Path Generators

Designing a path generating linkage means designing a linkage to guide a point path in some particular manner.  We specifically have worked on designing six-bars.  Much work has been completed on four-bar path generators, but it wasn’t until 1992 that Wampler et al. [1] found a complete solution set of 4,326 to the general 9-point problem. We generalized their work to six-bars which can achieve as many as 15 points, however, the number of solutions to the design equations greatly increased. The smallest degree computed thus far is 2.28×1046.

This work diverges from solving the most general case and instead focuses on inverting six-bar function generators into six-bar path generators. How this works is by reformulating the problem as the control of a 2R chain. A 2R chain can be specified and its end effector can be moved to any point within its workspace where its joint angles can be calculated. Moving the the end effector through 11 points provides 11 joint angle pairs which can be coordinated by six-bar function generators. Attaching these six-bar function generators onto the 2R chains is equivalent to kinematic inversion. Specifically, we focus on Stephenson linkages due to their ability to create 11 point function generators. There are four ways to invert Stephenson function generators into Stephenson path generators which appears in the figures below.

Stephenson III function generator inverted to a Stephenson I path generator

Stephenson III function generator inverted to a Stephenson II path generator

Stephenson II function generator inverted to a Stephenson II path generator

Stephenson II function generator inverted to a Stephenson III path generator

This work makes use of the large parameter homotopy solution sets that we calculated during research on six-bar function generators. The kinematic requirements for path generation are formulated as a function generation problem and then solved as such. Solutions to function generator synthesis equations are then inverted back to path generation. This synthesis technique was used for the design of six-bar walking linkages.

[1] C. W. Wampler, A. J. Sommese, and A. P. Morgan, 1992. “Complete solution of the nine-point path synthesis problem for four-bar linkages,” Journal of Mechanical Design 114(1):153-159.

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