(eng): Standard Bayesian algorithms used for online filtering of states of hidden Markov models from noisy measurements assume an accurate knowledge of the measurement model in the form of a conditional probability density function. However, this knowledge is often unreachable in practice, and the used models are more or less misspecified, or it is too complex, making the resulting models intractable. This paper focuses on these issues from the particle filtering perspective. It adopts the principles of the approximate Bayesian filtering, where the particle weights are based on the (dis)similarity of the true measurements and the pseudo-measurements obtained by plugging the state particles directly into the measurement equation. Specifically, a new robust method for online tuning of the weighting kernel is proposed.