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Journal of Intelligent & Robotic Systems

, Volume 80, Supplement 1, pp 23–56 | Cite as

Continuous Path Smoothing for Car-Like Robots Using B-Spline Curves

  • Mohamed ElbanhawiEmail author
  • Milan Simic
  • Reza N. Jazar
Article

Abstract

A practical approach for generating motion paths with continuous steering for car-like mobile robots is presented here. This paper addresses two key issues in robot motion planning; path continuity and maximum curvature constraint for nonholonomic robots. The advantage of this new method is that it allows robots to account for their constraints in an efficient manner that facilitates real-time planning. B-spline curves are leveraged for their robustness and practical synthesis to model the vehicle’s path. Comparative navigational-based analyses are presented to selected appropriate curve and nominate its parameters. Path continuity is achieved by utilizing a single path, to represent the trajectory, with no limitations on path, or orientation. The path parameters are formulated with respect to the robot’s constraints. Maximum curvature is satisfied locally, in every segment using a smoothing algorithm, if needed. It is demonstrated that any local modifications of single sections have minimal effect on the entire path. Rigorous simulations are presented, to highlight the benefits of the proposed method, in comparison to existing approaches with regards to continuity, curvature control, path length and resulting acceleration. Experimental results validate that our approach mimics human steering with high accuracy. Accordingly, efficiently formulated continuous paths ultimately contribute towards passenger comfort improvement. Using presented approach, autonomous vehicles generate and follow paths that humans are accustomed to, with minimum disturbances.

Keywords

Path planning Path smoothing Nonholonomic robots C2 continuity Maximum curvature Real time planning 

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References

  1. 1.
    Reif, J.H.: Complexity of the mover’s problem and generalizations. In: Foundations of Computer Science, 1979., 20th Annual Symposium on, pp. 421–427 (1979)Google Scholar
  2. 2.
    Latombe, J.-C.: Motion Planning: A journey of robots, molecules, digital actors, and other artifacts. Int. J. Robot. Res. 18(11), 1119–1128 (1999). doi: 10.1177/02783649922067753 CrossRefGoogle Scholar
  3. 3.
    Choset, H.M.: Principles of Robot Motion, Theory, Algorithms, and Implementation. Prentice Hall of India (2005)Google Scholar
  4. 4.
    Brooks, R.A., Lozano-Perez, T.: A subdivision algorithm in configuration space for findpath with rotation. IEEE Trans. Syst. Man Cybern. 15(2), 224–233 (1985). doi: 10.1109/TSMC.1985.6313352 CrossRefGoogle Scholar
  5. 5.
    Canny, J.: A Voronoi method for the piano-movers problem. In: Robotics and Automation. Proceedings. 1985 IEEE International Conference on, pp. 530–535 (1985)Google Scholar
  6. 6.
    Khatib, O.: Real-time obstacle avoidance for manipulators and mobile robots. Int. J. Robot. Res. 5(1), 90–98 (1986). doi: 10.1177/027836498600500106 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Arkin, R.C.: Motor Schema Based Mobile Robot Navigation. Int. J. Robot. Res. 8(4), 92–112 (1989). doi: 10.1177/027836498900800406 CrossRefGoogle Scholar
  8. 8.
    Koren, Y., Borenstein, J.: Potential field methods and their inherent limitations for mobile robot navigation. In: Robotics and Automation, 1991. Proceedings., 1991 IEEE International Conference on, vol. 1392, pp. 1398–1404 (1991)Google Scholar
  9. 9.
    Elbanhawi, M., Simic, M.: Sampling-based robot motion planning: a review. IEEE Access 2, 56–77 (2014). doi: 10.1109/ACCESS.2014.2302442 CrossRefGoogle Scholar
  10. 10.
    Geraerts, R., Overmars, M.H.: Creating high-quality paths for motion planning. Int. J. Robot. Res. 26(8), 845–863 (2007). doi: 10.1177/0278364907079280 CrossRefGoogle Scholar
  11. 11.
    Laumond, J.P., Sekhavat, S., Lamiraux, F.: Guidelines in nonholonomic motion planning for mobile robots. In: J.P. Laumond (ed.) Robot Motion Planning and Control, vol. 229. Lecture Notes in Control and Information Sciences, pp. 1–53. Springer Berlin Heidelberg (1998)Google Scholar
  12. 12.
    Cheng, P.: Sampling-based motion planning with differential constraints. Ph.D. University of Illinois at Urbana-Champaign (2005)Google Scholar
  13. 13.
    Wallace, R., Stentz, A., Thorpe, C., Moravec, H., Whittaker, W., Kanade, T.: First Results in Robot Road Following. In: 1985, pp. 381–387Google Scholar
  14. 14.
    Antonelli, G., Chiaverini, S., Fusco, G.: A fuzzy-logic-based approach for mobile robot path tracking. IEEE Trans. Fuzzy Syst. 15(2), 211–221 (2007)CrossRefGoogle Scholar
  15. 15.
    Perez, J., Milanes, V., Onieva, E.: Cascade architecture for lateral control in autonomous vehicles. IEEE Intell. Transp. Syst. 12(1), 73–82 (2011). doi: 10.1109/TITS.2010.2060722 CrossRefGoogle Scholar
  16. 16.
    Jazar, R.N.: Mathematical theory of autodriver for autonomous vehicles. J. Vib. Control. 16(2), 253–279 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Marzbani, H., Ahmad Salahuddin, M.H., Simic, M., Fard, M., Jazar, R.N.: Steady-state dynamic steering. In: Frontiers in Artificial Intelligence and Applications, vol. 262 (2014)Google Scholar
  18. 18.
    Cheein, F.A., Scaglia, G.: Trajectory Tracking Controller Design for Unmanned Vehicles: A New Methodology. Journal of Field Robotics, n/a-n/a. doi: 10.1002/rob.21492 (2013)
  19. 19.
    Magid, E., Keren, D., Rivlin, E., Yavneh, I.: Spline-Based Robot Navigation. In: Intelligent Robots and Systems, 2006 IEEE/RSJ International Conference on, pp. 2296–2301 (2006)Google Scholar
  20. 20.
    Roth, S., Batavia, P.: Evaluating Path Tracker Performance for Outdoor Mobile Robots. Paper presented at the Automation Technology for Off-Road Equipment, Chicago, Illinois, USA, 26–27/07Google Scholar
  21. 21.
    Lau, B., Sprunk, C., Burgard, W.: Kinodynamic motion planning for mobile robots using splines. In: Intelligent Robots and Systems, 2009. IROS 2009. IEEE/RSJ International Conference on, pp. 2427–2433 (2009)Google Scholar
  22. 22.
    Gulati, S., Kuipers, B.: High performance control for graceful motion of an intelligent wheelchair. In: Robotics and Automation, 2008. ICRA 2008. IEEE International Conference on, pp. 3932–3938 (2008)Google Scholar
  23. 23.
    Berglund, T., Brodnik, A., Jonsson, H., Staffanson, M., Soderkvist, I.: Planning smooth and obstacle-avoiding b-spline paths for autonomous mining vehicles. IEEE Trans. Autom. Sci. Eng. 71), 167–172 (2010). doi: 10.1109/TASE.2009.2015886 CrossRefGoogle Scholar
  24. 24.
    Maekawa, T., Noda, T., Tamura, S., Ozaki, T., Machida, K.-i.: Curvature continuous path generation for autonomous vehicle using B-spline curves. Comput. Aided Des. 42(4), 350–359 (2010) doi: 10.1016/j.cad.2009.12.007 CrossRefGoogle Scholar
  25. 25.
    Sabelhaus, D., Röben, F., Meyer zu Helligen, L.P., Schulze Lammers, P.: Using continuous-curvature paths to generate feasible headland turn manoeuvres. Biosyst. Eng. 116(4), 399–409 (2013). doi: 10.1016/j.biosystemseng.2013.08.012 CrossRefGoogle Scholar
  26. 26.
    Girbés, V., Armesto, L., Tornero, J.: Path following hybrid control for vehicle stability applied to industrial forklifts. Robot. Auton. Syst. 0 (2014). doi: 10.1016/j.robot.2014.01.004
  27. 27.
    Xuan-Nam, B., Boissonnat, J.-d., Soueres, P., Laumond, J.P.: Shortest path synthesis for Dubins non-holonomic robot. In: Robotics and Automation, 1994. Proceedings., 1994 IEEE International Conference on, 8–13 1994, Vol. 1, pp. 2–7Google Scholar
  28. 28.
    Anderson, E.P., Beard, R.W., McLain, T.W.: Real-time dynamic trajectory smoothing for unmanned air vehicles. IEEE Trans. Control Syst. Technol. 13(3), 471–477 (2005). doi: 10.1109/TCST.2004.839555 CrossRefGoogle Scholar
  29. 29.
    Myung, H., Kuffner, J., Kanade, T.: Efficient Two-phase 3D Motion Planning for Small Fixed-wing UAVs. In: Robotics and Automation, 2007 IEEE International Conference on, pp. 1035–1041 (2007)Google Scholar
  30. 30.
    LaValle, S.: Planning Algorithms. Cambridge University Press (2006)Google Scholar
  31. 31.
    Suzuki, Y., Kagami, S., Kuffner, J.J.: Path Planning with Steering Sets for Car-Like Robots and Finding an Effective Set. In: Robotics and Biomimetics, 2006. ROBIO ’06. IEEE International Conference on, pp. 1221–1226 (2006)Google Scholar
  32. 32.
    Pivtoraiko, M., Knepper, R.A., Kelly, A.: Differentially constrained mobile robot motion planning in state lattices. J. Field Robot. 26(3), 308–333 (2009)CrossRefGoogle Scholar
  33. 33.
    Fraichard, T., Scheuer, A.: From Reeds and Shepp’s to continuous-curvature paths. Robot. IEEE Trans. 20(6), 1025–1035 (2004). doi: 10.1109/TRO.2004.833789 CrossRefGoogle Scholar
  34. 34.
    Wang, L.Z., Miura, K.T., Nakamae, E., Yamamoto, T., Wang, T.J.: An approximation approach of the clothoid curve defined in the interval [0, π/2] and its offset by free-form curves. Comput. Aided Des. 33(14), 1049–1058 (2001). doi: 10.1016/S0010-4485(00)00142-1 CrossRefGoogle Scholar
  35. 35.
    Meek, D.S., Walton, D.J.: An arc spline approximation to a clothoid. J. Comput. Appl. Math. 170(1), 59–77 (2004) doi: 10.1016/j.cam.2003.12.038 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Montes, N., Herraez, A., Armesto, L., Tornero, J.: Real-time clothoid approximation by Rational Bezier curves. In: In: Robotics and Automation, 2008. ICRA 2008. IEEE International Conference on, pp. 2246–2251 (2008)Google Scholar
  37. 37.
    McCrae, J., Singh, K.: Sketching piecewise clothoid curves. Comput. Graph. 33(4), 452–461 (2009) doi: 10.1016/j.cag.2009.05.006 CrossRefGoogle Scholar
  38. 38.
    Brezak, M., Petrovic, I.: Real-time approximation of clothoids with bounded error for path planning applications. IEEE Trans. Robot. PP(99), 1–9 (2013). doi: 10.1109/TRO.2013.2283928 CrossRefGoogle Scholar
  39. 39.
    Farin, G.: From conics to NURBS: A tutorial and survey. IEEE Comput. Graph. Appl. 12(5), 78–86 (1992). doi: 10.1109/38.156017 MathSciNetCrossRefGoogle Scholar
  40. 40.
    Piegl, L.: On NURBS: a survey. IEEE Comput. Graph. Appl. 11(1), 55–71 (1991). doi: 10.1109/38.67702 CrossRefGoogle Scholar
  41. 41.
    Farin, G.: Curves and Surfaces for CAGD. Computing. Morgan Kaufmann (2002)Google Scholar
  42. 42.
    Lepetič, M., Klančar, G., Škrjanc, I., Matko, D., Potočnik, B.: Time optimal path planning considering acceleration limits. Robot. Auton. Syst. 45(3–4), 199–210 (2003). doi: 10.1016/j.robot.2003.09.007 CrossRefGoogle Scholar
  43. 43.
    Jolly, K.G., Sreerama Kumar, R., Vijayakumar, R.: A Bezier curve based path planning in a multi-agent robot soccer system without violating the acceleration limits. Robot. Auton. Syst. 57(1), 23–33 (2009). doi: 10.1016/j.robot.2008.03.009 CrossRefGoogle Scholar
  44. 44.
    Schoenberg, I.J.: Contributions to the problem of approximation of equidistant data by analytic functions - b. on the problem of osculatory interpolation - a 2nd class of approximation formulae. Q. Appl. Math. 4(2), 112–141 (1946)MathSciNetGoogle Scholar
  45. 45.
    Thompson, S.E., Patel, R.V.: Formulation of joint trajectories for industrial robots using b-splines. IEEE Trans. Ind. Electron. 34(2), 192–199 (1987). doi: 10.1109/TIE.1987.350954 CrossRefGoogle Scholar
  46. 46.
    Dyllong, E., Visioli, A.: Planning and real-time modifications of a trajectory using spline techniques. Robotica 21(05), 475–482 (2003). doi: 10.1017/S0263574703005009 CrossRefGoogle Scholar
  47. 47.
    Hodgins, J.K., O’Brien, J.F., Tumblin, J.: Perception of human motion with different geometric models. IEEE Trans. Vis. Comput. Graph. 4(4), 307–316 (1998). doi: 10.1109/2945.765325 CrossRefGoogle Scholar
  48. 48.
    Schmid, A.J., Woern, H.: Path planning for a humanoid using NURBS curves. In: Automation Science and Engineering, 2005. IEEE international conference on, pp. 351–356 (2005)Google Scholar
  49. 49.
    Sungchul, J., Taehoon, K.: Tool-path generation for NURBS surface machining. In: American control conference, 2003. Proceedings of the 200, Vol. 2613, pp. 2614–2619 (2003)Google Scholar
  50. 50.
    Cheng, M.Y., Kuo, J.C.: Real-time NURBS command generators for CNC servo controllers. Int. J. Mach. Tools Manuf. 42(7), 801–813 (2002) doi: 10.1016/S0890-6955(02)00015-9 CrossRefGoogle Scholar
  51. 51.
    Zhang, Y., Bazilevs, Y., Goswami, S., Bajaj, C.L., Hughes, T.J.R.: Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow. Comput. Methods Appl. Mech. Eng. 196(29–30), 2943–2959 (2007). doi: 10.1016/j.cma.2007.02.009 MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Ma, W., Kruth, J.P.: NURBS curve and surface fitting for reverse engineering. Int. J. Adv. Manuf. Technol. 14(12), 918–927 (1998). doi: 10.1007/BF01179082 CrossRefGoogle Scholar
  53. 53.
    Piegl, L.A., Tiller, W.: Parametrization for surface fitting in reverse engineering. Comput. Aided Des. 33(8), 593–603 (2001). doi: 10.1016/S0010-4485(00)00103-2 CrossRefGoogle Scholar
  54. 54.
    Hughes, T.J.R., Reali, A., Sangalli, G.: Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: Comparison of p-method finite elements with k-method NURBS. Comput. Methods Appl. Mech. Eng. 197(49–50), 4104–4124 (2008). doi: 10.1016/j.cma.2008.04.006 MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Koyuncu, E., Inalhan, G.: A probabilistic B-spline motion planning algorithm for unmanned helicopters flying in dense 3D environments. In: Intelligent Robots and Systems, 2008. IROS 2008. IEEE/RSJ International Conference on, pp. 815–821 (2008)Google Scholar
  56. 56.
    Zhou, F., Song, B., Tian, G.: Bézier curve based smooth path planning for mobile robot. J. Inf. Comput. Sci. 8(12), 2441–2450 (2011)Google Scholar
  57. 57.
    Kwangjin, Y., Sukkarieh, S.: An analytical continuous-curvature path-smoothing algorithm. Robot. IEEE Trans. 26(3), 561–568 (2010). doi: 10.1109/TRO.2010.2042990 CrossRefGoogle Scholar
  58. 58.
    Kwangjin, Y., Jung, D., Sukkarieh, S.: Continuous curvature path-smoothing algorithm using cubic Bezier spiral curves for non-holonomic robots. Adv. Robot. 27(4), 247–258 (2013). doi: 10.1080/01691864.2013.755246 CrossRefGoogle Scholar
  59. 59.
    Walton, D.J., Meek, D.S., Ali, J.M.: Planar G2 transition curves composed of cubic Bézier spiral segments. J. Comput. Appl. Math. 157(2), 453–476 (2003). doi: 10.1016/s0377-0427(03)00435-7 MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Huh, U.-Y., Chang, S.-R.: A G 2 continuous path-smoothing algorithm using modified quadratic polynomial interpolation. Int. J. Adv. Robot. Syst. 25(11) (2014). doi: 10.5772/57340
  61. 61.
    Piazzi, A., Bianco, C.G.L., Romano, M.: η3-Splines for the smooth path generation of wheeled mobile robots. robotics-splines for the smooth path generation of wheeled mobile robots. Robot. IEEE Trans. 23(5), 1089–1095 (2007). doi: 10.1109/TRO.2007.903816 CrossRefGoogle Scholar
  62. 62.
    Pan, J., Zhang, L., Manocha, D.: Collision-free and smooth trajectory computation in cluttered environments. Int. J. Robot. Res. 31(10), 1155–1175 (2012). doi: 10.1177/0278364912453186 CrossRefGoogle Scholar
  63. 63.
    Nikolos, I.K., Valavanis, K.P., Tsourveloudis, N.C., Kostaras, A.N.: Evolutionary algorithm based offline/online path planner for UAV navigation. Systems, Man, and Cybernetics, Part B. Cybern. IEEE Trans. 33(6), 898–912 (2003). doi: 10.1109/TSMCB.2002.804370 CrossRefGoogle Scholar
  64. 64.
    Guarino Lo Bianco, C.: Minimum-jerk velocity planning for mobile robot applications. Robot. IEEE Trans. 29(5), 1317–1326 (2013). doi: 10.1109/TRO.2013.2262744 CrossRefGoogle Scholar
  65. 65.
    Kunz, T., Stilman, M.: Time-optimal trajectory generation for path following with bounded acceleration and velocity. Robotics: Science and Systems, p 209 (2013)Google Scholar
  66. 66.
    Velenis, E., Tsiotras, P.: Minimum-time travel for a vehicle with acceleration limits: theoretical analysis and receding-horizon implementation. J. Optim. Theory Appl. 138(2), 275–296 (2008). doi: 10.1007/s10957-008-9381-7 MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Johnson, J., Hauser, K.: Optimal acceleration-bounded trajectory planning in dynamic environments along a specified path. In: Robotics and Automation (ICRA), 2012 IEEE International Conference on, pp. 2035–2041 (2012)Google Scholar
  68. 68.
    Elbanhawi, M., Simic, M., Jazar, R.: Continuous-curvature bounded trajectory planning using parametric splines. In: Frontiers in Artificial Intelligence and Applications, vol. 262, pp. 513–522 (2014)Google Scholar
  69. 69.
    Kelly, A., Stentz, A.: Rough terrain autonomous mobility—part 1: A theoretical analysis of requirements. Auton. Robot. 2(5), 129–161 (1998). doi: 10.1023/A:1008801421636 CrossRefGoogle Scholar
  70. 70.
    Ahmed, F., Deb, K.: Multi-objective path planning using spline representation. In: Robotics and Biomimetics (ROBIO), 2011 IEEE International Conference on, pp. 1047–1052 (2011)Google Scholar
  71. 71.
    De Boor, C.: On calculating with B-splines. J. Appro. Theory 6(1), 50–62 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Barsky, B.A., Derose, T.D.: Geometric continuity of parametric curves: constructions of geometrically continuous splines. IEEE Comput. Graph. Appl. 10(1), 60–68 (1990). doi: 10.1109/38.45811 CrossRefGoogle Scholar
  73. 73.
    Jan, G.E., Sun, C.C., Tsai, W.C., Lin, T.H.: An O(n log n) Shortest Path Algorithm Based on Delaunay Triangulation, Mechatronics. IEEE/ASME Trans. PP(99), 1–7 (2013). doi: 10.1109/TMECH.2013.2252076 CrossRefGoogle Scholar
  74. 74.
    Likhachev, M., Ferguson, D., Gordon, G., Stentz, A., Thrun, S.: Anytime search in dynamic graphs. Artif. Intell. 172(14), 1613–1643 (2008). doi: 10.1016/j.artint.2007.11.009 MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Bruce, J.R., Veloso, M.M.: Safe multirobot navigation within dynamics constraints. IEEE Proc. 94(7), 1398–1411 (2006). doi: 10.1109/JPROC.2006.876915 CrossRefGoogle Scholar
  76. 76.
    LaValle, S.M.: Planning Algorithms. Cambridge University Press (2006)Google Scholar
  77. 77.
    LaValle, S.M.: Rapidly-exploring random trees: A new tool for path planning. In. Iowa state university (1998)Google Scholar
  78. 78.
    Wein, R., van den Berg, J., Halperin, D.: Planning high-quality paths and corridors amidst obstacles. Int. J. Robot. Res. 27(11-12), 1213–1231 (2008). doi: 10.1177/0278364908097213 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Mohamed Elbanhawi
    • 1
    Email author
  • Milan Simic
    • 1
  • Reza N. Jazar
    • 1
  1. 1.School of Aerospace, Mechanical, and Manufacturing Engineering (SAMME)RMIT University.Bundoora East CampusMelbourneAustralia

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