The Synthesis of Three‐Degree‐of‐Freedom Planar Parallel Mechanisms with Revolute Joints (3‐RRR) for an Optimal Singularity‐Free Workspace

Abstract

In this paper, a method is presented for the synthesis of 3‐RRR planar parallel mechanisms. The method uses a genetic algorithm while considering three different design criteria: the optimization of the mechanism workspace to approach a prescribed workspace, the maximization of the mechanism's dexterity, and the avoidance of singularities inside the mechanism workspace. It is shown that, for a given mechanism, some working modes do not have any corresponding singularity curves located inside the mechanism workspace. Furthermore, a case is presented where, for a given orientation range of the mechanism's end‐effector, there are no parallel singularities located inside the workspace. Finally, two methods are described and compared to deal with the nonuniform units of the mechanism's Jacobian matrix during the dexterity computation. © 2004 Wiley Periodicals, Inc.

Quantitative Dexterous Workspace Comparison of Serial and Parallel Planar Mechanisms

“…Examining the right side of equation (9), and reducing $ 1,i in equation (3), the orthogonal product $ r,i ⊗ $ 1,i may be expressed as: (10) Therefore, writing equation (9) three times corresponding to each of the mechanism's limbs yields the following direct (J x ) and inverse (J q ) Jacobians expressed as: (11) ( 12) The results of J x and J q correspond exactly with those obtained by (Tsai, 1999) through the cross product method and by (Arsenault & Boudreau, 2004) through calculus. The resulting overall Jacobian matrix J = J q −1 J x is a square 3 × 3 matrix.…”

“…As the inverse displacement solution of this manipulator are previously published, no further discussion on the subject will be provided here. The Jacobian formulation provided for this manipulator in (Gosselin, 1988) and (Arsenault & Boudreau, 2004) is developed by differentiating the various inverse displacement equations, with respect to time. In (Tsai, 1999), the Jacobian matrix was obtained through the method of cross-products.…”

“…This provides full knowledge as to the position of the end effector platform and points A i . The solution for each limb's pose may be obtained by completing the inverse displacement solution provided in (Tsai, 1999) or (Arsenault & Boudreau, 2004) where points A i are known. The solution leads to two solutions for each limb.…”

“…The solution leads to two solutions for each limb. In (Arsenault & Boudreau, 2004), these are referred to as working modes. The different solutions correspond to either elbow up or elbow down configurations of each limb.…”

“…Singularity analysis of the 3-RRR manipulator has been explored extensively in (Tsai, 1999;Bonev & Gosselin, 2001;Arsenault & Boudreau, 2004). Essentially two singularities exist for this manipulator.…”