##### 1. Quick updates (January 1st – February 9th)

- Fixed the integral controller for yaw and position.
- Analyzed the horizontal sweep experiments for force and torque (they are very promising)
- Gave the prelim
- Moving average to smoothen sensor noise.
- Tried to incorporate a theoretical model for velocity field below a quad (still working on this)
- Wrote a script to get forces and torques on a quad from the velocity field
- Cleared the prelim
- Conducted velocity field experiments for the miniquad and tried fitting a model to it

##### 2. Detailed updates

###### A. Integral controller

The position and yaw integrator has been added in codes QuadcopterPositionController.hpp and QuadcopterAttitudeController.hpp respectively. Each vehicle now has its own variable for storing the integrated value and this works in experiments (except for the miniquad (MQ) giving some oscillations, but that is probably a tuning or hardware problem because the largequad (LQ) did not oscillate).

###### B. Horizontal sweep experiments

A description of the experiment:

- The miniquad (MQ) is on the top and is held fixed at (0, 0, 3.5)
- The largequad (LQ) on the bottom is first brought to (0, -2, desiredHeight).
- After 5 seconds, the LQ is commanded to move in positive y-direction at 0.2 m/s. This is done until it reaches (0, +2, desiredHeight). (This takes 20 seconds.)
- 5 seconds after that, both the quads land.
- desiredHeight can be anything. In my experiments, these are 3.0, 2.5, 2.0, 1.5 and 1.0 giving vertical separations of 0.5, 1.0, … 2.5 respectively.

A sample force plot for vertical separation of 1.0 m is shown below,

It can clearly be seen that there is a significant force when the LQ is within 0.5 m of the MQ in the horizontal direction. This variation can be approximated with a gaussian distribution which is covered in the theoretical model.

A sample roll torque plot for all vertical separations is shown below,

This can be explained by imagining the motion of the LQ below the MQ. When it approaches the MQ, there is more air flow on the part of the LQ closer to MQ, hence it would get a positive roll torque. The torque magnitude depends on 2 things: value of the velocity of air flow, and gradient of the velocity across the LQ. Hence it achieves a peak, but dies down at y=0, because the velocity field is symmetric on the LQ.

###### C. Moving average

These are required to get a neat trend rather than points all over the plot we can’t make sense of. To give an example, here is the torque plot without using moving average,

The rate gyro does not have as much noise as the accelerometer and hence we can at least see the trend in torque.

Here is a plot of the forces without moving average (same experiment as shown in previous section),

The noise level (~0.2 N) is comparable to the maximum force (~0.6 N) itself.

###### D. Theoretical Model

I have read multiple papers on getting velocity fields and each one has their own take on how to represent it. In general a velocity field below a quad is axisymmetric and hence can be written in cylindrical coordinates as , where z is the axial separation, and r the horizontal (or radial) separation.

Further, in most papers. What changes across the papers is $V(z)$ and $k(z)$. Some papers have an inverse relationship , while others have a linear relationship . It seems that each one uses a function appropriate for their application.

According to Navier-Stokes analytical solution, the relationship should be proportional to 1/z.

###### E. Velocity field experiment

A miniquad was held sideways such that its z-axis aligns with the lab y-axis. It was run at various thrusts and velocity was measured at different locations.

The results for axial velocity (r=0) are shown below. The solid line is a least-squares fit assuming an exponential decay model:. , where is the induced velocity given by the actuator disk theory.

It predicts well in the far-field (> 1m) but doesn’t work well in the near-field because the flow is still accelerating.

Similarly, velocities varying with r coordinate, with z fixed:

Solid lines are a gaussian fit. The standard deviations turn out to be proportional with the axial separations.

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

- Fix a model for velocity
- Compare the theoretical predictions with experimental results
- Conduct more proximity flight experiments
- Conduct a velocity field test experiment with better testing rig (assigned to Trey)
- Continue writing the paper