Intro to 2D Dynamics

C3 L3 01 Intro To 2D Dynamics V1

Review of Vehicle Dynamics

1D Dynamics

We can express the vertical acceleration \ddot{z} in terms of the vehicle's thrust u_1 (which we control), as follows:

\ddot{z} = g - \frac{u_1}{m}

To emphasize how directly we control acceleration we can rewrite this as:

\ddot{z} = \bar{u}_1

where

\bar{u}_1 = g - \frac{u_1}{m}

2D Dynamics

We can express the equations of motion of a drone in two dimensions in terms of the collective thrust u_1 and the commanded moment about the x axis u_2 as follows:

\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}

Rotor Physics

But where do these u_1 and u_2 come from?

Recall that the force produced by a propeller is related to the rotational velocity \omega of that propeller through the following equation:

F = k_f \omega^2

This means that the total collective thrust u_1 can be expressed as follows:

u_1 = k_f ( \omega_1^2 + \omega_2^2)

To get u_2, recall that a rotor at a perpendicular distance l from the x-axis will produce a moment M_x about the x-axis given by:

\begin{aligned} M_x &= l \times F \\ M_x &= l k_f \omega^2 \end{aligned}

and remember that this quantity is positive if it causes clockwise accelerations and negative if it causes counter-clockwise accelerations.

This means the total moment about the x-axis, u_2, is given by the following:

u_2 = lk_f (\omega_1^2 - \omega_2^2)

For the rest of this lesson we're going to ignore the rotational velocities of the individual rotors and assume that we can control u_1 and u_2 directly.

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