Using Fuzzy Simplicial Sets as Approximated Atlases

[cmuell89]

Hello,

I am currently working on a project makes use of Constrained Motion Planning techniques for robotics. Very briefly, constrained motion planning techniques generate a trajectory of points that represents a geodesic path along some implicit manifold embedded in the configuration/planning space of a robotic agent. So for example, a 7 degree of freedom arm can be thought of as a point object in 7-dimensional configuration space with each dimension representing a joint angle. A constraint of the motion of the robotic agent implicitly defines some subset of points that are constraint valid (keep a cup upright etc,.). The challenge is that this manifold is often not analytically definable. There are methods to take a random point in your configuration space and project it onto the manifold using a gradient descent style approach. I

It would be very useful to come up with a quick way to generate an Atlas-like representation of this constraint manifold by projecting a large number of points onto the manifold offline and then use this Atlas for faster planning events in the future. This has been explored in the following paper: http://www.leonardjaillet.com/Research_files/Isrr11_jaillet_AtlasRRT.pdf

My question is if it would be possible to use the fuzzy simplicial set as an approximation of an Atlas representation of this implicit manifold and to potentially use it to produce approximately on manifold points for motion planning.