This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS: SYSTEMS 1 Multicriteria Design Optimization of a Parallel Ankle Rehabilitation Robot: Fuzzy Dominated Sorting Evolutionary Algorithm Approach Prashant K. Jamwal, Member, IEEE, and Shahid Hussain Abstract—Parallel robots, owing to their increased stiffness, accuracy, and compactness, are preferred over their serial coun- terparts in applications involving higher torques and precision such as robot-assisted physical therapy. However, their design is complex and calls for obtaining a tradeoff between sev- eral conflicting objectives such as the minimization of actuator forces versus the maximization of workspace while maintain- ing a close to unity condition number, etc. While evolutionary algorithms have been proposed in the literature for simulta- neous optimization of many objectives, they have been found to be inefficient in dealing with a large number of objec- tives. We propose a fuzzy logic-based sorting approach in this paper which effectively replaces the concept of nondom- inated sorting and provides a better discrimination between solutions and clear termination logic. The proposed sorting algo- rithm has been evaluated against the existing nondominated sorting genetic algorithm II in the pretext of design optimiza- tion of a parallel ankle rehabilitation robot. The proposed fuzzy-based approach is able to provide a better discrimina- tion among solutions and, thereby an improved parallel ankle robot design. Index Terms—Evolutionary algorithms (EAs), fuzzy logic, multicriteria optimization, parallel robots. I. I NTRODUCTION R OBOTIC platforms that impart repetitive physical ther- apy to patients suffering from ankle joint disorders have been proposed. Most of these robotic platforms are based on parallel mechanisms. This is because parallel robots offer higher stiffness and precision compromising on workspace which fits quite well with the requirements of an ankle robot. However, most of the proposed robots [1]–[8] are inspired by the Stewart mechanism and, therefore, look like a table top where the foot of the subject is secured and retained for various rehabilitation treatments. Subsequent to further investigations a new design of ankle robot is proposed by Jamwal et al. [9], [10]. The proposed ankle robot (Fig. 1) is Manuscript received October 27, 2014; accepted June 13, 2015. This paper was recommended by Associate Editor Z. Liu. (Corresponding author: Shahid Hussain.) P. K. Jamwal is with the Department of Electrical and Electronics Engineering, Nazarbayev University, Astana 010000, Kazakhstan (e-mail: prashant.jamwal@nu.edu.kz). S. Hussain is with the School of Mechanical, Materials and Mechatronics Engineering, University of Wollongong, Wollongong, NSW 2522, Australia (e-mail: shussain@uow.edu.au). 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/TSMC.2015.2478389 Fig. 1. Parallel ankle rehabilitation robot. a parallel robot worn by patients which suitably clamps on their shin bone. Design of this robot is critical owing to the fact that it was to be actuated using four parallel actuators. Nevertheless, six important design objectives were identi- fied encompassing the ankle robot’s kinematic, actuation, and structural performance requirements [11]. It is important to mention here that the design analysis of parallel robots has been an active area of research in the past, and prior to our ankle robot design optimization, little related research had been accomplished. Parallel robots should be designed to reduce the adverse effects of simultaneous and parallel action of their actuators [12], [13]. Adverse effects such as reduced workspace, increased singularity, and higher actua- tor force requirements can be taken care of by selecting the appropriate geometrical parameters of the robot. Apart from using the trial and error approach, the issues of workspace and singularity have been optimized in the past using the exhaustive search minimization method [14], [15] for a lin- ear delta robot. Several performance indices (PIs) for parallel robots have been defined by different researchers and details of these indices are provided in [16]. Design of a robotic arm exoskeleton developed on the basis of a parallel mech- anism has been optimized to achieve high force using the low mass of the exoskeleton [17]. A spherical parallel robot design has been analyzed and optimized using the geomet- rical approach [18]. A new geometric method called perfor- mance atlas is introduced in [19] to optimize the transmission 2168-2216 c  2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.