Stabilization and Trajectory Tracking in Discrete-Time of an Autonomous Four Rotor Mini-Rotorcraft

In this paper, we present a stabilization and trajectory tracking controller in discrete –time for an autonomous four rotor mini-rotorcraft. The control algorithm proposed is based on the feedback linearization input-output for nonlinear discrete-time systems. The dynamic model of the autonomous four rotor rotorcraft is obtained via Lagrange formalism. Simulation results are presented to illustrate the performances of this controller. Introduction The miniature and autonomous flying robots arouse a growing interest in the civil and military domains; the fields of application of these vehicles are vast. We can state the ecological exploration mission and air cartography, search and rescue, surveillance and remote inspection. In this paper, we are particularly interested in controlling a mini rotorcraft having four rotors. Advantages of using a multi-rotor helicopter are the increased payload capacity and high manoeuvrability and the fact that the two motors turn in the clockwise direction whereas the two other turn in the opposite clockwise direction, gyroscopic effects and aerodynamic torques tend to cancel in trimmed flight. Disadvantages are the increased helicopter weight and increased energy consumption due to the extra motors. A control strategy based on the backstepping techniques proposed in [4] for configuration stabilization of quasistationary flight conditions of a four rotor vertical takeoff and landing (VTOL). The stabilization problem of a four rotor rotorcraft is also studied and tested in [2] where the nested saturation control algorithm is used. In [5], flatness and motion planning are combined to solve the point to point control problem with a predefined path of the four rotor rotorcraft. In this paper we present a discrete-time controller design of a four rotor helicopter modelled via Lagrange approach, the control strategy is based on the feedback linearization input-output for nonlinear discrete-time systems using the Euler approximate discrete-time model of the Lagrangian model when the sampling period should typically be sufficiently small in order to get a good approximation of the exact discrete-time model. Discrete-time designs are important because most controllers are implemented using digital computers with A/D and D/A converters (simpler and zero-order hold) and the digital controllers based on the Euler approximate discrete-time model may outperform discretized continuous-time controllers. The paper is organized as follows: Section II presents the dynamical model of the rotorcraft. Section III gives the controller synthesis. The simulation results are shown in section IV. The conclusions are finally given in section V. II. Dynamic model of autonomous rotorcraft In this section, a Lagrangian model is derived for the four rotor helicopter in generalized coordinates [1, 2, 6]. Consider figure 1. Let ) , , , ( Z Y X O e e e O = R denote an inertial frame. Let ) , , , ( 3 2 1 e e e G G = R denote a body fixed frame attached to the helicopter. The position of the center of mass of the rotorcraft relative to the frame O R is denoted ) , , ( Z Y X = ξ and ) , , ( φ θ ψ η = are the Euler angles (yaw, pitch and roll). Figure. 1. Frames of the four rotor rotorcraft with thrust inputs. The total thrust produced by the four rotors is given by: 4 3 2 1 1 f f f f u + + + = where is the thrust generated by the rotor and 2 i i i b f ω = 4 , 3 , 2 , 1 = i i ω is the angular speed generated by the motor and is a parameter. The torques acting of the four rotors helicopter result from the action of the thrust forces difference of each pair of rotors are denoted by: i M 0 > i b ) , , ( φ θ ψ τ τ τ τ = around axes , and respectively, with 1 e 2 e 3 e d f f ) ( 4 2 − = φ τ , d f f ) ( 1 3 − = θ τ and κ τψ ) ( 3 1 4 2 f f f f − − + = where represents the distance from the rotors to the center of gravity of the helicopter and d 0 > κ is a constant. In the sequel we use the state space representation with the following notations: T x x x x ] [ 5 3 1 1 = = ξ , , T x x x x ] [ 6 4 2 2 = = ξ& T x x x x ] [ 11 9 7 3 = =η , T x x x x ] [ 12 10 8 4 = =η& Let denote the state variable, ) , , , ( 4 3 2 1 x x x x x = ) , ( 1 τ u u = denote the control input and denote the output of the rotorcraft. ) , ( ) , ( 3 1 2 1 x x y y y = = The Euler discretization of Lagrangian model [6] using the state space representation gives: