【Paper】2013_ An efficient model predictive control scheme for an unmanned quadrotor helicopter

Abdolhosseini M, Zhang Y M, Rabbath C A. An efficient model predictive control scheme for an unmanned quadrotor helicopter[J]. Journal of intelligent & robotic systems, 2013, 70(1-4): 27-38.

1 Introduction

2 An Efficient Model Predictive Control Formulation

3 Dynamical Equations of Quadrotor Helicopter

3.1 Nonlinear Model of a Quadrotor Helicopter

Balance based on force and moment , See [12], The motion control dynamic equation of a four rotor helicopter in the ground fixed coordinate system can be expressed as :

$$\ddot{x}=\frac{(\sin \psi \sin \varphi +\cos \psi \sin \theta \cos \varphi )u_1-K_1\dot{x}}{m}\text{\hspace{1em}(21)}$$

$$\ddot{y}=\frac{(\sin \psi \sin \theta \cos \varphi -\cos \psi \sin \varphi )u_1-K_2\dot{y}}{m}\text{\hspace{1em}(22)}$$

$$\ddot{z}=\frac{(\cos \varphi \cos \theta )u_1-K_3\dot{z}}{m}-g\text{\hspace{1em}(23)}$$

$$\ddot{\varphi }=\frac{(u_{3}l-K_4\dot{\varphi })}{I_x}\text{\hspace{1em}(24)}$$

$$\ddot{\theta }=\frac{(u_{2}l-K_5\dot{\theta })}{I_y}\text{\hspace{1em}(25)}$$

$$\ddot{\psi }=\frac{(u_{4}c-K_6\dot{\psi })}{I_z}\text{\hspace{1em}(26)}$$

among $K_i,i=1,2,\cdots ,6$ Is the drag coefficient related to air resistance ,$l$ Is the distance from the center of gravity of the four rotors to the center of each propeller ,$c$ Is the thrust - Moment scale factor . Be careful , The drag coefficient is negligible at low speeds . Again ,$I_x,I_y,I_z$ Indication edge $x,y,z$ The moment of inertia in the direction . Convenient for calculation , System input $u_i,i=1,2,3,4$ Defined as :
$$\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\\ u_4\end{array}\right]=\left[\begin{array}{cccc}1& 1& 1& 1\\ 0& -1& 0& 1\\ -1& 0& 1& 0\\ 1& -1& 1& -1\end{array}\right]\left[\begin{array}{c}{F}_1\\ F_2\\ F_3\\ F_4\end{array}\right]\text{\hspace{1em}(27)}$$

The drive of a four rotor helicopter is a brushless DC motor . Imposed PWM The relationship between input and generated thrust is :
$$F_i=K_{motor}\frac{{\omega }_{motor}}{s+{\omega }_{motor}}{u}_{PWM}\text{\hspace{1em}(28)}$$

among $K_{motor}$ Positive gain ,${\omega }_{motor}$ For actuator bandwidth . surface 1 Contains the nominal values of the four rotor helicopter system parameters .

3.2 Model Reduction to Minimize Computational Demands

As mentioned earlier , Because fast dynamic systems require relatively high update rates , The success of predictive control in aviation applications highly depends on the real-time computing power of airborne computers . Because in almost all such applications , The available airborne computing power is limited , This is partly due to weight considerations , therefore , Any effort to reduce the computational burden is crucial , In order to make MPC Applied to aviation system , Especially the feasibility of driverless cars .

So , This paper attempts to decouple the six degree of freedom motion control dynamic equations of the four rotor , The system is described by four second order differential equations , The translational longitudinal displacement $x$ With rotary pitching motion $\theta$ coupling , Will translate laterally $y$ With rotary rolling motion $\phi$ coupling , Along the normal axis $z$ The direction of translation and vertical displacement are handled separately , And independent of the other two displacements . in other words ：
$$\ddot{x}=\frac{u_1\sin \theta }{m};\ddot{y}=\frac{u_1\sin \varphi }{m};\ddot{z}=\frac{u_1}{m}-g\text{\hspace{1em}(29)}$$

$$\ddot{\varphi }=\frac{u_{3}l}{I_x};\ddot{\theta }=\frac{u_{2}l}{I_y};\ddot{\psi }=\frac{u_{4}c}{I_z}\text{\hspace{1em}(30)}$$

such , The dimension of system matrix involved in iterative calculation , And a fraction of a second in a time step , Will be reduced by a third or less ; otherwise , The corresponding six degree of freedom motion of a four rotor helicopter is directly considered, including 14 A matrix of orders of magnitude ( Each degree of freedom corresponds to two matrices plus the matrix of the DC motor ). This separate processing of motion mode greatly affects the execution time of on-board calculation . Besides , About yaw motion $\psi$, Assume zero yaw angle is always maintained ; This can be achieved by integrating an independent reaction wheel mechanism , Instead of four DC motors taking over to control the yaw motion .

With this new subset of equations ,$\sin \theta ,\sin \varphi ,\frac{u_1}{m}g$ Will be taken as their corresponding equations （29） Control variable or input of . in other words ,$u_1$ It's based on （29） The third equation required for flight at medium and stable level is used for preliminary calculation . Then substitute this value into （29） In the first and second equations of , As a constant ( Within the forecast range ), Retain $\sin \theta$ and $\sin \varphi$ As the only manipulation variable . then , The new version of the equation is discretized with an appropriate discrete time step , Maintain the dynamic characteristics of the four rotor system . This rate may vary from equation to equation , This depends on how flexible the system is along this axis .

3.3 Validation of the Simplified Decoupled Model vs. the Elaborate Coupled Model