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Research Papers

Generation of Under-Actuated Manipulators With Nonholonomic Joints From Ordinary Manipulators

[+] Author and Article Information
Patrick Grosch

Institut de Robòtica i Informàtica Industrial (CSIC-UPC), Llorens Artigas 4-6, 2 planta, 08028 Barcelona, Spainpgrosch@iri.upc.edu

Raffaele Di Gregorio

Department of Engineering, University of Ferrara, Via Saragat 1, 44100 Ferrara, Italyrdigregorio@ing.unife.it

Federico Thomas

Institut de Robòtica i Informàtica Industrial (CSIC-UPC), Llorens Artigas 4-6, 2 planta, 08028 Barcelona, Spainfthomas@iri.upc.edu

The degrees-of-freedom (DOF) of the configuration space, also called configuration (or finite) DOF (5), are the minimum number of geometric parameters necessary to uniquely identify the configuration of the mechanical system (6). They may be different from the instantaneous DOF, also called velocity DOF (5), of the same mechanical system.

In general, two rollers whose axes locate with the sphere center two different planes constrain the sphere to rotate around the intersection line between the two planes, whereas three rollers whose axes locate with the sphere center three different planes lock the sphere.

The presence of nonholonomic constraints does not change the configuration DOF (5-6). It only affects the instantaneous DOF of the mechanism. Hereafter, the acronym dof used alone will mean configuration DOF.

Each U joint is constituted by two revolute pairs: one adjacent to the end effector, and the other not adjacent to the end effector.

System 11 does not model the mobility limitations due to the physical constitution of the real joints and to the real sizes of the links. Such limitations bound the workspace and, when correctly modeled, yield type-I singularities.

The “Lie algebra” of a set of vector fields is the linear span of all Lie products of all degrees of vector fields belonging to that set (27).

It is worth noting that, if the dimension of $span(v1,v2,v3,[v1,v2],[v1,v3],[v2,v3])$ is six, all the Lie products of any degree in ${v1,v2,v3}$ must belong to $span(v1,v2,v3,[v1,v2],[v1,v3],[v2,v3])$; thus, all the reachable configurations $x$ satisfy the condition $(x−x0)∊Span(v1,v2,v3,[v1,v2],[v1,v3],[v2,v3])$.

J. Mechanisms Robotics 2(1), 011005 (Nov 19, 2009) (8 pages) doi:10.1115/1.4000527 History: Received February 28, 2009; Revised August 23, 2009; Published November 19, 2009

Abstract

This paper shows how to generate underactuated manipulators by substituting nonholonomic spherical pairs for (holonomic) spherical pairs in ordinary (i.e. not underactuated) manipulators. As a case study, an underactuated manipulator, previously proposed by one of the authors, is demonstrated to be generated, through this pair substitution from an inversion of the 6-3 fully parallel manipulator. Moreover, the kinetostatic analysis of this underactuated manipulator is reconsidered, and a simple and compact formulation is obtained. The results of this kinetostatic analysis can be used both in the design of the underactuated manipulator and in its control.

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Figures

Figure 1

3D CAD model of one out of many different manufacturing schemes for the nS pair: (1) sphere, (2) roller (carried out through a roller bearing), (3) spherical bearing (there are three spherical bearings (only two are visible) that forbid the translation of the sphere), (4) preload adjust screw, and (5) parallelism adjust screws

Figure 2

6-3 FPM

Figure 3

UPnS limb (right) generated by substituting an nS pair for the S pair in the two UPS limbs with coalesced S pairs (left)

Figure 4

Underactuated manipulator with topology 3-nSPU: (a) kinematic model, (b) 3D CAD model of a manufacturing scheme

Figure 5

i-th limb of type nSPU: notations

Figure 6

i-th limb of type nSPU: free-body diagram

Figure 7

Singularity locus (Eq. 26) for the 3-nSPU geometry of the numerical example and end-effector orientation fixed to XZX Euler angles’ values (0, 1, 0) radians: (a) 3D view and (b) top view

Figure 8

Locus of the Eq. 35 roots for the 3-nSPU geometry of the numerical example and end-effector orientation fixed to XZX Euler angles’ values (0, 1, 0) radians: (a) 3D view and (b) top view

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