A coordinate-free approach to instantaneous kinematics of two rigid objects with rolling contact and its implications for trajectory planning

This paper adopts a coordinate-free approach to investigate the kinematics of rigid bodies with rolling contact. A new equation of angular velocity of the moving body is derived in terms of the magnitude of rolling velocity and two sets of geometric invariants belonging to the respective contact curves. This new formulation can be differentiated up to any order. Furthermore, qualitative information about trajectory planning can be deduced from this equation if the characteristics of rolling objects and the motion are taken into consideration

Configuration Controllability of Simple Mechanical Control Systems

In this paper we present a definition of 'configuration controllability' for mechanical systems whose Lagrangian is kinetic energy with respect to a Riemannian metric minus potential energy. A computable test for this new version of controllability is derived. This condition involves an object that we call the symmetric product. Of particular interest is a definition of 'equilibrium controllability' for which we are able to derive computable sufficient conditions. Examples illustrate the theory

Robotic Contact Juggling

We define "robotic contact juggling" to be the purposeful control of the motion of a three-dimensional smooth object as it rolls freely on a motion-controlled robot manipulator, or "hand." While specific examples of robotic contact juggling have been studied before, in this paper we provide the first general formulation and solution method for the case of an arbitrary smooth object in single-point rolling contact on an arbitrary smooth hand. Our formulation splits the problem into four subproblems: (1) deriving the second-order rolling kinematics; (2) deriving the three-dimensional rolling dynamics; (3) planning rolling motions that satisfy the rolling dynamics; and (4) feedback stabilization of planned rolling trajectories. The theoretical results are demonstrated in simulation and experiment using feedback from a high-speed vision system.Comment: 16 pages, 14 figures. | Supplemental Video: https://youtu.be/QT55_Q1ePfg | Code: https://github.com/zackwoodruff/rolling_dynamic

On the Experiments about the Nonprehensile Reconfiguration of a Rolling Sphere on a Plate

A method to reconfigure in a nonprehensile way the pose (position and orientation) of a sphere rolling on a plate is proposed in this letter. The nonholonomic nature of the task is first solved at a planning level, where a geometric technique is employed to derive a Cartesian path to steer the sphere towards the arbitrarily desired pose. Then, an integral passivity-based control is designed to track the planned trajectory. The port-Hamiltonian formalism is employed to model the whole dynamics. Two approaches to move the plate are addressed in this paper, showing that only one of them allows the full controllability of the system. A humanoid-like robot is employed to bolster the proposed method experimentally

Passive force closure and its computation in compliant-rigid grasps

The classical notion of force closure is formulated for multifingered hands, where the fingers actively apply any desired force consistent with friction constraints at the contacts. This paper considers a simpler notion of passive force closure, where each finger obeys some force-displacement law that depends on the finger's joint parameters. The fingers apply initial preload grasping forces, and the grasped object is stabilized against external disturbances by the automatic response of the grasping fingers. After motivating the usefulness of passive force closure, we characterize the conditions for its existence. Then we introduce the passive stability set, defined as the collection of external wrenches that can be passively resisted by a given grasp. We introduce a class of grasp arrangements where the grasping mechanism is compliant while the grasped object is rigid. Such compliant-rigid systems are common, and for these systems the passive closure set can be computed in closed form. Simulation results demonstrate the computation of the passive closure set for two and three-finger planar grasps

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

Computational Modeling, Visualization, and Control of 2-D and 3-D Grasping under Rolling Contacts

This chapter presents a computational methodology for modeling 2-dimensional grasping of a 2-D object by a pair of multi-joint robot fingers under rolling contact constraints. Rolling contact constraints are expressed in a geometric interpretation of motion expressed with the aid of arclength parameters of the fingertips and object contours with an arbitrary geometry. Motions of grasping and object manipulation are expressed by orbits that are a solution to the Euler-Lagrange equation of motion of the fingers/object system together with a set of first-order differential equations that update arclength parameters. This methodology is then extended to mathematical modeling of 3-dimensional grasping of an object with an arbitrary shape. Based upon the mathematical model of 2-D grasping, a computational scheme for construction of numerical simulators of motion under rolling contacts with an arbitrary geometry is presented, together with preliminary simulation results. The chapter is composed of the following three parts. Part 1 Modeling and Control of 2-D Grasping under Rolling Contacts between Arbitrary Smooth Contours Authors: S. Arimoto and M. Yoshida Part 2 Simulation of 2-D Grasping under Physical Interaction of Rolling between Arbitrary Smooth Contour Curves Authors: M. Yoshida and S. Arimoto Part 3 Modeling of 3-D Grasping under Rolling Contacts between Arbitrary Smooth Surfaces Authors: S. Arimoto, M. Sekimoto, and M. Yoshid