Open-Loop Quadrotor Flight Dynamics Identification ...

Open-Loop Quadrotor Flight Dynamics Identication in Frequency Domain via Closed-Loop Flight Testing Philipp Niermeyer∗ , Thomas Raer† and Florian Holzapfel‡ Institute of Flight System Dynamics, Technische Universität München, 85748, Germany

The paper presents the identication of a quadrotor's bare-airframe dynamics in hover by employing frequency domain methods commonly applied to large-scale rotorcraft and xed-wing aircraft. Flight tests are conducted with a combination of manual and automated inputs to enable excitation over a wide frequency range to achieve universal model applicability. In contrast to other publications on the topic of micro aerial vehicle system identication, all ight tests are performed outdoors where only inertial MEMS sensors and GNSS measurements are available. Our approach relates physical system parameters of the bare-airframe to the linear control and stability derivatives which are estimated by the identication process. The identied models are validated in terms of their time domain behavior by injecting step perturbations to the motor commands. The obtained information of the quadrotor's bare-airframe dynamics is used for controller synthesis and for validation of high-delity physical simulations.

Nomenclature (I G )BB ω F, G, H0 , H1 fR i FP lat,i Pi F P,i VG A i VP A δ γ2 ωmot Φ Ψ ρ Θ d G xy (f )

= diag([Ixx Iyy Izz ]), quadrotor inertia = [p q r]T , body-xed rotational rates System matrices as dened by CIFER

=

[fxR

fyR

around CG

c

fzR ]T , accelerometer measurements in the reference point

Force on i-th rotor because of lateral air inow Overall propulsive force

= [uA

vA

wA ]T ,

aerodynamic velocity in the center of gravity

Aerodynamic velocity of the point

= [δx

δy

δz

δ T ]T ,

Pi

axis controls

Coherence Motor rotational rate in rad/s Bank angle Heading Air density Pitch angle Cross spectral density estimate between signals

x

and

y;

called auto spectral density for

x=y d H xy (f ) Tb cH cP cT D g

Frequency response estimate from input

x

to output

y

Estimated transfer function Propeller coecient to account for lateral air inow Propeller power coecient Propeller thrust coecient Propeller diameter Gravitational acceleration

∗ PhD candidate, Institute of Flight System Dynamics, [email protected], AIAA Student Member. † PhD candidate, Institute of Flight System Dynamics, thomas.ra[email protected]. ‡ Professor, Institute of Flight System Dynamics, [email protected], AIAA Senior Member.

1 of 14 American Institute of Aeronautics and Astronautics

J kωδ knδ Mi,shaf t ni ni,cmd T Ti xGPi

Cost function

= 2πknδ ,

scaling from motor speed controller commands to rad/s

scaling from motor speed controller commands to revolutions per second Shaft moment at the i-th rotor Rotational rate of the i-th rotor in revolutions per second Motor speed controller command within 1  200 Model transfer function Thrust of the i-th rotor

= [xGPi

y GPi

z GPi ]T ,

distance vector from CG to point

Pi

I. Introduction Quadrotors are applied as a test platform for a variety of nonlinear and adaptive control algorithms at 1, 2, 3

the Institute of Flight System Dynamics of the Technical University Munich.

A physics-based nonlinear

model, which includes accurate motor dynamics, serves as basis for control design and ight simulation. Though a lot of eort has been taken to determine the vast amount of required parameters like propeller coecients, aerodynamic force coecients, masses and inertias, many uncertainties occur in such a bottomup modeling approach.

This work addresses the problem top-down: The dynamics of the overall system

are identied to yield a model of the multirotor's rigid body responses to motor command inputs. While Micro Aerial Vehicles (MAVs) have been successfully identied using motion capture systems and time 4

domain methods,

the procedure presented within this work relies on inertial MEMS measurements and

frequency domain identication methods successfully applied to substantially heavier Unmanned Aerial Vehicles (UAVs).

5

Figure 1 outlines the necessary steps to nd valid non-parametric and parametric models of

the quadrotor dynamics via the basic workow from Ref. 6. We conduct ight tests with well dened inputs to create a solid time history base for identication c 6

and validation. Then, CIFER

 a system identication and validation tool which was developed at the

NASA/Ames Research Center  is used to perform the computation of frequency responses which represent a non-parametric model of the vehicle dynamics.

A parametric model that matches the non-parametric

model best in terms of its frequency responses is then determined by minimizing a cost function.

The

model structure and initial parameters for this process are found by investigating the physics of multirotor systems.

7

As the best match often shows to be insensitive to some of the parameters, the cost function

minimization requires a few iterations to avoid over-parametrization.

The next step compares the ight

data from validation ights to simulations of the nal model when excited by the same inputs. It will be shown that time domain behavior of the actual quadrotor is predicted with sucient accuracy such that the modeling process is nished. Note the main dierences between this paper and approach in Ref. 8, where only closed-loop transfer function models of the dynamics in attitude control mode are identied. The results depend on the applied controller such that conclusions on the open-loop bare-airframe dynamics cannot easily be drawn.

This

limits model applicability mainly to outer-loop controller design. Contrary, we aim at the identication of state space models of the unstable quadrotor dynamics to relate physical eects to the identied parameters. This extends model applicability to rotational and attitude controller design as well as validation of the physics-based nonlinear model. This work is structured as follows:

Section II develops the equations of motions for multirotors and

identies the inuence of the physical system parameters to the stability derivatives of a linear quadrotor model in hover.

The principles of the applied method for system identication are explained in short in

section III. In Section IV, the full set of required frequency responses is estimated from numerous outdoor ight tests and a parametric quadrotor model is estimated.

In Section V, the parametric model of the

quadrotor is validated in the time domain to complete the identication process.

II. Multirotor Rigid Body Dynamics This section develops the equations of motion for a multirotor system, based on a simple propeller model. Apart from the inertial frame

Pi

I

and the North East Down frame

0,

the body-xed and propulsive frames

are used in the following. The origin of the body-xed frame is given by the reference point, the origin

2 of 14 American Institute of Aeronautics and Astronautics

Figure 1. Overview of the procedure for quadrotor identication of each

Pi -frame

is the point

Pi

which lies on the rotor shaft of the respective motor in the propeller plane.

Figure 2 visualizes the directions of the body-xed and propulsive frames as well as the rotating directions of the motors. The motors are consecutively numbered starting with the front motor and advancing clockwise when viewed from above (rotation along the positive Within this section,

VP A

zB -axis).

refers to the aerodynamic velocity of point

xB , yB and zB point P2 relative

aerodynamic velocities of the center of gravity in

p, q and r. The distance of a  T = xP1 P2 y P1 P2 z P1 P2 . Individual

rate are denoted denoted

r

P1 P2

P.

Scalars

uA , vA

and

wA

are the

direction respectively. Pitch, roll and yaw to a point

P1

in body-xed coordinates is

vector components are always written in

their reference frame is not explicitly mentioned.

Figure 2. Motor rotational directions and frame denitions

3 of 14 American Institute of Aeronautics and Astronautics

xB -frame

if

A. Forces and Moments The

i-th

rotor's propeller thrust

Ti

acting in negative

xB -direction

is given by.

9

Ti = cT ρn2i D4 .

(1)

v0 ni D , where v0 is the air inow velocity perpendicular to the rotor plane. This coecient is usually determined experimentally and

cT

Note that the thrust coecient

is a function of the propeller advance ratio

stored as a look-up table. The rotational rate all rotors and

D

ni

J=

 given in revolutions per second  is dened positive for

denotes the propeller diameter. An additional force of

 r



i FP lat,i =

i −uP A 3 Pi  cH ρ ni D −vA 

·

0

2  2 Pi i uP + vA A

Pi

V A

(2)

is assumed to occur at each rotor because of a lateral air inow. It scales proportionally with the propeller constant

cH .

Conclusively, the

i-th

i FP P,i 

rotor's overall propulsive force





is

0  Pi   i FP = F + 0  .   P,i lat,i −Ti

(3)

The torque on the motor shaft which results from air friction acting on the propeller is given by

Mi,shaf t = (−1)

−i

9

cP 2 5 ρn D 2π i

(4)

Together with the moment generated by the propulsive forces from Eq. (3), we yield the overall propulsive moment generated by the

i-th

rotor around the Center of Gravity (CG)



0 0



  GPi i × FP MG  P,i = r P,i +  Mi,shaf t

(5)

B. Linearization around Hover State Following pertinent kinematic and kinetic principles, results from II.A are used to obtain the analytical nonlinear quadrotor dynamics. We assume a wind free case, a at and non-rotating earth and linearize that

δ as   0.5 n1,cmd   0  n2,cmd   . 0.25 n3,cmd 

model around hover state. Furthermore, we dene the axis controls







δx 0 −0.5    0  δ   0.5 δ =  y =   δz  −0.25 0.25 δT 0.25 0.25

0 −0.5 −0.25 0.25

0.25

(6)

n4,cmd

This denition decouples the input-output behavior of the resulting linear quadrotor model which is then given by

   u˙ Xu     v˙   0    w˙   0     p˙   0        q˙  = Mu     r˙   0    Φ˙   0    ˙  Θ  0 ˙ Ψ 0

0 Yv 0 Lv 0 0 0 0 0

0 0 Zw 0 0 0 0 0 0

0 Yp 0 Lp 0 0 1 0 0

Xq 0 0 0 Mq 0 0 1 0

0 0 0 0 0 Nr 0 0 1

0 g 0 0 0 0 0 0 0

−g 0 0 0 0 0 0 0 0

   0 u 0    0  v   0       0  w   0    0   p  Lδx    0  q  +  0       0 r  0    0  Φ  0    0 Θ  0 0 Ψ 0

0 0 0 0 Mδy 0 0 0 0

4 of 14 American Institute of Aeronautics and Astronautics

0 0 0 0 0 Nδz 0 0 0

 0  0     ZδT   δx  0     δy  0  .   δz  0   δT 0    0  0

(7)

An equivalent model has already been given in Ref. 7.

Due to the structure of the system matrices, the

dynamical system (7) can be broken down into four dynamically decoupled systems for roll axis (states and input (state

w

δy ),

pitch axis (states

and input

u,q ,Θ

and input

δx ),

yaw axis (states

r,Ψ

and input

δz )

v ,p,Φ

and vertical motion

δT ).

The stability and control derivatives of the roll and pitch axis in Eq. (7) are dened as follows:

4 cH ρnh D3 m   2 z|z| 3 Xq = −Yp = − cH ρnh D √ +z m z 2 + l2 4 cH ρnh D2 z Mu = −Lv = − Iyy       p δcT 2 Mq = L p = − cH ρnh D3 |z| z 2 + l2 + z 2 + l2 ρnh D3 Iyy δJ h 4 Mδy = Mδx = knδ lcT,h ρnh D4 . Iyy Xu = Yv = −

Subscript

h

hereby denotes values in hover state and

knδ

(8a) (8b)

(8c)

(8d)

(8e)

represents the scaling from motor speed controller

z

command to rotational rate in revolutions per second. Distances

and

l

can be read from Figure 2. The

yaw axis' derivatives are

4 cH ρnh D3 l2 Iyy 4 cP = knδ ρnh D5 . Izz π

Nr = −

(9a)

Nδz

(9b)

The vertical motion's derivatives are

  4 δcT ρnh D3 m δJ h 2 = knδ cT,h ρnh D4 . m

Zw =

(10a)

ZδT

(10b)

Equations (8a-10b) are the parameters to be determined experimentally in section IV.

III. Frequency Domain Identication Method This section reviews the applied methods for frequency domain identication of aircrafts and rotorcrafts c

which are proposed in Ref. 6 and which are implemented in CIFER

accordingly. Reconsider the workow

overview in Figure 1. The linear model which was derived in subsection II.B suggests that no axis cross-coupling is present, an assumption that will be veried experimentally in section IV. This section therefore only reviews the c

concepts for SISO system identication with CIFER

.

A. Frequency Response Estimation In order to obtain the frequency responses from time history records of system inputs and outputs, the auto 10

and cross spectral densities of the signals are computed.

A main challenge in the computation of these

spectral densities is the choice of a proper window size, which has to comply the frequency range of interest while directly aecting bias and random errors in the estimate. Using composite windowing,

6

thus merging

the results obtained with dierent window sizes, generates better results than using a single window size.

y

known, the frequency response estimate

d d d H xy (f ) = Gxy (f )/Gxx (f ).

(11)

With auto- and cross spectral densities of input

x

and output

is easily calculated using

d d H xy (f ) represents the frequency response estimate from input x to output y and Gxy (f ) denotes cross-spectral density between signals x and y . The auto spectral density is computed equivalently. The

In Eq. (11), the

5 of 14 American Institute of Aeronautics and Astronautics

estimate is free of bias errors if the system dynamics are linear and uncorrelated noise is only present at the system output (measurement noise, gusts). To quantify the linear dependence of output

2 γxy =

The larger the coherence input excitation.

2 γxy ,

x

to input

y,

a coherence function is dened:

2 d |G xy (f )| . d d |G xx (f )||Gyy (f )|

(12)

the more the output measurement is the response of a linear system to the

It becomes unity for perfectly linear system behavior.

Bias and random errors become

visible in a decrease of the coherence values at the aected frequencies. Note that the assumption of uncorrelated noise has to be dropped if the system output is fed back to the system input, as the case in our setup. With increasing output noise power with respect to the power of external excitation, the quality of the estimated frequency response drops.

In a worst case scenario,

where the system input is only comprised of feedback, the frequency response estimate depends only on the 6

controller transfer function, not the system to be identied.

The authors decided to assess the eects of

the feedback loop in simulations of the actual experiment. A result was that bias errors are expected mainly at lower frequencies (those within controller bandwidth) and that they increase with the level of feedback while small coherence values still indicate the occurrence. With that in mind, special care is taken when considering the lower end of frequency responses which are estimated in section IV.

B. Identication of Model Parameters The algorithm iterates model parameter values, aiming at minimization of a cost function that evaluates the match between model and measurement. For SISO systems, Ref. 6 introduces the cost function

J=

ωn 20 X Wγ [Wg (|Tb| − |T |)2 + Wp (6 Tb − 6 T )2 ], nω ω

(13)

1

b ) for the Tb is the estimated transfer function which corresponds to the frequency response estimate H(f T is the parameter dependent model transfer function. Weight Wγ is coherence dependent to decrease the inuence of less reliable estimates whereas weights Wg and Wp are chosen constant. The number of frequency points ω1 ...ωn considered is denoted nω . A cost function value of J ≤ 100 is considered a fair match, an optimization that leads to J ≤ 50 implicates a

where

currently considered input/output combination and

6

model which is hardly distinguishable from the ight test data.

For MIMO systems, the cost function to

minimize is the average cost function of all frequency responses involved, thus

Javg =

1 nT F

n TF X

The optimization problem is nonlinear in the parameters. Bounds (CRs) are used. Values larger than

20%

Jl .

(14)

l=1 To assess parameter sensitivity, Cramer Rao

suggest insucient sensitivity of the cost function to the 6

parameter. If the cost function increase is negligible, the parameter should be removed by setting it to zero.

C. Model Validation In order to verify that the parametric model is eligible to represent the actual system in the time domain, ight test time responses and model time responses to the same inputs are compared. Note that these inputs have to dier from those used for model identication. The model response is gained by integrating the state space equations, which are appended with correction terms to account for miscalculated trim inputs, bias in 6

the output measurements or constant external disturbances to the state derivatives. is refrained from judging model validity using the cost guidelines,

6

Within this paper, it

as the author considers it problematic to

transfer them from large-scale, slow stable systems to small-scale, fast unstable systems.

6 of 14 American Institute of Aeronautics and Astronautics

IV. Identication of the Linear Parametric Quadrotor Model A. Collecting Time History Data The quadrotor system to be identied is an AscTec Hummingbird, as depicted in Figure 3. In order to provide additional processing power and signal logging capabilities, the stock ight control system is extended by

R

a custom control computer that is based on a Gumstix

computer-on-module and a real-time GNU/Linux

operating system. It has access to all sensor data, processes the Simulink

R

generated control algorithms and

feeds back direct motor RPM commands to the AscTec Autopilot. Being a benecial condition for system identication, a sample rate of 1000 Hz is retained throughout the whole signal chain, including the data logging to a microSD card.

Figure 3. Ascending Technologies Hummingbird extended by a custom ight control computer Figure 4 outlines the ight test setup to collect time history data.

Two dedicated input signals are

chosen: For low frequency excitation up to 2 Hz, the pilot is advised to y sweep inputs. A signal injector 6

on board the quadrotor excites frequencies between 1 Hz and 30 Hz by injecting an exponential sweep

of

30 seconds duration to the system's axis-input. Pitch, yaw and thrust axes are excited one after the other. The data of two experiments with manual and two experiments with automatic excitation are concatenated c

before transformation to frequency domain and computation of the frequency responses through CIFER

.

By including only the experimental data which produced the best coherence values, the frequency response data base with relevant frequency ranges given in Table 1 was created. We nd excellent coherence values for all responses, but especially for the yaw and vertical axis in Figure 6. Large sections of the given frequency 6

ranges exhibit coherence values well above the guideline

of

γ 2 ≥ 0.6.

Table 1. Input/output pairs, frequency ranges with γ 2 >= unit rad s

0.6;

δy δz δT

p

q

r

fxR

fyR

fzR



0.6100



0.3110







1.2170

















0.3100

B. Roll and Pitch Axis q(s)/δy (s) and fxR (s)/δy (s) can be included in cost function R minimization. Note that fx is not a state, but can be considered a system output which is computed from states and state derivatives as fxR = u˙ − z RG q˙ + gΘ. (15)

From Table 1, we read that the responses

The unknown distance from reference point to CG can be included in the parameter estimation process via state transformation only, because uncertain parameters in the output matrices are not allowed by CIFER

c

.

We dene the new state

qT = sgn(z RG ) · 100z RG · q, 7 of 14 American Institute of Aeronautics and Astronautics

(16)

Figure 4. Flight test setup such that we yield the modied pitch dynamics

 x = uG

with states

c

fed to CIFER

qT

Θ

T

(17a) (17b)

and outputs

 y= q

fxR

T

. The system matrices, which can directly be

, are



Xu  F = 100z RG Mu 0 ! 0 0 0 H0 = , 0 0 g F

The elements of system matrices following.

x˙ = F x + Gδy y = H 0 x + H 1 x, ˙

and

 −g  0 , 0

Xq 100z RG

Mq 1 100z1RG

H1 = G

0 1





0

  G = 100z RG Mδy  0 ! 0 1 . −0.01 0

(18a)

(18b)

c

are the parameters which are estimated by CIFER

Resulting parameter values and uncertainties are given in Table 2.

in the

Note that all identied

parameters feature CR bounds smaller than 10 %, indicating an impressively accurate model match. The origin of the two input/output time delays is further investigated in this section's last paragraph. From Table 2, all pitch derivatives as well as a the distance between reference point and CG can be determined. The respective pitch dynamics feature a fast unstable oscillatory mode at a relative damping of

ζ = −0.487

and a stable non-oscillatory mode at

ω = 3.98

ω = 3.87

rad/s with

rad/s.

Cost function values for the respective optimization are given in Table 5. As a magnitude drop in the acceleration response

fxR (s)/δy

occurs at about 60 rad/s, which can not be reproduced with the chosen simple

three DOF model, the response

fxR (s)/δy (s) was only included in cost function optimization between 0.5 and

60 rad/s. Figure 6 displays the parametric model responses versus the measured ight test responses. For both the pitch rate and acceleration response the tracking is highly satisfactory, just as one would expect from the low cost function values (Table 5). Small random mismatch in the pitch rate response of about 3 dB and

±

±

15 deg is uncritical. Even smaller mismatch can be observed in the acceleration response. The

only discrepancy remains the magnitude drop in the acceleration response beyond 60 rad/s which is likely to originate from sensor dynamics.

8 of 14 American Institute of Aeronautics and Astronautics

C. Yaw Axis The linearization of the quadrotor's rigid body dynamics led to the assumption of a simple rst order system for the yaw axis. The measured yaw responses however cannot be accurately reproduced with a rst order lag. Consequently, the motor dynamics have to be considered to update the model structure. The easiest way to create a linear motor model which is suitable for frequency domain identication is to assume a proportional motor speed controller to command a motor moment. propulsive yaw moment

NP .

The sum of all four motor moments equals the

By subtracting the air friction moment

Naero ,

one obtains the moment which

accelerates the quadrotor body. The propeller air friction moment is given by the linearization of Eq. (4) around hover motor angular speed

ωh .

The signal ow is shown in Figure 5. Note that the motor speed feedback to the controller would normally be

ω−r

because it is measured in the body-xed frame. As

neglected. The constant factor

kωδy

to the motor speed controller, to angular rate.

Cmot

ω  r,

this kinematic relationship can be easily

scales from the integer signal between 1 and 200, which is transmitted The motor controller is represented by the constant gain

and the motor inertia including the propeller is given by

Imot .

The state space system which follows

from the block diagram of Figure 5 is

ω˙ mot r˙

! =

Mmot,ω Nω

0 Nr

!

ωmot r

! +

where the relation between physical parameters and the derivatives

Mmot,δz −kωδ Nω

! δz ,

Mmot,ω , Nω , Nr , Mmot,δz

(19)

and

Nω

can be

read from Figure 5. The parameter estimates are summarized in Table 3. All CR bounds are less then 20 % which indicates accurately identied derivatives. The identied model match is stunning, which is consistent with the low cost function value of 34.37 (Table 5).

Figure 5. Yaw axis model including simplied motor dynamics, linearized around hover

D. Vertical Motion From the rigid body vertical motion model, we assume rst order lag dynamics

ω˙ = Zω ω + ZδT δT The non-parametric model corresponds well to those dynamics, thus only the two derivatives have to be identied.

c

CIFER

(20)

Zw

and

ZδT

was able to estimate both parameters with small CR bounds (Table 4)

and the very low cost function value (Table 5) coincides with an excellent match between model and ight response as displayed in Figure 6.

9 of 14 American Institute of Aeronautics and Astronautics

Table 3. Yaw model parameter estimates

Table 2. Pitch model parameter estimates Parameter

Xu 100z RG Mu Xq 100z RG

Mq 1 100z RG 100z RG Mδy τqδy τfx δy

Value

CRi

[%]

Parameter

−0.2090 8.207 0 0 0.7416 2.977 0.02423 0.05791

7.818 8.770 n/A n/A 7.618 6.025 6.090 3.721

Mmot,ω Nω Nr Mmot,δz τδz

Table 4. Vertical motion model parameter estimates Parameter

Zw ZδT τδT

CRi

Value

−0.5934 −0.1692 0.05202

CRi

Value

−42.21 −0.5541 0 152.1 0.01313

[%]

12.94 8.242 n/A 13.71 6.782

Table 5. Optimal cost function values of pitch, yaw and thrust axis

[%]

Response

11.97 4.242 4.402

q(s)/δy (s) R fx (s)/δy (s) r/δz (s) R fz (s)/δz (s)

Cost

Fitting Range [rad/s]

56.44 46.68 34.37 38.79

0.5100 0.560 1.7170 0.3260

E. Input/output Time Delay Considerations In total, four equivalent time delays have been identied: one for each frequency response which was included in cost function minimization.

Each time delay accumulates the dynamics not considered in the model

equations. Figure 7 breaks down the single cumulative delay of each response to multiple equivalent time delays and displays the relationships between them. The blocks labeled pitch, yaw and vertical contain the dynamical model equations. The additional dynamics block accounts for the time delays caused by digital data processing and higher order dynamics such as instationary aerodynamics, or structural dynamics. As motor dynamics have only been modeled for the yaw movement, there is no equivalent time delay

τmot

in

the yaw axis to account for motor dynamics. With an estimated equivalent time delay of the rotational sensors of

τrot ≈ 3

ms (based on data sheet

information and earlier experiments), the equivalent time delay for the accelerometer dynamics is

τacc = τfxR δy − τqδy + τrot = 47.94ms − 24.23ms + 3ms = 36.7ms

(21)

For Eq. (21) to hold, it has been assumed that the additional dynamics and motor dynamics for pitch and thrust excitation result in similar time delays. This equivalent accelerometer time delay is fairly large, disqualifying direct accelerometer feedback for many control applications.

V. Validation of the Linear Parametric Quadrotor Model This section validates the models which have been identied in section IV by comparing the time domain responses of model and ight test. Automatically generated doublet inputs are used as excitation signals.

A. Open-Loop Model While a control loop stabilizes the vehicle during ight tests, the model is excited open-loop.

For the

quadrotor system at hand, this approach is extremely challenging because of the following issues:

•

The slightest modeling errors or external inputs lead to divergence of the unstable pitch axis model's time response. To counteract this problem, only very short validation signals can be used.

•

High power vibration noise interferes both input and output measurements.

10 of 14 American Institute of Aeronautics and Astronautics

q(s)/δy (s)

fxR (s)/δy (s)

r(s)/δz (s)

fzR (s)/δT (s)

Figure 6. Quadrotor frequency responses: Flight test and parametric model •

High exposure of the time response to process noise such as wind gusts because of the small system inertia and mass.

Especially the rst and third item on that list showed to cause problems for pitch axes validation. Of many collected validation time histories of dierent amplitudes, only a few have been been found not to result in model divergence. Figure 8 compares ight test results to the model response when excited open-loop. Note that the axis controls in pitch and yaw are comprised of both feedback and the injected doublet while the vertical axis is excited open loop. To remove high frequency noise from the measurements, input and output data was low-pass ltered Bias and shift

6

6

by a rst order low-pass with a -3 dB bandwidth of 100 rad/s prior to model simulation.

in the measurements was automatically corrected by CIFER

11 of 14 American Institute of Aeronautics and Astronautics

c

.

Figure 7. Identied time delays overview Pitch axis model, pitch rate

Yaw axis model, yaw rate

q(t)

Pitch axis model, x-axis acceleration

r(t)

Vertical axis model, z-acceleration

fxR (t)

fzR (t)

Figure 8. Quadrotor time history responses to doublet injection at axis controls level: ight test and open-loop parametric model The dynamics of the pitch rate seems to be adequately modeled for the rst step in the input signal at

t = 0.5

s (Figure 8). The second step induces an overreaction of the model response, probably due to

the unmodeled nonlinear motor dynamics which become apparent for such large step amplitudes. The high absolute values in the pitch rate also suggest that the gyro measurements might be corrupted, as it is close to the maximum rate of

±300

deg/s. The last step at

t=2

s is modeled correctly except for the bias and

the succeeding divergence of the model response, issues that have been discussed in the last paragraph. The bad model validity for the step at

t = 1.25

s becomes visible in the acceleration response as well.

Other than that, the model predicts the acceleration quite well, even though bias errors and model divergence start to be an issue within the last second of the record.

Tracking for both the yaw and vertical motion

model response is highly satisfactory.

B. Closed-Loop Pitch Axis Model Motivated by the problems experienced in open-loop validation of the pitch dynamics, closed-loop system behavior is examined in short. The system is simulated with a PI rate controller matching the one used for data recording. No state derivative biases were determined to ease implementation, only an output bias was

12 of 14 American Institute of Aeronautics and Astronautics

computed and applied to the system. Results are shown in Figure 9. Pitch rate

Input

q(t)

δy (t)

X-axis acceleration

fxR (t)

Figure 9. Quadrotor time history responses to commanded pitch rate steps: ight test and closed-loop parametric model The rate response gives almost a perfect match, but may be misleading in terms of bare-airframe model validity:

It is quite insensitive to model parameter changes as it is largely dominated by the controller.

However, note the accurate identication of the time delay between pitch command and the pitch response. Overall, the underlying closed loop pitch dynamics model will help outer loop controller design in the future. The model error of the axis control, which is comprised of feedback and pitch command, is tracked satisfactory. This is no surprise considering the small errors between model and ight test pitch rates. The accurate match of the acceleration responses is much more important. Especially note the peaks in both model and ight test which occur at each step in the pitch command, conrming that the distance between center of gravity and reference point of the measurement has been determined with large accuracy.

VI. Conclusion This paper presented the successful determination of the ight dynamics of the Ascending Technologies Hummingbird around hover state. At rst, the dynamics of multirotors were investigated from physical considerations to provide insight into the dynamics to be identied and to nd a linearized model structure to form the basis for parametric system identication. The relationships between physical parameters, state- and control-derivatives provide system understanding and support simulation model tuning in the future. A ight-test setup was introduced which enables the collection of in-ight time data that is suitable to c

be used by the CIFER

system identication tool for response estimation. Based on the on-axis responses,

three individual parametric state space models were found: one for the identical roll- and pitch dynamics, one for the yaw axis and one for vertical motion. While it was not possible to identify motor parameters in the pitch axis or vertical motion, linearized motor dynamics dominate the yaw axis model. All models have been validated in time domain.

Though perturbations through measurement noise,

vibration and gusts together with the fast unstable modes of the dynamics were a challenge, satisfactory

13 of 14 American Institute of Aeronautics and Astronautics

ight-test time history tracking was achieved with the parametric models. Future work deals with the inclusion of known sensor dynamics instead of accounting for those dynamics with the equivalent time delay. Furthermore, the physical simulation model can be tuned using the identied parameters and their physical equivalents.

References 1 Wang, J., Raer, T., and Holzapfel, F., Nonlinear Position Control Approaches for Quadcopters Using a Novel State Representation,

AIAA Guidance, Navigation, and Control Conference ,

Guidance, Navigation, and Control and Co-located

Conferences, American Institute of Aeronautics and Astronautics, 2012.

2 Wang, J., Holzapfel, F., Xargay, E., and Hovakimyan, N., Non-cascaded Dynamic Inversion Design for Quadrotor Position Control with L1 Augmentation,

European Control Conference , 2013.

3 Achtelik, M., Bierling, T., Wang, J., Höcht, L., and Holzapfel, F., Adaptive Control of a Quadcopter in the Presence of large/complete Parameter Uncertainties,

Infotech@Aerospace , Infotech@Aerospace Conferences, American Institute of

Aeronautics and Astronautics, 2011.

4 Gremillion, G. and Humbert, J., System Identication of a Quadrotor Micro Air Vehicle,

Mechanics Conference , AIAA, 2010.

AIAA Atmospheric Flight

5 Mettler, B., Tischler, M. B., and Kanade, T., System Identication of Small-Size Unmanned Helicopter Dynamics,

American Helicopter Society 55th Forum , 1999. 6 Tischler, M. and Remple, R. K., Aircraft And Rotorcraft System Identication: Engineering Methods With Flight-test Examples , American Institute of Aeronautics and Astronautics, 2006. 7 Bristeau, P.-J., Martin, P., Salaün, E., and Petit, N., The Role of Propeller Aerodynamics in the Model of a Quadrotor UAV,

European Control Conference , Budapest, 2009.

8 Wei, W., Schwartz, N., and Cohen, K., Frequency-Domain System Identication and Simulation of a Quadrotor Controller,

AIAA Modeling and Simulation Technologies Conference ,

AIAA SciTech, American Institute of Aeronautics and

Astronautics, 2014.

9 Aviv, R. and Gur, O., Propeller Performance at Low Advance Ratio, 441.

10 Bendat, J. S. and Piersol, A. G.,

Journal of Aircraft , Vol. 42, No. 2, 2005, pp. 435

Engineering applications of correlation and spectral analysis ,

ed., 1993.

14 of 14 American Institute of Aeronautics and Astronautics

Wiley, New York, 2nd