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.