This paper provides a genetic algorithm based approach to calculate the optimal placement of receivers in a 3D position estimation system that uses the difference in the time-of-arrivals (TOA) of an ultrasonic wave from a transmitter to the different receivers fixed in 3D space. This is a different approach to traditional systems that use the actual time-of-flights (TOF) from the transmitter to the different receivers and triangulate the position of the transmitter. The new approach makes the system more accurate, makes the transmitter independent of the receivers and does not require the need of calculating the time delay term that is inherent in traditional systems due to delays caused by the electronic circuitry. This paper presents a thorough analysis of receiver configurations in the 2D and 3D system that lead to singularities, i.e. locations of receivers that lead to formulations that can not be solved due to a shortage of information. It provides guidelines of where not to place receivers, and further, presents a detailed analysis of locations that are optimal, i.e. locations that lead to the most accurate estimation of the transmitter positions. The results presented in this paper are not only applicable to ultrasonic systems, but all systems that use wave theory, e.g. infrared, laser, etc. This work finds applications in virtual reality cells, robotics, guidance of indoor autonomous vehicles and vibration analysis.
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