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We aim to experimentally validate the performance of our geometric controller. In practice, we expect that the almost global controller is sufficient since it only has one unstable attitude (at the antipodal point) with zero velocity. This means that real systems will likely not get stuck at the unstable equilibrium simply due to system noise.
The initial flight test of our geometric attitude controller compared to the standard quaternion based controller on the PX4 flight stack. Both controllers use the same position controller to determine the desired attitude (a 90degree rotation).
In this work, we implemented the geometric controller on the PX4 flight stack. This task was done with the help of Cheng (a master's student) under my guidance. The standard controller has 3 proportional gains: roll, pitch, and yaw. Our controller only requires 2 gains: the attitude error and the angular velocity error. In the test above, we left the gains unchanged for our geometric controller. As one can see, the geometric controller appears to be more stable before doing the rotation. However, the geometric controller overshoots and has to readjust after the rotation. We can find out why by looking at the data log below.
Standard Controller
Plot showing the norm of the attitude error of the standard attitude controller performing a 90 degree rotation. The gains are well tuned as the error converges exponentially without any overshoot. The rotate command was sent at 36.24s. The controller took 1.23s to complete the rotation.
Geometric Controller
Plot showing the norm of the attitude error of the geometric attitude controller performing a 90 degree rotation. The convergence rate seems to be exponential, but there is overshoot and the quadrotor has to adjust once to correct it. The rotate command was sent at 35.07s. The controller took 0.76s to first reach the desired attitude before overshooting.
Standard Controller
Plot showing the norm of the position error of the standard attitude controller performing a 90 degree rotation. The position error is larger than the geometric controller before the rotate command was sent at 36.24s. Note that the sudden increase in error is due to us landing the quadrotor.
Geometric Controller
Plot showing the norm of the position error of the geometric attitude controller performing a 90 degree rotation. The geometric controller seems to be more stable at holding the position than the standard controller. Note that the sudden increase in error is due to us landing the quadrotor.
From the above plots, we see that the geometric controller seems to be more stable in holding the position at steady state (before rotate command) and convergences faster. However, the geometric attitude controller experiences an overshoot which results in a slightly larger position error as it completes the rotation. This result is unexpected since the theory states that the controller should converge exponentially without overshoots. To find out why, we look at the control torque requirements below.
Geometric Controller
Plot showing the control torque requirements for the geometric attitude controller as it performs a 90 degree rotation. The solid lines indicates the desired torques along the roll, pitch, and yaw axes, while the dashed lines indicates the hardware/software limits.
The geometric controller with the same gains as the standard controller desires more torque than the motors/software allows. This results in faster convergence, but the physical system is unable to follow the motor commands which explains the overshoot. Due to the COVID-19 pandemic, hardware experiments were postponed. However, we are able to test the PX4 flight stack virtually on Gazebo/ROS. First, we have to determine which gains to use such that the torque limits are not reached and this can be done simply using our framework.
Plot showing the feasible gain pairs that will guarantee exponential convergence with some minimum convergence rate.
The initial geometric controller gains were set to k_d = 7, k_v = 7. However, the controller using that gain combination requires more torque than physically possible from the motors. Testing the feasible gain options, the gain pair k_d = 1.4, k_v = 2.5 produced the desired results. Note that this is not the only possible option or most optimal. In fact, any gain pairs with non-zero \beta (z-axis) will converge exponentially.
The desired control torque for the geometric attitude controller performing a 90 degree rotation with gains k_d = 1.4, k_v = 2.5. None of the control torque exceeds the maximum bounds. In fact, the gains can be increased for faster convergence
The norm of the attitude error now convergences exponentially with no overshoot for the geometric controller with gains k_d = 1.4, k_v = 2.5. However with the weaker controller, the attitude converges in about 3s.
The process to select new gains was done simply by looking at the feasible gain pair plot. There was no need to randomly test gains and hope that the system converged. The only limitations were due to the physical hardware not being able to meet the demand of the controller. Even in that case, the system still reacted quickly and converged.
The new gains produced nice exponential convergence results, however the convergence rate could be improved. Our next step is to construct a framework to automatically test feasible gain pairs that do not exceed the torque limits. This process will be done in simulation and then transfer to the real quadrotor. In addition, we are working towards proving and implementing a full position and attitude controller using the framework found in our global controller results.
This experiment is a great accomplishment (even at the early stages) because it showed that the theoretical results can be implement directly onto real systems without any changes. This has always been our focus, because sometimes there is a great disconnect between research and applications.
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