Geometric Analysis of the Kinematic Sensitivity of Planar Parallel Mechanisms

Abstract

The kinematic sensitivity is a unit-consistent measure that has been recently proposed as a mechanism performance index to compare robot architectures. This paper presents a robust geometric approach for computing this index for the case of planar parallel mechanisms. The physical meaning of the kinematic sensitivity is investigated through different combinations of the Euclidean and infinity norms and by means of several illustrative examples. Finally, this paper opens some avenues to the dimensional synthesis of parallel mechanisms by exploring the meaning of the global kinematic sensitivity index.

PPCOF: A Robust Proportion-Preserving Composite Objective Function for Scale-Invariant Multi-Objective Optimization

“…More clearly, if the solution found upon utilizing the PPCOF is not dominated by any of the ones obtained by implementing Pareto-solutions-based MOO on the same case study, and is preferable, or at least close, to them, then it means that the PPCOF has been devised, such that it could accomplish its missions that were sought from the outset. The relations and concepts required for dealing with the case study are explained in detail in studies such as [51]. Nevertheless, for quick reference, they are brie y recalled in what follows.…”

“…From the study conducted in [55], it can be inferred that, geometrically, kinematic sensitivity is the maximum displacement or rotation of the moving platform under a unit-norm change in the joint space, which, for a PPM, can be formulated as follows: Two types of norms, namely 2-and 1-norm, are frequently used in practice, since a geometric interpretation could be associated with them, which leads to four possible combinations of formulating the kinematic sensitivity, where [35,36] report that c = 1…”

“…(18), constraint k _ k c 1 for c = 1 can be rewritten as follows: kKxk 1 1; (19) which geometrically represents a polyhedron spanned by 2 n corners [35,36,55]. Furthermore, x = [x y ] T indicates the pose of the moving platform in the Cartesian space.…”

“…n = 3, which means that the polygon has eight corners, thus one should consider four, namely, (x i ; y i ), i = 1; : : : ; 4. Then, the kinematic sensitivity indices can be formulated as follows [34,35] Given the fact that the above parameters always take nonnegative values, they could be con ned within the interval from zero to one by taking advantage of the following substitutions:…”

“…Noticing that, as aforementioned, in most practical contexts, di erent objective functions are reported in non-homogeneous units or with di erent scales, using the foregoing strategy, the e ect of these complications is overcome, and the composite objective function is made robust against the changes of the scales by which the objective functions are represented, and each of them plays a role as signi cant as the others in guiding the algorithm through the search process. Besides, the practical reliability of the PPCOF is demonstrated via applying it to a use-case, namely, the MOO of the point-displacement and rotational kinematic sensitivity [34,35] and workspace of eight Planar Parallel Mechanisms (PPMs), and comparing its e ciency with that of NSGA-II, which has been implemented on the same case-study in [36].…”