IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 3, JUNE 2013 1161 Antagonistic Stiffness Optimization of Redundantly Actuated Parallel Manipulators in a Predefined Workspace Hyunpyo Shin, Sungcheul Lee, Jay. I. Jeong, and Jongwon Kim Abstract—An optimization procedure regarding shape of us- able workspace and antagonistic stiffness is developed for a redun- dantly actuated parallel manipulator. The kinematic parameters such as configuration of the mechanism and length of linkages are optimized to maximize and equal out antagonistic stiffness of the redundantly actuated manipulator when the shape of us- able workspace is given as a rectangle. The proposed procedure is verified by simulation and experiment with a 2-DOF planar ma- nipulator. In the experiment, the principal axes of displacements of ellipses are measured when the corresponding external forces are imposed on the end-effector. The simulation and experimental results show that the proposed procedure is valid for designing the redundantly actuated parallel manipulator with maximum antag- onistic stiffness. Index Terms—Antagonistic stiffness, mechanism design of ma- nipulators, parallel robots, redundant robots. I. INTRODUCTION R EDUNDANTLY actuated parallel manipulators (or over- actuated parallel manipulators) have many advantages such as enlarged dexterous workspace and higher stiffness com- pared to the nonredundant analogs. The kinematic singularity region inside the workspace can mostly be eliminated in re- dundant actuation and the workspace can be enlarged [1]. In addition, internal preload control can enhance the antagonistic stiffness of the mechanism [2]–[5]. In order to design the redundantly actuated parallel mecha- nism, the determination of kinematic parameters of the mech- anism and number of excessive actuators, that is force redun- dancy, should be carefully considered in the kinematic design process. Manuscript received July 3, 2011; revised November 22, 2011; accepted March 27, 2012. Date of publication May 30, 2012; date of current version Jan- uary 18, 2013. Recommended by Technical Editor G. Schitter. This work was supported in part by the second stage of the Brain Korea 21 Program of Seoul National University, in part by the research program of Kookmin University in Korea, and in part by the Korean Ministry of Knowledge and Economy through research projects titled, “Development of high speed ecological finishing pro- cess for precision and micro pattern products,” and “Development of Intelligent Green-car Powertrains.” H. Shin and J. Kim are with the School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-744, Korea (e-mail: hpshin@snu.ac.kr; jongkim@rodel.snu.ac.kr). S. Lee is with the Korea Institute of Machinery and Materials, Daejeon 305- 343, Korea (e-mail: sclee@kimm.re.kr). J. I. Jeong is with the School of Mechanical and Automotive Engineering, Kookmin University, Seoul 136-702, Korea (e-mail: jayjeong@kookmin.ac.kr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2012.2198224 In previous works, the kinematic parameter optimization of the workspace and stiffness for nonredundantly actuated parallel mechanisms has been examined. Liu and Wang [6] in- troduced performance indexes such as the global conditioning index, global velocity index, global payload index, and global stiffness index within the framework of the performance atlas of the five-bar parallel manipulator. They proposed the maxi- mal inscribed circle for practical applications and the maximal inscribed workspace to perform kinematic optimization [7]. Re- garding workspace analysis and optimization, Kosinska et al. [8] designed a set of kinematic parameters of a 3-DOF spatial ori- entation manipulator using constraint equations to produce a specified workspace. Ceccarelli and Lanni [9] optimized a 3-R serial manipulator that maximizes the workspace and minimizes the size of the manipulator within limit constraints of a prede- fined workspace. Carbone and Ceccarelli [10] also suggested indices for stiffness performance evaluation. In cases of the redundantly actuated parallel mechanisms, Lee et al. [11] discussed isotropy of the stiffness and gradient of the isotropy in the optimization of a redundantly actuated five-bar parallel manipulator. Although they considered antag- onistic stiffness, the workspace and its shape were not included in the optimization procedure. Kurtz et al. [12] performed a uniformity of the dexterity and an actuator force optimization of a redundantly actuated parallel manipulator. Chakarov [13] discussed antagonistic stiffness of a parallel manipulator and he showed that the maximum compliance in a random direction could be reduced by controlling the internal preload of linear actuators. The optimization process of antagonistic stiffness is not easy even though shape and size of the workspace is predefined as design constraint. In the optimization process, the various kine- matic parameters are tested and one optimal parameter set is determined to maximize or equalize the antagonistic stiffness of the mechanism. The shape and size of the entire workspace, however, could be changed dramatically by small deviation of the kinematic parameters. This means that the position of the us- able workspace should be rearranged as the change of the entire workspace. Moreover, the antagonistic stiffness of the mecha- nism must be recalculated with every candidate of kinematic parameters. Thus, the magnitude and isotropy of the antago- nistic stiffness should be considered in design process of the mechanism as well as the workspace of the mechanism. With an initial kinematic design, an optimization process for kinematic parameters and degrees of force redundancy usually follows. The magnitude of antagonistic stiffness needs to be 1083-4435/$31.00 © 2012 IEEE