A Survey on Advances in the Theory of Computational Robotics

  • John H. Reif


This paper describes work on the computational complexity of various movement planning problems relevant to robotics. This paper is intended only as a survey of previous and current work in this area. The generalized mover’s problem is to plan a sequence of movements of linked polyhedra through 3-dimensional Euclidean space, avoiding contact with a fixed set of polyhedra obstacles. We discuss our and other researchers’ work showing generalized mover’s problems are polynomial space hard. These results provide strong evidence that robot movement planning is computationally intractable, i.e., any algorithm requires time growing exponentially with the number of degrees of freedom. We also briefly discuss the computational complexity of four other quite different types of movement problems: (1) movement planning in the presence of friction, (2) minimal movement planning, (3) dynamic movement planning with moving obstacles and (4) adaptive movement planning problems.


Movement Planning Polynomial Space Robot Movement Planning Cylindric Algebraic Decomposition Polygonal Obstacle 


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  1. [1]
    J.H. Reif, “Complexity of the Mover’s Problem,” Proc. 20th IEEE Symposium on Foundations of Computer Science, San Juan, Puerto Rico, pp. 421–427, 1979, also appearing in Planning, Geometry and Complexity of Robot Planning J. Schwartz, ed., Ablex Pub., Norwood, N.J. 1985.Google Scholar
  2. [2]
    J.H. Reif and M. Sharir, “Motion Planning in the Presence of Moving Obstacles,” 26th IEEE Symposium on Foundations of Computer Science, Portland, Oregon, October 1985.Google Scholar
  3. [3]
    M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness Freedman and Co., San Francisco, 1979.Google Scholar
  4. [4]
    J.E. Hoperoft, D.A. Joseph and S.H. Whitesides, “On the Movement of Robot Arms in 2-dimensional Bounded Regions,” Proc. 23rd IEEE Symposium on Foundations of Computer Science, Chicago, Il., pp. 280–289, 1982.Google Scholar
  5. [5]
    J.E. Hoperoft, J.T. Schwartz and M. Sharir, “On the Complexity of Motion Planning for Multiple Independent Objects: PSPACE Hardness of the Warehouseman’s Problems,” TR-103, Courant Institute of Mathematics, Feb. 1984.Google Scholar
  6. [6]
    P. Spirakis and C. Yap, “Strong NP-Hardness of Moving Many Discs,” to appear, Information Processing Letters, August 1985.Google Scholar
  7. [7]
    N.J. Nilsson, “A Mobile Automation: An Application of Artificial Intelligence Techniques,” Proceedings IJCAI-69, pp. 509–520, 1969.Google Scholar
  8. [8]
    R. Paul, “Modelling Trajectory Calculation and Servicing of a Computer Controlled Arm,” Ph.D. Thesis, Stanford University, Nov. 1972.Google Scholar
  9. [9]
    S. Udupa, “Collision Detection and Avoidance in Computer Controlled Manipulators,” Ph.D. Thesis, Cal. Inst. Tech., 1977.Google Scholar
  10. [10]
    C. Widdoes, “A Heuristic Collision Avoider for the Stanford Robot Arm,” Stanford CS Memo 227, June 1974.Google Scholar
  11. [11]
    T. Lozano-Pérez and M. Wesley, “An Algorithm for Planning Collision-Free Paths Among Polyhedral Obstacles,” CA CM, vol. 22, pp. 560–570, 1979.Google Scholar
  12. [12]
    P.G. Comba, “A Procedure for Detecting Intersections of Three-Dimensional Objects,” J. ACM, vol. 15, No. 3, pp. 354–366, July 1968.MATHGoogle Scholar
  13. [13]
    J.T. Schwartz and M. Sharir, “On the Piano Mover’s Problem: I. The Special Case of a Rigid Polygonal Body Moving Amidst Polygonal Barriers,” Comm. Pure Applied Mathematics, vol. XXXVI, pp. 345–398, 1983.Google Scholar
  14. [14]
    C. O’Dunlaing, M. Sharir and C.K. Yap, “Retraction: A New Approach to Motion Planning,” Proc. 15th ACM Symposium on the Theory of Computing, Boston, Ma., pp. 207–220, 1983.Google Scholar
  15. [15]
    J.T. Schwartz and M. Sharir, “On the Piano Mover’s Problem: II. General Techniques for Computing Topological Properties of Real Algebraic Manifolds,” Adv. Applied Mathematics, vol. 4, pp. 298, 351, 1983.Google Scholar
  16. [16]
    G.E. Collins, “Quantifier Elimination for Real Closed Fields by Cylindric Algebraic Decomposition,” Proc. 2nd GI Conference on Automata Theory and Formal Languages Springer-Verlag, LNCS 35, Berlin, pp. 134–183, 1975.Google Scholar
  17. M. Ben-Or, D. Kozen and J.H. Reif, “Complexity of Elementary Algebra and Geometry,” 16th Symposium on Theory of Computing, 1984, also to appear in J. Computer and System Sciences,1985.Google Scholar
  18. [18]
    V.T. Rajan and J.T. Schwartz, work in progress, 1985.Google Scholar
  19. [19]
    G. Miller and J.H Reif, “Robotic Movement Planning in the Presence of Friction is Undecidable,” to appear, 1985.Google Scholar
  20. [20]
    B. Chazelle, “A Theorem on Polygon Cutting with Applications,” Proc. 23rd IEEE Symposium on Foundations of Computer Science, Chicago, Il., pp. 339–349, 1982.Google Scholar
  21. [21]
    M. Sharir and A. Schorr, “On the Shortest Path in Polyhedral Spaces,” Proc. 16th ACM Symposium on the Theory of Computing, Washington, D.C., pp. 144–153, 1984.Google Scholar
  22. [22]
    J.H. Reif and J. Storer, “Shortest Paths in Euclidean Space with Polyhedral Obstacles,” Center for Research in Computing Technology, Harvard University, TR-05–85, May 1985.Google Scholar
  23. [23]
    M. Brady, J.M. Hollerbach, T.L. Johnson, T. Lozano-Pérez and M.T. Mason (eds.), Robot Motion: Planning and Control M.I.T. Press, 1983.Google Scholar
  24. [24]
    H.R. Lewis and C.H. Papadimitriou, “Symmetric Space Bounded Computation,” Theor. Comput. Sci., vol. 19, pp. 161–187, 1982.MathSciNetMATHGoogle Scholar
  25. [25]
    W.J. Savitch, “Relationships Between Nondeterministic and Deterministic Tape Complexities,” J. Computer Sci., vol. 4, pp. 177–192, 1970.MathSciNetMATHGoogle Scholar
  26. [26]
    J.T. Schwartz and M. Sharir, “On the Piano Mover’s Problem: III. Coordinating the Motion of Several Independent Bodies: The Special Case of Circular Bodies Moving Amidst Polygonal Barriers,” The International Journal of Robotics Research, vol. 2, No. 3, pp. 46–75, fall 1983.Google Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • John H. Reif
    • 1
  1. 1.Aiken Computation Laboratory Division of Applied SciencesHarvard UniversityCambridgeUSA

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