An Algorithmic Approach to Some Problems in Terrain Navigation

Abstract Recent advances in the field of computational geometry have provided efficient algorithms for a variety of shortest path problems. Many problems in the field of terrain navigation can be cast as optimal path problems in a precise geometric model. With such a model one can develop and analyze algorithms for the solution of the original problem and can gain insights into how to design more efficient heuristics to deal with more complex problems. We examine the path planning problem in which we are given a “map” of a region of terrain and we are expected to find optimal paths from one point to another. This, for example, is a task which must be done repeatedly for the guidance of an autonomous vehicle. We examine how to formulate some path planning problems precisely, and we report algorithms to solve certain special cases.

[1]  Joseph S. B. Mitchell.  An Autonomous Vehicle Navigation Algorithm , 1984, Other Conferences.

[2]  Joseph S. B. Mitchell,et al.  The weighted region problem , 1987, SCG '87.

[3]  L. Paul Chew,et al.  Planning the shortest path for a disc in O(n2log n) time , 1985, SCG '85.

[4]  Nageswara S. V. Rao,et al.  Robot navigation in an unexplored terrain , 1986, J. Field Robotics.

[5]  S. Sitharama Iyengar,et al.  Robot navigation in unknown terrains using learned visibility graphs. Part I: The disjoint convex obstacle case , 1987, IEEE Journal on Robotics and Automation.

[6]  W. Böge,et al.  Quantifier Elimination for Real Closed Fields , 1985, AAECC.

[7]  David M. Keirsey,et al.  Planning Strategic Paths Through Variable Terrain Data , 1984, Other Conferences.

[8]  Michael Zyda,et al.  Solving global two-dimensional routing problems using snell's law and a search , 1987, Proceedings. 1987 IEEE International Conference on Robotics and Automation.

[9]  D. T. Lee,et al.  The Power of Geometric Duality Revisited , 1985, Inf. Process. Lett..

[10]  Der-Tsai Lee.  Proximity and reachability in the plane. , 1978 .

[11]  Bruce G. Baumgart.  A polyhedron representation for computer vision , 1975, AFIPS '75.

[12]  Emo WELZL,et al.  Constructing the Visibility Graph for n-Line Segments in O(n²) Time , 1985, Inf. Process. Lett..

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

[14]  David M. Mount,et al.  An Output Sensitive Algorithm for Computing Visibility Graphs , 1987, FOCS.

[15]  Joseph S. B. Mitchell.  Shortest Paths Among Obstacles, Zero-Cost Regions, and Roads , 1987 .

[16]  Ivan E. Sutherland.  A Method for Solving Arbitrary-Wall Mazes by Computer , 1969, IEEE Transactions on Computers.

[17]  Takao Asano,et al.  Shortest Path Between Two Simple Polygons , 1987, Inf. Process. Lett..

[18]  Leonidas J. Guibas,et al.  Optimal Shortest Path Queries in a Simple Polygon , 1989, J. Comput. Syst. Sci..

[19]  John F. Canny,et al.  New lower bound techniques for robot motion planning problems , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[20]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[21]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[22]  Jeffrey J. P. Tsai,et al.  Route planning for intelligent autonomous land vehicles using hierarchical terrain representation , 1987, Proceedings. 1987 IEEE International Conference on Robotics and Automation.

[23]  Varol Akman.  Unobstructed Shortest Paths in Polyhedral Environments , 1987, Lecture Notes in Computer Science.

[24]  D. T. Lee,et al.  Rectilinear shortest paths with rectangular barriers , 1985, SCG '85.

[25]  Christos H. Papadimitriou,et al.  An Algorithm for Shortest-Path Motion in Three Dimensions , 1985, Inf. Process. Lett..

[26]  Charles E. Thorpe,et al.  Path Relaxation: Path Planning for a Mobile Robot , 1984, AAAI.

[27]  Leonidas J. Guibas,et al.  Linear time algorithms for visibility and shortest path problems inside simple polygons , 2011, SCG '86.

[28]  S. Sitharama Iyengar,et al.  Robot navigation algorithms using learned spatial graphs , 1986, Robotica.

[29]  S. Kauffman.  Algorithmic approach to intelligent robot mobility , 1983 .

[30]  Joseph S. B. Mitchell,et al.  The Discrete Geodesic Problem , 1987, SIAM J. Comput..

[31]  Neil C. Rowe,et al.  Exploiting capability constraints to solve global two-dimensional path planning problems , 1986, Proceedings. 1986 IEEE International Conference on Robotics and Automation.

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

[33]  Joseph S. B. Mitchell,et al.  Shortest Rectilinear Paths Among Obstacles , 1987 .

[34]  Rodney A. Brooks,et al.  Natural decomposition of free space for path planning , 1985, Proceedings. 1985 IEEE International Conference on Robotics and Automation.

[35]  Kenneth L. Clarkson,et al.  Rectilinear shortest paths through polygonal obstacles in O(n(logn)2) time , 1987, SCG '87.

[36]  Chak-Kuen Wong,et al.  Distance problems in computational geometry with fixed orientations , 1985, SCG '85.

[37]  Joseph O'Rourke.  Finding A Shortest Ladder Path: A Special Case , 1987 .

[38]  S. Sitharama Iyengar,et al.  On terrain acquisition by a finite-sized mobile robot in plane , 1987, Proceedings. 1987 IEEE International Conference on Robotics and Automation.

[39]  Kenneth L. Clarkson,et al.  Approximation algorithms for shortest path motion planning , 1987, STOC.

[40]  Larry S. Davis,et al.  Multiresolution path planning for mobile robots , 1986, IEEE J. Robotics Autom..

[41]  Joseph S. B. Mitchell,et al.  Path planning in 0/1/ weighted regions with applications , 1988, SCG '88.

[42]  Joseph O'Rourke,et al.  Shortest Paths on Polyhedral Surfaces , 1985, STACS.

[43]  Francis Quek,et al.  A Decision System For Autonomous Robot Navigation Over Rough Terrain , 1985, Other Conferences.

[44]  Andrew K. C. Wong,et al.  Structuring Free Space as a Hypergraph for Roving Robot Path Planning and Navigation , 1987, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[45]  C. R. Weisbin,et al.  The 3-D world modeling with updating capability based on combinatorial geometry , 1987 .

[46]  J. Reif,et al.  Shortest Paths in Euclidean Space with Polyhedral Obstacles. , 1985 .

[47]  Rodney A. Brooks,et al.  Solving the Find-Path Problem by Good Representation of Free Space , 1983, Autonomous Robot Vehicles.

[48]  D. T. Lee,et al.  Euclidean shortest paths in the presence of rectilinear barriers , 1984, Networks.

[49]  Micha Sharir,et al.  On Shortest Paths in Polyhedral Spaces , 1986, SIAM J. Comput..

[50]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[51]  David W. Payton,et al.  Planning and reasoning for autonomous vehicle control , 1987 .

[52]  A.C.-C. Meng.  Free space modeling and geometric motion planning under unexpected obstacles , 1988, [1988] Proceedings. The Fourth Conference on Artificial Intelligence Applications.

[53]  Hanan Samet,et al.  Storing a collection of polygons using quadtrees , 1985, TOGS.

[54]  Helman I Stern.  A routing algorithm for a mobile search robot , 1986 .

[55]  Charles Elving Thorpe,et al.  Fido: vision and navigation for a robot rover , 1984 .

[56]  Robert F Richbourg.  Solving a class of spatial reasoning problems: minimal-cost path planning in the Cartesian plane. , 1987 .

[57]  Simeon C. Ntafos,et al.  Optimum Watchman Routes , 1988, Inf. Process. Lett..

[58]  Rodney A. Brooks,et al.  A subdivision algorithm in configuration space for findpath with rotation , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[59]  Hans P. Moravec.  Obstacle avoidance and navigation in the real world by a seeing robot rover , 1980 .

[60]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[61]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[62]  David G. Kirkpatrick,et al.  Optimal Search in Planar Subdivisions , 1983, SIAM J. Comput..