#!/usr/bin/env python
# coding: utf-8

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# ## Quadcopter
# 
# The quadcopter is a rigid body with 6 degrees of freedom. It can move in the x, y, and z directions, and it can rotate about the x, y, and z axes. The motion of the quadcopter is governed by the following equations:
# 
# $$
# \begin{aligned}
# \dot{x} &= v_x \\
# \dot{y} &= v_y \\
# \dot{z} &= v_z \\
# \dot{v}_x &= \frac{1}{m}(T_1 - T_3) \\
# \dot{v}_y &= \frac{1}{m}(T_2 - T_4) \\
# \dot{v}_z &= \frac{1}{m}(T_3 + T_4 - T_1 - T_2) \\
# \dot{\theta}_x &= \omega_x \\
# \dot{\theta}_y &= \omega_y \\
# \dot{\theta}_z &= \omega_z \\
# \dot{\omega}_x &= -\frac{k_x}{I_x}(\theta_x - \theta_r) \\
# \dot{\omega}_y &= -\frac{k_y}{I_y}(\theta_y - \theta_r) \\
# \dot{\omega}_z &= -\frac{k_z}{I_z}(\theta_z - \theta_r)
# \end{aligned}
# $$
# 
# where:
# 
# - x, y, and z are the position of the quadcopter in the global coordinate system
# - $\theta_i$ are the angles of rotation of the quadcopter about the x, y, and z axes
# - $\omega_i$ are the angular velocities of the quadcopter about the x, y, and z axes
# - $T_i$ are the thrust forces on the four rotors
# - $k_i$ are the stiffness coefficients of the rotors
# - $\theta_r$ is the desired angle of rotation of the rotors

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