Linearizing a 2-D quadrotor dynamic model

[Michael Wu]

Hi all!

I am taking an online course on aerial robotics and am currently on the topic of linearizing a 2-D quadrotor dynamic model. See slide (link below):

The equations under "linearized dynamics" are derived using the equilibrium hover configuration (e.g. y = y₀, z = z₀, Φ₀ = 0, u_1,0 = mg, u_2,0 = 0) and the fact that at hover, sin(Φ) ~ Φ and cos(Φ) ~ 1.

So, linearized

y_ddot = (-u₁/m)*sin(Φ) = (-mg/m)*Φ = -gΦ

This makes sense to me. But how come z_ddot isn't 0? and Φ_ddot isn't 0? Shouldn't

z_ddot = -g + (mg)/m*(1) = 0?
Φ_ddot = (0)/I_xx = 0?

Thank you!

 

[Achy47]

Hi there,

Thank you for your question! Linearizing a 2-D quadrotor dynamic model can be a complex topic, so I'm glad you're taking an online course to learn more about it. The equations under "linearized dynamics" are derived using the equilibrium hover configuration, as you mentioned. This means that the quadrotor is in a state of equilibrium, meaning that there are no external forces acting on it and it is not accelerating in any direction.

In the case of z_ddot, it is not 0 because there is a gravitational force acting on the quadrotor. This force is represented by the -mg term in the equation, which is why z_ddot is equal to -g. Even though the quadrotor is in equilibrium, it is still experiencing a downward force due to gravity.

Similarly, for Φ_ddot, it is not 0 because there is a moment acting on the quadrotor due to the thrust from the rotors. This moment is represented by the u2 term in the equation. While the quadrotor is not accelerating in the Φ direction, it is still experiencing a moment that keeps it in equilibrium.

I hope this helps clarify things for you. Keep up the good work in your course and don't hesitate to ask any further questions!