We present a new methodology for exact robot motion planning and control that unifies the purely kinematic path planning problem with the lower level feedback controller design. Complete information about the freespace and goal is encoded in the form of a special artificial potential function-a navigation function-that connects the kinematic planning problem with the dynamic execution problem in a provably correct fashion. The navigation function automatically gives rise to a bounded-torque feedback controller for the robot's actuators that guarantees collision-free motion and convergence to the destination from almost all initial free configurations. Since navigation functions exist for any robot and obstacle course, our methodology is completely general in principle. However, this paper is mainly concerned with certain constructive techniques for a particular class of motion planning problems. Specifically, we present a formula for navigation functions that guide a point-mass robot in a generalized sphere world. The simplest member of this family is a space obtained by puncturing a disc by an arbitrary number of smaller disjoint discs representing obstacles. The other spaces are obtained from this model by a suitable coordinate transformation that we show how to build. Our constructions exploit these coordinate transformations to adapt a navigation function on the model space to its more geometrically complicated (but topologically equivalent) instances. The formula that we present admits sphere-worlds of arbitrary dimension and is directly applicable to configuration spaces whose forbidden regions can be modeled by such generalized discs. We have implemented these navigation functions on planar scenarios, and simulation results are provided throughout the paper. Disciplines Robotics Comments

In this paper, the authors describe the design and control of RHex, a power autonomous, untethered, compliant-legged hexapod robot. RHex has only six actuators-one motor located at each hip-achieving mechanical simplicity that promotes reliable and robust operation in real-world tasks. Empirically stable and highly maneuverable locomotion arises from a very simple clock-driven, openloop tripod gait. The legs rotate full circle, thereby preventing the common problem of toe stubbing in the protraction (swing) phase. An extensive suite of experimental results documents the robot's significant "intrinsic mobility"-the traversal of rugged, broken, and obstacle-ridden ground without any terrain sensing or actively controlled adaptation. RHex achieves fast and robust forward locomotion traveling at speeds up to one body length per second and traversing height variations well exceeding its body clearance.

Planar, underactuated, biped walkers form an important domain of applications for hybrid dynamical systems. This paper presents the design of exponentially stable walking controllers for general planar bipedal systems that have one degree-of-freedom greater than the number of available actuators. The within-step control action creates an attracting invariant set-a two-dimensional zero dynamics submanifold of the full hybrid model-whose restriction dynamics admits a scalar linear time-invariant return map. Exponentially stable periodic orbits of the zero dynamics correspond to exponentially stabilizable orbits of the full model. A convenient parameterization of the hybrid zero dynamics is imposed through the choice of a class of output functions. Parameter optimization is used to tune the hybrid zero dynamics in order to achieve closed-loop, exponentially stable walking with low energy consumption, while meeting natural kinematic and dynamic constraints. The general theory developed in the paper is illustrated on a five link walker, consisting of a torso and two legs with knees. KeywordsBipeds, hybrid systems, Poincaré sections, underactuated system, zero dynamics This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. Abstract-Planar, underactuated, biped walkers form an important domain of applications for hybrid dynamical systems. This paper presents the design of exponentially stable walking controllers for general planar bipedal systems that have one degree-of-freedom greater than the number of available actuators. The within-step control action creates an attracting invariant set-a two-dimensional zero dynamics submanifold of the full hybrid model-whose restriction dynamics admits a scalar linear time-invariant return map. Exponentially stable periodic orbits of the zero dynamics correspond to exponentially stabilizable orbits of the full model. A convenient parameterization of the hybrid zero dynamics is imposed through the choice of a class of output functions. Parameter optimization is used to tune the hybrid zero dynamics in order to achieve closed-loop, exponentially stable walking with low energy consumption, while meeting natural kinematic and dynamic constraints. The general theory developed in the paper is illustrated on a five link walker, consisting of a torso and two legs with knees.

This paper concerns the construction of a class of scalar valued analytic maps on analytic manifolds with boundary. These maps, which we term navigation functions, are constructed on an arbitrary sphere world-a compact connected subset of Euclidean n-space whose boundary is formed from the disjoint union of a finite number of (n − l)-spheres. We show that this class is invariant under composition with analytic diffeomorphisms: our sphere world construction immediately generates a navigation function on all manifolds into which a sphere world is deformable. On the other hand, certain well known results of S. Smale guarantee the existence of smooth navigation functions on any smooth manifold. This suggests that analytic navigation functions exist, as well, on more general analytic manifolds than the deformed sphere worlds we presently consider.

We report on our efforts to develop a sequential robot controllercomposition technique in the context of dexterous "batting" maneuvers. A robot with a flat paddle is required to strike repeatedly at a thrown ball until the ball is brought to rest on the paddle at a specified location. The robot's reachable workspace is blocked by an obstacle that disconnects the free space formed when the ball and paddle remain in contact, forcing the machine to "let go" for a time to bring the ball to the desired state. The controller compositions we create guarantee that a ball introduced in the "safe workspace" remains there and is ultimately brought to the goal. We report on experimental results from an implementation of these formal composition methods, and present descriptive statistics characterizing the experiments.

Examination of total energy shows that the global limit behavior of a dissipative mechanical system is essentially equivalent to that of its constituent gradient vector field. The class of "navigation functions" is introduced and shown to result in "almost global" asymptotic stability for closed loop mechanical control systems upon which a navigation function has been imposed as an artifical potential energy. Two examples from the engineering literature -satellite attitude tracking and robot obstacle avoidance -are provided to demonstrate the utility of these observations. For more information: Kod*Lab Disciplines Electrical and Computer Engineering | Engineering | Systems EngineeringComments

Step length as a function of derived from 2s ϭ 2v / reveals the condition for the onset of swimming for տ 0.6 as s/R Ϸ 1. The solid lines and symbols are for values of 0. 580, 0.590, 0.600, 0.611, 0.616, 0.622, and 0.633.

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