Quaternion Error Based Optimal Attitude Control Applied to Pinpoint Landing

Abstract

The problem of Unmanned Aerial Vehicles (UAV) trajectory tracking has been studied widely and in depth, including the application of optimal control techniques to this problem, which has also been covered frequently in literature, although mostly to simplified models of the UAV and therefore with limited range of operation. The use of more complex mathematical models for UAV control, on the other hand, is less common and practically all the available research makes use of sub-optimal control techniques. Furthermore, the application of optimal control techniques to more accurate UAV models has seldom been attempted before, and even less for quaternion attitude-based models in particular. In this paper we propose the use of two proper optimal control algorithms that rely on attitude quaternion error and its dynamics. These two control algorithms share a feature which is that although the underlying linearized model on which they are based is model-independent, the LQR cost matrices are dependent on both the particularities of the autonomous vehicle and on the UAV state, and more specifically, on the angular and linear velocities. Therefore, the tuning of these controllers becomes the tuning of cost matrices, which is a priori a simpler task than adapting a more complex controller algorithm to pinpoint landing. On the other hand, these control algorithms are based on exact linearization underlying models that, for close tracking conditions, behave effectively like linear time invariant systems. The advantage of this is two-fold. On one hand, the calculation requirements for these controllers are lighter and, more importantly, the adaption of nonlinearities specific to UAV pinpoint landing is relatively simpler. The authors of this research consider these two features to be highly useful in pinpoint landing control. NomenclatureB = Body frame [χ] BI = [φ, θ, ψ] = Euler angles from B to I δ c , δ r , δ p , δ y = Thrust, roll, pitch, yaw commands e = Error (generic) [F A ] B = [F Ax , F Ay , F Az ] = Aerodynamic forces (expressed in B) [F P ] B = [F P x , F P y , F P z ] = Propulsive forces (expressed in B) g = Gravitational acceleration due to the Earth I = Inertial frame I B Bcm B = diag {[I xx , I yy , I zz ]} = Inertial tensor of Bcm with respect to B (expressed in B) m = Mass of UAV ω = Linear control bandwidth I ω B B = [p, q, r] = Angular rate from B to I (expressed in B) [q] BI = [q 0 , q 1 , q 2 , q 3 ] = Quaternion vector from B to I r Bcm/Io I = [x, y, z] = Position of Bcm from Io (expressed in I) v I Bcm B = [u, v, w] = Velocity of Bcm with respect to I (expressed in B) [x c , y c , z c , ψ c ] = Position and yaw commanded setpoints ζ = Damping ratio 1 Control