Industrial and Urban Applications of Eulerian and Chinese Walks

Industrial and Urban Applications of Eulerian and Chinese Walks

Khalil Amine (University of Sidi Mohammed Ben Abdellah, Morocco) and Rima Djellab (University of Hadj Lakhdar, Algeria)
DOI: 10.4018/978-1-4666-2661-4.ch022
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Eulerian walks are paths that visit each edge once in a connected graph. When the extremities of the walk are confused, then it is called Eulerian cycle or closed Eulerian walk. Introduced by Euler in 1736, Eulerian cycle concept was the historical beginning of the Graph theory. On account of the difficulty to get an Eulerian walk in a nonspecific graph, many problems were formulated with the aim to find a “weak Eulerian” walk. Thus, Chinese walk and cycle concept appeared consisting of visiting all edges of a connected graph at least one time. This concept was introduced due to the Kwan studies for postman problem. Furthermore, as each graph admits a Chinese walk, and a cycle respectively, postman studies has been of use in many modeling formulations and has given a wide range of applications relevant to transportation, urban planning, and industrial manufacturing among others. Laser or water cutting presents a technology for industrial manufacturing consisting of using water or laser to cut (metallic) materials for producing tools where it is plausible to follow Chinese walks to ensure efficiency.
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Industrial Manufacturing Applications

In manufacturing area, there is a common use of robots; because of their low cost as well as many safety concerns. Robots are subject to follow monitored scenarios to do automatic tasks. Doing so, manufacturers save time and cost; then the manufacturing production increases. In this field, many techniques, as laser or water-jet cutting, are based on repeatable and programmable robot’s arm movements.

Surface Painting

A simple case of introducing Eulerian walk in programming automatic arm is the painting of surfaces, like in cars manufacturing, especially in automotive car body painting. The task consists of using an atomizer over a painting surface, with constant speed and according to a specified trajectory. Linear spray system provides a uniform coating by a back and forth movement above materials. The movement is actually not a surface “scanning” but an Eulerian walk as described in Figure 1. This walk gives two opportunities; namely, no paint loss (so less cost) and coating uniformity.

Figure 1.

Painting walk regarded as an Eulerian walk

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