Abstract—External fixators are widely used in the area of orthopedics to manage deformities and fractures. The historical trend in obtaining a mechanical infrastructure has shifted from simple devices like pins, rods, and hinges to more sophisticated frames involving parallel manipulators. Despite advantages of such robotic frames especially in removing stiffness issues, there are still problems associated with the reliable deployment of these modern devices. Problems like singularity, possibilities of guiding fragments along many different trajectories have been handled in this paper. A method has been shown how to detect singularity in displaced, aligned, and intermediate positions of the fragments. How the robotic system could be actuated according to different functions have been investigated, in addition to their effects on trajectories of distal fragment end. These considerations have been demonstrated on numerical examples. I. INTRODUCTION The history of external fixators dates back to Hippocrates’ times when tibia fractures were treated by these devices 2500 years ago [1]. After some successful applications, its use was almost abandoned due to unsuccessful results caused by unconscious implementations. By biomechanical studies it regained wide acceptance [2, 3, 4]. At first, constructing a mechanical setup to serve as a fixator started in the form of serial manipulators which consisted of simple elements like rods, pins and hinges with one degree of freedom whereby complicated planning and stiffness issues dominated [4, 5, 6, 7]. These issues were overcome by a new generation of fixator devices that involved parallel platforms like Stewart – Gough Platform with six degrees of freedom [8]. The first successful application was observed in the so called Taylor’s Spatial Frame [9]. Despite wide practice utilizing parallel manipulators as fixators in the world, to our best knowledge, the issue of singularity has never been raised in orthopedic implementations. In view of the fact that the function of the robotic frame is not only to hold the fragments in stable equilibrium but also to move them from the displaced configuration to the final aligned configuration securely upon suitable actuation, the question of singularity becomes of great importance for reliable use of these devices. During this motion, how the end of the distal fragment moved was not questioned and remained also unnoticed. In actuating the frame, it is traditionally a common practice to take it for granted the selection of the rod lengths according to a linear model. Thus, the relationship between how the rod length selection affected the trajectory of the distal fragment end was underestimated. In this work, the issues mentioned above have been handled. After obtaining the nonlinear displacement function governing the motion in the fixator, it has been reduced from a multi-solution form to a single-root form pointing to a well-defined fixator position by linearization. The possibilities of solving the resulting equation have been investigated leading to the sources of singularities. It is shown that actuation of the robotic system according to different models will greatly enhance the possibilities of moving fragment ends along different paths. Numerical examples are provided to illustrate the importance of the issues mentioned. II. METHOD The robotic system in question here is a (6-6) type Stewart platform consisting of two rings which are connected by six rods through twelve spherical pairs or six spherical pairs on top and six universal pairs at the bottom. The fixed positions of joints on top ሺ ܣ ௜ ǡ  ൌ ͳ െ ͸ሻ and bottom rings ሺ ܤ ௜ ǡ  ൌ ͳ െ ͸ሻ with radii ሺሻ and ሺ ଵ ሻ are defined by angular parameters ሺ ߙ ௜ ǡ  ൌ ͳ െ ͸ሻ and ሺ ߝ ௜ ǡ  ൌ ͳ െ ͸ሻ, respectively (Fig. 1). The given position of the bottom ring is defined with respect to two coordinate systems, the origin of one (world coordinate system) located at the center of the top ring (xyz – system), the other located at the center of the bottom ring (moving uvw – system) by the following matrix:  ܣ ௫௬௭ ௨௩௪ ൧ ൌ ሾ ௨  ௩  ௪ ሿൌ൥  ௨௫  ௩௫  ௪௫  ௨௬  ௩௬  ௪௬  ௨௫  ௩௭  ௪௭ ൩ Also known is how a distal fragment is rigidly attached to the bottom ring by the following matrix and its coordinates on the ring plane being given by ሺ ݎ ௨ ǡ ݎ ௩ ǡ Ͳሻ ሾ ܣ ௨௩௪ ௨ᇱ௩ᇱ௪ᇱ ሿ ൌ ሾ ௨ᇱ  ௩ᇱ  ௪ᇱ ሿൌ൥  ௨ᇱ௫  ௩ᇱ௫  ௪ᇱ௫  ௨ᇱ௬  ௩ᇱ௬  ௪ᇱ௬  ௨ᇱ௫  ௩ᇱ௭  ௪ᇱ௭ ൩ whereby ݑ ᇱ ݒ ᇱ ݓ ᇱ coordinate system fixed to the distal fragment at its end in such a way that axis ݓ ᇱ and that of Displacement Analysis of Robotic Frames for Reliable and Versatile Use as External Fixator ø. D. Akçalﾕ, E. Avúar, Member, IEEE, M. K. Ün, A. Aydﾕn, Member, IEEE, T. øbrikçi, H. Mutlu, Ö. S. Biçer, C. Özkan, A. Durmaz Manuscript received March 26, 2014. Research is supported by the Scientific and Technological Research Council of Turkey (TÜBøTAK) under grant no: 112M406. ø. D. Akçalﾕ is with MACTIMARUM Resarch & Application Center, Çukurova University, Adana, Turkey (e-mail: idakcali@cu.edu.tr) E. Avúar, M. K. Ün, A. Aydﾕn, and T. øbrikçi are with Electrical and Electronics Engineering Department, Çukurova University, Adana, Turkey (e-mails: ercanavsar@cu.edu.tr, keremun@cu.edu.tr, aaydin@cu.edu.tr, ibrikci@cu.edu.tr) . H. Mutlu is with Mechanical Engineering Department, Mersin University, Mersin, Turkey (e-mail: hmutlu@mersin.edu.tr) Ö. S. Biçer and C. Özkan are with Faculty of Medicine, Orthopedics Department, Çukurova University, Adana, Turkey (e-mail: sbicer@cu.edu.tr) A. Durmaz is with Mechanical Engineering Department, Çukurova University, Adana, Turkey (e-mail: atakandurmaz@gmail.com) 978-1-4799-3669-4/14/$31.00 © 2014 IEEE 180 The 4th Annual IEEE International Conference on Cyber Technology in Automation, Control and Intelligent Systems June 4-7, 2014, Hong Kong, China