Kinematics, Polynomials, and Computers—A Brief History

Abstract

Journal of Mechanisms and Robotics Editorial Kinematics, Polynomials, and Computers—A Brief History As we move into the adolescent years of the 21st century, allow me to discuss where research in mechanisms and robotics has been as a prelude to considering where it is going. Polynomials Mechanisms have been characterized by the curves that they trace since the time of Archimedes 关1兴. In the 1800s, Reuleaux, Kennedy, and Burmester formalized this by applying the descrip- tive geometry of Gaspard Monge to the analysis and synthesis of machines 关2兴. Watt invented a straight-line linkage to convert the linear expansion of steam into the rotation of the great beam, making the steam engine practical 共Fig. 1兲, and captured the imagination of the mathematician Chebyshev, who introduced the mathematical analysis and synthesis of linkages. About the same time, Sylvester, who introduced the Sylvester resultant for the solution of polynomial equations, went on to lecture about the importance of the Peaucellier linkage, which generates a pure lin- ear movement from a rotating link 关3兴. Influenced by Sylvester, Kempe developed a method for designing a linkage that traces a given algebraic curve 关4兴 that even now inspires research at the intersection of geometry 关5,6兴 and computation. In the mid-1950s, Denavit and Hartenberg introduced a matrix formulation of the loop equations of a mechanism to obtain poly- nomials that defined its movement 关7兴. During a speech in 1972, Freudenstein famously used the phrase “Mount Everest of kine- matics” to describe the solution of these polynomials for the 7R spatial linkage 关8兴. In this context, the “solution” is not a single root but an algorithm that yields all of the roots of the polynomial system, which in turn defines all of the configurations of the link- age for a given input. It was immediately recognized that the 7R analysis problem was equivalent to solving the inverse kinematics for a general robot manipulator to obtain the configurations that are available to pick up an object. By the end of the 1970s, Duffy 关9兴 formulated an efficient set of equations for this problem, but it was not until the late 1980s when the degree 16 polynomial that yields the 16 robot configurations was obtained by Lee and Liang 关10兴. By the mid-1990s, computer algebra and sparse resultant tech- niques were the most advanced tools for formulating and solving increasingly complex arrays of polynomials obtained in the study of mechanisms and robotics systems 关11,12兴. In 1996, Husty used computer algebra to reduce eight quadratic equations in eight soma coordinates that locate the end-effector of a general six- legged Stewart platform to a degree 40 polynomial 关13兴, which allowed the calculation of the 40 configurations of the system. Computers In 1959, Freudenstein and Sandor 关14兴 used the newly devel- oped digital computer and the loop equations of a linkage to de- termine its dimensions, initiating the computer-aided design of mechanisms. Within 2 decades, the computer solution of the equa- Fig. 1 Watt’s linkage transforms the rotational motion of the great beam into the linear motion of the cylinder Journal of Mechanisms and Robotics Copyright © 2011 by ASME FEBRUARY 2011, Vol. 3 / 010201-1 Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 08/18/2015 Terms of Use: http://www.asme.org/about-asme/terms-of-use