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Indoor positioning systems (IPSs) locate objects in closed structures such as office buildings, hospitals, stores, factories, and warehouses, where Global Positioning System devices generally do not work. Most available systems apply wireless concepts, optical tracking, and/or ultrasound. This paper presents a standalone IPS using radio frequency identification (RFID) technology. The concept is based on an object carrying an RFID reader module, which reads low-cost passive tags installed next to the object path. A positioning system using a Kalman filter is proposed. The inputs of the proposed algorithm are the measurements of the backscattered signal power propagated from nearby RFID tags and a tag-path position database. The proposed algorithm first estimates the location of the reader, neglecting tag-reader angle-path loss. Based on the location estimate, an iterative procedure is implemented, targeting the estimation of the tag-reader angle-path loss, where the latter is iteratively compensated from the received signal strength information measurement. Experimental results are presented, illustrating the high performance of the proposed positioning system.