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# Periodic Multi-Agent Path Planning

Kazumi Kasaura*, Ryo Yonetani*, Mai Nishimura*
*OMRON SINIC X Corporation
Presented at AAAI Conference on Artificial Intelligence (AAAI) 2023 January 25, 2023

## Transcript

Kazumi Kasaura, Ryo Yonetani, Mai Nishimura OMRON SINIC X

Streams of Agents Streams of agents enter at random times and move to goals Traffic congestion occurs We aim to improve the throughput

We assume that agents appear periodically and generate a periodic plan beforehand and assign periodic trajectories to agents →Higher throughput

periodic plans, we solve periodic MAPP, a variant of Multi-Agent Path Planning (MAPP). Solution Problem

Given: Environment E Pairs (𝑠1 , 𝑔1 ), … , (𝑠𝑁 , 𝑔𝑁 ) of starts and goals Want: Collision-free trajectories for agents appearing periodically with a user-defined period Objective: Period that is as small as possible

same starts have not necessarily the same trajectory We fix cycle 𝑀 and plan 𝑀 trajectories for each pair M=1 M=2

Period, 𝑟: Radius of Agents A set of 𝑁𝑀 trajectories γ𝑛,𝑚 : 0, 𝑇𝑛,𝑚 → 𝐸 (1 ≤ 𝑛 ≤ 𝑁, 0 ≤ 𝑚 < 𝑀 ) s. t. • 𝛾𝑛,𝑚 0 = 𝑠𝑛 and 𝛾𝑛,𝑚 (𝑇𝑛,𝑚 ) = 𝑔𝑛 (start and goal condition) • 𝑑𝛾 𝑑𝑡 ≤ 𝑣𝑚𝑎𝑥 (Maximum velocity) • 𝑑𝑖𝑠𝑡𝐸 𝛾𝑛,𝑚 𝑡 ≥ 𝑟 (Clearance from boundaries) • 𝑚 − 𝑚′ 𝜏 + 𝑡 − 𝑡′ ∈ 𝑀𝜏𝒁 and 𝑛, 𝑚, 𝑡 ≠ 𝑛′, 𝑚′, 𝑡′ → 𝛾𝑛,𝑚 𝑡 − 𝛾𝑛′,𝑚′ (𝑡′) ≥ 2𝑟 (Collision-free)

MAPP Generating an initial plan with a large period → optimizing the plan while decreasing the period Initial plan Optimized plan

MAPP Generating an initial plan with a small agent radius → optimizing the plan while increasing the radius to the original Initial plan Optimized plan

the plan optimization as continuous optimization problem. 𝛾𝑛,𝑚 is represented by sequences of timed locations 𝑥𝑛,𝑚,0 , 0 , 𝑥𝑛,𝑚,0 , Δ𝑡𝑛,𝑚 , … , (𝑥𝑛,𝑚,𝐾 , 𝐾Δ𝑡𝑛,𝑚 ) s. t. • 𝑥𝑛,𝑚,0 = 𝑠𝑛 and 𝑥𝑛,𝑚,𝐾 = 𝑔𝑛 (start and goal condition) • 𝑣𝑛,𝑚,𝑘 ≔ 𝑥𝑛,𝑚,𝑘+1−𝑥𝑛,𝑚,𝑘 Δ𝑡𝑛,𝑚 ≤ 𝑣𝑚𝑎𝑥 (Maximum velocity) • 𝑑𝑖𝑠𝑡𝐸 𝑥𝑛,𝑚,𝑘 ≥ 𝑟 (Clearance from boundaries)

r 𝑡 ≔ 𝑡 − 𝑡 𝑀𝜏 𝑀𝜏 For all 𝑛, 𝑚, 𝑘, 𝑛’, 𝑚’, 𝑘’ s. t. 0 ≤ r 𝑚 − 𝑚′ 𝜏 + 𝑘∆𝑡𝑛,𝑚 − 𝑘′∆𝑡𝑛′,𝑚′ < ∆𝑡𝑛′,𝑚′ and (𝑛, 𝑚, 𝑘) ≠ (𝑛′, 𝑚′, 𝑘′), 𝑑𝑛,𝑚,𝑘,𝑛′,𝑚′,𝑘′ ≔ 𝑥𝑛,𝑚,𝑘 − ( 1 − 𝛼 𝑥𝑛′,𝑚′,𝑘′ + 𝛼𝑥𝑛′,𝑚′,𝑘′+1 ) ≥ 2𝑟, where 𝛼 ≔ r 𝑚 − 𝑚′ 𝜏 + 𝑘∆𝑡𝑛,𝑚 − 𝑘′∆𝑡𝑛′,𝑚′ Δ𝑡𝑛′,𝑚′ (Collision-free) (cf. prev. with 𝑡 = 𝑘Δ𝑡𝑛,𝑚 , 𝑡′ = 𝑘′Δt𝑛′,𝑚′ + 𝛼)

penalty function Objective: 𝜏 − 2𝑟 𝑣𝑚𝑎𝑥 2 + 𝜎𝑟 𝑟 − 𝑟0 2 + 𝜎𝑡 𝐾 σ 𝑣𝑛,𝑚,𝑘 2 + 𝜎𝑣 𝐾 ෍ max 0, 𝑣𝑛,𝑚,𝑘 − 𝑣𝑚𝑎𝑥 2 + 𝜎𝑜 𝐾 ෍ max 0, 𝑑𝑖𝑠𝑡𝐸 𝑥𝑛,𝑚,𝑘 −1 − 𝑟−1 2 + 𝜎𝑐 𝐾 ෍ max 0, 𝑑𝑛,𝑚,𝑘,𝑛′,𝑚′,𝑘′ − (2𝑟)−1 2 . where 𝜎𝑟 , 𝜎𝑡 , 𝜎𝑣 , 𝜎𝑜 and 𝜎𝑐 are constants. 𝜎𝑟 , 𝜎𝑣 , 𝜎𝑜 , 𝜎𝑐 → ∞ and 𝜎𝑡 → 0

Online MAPP • Pairs of starts and goals are given previously • Agents appear at random times • Agents can wait in queue before entering

(c) (d) (e) (f)

Plans 𝑀 ∈ {1, 2, 3} Initial plans: 𝑀 agents of one direction pass, 𝑀 agents of another direction pass alternately M=1 M=2 M=3

showing the periods of optimized plans

M=1 M=2 M=3

M=1 M=2 M=3

M=1 M=2 M=3

M=1 M=2 M=3

M=1 M=2 M=3

M=1 M=2 M=3