Biharmonic navigation using radial basis functions

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.

[1]  Steven M. LaValle,et al.  RRT-connect: An efficient approach to single-query path planning , 2000, Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065).

[2]  G. Manolis,et al.  Recent Advances in Boundary Element Methods , 2009 .

[3]  Leif Kobbelt,et al.  Geometric fairing of irregular meshes for free-form surface design , 2001, Comput. Aided Geom. Des..

[4]  Du Xin,et al.  Neural network and genetic algorithm based global path planning in a static environment , 2005 .

[5]  Yu-Chu Tian,et al.  Robot path planning in dynamic environments using a simulated annealing based approach , 2008, 2008 10th International Conference on Control, Automation, Robotics and Vision.

[6]  Dushyant Rao,et al.  Large-scale cost function learning for path planning using deep inverse reinforcement learning , 2017, Int. J. Robotics Res..

[7]  Ralf Salomon,et al.  Implementation of Path Planning using Genetic Algorithms on Mobile Robots , 2006, 2006 IEEE International Conference on Evolutionary Computation.

[8]  Jun-Hyeong Do,et al.  Intelligent Interaction For Human-Friendly Service Robot in Smart House Environment , 2008 .

[9]  Nils J. Nilsson,et al.  A Formal Basis for the Heuristic Determination of Minimum Cost Paths , 1968, IEEE Trans. Syst. Sci. Cybern..

[10]  S. LaValle.  Rapidly-exploring random trees : a new tool for path planning , 1998 .

[11]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[12]  Jean-Claude Latombe,et al.  On the Probabilistic Foundations of Probabilistic Roadmap Planning , 2006, Int. J. Robotics Res..

[13]  Vladimir Dobrokhodov.  Cooperative Path Planning of Unmanned Aerial Vehicles , 2011 .

[14]  Oussama Khatib,et al.  Real-Time Obstacle Avoidance for Manipulators and Mobile Robots , 1986 .

[15]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[16]  Michael Buro,et al.  Efficient Triangulation-Based Pathfinding , 2006, AAAI.

[17]  Ellips Masehian,et al.  A voronoi diagram-visibility graph-potential field compound algorithm for robot path planning , 2004 .

[18]  Emilio Frazzoli,et al.  Sampling-based algorithms for optimal motion planning , 2011, Int. J. Robotics Res..

[19]  Lei Ni.  Poisson equation and Hermitian-Einstein metrics on vector bundles over complete noncompact Kahler manifolds , 2002 .

[20]  Emilio Frazzoli,et al.  A Survey of Motion Planning and Control Techniques for Self-Driving Urban Vehicles , 2016, IEEE Transactions on Intelligent Vehicles.

[21]  Jonathan Schaeffer,et al.  Any-Angle Path Planning for Computer Games , 2011, AIIDE.

[22]  Luis Merino,et al.  Learning Human-Aware Path Planning with Fully Convolutional Networks , 2018, 2018 IEEE International Conference on Robotics and Automation (ICRA).

[23]  Jiaping Wang,et al.  Poisson equation on complete manifolds , 2017, Advances in Mathematics.

[24]  Yong-Jin Liu,et al.  Human experience–inspired path planning for robots , 2018 .

[25]  Cui Hu-tao.  Planetary Rover Path-planning Based on Ant Colony Optimization Algorithm , 2006 .

[26]  Hong Wang,et al.  A Motion Planning and Tracking Framework for Autonomous Vehicles Based on Artificial Potential Field Elaborated Resistance Network Approach , 2020, IEEE Transactions on Industrial Electronics.

[27]  Lei Ni.  The Poisson equation and Hermitian-Einstein metrics on holomorphic vector bundles over complete noncompact Kähler manifolds , 2002 .

[28]  Peter Li,et al.  Green's functions, harmonic functions, and volume comparison , 1995 .

[29]  C. Loewner.  On generation of solutions of the biharmonic equation in the plane by conformal mappings. , 1953 .

[30]  Miloš Šeda,et al.  Roadmap methods vs. cell decomposition in robot motion planning , 2007 .

[31]  Lei Ni,et al.  Poisson Equation, Poincaré-Lelong Equation and Curvature Decay on Complete Kähler Manifolds , 2001 .

[32]  Xu-Qian Fan.  On the Existence of Solutions of Poisson Equation and Poincaré–Lelong Equation , 2006 .

[33]  Tomás Lozano-Pérez,et al.  An algorithm for planning collision-free paths among polyhedral obstacles , 1979, CACM.

[34]  Qiang Huang,et al.  RimJump: Edge-based Shortest Path Planning for a 2D Map , 2018, Robotica.

[35]  S. Yau,et al.  On the parabolic kernel of the Schrödinger operator , 1986 .

[36]  Anthony Stentz.  Optimal and efficient path planning for partially-known environments , 1994 .

[37]  Jianda Han,et al.  Path Planning of UGV Based on Bézier Curves , 2019, Robotica.

[38]  Marina L. Gavrilova,et al.  Roadmap-Based Path Planning - Using the Voronoi Diagram for a Clearance-Based Shortest Path , 2008, IEEE Robotics & Automation Magazine.

[39]  Roland Siegwart,et al.  Smooth path planning in constrained environments , 2009, 2009 IEEE International Conference on Robotics and Automation.

[40]  Shuzhi Sam Ge,et al.  New potential functions for mobile robot path planning , 2000, IEEE Trans. Robotics Autom..

[41]  Rahul Kala,et al.  Autonomous Social Robot Navigation using a Behavioral Finite State Social Machine , 2020, Robotica.

[42]  Lydia E. Kavraki,et al.  Probabilistic roadmaps for path planning in high-dimensional configuration spaces , 1996, IEEE Trans. Robotics Autom..

[43]  Michael W. Otte.  A Survey of Machine Learning Approaches to Robotic Path-Planning , 2009 .

[44]  R. Duffin.  The maximum principle and biharmonic functions , 1961 .

[45]  Ahmad A. Masoud,et al.  Robot navigation using a pressure generated mechanical stress field: "the biharmonic potential approach" , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.

[46]  Han-Pang Huang,et al.  Dynamic visibility graph for path planning , 2004, 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566).

[47]  Du Xin,et al.  Neural network and genetic algorithm based global path planning in a static environment , 2005 .

[48]  Bengt Fornberg,et al.  Solving PDEs with radial basis functions * , 2015, Acta Numerica.

[49]  Stéphane Puechmorel,et al.  Automatic Conflict Solving using Biharmonic Navigation Functions , 2012 .

[50]  Xu-Qian Fan.  Poisson Equation on Some Complete Noncompact Manifolds , 2007 .

[51]  Wenyong Gong,et al.  Probabilistic model based path planning , 2021 .

[52]  Thierry Siméon,et al.  Transition-based RRT for path planning in continuous cost spaces , 2008, 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems.