Actuator Fault-Tolerant Control Architecture for Multirotor Vehicles in Presence of Disturbances

Abstract

In this article we present an actuator fault-tolerant control architecture for the attitude and altitude tracking problem of multirotor aircrafts, under the effects of unknown drag coefficients and external wind. The tracking problem is faced by splitting it into two sub-problems, namely control law and control allocation. The control law is designed in terms of desired forces and moments which should act on the system, it does permit to exploit possible estimations of the disturbances acting on the vehicle and does not depend on the multirotor configuration. The control allocation, instead, optimally solves the redistribution of the control efforts among the motors according to the specific multirotor configuration, moreover it can actively cope with actuator faults whenever their estimations are available. Numerical simulations based on realistic scenarios confirm that the control architecture permits to solve the attitude and altitude tracking problem, despite the effects of faults and disturbances on the system.

Appendices

Appendix A: Gyroscopic effect

For each multirotor configuration, the gyroscopic effect can be modelled as $\tau ^{gyro}_{p}=-J_{r}q\varOmega$ and $\tau ^{gyro}_{q} = J_{r} p \varOmega$. The scalar Ω is a linear function of the propellers’ speed, depending on the rotor configuration, as explained in the following.

Considering a multirotor configuration with n actuators, each actuator’s angular speed is Ω_i and produces a force f_i = b|Ω_i|Ω_i. Thus, the angular speed can be obtained in function of f_i, since $\varOmega _{i} = f_{i}/(b\sqrt {|f_{i}|})$. Defining the vector

$$ \begin{array}{@{}rcl@{}} \bar{\varOmega} &=& \left[ \begin{array}{cccc} \varOmega_{1} & \varOmega_{2} & {\ldots} & \varOmega_{n} \end{array} \right]^{T}\\ &=& \left[ \begin{array}{cccc} f_{1}/(b\sqrt{|f_{1}|}) & f_{2}/(b\sqrt{|f_{2}|}) & {\ldots} & f_{n}/(b\sqrt{|f_{n}|}) \end{array} \right]^{T} \end{array} $$

(48)

then, for each configuration, we can write the gyroscopic effect in the form $\varOmega = b_{gyro}\bar {\varOmega }$, where $b_{gyro}\in \mathbb {R}^{n}$ is a row vector of length n, namely

$$ \begin{array}{@{}rcl@{}} \text{standard quadrotor}\quad\! b_{gyro} \!&=&\!\left[\begin{array}{cccc} \hphantom{-}\!1 \>&\> \!-1 \>&\> \hphantom{-}\!1 \>&\> \!-1 \end{array}\right] \end{array} $$

(49)

$$ \begin{array}{@{}rcl@{}} \text{standard hexarotor}\!\quad b_{gyro} \!&=&\!\left[\begin{array}{cccccc} \hphantom{-}\!1 \>&\> \!-1 \>&\> \hphantom{-}\!1 \>&\> \!-1 \>&\> \hphantom{-}\!1 \>&\> \!-1 \end{array}\!\right] \end{array} $$

(50)

$$ \begin{array}{@{}rcl@{}} \text{coaxial hexarotor}\!\quad b_{gyro} \!&=&\!\left[\begin{array}{cccccc} \!-1 \>&\> \hphantom{-}\!1 \>&\> \!-1 \>&\> \hphantom{-}\!1 \>&\> \!-1 \>&\> \hphantom{-}\!1 \end{array}\!\right] \end{array} $$

(51)

$$ \begin{array}{@{}rcl@{}} \text{coaxial octarotor}\!\quad b_{gyro} \!&=&\!\left[\begin{array}{cccccccc} \!-1 \>&\> \hphantom{-}\!1 \>&\> \hphantom{-}\!1 \>&\> \!-1 \>&\> \!-1 \>&\> \hphantom{-}\!1 \>&\> \hphantom{-}\!1 \>&\> \!-1 \end{array}\!\right]\\ \end{array} $$

(52)

Since we want to provide a unique control law for each type of multirotor and since its effects is relatively small, we consider Ω as a disturbance in the control law design phase.

Appendix B: Wind effect

The nominal model is heavily perturbed by the presence of aerodynamic effects. In particular, the multirotor is subject to the effect of the relative air flow with respect to the body frame, hence both the speed of the multirotor and the presence of wind contribute to aerodynamic disturbances. The main effects are frame drag and blade flapping.

The frame drag has a significant effect on the multirotor’s dynamics [36], and it can be described by the classical relation

$$ F_{drag}=\frac{1}{2}\rho C_{d} A \|V\| V $$

(53)

where ρ is the air density, C_dA is the equivalent flat plate area seen by the wind, and $V\in \mathbb {R}^{3}$ is the relative air speed. The equivalent flat plate area has been experimentally estimated in [37]. The flat plate area is, in principle, dependent on the attitude of the multirotor; for the purpose of this article, it has been approximated with a constant. This relation gives a force vector which has the same direction and orientation of the wind speed. Note also the quadratic dependence of the frame drag with respect to the relative air speed. The airframe lift could be modelled analogously, but it has been neglected because its influence is limited if compared with the drag [38]. Finally, for each multirotor configuration, an angular drag can be modelled as well in terms of a linear function, namely $\tau ^{drag}_{p} = - k_{rp}p$, $\tau ^{drag}_{q} = - k_{rq}q$ and $\tau ^{drag}_{r} = - k_{rr}r$, where the constants k_rp, k_rq and k_rr are unknown.

When rotors translate due to the multirotor’s movement in the horizontal plane, the relative air speed on the blade tip is different in the forward and backward direction, inducing a difference in the lift force. It follows that the rotor path is tilted by those lift forces. Following the model proposed in [39], we introduce the rotor velocity vector

$$ v_{ri}={V}+\omega\times d_{i} $$

(54)

where v_ri is the linear relative velocity vector of the whole i-th rotor, $\omega =\left [\begin {array}{lll} p & q & r \end {array}\right ]^{T}$ is the angular velocity vector of the multirotor, and d_i is the position vector of the i-th rotor in the body fixed frame. The rotor advance ratio (i.e., the inverse of the tip speed ratio) is defined as

$$ \mu_{ri}=\frac{\| v_{r(1,2)i} \|}{{\varOmega}_{i} R} $$

(55)

where v_r(1,2)i is the vector whose entries are the first two elements of v_ri, Ω_i is the angular speed of the i-th propeller and R is the radius of the propeller. The azimuthal direction of motion can be written in terms of four quadrant inverse tangent

$$ \psi_{ri}=\text{atan}\left( {v_{r(2)i}},{v_{r(1)i}}\right) $$

(56)

The first harmonic approximation of the longitudinal $\left (\alpha _{1si}\right )$ and lateral $\left (\beta _{1si}\right )$ flapping angles can be obtained by means of the rotation

$$ \left[\begin{array}{l} \alpha_{1si} \\ \beta_{1si} \end{array}\right] = \left[\begin{array}{ll} \cos(\psi_{ri}) & -\sin(\psi_{ri}) \\ \sin(\psi_{ri}) & \cos(\psi_{ri}) \end{array}\right] \left[\begin{array}{l} u_{1si} \\ v_{1si} \end{array}\right] $$

(57)

where [39, 40]

$$ v_{1si}=\frac{1}{1+\frac{\mu_{ri}^{2}}{2}}\frac{4}{3}\left( \frac{C_{T}}{\sigma}\frac{2}{3}\frac{\mu_{ri}\gamma}{a_{0}}+\mu_{ri}\right) $$

(58)

$$ u_{1si}=\frac{1}{1-\frac{\mu_{ri}^{2}}{2}}\mu_{ri}\left( 4\theta_{t} - 2\lambda^{2}_{hi} \right) $$

(59)

In particular, flapping angles depend on the solidity ratio σ, the non-dimensionalized near hover inflow λ_hi, the Lock number γ and the slope of the lift curve per radian a₀. The solidity ratio σ of the propeller is given by

$$ \sigma=\frac{A_{b}}{A} $$

(60)

where A_b is the area of the blades and A is the area swept by the propeller. The non-dimensionalized near hover inflow λ_hi of the i-th rotor is approximated by $\lambda _{hi}=\sqrt {C_{T}/2}$, where C_T is the non-dimensionalized thrust coefficient, which can be obtained by the knowledge of the thrust coefficient b. The Lock number γ is related to the ratio between the aerodynamic to centrifugal forces

$$ \gamma=\frac{\rho a_{0} c R^{4}}{J_{r}} $$

(61)

where ρ is the air density, c is the chord of the blade and J_r is the rotor inertia.

The effect of the blade flapping on the multirotor dynamics can be modelled considering the skewness of the plane of rotation of each propeller. Overall, the effect of blade flapping and frame drag Eq. 53 can be expressed as:

$$ \left[\begin{array}{l}f_{x}^{wind} \\ f_{y}^{wind} \\ f_{z}^{wind} \end{array}\right] = \sum\limits_{i}\! \left\{\!b {\varOmega_{i}^{2}} \left[\!\begin{array}{l} -\sin(\alpha_{1si}) \\ -\cos(\alpha_{1si})\sin(\beta_{1si}) \\ \cos(\alpha_{1si})\cos(\beta_{1si})-1 \end{array}\!\right]\!\right\} + F_{drag} $$

(62)

$$ \left[\begin{array}{l}\tau_{p}^{wind} \\ \tau_{q}^{wind} \\ \tau_{r}^{wind} \end{array}\right] = \sum\limits_{i} \left\{d_{i} \times b {\varOmega_{i}^{2}} \left[\begin{array}{l} -\sin(\alpha_{1si}) \\ -\cos(\alpha_{1si})\sin(\beta_{1si}) \\ \cos(\alpha_{1si})\cos(\beta_{1si})-1 \end{array}\right]\right\} $$

(63)

The parameters used in this article are reported in Table 2.

Table 2 Physical parameters of the quadrotor