Single-Axis Gyroscope Scale Factor Estimation for a First-Order Gauss–Markov Drift Model

This paper presents a dynamic calibration approach for scale factor drift model identification applied to a single-axis gyroscope. Specifically, maximum likelihood estimation (MLE) is used by maximizing an innovations-based log-likelihood function (LLF) of the system parameters, derived from the steady-state Kalman filter equations. By expressing the innovations in terms of the measurements, the LLF is shown to be a quadratic polynomial function, which allows for analytical evaluation of the LLF’s gradient as well as the constrained Cramér–Rao lower bound (CCRB). The scale factor instability is expressed by a first-order Gauss–Markov (GM1) dynamic model which is characterized by an inverse time constant and process noise variance. The gyroscope is also prone to static noise parameters: bias instability (GM1 bias drift) and measurement noise. It is shown that the LLF of static and dynamic noise parameters cannot be directly maximized due to a mathematical ambiguity and severe ill-conditioning. Therefore, a two-stage procedure leading to a constrained optimization problem is proposed: the first stage consists of static calibration (zero input), followed by a dynamic calibration (non-zero known input) which utilizes the static noise parameter estimates from the first stage. The estimator is shown to be statistically efficient and well-conditioned compared to a combined MLE identification of static and dynamic noise parameters.