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Research Papers: Design of Mechanisms and Robotic Systems

# Kinematically Redundant Planar Parallel Mechanisms for Optimal Singularity Avoidance

[+] Author and Article Information
Mats Isaksson

Electrical and Computer Engineering Department,
Fort Collins, CO 80523
e-mail: mats.isaksson@gmail.com

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 11, 2016; final manuscript received December 21, 2016; published online February 8, 2017. Assoc. Editor: Oscar Altuzarra.

J. Mech. Des 139(4), 042302 (Feb 08, 2017) (9 pages) Paper No: MD-16-1501; doi: 10.1115/1.4035677 History: Received July 11, 2016; Revised December 21, 2016

## Abstract

A parallel mechanism possesses several advantages compared to a similar-sized serial mechanism, including the potential for higher accuracy and reduced moving mass, the latter enabling increased load capacity and higher acceleration. One of the most important issues affecting a parallel mechanism is the potential of parallel singularities. Such configurations strongly affect the performance of a parallel mechanism, both in the actual singularity and in its vicinity. For example, both the stiffness of a mechanism and the efficiency of the power transmission to the tool platform are related to the closeness to singular configurations. A mechanism with a mobility larger than the mobility of its tool platform is referred to as a kinematically redundant mechanism. It is well known that introducing kinematic redundancy enables a mechanism to avoid singular configurations. In this paper, three novel kinematically redundant planar parallel mechanisms are proposed. All three mechanisms provide planar translations of the tool platform in two degrees-of-freedom, in addition to infinite rotation of the platform around an axis normal to the plane of the translations. The unique feature of the proposed mechanisms is that, with the appropriate inverse kinematics solutions, all configurations in the entire workspace feature optimal singularity avoidance. It is demonstrated how it is sufficient to employ five actuators to achieve this purpose. In addition, it is shown how including more than five actuators significantly reduces the required actuator motions for identical motions of the tool platform, thereby reducing the cycle times for typical applications.

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## Figures

Fig. 1

Different approaches to employing redundancy to provide the non-redundant mechanism in (a) with infinite rotation of the tool platform (black). The joints R (solid circle) and R (hollow circle) are actuated and passive revolute joints, respectively, while the joints P are actuated prismatic joints. In order to reduce the figure size, only a section of the circular guide-way in (f) is shown: (a) nonredundant, (b) redundantly actuated, (c) redundantly actuated, (d) kinematically redundant, (e) kinematically redundant, and (f) kinematically redundant.

Fig. 2

Three kinematically redundant planar parallel mechanisms providing planar translations in two DOFs in addition to infinite rotation around an axis perpendicular to the plane of the translations. The upper image in each figure provides a top view of the mechanism, while the lower image provides an approximate side view. As can be seen from the side views, the mechanisms are designed to avoid any collisions between different sections except for collisions between the manipulated platform and the guide-ways. The number of actuators in (a), (b), and (c) are five, six, and seven, respectively.

Fig. 3

(a) Notation for determining the OTI of the tool platform. The output twist $̂Oi of each chain is a zero pitch twist through the corresponding instantaneous center of rotation ICi, while fi are the direction vectors of the zero pitch actuation wrenches$̂Ai. The common perpendicular between $̂Ai and$̂Oi is denoted by ρOi, while the maximum value of ρOi is denoted by ρOmaxi. (b) Notation for determining the OTI of the closed-loop subchain of the mechanisms in Figs. 2(b) and 2(c). The used notations are the same as in (a).

Fig. 4

Simulations of the mechanisms in Fig. 2. (a) Pure rotation of the mechanism in Fig. 2(a). (b) Pure rotation of the mechanisms in Figs. 2(b) and 2(c). (c) Square path by the mechanism in Fig. 2(a). (d) Square path by the mechanism in Fig. 2(b). (e) Square path by the mechanism in Fig. 2(c). (f) Square path by the mechanisms in Figs. 2(a)2(c).

Fig. 5

Simulations of a square path and a platform rotation for the described 3-RPR and 4-RPR mechanisms. For the 3-RPR mechanism, κ is given by Eq. (25) and for the 4-RPR mechanism by Eq. (26). (a) Three actuators, rotation, (b) three actuators, square, (c) four actuators, rotation, and (d) four actuators, square.

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