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Research Papers

Series Solution for Finite Displacement of Single-Loop Spatial Linkages

[+] Author and Article Information
Paul Milenkovic

 Department of Electrical and Computer Engineering, University of Wisconsin-Madison,1415 Engineering Drive, Madison, Wisconsin 53706phmilenk@wisc.edu

J. Mechanisms Robotics 4(2), 021016 (Apr 25, 2012) (8 pages) doi:10.1115/1.4006193 History: Received August 13, 2011; Revised December 27, 2011; Published April 25, 2012; Online April 25, 2012

The kinematic differential equation for a spatial point trajectory accepts the time-varying instantaneous screw of a rigid body as input, the time-zero coordinates of a point on that rigid body as the initial condition and generates the space curve traced by that point over time as the solution. Applying this equation to multiple points on a rigid body derives the kinematic differential equations for a displacement matrix and for a joint screw. The solution of these differential equations in turn expresses the trajectory over the course of a finite displacement taken by a coordinate frame in the case of the displacement matrix, by a joint axis line in the case of a screw. All of the kinematic differential equations are amenable to solution by power series owing to the expression for the product of two power series. The kinematic solution for finite displacement of a single-loop spatial linkage may, hence, be expressed either in terms of displacement matrices or in terms of screws. Each method determines coefficients for joint rates by a recursive procedure that solves a sequence of linear systems of equations, but that procedure requires only a single factorization of a 6 by 6 matrix for a given initial posture of the linkage. The inverse kinematics of an 8R nonseparable redundant-joint robot, represented by one of the multiple degrees of freedom of a 9R loop, provides a numerical example of the new analytical technique.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Labeling of links and joints in a single-loop linkage

Grahic Jump Location
Figure 2

Two-axis nonsingular robotic wrist

Grahic Jump Location
Figure 3

Nonsingular wrist at maximum deflection: Top: 50  deg actuation of one base joint for a total deflection of 100  deg in yaw. Middle: 50  deg and -50  deg actuation on the two base joints for maximum deflection in yaw and pitch in combination. Bottom: -50  deg of second base joint for a total deflection of -100  deg in pitch.

Grahic Jump Location
Figure 4

Single-loop kinematic chain comprised of a 3-axis robotic arm, 1-axis forearm roll, 2-axis robotic wrist, and 1-axis of commanded tool motion about a workpiece

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