Here is an update on the work that I’ve been doing. Just got back from a course on systems identification using a program called CIFER (Comprehensive Identification from Frequency Responses). I posted previously the pitch and roll frequency responses. I had also done frequency sweeps in the yaw axis but had not posted the results. I had the data with me and wanted to put what I had just learned to use. This course walked us through what was important about gathering the data (instrumentation, data consistency, frequency sweeps…), developing the bode plots that show the relationships between the important control inputs and outputs (states), and lastly how to develop a 6 degree of freedom model from the frequency response data. One thing that I learned/didn’t realize is how uncoupled multicopters are compared to helicopters. Helicopters become even more difficult to model going from a teetering rotor head to something that had geometric or effective offset (i.e. flap hinges displaced from the center of rotation or no flap hinges). So coming up with a model (set of transfer functions) that represent a multicopter is not too difficult. What remains to be seen is how scalable they would be to different multicopter size and configurations.
I used my frequency sweeps for the Y6 in roll and yaw to come up with a bare airframe model for the lateral/directional equations of motion. So you have 3 equations of motion for rigid body motion instead of the 6. SO the lateral directional equations are the lateral force, roll moment, and yaw moment equations.
I used CIFER to fit the following frequency responses, roll rate § to Rout (roll mixer output), lateral acceleration (AY) to Rout (roll mixer output), body axis lateral translation velocity (V) to Rout (roll mixer output) and yaw rate ® to Yout (yaw mixer output). It also determined the modes of motion for the system and stability and control derivatives. Here are the plots of the frequency responses with the model laid over it.
first chart is V to Rout and P to Rout
next chart is AY to ROUT and R to YOUT
Generally the model is fitted for frequencies from 1.5 rad/s to 15-20 rad/s. It captures the unstable long term mode, the yaw short term mode, and the roll mode. There was no roll damping derivative which leads me to believe the roll mode is effectively acceleration command in the short term. Here are the transfer functions
So in order to develop a full 6 DOF model, I will need to do a sweep in the vertical axis with the collective (that’s throttle for the multi’s). Then I can use the same process. So this information could be used to develop better a better control system however these equations only apply to the Y6. So a lot more work would need to be done to understand how to build a generic model of a multicopter or helicopter and what physical attributes for that type of aircraft play into the eigenvalues or the stability and control derivatives.