Path planning for autonomous ground vehicles based on quintic trigonometric Bézier curve

Path planning based on quintic trigonometric Bézier curve


Path planning is one of the essential steps for autonomous ground vehicles or even wheeled mobile robots. This paper proposes a general framework for the path planning using quintic trigonometric Bézier curve with two shape parameters and \(C_{3}\) continuity. We express that when there are obstacles, predefined path can be adjusted by only using shape parameters without altering any obstacle. Additionally, the velocity, lateral acceleration, longitudinal and lateral jerks of the predefined cubic and quintic Bézier, and cubic and quintic trigonometric Bézier paths are compared. Also, a path surface for autonomous ground vehicles can be generated using developable surfaces.

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  1. 1.

    Villani L, Siciliano B, Sciavicco L, Oriolo G (2009) Robotics (modelling, planning and control). Springer, London

    Google Scholar 

  2. 2.

    Xu X, Cai Y, Chen L, Qin D, Xie J (2019) A path and velocity planning method for lane changing collision avoidance of intelligent vehicle based on cubic 3-d Bézier curve. Adv Eng Softw 132:65–73

    Article  Google Scholar 

  3. 3.

    Luo Y, Li H, Wu J (2019) Collision-free path planning for intelligent vehicles based on Bézier curve. IEEE Access 7:123334–123340

    Article  Google Scholar 

  4. 4.

    Lv C, Ji XW, Liu YH, He XK, Liu YL (2018) Emergency steering control of autonomous vehicle for collision avoidance and stabilization. Veh Syst Dyn 57(8):1163–1187.

  5. 5.

    Achour N (2011) Mobile robots path planning using genetic algorithms. In: ICAS 2011 Seventh international conference on autonomic and autonomous systems

  6. 6.

    Zheng G, Ma Y, Perruquetti W (2013) Cooperative path planning for mobile robots based on visibility graph. In: Proceedings of the 32nd Chinese control conference

  7. 7.

    Uchiyama N, Simba KR, Sano S (2014) Real-time obstacle-avoidance motion planning for autonomous mobile robots. In: 2014 4th Australian control conference (AUCC), Canberra, ACT, pp 267–272

  8. 8.

    Klanc̃ar G, S̃krjanc I, Matko D, Lepetic̃ M, Potoc̃nik B (2003) Time optimal path planning considering acceleration limits. Robotics Autom Syst 45(3–4):199–210

    Google Scholar 

  9. 9.

    Hung GT, Liang TC, Liu JS, Chang YZ (2005) Practical and flexible path planning for car-like mobile robot using maximal-curvature cubic spiral. Robotics Autom Syst 52(4):312–335

    Article  Google Scholar 

  10. 10.

    He Y, Hu Y, Li D, Han J (2019) Path planning of UGV based on Bézier curves. Robotica 37(6):969–997

    Article  Google Scholar 

  11. 11.

    Victor S, Cassany L, Moze M, Aioun F, Guillemard F, Moreau J, Melchior P (2019) Reactive path planning in intersection for autonomous vehicle. IFAC-PapersOnLine 52(5):109–114

    Article  Google Scholar 

  12. 12.

    Korzeniowski D, Ślaski G (2016) Method of planning a reference trajectory of a single lane change manoeuver with Bézier curve. In: IOP Conf., Mater. Sci. Eng., 148(1)

  13. 13.

    Park H, Kim JH, Kim S, Bae I, Moon J (2013) Path generation and tracking based on a Bézier curve for a steering rate controller of autonomous vehicles. In: IEEE conference on intelligent transportation systems, pp 436–441

  14. 14.

    Nejad HTN, Do QH, Han L, Yashiro H, Mita S (2010) Bézier curve based path planning for autonomous vehicle in urban environment. In: Proceedings intelligent vehicles symposium, pp 1036–1042

  15. 15.

    Curry R, Choi JW, Elkaim G (2009) Path planning based on bézier curve for autonomous ground vehicles. In: Proceedings Proc. Adv. Elect. Electron. Eng.-IAENG Special World Congr. Eng. Comput. Sci., pp 158–166

  16. 16.

    Bulut Vahide (2019) Differential geometry of autonomous wheel-legged robots. Eng Comput 37(2):615–637

    Article  Google Scholar 

  17. 17.

    Chang P, Lin C, Luh J (1983) Formulation and optimization of cubic polynomial joint trajectories for industrial robots. IEEE Trans Autom Control 28(12):1066–1074

    Article  Google Scholar 

  18. 18.

    Aribowo W, Terashima K (2014) Cubic spline trajectory planning and vibration suppression of semiconductor wafer transfer robot arm. Int J Autom Technol 8(2):265–274

    Article  Google Scholar 

  19. 19.

    Paulos E (1998) On-line collision avoidance for multiple robots using b-splines. Technical report, University of California, Berkeley, Computer Science Division

  20. 20.

    Schoenberg IJ (1964) On trigonometric spline interpolation. J Math Mech 13(5):795–825

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Tan X, Zhu Y (2019) Quasi-quintic trigonometric Bézier curves with two shape parameters. Comput Appl Math 38(4)

  22. 22.

    Ramli A, Misro MY, Ali JM (2018) Quintic trigonometric Bézier curve and its maximum speed estimation on highway designs. In: AIP conference proceedings, vol 1974

  23. 23.

    Han XL, Wu XQ, Luo S (2008) Quadratic trigonometric polynomial Bézier curves with a shape parameter. J Eng Graph 29:82–87

    Google Scholar 

  24. 24.

    Huang XL, Han XA, Ma YC (2009) The cubic trigonometric Bézier curve with two shape parameters. Appl Math Lett 22:226–231

    MathSciNet  Article  Google Scholar 

  25. 25.

    Huang XL, Han XA, Ma YC (2010) Shape analysis of cubic trigonometric Bézier curves with a shape parameter. Appl Math Comput 217:2527–2533

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Han XL, Zhu YP (2014) Total positivity of the cubic trigonometric Bézier basis. J Appl Math 1–5:2014

    MATH  Google Scholar 

  27. 27.

    Xie W, Li J (2018) C\(_{3}\) cubic trigonometric b-spline curves with a real parameter. J Natl Sci Found Sri 46(1):89–94

    Google Scholar 

  28. 28.

    Han J, Zhu YP, Han XL (2012) Quartic trigonometric Bézier curves and shape preserving interpolation curves. J Comput Inform Syst 8:905–914

    Google Scholar 

  29. 29.

    Ali JM, Miscro MY, Ramli A (2017) Quintic trigonometric Bézier curve with two shape parameters. Sains Malays 46:825–831

    Article  Google Scholar 

  30. 30.

    Liu J-S, Wu C-S, Chiu Z-Y (2018) Time-optimal trajectory planning along parametric polynomial lane-change curves with bounded velocity and acceleration: simulations for a unicycle based on numerical integration. Model Simul Eng

  31. 31.

    Shifrin T (2015) Differential Geometry: a first course in curves and surfaces. University of Georgia

  32. 32.

    Michael Tsirlin (2017) Jerk by axes in motion along a space curve. J Theor Appl Mech 55(4):1437–1441

    Google Scholar 

  33. 33.

    Wei J, Liu J (2010) Generating minimax-curvature and shorter \(\eta ^{3}\)-spline path using multi-objective variable-length genetic algorithm. In: 2010 International conference on networking, sensing and control (ICNSC), pp 319–324

  34. 34.

    Kreyszig E (1991) Differential geometry. Dover Publications, New York

    MATH  Google Scholar 

  35. 35.

    Pottmann H, Farin G (1995) Developable rational Bézier and b-spline surface. J Comput Aided Geom Des 12(5):513–531

    Article  Google Scholar 

  36. 36.

    Hoschek J (1983) Dual Bézier curves and surfaces. In: Barnhill RE, Boehm W (eds) Surfaces in computer aided geometric design. North-Holland, Amsterdam, pp 147–156

    Google Scholar 

  37. 37.

    Gang H, Wu J (2019) Generalized quartic h-Bézier curves: construction and application to developable surfaces. Adv Eng Softw 138

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Correspondence to Vahide Bulut.

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Bulut, V. Path planning for autonomous ground vehicles based on quintic trigonometric Bézier curve. J Braz. Soc. Mech. Sci. Eng. 43, 104 (2021).

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  • Path planning
  • Quintic trigonometric Bézier curve
  • Developable surface