Path planning for deformable linear objects

We present a new approach to path planning for deformable linear (one-dimensional) objects such as flexible wires. We introduce a method for efficiently computing stable configurations of a wire subject to manipulation constraints. These configurations correspond to minimal-energy curves. By restricting the planner to minimal-energy curves, the execution of a path becomes easier. Our curve representation is adaptive in the sense that the number of parameters automatically varies with the complexity of the underlying curve. We introduce a planner that computes paths from one minimal-energy curve to another such that all intermediate curves are also minimal-energy curves. This planner can be used as a powerful local planner in a sampling-based roadmap method. This makes it possible to compute a roadmap of the entire "shape space," which is not possible with previous approaches. Using a simplified model for obstacles, we can find minimal-energy curves of fixed length that pass through specified tangents at given control points. Our work has applications in cable routing, and motion planning for surgical suturing and snake-like robots

[1]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[2]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[3]  M. L. Chambers.  The Mathematical Theory of Optimal Processes , 1965 .

[4]  B. O'neill.  Elementary Differential Geometry , 1966 .

[5]  Berthold K. P. Horn.  The Curve of Least Energy , 1983, TOMS.

[6]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[7]  Michael Kallay.  Plane curves of minimal energy , 1986, TOMS.

[8]  M. Kallay.  Method to approximate the space curve of least energy and prescribed length , 1987 .

[9]  Yoshihiko Nakamura,et al.  Optimal Redundancy Control of Robot Manipulators , 1987 .

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

[11]  Gilbert Strang,et al.  Wavelets and Dilation Equations: A Brief Introduction , 1989, SIAM Rev..

[12]  J. Dennis,et al.  Direct Search Methods on Parallel Machines , 1991 .

[13]  Joseph K. Kearney,et al.  A Case Study of Flexible Object Manipulation , 1991, Int. J. Robotics Res..

[14]  Guido H. Brunnett,et al.  1. Properties of Minimal-Energy Splines , 1992, Curve and Surface Design.

[15]  Carlo H. Séquin,et al.  Functional optimization for fair surface design , 1992, SIGGRAPH.

[16]  Weimin Han,et al.  2. Minimal-Energy Splines with Various End Constraints , 1992, Curve and Surface Design.

[17]  A. Gray.  Modern Differential Geometry of Curves and Surfaces , 1993 .

[18]  Juan C. Meza,et al.  OPT++: An object-oriented class library for nonlinear optimization , 1994 .

[19]  Jean-Claude Latombe,et al.  Geometric Reasoning About Mechanical Assembly , 1994, Artif. Intell..

[20]  Gregory S. Chirikjian,et al.  A modal approach to hyper-redundant manipulator kinematics , 1994, IEEE Trans. Robotics Autom..

[21]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[22]  Remco C. Veltkamp,et al.  Modeling 3D Curves of Minimal Energy , 1995, Comput. Graph. Forum.

[23]  Tsai-Yen Li,et al.  Assembly maintainability study with motion planning , 1995, Proceedings of 1995 IEEE International Conference on Robotics and Automation.

[24]  Remco C. Veltkamp,et al.  Interactive design of constrained variational curves , 1995, Comput. Aided Geom. Des..

[25]  J. Angeles,et al.  The inverse kinematics of hyper-redundant manipulators using splines , 1995, Proceedings of 1995 IEEE International Conference on Robotics and Automation.

[26]  B. Faverjon,et al.  Probabilistic Roadmaps for Path Planning in High-Dimensional Con(cid:12)guration Spaces , 1996 .

[27]  Rajeev Motwani,et al.  Path Planning in Expansive Configuration Spaces , 1999, Int. J. Comput. Geom. Appl..

[28]  Dominik Henrich,et al.  Direct and Inverse Simulation of Deformable Linear Objects , 2000 .

[29]  Scott Alan Hutchinson,et al.  Toward real-time path planning in changing environments , 2000 .

[30]  A. Pressley.  Elementary Differential Geometry , 2000 .

[31]  Lydia E. Kavraki,et al.  Planning Paths for Elastic Objects under Manipulation Constraints , 2001, Int. J. Robotics Res..

[32]  Joe Warren,et al.  Subdivision Methods for Geometric Design: A Constructive Approach , 2001 .

[33]  S. LaValle,et al.  Randomized Kinodynamic Planning , 2001 .

[34]  Dinesh K. Pai,et al.  STRANDS: Interactive Simulation of Thin Solids using Cosserat Models , 2002, Comput. Graph. Forum.

[35]  Lydia E. Kavraki,et al.  Simulated knot tying , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[36]  Nancy M. Amato,et al.  Probabilistic roadmap motion planning for deformable objects , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[37]  Jean-Claude Latombe,et al.  Real-time knot-tying simulation , 2004, The Visual Computer.

[38]  Lydia E. Kavraki,et al.  Using Motion Planning for Knot Untangling , 2004, Int. J. Robotics Res..

[39]  Lydia E. Kavraki,et al.  Path planning for minimal energy curves of constant length , 2004, IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04. 2004.

[40]  Hidefumi Wakamatsu,et al.  Static Modeling of Linear Object Deformation Based on Differential Geometry , 2004, Int. J. Robotics Res..

[41]  O. C. Zienkiewicz,et al.  The Finite Element Method: Its Basis and Fundamentals , 2005 .

[42]  Lydia E. Kavraki,et al.  Path Planning for Variable Resolution Minimal-Energy Curves of Constant Length , 2005, Proceedings of the 2005 IEEE International Conference on Robotics and Automation.

[43]  Hidefumi Wakamatsu,et al.  Manipulation Planning for Knotting/Unknotting and Tightly Tying of Deformable Linear Objects , 2005, Proceedings of the 2005 IEEE International Conference on Robotics and Automation.

[44]  Lydia E. Kavraki,et al.  Motion Planning in the Presence of Drift, Underactuation and Discrete System Changes , 2005, Robotics: Science and Systems.

[45]  Howie Choset,et al.  Principles of Robot Motion: Theory, Algorithms, and Implementation ERRATA!!!! 1 , 2007 .

[46]  Dinesh Manocha,et al.  Path Planning for Deformable Robots in Complex Environments , 2005, Robotics: Science and Systems.

[47]  Dominik Henrich,et al.  Manipulation of Deformable Linear Objects: From Geometric Model Towards Program Generation , 2005, Proceedings of the 2005 IEEE International Conference on Robotics and Automation.

[48]  Hidefumi Wakamatsu,et al.  Dynamic Modeling of Linear Object Deformation based on Differential Geometry Coordinates , 2005, Proceedings of the 2005 IEEE International Conference on Robotics and Automation.

[49]  Mitul Saha,et al.  Motion Planning for Robotic Manipulation of Deformable Linear Objects , 2006, ICRA.

[50]  A. Gray,et al.  Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition (Studies in Advanced Mathematics) , 2006 .