Use of the Jacobian to Verify Smooth Movement in Watt I and Stephenson I Six-Bar Linkages

Abstract

In this paper we present an algorithm applicable to the Watt I and Stephenson I six-bar linkage designs which determines if a candidate linkage moves smoothly through a desired range of input angles. Intended for use with a synthesis routine our algorithm uses the Jacobian of the linkage to determine if a linkage moves smoothly by identifying the continuous existence of a desired branch of a single circuit for all input angles within a bounded range. With the constraint that the input angle must be contained within the four-bar loop, the determinant of the Jacobian is factored into components that represent the individual linkage loops. The algorithm starts with a linkage of a known configuration which reaches one of the desired task positions to establish the set of signs for these determinants and this set is tracked for consistency throughout the bounded range of input angles. Linkages with defects that exist in a narrow range of input angles are addressed by numerically identifying the input angles which correspond to the minimal value of the Jacobian determinants. We verify that at these input angles the linkage remains on the same branch of the same circuit. A Watt I and Stephenson I six-bar linkage that passes this test will move smoothly through the desired range of input angles. Examples using Mathematica demonstrate the application of the algorithm on both the Watt I and Stephenson I linkage types.

Synthesis of Useful Eight-Bar Linkages as Constrained 6R Loops

“…The design procedure uses an analysis algorithm described in Parrish and McCarthy [15,16] for forward kinematics of multiloop linkage systems. The algorithm uses an automated loop generation technique based on Graph theory.…”

“…For that we use the procedure developed by Parrish et al(2013) [15,16]. Plecnik and McCarthy (2012) [17] demonstrated the effectiveness of random variation of the task positions within tolerance zones in order to increase the number of candidate linkages, for evaluation in the design of a 5SS spatial platform.…”