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Research Papers

# The Kinetostatic Conditioning of Two-Limb Schönflies Motion Generators

[+] Author and Article Information
Jean-François Gauthier

Axium Inc., Montreal, QC, H3A 2T5, Canadajean-francois.gauthier@axiumsolutions.com

Jorge Angeles

McGill University, Montreal, QC, H3A 2T5, Canadaangeles@cim.mcgill.ca

Scott B. Nokleby

University of Ontario Institute of Technology, Oshawa, ON, L1H 7K4, Canadascott.nokleby@uoit.ca

Alexei Morozov

JITECH Inc., Pointe Claire, QC, H3A 2T5, Canadaalexei@jitech.ca

Joint axes will be referred to as $Ji$ when associated with angle $θJi$.

One cannot speak of singular Jacobians here because the two Jacobians are rectangular, and hence, none admits a determinant.

The power developed by the motors is identical to that developed by the actuated joints under static conservative conditions; however, the motor torques and the motor rates are derived from their actuated-joint counterparts upon considering the gear-drive transmission ratios.

Although a four-axis force sensor would suffice in the case at hand, such a sensor may not be available on the market for a while.

In fact, it is neither obvious that the right-hand side of Eq. 62 is the transpose of the same side of Eq. 62, but the relation of interest can be found with the aid of the MIL as well.

J. Mechanisms Robotics 1(1), 011010 (Aug 05, 2008) (12 pages) doi:10.1115/1.2960544 History: Received May 24, 2008; Revised June 27, 2008; Published August 05, 2008

## Abstract

This paper introduces a study on the kinetostatic conditioning of two-limb Schönflies motion generators. These are robots capable of producing the motions undergone by the end-effector of what is known as selective-compliance assembly robot arm (SCARA) systems, which can best be described as the motions of the tray of a waiter: three independent translations plus one rotation about an axis of fixed orientation. SCARA systems are usually understood as four-axis serial robots, Schönflies motion generators being a generalization thereof, that encompass first and foremost parallel architectures. Kinetostatic conditioning is understood here in connection with the condition number of each of the two Jacobian matrices of the parallel robot under study. After a brief introduction on the geometry and the kinematics of two-limb parallel systems, the kinetostatics of this class of robots is discussed; whence, the calculation of the kinetostatic conditioning of these robots is undertaken. The motivation behind this work is the need to understand an unstable behavior of the prototype in a substantial part of its workspace, which is attributed to poor conditioning. A main result of this paper is the correlation between the shortest dimension of the robot kinematic chain and the characteristic length, which hints to the need of specifying the range of the characteristic length when optimizing the dimensions of robots of the class studied here, a result that may equally hold for parallel robots in general.

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## Figures

Figure 1

The first McGill SMG prototype

Figure 2

The unlimited-rotation moving platform

Figure 3

Front view of the kinematic chain of the SMG with the two leg-planes coincident with the X-Z plane

Figure 4

Top view of the kinematic chain of the SMG at an arbitrary posture

Figure 5

Zoom-in of top view of the kinematic chain of the SMG at an arbitrary posture

Figure 6

Vector representation of the kinematic chain of the SMG

Figure 7

Two parallel singularity configurations: (a) when ρ̃I3=−ρ̃I3 with θI1=0deg, θII1=180deg, and θI3=θII3=0deg; and (b) when ρ̃I3=ρ̃I3 with θI1=0deg, θII1=180deg, and θI3=θII3=90deg

Figure 8

Serial singularity configurations for Leg I: (a) when vJ=0 and (b) when ΔJ=0

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