Complete Solution of the Nine-Point Path Synthesis Problem for Four-Bar Linkages

Abstract

The problem of finding all four-bar linkages whose coupler curve passes through nine prescribed points has been a longstanding unsolved problem in kinematics. Using a combination of classical elimination, multihomogeneous variables, and numerical polynomial continuation, we show that there are generically 1442 nondegenerate solution along with their Roberts cognates, for a total of 4326 distinct solutions. Moreover, a computer algorithm that computes all solutions for any given nine points has been developed.

Comparison of probabilistic algorithms for analyzing the components of an affine algebraic variety

“…Each equation in the system is degree seven in the variables q, so the n-point problem has total degree 7 (n−1) . From [WMS92], for n = 9, we expect 8652 isolated roots that appear in a six-way symmetry group, meaning that the solutions appear in 1442 distinct orbits. Notice that 8652 is much smaller than the total degree of 7 8 , so the problem has a very special structure compared to systems of general seventhdegree equations.…”

“…The maximum number of general points that can be interpolated exactly is nine, as this equals the number of design parameters in a four-bar linkage. The associated nine-point problem was first posed by Alt in 1923 [Alt23] and complete solution lists (for various sets of nine given points) were first computed in 1992 [WMS92,WMS97]. One can also consider interpolating fewer than nine points, in which case the solution sets are positive-dimensional, giving the designer the ability to choose a design that satisfies other concerns that may affect the suitability of the four-bar.…”

Design of Stephenson linkages that guide a point along a specified trajectory

“…Roth and Freudenstein (1963) presented a numerical solution to the problem 2 of designing a four-bar linkage that reached nine task points on a trajectory, also see Wampler et al (1992). Hain (1967) described graphical methods for point path synthesis using six-bar linkages.…”

Multi-loop Position Analysis via Iterated Linear Programming

“…This method has been responsible for the first solutions of many long-standing problems in Kinematics. For example, using them, Tsai and Morgan first showed that the inverse kinematics of the general 6R manipulator has sixteen solutions [22], Raghavan showed that the direct kinematics of the general Stewart-Gough platform can have forty solutions [23], and Wampler et al solved nine-point path synthesis problems for four-bar linkages [24].…”