Assessment and Future Directions of Nonlinear Model Predictive ...

556 S. Gros et al.3E 31hdGE 2GdE 1E 2e 3η 1 η2E 3φE 2φe E 1 e 212Fig. 2. (E 1,E 2,E 3) is a frame attached to the VTOL at its center of mass G. h isthe vertical distance between center of mass and propeller center. d is the horizontaldistance between center of mass and propeller axis. The transformation (e 1,e 2,e 3) →(E 1,E 2,E 3) is defined by (i) rotation of angle η 1 around axis 1, (ii) rotation of angleη 2 around axis 2, (iii) rotation of angle φ around axis 3.4 rotate clockwise. The angle of attack (AoA) of the blades and the positionsof the propellers are fixed relative to the structure. The VTOL is controlled bymeans of the four motor torques. The states of the system are:X = [ xyzη 1 η 2 φ ẋ ẏ ż ˙η 1 ˙η 2 ˙φ ˙ρ1 ˙ρ 2 ˙ρ 3 ˙ρ 4](19)Variables [ xyz ] give the position of the center of gravity G in [m] within thelaboratory referential (e 1 ,e 2 ,e 3 ). Variables [ η 1 η 2 φ ] give the angular attitudeof the structure in [rad], with the transformation from the laboratory referentialto the VTOL referential, (e 1 ,e 2 ,e 3 ) → (E 1 ,E 2 ,E 3 ), being described by thematrix Φ(η 1 ,η 2 ,φ)=R e3 (φ)R e2 (η 2 )R e1 (η 1 ), where R e (α) is a rotation of angleα around the basis vector e.Variable ˙ρ k is the speed of the propeller k in [rad/s]. The model of the VTOLcan be computed by means of analytical mechanics. The aerodynamical forcesand torques generated by the propellers are modeled using the standard squaredvelocity law. The resulting model is rather complicated and will not be explicitedhere. The reader is referred to [9] for details. The model is nonlinear, and itslocal dynamics are strongly time varying.A simplified model can be computed by removing certain non-linearities,which is well justified in practice. Introducing the notationsv 1 =4∑˙ρ 2 k v 2 =k=1v 4 = ˙ρ 2 2 − ˙ρ2 4 v 5 =4∑(−1) k ˙ρ 2 k v 3 =˙ρ 2 1 − ˙ρ2 3k=14∑˙ρ kk=1