SUMMARYThis paper presents the computation of the safe working zone (SWZ) of a parallel manipulator having three degrees of freedom. The SWZ is defined as a continuous subset of the workspace, wherein the manipulator does not suffer any singularity, and is also free from the issues of link interference and physical limits on its joints. The proposed theory is illustrated via application to two parallel manipulators: a planar 3-R̲RR manipulator and a spatial manipulator, namely, MaPaMan-I. It is also shown how the analyses can be applied to any parallel manipulator having three degrees of freedom, planar or spatial.
It is natural to employ an optimization algorithm for the approximate kinematic synthesis of linkages. The hope is to find some superior points in the design space that indicate dimensions that are practically useful. One way to achieve this is to find all minima of an objective, then to filter them so the best remain. However, the prospect of finding all minima is bleak unless the optimization problem at hand is particularly small. In this work, we show how to find nearly all minima for a large optimization problem using polynomial homotopy continuation in the approximate synthesis of a four-bar path generator. The system at hand has a Bézout bound of 543,848,665 and a Schnabel estimate to the maximum number of stationary points of 6 · (303,249 ± 713), within a 95% confidence interval. At least with regards to mechanism synthesis, this work represents the largest scale deployment to date of homotopy continuation to solve an unconstrained optimization problem. The challenges of scaling and suggestions for design are given. Example usage for the design of a leg mechanism is given. On the mechanism design front, this is the first presentation of a nearly complete (within the limitations of numerical discernment) solution of the general four-bar optimal path synthesis problem.
The kinematic synthesis equations of fairly simple planar linkage topologies are vastly nonlinear. This indicates that a large number of solutions exist, and hence a large number of design candidates might be present. Recent algorithms based in polynomial homotopy continuation have enabled the computation of entire solution sets that were previously not possible. These algorithms are based on a technique that stochastically accumulates finite roots and guarantees the exclusion of infinite roots. Here we apply the Cyclic Coefficient Parameter Continuation (CCPC) method to obtain for the first time the complete solution of a Stephenson III six-bar that traces a path and coordinates the angle of its input link along that path. Linkages of this type, called timed curve generators, are particularly useful for controlling the motion of an end effector point and influencing its transmission properties from a rotary input. For a numerically general version of the synthesis equations, we computed an approximately complete set of 1,017,708 solutions that divides into subsets of four according to the Stephenson III cognate structure. This numerically generic solution set essentially represents a design tool. It can be used in conjunction with a parameter homotopy to efficiently obtain all isolated roots of other systems of this same structure that correspond to a specific synthesis task. This is demonstrated with two example synthesis tasks.
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