An Intuitive Approach to Inertial Sensor Bias Estimation

A simple approach to gyro and accelerometer bias estimation is proposed. It does not involve Kalman filtering or similar formal techniques. Instead, it is based on physical intuition and exploits a duality between gimbaled and strapdown inertial systems. The estimation problem is decoupled into two separate stages. At the first stage, inertial system attitude errors are corrected by means of a feedback from an external aid. In the presence of uncompensated biases, the steady-state feedback rebalances those biases and can be used to estimate them. At the second stage, the desired bias estimates are expressed in a closed form in terms of the feedback signal. The estimator has only three tunable parameters and is easy to implement and use. The tests proved the feasibility of the proposed approach for the estimation of low-cost MEMS inertial sensor biases on a moving land vehicle.

💡 Research Summary

The paper presents a novel, intuitive method for estimating gyroscope and accelerometer biases in low‑cost MEMS‑based attitude and heading reference systems (AHRS) without resorting to Kalman filtering or other formal state‑estimation techniques. The approach exploits the classic “virtual platform” concept: an external aid (e.g., GPS‑derived specific force) provides a reference acceleration, and the difference between this reference and the measured specific force on the platform generates a corrective “torque” u. This torque is applied to the virtual platform to drive attitude errors to zero, but when sensor biases are present the torque does not vanish; instead, it reaches a steady‑state value that directly reflects the underlying biases.

The estimator is organized into two stages. In the first stage, the external aid is used to correct attitude errors via a proportional feedback loop with gain k (or equivalently a time constant τ_k). The resulting torque u is recorded continuously. In the second stage, analytical relationships derived from the strapdown dynamics link the torque u and the vehicle’s body angular rate ω_b to the sensor biases. In steady‑state, the torque satisfies a linear equation (Eq. 11) of the form u = –g_b + (1/τ_k) ω_b · b, where g_b represents the gyroscope bias vector. When the vehicle travels in a straight line (ω_b≈0), the torque reduces to u = –g_b, providing a direct estimate of the gyroscope bias. When the vehicle executes a turn, the term involving ω_b becomes non‑zero, allowing the accelerometer bias a_b to be solved (Eqs. 13‑14). The method explicitly states the observability conditions: gyroscope biases are observable during straight‑line motion, while accelerometer biases require non‑zero angular rates (turns). Consequently, a practical test trajectory must contain both straight and turning segments.

Only three tunable parameters are required: (i) the attitude‑correction time constant τ_k, (ii) the gyroscope‑bias filter time constant τ_g, and (iii) the accelerometer‑bias filter time constant τ_a. These parameters have clear physical meanings: τ_k determines how quickly the virtual platform aligns with the reference, while τ_g and τ_a control the low‑pass filtering of the bias estimates, trading off convergence speed against noise rejection.

The algorithm was implemented in the firmware of a Topcon GPS/GLONASS receiver equipped with an integrated MEMS IMU. Tests were performed on two agricultural/construction tractors (John Deere 5515 wheel tractor and Caterpillar Challenger tracked tractor). The attitude‑correction constant was set to τ_k = 4 s, and both bias filters were given τ_g = τ_a = 40 s. Test routes combined straight‑line segments and 90° turn arcs to satisfy the observability requirements.

Results show that the gyroscope‑bias filter converged in roughly 120 s (≈3 τ_g), achieving root‑mean‑square (RMS) errors of 0.01 deg/s for the wheel tractor and 0.02 deg/s for the tracked tractor—substantially better than previously reported symmetry‑preserving filters (≈0.5 deg/s). Accelerometer‑bias estimation, inherently weaker in observability, required about 300 s to settle (≈7.5 τ_a) and yielded RMS errors of 0.04 m/s². Artificial biases injected into the accelerometer data were accurately recovered, confirming the method’s validity. Moreover, the estimator proved robust to IMU installation misalignments up to 10–15°, as such misalignments merely redefine the body frame without degrading bias recovery.

In summary, the proposed bias estimator offers a computationally lightweight, physically intuitive alternative to Kalman‑filter‑based solutions for low‑cost AHRS. It requires only three easily interpretable parameters, provides closed‑form bias estimates, and demonstrates reliable performance in real‑world vehicle tests. While the technique is well‑suited for precision agriculture, construction, and other applications where MEMS sensors dominate, extending it to tactical‑grade or navigation‑grade inertial systems would demand additional modeling of Earth rotation, gravity variations, and possibly reduced correction gains. In those high‑performance contexts, traditional Kalman filtering may remain the preferred approach.