Necessary and Sufficient Conditions for Observability of SLAM-Based TDOA Sensor Array Calibration and Source Localization

Sensor array-based systems, which adopt time difference of arrival (TDOA) measurements among the sensors, have found many robotic applications. However, for existing frameworks and systems to be useful, the sensor array needs to be calibrated accurately. Of particular interest in this article are microphone array-based robot audition systems. In our recent work, by using a moving sound source, and the graph-based formulation of simultaneous localization and mapping (SLAM), we have proposed a framework for joint sound source localization and calibration of microphone array geometrical information, together with the estimation of microphone time offset and clock difference/drift rates. However, a thorough study on the identifiability question, termed observability analysis here, in the SLAM framework for microphone array calibration and sound source localization, is still lacking in the literature. In this article, we will fill the abovementioned gap via a Fisher information matrix approach. Motivated by the equivalence between the full column rankness of the Fisher information matrix and the Jacobian matrix, we leverage the structure of the latter associated with the SLAM formulation, and present necessary and sufficient conditions guaranteeing its full column rankness, which lead to parameter identifiability. We have thoroughly discussed the 3-D case with asynchronous (with both time offset and clock drifts, or with only one of them) and synchronous microphone array, respectively. These conditions are closely related to the motion varieties of the sound source and the microphone array configuration, and have intuitive and physical interpretations. Based on the established conditions, we have also discovered some particular cases where observability is impossible. Connections with calibration of other sensors will also be discussed, amongst others. To our best knowledge, this is the first systematic work on observability analysis of SLAM-based microphone array calibration and sound source localization. The tools and concepts used in this article are also applicable to other TDOA sensing modalities such as ultrawide band (UWB) sensors.