FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES A.ABRAM AND R.GHRIST Also “Robot navigation functions on manifolds with boundary” – Daniel Koditschek and Elon Rimon.

2  Motivation: Consider an automated factory with a cadre of Robots. Figure1: Two Robots finding their way from start points to destination. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

3  Configuration Spaces: We define.  What looks like? -. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

4  Why homeomorphic to ?  homeomorphic to. Figure2: may be represented as a b FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

5  How does configuration space helps us with robot motion planning problem?  Safe control scheme using vector field on configuration space. Figure3: A Vector Field in Configuration space translates to robot motion. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

6  Navigation function:  It can be shown that all initial conditions away from a set of zero measure are successfully brought to by. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

7  Remark: Global attracting equilibrium state is topologically impossible. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

8  One particular solution: Koditschek and Rimon.  Composition of repulsive and attractive potentials. Figure4: “Attractive” and “repulsive” potentials produce navigation function. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

9  Koditschek and Rimon in more detail[1]:  Sphere World (for ) - Sphere World Boundary - Obstacle Repulsive Attractive Total: … → FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

10  Koditschek and Rimon in more detail[2]: K=3K=4K=6 Figure5: Koditschek and Rimon Navigation function. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

11  Navigation properties are invariant under deformation.  So this solution is valid for any manifold to which Sphere World is deformable. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

12  Robots moving about a collection of tracks embedded in the floor. Figure6: “Robots on an graph”. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

13  Example: Figure7: Realization of. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

14  It is hard to visualize even simple conf. spaces.  Discretized configuration space Figure8: Even Simple graph leads to complicated configuration spaces. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

15  Discretized configuration space  We can think of this as imposing the restriction that any path between two robots must be at least one full edge apart. Figure9: Excluded Configurations [left] Closure of edge [center] Remaining Configurations [right]. closure FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

16  Example: 0-cells 1-cells 2-cells Figure10: Realization of. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

17  Using same strategy it is easy to apprehend those spaces. Some interesting results appear.  Those are rather surprising results: Figure11: homeomorphic to closed orientable manifold g = 6. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

18  How close are those discretized spaces to original ones? FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

19  How this works - Figure11: Graph that does not comply [upper] graph that complies [lower] with the theorem. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

20  Another powerful result: FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

21  How this works[1] - FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

22  How this works[2] - P = 5 Figure12: Topological structure (homology class) of 5-prone tree. FINDING TOPOLOGY IN A FACTORY: CONFIGURATION SPACES - Presentation by Gregory Naitzat

23 Thank You!