Optimal Multi-Robot Path Planning on Graphs: Structure, Complexity, Algorithms,

and Applications

CASE 2016 Multi-Robot Workshop

Jingjin Yu

Computer Science | Rutgers

Outline

Problem Statement Structural Properties of Optimal Formulations Complexity of Non-Optimal Fomulations Complexity of Optimal Formulations

General Graphs Planar Graphs

An ILP Based Novel Solution Application to the Continuous Domain Conclusion The microMVP Platform

Problem Statement

Forbidden moves

1 2

1 2𝐺 = (𝑉, 𝐸)

1

3 4

2

𝑋𝐼

13

4

2

𝑋𝐺

MPP Problem: (𝐺, 𝑋𝐼, 𝑋𝐺), solution: collision free 𝑃 = {𝑝1, … , 𝑝𝑛}

Optimality objectives (minimization):

Max time (makespan): min𝑃∈𝒫

max𝑝𝑖∈𝑃

𝑡𝑖𝑚𝑒(𝑝𝑖)

Total time: min𝑃∈𝒫

σ𝑝𝑖∈𝑃𝑡𝑖𝑚𝑒(𝑝𝑖)

Max distance: min𝑃∈𝒫

max𝑝𝑖∈𝑃

𝑙𝑒𝑛𝑔𝑡ℎ(𝑝𝑖)

Total distance: min𝑃∈𝒫

σ𝑝𝑖∈𝑃𝑙𝑒𝑛𝑔𝑡ℎ(𝑝𝑖)

ApplicationsApplications

Structural Properties of Optimal Formulations

1

1

32

23

𝑥

Makespan Total time

Clockwise

Counterclockwise

𝑥 + 1

𝑥 + 4 𝑥 + 12

2𝑥 + 3

Total distance Total time

Left path only

Using right path

4𝑥 + 14

4𝑥 + 134𝑥 + 10

4𝑥 + 8

2 3

41

1

3

4

2

𝑥 𝑥

Theorem. A pair of the four MPP objectives on makespan, total time, max distance, and total distance demonstrates a Pareto-optimal structure.

Y-LaValle, Arxiv 1507.03289

Complexity of Non-Optimal Formulation

Feasibility is not guaranteed

Theorem. Feasibility of MPP can be decided in linear time. Moreover, a solution for a feasible instance can be computed in 𝐜𝐮𝐛𝐢𝐜 time.

1

4

325

6

7

9

8

256

97

1 8

3

4

?

1 2 2 1×

What about finding optimal solutions?

Y-Rus, WAFR’14

Intractability of Time Optimal MPP

𝒄𝟑𝒄𝟐𝒄𝟏

Theorem. Min Makespan MPP is NP-hard.

𝑥1 ∨ ¬𝑥3 ∨ 𝑥4 ∧ ¬𝑥1 ∨ 𝑥2 ∨ ¬𝑥4 ∧ (¬𝑥2 ∨ 𝑥3 ∨ 𝑥4)3SAT

Y-LaValle, AAAI ’13

𝑥1

𝑥4

𝑥3

𝑥2

𝑐1 𝑐2 𝑐3

𝑛 = 4 variables𝑚 = 3 clauses

𝒄𝟏 𝒄𝟐 𝒄𝟑

𝑚

Min Makespan MPP

𝒙𝟒

𝒙𝟑

𝒙𝟐

𝒙𝟏 𝒙𝟏

𝒙𝟐

𝒙𝟑

𝒙𝟒

Intractability Distance Optimal MPP

Theorem. MPP is NP-hard when optimizing min makespan, min total time, minmax distance, and min total distance.

NP-hardness of distance optimal MPP is slightly more tricky…

Y-LaValle, Arxiv 1507.03289

The Planar Case (𝐺 is Planar)

Planar Monotone 3-SAT

Theorem. Optimal Planar MPP (PMPP) is NP-hard for min makespan,min total time, min max distance, and min total distance objectives.

𝑥1 ∨ 𝑥4 ∨ 𝑥5∧ (𝑥2 ∨ 𝑥3)∧ ¬𝑥1 ∨ ¬𝑥2 ∨ ¬𝑥3∧ ¬𝑥3 ∨ ¬𝑥4 ∨ ¬𝑥5

𝑐1𝑐2𝑐3𝑐4

Planar MPP

𝒄𝟒𝒄𝟑

𝒄𝟐

𝒄𝟏

𝒄𝟏

𝒄𝟐

𝒄𝟑 𝒄𝟒

Y, IEEE RA-L, 2016

Practical Implications

Polynomial time exact solution Polynomial time suboptimal solutions?

Engineering the environment helps Two way, multi-lane roads Elevated intersections

Optimal MPP and PMPP are often NP-hard

𝒄𝟒𝒄𝟑

𝒄𝟐

𝒄𝟏

𝒄𝟏

𝒄𝟐

𝒄𝟑 𝒄𝟒

We offer rigorous, quantitative justifications of these phenomena through complexity theory

Approaches Based on Discrete Search

Discrete approaches are mostly A* based The global search space is 𝐺 𝑛

Differ on the handling of robot interactions

Local Repair A* (LRA*) [Zelinsky, IEEE TRA’92] Planning for each robot and resolving conflicts locally as they appear

Windowed Hierarchical Cooperative A* (WHCA*) [Sliver, AIIDE’05] Local space-time window for handling multi-robot interactions

IDA* based [Sharon-Stern-Felner-Sturtevant, AAAI’12] Iteratively handle more robot-robot interaction

Maximum Group Size (MGSn) [Standley-Korf, IJCAI’11] Grouping robots into larger bundles as necessary The authors pushed a highly effective implementation

Many additional approaches: [Ryan’08], [van Den Berg et al.‘09], [Wagner-Choset’11], [Boyarski et al.’15],…

𝑛 independent robots: 𝒏|𝑮| joint search space: 𝑮 𝒏

An Integer Programming Based Novel Approach

Key idea: time expansion

1 1’ 2 2’t = 0 T=4 4’3 3’

𝑟1 𝑟2′

𝑟2𝑟1′

12

3

4

2

1

3,4

Theorem. Fixing a natural number 𝑇, a MPP instance admits a solution with atmost 𝑇 time steps if and only if the corresponding time-expanded networkwith 𝑇 periods admits a solution consisting of disjoint paths.

Y-LaValle, WAFR’12

ILP Approach: The Constraints

𝑢 𝑣

Meet-on-edge

……

……

……

……

……

……

𝑢

𝑤

𝑣

𝑡 𝑡 + 1

1≤𝑖≤𝑛

(𝑥𝑢𝑣,𝑡,𝑡+1𝑖 + 𝑥𝑣𝑣,𝑡,𝑡+1

𝑖 + 𝑥𝑤𝑣,𝑡,𝑡+1𝑖 ) ≤ 1

1 2

……

……

……

……

𝑢

𝑣

𝑡 𝑡 + 1

1≤𝑖≤𝑛

(𝑥𝑢𝑣,𝑡,𝑡+1𝑖 + 𝑥𝑣𝑢,𝑡,𝑡+1

𝑖 ) ≤ 1

𝑢 𝑤𝑣

Meet-on-vertex

1 2

Algorithm for Min MakespanY-LaValle, ICRA’13, TRO-16, in press, online

Pick an initial 𝑇

No 𝑇 = 𝑇 + 1

Yes

Return the path set

Feasible? Run optimizer

Additional heuristics Reachability analysis Divide and conquer

Other objectives (total time, max distance, total distance)

Build the time-expanded network

1 1’0 2 2’ 3 3’ 4 4’

Set up an ILP model

𝑚𝑎𝑥

𝑖=1

𝑛

𝑥𝑖,𝑖 , 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜

∀𝑒𝑗 ,

𝑖=1

𝑛

𝑥𝑖,𝑗 ≤ 1

∀𝑣,

𝑒𝑗∈𝛿+(𝑣)

𝑥𝑖,𝑗 =

𝑒𝑗∈𝛿−(𝑣)

𝑥𝑖,𝑗

Performance – Makespan

24x18 grid, with some vertices randomly removed to simulate obstacles

Exact solution Near-optimal solution

44% robot density

Performance – Total Time and Total Distance

Min Total Time Min Total Distance

We Can Solve Some Tough Problems…

1025 states> 104 branching factor

13 17 144 23

1 22 129 7

11 16 815 21

25 24 196 20

10 3 25 18

1 2 43 5

6 7 98 10

11 12 1413 15

16 17 1918 20

21 22 2423 25

A 7-step min makespan plan

Generalization to Continuous DomainY-Rus, ISRR’15

Lattice overlay Restore connectivity

Snapping Trajectory planning Path smoothing

Conclusion

Contributions

Structure and complexity

MPP and PMPP appear NP-hard in general

Algorithmic solution

Effective integer programming based solution approach

Extensible to continuous problems

Future work

Algorithmic solutions for more realistic setup

Continuously appearing start and goal locations

Various constraints

Effective environment design

“Almost planar” design with guaranteed traffic throughput

Planar graph for doing the same?

Remove the ILP dependency

microMVP (micro Multi-Vehicle Platform)

An open platform targeting robotics research and education Low cost (currently <$100 per vehicle, $100 for the sensing “platform”)

Easy assembly 20min to build a vehicle – enabled by 3D printing

1 minute to setup the platform

Small scale (in my backpack!)

Suitable for deployment everywhere

Open (very soon)