Left Equivariant Filter with Autonomous Error Dynamics for Inertial Integrated Navigation Systems with Biases

A left equivariant filter to biased inertial integrated navigation system is proposed. The geometric property of the kinematics of inertial integrated navigation system has been leveraged to design filters with better property. The equivariant property has played a key role in the filter design of the kinematic systems on Lie groups. However, there are few works have considered the biases of the low cost inertial measurement unit (IMU) as they are difficult to be embraced into the kinematics on the Lie group. The recent equivariant filter provides a fully geometric Extended Kalman Filter (EKF)-like for a inertial navigation system including bias state, where a symmetry group defined by exploiting the semi-direct product structure is the key to the successful equivariant property. But, the resulting filter still involves the navigation state via the bias and is only suitable for relative measurement such as visual odometry. In this paper, we proposed a novel equivariant filter for biased inertial integrated navigation incorporating global measurement. The equivariance of the biased inertial integrated navigation system is substantiated by exploiting a novel direct product symmetry and the associated left group action. The resulting filter is also a fully geometric framework and leads to a autonomous error dynamics which is better than previous. The performance of the proposed filter is evaluated with field experiments of global navigation satellite system/ inertial navigation system, and demonstrates increased robustness to a wide range of erroneous initial conditions, and improved accuracy when compared with the other EKF-like approaches.