Smooth path planning via cubic GHT-Bézier spiral curves based on shortest distance, bending energy and curvature variation energy

Samia BiBi; Md Yushalify Misro; Muhammad Abbas; Samia BiBi; Md Yushalify Misro; Muhammad Abbas

doi:10.3934/math.2021501

AIMS Mathematics

2021, Volume 6, Issue 8: 8625-8641. doi: 10.3934/math.2021501

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Research article

Smooth path planning via cubic GHT-Bézier spiral curves based on shortest distance, bending energy and curvature variation energy

1.
School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, Malaysia
2.
Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan

Received: 25 January 2021 Accepted: 27 April 2021 Published: 07 June 2021
MSC : 53A04, 49Q10

In this article, an algorithm to obtain a smooth path (free from obstacles) that can be optimized by shortest path distance, bending energy and curvature variation energy will be presented. Previously, scholars used various tools to generate smooth path such as Clothoid, Log-Aesthetic curves (LACs), and Bézier curves. The limited number of solutions from the aforementioned curves become one of the drawback to generate smooth path planning. Therefore, providing a number of solutions that can generate smooth path planning becomes the objective of this study. In this paper to generate a smooth path, five templates of spiral transition curves having three different shape parameters with monotone curvature (either increase or decrease) by cubic GHT-Bézier curves are proposed. Moreover, few examples of path planning technique via cubic GHT-Bézier spiral curve to show the flexibility of smooth path by minimization of the shortest path (minimum arc length) L, bending energy E and curvature variation energy V are presented. The superiority of cubic GHT-Bézier spiral path smoothing techniques as compared to Clothoid and LACs is also demonstrated.
- cubic GHT-Bézier curve,
- shape parameters,
- spiral curves,
- monotone curvature,
- minimum arc length,
- bending energy,
- curvature variation energy
Citation: Samia BiBi, Md Yushalify Misro, Muhammad Abbas. Smooth path planning via cubic GHT-Bézier spiral curves based on shortest distance, bending energy and curvature variation energy[J]. AIMS Mathematics, 2021, 6(8): 8625-8641. doi: 10.3934/math.2021501

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Abstract

In this article, an algorithm to obtain a smooth path (free from obstacles) that can be optimized by shortest path distance, bending energy and curvature variation energy will be presented. Previously, scholars used various tools to generate smooth path such as Clothoid, Log-Aesthetic curves (LACs), and Bézier curves. The limited number of solutions from the aforementioned curves become one of the drawback to generate smooth path planning. Therefore, providing a number of solutions that can generate smooth path planning becomes the objective of this study. In this paper to generate a smooth path, five templates of spiral transition curves having three different shape parameters with monotone curvature (either increase or decrease) by cubic GHT-Bézier curves are proposed. Moreover, few examples of path planning technique via cubic GHT-Bézier spiral curve to show the flexibility of smooth path by minimization of the shortest path (minimum arc length) L, bending energy E and curvature variation energy V are presented. The superiority of cubic GHT-Bézier spiral path smoothing techniques as compared to Clothoid and LACs is also demonstrated.

References

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