Transmission electron microscopes (TEMs) enable atomic-scale imaging but suffer from aberrations caused by lens imperfections and environmental conditions, reducing image quality. These aberrations can be compensated by adjusting electromagnetic lenses, but this requires accurate estimates of the aberration coefficients, which can drift over time. This paper introduces a method for the estimation of aberrations in TEM by leveraging the relationship between an induced tilt of the electron beam and the resulting image shift. The method uses a Kalman filter (KF) to estimate the aberration coefficients from a sequence of image shifts, while accounting for the drift of the aberrations over time. The applied tilt sequence is optimized by minimizing the trace of the predicted error covariance in the KF, which corresponds to the A-optimality criterion in experimental design. We show that this optimization can be performed offline, as the cost criterion is independent of the actual measurements. The resulting non-convex optimization problem is solved using a gradient-based, receding-horizon approach with multi-starts. Additionally, we develop an approach to estimate specimen-dependent noise properties using expectation maximization (EM), which are then used to tailor the tilt pattern optimization to the specific specimen being imaged. The proposed method is validated on a real TEM set-up with several optimized tilt patterns. The results show that optimized patterns significantly outperform naive approaches and that the aberration and drift model accurately captures the underlying physical phenomena. A direct comparison with the widely used Zemlin tableau shows that the proposed method achieves comparable or higher image quality on amorphous specimens, while additionally extending to non-amorphous specimens where the Zemlin tableau cannot operate.