Composing Navigation Functions on Cartesian Products of Manifolds with Boundary

Given two compact, simply connected manifolds with boundary, and a navigation function (NF) on each manifold, this paper presents a simple composition law that yields a new NF on the cross product space. The method provides tunable “hooks” for shaping the new potential function while still guaranteeing obstacle avoidance and essentially global convergence. The composition law is associative, and successive compositions fold into a single, computational simple expression, enabling the practical construction of NFs on the Cartesian product of several manifolds.

[1]  Naomi Ehrich Leonard,et al.  Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem , 2000, IEEE Trans. Autom. Control..

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

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

[4]  Daniel E. Koditschek,et al.  The construction of analytic diffeomorphisms for exact robot navigation on star worlds , 1989, Proceedings, 1989 International Conference on Robotics and Automation.

[5]  Daniel E. Koditschek,et al.  Distributed real-time control of a spatial robot juggler , 1992, Computer.

[6]  S. Shankar Sastry,et al.  Vision-based follow-the-leader , 2003, Proceedings 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2003) (Cat. No.03CH37453).

[7]  D. Koditschek,et al.  The construction of analytic diffeomorphisms for exact robot navigation on star worlds , 1991 .

[8]  Daniel E. Koditschek,et al.  Sequential Composition of Dynamically Dexterous Robot Behaviors , 1999, Int. J. Robotics Res..

[9]  Howie Choset,et al.  Composition of local potential functions for global robot control and navigation , 2003, Proceedings 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2003) (Cat. No.03CH37453).

[10]  Naomi Ehrich Leonard,et al.  Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping , 2001, IEEE Trans. Autom. Control..

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

[12]  Noah J. Cowan,et al.  Geometric visual servoing , 2005, IEEE Transactions on Robotics.

[13]  Phillip J. McKerrow,et al.  Introduction to robotics , 1991 .

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

[15]  Daniel E. Koditschek,et al.  Phase Regulation of Decentralized Cyclic Robotic Systems , 2002, Int. J. Robotics Res..

[16]  B. Dundas,et al.  DIFFERENTIAL TOPOLOGY , 2002 .

[17]  Daniel E. Koditschek.  The Control of Natural Motion in Mechanical Systems , 1991 .

[18]  Daniel E. Koditschek,et al.  Level sets and stable manifold approximations for perceptually driven nonholonomically constrained navigation , 2004, 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566).

[19]  Oussama Khatib,et al.  Commande dynamique dans l''espace op'erational des robots ma-nipulaters en pr'esence d''obstacles , 1980 .

[20]  Daniel E. Koditschek,et al.  Planning and Control of Robotic Juggling and Catching Tasks , 1994, Int. J. Robotics Res..

[21]  R. Olfati-Saber,et al.  Collision avoidance for multiple agent systems , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).