Quadrotor helicopter control problem

Quadrotor helicopter control problem
State dimension:	1
Differential states:	6
Continuous control functions:	1
Discrete control functions:	3
Interior point equalities:	6

The mixed-integer optimal control problem of a quadrotor helicopter in two dimensions is taken from (Link: Gillula et al.).

and from (Link: Vasudevan et al.). The evolution

of the quadrotor can be defined with respect to a fixed two dimensional reference frame using six dimensions, where the first three dimensions represent the position along a horizontal axis, the position along the vertical axis, and the roll angle of the helicopter, respectively, and the last three dimensions represent the time derivative of the first three dimensions.

Mathematical formulation

The mixed-integer optimal control problem is given by

$\begin{array}{llclr} \displaystyle \min_{x,u, w} & 5(x_1(t_f)-6)^2&+&5(x_3(t_f)-1)^2+(\sin(x_5(t_f)0.5))^2 +\int\limits_{t_0}^{t_f} 5u(\tau)^2 \ d \tau \\[1.5ex] \mbox{s.t.} & \dot{x}_1 & = & x_2(t), \\ & \dot{x}_2 & = & g \sin( x_5(t)) + w_1(t)u(t)\frac{\sin(x_5(t))}{M}, \\ & \dot{x}_3 & = & x_4(t), \\ & \dot{x}_4 & = & g \cos( x_5(t))-g+ w_1(t)u(t)\frac{\cos(x_5(t))}{M}, \\ & \dot{x}_5 & = & x_6(t), \\ & \dot{x}_6 & = & -w_2(t)L \frac{u(t)}{I}+w_3(t)L \frac{u(t)}{I} \\[1.5ex] & x(0) &=& (0, 0, 1, 0 , 0, 0)^T, \\ & w_i(t) &\in& \{0, 1\}, i=1,\ldots,3 \\ & \sum\limits_{i=1}^{3}w_i(t) &=& 1, \\ & u(t) & \in& [0,0.001], \quad t\in[t_0,t_f],\\ & x_3(t) & \geq & 0, \quad t\in[t_0,t_f]. \end{array}$

Parameters

These fixed values are used within the model.

$\begin{array}{rcl} [t_0, t_f] &=& [0, 7.5],\\ (g, M, L, I) &=& (9.8, 1.3, 0.305, 0.0605), \end{array}$

Reference Solutions

A reference solution can be found in Vasudevan et al. based on the embedding transformation technique for switched systems.

Variants

There are several alternative formulations and variants of the above problem, in particular

• Quadrotor (binary variant): The quadrotor helicoptor problem, where the continuous control is replaced via partial outer convexification by binary controls.