Skip to main content
SpringerLink
Log in

Optimal Smooth Paths Based on Clothoids for Car-like Vehicles in the Presence of Obstacles

  • Regular Papers
  • Robot and Applications
  • Published:
International Journal of Control, Automation and Systems Aims and scope Submit manuscript

Abstract

Automated Guided Vehicles are increasingly used for material transfer in factory and warehouse environments amongst humans and human operated vehicles. Safe and efficient operation is challenging when there is a mix of human and automated traffic as fixed guide paths can become blocked more frequently. In this work we aim to show smooth and efficient paths based on clothoid curves can be used to automatically plan diversions which can be traversed at high speed by automated fork-lift vehicles, which are car-like in the sense they have a limited turning radius and angular acceleration. The approach, based on numerical optimisation within convex region constraints is described in detail, and numerical results for curvature and sharpness are compared to a cubic spline on a small number of simulated environments. The clothoid spline is less affected, in terms of its objective function, by a shift in the obstacle boundaries than a cubic spline, for obstacle shifts below 0.5m. The clothoid spline takes longer to converge for but the output path has attractive qualities like lower peak sharpness, enabling high speed operation. The method is therefore most useful for applications where path quality is important and updates are required less frequently. Changing the objective function by increasing weighting parameter b allowed the path shape to be tuned to reduce the peak sharpness, at the cost of increasing the total length. With b > 100, convergence was poor because parts of the spline were pushed outside the assigned region, an artefact arising from the constraints only being enforced at the start and end of each segment. The analytical Jacobian of the constraints was effective at reducing the number of function evaluations to reach convergence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. Digani, F. Caramaschi, L. Sabattini, C. Secchi and C. Fantuzzi “Obstacle avoidance for industrial AGVs,” Proc. of IEEE 10th International Conference on Intelligent Computer Communication and Processing, ICCP 2014, pp-227–232, 2014.

  2. B. Paden, M. Cap, S. Z. Yong, D. Yershov, and E. Frazzoli, “A survey of motion planning and control techniques for self-driving urban vehicles,” http://arxiv.org/abs/1604.07446, pp. 1–27, 2016.

  3. I. F. Vis, “Survey of research in the design and control of automated guided vehicle systems,” European Journal of Operational Research, vol. 179, no. 3, pp. 677–709, 2006.

    Article  MathSciNet  Google Scholar 

  4. N. Boysen, R. de Koster, and F. Weidinger, “Warehousing in the e-commerce era — A Survey,” European Journal of Operational Research, vol. 277, no. 2, pp. 396–411, 2019.

    Article  MathSciNet  Google Scholar 

  5. S. M. LaValle and D. Leidner, “Motion Planning,” in Planning Algorithms, ch. 3–8, pp. 81–412, Cambridge University Press, 2006

  6. C. Katrakazas, M. Quddus, W.-H. Chen, and L. Deka, “Real-time motion planning methods for autonomous on-road driving: State-of-the-art and future research directions,” Transportation Research Part C: Emerging Technologies, vol. 60. pp. 416–442, 2015.

    Article  Google Scholar 

  7. J. Moreau, P. Melchior, S. Victor, L. Cassany, M. Moze, F. Aioun, and F. Guillemard, “Reactive path planning in intersection for autonomous vehicle,” IFAC-PapersOnLine, vol. 52, no. 5, pp. 109–114, 2019.

    Article  Google Scholar 

  8. A. Chebly, R. Talj, A. Charara, A. Chebly, R. Talj, A. Charara, M. Planning, C. Alia, T. Reine, and C. Ali, “Maneuver planning for autonomous vehicles with clothoid tentacles for local trajectory planning,” Proc. of IEEE 20th International Conference on Intelligent Transportation Systems (ITSC), 2017.

  9. E. Bertolazzi and M. Frego, “Interpolating clothoid splines with curvature continuity,” Mathematical Methods in the Applied Sciences, vol 41, no. 4, pp. 1723–1737, 2018.

    Article  MathSciNet  Google Scholar 

  10. E. Lambert, R. Romano, and D. Watling, “Optimal Path Planning with Clothoid Curves for Passenger Comfort,” Proceedings of the 5th International Conference on Vehicle Technology and Intelligent Transport Systems (O. Gusikhin and M. Helfert, eds.), (Heraklion, Crete), pp. 609–615, SCITEPRESS, 2019.

  11. M. Frego, E. Bertolazzi, F. Biral, D. Fontanelli, and L. Palopoli, “Semi-analytical minimum time solutions for a vehicle following clothoid-based trajectory subject to velocity constraints,” Proc. of European Control Conference, ECC 2016, pp. 2221–2227, 2017.

  12. V. Girbés, L. Armesto, and J. Tornero, “Path following hybrid control for vehicle stability applied to industrial forklifts,” Robotics and Autonomous Systems, vol. 62, no. 6, pp. 910–922, 2014.

    Article  Google Scholar 

  13. M. Pivtoraiko, R. A. Knepper, and A. Kelly, “Differentially constrained mobile robot motion planning in state lattices,” Journal of Field Robotics, vol. 26, no. 3, pp. 308–333, 2009.

    Article  Google Scholar 

  14. J. Minguez, F. Lamiraux, and J.-P. Laumond, “Motion planning and obstacle avoidance,” in Springer Handbook of Robotics (B. Siciliano and O. Khatib, eds.), no. 2, ch. 47, pp. 1177–1202, Springer, 2nd ed., 2016.

  15. Y. Kuwata, J. Teo, G. Fiore, S. Karaman, E. Frazzoli, and J. P. How, “Real-time motion planning with applications to autonomous urban driving,” IEEE Transactions on Control Systems Technology, vol. 17, no. 5, pp. 1105–1118, 2009.

    Article  Google Scholar 

  16. S. M. laValle and J. J. Kuffner, “Rapidly-exploring random trees: Progress and prospects,” Proc. of4th Workshop on Algorithmic and Computational Robotics: New Directions, pp. 293–308, 2000.

  17. S. Yoon, D. Lee, J. Jung, and D. H. Shim, “Spline-based RRT* using piecewise continuous collision-checking algorithm for car-like vehicles,” Journal of Intelligent and Robotic Systems: Theory and Applications, vol. 90, mo. 34, pp. 537–549, 2018.

    Article  Google Scholar 

  18. R. Deits and R. Tedrake, “Efficient mixed-integer planning for UAVs in cluttered environments,” Proc. of IEEE International Conference on Robotics and Automation (ICRA), pp. 42–49, 2015.

  19. C. Rosmann, F. Hoffmann, and T. Bertram, “Kinody-namic trajectory optimization and control for car-like robots,” Proc. of IEEE International Conference on Intelligent Robots and Systems, vol. 2017-Septe, pp. 5681–5686, 2017.

    Google Scholar 

  20. J. C. Kim, D. S. Pae, and M. T. Lim, “Obstacle avoidance path planning based on output constrained model predictive control,” International Journal of Control, Automation and Systems, vol. 17, no. 11, pp. 2850–2861, 2019.

    Article  Google Scholar 

  21. M. Yue, X. Wu, L. Guo, and J. Gao, “Quintic polynomial-based obstacle avoidance trajectory planning and tracking control framework for tractor-trailer system,” International Journal of Control, Automation and Systems, vol. 17, no. 10, pp. 2634–2646, 2019.

    Article  Google Scholar 

  22. R. Levien, “The Elastica: A Mathematical History,” Tech. Rep., University of California, Bekeley, 2008.

    Google Scholar 

  23. S. Gim, L. Adouane, S. Lee, and J. P. Dérutin, “Clothoids composition method for smooth path generation of carlike vehicle navigation,”, Journal of Intelligent and Robotic Systems: Theory and Applications, vol. 88, no. 1, pp. 129–146, 2017.

    Article  Google Scholar 

  24. J. Henrie and D. Wilde, “Planning Continuous Curvature Paths Using Constructive Polylines,” Journal of Aerospace Computing, Information, and Communication, vol. 4, no. 12, pp. 1143–1157, 2007.

    Article  Google Scholar 

  25. Path Planning for Autonomous Vehicles Using Clothoid Based Smoothing of A* Generated Paths and Optimal Control, PhD Thesis, KTH Royal Institute of Technology, 2017.

  26. T. Fraichard and A. Scheuer, “From reeds and shepp’s to continuous-curvature paths,” IEEE Transactions on Robotics, vol. 20, no. 6, pp. 1025–1035, 2004.

    Article  Google Scholar 

  27. F. Lamiraux and J. P. Laumond, “Smooth motion planning for car-like vehicles,” IEEE Transactions on Robotics and Automation, vol. 17, no. 4, pp. 498–502, 2001.

    Article  Google Scholar 

  28. M. Fliess, J. Levine, P. Martin, and P. Rouchon, “Flatness and defect of non-linear systems: Introductory theory and examples,” International Journal of Control, vol. 61, no. 6, pp. 1327–1361, 1995.

    Article  MathSciNet  Google Scholar 

  29. J. A. Reeds and L. A. Shepp, “Optimal paths for a car that goes forwards and backwards,” Pacific Journal of Mathematics, vol. 145, no. 2, pp. 367–393, 1990.

    Article  MathSciNet  Google Scholar 

  30. S. Gim, Flexible and Smooth Trajectory Generation Based on Parametric Clothoids for Nonholonomic Car-like Vehicles, PhD Thesis, Université Clermon Auvergne, 2017.

  31. R. Gobithaasan, Y. Wei, K. Miura, and M. Shanmugavel, “Optimal path smoothing with log-aesthetic curves based on shortest distance, minimum bending energy and curvature variation energy,” Proceedings of CAD’19, (Singapore), pp. 397–402, 2019.

  32. J. E. M. Solanes, L. Armesto, J. Tornero, P. Muñoz-Benavent, and V. Girbés, “Mobile robot obstacle avoidance based on quasi-holonomic smooth paths,” Proc. of Joint of the 13th Annual Conference on Towards Autonomous Robotic Systems, TAROS 2012 and the 15th Annual FIRA RoboWorld Congress, vol. 7429, pp. 152–163, 2012.

    Google Scholar 

  33. J. A. Silvan and V. Grassi, “Clothoid-based global path planning for autonomous vehicles in urban scenarios,” Proceedings — IEEE International Conference on Robotics and Automation, pp. 4312–4318, 2018.

  34. P. F. Lima, M. Trincavelli, J. Martensson, and B. Wahlberg, “Clothoid-based model predictive control for autonomous driving,” Proc. of European Control Conference, ECC 2015, pp. 2983–2990, 2015.

  35. M. G. Plessen, P. F. Lima, J. Martensson, A. Bemporad, and B. Wahlberg, “Trajectory planning under vehicle dimension constraints using sequential linear programming,” Proc. of IEEE Conference on Intelligent Transportation Systems, Proceedings, ITSC, vol. 2018-March, pp. 1–6, 2018.

    Google Scholar 

  36. D. Shin, S. Singh, and W. Wittaker, “Path generation for a robot vehicle using composite clothoid segments,” IFAC Proceedings Volumes, vol. 25, no. 6, pp. 443–448, 1992.

    Article  Google Scholar 

  37. D. J. Walton and D. S. Meek, “A controlled clothoid spline,” Computers and Graphics, vol. 29, no. 3, pp. 353–363, 2005.

    Article  Google Scholar 

  38. S. Thrun and A. Buecken, “Integrating grid-based and topological maps for mobile robot navigation,” Proceedings of the National Conference on Artificial Intelligence, vol. 2, no. August, pp. 944–950, 1996.

    Google Scholar 

  39. S. Thrun, W. Burgard, and D. Fox, Probabilistic Robotics, MIT Press, Cambridge, Mass., 2005.

    MATH  Google Scholar 

  40. S. M. LaValle and D. Leidner, “Chapter 6: Combinatorial Motion Planning,” in Planning Alogrithms, ch. 6, pp. 249–310, Cambridge University Press, 2006.

  41. R. Deits and R. Tedrake, “Computing large convex regions of obstacle-free space through semidefinite programming,” Springer Tracts in Advanced Robotics, vol. 107, pp. 109–124, 2015.

    Article  MathSciNet  Google Scholar 

  42. T. Gu, Improved Trajectory Planning for On-Road Self-Driving Vehicles Via Combined Graph Search, Optimization & Topology Analysis, Doctor of Philisophy, Carnegie Mellon University, 2017.

  43. Hyster, “E30-40HSD Technical Guide,” http://www.hyster.com, August 2020.

  44. R. Baird, An Autonomous Forklift Research Platform for Warehouse Operations, Master’s thesis, Massachusetts Institude of Technology, 2018.

  45. G. Raballan, “How do differing standards increase trade costs? THE CASE OF PALLETS Ga “el,” World Bank Policy Research Working Paper 3519, pp. 1–20, 2005.

  46. R. Bostelman, R. Nocross, J. Falco, and J. Marvel, “Development of standard test methods for unmanned and manned industrial vehicles used near humans,” Multisensor, Multisource Information Fusion: Architectures, Algorithms, and Applications, vol. 8756, p. 87560P, 2013.

    Google Scholar 

  47. M. P. Kelly, “Transcription Methods for Trajectory Optimization: a beginners tutorial,” Tech. Rep., 2017. http://arxiv.org/abs/1707.00284

  48. “MATLAB Help Center — Choosing the Algorithm.,” https://uk.mathworks.com/help/optim/ug/choosing-the-algorithm.html, August 2020.

  49. L. F. Shampine, “Vectorized adaptive quadrature in MAT-LAB,” Journal of Computational and Applied Mathematics, vol. 211, no. 2, pp. 131–140, 2008.

    Article  MathSciNet  Google Scholar 

  50. M. A. Cooper, J. F. Raquet, and R. Patton, “Range information characterization of the Hokuyo UST-20LX LIDAR sensor,” Phtonics, vol. 5, no. 2, 2018.

    Google Scholar 

  51. M. Grant and S. Boyd, “CVX: Matlab Software for Disciplined Convex Programming, version 2.1.,” http://cvxr.com/cvx

  52. K. J. Waldron and J. Schmiedeler, “Kinematics,” in Springer Handbook of Robotics, ch. 2, pp. 11–36, Springer, 2016.

Download references

Author information

Authors and Affiliations

  1. Institute for Transport Studies, University of Leeds, 34-40 University Road, Leeds, LS2 9JT, UK

    Edward Derek Lambert, Richard Romano & David Watling

Authors
  1. Edward Derek Lambert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Richard Romano
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Watling
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edward Derek Lambert.

Additional information

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Edward Derek Lambert received his MEng in Engineering Science from the University of Oxford, graduating with a first class degree in 2013. After some time in industry, he began an EPSRC Doctoral Training Partnership Studentship in 2018, working towards a PhD degree at the Institute for Transport Studies at the University of Leeds. His research interests include optimisation-based path planning and multiple vehicle motion coordination.

Richard Romano has over twenty five years of experience developing and testing AVs and ADAS concepts and systems which began with the Automated Highway Systems (AHS) project while he directed the Iowa Driving Simulator in the early 1990’s. He recieved his BASc and MASc in Engineering Science and Aerospace Engineering respectively from the University of Toronto, Canada and a PhD in Motion Drive Algorithms for Large Excursion Motion Bases, Industrial Engineering from the University of Iowa, USA. In addition to a distinguished career in industry he has supervised numerous research projects and authored many journal papers. In 2015 he was appointed a Professor of Driving Simulation at the Institute for Transport Studies, University of Leeds, UK. His research interests include the development, validation and application of transport simulation to support the human-centred design of vehicles and infrastructure.

David Watling’s primary research focus is the development of mathematical models and methods for analysing transport systems, especially those that represent the interactions between travellers’ decision-making and the physical infrastructure. He has particularly developed methods for modelling, simulating or optimizing transport networks with random, dynamic or unreliable elements. With a B.Sc. degree in mathematics from the University of Leeds and a Ph.D. from the Department of Probability and Statistics at the University of Sheffield, he has held the post of Centenary Chair of Transport Analysis at the University of Leeds since its instigation in 2004, where he is the co-leader of the Spatial Modelling and Dynamics group in the Institute for Transport Studies.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lambert, E.D., Romano, R. & Watling, D. Optimal Smooth Paths Based on Clothoids for Car-like Vehicles in the Presence of Obstacles. Int. J. Control Autom. Syst. 19, 2163–2182 (2021). https://doi.org/10.1007/s12555-020-0179-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12555-020-0179-1

Keywords

Search

Navigation