This article presents an up-to-date tutorial review of nonlinear Bayesian estimation. State estimation for nonlinear systems has been a challenge encountered in a wide range of engineering fields, attracting decades of research effort. To date, one of the most promising and popular approaches is to view and address the problem from a Bayesian probabilistic perspective, which enables estimation of the unknown state variables by tracking their probabilistic distribution or statistics (e.g., mean and covariance) conditioned on the system's measurement data. This article offers a systematic introduction of the Bayesian state estimation framework and reviews various Kalman filtering (KF) techniques, progressively from the standard KF for linear systems to extended KF, unscented KF and ensemble KF for nonlinear systems. It also overviews other prominent or emerging Bayesian estimation methods including the Gaussian filtering, Gaussian-sum filtering, particle filtering and moving horizon estimation and extends the discussion of state estimation forward to more complicated problems such as simultaneous state and parameter/input estimation.

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Tensile testing of soft, slippery biological materials is a challenging task due to the difficulties associated with the gripping method and accurate measurement of axial and lateral strains. In this manuscript, the above issues were effectively resolved by using a shoulder-supported tensile specimen and digital image correlation (DIC) technique, respectively. The tensile response of agarose gel with concentration ranging from 1.5 to 4.0 wt% was determined using the above method. Unlike the previous literature where the tensile strain was obtained from machine crosshead displacement, the DIC technique utilized a speckle pattern introduced into the gage area to obtain full-field maps of axial and lateral strains. It is found that the tensile strength and modulus of agarose gel increases linearly with an increase in agarose concentration. The Poisson's ratio was determined to be around 0.5 for virgin specimens and it decreased slightly with axial strain, possibly due to the loss of water during deformation.

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