# 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? – Brian Lynch Oct 27 '15 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. – CroCo Nov 26 '15 at 4:20