The ETH video demonstrating a new algorithm to save a quadcopter after a single motor or prop failure didn’t reveal the algorithm. It did get me thinking about the quadcopter dynamics after a motor or prop failure. My analysis starts with the model above, and the equations in “Quadrotor Dynamics and Control” by Randal W. Beard, Brigham Young University, February 19, 2008, available at
http://image.ednchina.com/GROUP/uploadfile/201304/20130429210226589.pdf
The PDF attachment to this blog post shows why spinning up the body axis yaw rate after a motor or prop failure helps with quadcopter controllability, and points to potential new control laws that use controlled body axis yaw rate to allow cross-coupling between pitch and roll control. My next step is to try to work out a specific, simple motor or prop loss recovery algorithm. I haven't taken it that far yet, but there are a lot of sharp, creative and inventive people in this community. If it looks like this is on the right track, let's see what we can do with it.
I wrote up the analysis in Word 2010 and its useful equation editor. I then discovered that it's not easy to copy and paste those results into my blog! I looked at converting everything over to JPEGs to paste it up as images, but the resolutions on some of the equations suffered. The best way to share the results is to convert it to a PDF document, the attached file. I look forward to your peer review - maybe I made a mistake or overlooked something.
Blog%20Quad%20John%20Malley%20%20February%209%202014%20Post%20on%20Spinning%20Quads.pdf
John
Comments
@André Bertelsen - Thanks for pointing me to L1-AC and the on-line lectures. Google shows there's activity in applying it to quadrotors as well as fixed-wing UAVs. I read two papers this weekend with strikingly different takes on L1-AC The first is from Buddy Michini and Jonathan P. How at MIT in the USA, http://acl.mit.edu/papers/L1GNC09.pdf . The second is a critique of L1-AC by Saeid J. Jafari, Petros A Ioannou and Lael Rudd, at http://arc.aiaa.org/doi/abs/10.2514/6.2013-4513 No matter what, they would agree that Model Referenced Adaptive Control approaches might work wherever L1-AC can be applied.
@ArthurK - Still thinking about your suggestion.
Looks good, you should look into L1 adaptive control - a good lectur about it here:
https://www.youtube.com/watch?v=w8L2JdmpccE
I'm currently making my masters about an L1 controller on a quadcopter - hoping to make it run on the APM (dont know if there is enought CPU)
I have been looking at the ETH video too when it was announced. However I fail to see how this could be special. What I see is a quad that is converted to a rotor disk (or better cone) than is controlled by precession forces. Just like a helicopter rotor with cyclic control.
The control laws at work here (assumed Fr is lost as in John's example):
Lift = Ff + Fb + Fl
Cone angle = f(Fl)
Yaw rate = f(Tl - Tf - Tb)
(inertial frame) pitch = Fl cos (yaw) + (Fb - Ff) sin(yaw)
(inertial frame) roll = Fl sin(yaw) + (Ff - Fb) cos(yaw)
Due to precession the inertial frame roll and pitch forces are 90 ahead of the force produced by each motor. Just the same as in any helicopter.
Lift control remains the same as in normal flight mode.
The yaw rate is needed to stabilize the system. The higher the rate the higher the precession forces can be. But we need to limit the yaw rate to prevent gyro saturaton. Tl is used to control the yaw rate. The more power on the left motor the lower the yaw rate (balances against front and back motors). The cone angle depends on the yaw rate and Fl. The higher Fl the steeper the cone angle. To obtain a controllable airframe the cone angle needs to be between a minimum and maximum value. The yaw rate needs to be between a minimum and maximum value as well. Depending on the inertia, gyro limits and motor parameters either the yaw rate or cone angle has to be controlled to a fixed value.
Roll and pitch can no longer be controlled in body frame but only in inertial frame. By changing the Ff Fb Fl values in a cyclic manner attitude control can be obtained.
Off the top of my head, I don't know. We would want to look at the moments of inertia for the tricopter and the body axis attitude accelerations in terms of the cross-products of pitch and roll rates when the tricopter has a body-axis yaw rate.
curious, in theory could we could build a tricopter without a servo/pitch then?
Yes!
Thats great news.
Well done.