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The purpose of this paper is to present the design, development and testing of an innovative instrument called GraviMob, which allows performing dynamic measurements of underwater gravity anomalies. After recalling the interest in underwater gravimetry, we describe the system, the core of which consists of triads of accelerometers rigidly attached to an Autonomous Underwater Vehicle (AUV). The article also presents the mathematical methods for estimating the east, north and vertical components of the local gravity vector. An unscented Kalman filter, integrating AUV position and orientation data, performs estimation of gravity in a frame adapted to its interpretation. To assess its performance, GraviMob was tested in the Mediterranean Sea during the year 2016. A comparison of the surface gravimetric signal previously acquired by the French Navy indicates that the maximum discrepancy between the vertical gravity component and its reference is below 4 mGal. Components of the vertical deflection calculated from GraviMob's measurements were compared with those calculated from recent gravity field models. While a remarkable agreement was found on the north component, there remains a discrepancy (7 arcsec) on the east component which can be largely reduced by refining the estimation of the orientation of GraviMob's sensitive axes in the AUV. [ABSTRACT FROM AUTHOR]