Assignment Algorithms for Variable Robot Formations Srinivas Akella - PowerPoint PPT Presentation

Assignment Algorithms for Variable Robot Formations Srinivas Akella Department of Computer Science University of North Carolina at Charlotte

Droplets, Lights, Action! • Light-actuated digital microfluidic systems [Chiou et. al. 08, Pei et. al. 10] • Move droplets by projecting light on continuous photoconductive surface Pei and Wu, 2010

Variable Goal Formations • Problem: Find optimal assignments for team of n robots with variable goal formations • Can change scale or translate goal formation • Motivation: Drone formations Chemical droplets in in cluttered spaces light-actuated lab-on-chip

Problem • Given initial robot formation P and specified shape S, find optimal assignment for variable goal formation Q

Related Work • Fixed assignment, compute variable goal formations – Derenick and Spletzer (2007) • Fixed goal formation, compute assignment, coordinate motions – Kloder and Hutchinson (2006), Turpin, Michael, Kumar (2014) – Luna and Bekris (2011), Yu and Lavalle (2013), Solovey and Halperin (2015) • Assignment problems with cost uncertainty – Liu and Shell (2011)

Linear Bottleneck Assignment Problem (LBAP) • Minimizes maximum cost of any task • Can be solved by Threshold algorithm • LBAP property: Optimal assignment depends only on order of costs, not actual values

Scaled Goal Formation Problem • Given: n robots in initial formation P = {p i } and desired shape S={s j } for goal • Find: assignment X and goal formation Q={q j } that – minimize maximum travel distance, so – Q is a scaled copy of S

Assignment for Scaled Goal Formations • LBAP for Scaled Goal Formations

Cost Curves for Scaled Goal Formations

Algorithm Outline • Idea: Exploit geometric structure with LBAP cost order property 1. Compute equivalence classes of formation parameters (with invariant cost order) 2. For each equivalence class, compute optimal LBAP solution 3. Best solution over all equivalence classes is global optimum

Example: Scaled Goal Formation • Three robot example: Optimal solutions for three equivalence classes Global optimum

Incremental Updates of Optimal Assignment Solution • When two cost curves intersect, they swap positions in the cost order • Can “warm start”: Find new optimal solution by updating previous optimal solution in O(n 2 ) • There are O(n 4 ) equivalence classes, and it takes O(n 2 ) to incrementally update the LBAP for each class • Complexity: O(n 6 )

Assignment for Translated Goal Formations • Find the optimal assignments and translation d=(d x , d y ) that minimize the maximum cost • q j = s j + d • Formulate as LBAP

Arrangement of Surfaces • Parabolic cost surfaces • Arrangement of surfaces gives equivalence classes

Arrangement of Hyperplanes • An arrangement A(H) induced by the set of hyperplanes H is the convex subdivision of space defined by the hyperplanes H

Arrangement of Hyperplanes • A cell lies at depth i if there are exactly i planes above the cell.

Arrangement of Hyperplanes • Level i of arrangement is the boundary of the union of cells at depths zero, one, up to i-1.

Equivalence Classes for Translation Costs • Rewrite cost as distance from r ij sites: • Can now compute equivalence classes from order-k Voronoi diagrams of r ij sites via arrangement of associated tangent planes

Unit Paraboloids and Voronoi Diagrams • Lift points up to unit paraboloid and use tangent planes to track order of distances to sites de Berg et al. 2008

Unit Paraboloids and Voronoi Diagrams • Lift points up to unit paraboloid and use tangent planes to track order of distances to sites de Berg et al. 2008

Unit Paraboloids and Voronoi Diagrams • Lift points up to unit paraboloid and use tangent planes to track order of distances to sites Projecting level 1 of arrangement gives Voronoi diagram! de Berg et al. 2008

Order-k Voronoi Diagrams • Order-1 Voronoi diagram: each cell contains the points closest to one site, i.e., standard Voronoi diagram de Berg et al. 2008 • Order-k Voronoi diagram: Partitions space according to k closest sites of N sites, for some 1 ≤ k ≤ N -1. So, in an order-2 Voronoi diagram, each cell contains points closest to an unordered pair of sites. • Can obtain order-k Voronoi diagram by projecting level k of A(H).

Arrangement of Planes • Planes (associated with sites) are tangent to unit paraboloid located at origin • Projecting level k of A(H) gives order-k Voronoi diagram

Overlay of order-k Voronoi Diagrams • Overlay of order-1 through order-(N-1) Voronoi diagrams on d x d y -plane partitions plane into convex cells. • Each cell has an invariant cost ordering of sites based on distances of points in cell to the sites. Four sites

Equivalence Classes for Translated Goal Formations • Each cell has an invariant ordering of sites based on distances of points in cell to sites. • Solve LBAP and find best translation in each cell Equivalence classes for nine sites

Example: Translated Goal Formation • Optimal solution for three robots

Example: Translated Goal Formation • Optimal solution for three robots

Example: Translated Goal Formation • Optimal solution for three robots

Complexity • For n robots, there are O(n 8 ) cells • Takes O(n 2 ) time to solve LBAP and then QP at each cell • Overall complexity: O(n 10 )

Future Work • Combine scaling, translation, orientation of goal formations • Extend to 3D formations • Improve computational complexity • Collision-free coordination of robots • Formations with heterogeneous robots

Acknowledgments • Thanks to Saurav Agarwal, Danny Halperin, Zhiqiang Ma, Erik Saule. • Supported in part by NSF Awards IIS-1547175 and IIP- 1439695