بزرگ‌ترين دايره ي عاري از تكينگي در فضاي كاري ربات‌هاي موازي صفحه‌يي، با استفاده از تحليل بازه‌يي و جابه‌جايي مرزها

DETERMINATION OF THE MAXIMAL SINGULARITY-FREE CIRCLE IN THE WORKSPACE OF PLANAR PARALLEL MECHANISMS USING INTERVAL ANALYSIS AND CONSTRUCTIVE GEOMETRIC APPROACH

در اين مطالعه الگوريتمي قاعده مند، براساس مفهوم تحليل بازه‌يي، ارايه مي‌شود تا به‌وسيله‌ي آن بتوان بزرگ‌ترين دايره‌ي عاري از تكينگي در فضاي كاري ربات‌هاي موازي صفحه‌يي را تعيين كرد. اين الگوريتم براي بهينه‌سازي فضاي كاري ربات‌هاي سه درجه آزادي كارآمد است و به‌عنوان مطالعه‌ي موردي سازوكار موازي 3-RPR بررسي شده است. دو روشِ ارايه شده بدين منظور مبتني بر تحليل بازه‌يي و روش جابه‌جايي هندسي هستند. در روش تحليل بازه‌يي، الگوريتم اصلي نهايتاً منتهي به يافتن بزرگ‌ترين دايره‌ي عاري از تكينگي در فضاي كاري كل ربات مي‌شود. علاوه بر آن، روش جابه‌جايي هندسي ارايه شده مي‌تواند در نرم افزار‌هاي CAD و نيز در ابزارهاي محاسباتي پياده‌سازي شود. سهم اعظم اين مقاله به يافتن فضاي كاري عاري از تكينگي با در نظر گرفتن فضاي كاري ناشي از محدوديت‌هاي مكانيكي در عملگرها اختصاص دارد ــ مطلبي كه اين مطالعه را از ساير مطالعات مشابه متمايز مي‌كند.

This paper proposes a systematic algorithm, based on the interval analysis concept, in order to optimize the maximal singularity-free circle within the workspace of 3-DOF planar parallel mechanisms. A 3-RPR parallel mechanism is considered as the case study. Two approaches are presented, namely, interval analysis with four algorithms and a geometric constructive approach. A new concept in applying interval analysis is introduced, which could be of great help in other optimization contexts. The proposed algorithm in the interval analysis section is divided into four sub-algorithms, which eases the understanding of the main concept and leads to a more effective and robust algorithm to solve the problem. First, the workspace of the mechanism is obtained, using the branch and prune method. Then, by the same token, the singularity locus of the mechanism under study will be obtained. Afterwards, the maximal singularity-free circle for a constant orientation and a prescribed guess box is found. Finally, the maximal singularity-free circle for all orientations of the end-effector is achieved. Moreover, a geometric approach, referred to as the offset curve approach, is presented, which can be implemented either in CAD software or in a computer algebra system. This method is quite fast and can be applied to more complicated mechanisms. As it is a geometric approach, it can be easily extended to higher degrees of freedom parallel robots. The main contribution of this work can be regarded as a combination of the maximal singularity-free circle with the workspace analysis, upon considering the stroke of the actuators, as additional constraints to the latter problem. The results of this paper reveal that the singularity-free circle of any parallel mechanism can be readily obtained upon following the proposed algorithm, and, as a case study, a 3-DOF planar parallel mechanism is presented.