Branch Reconfiguration of Bricard Linkages Based on Toroids Intersections: Plane-Symmetric Case

Abstract

This paper for the first time reveals a set of special plane-symmetric Bricard linkages with various branches of reconfiguration by means of intersection of two generating toroids, and presents a complete theory of the branch reconfiguration of the Bricard plane-symmetric linkages. An analysis of the intersection of these two toroids reveals the presence of coincident conical singularities, which lead to design of the plane-symmetric linkages that evolve to spherical 4R linkages. By examining the tangents to the curves of intersection at the conical singularities, it is found that the linkage can be reconfigured between the two possible branches of spherical 4R motion without disassembling it and without requiring the usual special configuration connecting the branches. The study of tangent intersections between concentric singular toroids also reveals the presence of isolated points in the intersection, which suggests that some linkages satisfying the Bricard plane-symmetry conditions are actually structures with zero finite degrees-of-freedom (DOF) but with higher instantaneous mobility. This paper is the second part of a paper published in parallel by the authors in which the method is applied to the line-symmetric case.

A synthesis method for 1-DOF mechanisms with a cusp in the configuration space

“…In this paper, this idea was extended to the spatial case in order to design spatial 1-DOF cusp mechanisms. It was found that a simple way of dealing with curves traced by spatial mechanisms is the use of intersection of surfaces generated [26,28,29,44] by kinematic dyads. Using this method, three examples were presented.…”

“…Therefore, in this paper a method for the design of spatial 1-DOF mechanisms with cusp singularities is presented as a means to generate test cases facilitating such research. This method is based in the intersection of surfaces [26][27][28][29]. In addition, a planar mechanism with a higher-order cusp singularity is presented using the same idea applied by Connelly and Servatius [8] in the design of the double-Watt mechanism.…”

“…A special line-and plane-symmetric Bricard mechanism presenting the same reconfiguration to a spherical 4-bar linkage was presented in [6]. The line-symmetric Bricard linkages obtained as the intersection of two toroids was first analyzed in [29].…”

“…It is important to mention, that since P 1 ∈ sing(S A 1 ) ∩ sing(S B 1 ), E 1 is unable to move from one branch to another at P 1 , see [29] for an in-depth analysis of this singularity. Therefore, unlike in the Koenigs joint, in this Bricard mechanism the two branches of the configuration space, V , related to the two branches of C 1 intersecting at P 1 are not intersecting when E 1 = P 1 and, thus,…”

Kinematics and Constraints of the Exechon Robot Accounting Offsets Due to Errors in the Base Joint Axes

“…Point A 2b which in the ideal Exechon robot is fixed and coincident with A 2a is now moving in a toroid S [22,23,24] as shown in figure 7. The common perpendicular between axes S 21 and S 22 intersects the later at point A 2c .…”