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Precise position estimation has always been a challenging but highly requested task in many technical problems. The time-difference of arrival (TDOA) based local position measurement system LPM uses the well-known Bancroft algorithm, which computes a closed-form solution to the non-linear range measurement equations. A critical issue of this computation method is that outliers in the measurements will decrease the quality of the position estimate significantly. In this contribution a least median of squares (LMS) algorithm for position estimation is developed which delivers an appropriate position estimate even if the raw data contain corrupted measurements.