##### 1. Quick weekly updates

- Conducted MQ-LQ experiments
- Analyzed torque disturbances
- Obtained frequency domain response of forces

##### 2. Detailed updates

###### MQ-LQ Experiments

- Astonishingly, the miniquad could hover below the large quad for as low as 1 m. It did not crash for any of the proximity flight experiments.
- The experimental data shows that the value lies between 0.25-0.35 N. Attached below is a plot for the same:

- The forces show a sharply decreasing tend with increasing separation. The commanded setpoint separations were maintained well by the miniquad (It settled about 0.3-0.5 m below the setpoint, but was not disturbed too much).
- Actuator disk theory predicted a very high value of force: 3.3 N for large quad on top of miniquad. Here is how it was obtained
- Let the weight of the top quad be ‘W’. Then each propeller provides a thrust of ‘W/4’ in hover. Using this thrust ‘T’, and the area of the propeller ‘A’, we can get the flow velocity () as ( being the density of air)
- Substituting W=6.4 N, and A = 0.0324 m^2, we get = 8.96 m/s. Using this velocity as the oncoming flow for the miniquad, and assuming this acts as a drag force on the projected area of the miniquad (84.5 cm^2), we get 3.3 N as the force.

###### Torque Disturbances

- A script was written to characterize the torque disturbances using the angular accelerations and motor forces. The script was implemented on both sets of experiments (LQ-MQ and MQ-LQ).
- The thing about torques is that they are pretty non-intuitive. It is difficult to say what values of torque you would expect, or what level of disturbance will a specific torque cause. In any case, shown below are the plots (6 of them, so please bear with me or just skip them).

1. LQ-MQ

a) Roll

b) Pitch

c) Yaw

- The pitch and roll torques are around 0.05 Nm on the bottom quad (LQ), and of the order of 0.002 Nm on the top quad (MQ).
- As mentioned before, torque values are non-intuitive. A way to think about it is
**imagining a weight being placed at one of the motors of the quads**. For example, the large quad has an arm length of 0.166 m, so 0.05 Nm torque is equivalent to a weight of 30 g placed at one motor. This is significant enough to cause angular motion, but small enough to easily overcome by increasing/decreasing the RPM of motors. All we need to know is the quantity of this change, and at what relative location to implement this. - For the MQ here, it is like placing a 3 g weight at the motors which can be ignored (and even mitigated if we are that sensitive about the performance).
- Yaw torques are much lesser than pitch and roll torques which is as expected because direct push by air flow cannot cause a yaw torque. Yaw disturbances also don’t affect the performance much as compared to roll and pitch disturbances.

2. MQ-LQ

a) Roll

b) Pitch

c) Yaw

- This pitch and roll torques now have values around 0.005 Nm for the MQ (bottom vehicle) and about 0.02 Nm for the LQ.
- Equivalent weights – 8 g for MQ, and 12.5 g for LQ.
- As before, for the top quad, the weight can be ignored but can include to improve performance.
- For the bottom quad, 8 g weight is significant as each motor produces about 35 g during hover. However, this additional thrust can be produced easily by the motors and hence we can avoid loss of control.
- Once again, yaw disturbances are way below the pitch and roll torques, and hence we can safely ignore them.

###### Frequency content of disturbances

A frequency response was obtained to check whether the disturbances have a periodic nature of any sort, or if they are just white noise. We get a frequency response using fourier transform to move from time domain to frequency domain. Once in, the response shows magnitudes at various frequencies, which helps in realizing what kind of periodic nature the signal/disturbance has.

(DC component has been eliminated for all the frequency domain plots shown below)

Shown below is a plot of aerodynamic force (on the bottom quad) vs. time, and that of the frequency content of the force for the LQ-MQ experiment at 1.0 m separation.

- We have high magnitudes for frequencies between 0-2 Hz. This suggests that either the forces are periodic with that frequency, or that the quad enters and exits the downwash regime with that frequency. In either case, the frequency is much lower compared to our sensor/estimator frequencies and hence can be mitigated easily.
- For frequencies beyond 2 Hz, the magnitude is pretty small and fluctuates a lot. Imagine trying to characterize noise using frequency domain analysis, and the frequency content itself is noisy. Pretty frustrating. But since the magnitudes are low, we can again safely ignore them.

I am attaching the plot for the MQ-LQ experiment (1.0 m separation) just for reference. Similar conclusions hold.

Frequency content of the torques is also very similar. Not attaching those for the sake of brevity.

##### 3. Planned work for next week

Since the empirical approach (analyzing sensor data) has not resulted in great conclusions so far, I am planning to take a theoretical approach. I will start by studying advanced models for flows through propellers. There is an excellent paper on distance dependent flow profile of a propeller downwash and another great paper on empirical model of UAV downwash.

- I plan to study these papers and build a conceptual model.