Trajectories in same homotopy classses

Trajectories in different homotopy classses

Definition of Homotopy Class

Set of trajectories joining same start and goal points that can be smoothly deformed into one another without intersecting obstacles

Deploying multiple agents:•Searching/exploring the map•Pursuing an agent with uncertain paths

Motivational Examples

initial

final

??

?

? start

goal

Predicting:• Possible paths of an agent

with uncertainty in behaviors• Possible paths taken by an

agent when only start and final positions are known

Constraints:• Avoid high-risk regions

and homotopy classes• Follow a known

homotopy class in order to perform certain tasks

• Geometric approach [Hershberger et al.; Grigoriev et al.]

- Not well-suited for graph representation - Inefficient for planning with homotopy class

constraints• Triangulation based method [Demyen et al.]- Not suitable for non-Euclidean cost functions- Requires triangulation-based discretization

schemes.- Complexity increases significantly if environment

contains many small obstacles .- Cannot be easily used with an arbitrary graph

search and arbitrarily discretization.

Homotopy class in literature

Our Approach – Exploit Theorems from Complex Analysis To plan least-cost paths for arbitrary cost functions (not necessarily Euclidean distances) within a particular homotopy class or while avoiding certain homotopy classes.

To develop efficient representation of homotopy classes that supports planning using arbitrary discretizations and graph representations, (uniform discretization, unstructured discretization , triangulation, visibility graph, etc.) and using any standard graph search algorithm. (Dijkstra’s, A*, D*, ARA*, etc.)

Goal: Homotopy class constraint

Basic Principle:

Re

Im

Represent the X-Y plane by a complex planei.e. A point (x,y) is represented as z = x + iy

ζ1

ζ2

ζ3

Place “representative points”, ζi,inside significant obstacles

(we can ignore small obstacles which we don’t want to contribute towards homotopy classes)

Define an Obstacle Marker function suchthat it is Complex Analytic everywhere,

except for having poles (singularities)at the representative points

f0 , for example, can be anyarbitrary polynomial in z

Complex Analytic Function ≡ Complex Differentiable F(z) ≡ F (x + iy) ≡ u(x, y) + i v(x, y)Equivalently, F ( ) = ( )with u, v following certain properties ( 2u = 2v = 0) which are guaranteed when x & y are implicitly used within z in construction of F.

xy

u(x,y)v(x,y)

ζ1

ζ2

ζ3

τ1

τ2

τ3

τ1τ2 τ3

= ≠

A direct consequence of Cauchy Integral Theorem and Residue Theorem

The value of uniquely definesthe homotopy class of a trajectory τ

Computing L(e) for any straight line segment e (e.g. edge of a graph laid down on the environment)

z1

z2

Can be computed numerically by further discretizing e

If e is “small”, and f0 an order-N-1 polynomial, this can be computedanalytically in a fast and efficient way. kl = argminkl

L-augmented graph

Augment each node, z, with distinct L-values of trajectories leading to it from start.

Integrating L-values along paths by adding up L-values of the edges

Cost function remains same

Given the graph laid upon the environment,

we construct GL by augmenting each state z with L-value

of trajectory leading to it from start coordinate

Insight into graph topology

Experimental Analysis

Homotopy class exploration

“Visibility” constraint translates tohomotopy class constraint

Non-Euclidean cost function

Planning in X-Y-TimeNo homotopy class constraint:

A homotopy class blocked:

Exploring 20 homotopy classes in a 1000x1000 uniformly discritized environment

Implementation on a Visibility Graph (polygonal obstacles)

Conclusions:Developed a compact and efficient representation of homotopy classes, usingwhich homotopy class constraints can beimposed on existing graph search-based planning methods.

Acknowledgements We gratefully acknowledge support from ONR grant no. N00014-09-1-1052, NSF grant no. IIS-0427313, ARO grant no. W911NF-05-1-0219, ONR grants no. N00014-07-1-0829 and N00014-08-1-0696, and ARL grant no. W911NF-08-2-0004.

A – set of allowed homotopy classesB – set of blocked homotopy classes

Search-based Path Planning with Homotopy Class ConstraintsSubhrajit Bhattacharya | Vijay Kumar | Maxim Likhachev GRAS

PLABORATORY

University of

Pennsylvania

e

zs zg

z1

z2

ζ1 unique goal statestart

{zs , 0+0i}

ζ1start

e1

e2

e3

e4

{z2 , L(e1)}

{zg , L(e1)+L(e3)}e1

e2

e3

e4

(z1 , L(e2))

{zg , L(e2)+L(e4)}≠

G

G L

Goal states are distinguished based on the path taken to reach it

z in G

{z, L(zs→z)}in G L

Illustration of the effect of augmenting L-values with state coordinates:

Addendum

For the simple cases in 2-dimensions we have not distinguished between homotopy and homology. The distinction however does exist even in 2-d. See our more recent [AURO 2012] paper or [RSS 2011] paper for a comprehensive discussion on the distinction between homotopy and homology, examples illustrating the distinction, and its implications in robot planning problems.

[AURO 2012] Subhrajit Bhattacharya, Maxim Likhachev and Vijay Kumar (2012) "Topological Constraints in Search-based Robot Path Planning". Autonomous Robots, 33(3):273-290, October, Springer Netherlands. DOI: 10.1007/s10514-012-9304-1.

[RSS 2011] Subhrajit Bhattacharya, Maxim Likhachev and Vijay Kumar (2011) "Identification and Representation of Homotopy Classes of Trajectories for Search-based Path Planning in 3D". [Original title: "Identifying Homotopy Classes of Trajectories for Robot Exploration and Path Planning"]. In Proceedings of Robotics: Science and Systems. 27-30 June.