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

Abstract

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 jm 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 jm 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 jm is given and demonstrates that the optimal shape is in fact an equilateral triangle

Rotation Symmetry Axes and the Quality Index in a 3D Octahedral Parallel Robot Manipulator System

“…): 1 $ = aA , 2 $ = dD , 3 $ = bB , 4 $ = eE , 5 $ = cC and 6 $ = fF . In this paper the quality index Hunt, 1998, Lee, Duffy andKeler, 1999) is defined in a new form, without using absolute values, as:…”

“…(7) takes into account the sign of the determinant of the Jacobian matrix J , and that is why it is bounded between -1 and 1, which (depending on the particular configuration) may differ in sign from the quality index defined by Hunt, 1998, andKeler, 1999. A zero value of the quality index indicates singularities and the value 1 (or -1) for the quality index determines the best configuration of the robot.…”

“…Several kinematic perfomance indices have been introduced (Yoshikawa, 1985, Angeles and Lopez-Cajun, 1992, Lee, Duffy and Hunt, 1998, Lee, Duffy and Keler, 1999. In particular the so-called quality index Hunt, 1998, Lee, Duffy andKeler, 1999) has been used widely. The values of the index are usually plotted for a range of movements of the platform within the workspace.…”

New performance indices for 6-dof UPS and 3-dof RPR parallel manipulators

“…Therefore there exists no local maximum over the definition interval and S K has no optimal configuration with respect to this index. If S K is evaluated by the index of Lee and Duffy [7], we get a one parametric set of optimal configurations (see Fig. 3 (b)).…”

“…But the computation of |det(J)| + is a nonlinear task and was only done for manipulators with planar base and platform with very special geometries. Only for these manipulators |det(J)| + can be interpreted geometrically as the volume of the framework (see [7,9]). Moreover this approach was applied to 3-dof RPR manipulators in [8] and extended to redundant 6-dof UPS manipulators in [21].…”