Since this is my main area of research, let me attempt an answer (which inevitably got quite long!). The short answer is that I do not know of any specific instances where topological complexity has been used to solve robotics problems, but I know that roboticists are interested in the concept, and am hopeful that such instances may occur in the future.

Let me start by saying that, in addition to the nice survey article by Farber you link to, you should also be familiar with his original papers on the subject, which perhaps provide a bit more motivation than the survey article:

*Michael Farber*, MR 1957228 **Topological complexity of motion planning**, *Discrete Comput. Geom.* **29** (2003), no. 2, 211--221.

*Michael Farber*, MR 2074919 **Instabilities of robot motion**, *Topology Appl.* **140** (2004), no. 2-3, 245--266.

Let me briefly recall the main definition and attempt to explain why a roboticist should be interested. Let $X$ be a path-connected space, which we view as the configuration space of some mechanical system. A *motion planner* in $X$ is a section $s:X\times X\to X^I$ of the free path fibration
$$
\pi: X^I\to X\times X,\qquad \pi(\gamma) = (\gamma(0),\gamma(1))
$$
which evaluates a path at its pair of initial and final points. Hence it is a rule for how to get from $A$ to $B$, such as is required for the robot to perform tasks.

It turns out that a *continuous* motion planner in $X$ exists if and only if $X$ is contractible (an easy exercise in homotopy theory). Therefore motion planners usually have discontinuities forced upon them by the topology of the configuration space.

Now, the roboticist may seek to minimize such instabilities, to produce motion planners which are optimally elegant, or robust to noise. One way to do this would be to find a minimal partition of the domain $X\times X$ into smaller pieces, on each of which we have a continuous motion planner. This motivates the following:

**Definition:** The *topological complexity* of $X$, denoted $TC(X)$, is the minimal integer $k$ such that $X\times X=U_0\cup U_1\cdots \cup U_k$ for some open sets $U_i\subseteq X\times X$, on each of which there exists a continuous local section $s_i:U_i\to X^I$ of the free path fibration.

Here we give the definition in terms of open covers of $X\times X$; Farber shows that this is equivalent to finding a partition of $X\times X$ into nice pieces (say, ENRs). We have also normalized so that $TC(*)=0$, as is customary in a lot of the newer papers on the subject.

It turns out that $TC$ is a homotopy invariant, and therefore interesting to algebraic topologists. It is closely related to more classical invariants, such as the Lusternik-Schnirelmann category and the immersion dimension of real projective spaces. Obtaining good estimates can involve large chunks of classical algebraic topology, including cohomology algebras and operations, obstruction theory, and (most recently) Hopf invariants. It is an instance of a more general concept called the *sectional category* of a fibration, originally defined and studied by Albert Schwarz under the name *genus*.

Returning to your actual questions: Another way to estimate $TC(X)$ from above would be to exhibit an actual motion planner with the fewest possible domains of continuity. This is one way in which the study of $TC$ by mathematicians could benefit the robotics community: by producing motion planners which are "topologically optimal" for configuration spaces of interest.

Of course, Farber's approach only takes into account the topology of $X$, and not its geometry. In applications, one might be interested in finding paths between configurations which minimize the distance or energy. There is a recent article by Zbigniew Błaszczyk and José Carrasquel which attempts to reconcile this with Farber's approach.

You may also be interested in recent work of Petar Pavesic, which investigates topological complexity of the forward kinematic map. This is perhaps closer to applications, as it is concerned with finding a path from a given configuration to a given pose of the end-effector (of, say, a robot arm).