【Paper】2016_ A Learning-Based Fault Tolerant Tracking Control of an Unmanned Quadrotor Helicopter

Liu, Z., Yuan, C., Zhang, Y. et al. A Learning-Based Fault Tolerant Tracking Control of an Unmanned Quadrotor Helicopter. J Intell Robot Syst 84, 145–162 (2016).

1 Introduction

2 Description and Dynamics of the Unmanned Quadrotor Helicopter

Pictured 1 Shown , thrust ($u_1,u_2,u_3,u_4$) It consists of four parts respectively arranged in the front corner 、 Back corner 、 The propeller driven by the independent motor in the left and right corners . The front and rear motors rotate clockwise , The left and right motors rotate counterclockwise . The resulting thrust is $z_B$ Always up in the direction .

therefore ,1） The vertical translation can be realized by directly distributing the same number of control signals to each motor ; 2） Horizontal translation requires the four rotor helicopter to roll or pitch in advance , This allows for forward or lateral movement . Besides , Roll and pitch rotation can be achieved by assigning different amounts of control signals to the opposite motors , This forces the four rotor helicopter to tilt towards the slowest motor [1].

2.1 Nonlinear Model of the Unmanned Quadrotor Helicopter

utilize [28] and [29] Four rotor helicopter model in , The commonly used dynamic models of four rotor helicopters in the earth fixed coordinate system can be :
$$\{\begin{array}{rl}\ddot{x}& =\frac{(\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi )u_1(t)-K_1\dot{x}}{m}\\ \ddot{y}& =\frac{(\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi )u_1(t)-K_2\dot{y}}{m}\\ \ddot{z}& =\frac{(\cos \varphi \cos \theta )u_z(t)-K_3\dot{z}}{m}-g\\ \ddot{\varphi }& =\frac{u_3(t)-K_4\dot{\varphi }}{I_x}\\ \ddot{\theta }& =\frac{u_2(t)-K_5\dot{\theta }}{I_y}\\ \ddot{\psi }& =\frac{u_4(t)-K_6\dot{\psi }}{I_z}\end{array}\text{\hspace{1em}(1)}$$

Acceleration and lift / The relationship between moments is expressed as :
$$\begin{array}{rl}\left[\begin{array}{c}{u}_z(t)\\ u_{\theta }(t)\\ u_{\varphi }(t)\\ u_{\psi }(t)\end{array}\right]& =\left[\begin{array}{cccc}1& 1& 1& 1\\ L& -L& 0& 0\\ 0& 0& L& -L\\ C& C& -C& -C\end{array}\right]\left[\begin{array}{c}{u}_{c1}(t)\\ u_{c2}(t)\\ u_{c3}(t)\\ u_{c4}(t)\end{array}\right]\end{array}\text{\hspace{1em}(2)}$$

Each motor is modulated by its corresponding pulse width (PWM) Signal control , The relationship is defined as :
$$u_i(t)=K_m\frac{{\omega }_m}{s+{\omega }_m}{u}_{ci}(t)\text{\hspace{1em}(3)}$$

2.2 Linearized Model of the Unmanned Quadrotor Helicopter

Assumption 1

It is assumed that the four rotor unmanned helicopter is in hover during operation [20], This indicates that in the vertical direction $u_z\approx mg$. The change range of pitch angle and roll angle is also relatively small , Yes $\sin \varphi \approx \varphi ,\sin \theta \approx \theta$, And there is no heading angle change $\psi \approx 0$. in addition , When the UAV moves very slowly , The drag coefficient is negligible .

Then according to the above assumption 1, Can simplify formula （1） Turn into

$$\begin{array}{r}\{\begin{array}{rl}& \ddot{x}=\theta g\\ & \ddot{y}=-\varphi g\\ & \ddot{z}=u_z(t)/m-g\\ & I_x\ddot{\theta }=u_{\theta }(t)\\ & I_y\ddot{\varphi }=u_{\varphi }(t)\\ & I_z\ddot{\psi }=u_{\psi }(t)\end{array}\end{array}\text{\hspace{1em}(4)}$$

further , When taking the UAV and its actuator as a whole , Actuator dynamics can be neglected in the design of control system , And will not cause significant residual error , This is because the time constant of the actuator is much smaller than that of the UAV [30].

So the formula （3） It can be simplified to $K_m\frac{{\omega }_m}{s+{\omega }_m}\approx K_m$, This can still be used to describe the effectiveness of control behavior . therefore , The formula （2） Can be written as
$$\begin{array}{rl}\left[\begin{array}{c}{u}_z(t)\\ u_{\theta }(t)\\ u_{\varphi }(t)\\ u_{\psi }(t)\end{array}\right]& =\left[\begin{array}{cccc}{K}_m& K_m& K_m& K_m\\ K_{m}L& -K_{m}L& 0& 0\\ 0& 0& K_{m}L& -K_{m}L\\ K_{m}C& K_{m}C& -K_{m}C& -K_{m}C\end{array}\right]\left[\begin{array}{c}{u}_{c1}(t)\\ u_{c2}(t)\\ u_{c3}(t)\\ u_{c4}(t)\end{array}\right]\end{array}\text{\hspace{1em}(5)}$$

The Euler angular acceleration is mapped to the propeller speed .