IROS 2013 talk

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, Italyhttp://webuser.unicas.it/lai/robotica

⊕CNRS at IRISA and Inria Bretagne Atlantique, Francehttp://www.irisa.fr/lagadic

≀Max Planck Institute for Biol. Cybernetics, Germanyhttp://www.kyb.mpg.de/research/dep/bu/hri/

IROS 2013

Antonelli CataldiRobuffoChiaverini Franchi Tokyo, 5 November 2013

Trajectory tracking control for quadrotor

Adaptive with respect to

uncertainty in total massuncertainty in Center Of Mass (CoM)presence of 6-DOF external disturbances

Assumption: closed-loop orientation dynamics faster thantranslational one

Stability analysis & numerical simulations1

Experimental results

1Antonelli, Arrichiello, Chiaverini, Robuffo Giordano, “Adaptivetrajectory tracking for quadrotor MAVs in presence of parameteruncertainties and external disturbances”, AIM 2013


Kinematics

earth-fixed

O x

z

y

η1

body-fixed

Ob

xb

zb

yb

u, surge

w, heave

v, sway

p, roll

r, yaw

q, pitch η1 =[

x y z]T

η2 =[

φ θ ψ]T

ν1 = RBI η1

ν2 = T (η2)η2


Dynamics

Mathematical model expressed in body-fixed frame

Mν +C(ν)ν + τW + g(RBI ) = τ ,

beyond the common terms, we model

τW =

[

RBI O3×3

O3×3 RBI

]

γW

γW ∈R6 external disturbance constant in the inertial frame (wind)


Dynamics -2-

Exploiting the linearity in the parameters

Φ(ν,ν,RBI )γ = τ

and rewriting with respect to the inertial frame while separating thexy dynamics from z:

[

Φxy(η, η, η)φz(η, η, η)

]

γ = RIBτ 1

with γ ∈ R16:

mass (1 parameter)

first moment of inertia (3 p.)

inertia tensor (6 p.)

external disturbance (6 p.)

τ =

[

τ 1

τ 2

]

=

00Z

K

M

N


Thrust

Assuming CoM coincident with Ob

xy

z

O

xbyb zb

Ob

f1

f2

f3

f41

2

3

4

ωt,1

ωt,2

ωt,3

ωt,4

τt,1

τt,2

τt,3

τt,4

l

fi = bω2t,i

τt,i = dω2t,i

τ 1 =

00

4∑

i=1

fi

τ 2 =

l(f2 − f4)l(f1 − f3)

−τt,1 + τt,2 − τt,3 + τt,4


Mapping from the angular velocities to the force-torques

Assuming CoM of coordinates rC :

Z

K

M

N

= Bv

ω2t,1

ω2t,2

ω2t,3

ω2t,4

with

Bv=

b b b b

0 b(l + rC,y) 0 −b(l − rC,y)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


Inverse mapping

Any controller determines a control action[

Zc Kc Mc Nc

]Tfurther

projected onto the motor input u ∈ R4

u = B−1v

Zc

Kc

Mc

Nc

where B−1v ∈ R

4×4 is

B−1v =

l − rC,x

4bl0

1

2bl−l − rC,x

4dll − rC,y

4bl

1

2bl0

l − rC,y

4dll + rC,x

4bl0 −

1

2bl−l + rC,x

4dll + rC,y

4bl−

1

2bl0

l + rC,y

4dl


Current inverse mapping

When the CoM position estimate rC is affected by an error, the real

mapping becomes

Z

K

M

N

= Bv|rC B−1v

∣

∣

rC

Zc

Kc

Mc

Nc

=

1 0 0 0rC,y

21 0

brC,y

2drC,x

20 1 −

brC,x

2d0 0 0 1

Zc

Kc

Mc

Nc

wrong CoM estimate ⇒ a coupling from altitude and yaw control

actions onto roll and pitch dynamics


Controller block diagram

η1d

ψdφd, θd

Zc

posor

Kc

Mc

Nc

B−1

v

umotors

w2

t,iBv

Z

K

M

N

τW

η

plant

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


Altitude controller

error

z = zd − z ∈ R

sz = ˙z + λz z ∈ R

full version

Z =1

cosφ cos θ(φzγ + kvzsz)

˙γ = K−1γ,zφ

Tz sz

with γ ∈ R16

reduced version

Z =1

cosφ cos θ(γz + kvzsz)

˙γz = k−1γ,zsz

with γz ∈ 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,Kγ,z > O, kγ,z > 0)


Horizontal controller

error

ηxy =[

xd − x yd − y]T

∈ R2

sxy = ˙ηxy + λxyηxy ∈ R2

full version

virtual input solutions of:

[

cφsθ−sφ

]

=1

ZRz (Φxyγ + kv,xysxy)

˙γ = K−1γ,xyΦ

Txysxy

with γ ∈ R16

reduced version

virtual input solutions of:

[

cφsθ−sφ

]

=1

ZRz

(

γxy + kv,xysxy)

˙γxy = k−1γ,xysxy

with γxy ∈ 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,Kγ,xy > O, kγ,xy > 0)


Orientation controller

The inputs are the desired roll, pitch and yawThe commanded forces map onto the real ones according to

K = Kc +rC,y

2Zc +

brC,y

2dNc

M = Mc +rC,x

2Zc −

brC,x

2dNc

N = Nc

Neither the altitude nor the yaw control loop are affected by rC , thus both Zc

and Nc convergence to a steady state valueRoll and pitch control can be designed by considering the estimation error asan external, constant, disturbance:

K = Kc +1

2

(

Zc +b

dNc

)

rC,y

M = Mc +1

2

(

Zc −b

dNc

)

rC,x

The disturbance value is unknown and its effect may be compensated by

resorting to several adaptive control laws well known in the literature


CoM estimation

PD control for roll and pitch =⇒ steady-state error because of thewrong CoM estimateA simple integral action can counteract this effect resulting a zerosteady-state error

[

˙rC,x

˙rC,y

]

= −krC

[

θd − θ

φd − φ

]

, krC > 0

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


Stability analysis

Altitude controller: let γ = γ − γ and consider the Lyapunov function

V (sz, γ) =m

2s2z +

1

2γTKγ,zγ

Along the system trajectories

V (sz, γ) = sz(

mzd −mz +mλz ˙z)

− γTKγ,z˙γ

= sz (φzγ − cosφ cos θZ)− γTKγ,z˙γ = −kvzs

2z ≤ 0

State trajectories are boundedAsymptotic stability can be further proven by resorting to Barbalat’sLemma as in classical adaptive control schemesSimilar machinery for the horizontal controller case



Experiments run at the Max Planck Institute of Tubingen, Germany

additionalweight

case weight adaptive

a) no nob) no yesc) yes nod) yes yes

gain a/c b/d

λz 3 3kvz 5.5 5.5kγ,z 0 1.5λxy 3 3kv,xy 3 3kγ,xy 0 1kv,ϕθψ 1 1kv,ϕθψ 1 1krC 0 .1



desired trajectory

0 20 40 60 80 100 120 140

−1.5

−1

−0.5

0

0.5

1

time [s]

η1,d[m

]



Norm of the 3D position errors for the cases a) (green) and b) (blue)(no weight)

0 20 40 60 80 100 120 1400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

time [s]

∥ ∥

η1,d−

η1

∥ ∥

[m]



Norm of the 3D position errors for the cases c) (green) and d) (blue)(weight)

0 20 40 60 80 100 120 1400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

time [s]

∥ ∥

η1,d−

η1

∥ ∥

[m]



Roll (top) and pitch (bottom) angles for cases c) (green) and d) (blue)

0 20 40 60 80 100 120 140−4

−2

0

2

0 20 40 60 80 100 120 140−5

0

5

10

time [s]

time [s]

ϕ[deg]

θ[deg]



Control forces for the cases c) (green) and d) (blue)

0 20 40 60 80 100 120 140

−16

−14

0 20 40 60 80 100 120 1400

0.5

0 20 40 60 80 100 120 140−0.4−0.2

00.2

0 20 40 60 80 100 120 140−0.03−0.02−0.01

00.01

time [s]

time [s]

time [s]

time [s]

Z[N

]K

[Nm]

M[N

m]

N[N

m]



Time history of the parameters estimates for the case d). Top:parameter γz, center: parameter γxy, bottom: parameter rC .

0 20 40 60 80 100 120 140−18

−16

−14

0 20 40 60 80 100 120 140−1

0

1

0 20 40 60 80 100 120 140−0.05

0

0.05

time [s]

time [s]

time [s]

γz[N

]γxy[N

]rC[m

]


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, Italyhttp://webuser.unicas.it/lai/robotica

⊕CNRS at IRISA and Inria Bretagne Atlantique, Francehttp://www.irisa.fr/lagadic

≀Max Planck Institute for Biol. Cybernetics, Germanyhttp://www.kyb.mpg.de/research/dep/bu/hri/

IROS 2013


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IROS 2013 talk