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Biharmonic navigation using radial basis functions

Published online by Cambridge University Press:  17 June 2021

Xu-Qian Fan  and
Wenyong Gong Open the ORCID record for Wenyong Gong [Opens in a new window]
Xu-Qian Fan
Affiliation:
Department of Mathematics, Jinan University, Guangzhou, China
Wenyong Gong*
Affiliation:
Department of Mathematics, Jinan University, Guangzhou, China
*
*Corresponding author. Email: gongwenyong@jnu.edu.cn
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Abstract

Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

Keywords

Path planningRadial basis functionBiharmonic function
Type
Reply
Information
Robotica , Volume 40 , Issue 3 , March 2022 , pp. 599 - 610
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Paden, B., Cáp, M., Yong, S. Z., Yershov, D. S. and Frazzoli, E., “A survey of motion planning and control techniques for self-driving urban vehicles,” IEEE Trans. Intell. Veh. 1(1), 33–55 (2016).CrossRefGoogle Scholar
Bien, Z. N., Lee, H., Do, J., Kim, Y., Park, K. and Yang, S., “Intelligent interaction for human-friendly service robot in smart house environment,” Int. J. Comput. Intell. Syst. 1(1), 77–93 (2008).Google Scholar
Gong, W., Xie, X. and Liu, Y., “Human experience–inspired path planning for robots,” Int. J. Adv. Rob. Syst. 15(1), 172988141875704 (2018).CrossRefGoogle Scholar
Malviya, V., Reddy, A. K. and Kala, R., “Autonomous social robot navigation using a behavioral finite state social machine,” Robotica 38(12), 2266–2289 (2020).CrossRefGoogle Scholar
Yao, Z., Zhang, W., Shi, Y., Li, M., Liang, Z., Li, F. and Huang, Q., “Rimjump: Edge-based shortest path planning for a 2D map,” Robotica 37(4), 641–655 (2019).CrossRefGoogle Scholar
Dobrokhodov, V., “Cooperative path planning of unmanned aerial vehicles,” J. Guidance Control Dyn. 34(5), 1601–1602 (2011).CrossRefGoogle Scholar
Hu, Y., Li, D., He, Y. and Han, J., “Path planning of UGV based on bézier curves,” Robotica 37(6), 969–997 (2019).CrossRefGoogle Scholar
Hart, P. E., Nilsson, N. J. and Raphael, B., “A formal basis for the heuristic determination of minimum cost paths,” IEEE Trans. Syst. Sci. Cybern. 4(2), 100–107 (1968).CrossRefGoogle Scholar
Dijkstra, E. W., “A note on two problems in connexion with graphs,” Numerische Mathematik 1(1), 269–271 (1959).CrossRefGoogle Scholar
Rufli, M., Ferguson, D. and Siegwart, R., “Smooth Path Planning in Constrained Environments,” IEEE International Conference on Robotics and Automation (2009) pp. 3780–3785.Google Scholar
Schneider, R. and Kobbelt, L., “Geometric fairing of irregular meshes for free-form surface design,” Comput. Aided Geom. Des. 18(4), 359–379 (2001).CrossRefGoogle Scholar
Guys, L., Puechmorel, S. and Lapasset, L., “Automatic conflict solving using biharmonic navigation functions,” Proc. Soc. Behav. Sci. 54(4), 1378–1387 (2012).CrossRefGoogle Scholar
Connolly, C. I., Burns, J. B. and Weiss, R., “Path Planning Using Laplace’s Equation,” International Conference on Robotics and Automation (1990) pp. 2102–2106.Google Scholar
Ge, S. S. and Cui, Y., “New potential functions for mobile robot path planning,” IEEE Trans. Rob. Autom. 16(5), 615–620 (2000).CrossRefGoogle Scholar
Gong, W., “Probabilistic model based path planning,” Phys. A Stat. Mech. Appl. 568, 125718 (2021).Google Scholar
Kavraki, L. E., Svestka, P., Latombe, J. and Overmars, M. H., “Probabilistic roadmaps for path planning in high-dimensional configuration spaces,” IEEE Trans. Rob. Autom. 12(4), 566–580 (1996).CrossRefGoogle Scholar
Kuffner, J. J. and Lavalle, S. M., “RRT-Connect: An Efficient Approach to Single-Query Path Planning,” International Conference on Robotics and Automation, vol. 2 (2000) pp. 995–1001.Google Scholar
Connolly, C. I., “Applications of Harmonic Functions to Robotics,” International symposium on Intelligent Control (1992) pp. 498–502.Google Scholar
Masoud, A. A., Masoud, S. A. and Bayoumi, M. M., “Robot Navigation Using a Pressure Generated Mechanical Stress Field: ‘The Biharmonic Potential Approach’,” IEEE International Conference on Robotics and Automation (1994) pp. 124–129.Google Scholar
Fornberg, B. and Flyer, N., “Solving PDES with radial basis functions,” Acta Numerica 24, 215–258 (2015).CrossRefGoogle Scholar
Paden, B., Čáp, M., Yong, S. Z., Yershov, D. and Frazzoli, E., “A survey of motion planning and control techniques for self-driving urban vehicles,” IEEE Trans. Intell. Veh. 1(1), 33–55 (2016).CrossRefGoogle Scholar
Huang, H.-P. and Chung, S.-Y., “Dynamic Visibility Graph for Path Planning,” 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No. 04CH37566), vol. 3 (IEEE, 2004) pp. 2813–2818.Google Scholar
Lozano-Pérez, T. and Wesley, M. A., “An algorithm for planning collision-free paths among polyhedral obstacles,” Commun. ACM 22(10), 560–570 (1979).CrossRefGoogle Scholar
Bhattacharya, P. and Gavrilova, M. L., “Roadmap-based path planning-using the voronoi diagram for a clearance-based shortest path,” IEEE Rob. Autom. Mag. 15(2), 58–66 (2008).CrossRefGoogle Scholar
Šeda, M., “Roadmap Methods vs. Cell Decomposition in Robot Motion Planning,” Proceedings of the 6th WSEAS International Conference on Signal Processing, Robotics and Automation (World Scientific and Engineering Academy and Society (WSEAS), 2007) pp. 127–132.Google Scholar
Demyen, D. and Buro, M., “Efficient Triangulation-based Pathfinding,” AAAI, vol. 6 (2006) pp. 942–947.Google Scholar
Kallmann, M., “Path Planning in Triangulations,” Proceedings of the IJCAI Workshop on Reasoning, Representation, and Learning in Computer Games (2005) pp. 49–54.Google Scholar
Yan, H., Wang, H., Chen, Y. and Dai, G., “Path Planning Based on Constrained Delaunay Triangulation,” 2008 7th World Congress on Intelligent Control and Automation (IEEE, 2008) pp. 5168–5173.Google Scholar
Jaillet, L., Cortés, J. and Siméon, T., “Transition-based RRT for Path Planning in Continuous Cost Spaces,” 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2008) pp. 2145–2150.CrossRefGoogle Scholar
LaValle, S. M., “Rapidly-exploring random trees: A new tool for path planning,” Technical report, Computer Science Department, Iowa State University (1998).Google Scholar
Karaman, S. and Frazzoli, E., “Sampling-based algorithms for optimal motion planning,” Int. J. Rob. Res. 30(7), 846–894 (2011).CrossRefGoogle Scholar
Hsu, D., Latombe, J.-C. and Kurniawati, H., “On the probabilistic foundations of probabilistic roadmap planning,” Int. J. Rob. Res. 25(7), 627–643 (2006).CrossRefGoogle Scholar
Kavraki, L., Svestka, P. and Overmars, M. H., “Probabilistic roadmaps for path planning in high-dimensional configuration spaces,” IEEE Trans. Rob. Autom. 12(4), 566–580 (1996).CrossRefGoogle Scholar
Huang, Y., Ding, H., Zhang, Y., Wang, H., Cao, D., Xu, N. and Hu, C., “A motion planning and tracking framework for autonomous vehicles based on artificial potential field-elaborated resistance network (apfe-rn) approach,” IEEE Trans. Ind. Electron. 67(2), 1376–1386 (2020).CrossRefGoogle Scholar
Khatib, O., “Real-time obstacle avoidance for manipulators and mobile robots,” Int. J. Rob. Res. 5(1), 90–98 (1986).CrossRefGoogle Scholar
Yap, P., Burch, N., Holte, R. C. and Schaeffer, J., “Any-Angle Path Planning for Computer Games,” Proceedings of the Seventh AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, AIIDE’11 (AAAI Press, 2011) pp. 201–207.Google Scholar
Stentz, A., “Optimal and Efficient Path Planning for Partially Known Environments,” In: Intelligent Unmanned Ground Vehicles (Springer, 1997) pp. 203–220.CrossRefGoogle Scholar
Koenig, S. and Likhachev, M., “D*lite,” Eighteenth National Conference on Artificial Intelligence, Menlo Park, CA, USA (American Association for Artificial Intelligence, 2002) pp. 476–483.Google Scholar
Otte, M., “A survey of machine learning approaches to robotic path-planning,” Int. J. Rob. Res. 5(1), 90–98 (2008).Google Scholar
Pérez-Higueras, N., Caballero, F. and Merino, L., “Learning Human-Aware Path Planning with Fully Convolutional Networks,” 2018 IEEE International Conference on Robotics and Automation (ICRA) (IEEE, 2018) pp. 1–5.CrossRefGoogle Scholar
Wulfmeier, M., Rao, D., Wang, D. Z., Ondruska, P. and Posner, I., “Large-scale cost function learning for path planning using deep inverse reinforcement learning,” Int. J. Rob. Res. 36(10), 1073–1087 (2017).CrossRefGoogle Scholar
Miao, H. and Tian, Y.-C., “Robot Path Planning in Dynamic Environments Using a Simulated Annealing Based Approach,” 2008 10th International Conference on Control, Automation, Robotics and Vision (IEEE, 2008) pp. 1253–1258.CrossRefGoogle Scholar
Burchardt, H. and Salomon, R., “Implementation of Path Planning Using Genetic Algorithms on Mobile Robots,” 2006 IEEE International Conference on Evolutionary Computation (IEEE, 2006) pp. 1831–1836.Google Scholar
Yue, F.-Z., Cui, P.-Y. and Cui, H.-T., “Planetary rover path-planning based on ant colony optimization algorithm,” Control and Decision 21(12), 1437 (2006).Google Scholar
Du, X., Chen, H.-H. and Gu, W.-K., “Neural network and genetic algorithm based global path planning in a static environment,” J. Zhejiang Univ. Sci. A 6(6), 549–554 (2005).Google Scholar
Masehian, E. and Amin-Naseri, M. R., “A voronoi diagram-visibility graph-potential field compound algorithm for robot path planning,” J. Rob. Syst. 21(6), 275–300 (2004).CrossRefGoogle Scholar
Duffin, R. J., “The maximum principle and biharmonic functions,” J. Math. Anal. Appl. 3(3), 399–405 (1961).CrossRefGoogle Scholar
Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 1983).Google Scholar
Loewner, C., “On generation of solutions of the biharmonic equation in the plane by conformal mappings,” Pac. J. Math. 3(2), 417–436 (1953).CrossRefGoogle Scholar
Manolis, G. D. and Polyzos, D., Recent Advances in Boundary Element Methods. A Volume to Honor Professor Dimitri Beskos (Springer, New York, 2009).CrossRefGoogle Scholar
Fan, X.-Q., “On the existence of solutions of poisson equation and poincaré - lelong equation,” Manuscripta Mathematica 120(4), 435–467 (2006).CrossRefGoogle Scholar
Fan, X.-Q., “Poisson equation on some complete noncompact manifolds,” Acta Mathematica Sinica-English Ser. 23(4), 623–638 (2007).CrossRefGoogle Scholar
Li, P. and Tam, L.-F., “Green’s functions, harmonic functions, and volume comparison,” J. Differ. Geom. 41(2), 277–318 (1995).CrossRefGoogle Scholar
Li, P. and Yau, S.-T., “On the parabolic kernel of the schrödinger operator,” Acta Mathematica 156(3–4), 153–201 (1986).CrossRefGoogle Scholar
Munteanu, O., Sung, C.-J. and Wang, J., “Poisson equation on complete manifolds,” Adv. Math. 348, 81–145 (2019).CrossRefGoogle Scholar
Ni, L., “The poisson equation and Hermitian–Einstein metrics on holomorphic vector bundles over complete noncompact kähler manifolds,” Indiana Univ. Math. J. 51(3), 679–704 (2002).CrossRefGoogle Scholar
Ni, L., Shi, Y. and Tam, L.-F., “Poisson equation, poincaré–lelong equation and curvature decay on complete kähler manifolds,” J. Differ. Geom. 57(2), 339–388 (2001).CrossRefGoogle Scholar