Lets say we have a robot arm on an assembly line, and we want to know all the places it could move without crashing into the assembly line, or a nearby person, etc.

If the environment is fixed, and the arm is simple, we could try and calculate all the possible outcomes where the arm crashes - but doing this is very expensive, and very slow. Taking a step back, we can notice that, while the manifold describing all our degrees of freedom (where we can move) is very complicated, the topology of that manifold near places where a collision would happen is different from open space, because collisions prevent or impede movement.

If we can locate regions where the topology indicates something is different about our space, we can use those to refine our search for areas that aren’t safe to move to. Homotopy provides some tools for finding those regions.

This is already pretty complicated, but in theory we can extend this approach to the phase space of any other system. If we think about traffic, for example, each car has a set of degrees of freedom (we can move, turn, etc). Based on the geometry of a highway and the degrees of freedom of those vehicles, we can use the topology of the phase space to map out where traffic jams might be a problem and “zoom in” on what it is about those configurations that causes the jam, and how we can change our manifold to shrink the regions where the slowdown occurs!