# Backstepping Integrator: changing the virtual control

given the following differential equation 2°ODE in the following form:

$\ddot{z}=-g + ( cos(\phi) cos(\theta))U_{1}/m$

found in many papers (example) and describing the dynamic model of a quadrotor (in this case I'm interested as an example only for the vertical axis $Z$) , I get the movement about $Z$ after integrating the variable $\ddot{z}$ two times. As control input I can control $U_{1}$, which represents the sum of all forces of the rotors.

A Backstepping Integrator (as in many of papers already implemented) defines a tracking error for the height $e_{z} = z_{desired} - z_{real}$ and for the velocity $\dot{e}_{z} = \dot{z}_{desired} - \dot{z}_{real}$ to build virtual controls.

Through the virtual controls one can find the needed valueof $U_{1}$ to drive the quadrotor to the desired height (see the solution later on)

But wait...as said above I need to track both: position error and velocity error.

Now I asked myself, how can I transform such equation and the corresponding virtual controls to track only the velocity??

In my code I need to develop an interface to another package which accepts only velocity inputs and not position information. I should be able to drive my quadrotor to the desired position using only velocity informations, tracking the error for the z displacement it not allowed.

The solution for the more general case looks like:

$U_{1}=(m/(cos(\phi)cos(\theta))*(e_{z} + \ddot{z}_{desired} + \alpha_{1}^{2}\dot{e}_{z} - \alpha_{1}^{2}e_{z} + g + \alpha_{2}\dot{e}_{z})$

for $\alpha_{1}, \alpha_{2} > 0$

I could simply put brutal the $\alpha_{1} = 0$ for not tracking the position on Z but I think that is not the correct way.

Maybe could you please point me in the right direction?

Regards

• Are you saying you want to control the $z$ position but you only have feedback regarding the velocity $\dot{z}$? Does that mean you are assuming some initial condition and estimating your position based on integration of the velocity? Oct 27, 2015 at 3:24
• @BrianLynch, I think he's looking for velocity control to overcome the limitation of the joystick's workspace. This is the recommended way to allow the quadrotor to fly over long distances. Nov 26, 2015 at 4:20