Underactuation

\begin{aligned} \\ \ddot{y} &= \frac{u_1}{m}\sin\phi \\ \ddot{z} &= g - \frac{u_1}{m}\cos\phi \\ \ddot{\phi} &= \frac{u_2}{I_{xx}} \end{aligned}

Since a 2D drone only has two control inputs, we can only simultaneously control two of its three degrees of freedom.

A real quadrotor has four controllable quantities (the four propeller rotational speeds) and 6 degrees of freedom ( x,y,z,\phi,\theta,\psi ).

How many of these degrees of freedom do you expect to be simultaneously controllable for a quadrotor?

2

3

4

5

6

SOLUTION:

4

When we're controlling a real 3D quad, we typically choose to control the three positional degrees of freedom, ( x,y,z ) and yaw ( \psi ).

But sometimes we choose to sacrifice one or more of these degrees of freedom in favor of roll and/or pitch control.

Take a look at this video for some interesting examples of maneuvers that require roll and/or pitch control.