AIM 2013 talk

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The paper presents an adaptive trajectory tracking control strategy for quadrotor Micro Aerial Vehicles. The proposed approach, while keeping the typical assumption of an orientation dynamics faster than the translational one, removes that of absence of external disturbances and of perfect symmetry of the vehicle. In particular, the trajectory tracking control law is made adaptive with respect to the presence of external forces and moments, and to the uncertainty of dynamic parameters as the position of the center of mass of the vehicle. A stability analysis as well as numerical simulations are provided to support the control design.

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AIM 2013 talk

  1. 1. Adaptive trajectory tracking for quadrotor MAVs in presence of parameter uncertainties and external disturbances Gianluca Antonelli†, Filippo Arrichiello†, Stefano Chiaverini†, Paolo Robuffo Giordano⊕, in alphabetical order †University of Cassino and Southern Lazio, Italy http://webuser.unicas.it/lai/robotica ⊕CNRS at IRISA, France http://www.irisa.fr/lagadic kindly presented by prof. Bruno Siciliano Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  2. 2. Trajectory tracking control for quadrotor Adaptive with respect to uncertainties in total mass uncertainties in Center Of Gravity (COG) presence of 6-DOF external disturbances Assumption: closed-loop orientation dynamics faster than translational one Stability analysis Numerical simulations Experimental results (not in the paper) Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  3. 3. 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 = RB I ˙η1 ν2 = Jk,o(η2) ˙η2 Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  4. 4. Dynamics Mathematical model expressed in body-fixed frame MRB ˙ν + CRB(ν)ν + τv,W + gRB(RB I ) = τv, beyond the common terms, we model τv,W = Φv,W (RI B)γv,W = RB I O3×3 O3×3 RB I γv,W whit γv,W ∈ R6 external disturbance constant in the inertial frame, e.g., wind Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  5. 5. Dynamics -2- Exploiting the linearity in the parameters Φv( ˙ν, ν, RB I )γv = τv and rewriting with respect to the inertial frame while separating the xy dynamics from z: Φxy(η, ˙η, ¨η) φz(η, ˙η, ¨η) γv = τv with γv ∈ R16: mass (1 parameter) first moment of inertia (3 p.) inertia tensor (6 p.) external disturbance (6 p.) with: τv = τ1 τ2 =         0 0 Z K M N         Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  6. 6. Thrust x y z O xb yb zb Ob f1 f2 f3 f4 1 2 3 4 ωt,1 ωt,2 ωt,3 ωt,4 τt,1 τt,2 τt,3 τt,4 l fi = bω2 t,i τt,i = dω2 t,i τ1 =      0 0 4 i=1 fi      τ2 =   l(f2 − f4) l(f1 − f3) −τt,1 + τt,2 − τt,3 + τt,4   Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  7. 7. Mapping from the angular velocities to the force-torques     Z K M N     = Bv     ω2 t,1 ω2 t,2 ω2 t,3 ω2 t,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     rC,x, rC,y being the COG coordinates COG influences the mapping from thrust generated from the motors to the vehicle forces/moments Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  8. 8. Inverse mapping Any controller determines a control action Zc Kc Mc Nc T further projected onto the motor input u ∈ R4 u = B−1 v     Zc Kc Mc Nc     where B−1 v ∈ R4×4 is B−1 v =           l − rC,x 4bl 0 1 2bl − l − rC,x 4dl l − rC,y 4bl 1 2bl 0 l − rC,y 4dl l + rC,x 4bl 0 − 1 2bl − l + rC,x 4dl l + rC,y 4bl − 1 2bl 0 l + rC,y 4dl           Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  9. 9. Current inverse mapping When the COG position estimate ˆrC is affected by an error, the real mapping becomes     Z K M N     = Bv|rC B−1 v ˆrC     Zc Kc Mc Nc     =        1 0 0 0 ˜rC,y 2 1 0 b˜rC,y 2d ˜rC,x 2 0 1 − b˜rC,x 2d 0 0 0 1            Zc Kc Mc Nc     wrong COG estimate ⇒ a coupling from altitude and yaw control actions onto roll and pitch dynamics Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  10. 10. Controller block diagram η1d ψd φd, θd Zc pos or   Kc Mc Nc   B−1 v u motors w2 t,i Bv     Z K M N     τv,W η plant Classical MAV control architecture with adaptation wrt parameters and compensation of the COG position Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  11. 11. Altitude controller error ˜z = zd − z ∈ R sz = ˙˜z + λz ˜z ∈ R full version Z = 1 cos φ cos θ (φz ˆγv + kpzsz) ˙ˆγv = K−1 γ,zφT z sz with ˆγv ∈ R16 reduced version Z = 1 cos φ cos θ (ˆγz + kpzsz) ˙ˆγz = k−1 γ,zsz with ˆγz ∈ R1 the reduced version designed to compensate only for persistent terms ⇒ null steady state error wrt a minimal set of parameters! (λz > 0, kpz > 0, Kγ,z > O, kγ,z > 0) Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  12. 12. Horizontal controller error ˜ηxy = xd − x yd − y T ∈ R2 sxy = ˙˜ηxy + λxy ˜ηxy ∈ R2 full version virtual input solutions of: cφsθ −sφ = 1 Z Rz (Φxy ˆγv + kp,xysxy) , ˙ˆγv = K−1 γ,xyΦT xysxy. with ˆγv ∈ R16 reduced version virtual input solutions of: cφsθ −sφ = 1 Z Rz ˆγxy + kp,xysxy ˙ˆγxy = k−1 γ,xysxy with ˆγxy ∈ R2 again: the reduced version compensates only for persistent terms ⇒ null steady state error wrt a minimal set of parameters! (λxy > 0, kp,xy > 0, Kγ,xy > O, kγ,xy > 0) Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 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 = Kc + ˜rC,y 2 Zc + b˜rC,y 2d Nc M = Mc + ˜rC,x 2 Zc − b˜rC,x 2d Nc N = Nc 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 = Kc + 1 2 Zc + b d Nc ˜rC,y M = Mc + 1 2 Zc − b d Nc ˜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 Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  14. 14. COG estimation PD control for roll and pitch =⇒ steady-state error because of the wrong COG estimate A simple integral action can counteract this effect resulting a zero steady-state error ˙ˆrC,x ˙ˆrC,y = −krC θd − θ φd − φ , krC > 0 As a byproduct, in absence of moment disturbance, the estimates (ˆrC,x, ˆrC,y) are driven towards the real COG offsets (rC,x, rC,y) Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  15. 15. Stability analysis Altitude controller: let ˜γv = γv − ˆγv and consider the Lyapunov function V (sz, ˜γv) = m 2 s2 z + 1 2 ˜γT v Kγ,z ˜γv Along the system trajectories ˙V (sz, ˜γv) = sz m¨zd − m¨z + mλz ˙˜z − ˜γT v Kγ,z ˙ˆγv = sz (φzγv − cos φ cos θZ) − ˜γT v Kγ,z ˙ˆγv = −kp,zs2 z ≤ 0 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 Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  16. 16. Simulation results Constant disturbance γv,W =         0.5 0.6 0 0 0 0         [N, Nm] displacement of 1 m in the 3 directions x, y and z 20 deg in yaw 0 2 4 6 8 10 12 14 16 18 0 0.5 1 1.5 0 2 4 6 8 10 12 14 16 18 0 5 10 15 20 norm of the 3D position error [m] norm of the yaw error [deg] time [s] time [s] proposed control law (blue-solid line) and its non adaptive version (red-dashed line) Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  17. 17. Simulation results 0 2 4 6 8 10 12 14 16 18 20 −10 −5 0 5 10 0 2 4 6 8 10 12 14 16 18 20 −10 −5 0 5 10 rollpitch[deg]rollpitch[deg] time [s] time [s] Roll and pitch angle; on the top the desired (gray) and real (blue) values for the adaptive version while on the bottom the desired (gray dashed) and real (red dashed) values for the non adaptive simulation Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  18. 18. Simulation results 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 13 14 15 16 17 18 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.2 −0.1 0 0.1 0.2 time [s] time [s] Z[N]τ2[Nm] Force along zb (top) and moments (bottom) by applying the proposed control law (blue-solid line) and its non adaptive version (red-dashed line) Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  19. 19. Simulation results 0 2 4 6 8 10 12 14 16 18 14.7 14.8 14.9 0 2 4 6 8 10 12 14 16 18 0 0.2 0.4 0.6 0 2 4 6 8 10 12 14 16 18 −0.1 −0.05 0 0.05 0.1 time [s] time [s] time [s]γzγxy rC,x,rC,y Simulated parameters (gray) and estimated ones (blue) Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  20. 20. Experimental results additional weight Experiments on a mikrokopter with an unknown weight attached Theory and numerical simulations confirmed Results on a forthcoming publication Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013
  21. 21. Adaptive trajectory tracking for quadrotor MAVs in presence of parameter uncertainties and external disturbances Gianluca Antonelli†, Filippo Arrichiello†, Stefano Chiaverini†, Paolo Robuffo Giordano⊕, in alphabetical order †University of Cassino and Southern Lazio, Italy http://webuser.unicas.it/lai/robotica ⊕CNRS at IRISA, France http://www.irisa.fr/lagadic kindly presented by prof. Bruno Siciliano Antonelli Arrichiello Chiaverini Robuffo-Giordano Wollongong, 12 July 2013

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