne a vehicle-carried Earth reference frame {E}

with the iEaxis pointing towards the North and jEaxis

pointing in the East direction. The kEcomplements the

reference frame and is aligned with the gravity vector. The

iEjE-plane is assumed to be tangent to the Earths surface.

Given the system as depicted in Fig. 2 a body-ﬁxed reference

frame FBis deﬁned in the center-of-gravity of the quadrotor.

The ibaxis is pointing forwards while the jbaxis points

towards the right. Again, the kbaxis complements the right-

handed system.

In order to keep the theoretical analysis as uncluttered as

possible, we make the following assumptions. The IMU is

to be located in the center-of-gravity of the vehicle and to

be perfectly aligned with the ibjb-plane. The accelerometers

are considered to be degraded by an unknown but constant

bias, which is a lumped term consisting of both electrical

noise and vibrations imposed onto the system by the motors.

Finally it is assumed that the gyroscopes are unbiased and

the contribution of the Coriolis terms are negligible.

The process model described by (1) and (5) assumes the

estimated wind components will persist until the next time

step, i.e. [˙

Wb

x˙

Wb

y˙

Wb

z]T=0. This assumption removes the

necessity of modelling the complex wind dynamics in real

time [7], [3]. It has to be noted that the wind component

along the kbaxis is not estimated as it is not directly

proportional to the body velocities. It may be shown that

Wb

zaffects to the total wind velocity through the rotor.

The aerodynamic power supplied to the air is a function

of the total wind velocity. Knowing the mechanical power

generated by the motor it would be possible to compute the

wind component along this kbaxis [16]. This is however

outside the scope of this work and left for future work.

The gyroscope measurements are assumed to be equal

to the angular rates of the quadrotor and are the inputs to

the system. The thrust force is considered to be directly

measurable with the accelerometer along the kbaxis [12].

The EKF relies on two distinct steps to estimate the state,

i.e. the prediction and measurement update steps. In the pre-

diction step the process model and model covariance are used

to predict the state for the next time step. In the measurement

update step the state predictions are corrected using the

measurement residual. The accelerometer measurements and

the GPS-derived body-ﬁxed ground speed components are

assumed to be the available measurements for the innovation,

which is modelled as follows

y=

ab

x

ab

y

−g

vb

GPS

=

−

µ

m(vb

x−Wb

x) +

β

x

−

µ

m(vb

y−Wb

y) +

β

y

ab

z+

β

z

vb

x

.(7)

It is assumed that the vibration-induced bias

β

is constant,

˙

β

=0, and imposed onto the accelerometer measurements

equally. This relation is therefore added to the process model.

Due to the inherent problems with latency and the output

frequency of the GPS module with respect to the periodic

frequency of the main control loop the body-ﬁxed ground

speed estimates are corrected every time a new measurement

becomes available.

IV. OBSERVABILITY ANALYSIS

It is well-known that observability is a necessary condition

for a nonlinear observer to converge. To determine if the

(a) Ground speed provided by the motion tracking system (blue) and

the ground speed computed by adding the predicted wind speed to

the measured airspeed from the accelerometers (red).

(b) Ground speed provided by the motion tracking system (blue) and

the ground speed computed by adding the predicted wind speed to

the measured airspeed from the accelerometers (red).

(c) Euler attitude angles, the roll angle (blue), the pitch angle (red)

and yaw angle (magenta).

(d) Predicted wind components along the body-ﬁxed x-axis (blue),

or the body-ﬁxed y-axis (red), and the norm of the predicted

wind vector (magenta). The calibrated wind tunnel airspeed is also

indicated (black).

Fig. 3: Measurements and predictions from the nonlinear observer given a calibrated wind tunnel airspeed of 3[m/s] with a

predicted drag-force coefﬁcient

µ

=0.20.

nonlinear system is observable, or whether there are unob-

servable states, we need to address nonlinear observability

theory. The observability rank condition will deﬁne if a

system is locally weakly observable [17]. This is not a

sufﬁcient, but a necessary condition for observability. Given

a nonlinear system

˙

x=f(x,u),

y=g(x)(8)

, in which x∈Rn,u∈Rmand y∈Rl, it can be shown that

the system is locally observable if for any arbitrary point x0

there exists a neighbourhood N(x0)in which every other x

is distinguishable from x0[17]. This condition implies that

at the point x0the rank of the observability matrix is equal

to n.

The observability matrix of a nonlinear system is deﬁned

by the gradient of the measurement equations and the gradi-

ents of the kLie-derivatives evaluated around x0

O=

∂

g(x)

∂

x

.

.

.

dLk

fg(x)

(9)

for k=1,...,n−1. For the system to be observable around

a point x0matrix Oshould be of rank n. This implies that

Oshould at least have nindependent rows or columns. The

observability of the nonlinear observer is analysed by using

a similar approach as in [12], [8], in which only the lon-

gitudinal dynamics of the quadrotor system are considered.

Referring to (5) we can reduce the system of equations to

˙

θ

˙u

˙

Wb

x

β

x

=

0

−gsin

θ

−

µ

m(u−Wb

x)

0

0

+q

1

0

0

0

(10)

while assuming ˙

θ

=qis the input to the system. Considering

the following measurements equation

y=ab

x=−

µ

m(vb

x−Wb

x) +

β

x(11)

the observability matrix may computed by taking the corre-

sponding Lie-derivatives

O=

0−

µ

m

µ

m1

g

µ

cos

θ

m

µ

2

m2−

µ

2

m20

g

µ

2cos

θ

m2−

µ

3

m3

µ

3

m30

g

µ

3cos

θ

m3

µ

4

m4−

µ

4

m40

(12)

Observability matrix Ois of rank 2 because the second

and third column are linearly dependent. The observability

matrix states that the difference between the ground speed,

wind speed and the measurement bias is unobservable, thus

additional measurement equations are required. By adding

the longitudinal GPS-derived body-ﬁxed ground speed com-

ponents the observability matrix is extended to

(a) Ground speed provided by the motion tracking system (blue) and

the ground speed computed by adding the predicted wind speed to

the measured airspeed from the accelerometers (red).

(b) Ground speed provided by the motion tracking system (blue) and

the ground speed computed by adding the predicted wind speed to

the measured airspeed from the accelerometers (red).

(c) Euler attitude angles, the roll angle (blue), the pitch angle (red)

and yaw angle (magenta).

(d) Predicted wind components along the body-ﬁxed x-axis (blue),

or the body-ﬁxed y-axis (red), and the norm of the predicted

wind vector (magenta). The calibrated wind tunnel airspeed is also

indicated (black).

Fig. 4: Measurements and predictions from the nonlinear observer given a calibrated wind tunnel airspeed of 5[m/s] with a

predicted drag-force coefﬁcient

µ

=0.17.

O=

0−

µ

m

µ

m1

g

µ

cos

θ

m

µ

2

m2−

µ

2

m20

g

µ

2cos

θ

m2−

µ

3

m3

µ

3

m30

g

µ

3cos

θ

m3

µ

4

m4−

µ

4

m40

0 1 0 0

−gcos

θ

−

µ

m

µ

m0

µ

mgcos

θµ

2

m2−

µ

2

m20

−

µ

2

m2gcos

θ

−

µ

3

m3

µ

3

m30

(13)

which is of rank 4 justifying the augmentation of the mea-

surement equations of the EKF. The system is considered

to be locally observable, but this is not a guarantee for

global observability. Note, however, that considering the

observability conditions posed in [12], [8] the proposed

observer is merely locally weakly observable except during

hover when 0=−gsin

θ

−

µ

m(u−Wb

x)(14)

as it becomes impossible to distinguish between

θ

and (u−

Wb

x), i.e. during unaccelerated ﬂight.

V. RESULTS

The experiment was conducted using a custom-build

quadrotor running two STM32F105RC 74 MHz Microcon-

troller units (MCUs) in parallel to share the computational

load. The ﬁrst MCU will run the Paparazzi1 autopilot soft-

ware while the second one carries out the operations required

to estimate the wind components at 512 Hz. An inter-

MCU communication bridge was established. An onboard

Fig. 5: Experimental set-up, the vehicle is shown to operate

in front of the cross section of the open jet wind tunnel.

IMU running at 512 Hz was used to estimate the attitude

and the body angular rates of the vehicle. To reduce the

adverse effect of vibrations onto the system, a second-order

Butterworth ﬁlter with a cut-off frequency of 1 Hz was used

to ﬁlter the IMU measurements.

The ground position and ground velocity estimates are

provided by an Optitrack motion tracking system. These

measurements are transmitted to the vehicle through a ded-

icated uplink at 120[Hz]. Even though the measurements of

this motion tracking system do not resemble the performance

of a typical low-cost GPS receiver, by resampling the discrete

signal the position and velocity data are purposely degraded.

The received GPS position signal is often perturbed by a

Gaussian noise term due atmospheric disturbances, so adding

such a perturbation to the motion tracking position signal

will increase the representability of the signal. A 10[Hz]

subsampling rate and Gaussian noise with a .1[m/s] standard

deviation were used during the experiment. To synchronise

the GPS with the two-samples delayed IMU measurements,

the GPS velocity was ﬁltered as well.

Previous research regarding the rotor drag of quadrotor

MAVs focussed on assessing the validity of the EKF and

determining the drag force coefﬁcient. Given that the states

of the system become unobservable during unaccelerated

ﬂight, the vehicle is commanded to transverse the cross

section of the wind tunnel, see Fig. 5. The requirement that

the quadrotor should also be moving is relaxed by the fact

that due to inaccuracies and uncertainty within the position

controller it is never truly stationary.

Fig. 6: Estimation results of the magnitude of the wind speed

vector for various calibration wind tunnel airspeeds. The

wind tunnel was set to an airspeed of 1[m/s] (blue),

µ

=0.25,

2[m/s] (red),

µ

=0.25, 3[m/s] (magenta),

µ

=0.2, 4[m/s]

(black),

µ

=0.2 and 5[m/s] (green) using

µ

=0.15.

Next, the performance of the nonlinear observer is anal-

ysed in a quasi-steady air ﬂow. The quadrotor is placed

in an open-jet windtunnel, as is shown in Fig. 1. The

experiment was performed at various airspeeds while keeping

the vehicles ibaxis pointing towards true north as is shown in

Fig. 5. The windtunnel has an offset angle of 60[deg] (1.05

[rad]) with respect to true north, implying that two distinct

axes of the accelerometer triad will measure the projection

of the wind vector onto the body-ﬁxed reference frame.

The results of two independent experiments, with wind

speeds of 3[m/s] and 5[m/s], are shown in Fig. 3 and 4 re-

spectively. There seems to be good correspondence between

the ground speed estimate provided by the GPS receiver and

the estimate derived from the accelerometers as is shown

in Fig. 3a or 4a. The latter requires to be corrected for the

wind speed, which is available from the nonlinear observer.

Discrepancies are assumed to be caused by mathematical

rounding errors and bias due to estimation errors. Also, as is

clearly apparent in Fig. 3, the standard deviation of the wind

estimate is considerable, while Fig. 4 shows that increasing

the wind speed increases this adverse effect. This is attributed

to the performance of the low-cost IMU in the presence of

strong vibrations. These vibrations are due to the unavoidable

mass-imbalance of the rotors and increased wind speed.

The results of the estimated wind speed components for

various calibrated wind tunnel airspeeds are summarised in

Fig. 6. The magnitude of the wind speed vector is shown

over time. Future work should focus on low-pass ﬁltering

the accelerometer signal while retaining the information.

Initially it was assumed that the drag-force coefﬁcient was

reasonably constant, however it was found that the drag-force

coefﬁcient was necessarily decreased for the measurements

to coincide with the predicted states coming from the EKF.

For example Fig. 3 was composed using

µ

=0.2, while Fig.

4 was found using

µ

=0.15. This inherently implies that for

a proper estimation of the wind components the drag-force

coefﬁcient should be estimated on-line. However, it is easily

shown that by adding the dynamic relation for the drag co-

efﬁcient to the reduced process model expressed in equation

(10) the system again becomes (partially) unobservable. It

will be impossible to see the difference between the drag

coefﬁcient, the ground speed and wind speed.

VI. CONCLUSIONS

The drag-force enhanced quadrotor model was augmented

with the wind components along the body-ﬁxed axis parallel

to the rotor plane. This model was used to create a nonlinear

observer capable of accurately predicting the wind compo-

nents using only IMU and GPS-velocity measurements. The

accuracy of the observer is limited by the performance of the

low-cost IMU. In future work, we will also study estimating

the wind speed component along the vertical axis, kb, using

the relation between the thrust and the aerodynamic power.

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