TITLE: Applications of Topological Complexity of configuration space
QUESTION [7 upvotes]: I'm starting to work on topological complexity of configuration spaces. 
Articles say that this field has applications in robotic and control theory. One of the important articles belongs to Michael Farber 
My questions are :
1) How topological complexity can help to robotic and control theory?
2) Does there exist any applied problems in robotic or control or other subjects that have been solved by notion of Topological complexity?   
Thanks

REPLY [3 votes]: (EDITED) 
It's a bit late to answer this. However, I have spent around a year working on this topic as an undergraduate and it was my dissertation's topic so I hope I can make my own tiny contribution. Topological Complexity can help you to understand the Gimbal Lock, which is strongly related to the incident happened in some of the Apollo Moon missions:


The explanation from the point of view of topological complexity goes as follows:
The configuration space of the spacecraft is (topologically):
$$X \colon = \mathbb{R}^3 \times SO(3)$$
and $TC(X)=4$ since $X$ is homotopy equivalent to $SO(3)$ and $TC(SO(3))=4$. The key to prove the last fact is a result proved by Michael Farber (thanks to Mark Grant for pointing out a previous mistake here) which says that if $X$ is a topological group, then the L-S category of $X$ and it's topological complexity are equal. (If you want I can send you the full proofs of all this facts -in Spanish-)
You can also use Topological Complexity to deal with robot arms in factories and so on....but I don't want to write a book as an answer.
Hope that this answers, at least partially, some of your questions.
Don't hesitate to ask for details or more information if you wish!
--------------------------EDIT-------------------------
The way I relate gimbal lock to TC is as follows.
First of all, gimbal lock is the loss of control of an airplane or spacecraft. And when restricted to drones, we see it as an overturning.  As you know better than me, $TC$ measures the stability of motion planning algorithms, and if $TC(X)$ is grater than one, then the motion planning algorithms over $X$ are unstable.   
Moreover, the configuration space of the drone is homotopy equivalent to $SO(3)$. And the motion planner needs to check periodically the inclination of the drone (which is alterated due to exterior circumstances -a storm or wind in case of a dron flying close to the land-) and considering the inclination that it should have, to correct it by changing the path in  $P(SO(3))=SO(3)^{I}$. The problem about $TC(SO(3))=4$ is that there is no continuous global section of the path fibration, and as a consequence, given a small change in the inlination of the drone, the algorithm can produce a completely wrong path. I have made some pictures to illustrate this. 


In the first picture we see the drone, (it is supposed to have suffered a small perturbation despite of I have pictured it very big in order to produce a better drawing). It should correct its inclination by rotating clockwise direction. However, due to the instabilities caused by $TC(SO(3))=4$ it rotates the other way around and it overturns, which means we have lost it since it can not recover from that position. 
I hope that this could help a bit. Feel free to criticize the ''arguments'' or ''explanations'' written above since they are just the way an undergraduate thought about gimbal lock and are far from rigorous or even serious.