I've noticed that almost all research being done with helicopter robots is done using quadcopters (four propellers). Why is there so little work done using tricopters in comparison? Or a different number of propellers? What about four propellers has made quadcopters the most popular choice?
At least in part quadrotors offer a nice balance between the complexity of the dynamics and power requirements. With traditional single rotor helicopters, control is a function of the orientation of the rotor which means you must change its orientation to change direction of the craft. This makes for very complex mechanical linkages comparatively speaking and it complicates the dynamics. With tri-copters the dynamics include an imbalance of the moments induced by the spinning of the rotors. With more than four rotors you get improved stability and some ability to handle failure, such as a motor going out, but you quickly run into a power problem. The more motors you need to drive the higher your power requirements and quadrotors are already very power hungry. This is a major issue in robotics in general. The quadrotor dynamics naturally balance the moments from the rotors and the mechanical linkages are simpler.
You need 4 degrees of freedom to control yaw, pitch, roll and thrust.
Four props is therefore the minimum number of actuators required. Tricoptors require a servo to tilt one or more rotors which is more mechanically complicated.
There is no restriction to only 4 props, hexa+ coptors are also very common.
Generally you want an even number of props unless you are tilting so the yaw forces balance out.
Choosing the exact number of propellers used involves many complicated tradeoffs. A single prop cannot be too large or the inertia makes the multicopter unstable (which is why you see more props instead of larger props for large multirotors).
Large propellers are far, far, more efficient than many small propellers which is why there is essentially a size cap on multicoptors (unless you go variable/collective pitch which would be stupid).
I think the main reason is that they're simply easier to build in a stable way. A 120º angle is harder to get right than a 90º angle.
Another thing that is a little easier to understand is how the relationship between propellers leads to different types of motion. Thinking about different propellers moving at different speeds and directions and how that affects robot motion is sort of intuitive, because you don't have to do a lot of trigonometry in your head.
Lastly, it's just a good compromise between stability/controllability and cost, since motors are usually one of the most expensive components for that kind of robot.
The mechanical answers above are correct. The inherent stability problems with single large motors are exchanged for dynamic comtrol over 12 dimensions of acceleration, yaw, pitch, roll which can be partially coupled (the translational amd rotational matrix) where one is presented with a simplified diagonal inertial frame to build a dynamic model with. In this model there is also an inverse relation between the radius of the quad and the translational and rotational agility. It becomes very easy to "dodge bullets" at very very small radii.
To answer the question How do you get pure yaw motion with a quadcoptor?, in comments to this answer, you get pure yaw in the following way:
North and South motors rotating the same speed but collectively at a higher (or lower) speed than East and West Motors which are also at the same speed.
It won't pitch or roll, it will yaw y'all. (Sorry)
Furthermore, in software one can control the copter after breaking off the north and south propellers at the expense of yaw control, the craft will continuously spin and as long as the frequency of software refresh rate is able to handle the speed of yaw rotation the copter remains exactly as stable (sort of) the dimension of acceleration is clipped and response or jerk is also somewhat clipped but it can move pitch and yaw just the same by compensating in software. (Desired yaw state becomes virtually coupled to the physical state)