Experimental validation of a new adaptive
control scheme for quadrotors MAVs
Gianluca Antonelli† , Elisabetta Cataldi† ,
P...
Trajectory tracking control for quadrotor

Adaptive with respect to
uncertainty in total mass
uncertainty in Center Of Mas...
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 ...
Dynamics

Mathematical model expressed in body-fixed frame
˙
M ν + C(ν)ν + τ W + g(RB ) = τ ,
I
beyond the common terms, we...
Dynamics -2Exploiting the linearity in the parameters
˙
Φ(ν, ν, RB )γ = τ
I
and rewriting with respect to the inertial fra...
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...
Mapping from the angular velocities to the force-torques
Assuming CoM of coordinates r C :
 
 2 
ωt,1
Z
2
K 
ωt,2 ...
Inverse mapping
Any controller determines a control action Zc Kc Mc Nc
projected onto the motor input u ∈ R4
 
Zc
 Kc ...
Current inverse mapping

ˆ
When the CoM position estimate r C is affected
mapping becomes

1
0
 
 
Z
Zc
 rC,y
˜
1
K ...
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

τ...
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 = ...
Horizontal controller
error
T

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

full version

reduced version...
Orientation controller
The inputs are the desired roll, pitch and yaw
The commanded forces map onto the real ones accordin...
CoM estimation

PD control for roll and pitch =⇒ steady-state error because of the
wrong CoM estimate
A simple integral ac...
Stability analysis
˜
ˆ
Altitude controller: let γ = γ − γ and consider the Lyapunov function
˜
V (sz , γ ) =

m 2 1 T
˜
˜
...
Experimental results
Experiments run at the Max Planck Institute of T¨bingen, Germany
u
case
a)
b)
c)
d)

additional
weigh...
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]

An...
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 [...
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]
...
Experimental results
Roll (top) and pitch (bottom) angles for cases c) (green) and d) (blue)

ϕ [deg]

2
0
−2
−4
0

20

40...
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...
Experimental results

ˆ
r C [m]

ˆ
γ xy [N]

γz [N]
ˆ

Time history of the parameters estimates for the case d). Top:
para...
Experimental validation of a new adaptive
control scheme for quadrotors MAVs
Gianluca Antonelli† , Elisabetta Cataldi† ,
P...
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IROS 2013 talk

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IROS 2013 talk

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

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