Mobility of bodies in contact. II. How forces are generated bycurvature effects

For part I, see ibid., p.696-708. The paper considers how forces are produced by compliance and surface curvature effects in systems where an object a is kinematically immobilized to second-order by finger bodies Al,...,Ak. A class of configuration-space based elastic deformation models is introduced. Using these elastic deformation models, it is shown that any object which is kinematically immobilized to first or second-order is also dynamically locally asymptotically stable with respect to perturbations. Moreover, it is shown that for preloaded grasps kinematic immobility implies that the stiffness matrix of the grasp is positive definite. The stability result provides physical justification for using second-order effects for purposes of immobilization in practical applications. Simulations illustrate the concepts

Mobility of bodies in contact. I. A 2nd-order mobility index formultiple-finger grasps

Using a configuration-space approach, the paper develops a 2nd-order mobility theory for rigid bodies in contact. A major component of this theory is a coordinate invariant 2nd-order mobility index for a body, B, in frictionless contact with finger bodies A1,...A k. The index is an integer that captures the inherent mobility of B in an equilibrium grasp due to second order, or surface curvature, effects. It differentiates between grasps which are deemed equivalent by classical 1st-order theories, but are physically different. We further show that 2nd-order effects can be used to lower the effective mobility of a grasped object, and discuss implications of this result for achieving new lower bounds on the number of contacting finger bodies needed to immobilize an object. Physical interpretation and stability analysis of 2nd-order effects are taken up in the companion pape

The Stability of Heavy Objects with Multiple Contacts

In both robot grasping and robot locomotion, we wish to hold objects stably in the presence of gravity. We present a derivation of second-order stability conditions for a supported heavy object, employing the tool of Stratified Morse theory. We then apply these general results to the case of objects in the plane

Constructing minimum deflection fixture arrangements using frame invariant norms

This paper describes a fixture planning method that minimizes object deflection under external loads. The method takes into account the natural compliance of the contacting bodies and applies to two-dimensional and three-dimensional quasirigid bodies. The fixturing method is based on a quality measure that characterizes the deflection of a fixtured object in response to unit magnitude wrenches. The object deflection measure is defined in terms of frame-invariant rigid body velocity and wrench norms and is therefore frame invariant. The object deflection measure is applied to the planning of optimal fixture arrangements of polygonal objects. We describe minimum-deflection fixturing algorithms for these objects, and make qualitative observations on the optimal arrangements generated by the algorithms. Concrete examples illustrate the minimum deflection fixturing method. Note to Practitioners-During fixturing, a workpiece needs to not only be stable against external perturbations, but must also stay within a specified tolerance in response to machining or assembly forces. This paper describes a fixture planning approach that minimizes object deflection under applied work loads. The paper describes how to take local material deformation effects into account, using a generic quasirigid contact model. Practical algorithms that compute the optimal fixturing arrangements of polygonal workpieces are described and examples are then presented

A stiffness-based quality measure for compliant grasps and fixtures

This paper presents a systematic approach to quantifying the effectiveness of compliant grasps and fixtures of an object. The approach is physically motivated and applies to the grasping of two- and three-dimensional objects by any number of fingers. The approach is based on a characterization of the frame-invariant features of a grasp or fixture stiffness matrix. In particular, we define a set of frame-invariant characteristic stiffness parameters, and provide physical and geometric interpretation for these parameters. Using a physically meaningful scheme to make the rotational and translational stiffness parameters comparable, we define a frame-invariant quality measure, which we call the stiffness quality measure. An example of a frictional grasp illustrates the effectiveness of the quality measure. We then consider the optimal grasping of frictionless polygonal objects by three and four fingers. Such frictionless grasps are useful in high-load fixturing applications, and their relative simplicity allows an efficient computation of the globally optimal finger arrangement. We compute the optimal finger arrangement in several examples, and use these examples to discuss properties that characterize the stiffness quality measure

Experiments in fixturing mechanics

This paper describes an experimental fixturing system wherein fixel reaction forces, workpiece loading, and workpiece displacements are measured during simulated fixturing operations. The system's configuration, its measurement principles, and tests to characterize its performance are summarized. This system is used to experimentally determine the relationship between workpiece displacement and variations in fixed preload force or workpiece loading. We compare the results against standard theories, and conclude that commonly used linear spring models do not accurately predict workpiece displacements, while a non-linear compliance model provides better predictive behavior

Towards planning with force constraints: on the mobility of bodies in contact

A configuration-space-based approach for analyzing the interactions and mobility of objects in quasi-static contact is described. This analysis is motivated by a class of articulated robot motion-planning problems that are not handled by current planning systems. Examples are a snake-like robot that crawls inside a tunnel by bracing against the tunnel walls, a limbed robot (analogous to a monkey) that climbs a trussed structure by pushing and pulling, and a dextrous robotic hand that moves its fingers along an object while holding it stationary. In these examples, one must plan the robot motion to satisfy high-level goals while maintaining quasistatic stability. Planning the hand-hold states (analogous to the hand-holds used by rock climbers between dynamically moving states) where the robot mechanism is at a static equilibrium is considered primarily. The results obtained are some of the first steps necessary to develop planning paradigms for this class of problems. Although the authors have the general class of quasistatic planning problems in mind, the authors use the language of grasping for discussion

The Construction of Analytic Diffeomorphisms for Exact Robot Navigation on Star Worlds

A Euclidean Sphere World is a compact connected submanifold of Euclidean n-space whose boundary is the disjoint union of a finite number of (n — 1) dimensional Euclidean spheres. A Star World is a homeomorph of a Euclidean Sphere World, each of whose boundary components forms the boundary of a star shaped set. We construct a family of analytic diffeomorphisms from any analytic Star World to an appropriate Euclidean Sphere World model. Since our construction is expressed in closed form using elementary algebraic operations, the family is effectively computable. The need for such a family of diffeomorphisms arises in the setting of robot navigation and control. We conclude by mentioning a topological classification problem whose resolution is critical to the eventual practicability of these results

New bounds on the number of frictionless fingers required to immobilize 2D objects

This paper develops new lower bounds on the number of frictionless fingers or fixtures which are required to immobilize planar objects. We study in detail the case of objects with smooth boundaries and polygonal objects. Analogous results for the case of piecewise smooth objects follow directly from the analysis presented herein. These results have obvious applications to fixture planning and grasp planning, as we show that it is possible to immobilize objects with fewer fingers than was previously thought possible