A 6R linkage reconfigurable between the line-symmetric Bricard linkage and the Bennett linkage
Introduction
There are two major linkage families among the various single-loop overconstrained spatial linkages: the Bennett-based one [1] and the Bricard-related one. In the former one, the Bennett linkage [2], [3] is used as the basic construct unit to form different overconstrained 5R and 6R linkages, such as Myard linkages [4], extended Myard linkage [5], Goldberg's 5R and 6R linkages [6], Waldron's hybrid 6R linkages [7], Yu and Baker's syncopated 6R linkage [8], generalised Goldberg 5R linkage and Wohlhart's double-Goldberg 6R linkage [9], mixed double-Goldberg 6R linkages [10] and so on. Linkages in the Bricard-related family usually contain certain symmetric geometry properties to enable mobility [11], [12], including three octahedral cases [13], three linkage cases [14], Altmann's 6R linkage [15], Wohlhart's hybrid 6R linkage [16] and so on. However, there is little interaction between the linkages in Bennett-based linkage family and those in Bricard-related one, except for the isomerization introduced by Wohlhart [17], which sets up the connection between the Wohlhart's double-Goldberg 6R linkage and the line-symmetric Bricard linkage.
Recent development in mechanism and machine design promotes the concept of reconfigurable mechanism, which has emerged into three major categories. The first is based on the re-assembly of identical or similar robotics modules [18], [19], [20], each of which is an integrated system of microprocessors, batteries, sensors, end-effectors, etc. The second is the metamorphic mechanism [21], [22], [23], which can generate different topologies for reconfigurations. The third is to incorporate certain bifurcation behaviours to the existing linkage's kinematic paths [24], [25], [26]. At the transit configurations, kinematotropy mechanism [25], [27] can change its global mobility. The mechanism reported by Kong and Huang [24] can change between two operation modes. And the multifunctional 7R mechanism can function as two different types of overconstrained linkages [26]. A comprehensive review about the current development, principles and strategies of the reconfigurable mechanisms was discussed in [28]. The potential of reconfiguration can be identified when two or more subgroups are involved in the construct of the mechanism [29]. The bifurcation behaviour of the double-subtractive-Goldberg 6R linkage was analysed in [30], where multiple linkage closures were found using both construct method and non-construct method. As the result, this Goldberg 6R linkage could be reconfigured among four different 6R linkages, whose kinematic curves form a closed loop through four bifurcation points. Recently, the Wohlhart's double-Goldberg 6R linkage was analysed and the operation form of a 4R linkage is successfully introduced to the linkage's bifurcation paths [31]. Here, the effort is made to construct reconfigurable mechanism under the third category.
In this paper, two spatial triangles and Bennett linkages are used as the basic elements to construct the 6R linkages with different reconfiguration capabilities. In Section 2, the kinematics of the spatial triangle, the Bennett linkage and the general line-symmetric Bricard linkage are introduced. 3 The asymmetric 6, 4 The line-symmetric 6 demonstrate the construct process of an asymmetric 6R linkage and a line-symmetric 6R linkage, as well as analyse their kinematic bifurcation for reconfiguration. Final conclusions are drawn in Section 5.
Section snippets
The spatial triangle structure
The spatial triangle is a single-loop structure enclosed by three spatial links and connected by three revolute joints. It was initially introduced and analysed by Yang [32] using dual quaternion method. Later, Mavroidis and Roth [33] studied the kinematics of the spatial polygons using matrix method and extended the research into screw polygons. And screw triangles were applied to unify the finite and infinitesimal kinematics [34]. The importance of spatial triangle was recently revisited by
The asymmetric 6R linkage
Two identical spatial triangles 123 and 456 are prepared for construction in Fig. 3, whose geometry conditions are set up as follows.
Take spatial triangle 123 for example, the geometry conditions on links 12 and 23, i.e. a12ST, α12ST, a23ST, α23ST, θ2ST and R2ST are pre-defined design parameters. The rest of the parameters related to link 31 can be derived with Eq.
The line-symmetric 6R linkage
As illustrated in Fig. 15, two identical spatial triangles are the same as those in Fig. 3, and the intermediate bridge is a Bennett linkage in line-symmetric joint-axis setup, whose geometry conditions are
In Fig. 16, after removing the common links and joints in dash lines, the rest form the second 6R linkage. Its geometry conditions are
Conclusion
In this paper, the feasibility of reconfiguration between the general line-symmetric Bricard linkage and the Bennett linkage is explored base on their symmetry property. By using spatial triangles and Bennett linkages in different symmetry joint-axis setups as the building blocks, two 6R linkages have been constructed.
The first 6R linkage is achieved by connecting two identical spatial triangles with a Bennett linkage in asymmetric joint-axis setup, and then removing the redundant links inside
Acknowledgement
C.-Y. Song would like to thank NTU for providing the University Graduate Scholarship during his PhD study. This work is financially supported by the Natural Science Foundation of China (Projects No. 51275334 and No. 51290293).
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