The Optimum Quality Index for the Stability of In-Parallel Planar Platform Devices

The paper investigates primarily the geometrical meaning of the determinant of the Jacobian (det j) of the three connector lines of a planar in-parallel platform device using reciprocity. A remarkably simple result is deduced. The maximum value of det j namely, det j m is simply one-half of the sum of the lengths of the sides of the moving triangular platform. Further, this result is shown to be independent of the location of the fixed pivots in the base. A dimensionless ratio λ = | det j |/det j m is defined as the quality index (0 ≤ λ ≤ 1) and it is proposed here to use it to measure closeness to a singularity. An example which determines the optimal design by comparing different shaped moving platforms having the same det j m is given and demonstrates that the optimal shape is in fact an equilateral triangle.