The construction of analytic diffeomorphisms for exact robot navigation on star worlds

The authors consider the construction of navigation functions on configuration spaces whose geometric expressiveness is rich enough for navigation amidst real-world obstacles. They describe a general methodology which extends the construction of navigation functions on sphere worlds to any smoothly deformable space. According to this methodology, the problem of constructing a navigation function is reduced to the construction of a transformation mapping a given space into its model sphere world. The transformation must satisfy certain regularity conditions guaranteeing invariance of the navigation function properties. The authors demonstrate this idea by constructing navigation functions on star worlds: n-dimensional star shaped subsets of E/sup n/ punctured by any finite number of smaller disjoint n-dimensional stars. This construction yields automatically a bounded torque feedback control law which is guaranteed to guide the robot to destination point from almost every initial position without hitting any obstacle.<<ETX>>

[1]  J. Schwartz,et al.  On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds , 1983 .

[2]  S. Smale.  On Gradient Dynamical Systems , 1961 .

[3]  Elon L. Lima.  The Jordan-Brouwer separation theorem for smooth hypersurfaces , 1988 .

[4]  Daniel E. Koditschek,et al.  Exact robot navigation using cost functions: the case of distinct spherical boundaries in E/sup n/ , 1988, Proceedings. 1988 IEEE International Conference on Robotics and Automation.

[5]  J. Milnor.  Topology from the differentiable viewpoint , 1965 .

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

[7]  Neville Hogan,et al.  Impedance Control: An Approach to Manipulation , 1984, 1984 American Control Conference.

[8]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[9]  O. Khatib,et al.  Real-Time Obstacle Avoidance for Manipulators and Mobile Robots , 1985, Proceedings. 1985 IEEE International Conference on Robotics and Automation.

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

[11]  Neville Hogan,et al.  Impedance Control: An Approach to Manipulation: Part I—Theory , 1985 .

[12]  D. Koditschek.  The Application of Total Energy as a Lyapunov Function for Mechanical Control Systems , 1989 .

[13]  D. Koditschek.  Strict Global Lyapunov Functions for Mechanical Systems , 1988, 1988 American Control Conference.

[14]  Fumio Miyazaki,et al.  Asymptotic Stability of Feedback Control Laws for Robot Manipulator , 1985 .

[15]  M. Morse.  The Existence of Polar Non-Degenerate Functions on Differentiable Manifolds , 1960 .

[16]  D. Koditschek,et al.  Robot navigation functions on manifolds with boundary , 1990 .