High-Density Automated Valet Parking with Relocation-Free Sequential Operations

Anonymous Authors
Anonymous Institute

Abstract

In this paper, we present DROP, high-Density Relocation-free sequential OPerations in automated valet parking. DROP addresses the challenges in high-density parking & vehicle retrieval without relocations. Each challenge is handled by jointly providing area-efficient layouts and relocation-free parking & exit sequences, considering accessibility with relocation-free sequential operations. To generate such sequences, relocation-free constraints are formulated as explicit logical conditions expressed in boolean variables. Recursive search strategies are employed to derive the logical conditions and enumerate relocation-free sequences under sequential constraints. We demonstrate the effectiveness of our framework through extensive simulations, showing its potential to significantly improve area utilization with relocation-free constraints. We also examine its viability on an application problem with prescribed operational order.

💡 Concept of DROP

DROP Concept ><

"DROP solves trade-off between area utilization and relocation-free exits."

🔎 Overview of DROP

Overview ><

📊 Experiment Results

  • Target Instances: $\texttt{15x12}$, $\texttt{20x16}$, and $\texttt{20x20}$
  • Key Results per Instance:
    • Generated unique layouts
    • Adjacency graphs for all unique layouts
    • Number of relocation-free exit sequences
    • Parking-exit sequence pairs (under prescribed operation order $\pi$)
    • Animated example of parkiing and exit sequence pairs (under $\pi$)

Note: The count of relocation-free parking matches the number of relocation-free exit sequences.

Instance 1: 15x12

Generated Unique Layouts

Unique Layouts for Instances 1 ><

Note:

  • A gray large rectangular box (and its surroundings): Parking lot (boundary)
  • A skyblue-colored rectangles: Parking stalls
  • Label on each stall: Stall ID (0-4)

3 unique layouts with packing capacity of 5 vehicles.

Adjacency Graphs

Adjacency Graphs for Instances 1 ><

The adjacency graphs for all 3 unique layouts. No layouts skipped in postprocessing.

The Number of Relocation-Free Exit Sequences

  Layout 1 Layout 2 Layout 3
$\vert\textit{exitSeqs}_\omega \vert$ 56 34 1

Note: Bold values indicate layouts with valid parking-exit sequence pairs under $\pi$.

The Parking-Exit Sequence Pairs with $\pi$

Operation order $\pi$ Layout 1 Layout 2 Layout 3
$[0,1,2,3,4]$ 8 2 0
$[1,2,3,4,0]$ 24 2 0
$[2,3,4,0,1]$ 48 4 0
$[3,4,0,1,2]$ 40 12 0
$[4,0,1,2,3]$ 16 26 0

Note: Each cell is the count of parking-exit sequence pairs per layout and operation order $\pi$.

  • Layouts 1, 2: Valid pairs for all operational orders ($\pi$)
  • Layout 3: No valid pairs identified

Animated Example of Parking-Exit Sequence Pairs with $\pi$

Parking-Exit Sequence Pairs for Instance 1 ><

Instance 2: 20x16

Generated Unique Layouts

Unique Layouts for Instances 2 ><

Note:

  • A gray large rectangular box (and its surroundings): Parking lot (boundary)
  • A skyblue-colored rectangles: Parking stalls
  • Label on each stall: Stall ID (0-4)

22 unique layouts with packing capacity of 10 vehicles.

Adjacency Graphs

Adjacency Graphs for Instances 2 ><

The adjacency graphs for all 22 unique layouts. No layouts skipped in postprocessing.

The Number of Relocation-Free Exit Sequences

  Layout 1 Layout 2 Layout 3 Layout 4 Layout 5 Layout 6 Layout 7 Layout 8 Layout 9 Layout 10 Layout 11 Layout 12 Layout 13 Layout 14 Layout 15 Layout 16 Layout 17 Layout 18 Layout 19 Layout 20 Layout 21 Layout 22
$\vert\textit{exitSeqs}_\omega \vert$ 3,816 5,760 348 9,382 172 5,475 6,480 159,465 4,320 18,120 2,070 3,418 90 4,500 896 120 6,249 4,214 1,352 6,594 85 896

Note: Bold values indicate layouts with valid parking-exit sequence pairs under $\pi$.

The Parking-Exit Sequence Pairs with $\pi$

Operation order $\pi$ Layout 4 Layout 8 Layout 14 Layout 20
$[0,1,2,3,4,5,6,7,8,9]$ 0 0 0 0
$[1,2,3,4,5,6,7,8,9,0]$ 2 0 0 0
$[2,3,4,5,6,7,8,9,0,1]$ 4 0 0 0
$[3,4,5,6,7,8,9,0,1,2]$ 0 0 0 0
$[4,5,6,7,8,9,0,1,2,3]$ 0 0 0 0
$[5,6,7,8,9,0,1,2,3,4]$ 0 14,400 0 0
$[6,7,8,9,0,1,2,3,4,5]$ 0 14,400 0 0
$[7,8,9,0,1,2,3,4,5,6]$ 92 7,392 0 0
$[8,9,0,1,2,3,4,5,6,7]$ 112 2,496 32 32
$[9,0,1,2,3,4,5,6,7,8]$ 82 534 0 44

Note: Each cell is the count of parking-exit sequence pairs per layout and operation order $\pi$.

  • Layouts 4, 8, 14, 20: Valid pairs for some operational orders ($\pi$)
  • The other layouts: No valid pairs identified

Animated Example of Parking-Exit Sequence Pairs with $\pi$

Parking-Exit Sequence Pairs for Instance 2 ><

Instance 3: 20x20

Generated Unique Layouts

Unique Layouts for Instances 3 ><

Note:

  • A gray large rectangular box (and its surroundings): Parking lot (boundary)
  • A skyblue-colored rectangles: Parking stalls
  • Label on each stall: Stall ID (0-4)

52 unique layouts with packing capacity of 12 vehicles.

Adjacency Graphs

Adjacency Graphs for Instances 3 ><

The adjacency graphs for 30 unique layouts. 22 layouts skipped in postprocessing.

The Number of Relocation-Free Exit Sequences

  Layout 1 Layout 2 Layout 3 Layout 4 Layout 5 Layout 6 Layout 7 Layout 8 Layout 9 Layout 10 Layout 12 Layout 13 Layout 14 Layout 15 Layout 16 Layout 18 Layout 20 Layout 21 Layout 22 Layout 23 Layout 24 Layout 26 Layout 28 Layout 29 Layout 30 Layout 31 Layout 32 Layout 37 Layout 39 Layout 40
$\vert\textit{exitSeqs}_\omega \vert$ 1,061,400 952,826 1,181,608 1,811,594 972,222 473,292 573,888 869,010 201,876 198,552 2,250,000 794,664 11,214,231 2,316,420 2,234,592 1,668,018 1,117,338 149,328 1,152,484 610,536 99,888 645,738 21,968 390,100 733,830 49,128 610,914 6,696 4,676 7,476

Note: Bold values indicate layouts with valid parking-exit sequence pairs under $\pi$.

The Parking-Exit Sequence Pairs with $\pi$

Operation order $\pi$ Layout 1 Layout 2 Layout 3 Layout 4 Layout 5 Layout 7 Layout 8 Layout 9 Layout 10 Layout 12 Layout 14 Layout 15 Layout 18 Layout 20 Layout 21 Layout 22 Layout 23 Layout 26 Layout 29 Layout 30 Layout 31
$[0,1,2,3,4,5,6,7,8,9,10,11]$ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
$[1,2,3,4,5,6,7,8,9,10,11,0]$ 0 16 0 72 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
$[2,3,4,5,6,7,8,9,10,11,0,1]$ 0 28 0 112 0 0 40 0 0 0 0 0 0 0 0 0 0 0 0 0 0
$[3,4,5,6,7,8,9,10,11,0,1,2]$ 0 0 0 132 132 0 108 0 108 0 0 0 0 0 0 0 0 0 0 0 0
$[4,5,6,7,8,9,10,11,0,1,2,3]$ 0 0 0 0 0 384 384 0 48 0 0 0 0 0 0 0 0 0 0 0 0
$[5,6,7,8,9,10,11,0,1,2,3,4]$ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
$[6,7,8,9,10,11,0,1,2,3,4,5]$ 0 0 16 2,880 0 0 0 0 0 2,880 518,400 8,640 0 0 48 0 0 120 0 0 0
$[7,8,9,10,11,0,1,2,3,4,5,6]$ 0 2,016 32 1,120 0 0 0 0 0 5,760 604,800 16,320 0 2,880 24 1,008 336 0 0 0 0
$[8,9,10,11,0,1,2,3,4,5,6,7]$ 1,536 672 512 8,960 0 48 4,352 48 0 4,512 370,944 12,672 4,608 288 96 2,784 768 0 0 0 0
$[9,10,11,0,1,2,3,4,5,6,7,8]$ 2,232 2,672 3,628 49,400 0 48 5,832 36 0 2,016 156,384 6,840 4,896 360 120 7,932 2,064 0 144 5,184 72
$[10,11,0,1,2,3,4,5,6,7,8,9]$ 1,200 1,156 560 14,400 0 176 4,864 56 0 336 48,672 2,784 1,984 160 40 2,004 264 0 132 2,304 8
$[11,0,1,2,3,4,5,6,7,8,9,10]$ 480 392 64 2,816 0 10 510 4 0 0 10,096 792 252 42 12 280 42 0 16 722 4

Note: Each cell is the count of parking-exit sequence pairs per layout and operation order $\pi$.

  • The layouts reported in the table: Valid pairs for some operational orders ($\pi$)
  • The other layouts (not reported in the table): No valid pairs identified

Animated Example of Parking-Exit Sequence Pairs with $\pi$

Parking-Exit Sequence Pairs for Instance 3 ><