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Formally, a quadcopter has four degrees of freedom. As you noted, at any given instant in time you can only apply force/moment in only four directions. …

The thrust vector is given by $f = qRg$ where $q$ is the magnitude of the thrust vector, $R$ is the rotation matrix for the current attitude and $g$ is the gravity vector in world coords, i.e. $[0~0~\ …

What you say ($\dot{z}_r$ = 0, $\ddot{z}_r=0$) is true in only the most basic systems. As a trivial counter example, consider systems where the desired height is moving: then it should be obvious that …

For the most common definition of roll, pitch, yaw used in UAVs, when pitch=90deg you encounter what's called gimbal Lock. Basically, when pitch=90deg you have infinite combinations of roll and yaw th …

Speaking from experience, smartphones (android in my case) do work as flight controllers but they have significant drawbacks The operating system gets in the way. You don't have root access and can' …

In a normal quadcopter configuration, if you tilt you will be pointing some of the lift force to the side. The only way to hover at a tilt, then, would be to somehow apply a counter force. … A standard quadcopter can't do that itself, so it would need be an external force (tether, wind, etc). …

1) I would try find out how fast your ESCs can update and use that as the PID update rate. Your current rate is definitely too slow (need at least 50Hz for marginal performance). 2) Putting a thresho …

First, you will need to test the communication range. Make a test app to talk between the phone and arduino/bluetooth shield and then walk around to see how far away they can still communicate. If it' …

I've implemented this algorithm before but I found a different paper easier to read. Try find the paper by Hamel and Mahoney (with Hamel listed as the first author). From that paper, which I don't h …

You've written your equations as if the weight vector you drew was positive, but then used a negative weight vector in your calculation. If you flipped $W$ around in your drawing you'd get $$ \begin{a …

In short, there is no particular need for synchronization of inner and outer loops. The inner loop should run "fast enough" to control attitude dynamics and the outer loop should run "fast enough" to …

The most common way to address this is to actually change the setpoint sent to the controller over time. For example, to get from point A to point B in T seconds, you would do $$ \begin{align} x^d(t) …

The first quadcopter probably only had a gyroscope to assist in roll/pitch/yaw control. … As you guessed, the second quadcopter was using an ultrasonic sensor to know it's height. It probably also had a more advanced gyroscope and algorithm to hold roll/pitch. …

By far, the most common solution to this problem is to use 2 separate controllers. The inner loop controller is what you already have: it tracks angle and height commands. The outer loop controller a …

First, I think you need to go back and look at your code. Gimbal lock is only a problem when you get very near (within a couple degrees) of 90. If you are seeing strange behavior at 45 degrees somethi …