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Periodic Multi-Agent Path Planning

Hziwara
January 25, 2023

Periodic Multi-Agent Path Planning

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

Hziwara

January 25, 2023
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  1. © OMRON Corporation All Rights Reserved Periodic Multi-Agent Path Planning

    Kazumi Kasaura, Ryo Yonetani, Mai Nishimura OMRON SINIC X
  2. © OMRON Corporation All Rights Reserved Motivation: Path Planning for

    Streams of Agents Streams of agents enter at random times and move to goals Traffic congestion occurs We aim to improve the throughput
  3. © OMRON Corporation All Rights Reserved Our proposal: Periodic Approximation

    We assume that agents appear periodically and generate a periodic plan beforehand and assign periodic trajectories to agents →Higher throughput
  4. © OMRON Corporation All Rights Reserved Periodic MAPP To obtain

    periodic plans, we solve periodic MAPP, a variant of Multi-Agent Path Planning (MAPP). Solution Problem
  5. © OMRON Corporation All Rights Reserved Problem definition: periodic MAPP

    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
  6. © OMRON Corporation All Rights Reserved Cycle Agents with the

    same starts have not necessarily the same trajectory We fix cycle 𝑀 and plan 𝑀 trajectories for each pair M=1 M=2
  7. © OMRON Corporation All Rights Reserved Formulation of conditions 𝜏:

    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)
  8. © OMRON Corporation All Rights Reserved Solution Strategy for Periodic

    MAPP Generating an initial plan with a large period → optimizing the plan while decreasing the period Initial plan Optimized plan
  9. © OMRON Corporation All Rights Reserved Solution Strategy for Periodic

    MAPP Generating an initial plan with a small agent radius → optimizing the plan while increasing the radius to the original Initial plan Optimized plan
  10. © OMRON Corporation All Rights Reserved Optimization Method We approximate

    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)
  11. © OMRON Corporation All Rights Reserved Optimization Method • Let

    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)
  12. © OMRON Corporation All Rights Reserved Optimization Method Constraints →

    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
  13. © OMRON Corporation All Rights Reserved Application to Online MAPP

    Online MAPP • Pairs of starts and goals are given previously • Agents appear at random times • Agents can wait in queue before entering
  14. © OMRON Corporation All Rights Reserved Experiments: Environments (a) (b)

    (c) (d) (e) (f)
  15. © OMRON Corporation All Rights Reserved Experiments: Cycles and Initial

    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
  16. © OMRON Corporation All Rights Reserved Experiments: Optimization Results Table

    showing the periods of optimized plans
  17. © OMRON Corporation All Rights Reserved Optimization Results: Environment (a)

    M=1 M=2 M=3
  18. © OMRON Corporation All Rights Reserved Optimization Results: Environment (b)

    M=1 M=2 M=3
  19. © OMRON Corporation All Rights Reserved Optimization Results: Environment (c)

    M=1 M=2 M=3
  20. © OMRON Corporation All Rights Reserved Optimization Results: Environment (d)

    M=1 M=2 M=3
  21. © OMRON Corporation All Rights Reserved Optimization Results: Environment (e)

    M=1 M=2 M=3
  22. © OMRON Corporation All Rights Reserved Optimization Results: Environment (f)

    M=1 M=2 M=3
  23. © OMRON Corporation All Rights Reserved Experiments in Online MAPP

    Comparison with first-come first-serve (FCFS) strategy Time intervals of agents: 1.0 + exponential distribution FCFS Our Method
  24. © OMRON Corporation All Rights Reserved Results: Throughput Our method

    achieved higher throughput than FCFS method in 5/6 environments.
  25. © OMRON Corporation All Rights Reserved Extension and Future Work

    • Simple geometry and kinematics for agents → More realistic models of agents by modifying constraints • Assumption that agents can trace planned trajectory exactly → Consideration for uncertainties • Investigation of values of cycle and initial plans
  26. © OMRON Corporation All Rights Reserved Conclusion • Definition of

    periodic MAPP • Collision-free trajectories for periodically appearing agents • Period is as small as possible • Planning for fixed cycle • A method to solve periodic MAPP • Generating an initial plan with relaxed constraints • Optimizing the plan by solving continuous optimization problem • To use a solution of periodic MAPP for online MAPP → higher throughput than the baseline