SlideShare a Scribd company logo
Experimental validation of a new adaptive
control scheme for quadrotors MAVs
Gianluca Antonelli† , Elisabetta Cataldi† ,
Paolo Robuffo Giordano⊕ , Stefano Chiaverini† , Antonio Franchi≀
† University

of Cassino and Southern Lazio, Italy
http://webuser.unicas.it/lai/robotica

⊕ CNRS at IRISA and Inria Bretagne Atlantique, France
http://www.irisa.fr/lagadic
≀ Max Planck Institute for Biol. Cybernetics, Germany
http://www.kyb.mpg.de/research/dep/bu/hri/

IROS 2013

Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
Trajectory tracking control for quadrotor

Adaptive with respect to
uncertainty in total mass
uncertainty in Center Of Mass (CoM)
presence of 6-DOF external disturbances

Assumption: closed-loop orientation dynamics faster than
translational one
Stability analysis & numerical simulations1
Experimental results

1
Antonelli, Arrichiello, Chiaverini, Robuffo Giordano, “Adaptive
trajectory tracking for quadrotor MAVs in presence of parameter
uncertainties and external disturbances”, AIM 2013
Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
Kinematics

body-fixed u, surge
xb
p, roll
Ob
q, pitch
η1
r, yaw
yb
v, sway
zb
w, heave

η1 =
η2 =
O
x
earth-fixed

x y z
φ θ ψ

ν 1 = RB η 1
I ˙
˙
ν 2 = T (η 2 )η 2

y
z

Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013

T
T
Dynamics

Mathematical model expressed in body-fixed frame
˙
M ν + C(ν)ν + τ W + g(RB ) = τ ,
I
beyond the common terms, we model
τW =
γ W ∈ R6 external disturbance

RB O 3×3
I
γW
O 3×3 RB
I
constant in the inertial frame (wind)

Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
Dynamics -2Exploiting the linearity in the parameters
˙
Φ(ν, ν, RB )γ = τ
I
and rewriting with respect to the inertial frame while separating the
xy dynamics from z:
˙ ¨
Φxy (η, η, η )
γ = RI τ 1
B
˙ ¨
φz (η, η, η )
with γ ∈ R16 :
mass (1 parameter)
first moment of inertia (3 p.)
inertia tensor (6 p.)

τ =

τ1
τ2

external disturbance (6 p.)
Antonelli Cataldi Robuffo Chiaverini Franchi


0
0
 
Z 
= 
K 
 
M 


N

Tokyo, 5 November 2013
Thrust
Assuming CoM coincident with Ob
f2
2
τt,3

l

f3

ωt,2

f4

yb z
b

ωt,4
τt,4

4

τt,2
f1

Ob

ωt,3
3

2
fi = bωt,i

ωt,1

xb

1
O
y

τt,1
x

z

Antonelli Cataldi Robuffo Chiaverini Franchi

2
τt,i = dωt,i

0
 0

τ1 =  4

i=1




fi







l(f2 − f4 )

l(f1 − f3 )
τ2 = 
−τt,1 + τt,2 − τt,3 + τt,4

Tokyo, 5 November 2013
Mapping from the angular velocities to the force-torques
Assuming CoM of coordinates r C :
 
 2 
ωt,1
Z
2
K 
ωt,2 
  = Bv 

M 
ω 2 
t,3
2
N
ωt,4
with



b
b
b
b

0
b(l + rC,y )
0
−b(l − rC,y )

Bv = 
b(l + rC,x )

0
−b(l − rC,x )
0
−d
d
−d
d
CoM influences the mapping from thrust generated from the
motors to the vehicle forces/moments
Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
Inverse mapping
Any controller determines a control action Zc Kc Mc Nc
projected onto the motor input u ∈ R4
 
Zc
 Kc 
u = B −1  
v 
Mc 
Nc
where B −1 ∈ R4×4 is
v

B −1
v

l − rC,x
 4bl
l − r
C,y


 4bl
= l + r
C,x

 4bl
 l + rC,y
4bl

0
1
2bl
0
1
−
2bl

Antonelli Cataldi Robuffo Chiaverini Franchi

1
2bl
0
1
−
2bl
0


l − rC,x
−
4dl 
l − rC,y 


4dl 
l + rC,x 

−
4dl 
l + rC,y 
4dl

Tokyo, 5 November 2013

T

further
Current inverse mapping

ˆ
When the CoM position estimate r C is affected
mapping becomes

1
0
 
 
Z
Zc
 rC,y
˜
1
K 
 Kc  
  = B v | B −1
   2
ˆ
r C v r C Mc  =  rC,x
M 
˜

0

2
N
Nc
0
0

by an error, the real

0
 
b˜C,y  Zc
r
 
0
2d   Kc 
b˜C,x  Mc 
r 
1 −

2d
Nc
0
1
0

wrong CoM estimate ⇒ a coupling from altitude and yaw control
actions onto roll and pitch dynamics

Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
Controller block diagram


Z
K 
 
M 
N


η 1d
ψd

Zc
pos φd , θd




Kc
Mc  B −1
v
or
Nc

u

motors

2
wt,i

τW
η

Bv
plant

Classical MAV control architecture with adaptation wrt parameters
and compensation of the CoM position

Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
Altitude controller
error
z = zd − z ∈ R
˜
˙
s z = z + λz z ∈ R
˜
˜

full version
1
ˆ
(φ γ + kvz sz )
cos φ cos θ z
˙
γ = K −1 φT sz
ˆ
γ,z z

Z =

ˆ
with γ ∈ R16

reduced version
1
(ˆz + kvz sz )
γ
cos φ cos θ
−1
= kγ,z sz

Z =
˙
γz
ˆ

ˆ
with γ z ∈ R1

the reduced version designed to compensate only for persistent
terms ⇒ null steady state error wrt a minimal set of parameters!
(λz > 0, kvz > 0, K γ,z > O, kγ,z > 0)
Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
Horizontal controller
error
T

˜
η xy = xd − x yd − y ∈ R2
˙
˜
sxy = η xy + λxy η xy ∈ R2
˜

full version

reduced version

virtual input solutions of:

virtual input solutions of:

1
cφ sθ
ˆ
Rz (Φxy γ + kv,xy sxy )
=
−sφ
Z
˙
γ = K −1 ΦT sxy
ˆ
γ,xy xy
ˆ
with γ ∈ R16

1
cφ sθ
ˆ
Rz γ xy + kv,xy sxy
=
−sφ
Z
−1
˙
γ xy = kγ,xy sxy
ˆ
ˆ
with γ xy ∈ R2

again: the reduced version compensates only for persistent terms
⇒ null steady state error wrt a minimal set of parameters!
(λxy > 0, kv,xy > 0, K γ,xy > O, kγ,xy > 0)
Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
Orientation controller
The inputs are the desired roll, pitch and yaw
The commanded forces map onto the real ones according to
K

=

M
N

=
=

b˜C,y
rC,y
˜
r
Zc +
Nc
2
2d
b˜C,x
rC,x
˜
r
Zc −
Nc
Mc +
2
2d
Nc
Kc +

˜
Neither the altitude nor the yaw control loop are affected by rC , thus both Zc
and Nc convergence to a steady state value
Roll and pitch control can be designed by considering the estimation error as
an external, constant, disturbance:
K

=

M

=

1
b
Zc + Nc rC,y
˜
2
d
1
b
Mc +
Zc − Nc rC,x
˜
2
d
Kc +

The disturbance value is unknown and its effect may be compensated by
resorting to several adaptive control laws well known in the literature
Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
CoM estimation

PD control for roll and pitch =⇒ steady-state error because of the
wrong CoM estimate
A simple integral action can counteract this effect resulting a zero
steady-state error
˙
θ −θ
rC,x
ˆ
= −krC d
,
˙
φd − φ
rC,y
ˆ

krC > 0

As a byproduct, in absence of moment disturbance, the estimates
(ˆC,x , rC,y ) are driven towards the real CoM offsets (rC,x , rC,y )
r
ˆ

Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
Stability analysis
˜
ˆ
Altitude controller: let γ = γ − γ and consider the Lyapunov function
˜
V (sz , γ ) =

m 2 1 T
˜
˜
s + γ K γ,z γ
2 z 2

Along the system trajectories
˙
˙
˙
˜
˜
z
z
˜
ˆ
V (sz , γ ) = sz m¨d − m¨ + mλz z − γ T K γ,z γ
˙
˜
= sz (φz γ − cos φ cos θZ) − γ T K γ,z γ = −kvz s2 ≤ 0
ˆ
z
State trajectories are bounded
Asymptotic stability can be further proven by resorting to Barbalat’s
Lemma as in classical adaptive control schemes
Similar machinery for the horizontal controller case

Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
Experimental results
Experiments run at the Max Planck Institute of T¨bingen, Germany
u
case
a)
b)
c)
d)

additional
weight

Antonelli Cataldi Robuffo Chiaverini Franchi

weight
no
no
yes
yes

gain
λz
kvz
kγ,z
λxy
kv,xy
kγ,xy
kv,ϕθψ
kv,ϕθψ
krC

a/c
3
5.5
0
3
3
0
1
1
0

Tokyo, 5 November 2013

adaptive
no
yes
no
yes
b/d
3
5.5
1.5
3
3
1
1
1
.1
Experimental results
desired trajectory
1

η1,d [m]

0.5

0

−0.5

−1

−1.5
0

20

40

60

80

100

120

140

time [s]

Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
Experimental results
Norm of the 3D position errors for the cases a) (green) and b) (blue)
(no weight)
0.35

η 1,d − η 1 [m]

0.3
0.25
0.2
0.15
0.1
0.05
0
0

20

40

60

80

time [s]

Antonelli Cataldi Robuffo Chiaverini Franchi

100

120

140

Tokyo, 5 November 2013
Experimental results
Norm of the 3D position errors for the cases c) (green) and d) (blue)
(weight)
0.35

η 1,d − η 1 [m]

0.3
0.25
0.2
0.15
0.1
0.05
0
0

20

40

60

80

time [s]

Antonelli Cataldi Robuffo Chiaverini Franchi

100

120

140

Tokyo, 5 November 2013
Experimental results
Roll (top) and pitch (bottom) angles for cases c) (green) and d) (blue)

ϕ [deg]

2
0
−2
−4
0

20

40

20

40

60

80

100

120

140

60

80

100

120

140

time [s]

θ [deg]

10
5
0
−5
0

time [s]

Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013
Experimental results

N [Nm] M [Nm] K [Nm]

Z [N]

Control forces for the cases c) (green) and d) (blue)
−14
−16
0

20

40

60

80

100

120

140

0
0

20

40

60

80

100

120

140

0.2
0
−0.2
−0.4
0

20

40

60

80

100

120

140

0.01
0
−0.01
−0.02
−0.03
0

20

40

60

80

100

120

140

time [s]

0.5

Antonelli Cataldi Robuffo Chiaverini Franchi

time [s]

time [s]

time [s]

Tokyo, 5 November 2013
Experimental results

ˆ
r C [m]

ˆ
γ xy [N]

γz [N]
ˆ

Time history of the parameters estimates for the case d). Top:
parameter γz , center: parameter γ xy , bottom: parameter r C .
−14
−16
−18
0

20

40

20

40

20

40

60

80

100

120

140

60

80

100

120

140

60

80

100

120

140

time [s]

1
0
−1
0

time [s]

0.05
0
−0.05
0

Antonelli Cataldi Robuffo Chiaverini Franchi

time [s]

Tokyo, 5 November 2013
Experimental validation of a new adaptive
control scheme for quadrotors MAVs
Gianluca Antonelli† , Elisabetta Cataldi† ,
Paolo Robuffo Giordano⊕ , Stefano Chiaverini† , Antonio Franchi≀
† University

of Cassino and Southern Lazio, Italy
http://webuser.unicas.it/lai/robotica

⊕ CNRS at IRISA and Inria Bretagne Atlantique, France
http://www.irisa.fr/lagadic
≀ Max Planck Institute for Biol. Cybernetics, Germany
http://www.kyb.mpg.de/research/dep/bu/hri/

IROS 2013

Antonelli Cataldi Robuffo Chiaverini Franchi

Tokyo, 5 November 2013

More Related Content

What's hot

VKI_RVAD_2005_Application_of_a_Lattice_Boltzmann_Code
VKI_RVAD_2005_Application_of_a_Lattice_Boltzmann_CodeVKI_RVAD_2005_Application_of_a_Lattice_Boltzmann_Code
VKI_RVAD_2005_Application_of_a_Lattice_Boltzmann_CodeNorbert Gruen
 
A new geotechnical method for natural slope exploration
A new geotechnical method for natural slope explorationA new geotechnical method for natural slope exploration
A new geotechnical method for natural slope exploration
Rasika Athapaththu
 
電路學 - [第五章] 一階RC/RL電路
電路學 - [第五章] 一階RC/RL電路電路學 - [第五章] 一階RC/RL電路
電路學 - [第五章] 一階RC/RL電路
Simen Li
 
Sloshing-aware MPC for upper stage attitude control
Sloshing-aware MPC for upper stage attitude controlSloshing-aware MPC for upper stage attitude control
Sloshing-aware MPC for upper stage attitude control
Pantelis Sopasakis
 
Camera calibration from a single image based on coupled line cameras and rect...
Camera calibration from a single image based on coupled line cameras and rect...Camera calibration from a single image based on coupled line cameras and rect...
Camera calibration from a single image based on coupled line cameras and rect...
Joo-Haeng Lee
 
Wk 12 fr bode plot nyquist may 9 2016
Wk 12 fr bode plot nyquist   may 9 2016Wk 12 fr bode plot nyquist   may 9 2016
Wk 12 fr bode plot nyquist may 9 2016
Charlton Inao
 
A Fast Implicit Gaussian Curvature Filter
A Fast Implicit Gaussian Curvature FilterA Fast Implicit Gaussian Curvature Filter
A Fast Implicit Gaussian Curvature Filter
Yuanhao Gong
 
電路學 - [第三章] 網路定理
電路學 - [第三章] 網路定理電路學 - [第三章] 網路定理
電路學 - [第三章] 網路定理
Simen Li
 
Anatomy of human motion
Anatomy of human motionAnatomy of human motion
Anatomy of human motion
Wangdo Kim
 
Recursive Compressed Sensing
Recursive Compressed SensingRecursive Compressed Sensing
Recursive Compressed Sensing
Pantelis Sopasakis
 
Note chapter9 0708-edit-1
Note chapter9 0708-edit-1Note chapter9 0708-edit-1
Note chapter9 0708-edit-1
Muhammad Solehin
 
射頻電子 - [第四章] 散射參數網路
射頻電子 - [第四章] 散射參數網路射頻電子 - [第四章] 散射參數網路
射頻電子 - [第四章] 散射參數網路
Simen Li
 
Control and Guidance law for Guided Bomb
Control and Guidance law for Guided BombControl and Guidance law for Guided Bomb
Control and Guidance law for Guided Bomb
Mohammed Osman
 
Section 4 3_the_scattering_matrix_package
Section 4 3_the_scattering_matrix_packageSection 4 3_the_scattering_matrix_package
Section 4 3_the_scattering_matrix_package
Jamal Kazazi
 
Spatial and temporal study of a mechanical and harmonic vibration by high spe...
Spatial and temporal study of a mechanical and harmonic vibration by high spe...Spatial and temporal study of a mechanical and harmonic vibration by high spe...
Spatial and temporal study of a mechanical and harmonic vibration by high spe...IAEME Publication
 
Master Thesis Presentation
Master Thesis PresentationMaster Thesis Presentation
Master Thesis Presentation
EvaBernat4
 
HMPC for Upper Stage Attitude Control
HMPC for Upper Stage Attitude ControlHMPC for Upper Stage Attitude Control
HMPC for Upper Stage Attitude Control
Pantelis Sopasakis
 
CMSI計算科学技術特論C (2015) ALPS と量子多体問題①
CMSI計算科学技術特論C (2015) ALPS と量子多体問題①CMSI計算科学技術特論C (2015) ALPS と量子多体問題①
CMSI計算科学技術特論C (2015) ALPS と量子多体問題①
Computational Materials Science Initiative
 

What's hot (20)

VKI_RVAD_2005_Application_of_a_Lattice_Boltzmann_Code
VKI_RVAD_2005_Application_of_a_Lattice_Boltzmann_CodeVKI_RVAD_2005_Application_of_a_Lattice_Boltzmann_Code
VKI_RVAD_2005_Application_of_a_Lattice_Boltzmann_Code
 
A new geotechnical method for natural slope exploration
A new geotechnical method for natural slope explorationA new geotechnical method for natural slope exploration
A new geotechnical method for natural slope exploration
 
Bode lect
Bode lectBode lect
Bode lect
 
電路學 - [第五章] 一階RC/RL電路
電路學 - [第五章] 一階RC/RL電路電路學 - [第五章] 一階RC/RL電路
電路學 - [第五章] 一階RC/RL電路
 
Sloshing-aware MPC for upper stage attitude control
Sloshing-aware MPC for upper stage attitude controlSloshing-aware MPC for upper stage attitude control
Sloshing-aware MPC for upper stage attitude control
 
Camera calibration from a single image based on coupled line cameras and rect...
Camera calibration from a single image based on coupled line cameras and rect...Camera calibration from a single image based on coupled line cameras and rect...
Camera calibration from a single image based on coupled line cameras and rect...
 
Wk 12 fr bode plot nyquist may 9 2016
Wk 12 fr bode plot nyquist   may 9 2016Wk 12 fr bode plot nyquist   may 9 2016
Wk 12 fr bode plot nyquist may 9 2016
 
A Fast Implicit Gaussian Curvature Filter
A Fast Implicit Gaussian Curvature FilterA Fast Implicit Gaussian Curvature Filter
A Fast Implicit Gaussian Curvature Filter
 
Ec gate 13
Ec gate 13Ec gate 13
Ec gate 13
 
電路學 - [第三章] 網路定理
電路學 - [第三章] 網路定理電路學 - [第三章] 網路定理
電路學 - [第三章] 網路定理
 
Anatomy of human motion
Anatomy of human motionAnatomy of human motion
Anatomy of human motion
 
Recursive Compressed Sensing
Recursive Compressed SensingRecursive Compressed Sensing
Recursive Compressed Sensing
 
Note chapter9 0708-edit-1
Note chapter9 0708-edit-1Note chapter9 0708-edit-1
Note chapter9 0708-edit-1
 
射頻電子 - [第四章] 散射參數網路
射頻電子 - [第四章] 散射參數網路射頻電子 - [第四章] 散射參數網路
射頻電子 - [第四章] 散射參數網路
 
Control and Guidance law for Guided Bomb
Control and Guidance law for Guided BombControl and Guidance law for Guided Bomb
Control and Guidance law for Guided Bomb
 
Section 4 3_the_scattering_matrix_package
Section 4 3_the_scattering_matrix_packageSection 4 3_the_scattering_matrix_package
Section 4 3_the_scattering_matrix_package
 
Spatial and temporal study of a mechanical and harmonic vibration by high spe...
Spatial and temporal study of a mechanical and harmonic vibration by high spe...Spatial and temporal study of a mechanical and harmonic vibration by high spe...
Spatial and temporal study of a mechanical and harmonic vibration by high spe...
 
Master Thesis Presentation
Master Thesis PresentationMaster Thesis Presentation
Master Thesis Presentation
 
HMPC for Upper Stage Attitude Control
HMPC for Upper Stage Attitude ControlHMPC for Upper Stage Attitude Control
HMPC for Upper Stage Attitude Control
 
CMSI計算科学技術特論C (2015) ALPS と量子多体問題①
CMSI計算科学技術特論C (2015) ALPS と量子多体問題①CMSI計算科学技術特論C (2015) ALPS と量子多体問題①
CMSI計算科学技術特論C (2015) ALPS と量子多体問題①
 

Viewers also liked

ICRA 2012 talk
ICRA 2012 talkICRA 2012 talk
ICRA 2012 talk
Gianluca Antonelli
 
State estimate
State estimateState estimate
State estimate
Gianluca Antonelli
 
ICRA 2013 talk 1
ICRA 2013 talk 1ICRA 2013 talk 1
ICRA 2013 talk 1
Gianluca Antonelli
 
Computers assignment
Computers assignmentComputers assignment
Computers assignmentirut1
 
IROS 2014 talk
IROS 2014 talkIROS 2014 talk
IROS 2014 talk
Gianluca Antonelli
 
IROS 2011 talk 1 (Filippo's file)
IROS 2011 talk 1 (Filippo's file)IROS 2011 talk 1 (Filippo's file)
IROS 2011 talk 1 (Filippo's file)
Gianluca Antonelli
 
La importancia de la lectura en la educación preescolar.docx
La importancia de la lectura en la educación preescolar.docxLa importancia de la lectura en la educación preescolar.docx
La importancia de la lectura en la educación preescolar.docxbesos3006
 
IROS 2011 talk 2 (Filippo's file)
IROS 2011 talk 2 (Filippo's file)IROS 2011 talk 2 (Filippo's file)
IROS 2011 talk 2 (Filippo's file)
Gianluca Antonelli
 
IROS 2012 talk
IROS 2012 talkIROS 2012 talk
IROS 2012 talk
Gianluca Antonelli
 
TRIDENT school 2012
TRIDENT school 2012TRIDENT school 2012
TRIDENT school 2012
Gianluca Antonelli
 
Citologija uvodhemijskisastavsept2011-111002155646-phpapp01
Citologija uvodhemijskisastavsept2011-111002155646-phpapp01Citologija uvodhemijskisastavsept2011-111002155646-phpapp01
Citologija uvodhemijskisastavsept2011-111002155646-phpapp01Svetlana Palikruseva
 
02hemijskisastavivogsveta 100916121631-phpapp01
02hemijskisastavivogsveta 100916121631-phpapp0102hemijskisastavivogsveta 100916121631-phpapp01
02hemijskisastavivogsveta 100916121631-phpapp01Svetlana Palikruseva
 
ICRA 2013 talk 2
ICRA 2013 talk 2ICRA 2013 talk 2
ICRA 2013 talk 2
Gianluca Antonelli
 
La importancia de la lectura en la educación preescolar.docx
La importancia de la lectura en la educación preescolar.docxLa importancia de la lectura en la educación preescolar.docx
La importancia de la lectura en la educación preescolar.docxbesos3006
 
2015 SIDRA School on Underwater Robotics. Introduction
2015 SIDRA School on Underwater Robotics. Introduction2015 SIDRA School on Underwater Robotics. Introduction
2015 SIDRA School on Underwater Robotics. Introduction
Gianluca Antonelli
 

Viewers also liked (17)

ICRA 2012 talk
ICRA 2012 talkICRA 2012 talk
ICRA 2012 talk
 
State estimate
State estimateState estimate
State estimate
 
Survey
SurveySurvey
Survey
 
ICRA 2013 talk 1
ICRA 2013 talk 1ICRA 2013 talk 1
ICRA 2013 talk 1
 
Computers assignment
Computers assignmentComputers assignment
Computers assignment
 
IROS 2014 talk
IROS 2014 talkIROS 2014 talk
IROS 2014 talk
 
IROS 2011 talk 1 (Filippo's file)
IROS 2011 talk 1 (Filippo's file)IROS 2011 talk 1 (Filippo's file)
IROS 2011 talk 1 (Filippo's file)
 
La importancia de la lectura en la educación preescolar.docx
La importancia de la lectura en la educación preescolar.docxLa importancia de la lectura en la educación preescolar.docx
La importancia de la lectura en la educación preescolar.docx
 
IROS 2011 talk 2 (Filippo's file)
IROS 2011 talk 2 (Filippo's file)IROS 2011 talk 2 (Filippo's file)
IROS 2011 talk 2 (Filippo's file)
 
IROS 2012 talk
IROS 2012 talkIROS 2012 talk
IROS 2012 talk
 
TRIDENT school 2012
TRIDENT school 2012TRIDENT school 2012
TRIDENT school 2012
 
Citologija uvodhemijskisastavsept2011-111002155646-phpapp01
Citologija uvodhemijskisastavsept2011-111002155646-phpapp01Citologija uvodhemijskisastavsept2011-111002155646-phpapp01
Citologija uvodhemijskisastavsept2011-111002155646-phpapp01
 
02hemijskisastavivogsveta 100916121631-phpapp01
02hemijskisastavivogsveta 100916121631-phpapp0102hemijskisastavivogsveta 100916121631-phpapp01
02hemijskisastavivogsveta 100916121631-phpapp01
 
5s
5s5s
5s
 
ICRA 2013 talk 2
ICRA 2013 talk 2ICRA 2013 talk 2
ICRA 2013 talk 2
 
La importancia de la lectura en la educación preescolar.docx
La importancia de la lectura en la educación preescolar.docxLa importancia de la lectura en la educación preescolar.docx
La importancia de la lectura en la educación preescolar.docx
 
2015 SIDRA School on Underwater Robotics. Introduction
2015 SIDRA School on Underwater Robotics. Introduction2015 SIDRA School on Underwater Robotics. Introduction
2015 SIDRA School on Underwater Robotics. Introduction
 

Similar to IROS 2013 talk

CELEBRATION 2000 (3rd Deployment) & ALP 2002 Generation of a 3D-Model of the ...
CELEBRATION 2000 (3rd Deployment) & ALP 2002Generation of a 3D-Model of the ...CELEBRATION 2000 (3rd Deployment) & ALP 2002Generation of a 3D-Model of the ...
CELEBRATION 2000 (3rd Deployment) & ALP 2002 Generation of a 3D-Model of the ...
gigax2
 
EENG519FinalProjectReport
EENG519FinalProjectReportEENG519FinalProjectReport
EENG519FinalProjectReportDaniel K
 
BMC 2012 - Invited Talk
BMC 2012 - Invited TalkBMC 2012 - Invited Talk
BMC 2012 - Invited Talk
BOUWMANS Thierry
 
Platoon Control of Nonholonomic Robots using Quintic Bezier Splines
Platoon Control of Nonholonomic Robots using Quintic Bezier SplinesPlatoon Control of Nonholonomic Robots using Quintic Bezier Splines
Platoon Control of Nonholonomic Robots using Quintic Bezier Splines
Kaustav Mondal
 
Charged-Current Pion Production in T2K
Charged-Current Pion Production in T2KCharged-Current Pion Production in T2K
Charged-Current Pion Production in T2K
Son Cao
 
Kk2518251830
Kk2518251830Kk2518251830
Kk2518251830
IJERA Editor
 
Kk2518251830
Kk2518251830Kk2518251830
Kk2518251830
IJERA Editor
 
A walk through the intersection between machine learning and mechanistic mode...
A walk through the intersection between machine learning and mechanistic mode...A walk through the intersection between machine learning and mechanistic mode...
A walk through the intersection between machine learning and mechanistic mode...
JuanPabloCarbajal3
 
PROPAN - Propeller Panel Code
PROPAN - Propeller Panel CodePROPAN - Propeller Panel Code
PROPAN - Propeller Panel Code
João Baltazar
 
Application of parallel hierarchical matrices and low-rank tensors in spatial...
Application of parallel hierarchical matrices and low-rank tensors in spatial...Application of parallel hierarchical matrices and low-rank tensors in spatial...
Application of parallel hierarchical matrices and low-rank tensors in spatial...
Alexander Litvinenko
 
“Solving QCD: from BG/P to BG/Q”. Prof. Dr. Attilio Cucchieri – IFSC/USP.
“Solving QCD: from BG/P to BG/Q”. Prof. Dr. Attilio Cucchieri – IFSC/USP.“Solving QCD: from BG/P to BG/Q”. Prof. Dr. Attilio Cucchieri – IFSC/USP.
“Solving QCD: from BG/P to BG/Q”. Prof. Dr. Attilio Cucchieri – IFSC/USP.
lccausp
 
Data sparse approximation of Karhunen-Loeve Expansion
Data sparse approximation of Karhunen-Loeve ExpansionData sparse approximation of Karhunen-Loeve Expansion
Data sparse approximation of Karhunen-Loeve Expansion
Alexander Litvinenko
 
Data sparse approximation of the Karhunen-Loeve expansion
Data sparse approximation of the Karhunen-Loeve expansionData sparse approximation of the Karhunen-Loeve expansion
Data sparse approximation of the Karhunen-Loeve expansion
Alexander Litvinenko
 
Slides
SlidesSlides
A study of the worst case ratio of a simple algorithm for simple assembly lin...
A study of the worst case ratio of a simple algorithm for simple assembly lin...A study of the worst case ratio of a simple algorithm for simple assembly lin...
A study of the worst case ratio of a simple algorithm for simple assembly lin...narmo
 
Stochastic Alternating Direction Method of Multipliers
Stochastic Alternating Direction Method of MultipliersStochastic Alternating Direction Method of Multipliers
Stochastic Alternating Direction Method of Multipliers
Taiji Suzuki
 
Dynamic modelling and optimal controlscheme of wheel inverted pendulum for mo...
Dynamic modelling and optimal controlscheme of wheel inverted pendulum for mo...Dynamic modelling and optimal controlscheme of wheel inverted pendulum for mo...
Dynamic modelling and optimal controlscheme of wheel inverted pendulum for mo...
ijctcm
 
Actuator Constrained Optimal Control of Formations Near the Libration Points
Actuator Constrained Optimal Control of Formations Near the Libration PointsActuator Constrained Optimal Control of Formations Near the Libration Points
Actuator Constrained Optimal Control of Formations Near the Libration Points
Belinda Marchand
 
Improving EV Lateral Dynamics Control Using Infinity Norm Approach with Close...
Improving EV Lateral Dynamics Control Using Infinity Norm Approach with Close...Improving EV Lateral Dynamics Control Using Infinity Norm Approach with Close...
Improving EV Lateral Dynamics Control Using Infinity Norm Approach with Close...
Valerio Salvucci
 
Multiband Transceivers - [Chapter 1]
Multiband Transceivers - [Chapter 1] Multiband Transceivers - [Chapter 1]
Multiband Transceivers - [Chapter 1]
Simen Li
 

Similar to IROS 2013 talk (20)

CELEBRATION 2000 (3rd Deployment) & ALP 2002 Generation of a 3D-Model of the ...
CELEBRATION 2000 (3rd Deployment) & ALP 2002Generation of a 3D-Model of the ...CELEBRATION 2000 (3rd Deployment) & ALP 2002Generation of a 3D-Model of the ...
CELEBRATION 2000 (3rd Deployment) & ALP 2002 Generation of a 3D-Model of the ...
 
EENG519FinalProjectReport
EENG519FinalProjectReportEENG519FinalProjectReport
EENG519FinalProjectReport
 
BMC 2012 - Invited Talk
BMC 2012 - Invited TalkBMC 2012 - Invited Talk
BMC 2012 - Invited Talk
 
Platoon Control of Nonholonomic Robots using Quintic Bezier Splines
Platoon Control of Nonholonomic Robots using Quintic Bezier SplinesPlatoon Control of Nonholonomic Robots using Quintic Bezier Splines
Platoon Control of Nonholonomic Robots using Quintic Bezier Splines
 
Charged-Current Pion Production in T2K
Charged-Current Pion Production in T2KCharged-Current Pion Production in T2K
Charged-Current Pion Production in T2K
 
Kk2518251830
Kk2518251830Kk2518251830
Kk2518251830
 
Kk2518251830
Kk2518251830Kk2518251830
Kk2518251830
 
A walk through the intersection between machine learning and mechanistic mode...
A walk through the intersection between machine learning and mechanistic mode...A walk through the intersection between machine learning and mechanistic mode...
A walk through the intersection between machine learning and mechanistic mode...
 
PROPAN - Propeller Panel Code
PROPAN - Propeller Panel CodePROPAN - Propeller Panel Code
PROPAN - Propeller Panel Code
 
Application of parallel hierarchical matrices and low-rank tensors in spatial...
Application of parallel hierarchical matrices and low-rank tensors in spatial...Application of parallel hierarchical matrices and low-rank tensors in spatial...
Application of parallel hierarchical matrices and low-rank tensors in spatial...
 
“Solving QCD: from BG/P to BG/Q”. Prof. Dr. Attilio Cucchieri – IFSC/USP.
“Solving QCD: from BG/P to BG/Q”. Prof. Dr. Attilio Cucchieri – IFSC/USP.“Solving QCD: from BG/P to BG/Q”. Prof. Dr. Attilio Cucchieri – IFSC/USP.
“Solving QCD: from BG/P to BG/Q”. Prof. Dr. Attilio Cucchieri – IFSC/USP.
 
Data sparse approximation of Karhunen-Loeve Expansion
Data sparse approximation of Karhunen-Loeve ExpansionData sparse approximation of Karhunen-Loeve Expansion
Data sparse approximation of Karhunen-Loeve Expansion
 
Data sparse approximation of the Karhunen-Loeve expansion
Data sparse approximation of the Karhunen-Loeve expansionData sparse approximation of the Karhunen-Loeve expansion
Data sparse approximation of the Karhunen-Loeve expansion
 
Slides
SlidesSlides
Slides
 
A study of the worst case ratio of a simple algorithm for simple assembly lin...
A study of the worst case ratio of a simple algorithm for simple assembly lin...A study of the worst case ratio of a simple algorithm for simple assembly lin...
A study of the worst case ratio of a simple algorithm for simple assembly lin...
 
Stochastic Alternating Direction Method of Multipliers
Stochastic Alternating Direction Method of MultipliersStochastic Alternating Direction Method of Multipliers
Stochastic Alternating Direction Method of Multipliers
 
Dynamic modelling and optimal controlscheme of wheel inverted pendulum for mo...
Dynamic modelling and optimal controlscheme of wheel inverted pendulum for mo...Dynamic modelling and optimal controlscheme of wheel inverted pendulum for mo...
Dynamic modelling and optimal controlscheme of wheel inverted pendulum for mo...
 
Actuator Constrained Optimal Control of Formations Near the Libration Points
Actuator Constrained Optimal Control of Formations Near the Libration PointsActuator Constrained Optimal Control of Formations Near the Libration Points
Actuator Constrained Optimal Control of Formations Near the Libration Points
 
Improving EV Lateral Dynamics Control Using Infinity Norm Approach with Close...
Improving EV Lateral Dynamics Control Using Infinity Norm Approach with Close...Improving EV Lateral Dynamics Control Using Infinity Norm Approach with Close...
Improving EV Lateral Dynamics Control Using Infinity Norm Approach with Close...
 
Multiband Transceivers - [Chapter 1]
Multiband Transceivers - [Chapter 1] Multiband Transceivers - [Chapter 1]
Multiband Transceivers - [Chapter 1]
 

More from Gianluca Antonelli

Current research activities in marine robotics at the Italian interuniversity...
Current research activities in marine robotics at the Italian interuniversity...Current research activities in marine robotics at the Italian interuniversity...
Current research activities in marine robotics at the Italian interuniversity...
Gianluca Antonelli
 
Workshop on Marine Robotics: vehicle-manipulator systems
Workshop on Marine Robotics: vehicle-manipulator systemsWorkshop on Marine Robotics: vehicle-manipulator systems
Workshop on Marine Robotics: vehicle-manipulator systems
Gianluca Antonelli
 
Underwater manipulation
Underwater manipulationUnderwater manipulation
Underwater manipulation
Gianluca Antonelli
 
Workshop Italy-Japan
Workshop Italy-JapanWorkshop Italy-Japan
Workshop Italy-Japan
Gianluca Antonelli
 
Control problems for floating-base manipulators
Control problems for floating-base manipulatorsControl problems for floating-base manipulators
Control problems for floating-base manipulatorsGianluca Antonelli
 
Control problems for floating-base manipulators
Control problems for floating-base manipulatorsControl problems for floating-base manipulators
Control problems for floating-base manipulators
Gianluca Antonelli
 
SIDRA 2012 talk
SIDRA 2012 talkSIDRA 2012 talk
SIDRA 2012 talk
Gianluca Antonelli
 

More from Gianluca Antonelli (7)

Current research activities in marine robotics at the Italian interuniversity...
Current research activities in marine robotics at the Italian interuniversity...Current research activities in marine robotics at the Italian interuniversity...
Current research activities in marine robotics at the Italian interuniversity...
 
Workshop on Marine Robotics: vehicle-manipulator systems
Workshop on Marine Robotics: vehicle-manipulator systemsWorkshop on Marine Robotics: vehicle-manipulator systems
Workshop on Marine Robotics: vehicle-manipulator systems
 
Underwater manipulation
Underwater manipulationUnderwater manipulation
Underwater manipulation
 
Workshop Italy-Japan
Workshop Italy-JapanWorkshop Italy-Japan
Workshop Italy-Japan
 
Control problems for floating-base manipulators
Control problems for floating-base manipulatorsControl problems for floating-base manipulators
Control problems for floating-base manipulators
 
Control problems for floating-base manipulators
Control problems for floating-base manipulatorsControl problems for floating-base manipulators
Control problems for floating-base manipulators
 
SIDRA 2012 talk
SIDRA 2012 talkSIDRA 2012 talk
SIDRA 2012 talk
 

Recently uploaded

This Is The First All Category Quiz That I Made
This Is The First All Category Quiz That I MadeThis Is The First All Category Quiz That I Made
This Is The First All Category Quiz That I Made
Aarush Ghate
 
Treasure Hunt Puzzles, Treasure Hunt Puzzles online
Treasure Hunt Puzzles, Treasure Hunt Puzzles onlineTreasure Hunt Puzzles, Treasure Hunt Puzzles online
Treasure Hunt Puzzles, Treasure Hunt Puzzles online
Hidden Treasure Hunts
 
A TO Z INDIA Monthly Magazine - JUNE 2024
A TO Z INDIA Monthly Magazine - JUNE 2024A TO Z INDIA Monthly Magazine - JUNE 2024
A TO Z INDIA Monthly Magazine - JUNE 2024
Indira Srivatsa
 
Meet Dinah Mattingly – Larry Bird’s Partner in Life and Love
Meet Dinah Mattingly – Larry Bird’s Partner in Life and LoveMeet Dinah Mattingly – Larry Bird’s Partner in Life and Love
Meet Dinah Mattingly – Larry Bird’s Partner in Life and Love
get joys
 
The Evolution of Animation in Film - Mark Murphy Director
The Evolution of Animation in Film - Mark Murphy DirectorThe Evolution of Animation in Film - Mark Murphy Director
The Evolution of Animation in Film - Mark Murphy Director
Mark Murphy Director
 
From Slave to Scourge: The Existential Choice of Django Unchained. The Philos...
From Slave to Scourge: The Existential Choice of Django Unchained. The Philos...From Slave to Scourge: The Existential Choice of Django Unchained. The Philos...
From Slave to Scourge: The Existential Choice of Django Unchained. The Philos...
Rodney Thomas Jr
 
Reimagining Classics - What Makes a Remake a Success
Reimagining Classics - What Makes a Remake a SuccessReimagining Classics - What Makes a Remake a Success
Reimagining Classics - What Makes a Remake a Success
Mark Murphy Director
 
Skeem Saam in June 2024 available on Forum
Skeem Saam in June 2024 available on ForumSkeem Saam in June 2024 available on Forum
Skeem Saam in June 2024 available on Forum
Isaac More
 
I Know Dino Trivia: Part 3. Test your dino knowledge
I Know Dino Trivia: Part 3. Test your dino knowledgeI Know Dino Trivia: Part 3. Test your dino knowledge
I Know Dino Trivia: Part 3. Test your dino knowledge
Sabrina Ricci
 
Young Tom Selleck: A Journey Through His Early Years and Rise to Stardom
Young Tom Selleck: A Journey Through His Early Years and Rise to StardomYoung Tom Selleck: A Journey Through His Early Years and Rise to Stardom
Young Tom Selleck: A Journey Through His Early Years and Rise to Stardom
greendigital
 
240529_Teleprotection Global Market Report 2024.pdf
240529_Teleprotection Global Market Report 2024.pdf240529_Teleprotection Global Market Report 2024.pdf
240529_Teleprotection Global Market Report 2024.pdf
Madhura TBRC
 
Meet Crazyjamjam - A TikTok Sensation | Blog Eternal
Meet Crazyjamjam - A TikTok Sensation | Blog EternalMeet Crazyjamjam - A TikTok Sensation | Blog Eternal
Meet Crazyjamjam - A TikTok Sensation | Blog Eternal
Blog Eternal
 
Tom Selleck Net Worth: A Comprehensive Analysis
Tom Selleck Net Worth: A Comprehensive AnalysisTom Selleck Net Worth: A Comprehensive Analysis
Tom Selleck Net Worth: A Comprehensive Analysis
greendigital
 
Hollywood Actress - The 250 hottest gallery
Hollywood Actress - The 250 hottest galleryHollywood Actress - The 250 hottest gallery
Hollywood Actress - The 250 hottest gallery
Zsolt Nemeth
 
Create a Seamless Viewing Experience with Your Own Custom OTT Player.pdf
Create a Seamless Viewing Experience with Your Own Custom OTT Player.pdfCreate a Seamless Viewing Experience with Your Own Custom OTT Player.pdf
Create a Seamless Viewing Experience with Your Own Custom OTT Player.pdf
Genny Knight
 
Maximizing Your Streaming Experience with XCIPTV- Tips for 2024.pdf
Maximizing Your Streaming Experience with XCIPTV- Tips for 2024.pdfMaximizing Your Streaming Experience with XCIPTV- Tips for 2024.pdf
Maximizing Your Streaming Experience with XCIPTV- Tips for 2024.pdf
Xtreame HDTV
 
Christina's Baby Shower Game June 2024.pptx
Christina's Baby Shower Game June 2024.pptxChristina's Baby Shower Game June 2024.pptx
Christina's Baby Shower Game June 2024.pptx
madeline604788
 
Panchayat Season 3 - Official Trailer.pdf
Panchayat Season 3 - Official Trailer.pdfPanchayat Season 3 - Official Trailer.pdf
Panchayat Season 3 - Official Trailer.pdf
Suleman Rana
 
Scandal! Teasers June 2024 on etv Forum.co.za
Scandal! Teasers June 2024 on etv Forum.co.zaScandal! Teasers June 2024 on etv Forum.co.za
Scandal! Teasers June 2024 on etv Forum.co.za
Isaac More
 

Recently uploaded (19)

This Is The First All Category Quiz That I Made
This Is The First All Category Quiz That I MadeThis Is The First All Category Quiz That I Made
This Is The First All Category Quiz That I Made
 
Treasure Hunt Puzzles, Treasure Hunt Puzzles online
Treasure Hunt Puzzles, Treasure Hunt Puzzles onlineTreasure Hunt Puzzles, Treasure Hunt Puzzles online
Treasure Hunt Puzzles, Treasure Hunt Puzzles online
 
A TO Z INDIA Monthly Magazine - JUNE 2024
A TO Z INDIA Monthly Magazine - JUNE 2024A TO Z INDIA Monthly Magazine - JUNE 2024
A TO Z INDIA Monthly Magazine - JUNE 2024
 
Meet Dinah Mattingly – Larry Bird’s Partner in Life and Love
Meet Dinah Mattingly – Larry Bird’s Partner in Life and LoveMeet Dinah Mattingly – Larry Bird’s Partner in Life and Love
Meet Dinah Mattingly – Larry Bird’s Partner in Life and Love
 
The Evolution of Animation in Film - Mark Murphy Director
The Evolution of Animation in Film - Mark Murphy DirectorThe Evolution of Animation in Film - Mark Murphy Director
The Evolution of Animation in Film - Mark Murphy Director
 
From Slave to Scourge: The Existential Choice of Django Unchained. The Philos...
From Slave to Scourge: The Existential Choice of Django Unchained. The Philos...From Slave to Scourge: The Existential Choice of Django Unchained. The Philos...
From Slave to Scourge: The Existential Choice of Django Unchained. The Philos...
 
Reimagining Classics - What Makes a Remake a Success
Reimagining Classics - What Makes a Remake a SuccessReimagining Classics - What Makes a Remake a Success
Reimagining Classics - What Makes a Remake a Success
 
Skeem Saam in June 2024 available on Forum
Skeem Saam in June 2024 available on ForumSkeem Saam in June 2024 available on Forum
Skeem Saam in June 2024 available on Forum
 
I Know Dino Trivia: Part 3. Test your dino knowledge
I Know Dino Trivia: Part 3. Test your dino knowledgeI Know Dino Trivia: Part 3. Test your dino knowledge
I Know Dino Trivia: Part 3. Test your dino knowledge
 
Young Tom Selleck: A Journey Through His Early Years and Rise to Stardom
Young Tom Selleck: A Journey Through His Early Years and Rise to StardomYoung Tom Selleck: A Journey Through His Early Years and Rise to Stardom
Young Tom Selleck: A Journey Through His Early Years and Rise to Stardom
 
240529_Teleprotection Global Market Report 2024.pdf
240529_Teleprotection Global Market Report 2024.pdf240529_Teleprotection Global Market Report 2024.pdf
240529_Teleprotection Global Market Report 2024.pdf
 
Meet Crazyjamjam - A TikTok Sensation | Blog Eternal
Meet Crazyjamjam - A TikTok Sensation | Blog EternalMeet Crazyjamjam - A TikTok Sensation | Blog Eternal
Meet Crazyjamjam - A TikTok Sensation | Blog Eternal
 
Tom Selleck Net Worth: A Comprehensive Analysis
Tom Selleck Net Worth: A Comprehensive AnalysisTom Selleck Net Worth: A Comprehensive Analysis
Tom Selleck Net Worth: A Comprehensive Analysis
 
Hollywood Actress - The 250 hottest gallery
Hollywood Actress - The 250 hottest galleryHollywood Actress - The 250 hottest gallery
Hollywood Actress - The 250 hottest gallery
 
Create a Seamless Viewing Experience with Your Own Custom OTT Player.pdf
Create a Seamless Viewing Experience with Your Own Custom OTT Player.pdfCreate a Seamless Viewing Experience with Your Own Custom OTT Player.pdf
Create a Seamless Viewing Experience with Your Own Custom OTT Player.pdf
 
Maximizing Your Streaming Experience with XCIPTV- Tips for 2024.pdf
Maximizing Your Streaming Experience with XCIPTV- Tips for 2024.pdfMaximizing Your Streaming Experience with XCIPTV- Tips for 2024.pdf
Maximizing Your Streaming Experience with XCIPTV- Tips for 2024.pdf
 
Christina's Baby Shower Game June 2024.pptx
Christina's Baby Shower Game June 2024.pptxChristina's Baby Shower Game June 2024.pptx
Christina's Baby Shower Game June 2024.pptx
 
Panchayat Season 3 - Official Trailer.pdf
Panchayat Season 3 - Official Trailer.pdfPanchayat Season 3 - Official Trailer.pdf
Panchayat Season 3 - Official Trailer.pdf
 
Scandal! Teasers June 2024 on etv Forum.co.za
Scandal! Teasers June 2024 on etv Forum.co.zaScandal! Teasers June 2024 on etv Forum.co.za
Scandal! Teasers June 2024 on etv Forum.co.za
 

IROS 2013 talk

  • 1. Experimental validation of a new adaptive control scheme for quadrotors MAVs Gianluca Antonelli† , Elisabetta Cataldi† , Paolo Robuffo Giordano⊕ , Stefano Chiaverini† , Antonio Franchi≀ † University of Cassino and Southern Lazio, Italy http://webuser.unicas.it/lai/robotica ⊕ CNRS at IRISA and Inria Bretagne Atlantique, France http://www.irisa.fr/lagadic ≀ Max Planck Institute for Biol. Cybernetics, Germany http://www.kyb.mpg.de/research/dep/bu/hri/ IROS 2013 Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 2. Trajectory tracking control for quadrotor Adaptive with respect to uncertainty in total mass uncertainty in Center Of Mass (CoM) presence of 6-DOF external disturbances Assumption: closed-loop orientation dynamics faster than translational one Stability analysis & numerical simulations1 Experimental results 1 Antonelli, Arrichiello, Chiaverini, Robuffo Giordano, “Adaptive trajectory tracking for quadrotor MAVs in presence of parameter uncertainties and external disturbances”, AIM 2013 Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 3. Kinematics body-fixed u, surge xb p, roll Ob q, pitch η1 r, yaw yb v, sway zb w, heave η1 = η2 = O x earth-fixed x y z φ θ ψ ν 1 = RB η 1 I ˙ ˙ ν 2 = T (η 2 )η 2 y z Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013 T T
  • 4. Dynamics Mathematical model expressed in body-fixed frame ˙ M ν + C(ν)ν + τ W + g(RB ) = τ , I beyond the common terms, we model τW = γ W ∈ R6 external disturbance RB O 3×3 I γW O 3×3 RB I constant in the inertial frame (wind) Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 5. Dynamics -2Exploiting the linearity in the parameters ˙ Φ(ν, ν, RB )γ = τ I and rewriting with respect to the inertial frame while separating the xy dynamics from z: ˙ ¨ Φxy (η, η, η ) γ = RI τ 1 B ˙ ¨ φz (η, η, η ) with γ ∈ R16 : mass (1 parameter) first moment of inertia (3 p.) inertia tensor (6 p.) τ = τ1 τ2 external disturbance (6 p.) Antonelli Cataldi Robuffo Chiaverini Franchi  0 0   Z  =  K    M   N Tokyo, 5 November 2013
  • 6. Thrust Assuming CoM coincident with Ob f2 2 τt,3 l f3 ωt,2 f4 yb z b ωt,4 τt,4 4 τt,2 f1 Ob ωt,3 3 2 fi = bωt,i ωt,1 xb 1 O y τt,1 x z Antonelli Cataldi Robuffo Chiaverini Franchi 2 τt,i = dωt,i  0  0  τ1 =  4  i=1   fi      l(f2 − f4 )  l(f1 − f3 ) τ2 =  −τt,1 + τt,2 − τt,3 + τt,4 Tokyo, 5 November 2013
  • 7. Mapping from the angular velocities to the force-torques Assuming CoM of coordinates r C :    2  ωt,1 Z 2 K  ωt,2    = Bv   M  ω 2  t,3 2 N ωt,4 with   b b b b  0 b(l + rC,y ) 0 −b(l − rC,y )  Bv =  b(l + rC,x )  0 −b(l − rC,x ) 0 −d d −d d CoM influences the mapping from thrust generated from the motors to the vehicle forces/moments Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 8. Inverse mapping Any controller determines a control action Zc Kc Mc Nc projected onto the motor input u ∈ R4   Zc  Kc  u = B −1   v  Mc  Nc where B −1 ∈ R4×4 is v  B −1 v l − rC,x  4bl l − r C,y    4bl = l + r C,x   4bl  l + rC,y 4bl 0 1 2bl 0 1 − 2bl Antonelli Cataldi Robuffo Chiaverini Franchi 1 2bl 0 1 − 2bl 0  l − rC,x − 4dl  l − rC,y    4dl  l + rC,x   − 4dl  l + rC,y  4dl Tokyo, 5 November 2013 T further
  • 9. Current inverse mapping ˆ When the CoM position estimate r C is affected mapping becomes  1 0     Z Zc  rC,y ˜ 1 K   Kc     = B v | B −1    2 ˆ r C v r C Mc  =  rC,x M  ˜  0  2 N Nc 0 0 by an error, the real  0   b˜C,y  Zc r   0 2d   Kc  b˜C,x  Mc  r  1 −  2d Nc 0 1 0 wrong CoM estimate ⇒ a coupling from altitude and yaw control actions onto roll and pitch dynamics Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 10. Controller block diagram  Z K    M  N  η 1d ψd Zc pos φd , θd   Kc Mc  B −1 v or Nc u motors 2 wt,i τW η Bv plant Classical MAV control architecture with adaptation wrt parameters and compensation of the CoM position Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 11. Altitude controller error z = zd − z ∈ R ˜ ˙ s z = z + λz z ∈ R ˜ ˜ full version 1 ˆ (φ γ + kvz sz ) cos φ cos θ z ˙ γ = K −1 φT sz ˆ γ,z z Z = ˆ with γ ∈ R16 reduced version 1 (ˆz + kvz sz ) γ cos φ cos θ −1 = kγ,z sz Z = ˙ γz ˆ ˆ with γ z ∈ R1 the reduced version designed to compensate only for persistent terms ⇒ null steady state error wrt a minimal set of parameters! (λz > 0, kvz > 0, K γ,z > O, kγ,z > 0) Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 12. Horizontal controller error T ˜ η xy = xd − x yd − y ∈ R2 ˙ ˜ sxy = η xy + λxy η xy ∈ R2 ˜ full version reduced version virtual input solutions of: virtual input solutions of: 1 cφ sθ ˆ Rz (Φxy γ + kv,xy sxy ) = −sφ Z ˙ γ = K −1 ΦT sxy ˆ γ,xy xy ˆ with γ ∈ R16 1 cφ sθ ˆ Rz γ xy + kv,xy sxy = −sφ Z −1 ˙ γ xy = kγ,xy sxy ˆ ˆ with γ xy ∈ R2 again: the reduced version compensates only for persistent terms ⇒ null steady state error wrt a minimal set of parameters! (λxy > 0, kv,xy > 0, K γ,xy > O, kγ,xy > 0) Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 13. Orientation controller The inputs are the desired roll, pitch and yaw The commanded forces map onto the real ones according to K = M N = = b˜C,y rC,y ˜ r Zc + Nc 2 2d b˜C,x rC,x ˜ r Zc − Nc Mc + 2 2d Nc Kc + ˜ Neither the altitude nor the yaw control loop are affected by rC , thus both Zc and Nc convergence to a steady state value Roll and pitch control can be designed by considering the estimation error as an external, constant, disturbance: K = M = 1 b Zc + Nc rC,y ˜ 2 d 1 b Mc + Zc − Nc rC,x ˜ 2 d Kc + The disturbance value is unknown and its effect may be compensated by resorting to several adaptive control laws well known in the literature Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 14. CoM estimation PD control for roll and pitch =⇒ steady-state error because of the wrong CoM estimate A simple integral action can counteract this effect resulting a zero steady-state error ˙ θ −θ rC,x ˆ = −krC d , ˙ φd − φ rC,y ˆ krC > 0 As a byproduct, in absence of moment disturbance, the estimates (ˆC,x , rC,y ) are driven towards the real CoM offsets (rC,x , rC,y ) r ˆ Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 15. Stability analysis ˜ ˆ Altitude controller: let γ = γ − γ and consider the Lyapunov function ˜ V (sz , γ ) = m 2 1 T ˜ ˜ s + γ K γ,z γ 2 z 2 Along the system trajectories ˙ ˙ ˙ ˜ ˜ z z ˜ ˆ V (sz , γ ) = sz m¨d − m¨ + mλz z − γ T K γ,z γ ˙ ˜ = sz (φz γ − cos φ cos θZ) − γ T K γ,z γ = −kvz s2 ≤ 0 ˆ z State trajectories are bounded Asymptotic stability can be further proven by resorting to Barbalat’s Lemma as in classical adaptive control schemes Similar machinery for the horizontal controller case Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 16. Experimental results Experiments run at the Max Planck Institute of T¨bingen, Germany u case a) b) c) d) additional weight Antonelli Cataldi Robuffo Chiaverini Franchi weight no no yes yes gain λz kvz kγ,z λxy kv,xy kγ,xy kv,ϕθψ kv,ϕθψ krC a/c 3 5.5 0 3 3 0 1 1 0 Tokyo, 5 November 2013 adaptive no yes no yes b/d 3 5.5 1.5 3 3 1 1 1 .1
  • 17. Experimental results desired trajectory 1 η1,d [m] 0.5 0 −0.5 −1 −1.5 0 20 40 60 80 100 120 140 time [s] Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 18. Experimental results Norm of the 3D position errors for the cases a) (green) and b) (blue) (no weight) 0.35 η 1,d − η 1 [m] 0.3 0.25 0.2 0.15 0.1 0.05 0 0 20 40 60 80 time [s] Antonelli Cataldi Robuffo Chiaverini Franchi 100 120 140 Tokyo, 5 November 2013
  • 19. Experimental results Norm of the 3D position errors for the cases c) (green) and d) (blue) (weight) 0.35 η 1,d − η 1 [m] 0.3 0.25 0.2 0.15 0.1 0.05 0 0 20 40 60 80 time [s] Antonelli Cataldi Robuffo Chiaverini Franchi 100 120 140 Tokyo, 5 November 2013
  • 20. Experimental results Roll (top) and pitch (bottom) angles for cases c) (green) and d) (blue) ϕ [deg] 2 0 −2 −4 0 20 40 20 40 60 80 100 120 140 60 80 100 120 140 time [s] θ [deg] 10 5 0 −5 0 time [s] Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013
  • 21. Experimental results N [Nm] M [Nm] K [Nm] Z [N] Control forces for the cases c) (green) and d) (blue) −14 −16 0 20 40 60 80 100 120 140 0 0 20 40 60 80 100 120 140 0.2 0 −0.2 −0.4 0 20 40 60 80 100 120 140 0.01 0 −0.01 −0.02 −0.03 0 20 40 60 80 100 120 140 time [s] 0.5 Antonelli Cataldi Robuffo Chiaverini Franchi time [s] time [s] time [s] Tokyo, 5 November 2013
  • 22. Experimental results ˆ r C [m] ˆ γ xy [N] γz [N] ˆ Time history of the parameters estimates for the case d). Top: parameter γz , center: parameter γ xy , bottom: parameter r C . −14 −16 −18 0 20 40 20 40 20 40 60 80 100 120 140 60 80 100 120 140 60 80 100 120 140 time [s] 1 0 −1 0 time [s] 0.05 0 −0.05 0 Antonelli Cataldi Robuffo Chiaverini Franchi time [s] Tokyo, 5 November 2013
  • 23. Experimental validation of a new adaptive control scheme for quadrotors MAVs Gianluca Antonelli† , Elisabetta Cataldi† , Paolo Robuffo Giordano⊕ , Stefano Chiaverini† , Antonio Franchi≀ † University of Cassino and Southern Lazio, Italy http://webuser.unicas.it/lai/robotica ⊕ CNRS at IRISA and Inria Bretagne Atlantique, France http://www.irisa.fr/lagadic ≀ Max Planck Institute for Biol. Cybernetics, Germany http://www.kyb.mpg.de/research/dep/bu/hri/ IROS 2013 Antonelli Cataldi Robuffo Chiaverini Franchi Tokyo, 5 November 2013