Dear Students,

In this article, I want to talk a bit more about mass moments of inertia applied to a UAV.

With regards to mass moments of inertia, if you want to calculate a very precise number, then since there are many components in the drone, you would have to compute the inertia of each component and you should take into account how far away it is from the main axis of the drone around which you compute your mass moment of inertia. When you design your drone in programs like Catia or Solidworks, then they usually give you that precise number.

Another way would be to measure it somehow with a device. If you know the torque that you apply to a drone around a certain axis, and you know the angular acceleration of the drone about that axis, then you can calculate your mass moment of inertia about that axis.

However, if you want to get a very rough estimate of the drone's mass moment of inertia quickly, then if you assume that the main body of the drone is two cylindrical rods that are perpendicular to each other, then it will look something like this as shown in the image below:

If you apply the fundamental Calculus equations of the mass moment of inertia to a cylindrical rod, then, in the image above, you can see the final equations for the rod's mass moments of inertia about each of its axes.

I have found a couple of articles that give you these derivations:

https://www.scienceabc.com/nature/universe/moment-of-inertia-calculate-rod.html

In the link above, the relevant part is (Axis through the center of mass)

And then another link:

http://hyperphysics.phy-astr.gsu.edu/hbase/icyl.html

Here, the relevant topic is: Moment of Inertia: Cylinder

In the image above, R is the radius of the cylinder, m is the mass of the rod, l is the length of the rod, and then the numbers 1 and 2, they are for the rod 1 and 2, respectively.

And so:

I_x_total = I_x1 + I_x2

I_y_total = I_y1 + I_y2

I_z_total = I_z1 + I_z2

If you want to go a bit more specific and you want to take into account the propellers when you compute I_z_total for example, then you can do it as shown in the image below:

You can see the inertia values for the propellers about their own z axes: J_bi. These values are hard to compute by hand because the blades contain all kinds of twists. The manufacturer of the blades should give you those values, or your blade design program. However, because of the fact that their z axes are located from the drone's z axis at a distance of l/2 meters, you have to use the parallel-axis theorem. What it means is that the contribution of a propeller to the total mass moment of inertia about the drone's z axis is its own J_bi (mass moment of inertia about its own z axis) + the mass of the propeller and its center point distance to the drone's z axis squared.

So,

I_z_total = I_z1 + I_z2 + 4*(J_b + m*(l/2)^2)

You have to multiply it by 4 because you have 4 propellers. Keep in mind though that it is a very rough estimate, but that would be the first approach.

Best,

Mark