Spiral transitions. (English) Zbl 1424.65012
Summary: Spiral curves are free from singularities and curvature extrema. These are used in path smoothing applications to overcome the abrupt change in curvature and super-elevation of moving object that occurs between tangent and circular curve. Line to circle spiral transition is made of straight line segment and curvature continuous spiral curve. It is extendible to other important types of transitions like line to line and circle to circle. Although the problem of line to circle transition has been addressed by many researchers, there is no comprehensive literature review available. This paper presents state-of-the-art of line to circle spiral transition, applications in different fields, limitations of existing approaches, and recommendations to meet the challenges of compatibility with today’s CAD/CAM soft wares, satisfaction of Hermite end conditions, approximation of discrete data for image processing, 3D path smoothness for an unmanned aerial vehicle (UAV), and arc-length parametrization. Whole discussion is concluded with future research directions in various areas of applications.
MSC:
| 65D17 | Computer-aided design (modeling of curves and surfaces) |
References:
| [1] | Benjamin Adler, Junhao Xiao, Jianwei Zhang. Autonomous exploration of urban environ-ments using unmanned aerial vehicles, Journal of Field Robotics, 31(6):912-939, 2014. |
| [2] | Mohammad Al Khawaldah, Andreas Nüchter. Enhanced frontier-based exploration for in-door environment with multiple robots, Advanced Robotics, 29(10):657-669, 2015. |
| [3] | Gianluca Antonelli, G Antonelli. Underwater robots, Springer, 2014. · Zbl 1049.93001 |
| [4] | Asghar, K., Zulfiqar Habib, Muhammad Hussain. Copy-move and splicing image forgery detection and localization techniques: a review, 1-27 (2016) |
| [5] | K G Baass. The use of clothoid templates in highway design, Transportation Forum, 1:47-52, 1984. |
| [6] | Jose Luis Blanco, Mauro Bellone, Antonio Gimenez-Fernandez. Tp-space rrt-kinematic path planning of non-holonomic any-shape vehicles, International Journal of Advanced Robotic Systems, 12, 2015. |
| [7] | David BL Bong, Koon Chun Lai, Annie Joseph. Automatic road network recognition and extraction for urban planning, International Journal of Applied Science, Engineering and Technology, 5(1):209-215, 2009. |
| [8] | JL Campbell, CM Richard, J Graham. Human factors guidelines for roadway systems, washington, dc. Technical report, NCHRP Report, 2008. |
| [9] | Anjan Chakrabarty, Jack W Langelaan. Energy-based long-range path planning for soaring-capable unmanned aerial vehicles, Journal of Guidance, Control, and Dynamics, 34(4):1002-1015, 2011. |
| [10] | Jie Chen, Guo-Jin Wang. A new type of the generalized bézier curves, Applied Mathematics-A Journal of Chinese Universities, 26(1):47-56, 2011. · Zbl 1240.65046 |
| [11] | Yonghua Chen, Fenghua Dong. Robot machining: recent development and future research issues, The International Journal of Advanced Manufacturing Technology, 66(9-12):1489-1497, 2013. |
| [12] | Howie M Choset. Principles of robot motion: theory, algorithms, and implementation, MIT press, 2005. · Zbl 1081.68700 |
| [13] | Licai Chu, Xiao-Ming Zeng. Constructing curves and triangular patches by beta functions, Journal of Computational and Applied Mathematics, 260:191-200, 2014. · Zbl 1293.65026 |
| [14] | Dominique David, Nathalie Sprynski. Method and device for acquisition of a geometric shape, November 17 2015. US Patent 9,188,422. |
| [15] | D A Dietz, B Piper, E Sebe. Rational cubic spirals, Computer-Aided Design, 40:3-12, 2008. |
| [16] | D W Dudley. Handbook of practical gear design, CRC Press, 1994. |
| [17] | Mohamed Elbanhawi, Milan Simic, Reza N Jazar. Continuous path smoothing for car-like robots using b-spline curves, Journal of Intelligent & Robotic Systems, 80(1):23-56, 2015. |
| [18] | Gerald Farin. Curves, Surfaces for CAGD: A Practical Guide. Morgan-Kaufmann, 5th edition, 2002. |
| [19] | Gerald Farin. Geometric Hermite interpolation with circular precision, Computer-Aided Design, 40(4):476-479, 2008. |
| [20] | Farouki, R. T., Construction of G1 planar hermite interpolants with prescribed arc lengths (2016) · Zbl 1418.65021 |
| [21] | R T Farouki, T Sakkalis. Pythagorean hodographs, IBM Journal of Research and Development, 34(5):736-752, 1990. |
| [22] | G M Gibreel, S M Easa, Y Hassan, I A El-Dimeery. State of the art of highway geometric design consistency, ASCE Journal of Transportation Engineering, 125(4):305-313, 1999. |
| [23] | T N T Goodman, D S Meek. Planar interpolation with a pair of rational spirals, Computational and Applied Mathematics, 201:112-127, 2007. · Zbl 1112.41001 |
| [24] | Goodman, T. N T.; Ong, B. H.; Unsworth, K., NURBS for Curve and Surface Design, 59-74 (1991), Philadelphia · Zbl 0772.41001 |
| [25] | H Guggenheimer. Differential Geometry, McGraw-Hill, New York, 1963. · Zbl 0116.13402 |
| [26] | D Guo, A Weeks, H Klee. Robust approach for suburban road segmentation in high-resolution aerial images, International Journal of Remote Sensing, 28(2):307-318, 2007. |
| [27] | Zulfiqar Habib. Spiral Function and Its Application in CAGD, VDM Verlag, Germany, 2010. ISBN: 978-3-639-24988-0. |
| [28] | Zulfiqar Habib, Manabu Sakai. G2 two-point Hermite rational cubic interpolation, International Journal of Computer Mathematics, 79(11):1225-1231, 2002. http://dx.doi.org/10.1080/00207160213938. · Zbl 1031.41009 |
| [29] | Zulfiqar Habib, Manabu Sakai. G2 planar cubic transition between two circles, International Journal of Computer Mathematics, 80(8):959-967, 2003. http://dx.doi.org/10.1080/0020716031000112367. · Zbl 1042.65018 |
| [30] | Habib, Z.; Sakai, M., Advances in Geometric Modeling, 133-150 (2004) |
| [31] | Zulfiqar Habib, Manabu Sakai. Shapes of planar cubic curves, Scientiae Mathematicae Japonicae, 59(1):133-138, 2004.:e9, 253-258, http://www.jams.or.jp/scmjol/9.html. · Zbl 1047.65008 |
| [32] | Zulfiqar Habib, Manabu Sakai. Spiral transition curves and their applications, Scientiae Mathematicae Japonicae, 61(2):195-206, 2005. e2004, 251-262, http://www.jams.or.jp/scmjol/2004.html. · Zbl 1074.65019 |
| [33] | Zulfiqar Habib, Manabu Sakai. G2 Pythagorean hodograph quintic transition between t-wo circles with shape control, Computer Aided Geometric Design, 24(5):252-266, 2007. http://dx.doi.org/10.1016/j.cagd.2007.03.004. · Zbl 1171.65335 |
| [34] | Zulfiqar Habib, Manabu Sakai. On PH quintic spirals joining two circles with one circle inside the other, Computer Aided Design, 39(2):125-132, 2007. http://dx.doi.org/10.1016/j.cad.2006.10.006. |
| [35] | Zulfiqar Habib, Manabu Sakai. Transition between concentric or tangent circles with a single segment of G2 PH quintic curve, Computer Aided Geometric Design, 25(4-5):247-257, 2008. http://dx.doi.org/10.1016/j.cagd.2007.10.006. · Zbl 1172.65313 |
| [36] | Zulfiqar Habib, Manabu Sakai. G2 cubic transition between two circles with shape control, Computational and Applied Mathematics, 223:133-144, 2009, http://dx.doi.org/10.1016/j.cam.2007.12.024. · Zbl 1156.65014 |
| [37] | Zulfiqar Habib, Manabu Sakai. Admissible regions for rational cubic spirals matching G2 Hermite data, Computer-Aided Design, 42(12):1117-1124, 2010, http://dx.doi.org/10.1016/j.cad.2010.07.006. |
| [38] | Zulfiqar Habib, Manabu Sakai. Cubic spiral transition matching G2 Hermite end condi-tions, Numerical Mathematics, Theory, Methods, Applications, 4(4):525-536, 2011. · Zbl 1265.65021 |
| [39] | Zulfiqar Habib, Manabu Sakai. Fairing arc spline and designing by using cubic Bézier spiral segments, Mathematical Modelling and Analysis, 17(2):141-160, 2012. · Zbl 1242.65032 |
| [40] | Zulfiqar Habib, Manabu Sakai. Fairing an arc spline and designing with G2 PH quintic spiral transitions, International Journal of Computer Mathematics, 90(5):1023-1039, 2013. · Zbl 1277.65011 |
| [41] | Zulfiqar Habib, Muhammad Sarfraz, Manabu Sakai. Rational cubic spline interpolation with shape control, Computers & Graphics, 29(4):594-605, 2005, http://dx.doi.org/10.1016/j.cag.2005.05.010. |
| [42] | P Hartman. The highway spiral for combining curves of different radii, Transactions of the American Society of Civil Engineers, 122:389-409, 1957. |
| [43] | P Henrici. Applied and Computational Complex Analysis, volume 1. Wiley, New York, 1988. · Zbl 0635.30001 |
| [44] | Josef Hoschek, Dieter Lasser. Fundamentals of Computer Aided Geometric Design (Trans-lation by L.L. Schumaker). A, K. Peters, Wellesley, MA, 1993. · Zbl 0788.68002 |
| [45] | Md Arafat Hossain, Israt Ferdous. Autonomous robot path planning in dynamic environ-ment using a new optimization technique inspired by bacterial foraging technique, Robotics and Autonomous Systems, 64:137-141, 2015. |
| [46] | Mathieu Huard, Rida T Farouki, Nathalie Sprynski, Luc Biard. C2 interpolation of s-patial data subject to arc-length constraints using pythagorean hodograph quintic splines, Graphical models, 76(1):30-42, 2014. |
| [47] | Uk-Youl Huh, Seong-Ryong Chang. A G2 continuous path-smoothing algorithm using mod-ified quadratic polynomial interpolation, International Journal of Advanced Robotic Systems, 11(25), 2014. |
| [48] | KG Jolly, R Sreerama Kumar, R Vijayakumar. A bezier curve based path planning in a multi-agent robot soccer system without violating the acceleration limits, Robotics and Autonomous Systems, 57(1):23-33, 2009. |
| [49] | Amna Khan, IRAM NOREEN, Zulfiqar Habib. On complete coverage path planning al-gorithms for non-holonomic mobile robots: Survey and challenges. Journal of Information Science & Engineering, 33(1), 2017. |
| [50] | Khan, A.; Noreen, I.; Ryu, H.; Doh, N. L.; Habib, Z., Online com-plete coverage path planning using two-way proximity search, 1-12 (2017) |
| [51] | Steven M LaValle. Planning algorithms, Cambridge university press, 2006. · Zbl 1100.68108 |
| [52] | Raphael Linus Levien. From spiral to spline: Optimal techniques in interactive curve de-sign, University of California, Berkeley, 2009. |
| [53] | Z Li, D S Meek. Smoothing an arc spline, Computers & Graphics, 29:576-587, 2005. |
| [54] | Li, Z.; Ma, L.; Zhao, M.; Mao, Z., Improvement construction for planar G2 transition curve between two separated circles, 358-361 (2006) · Zbl 1155.65314 |
| [55] | T C Liang, J S Liu, G T Hung, Y Z Chang. Practical and flexible path planning for car-like mobile robot using maximal-curvature cubic spiral, Robotics and Autonomous Systems, 52(4):312-335, 2005. |
| [56] | Lu, C.; Jiang, L., An iterative algorithm for G2 multiwise merging of Bézier curves, 352-361 (2016) · Zbl 1342.65092 |
| [57] | Lizheng Lu. Planar quintic G2 hermite interpolation with minimum strain energy, Journal of Computational and Applied Mathematics, 274:109-117, 2015. · Zbl 1296.65017 |
| [58] | Ellips Masehian, Davoud Sedighizadeh. Multi-objective robot motion planning using a parti-cle swarm optimization model, Journal of Zhejiang University SCIENCE C, 11(8):607-619, 2010. |
| [59] | Asif Masood, Muhammad Sarfraz. An efficient technique for capturing 2d objects, Computers & Graphics, 32(1):93-104, 2008. |
| [60] | D S Meek, D J Walton. The use of cornu spirals in drawing planar curves of controlled curvature, Computational and Applied Mathematics, 25:69-78, 1989. · Zbl 0662.65008 |
| [61] | D S Meek, D J Walton. The use of Cornu spirals in drawing planar curves of controlled curvature, Computational and Applied Mathematics, 25:69-78, 1989. · Zbl 0662.65008 |
| [62] | Dereck S Meek, Desmond J Walton. Approximation of discrete data by G1 arc splines, Computer-Aided Design, 24(6):301-306, 1992. · Zbl 0755.65010 |
| [63] | Ana Margarida Monteiro, Reha H Tütüncü, Luís N Vicente. Recovering risk-neutral proba-bility density functions from options prices using cubic splines and ensuring nonnegativity, European Journal of Operational Research, 187(2):525-542, 2008. · Zbl 1149.90114 |
| [64] | Niu, G.; Liu, L.; Chen, H.; Gao, Q., Set-point stabilization of nonholonomic mobile robot based on optimizing bezier curve (2008) |
| [65] | Iram Noreen, Amna Khan, Zulfiqar Habib. A comparison of rrt, rrt* and rrt*-smart path planning algorithms, International Journal of Computer Science and Network Security (IJCSNS), 16(10):20, 2016. |
| [66] | Iram Noreen, Amna Khan, Zulfiqar Habib. Optimal path planning using rrt* based ap-proaches: A survey and future directions, International Journal of Advanced Computer Science & Applications, 1(7):97-107, 2016. |
| [67] | M Pan, X Yang, J Tang. Research on interpolation methods in medical image processing, Journal of Medical Systems, 36(2):777-807, July 2012. |
| [68] | L Piegl, W Tiller. The NURBS book, Springer, 1995. · Zbl 0828.68118 |
| [69] | M Rajeswari, KS Gurumurthy, L Pratap Reddy, SN Omkar, J Senthilnath. Automatic road extraction based on normalized cuts and level set methods, Int J Comput App, 18:10-16, 2011. |
| [70] | Dimulyo Sarpono, Zulfiqar Habib, Manabu Sakai. Fair cubic transition between two circles with one circle inside or tangent to the other, Numerical Algorithms, 51(4):461-476, July 2009. http://dx.doi.org/10.1007/s11075-008-9252-1. · Zbl 1173.65007 |
| [71] | Dimulyo Sarpono, Zulfiqar Habib, Manabu Sakai. Fair cubic transition between two circles with one circle inside or tangent to the other, Numerical Algorithms, 51(4):461-476, July 2009. http://dx.doi.org/10.1007/s11075-008-9252-1. · Zbl 1173.65007 |
| [72] | D J Walton, D S Meek. Computer-aided design for horizontal alignment, ASCE Journal of Transporation Engineering, 115:411-424, 1989. |
| [73] | D J Walton, D S Meek. Clothoid splines, Computers & Graphics, 14:95-100, 1990. |
| [74] | D J Walton, D S Meek. A planar cubic Bézier spiral, Computational and Applied Mathematics, 72(1):85-100, 1996. · Zbl 0857.65019 |
| [75] | D J Walton, D S Meek. A Pythagorean hodograph quintic spiral, Computer-Aided Design, 28(12):943-950, 1996. |
| [76] | D J Walton, D S Meek. G2 curves composed of planar cubic and Pythagorean hodograph quintic spirals, Computer Aided Geometric Design, 15(6):547-566, 1998. · Zbl 0905.68147 |
| [77] | D J Walton, D S Meek. Planar G2 transition between two circles with a fair cubic Bézier curve, Computer-Aided Design, 31(14):857-866, 1999. · Zbl 1038.68579 |
| [78] | D J Walton, D S Meek. Planar G2 transition with a fair Pythagorean hodograph quintic curve, Computational and Applied Mathematics, 138:109-126, 2002. · Zbl 0994.65012 |
| [79] | D J Walton, D S Meek. A controlled clothoid spline, Computers & Graphics, 29:353-363, 2005. |
| [80] | D J Walton, D S Meek. G2 curve design with a pair of Pythagorean hodograph quintic spiral segments, Computer Aided Geometric Design, 24(5):267-285, 2007. · Zbl 1171.65357 |
| [81] | D J Walton, D S Meek. A further generalisation of the planar cubic Bézier spiral, Computational and Applied Mathematics, 236(11):2869-2882, 2012. · Zbl 1245.65020 |
| [82] | D J Walton, D S Meek, J M Ali. Planar G2 transition curves composed of cubic Bézier spiral segments, Computational and Applied Mathematics, 157(2):453-476, 2003. · Zbl 1030.65008 |
| [83] | H Wang, J Kearney, K Atkinson. Arc-length parameterized spline curves for real-time simulation, In Curve and Surface Design, pages 387-396, San Malo, France, June 2002. Proceedings of the 5th International Conference on Curves and Surfaces, Nashboro Press. http://www.cs.uiowa.edu/kearney/pubs/CurvesAndSurfacesArcLength.pdf. · Zbl 1043.65045 |
| [84] | Joab R Winkler. A unified approach to resultant matrices for bernstein basis polynomials, Computer Aided Geometric Design, 25(7):529-541, 2008. · Zbl 1172.65350 |
| [85] | Wu, W.; Yang, X., Geometric hermite interpolation by a family of intrinsically defined planar curves (2016) |
| [86] | Lianghong Xu, Jianhong Shi. Geometric hermite interpolation for space curves, Computer aided geometric design, 18(9):817-829, 2001. · Zbl 0984.68163 |
| [87] | J Yang, RS Wang. Classified road detection from satellite images based on perceptual or-ganization, International Journal of Remote Sensing, 28(20):4653-4669, 2007. |
| [88] | K Yang, S K Gan, S Sukkarieh. An efficient path planning and control algorithm for RUAV’s in unknown and cluttered environmnets, Journal of Intelligent Robot System, 57:101-122, 2010. · Zbl 1203.68283 |
| [89] | Kwangjin Yang, Sangwoo Moon, Seunghoon Yoo, Jaehyeon Kang, Nakju Lett Doh, Hong Bong Kim, Sanghyun Joo. Spline-based rrt path planner for non-holonomic robots, Journal of Intelligent & Robotic Systems, 73(1-4):763-782, 2014. |
| [90] | Kwangjin Yang, Salah Sukkarieh. An analytical continuous-curvature path-smoothing al-gorithm, IEEE Transactions on Robotics, 26(3):561-568, 2010. |
| [91] | Jun-Hai Yong, Shi-Min Hu, Jia-Guang Sun. A note on approximation of discrete data by G1 arc splines, Computer-Aided Design, 31(14):911-915, 1999. · Zbl 1039.68770 |
| [92] | Junku Yuh. Design and control of autonomous underwater robots: A survey, Autonomous Robots, 8(1):7-24, 2000. |
| [93] | Beong In Yun. A cumulative averaging method for piecewise polynomial approximation to discrete data, Applied Mathematical Sciences, 10(7):331-343, 2016. |
| [94] | Zhi-hao Zheng, Guo-zhao Wang. A note on pythagorean hodograph quartic spiral, Applied Mathematics-A Journal of Chinese Universities, 33(2):234-252, 2018. · Zbl 1399.65090 |
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