Backlash reduction using base proximal actuation redundancy for 3-RRR and 3-RPR planar parallel manipulators
Date
2012-12-24
Authors
Mao, Xu
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Abstract
The goal of the research of this Dissertation is using actuation redundancy to reduce backlash in parallel manipulators (PMs.) Initially, 3-RRR and 3-RPR PM layouts where 3 is the number of branches, R is a revolute joint and P is a prismatic joint, are introduced. Actuated joints will later be underlined in the PM desciptions. A method for determining PM working area for rotated payload platforms, based on a mechanism inversion, is presented.
Force solutions for non-redundantly actuated 3-RRR, 3-RRR, 3-RPR and 3-RPR PMs are formulated in terms of screw coordinates. The reciprocal product of screw coordinates is demonstrated to be invarient under changes in reference location and orientation. As examples, the PMs execute basic circle, logarithmic spiral and arc displacement and force trajectories. All non-redundantly-actuated PMs, encounter two backlash-prone zero-actuator-output configurations when executing any of the trajectories. Therefore, non-redundantly actuated PMs are found inadequate for precision applications.
Force-uncertainties, where PMs cannot sustain or apply forces in uncertain directions, are examined. For typically actuated 3-RRR and 3-RPR PMs, force uncertainties are identified using screw system arguments based on the existance of 3 actuated forces forming degenerate (rank = 2) planar pencils of forces. These degenerate force pose make arbitrary force and moment application impossible and cause singularities in the force solutions.
The working area of the 3-RRR PM is found compatible with all trajectories. This compatibility is due to zero minimum branch length being possible with the limitless angular displacements possible with stacked R joints. In comparison, the 3-RPR PM with minimum joint lengthes imposed on the P joints, has a smaller working area, and is not compatible with any of the trajectories. A P joint modification allowing relative length minimums of zero and a compatible working area identical to the 3-RRR PM, is considered.
To address inadequacies, symmetric actuation-redundant 3-RRR and 3-RPR PMs are considered. Pseudo (right Moore-Penrose) inverse of the 3×6 ARS (associated reciprocal screw) matrix is considered to solve for the required actuation. This solution, while providing a minimum 2-norm of the vector of required actuator outputs, does not reduce backlash-prone configurations with all actuators still having two backlash-prone zero-output configurations.
An algorithm for reducing backlash, using MATLAB’s constrained optimization routine FMINCON is applied. Minimizing the 2-norm of the vector of actuator outputs, subject to the backlash-free constraint of having outputs ≥ 0 or ≤ 0 depending on the initial values, is considered. Actuators providing the best conditioned ARS matices are utilized for the particular solutions.
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Keywords
Backlash reduction, Actuation redundancy, Planar parallel manipulators, Optimization