Asymptotically stable control for a nonlinear-based multirotor aerial vehicle model

The interest for multirotor aerial vehicles (MAVs) is currently growing due to their low cost, high manoeuvrability, simplified mechanics, capability to perform vertical take-off and landing as well as hovering flight. These characteristics make them a promising technology suitable for applications such as suveillance of indoor and urban environments. The present work faces the problem of controlling the attitude of a MAV by means of a linear feedback control which guarantees asymptotic stability when controlling a nonlinear dynamics. The simulations show the effectiveness of the method.


Introduction
Multirotor Aerial Vehicles (MAVs) have motivated many researches in different fields of knowledge such as sensors fusion [3], computer vision [7] and control strategies [6].Although there is a massive amount of concluded and ongoing research works on MAVs, the design of control laws for such vehicles still has challenges to be overcome.Most of those challenges are related to design control laws that guarantee asymptotic stability, mostly because this system is nonlinear.
Reference [2] presents two nonlinear control techniques, backstepping and sliding mode, in order to control the attitude of the MAV.On the other hand, reference [4] utilizes a nonlinear H ∞ controller to stabilize the rotational movements.Both strategies have a high degree of complexity.Therefore, it would be interesting to design a linear feedback control which also guarantees stability of system.
The Linear Quadratic Regulator (LQR) is an usual control strategy applied to MAVs [1], which requires a linear design model.Such linear model does not take into account the nonlinear phenomena that are present in the real operation of these vehicles.However, for a more effective control law, nonlinearities have to be considered in order to avoid unstable regions of operation.One of the nonlinearities included in the motion equations is the gyroscopic effects which come from the rotation of the rigid body and the four propellers.This paper uses the [5] theorem to stabilize the attitude of a MAV.This theorem showed that is possible to construct a linear feedback control law for a nonlinear dynamic that is globally asymptotically stable.a e-mail: igorap@ita.brb e-mail: davifc@ita.brc e-mail: mateusfvp@hotmail.comd e-mail: davists@ita.bre e-mail: jmbaltha@ita.br

MAV modeling
This section presents the equations of motion of a multirotor aerial vehicle (MAV), considering all the relevant effects necessary to be accounted for in ground-truth model for simulation-based evaluation of control laws.We start with preliminary definitions in Subsection 2.1, then we present the rotor dynamics in 2.2 and the MAV rotational dynamics in 2.3.

Preliminary definitions
We define two Cartesian coordinate systems (CCS) as illustrated in Figure 1.The body CCS, S B {X B , Y B , Z B } is attached to the vehicle's body with the origin at the vehicle's center of mass (CM), the X B axis pointing forward, the Z B axis pointing upward, perpendicular to the plane of the rotors, and the Y B axis completing a dextrogeous frame.The CCS, S G = {X G , Y G , Z G } is fixed to the ground at a known point O, with the Z G axis pointing upward, aligned with the local vertical.

Rotor dynamics
The thrust force f i and reaction torque τ i produced by each individual rotor are modeled, respectively, by the aerodynamic models for i = 1, ..., 4, where k f is the force coefficient, k τ is the torque coefficient, and ω i is the rotation speed of the ith rotor (positive in counterclockwise direction), whose dynamics can be modeled by the following first-order linear model: where ωi ∈ [0, ωmax ] is the rotation speed command for the ith rotor, k ω is the rotation coefficient, and τ ω is rotor time constant.

Resultant control thrust and torque
Although the formulation given here could be generalized to any multirotor vehicle configuration, for illustration purpose, we consider a quadrotor configuration with the longitudinal axis X B pointing between rotor 1 and rotor 2, and with 45 degree of separation angle between adjacent arms; we call it the "x" configuration.Moreover, we consider that rotor 1 and rotor 3 rotate in clockwise direction, while rotor 2 and rotor 4 rotate in counterclockwise direction.For the aforementioned configuration, one can show that the relationship between the individual rotor thrusts f i and the resultant thrust magnitude and torque vector is given by with where l is the length of each vehicle's arm.The superscript c in Eq. ( 5) stands for the control command defined by the operator and is used to distinguish from the other sources of force and torque that will be presented soon.By inverting Eq. ( 5), one can compute the command f f1 f2 f3 f4 where Ξ Γ −1 , Fc is the thrust magnitude command, and Tc is the control torque command.

Rotational motion
Representing the attitude using the Euler angles α φ θ ψ T in the rotation sequence 1-2-3, we have the following rotational kinematics equations: where Ω Ω x Ω y Ω z T is the S B representation of the vehicle's angular velocity w.r.t.S G and Assume that the vehicle has a rigid structure and S G is an inertial frame.Considering the existence of gyroscopic effects, due to the rotors, and disturbance torques, and using the Newton-Euler formulation, one can model the rotational dynamics of the MAV by where T c and T d are the S B representations of the resultant control torque and disturbance torque, respectively, e 3 [0 0 1] T , I r is the moment of inertia of the rotors w.r.t. the rotation axis, and J is the body inertia matrix.Consider that the vehicle has a symmetric structure with known mass m and inertia matrix in This paper makes use of the AR Drone 2 parameters in order to emulate a real platform.The values can be seen in Table 1.

State-space models
This section presents the nonlinear state-space model for rotation (Section 2).This model will be adopted for designing control laws by using the Rafikov and Balthazar's theorem.Firstly, note that the dynamics in Eq. ( 7) and (9) can be formulated as where x ∈ R n is the state vector, u ∈ R m is the control inputs, A ∈ R n×n and B ∈ R n×m are constant real matrices, and h(x) ∈ R n is a vector whose elements are nonlinear functions of x.

Control design
In this section, we present a control strategy to drive the MAV's nonlinear dynamics from a certain initial condition to a desired state.Following the control method proposed by [5], we formulate an optimal control problem that, given the nonlinear dynamics in Eq. ( 11), aims at minimizing the following LQR-like cost function: where R ∈ R m×m is a positive definite matrix and l(x) ∈ R is a function related to the weights of the dynamical states over time.Function l(x) will be selected as follows to account for the nonlinear terms in the dynamical model.
where Q ∈ R n×n and P ∈ R n×n are real symmetric positive definite matrices and P is the solution of the matrix algebraic Riccati equation Then, as long as l(x) is positive definite, the linear feedback control law is optimal in order to transfer the nonlinear system in Eq.
(2) from an initial condition x(0) to the origin while minimizing the cost function in Eq (3).Additionally, the controlled system is locally asymptotically stable in the neighborhood Γ 0 ⊂ Γ, Γ ⊂ R n , of the origin if x(0) ∈ Γ 0 .If Γ = R n , then the controlled system is globally asymptotically stable.

Conclusion
This paper has presented the attitude control of a MAV by designing the linear feedback controller.The linear feedback control problem was formulated under the optimal control theory viewpoint.Asymptotic stability of the closedloop nonlinear system is guaranteed by means of a Lyapunov function as presented in [5].

Fig. 3 .
Fig. 3. Positive function l(t), calculated for the attitude dynamics of the MAV.