Improving Inertial Velocity Estimation Through Magnetic Field Gradient-based Extended Kalman Filter

Abstract

In this paper we focus on the velocity estimation problem of a rigid body and how to improve it with magnetoinertial sensors-based theory. We provide a continuous-time model that describes the motion of the body and we augment it after by introducing a new magnetic field gradient equation instead of using its value directly as an input for the model, as done usually in the corresponding literature. We investigate the advantage of moving to higher order spatial derivatives of the magnetic field in the estimation of velocity. These derivatives are computed thanks to a determined arrangement of magnetometers array. Within this framework, a specific set configuration of Extended Kalman Filters (EKFs) is proposed to focus mainly on the estimation of velocity and attitude of the body, but includes also an estimation of the magnetic field and its gradient. Some simulations for a specific scenario are proposed to show the improvements that we bring to the velocity estimation.

Magnetic Field Gradient-Based EKF for Velocity Estimation in Indoor Navigation

“…By multiplying both sides of ( 9 ) by R and , respectively, the following equation is deduced, where is the first spatial derivative of , with the same form as ( 7 ), represented in . The reader can check [ 34 ] for more information on how is measured.…”

“…The state vector for this dynamic state-space model is , the input vector is , and the output (measurement) vector is . Recall that 7 elements of are sufficient to calculate all the tensor’s components [ 34 ]. The matrix is defined in ( 1 ), where is the estimated quaternion.…”

“…As , defined in ( 10 ), is measurable, it is possible to add to x . A first schema of the magnetic field gradient-based EKF was presented in [ 34 ]. The estimation approach was based on two EKFs, in cascade, as displayed in Figure 2 .…”

“…It is important to indicate that, unlike [ 34 ], in the new proposed model ( 11 ), Equation ( 13 ) representing the quaternion dynamics is discarded. Instead, attitude estimation is conducted using the gradient descent algorithm represented with the green block in Figure 1 .…”

“…In fact, the effect of attitude estimation can vary from one model to another (if it is included in a single EKF), which makes it impossible to conclude on the advantage of the proposed velocity estimation approach, as the results are also influenced by the varying errors on the attitude estimate (from one model to another and also between static and dynamic cases in motion), and the correlation in the covariance matrix. Furthermore, by adding the magnetic field gradient equation to the main EKF of [ 34 ], and eliminating the primary EKF, the process covariance matrix dimension increases from to . This makes it harder to properly tune Q , especially when taking into account the correlation between all the variables (it is brought to the attention of the reader that the magnetic field gradient equation is dependent on all the other variables in the model, and mainly q ).…”