QGD-OE: IMU Orientation Estimation Based on Gradient Descent in the Quaternion Field

Orientation estimation based on inertial measurement units (IMUs) has emerged as a promising solution for real-time orientation tracking. Quaternion numbers are frequently employed by estimation algorithms to represent orientation in 3-D space. In recent years, gradient descent-based algorithms have been extensively utilized for quaternion-based orientation estimation due to their simplicity and effectiveness. However, the real functions of quaternion variables are nonanalytic. Current state-of-the-art algorithms for orientation estimation based on gradient descent methods overcome this obstacle by transforming the problem from the quaternion domain into the real domain. In contrast, we leverage the mathematical definition of the quaternion gradient based on the generalized Hamilton-real (GHR) algebra to solve the orientation estimation optimization problem based on IMUs directly in the quaternion domain. More specifically, we derive the accelerometer and magnetometer gradient descents in the quaternion domain and propose the quaternion gradient descent orientation estimation (QGD-OE) algorithm to estimate orientation from these gradient descents. We compare our QGD-OE algorithm with two state-of-the-art orientation estimation algorithms. We find that the QGD-OE algorithm achieves higher accuracy, improved robustness, and shorter convergence time than state-of-the-art methods. The comparison highlights the deficiencies of transforming from the quaternion domain into the real domain and underscores the importance of conducting gradient descent and estimation optimization in the quaternion domain.