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
• nly on order of costs, not actual values

Scaled Goal Formation Problem

• Given: n robots in initial formation P = {pi}

and desired shape S={sj} for goal

• Find: assignment X and goal formation

Q={qj} 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

• rder)
• 2. For each equivalence class, compute
• ptimal 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

• ptimum

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(n2)

• There are O(n4) equivalence classes, and it

takes O(n2) to incrementally update the LBAP for each class

• Complexity: O(n6)

Assignment for Translated Goal Formations

• Find the optimal assignments and translation

d=(dx, dy) that minimize the maximum cost

• qj = sj + 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
• f cells at depths zero, one, up to i-1.

Equivalence Classes for Translation Costs

• Rewrite cost as distance from rij sites:
• Can now compute equivalence classes from
• rder-k Voronoi diagrams of rij 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

de Berg et al. 2008

Projecting level 1

• f arrangement

gives Voronoi diagram!

Order-k Voronoi Diagrams

• Order-1 Voronoi diagram: each cell contains the points closest to
• ne site, i.e., standard Voronoi diagram
• 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).

de Berg et al. 2008

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 dxdy-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(n8) cells
• Takes O(n2) time to solve LBAP and then

QP at each cell

• Overall complexity: O(n10)

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