Multi-kernel Correntropy-based Orientation Estimation of IMUs: Gradient Descent Methods

TL;DR

This work addresses robust, real-time orientation estimation for IMUs in disturbed environments. It introduces two MKCL-based algorithms, CGD and CDOE, by replacing the traditional LS objective in GD/DOE with the multi-kernel correntropy loss, aligning the objective with heavy-tailed noise distributions. The methods demonstrate improved robustness to external acceleration and magnetic disturbances while offering significantly lower computational complexity than Kalman-filter-based approaches, making them suitable for low-cost microprocessors. Extensive experiments on commercial and low-cost IMUs corroborate improved accuracy and robustness, with practical kernel bandwidth tuning guidance and a discussion of future enhancements such as adaptive bandwidth and variable-center kernels.

Abstract

This paper presents two computationally efficient algorithms for the orientation estimation of inertial measurement units (IMUs): the correntropy-based gradient descent (CGD) and the correntropy-based decoupled orientation estimation (CDOE). Traditional methods, such as gradient descent (GD) and decoupled orientation estimation (DOE), rely on the mean squared error (MSE) criterion, making them vulnerable to external acceleration and magnetic interference. To address this issue, we demonstrate that the multi-kernel correntropy loss (MKCL) is an optimal objective function for maximum likelihood estimation (MLE) when the noise follows a type of heavy-tailed distribution. In certain situations, the estimation error of the MKCL is bounded even in the presence of arbitrarily large outliers. By replacing the standard MSE cost function with MKCL, we develop the CGD and CDOE algorithms. We evaluate the effectiveness of our proposed methods by comparing them with existing algorithms in various situations. Experimental results indicate that our proposed methods (CGD and CDOE) outperform their conventional counterparts (GD and DOE), especially when faced with external acceleration and magnetic disturbances. Furthermore, the new algorithms demonstrate significantly lower computational complexity than Kalman filter-based approaches, making them suitable for applications with low-cost microprocessors.

Figures (11)

• Figure 1: The pdfs of $v_{A,k}$ and $v_{M,k}$ without and with disturbances. (a) and (c) show the pdfs of $v_{A,k}$ and $v_{M,k}$ when IMU is static and free of disturbances. (b) and (d) show the pdfs of $v_{A,k}$ and $v_{M,k}$ when IMU is with disturbances. The IMU used in this experiment is Xsens MTI-670 and the disturbed data is generated by the translation experiment as shown in Fig. \ref{['linear']}.
• Figure 2: The pdfs of $\mathcal{N}(0,1)$ and $p(\tilde{e}_{i,k})$ in \ref{['pdfe']} with different kernel bandwidths. The error $\tilde{e}_{i,k}$ is assumed to be bounded within $[-20,20]$ for $p(\tilde{e}_{i,k})$ and the coefficient $c_i$ is obtained by $c_i=1/\int_{-20}^{20} \exp(-\sigma_i^2(1-\exp(-\frac{\tilde{e}_{i,k}^2}{2\sigma_i^2}))) \mathrm{d} \tilde{e}_{i,k}$.
• Figure 3: A comparison of $\log \mathcal{L}_{CL}$ and $\log \mathcal{L}_{LS}$ under differ $p$.
• Figure 4: Objective functions and influence functions for $J_{LS}$ and $J_{CL}$ with different kernel bandwidths.
• Figure 5: The experimental setups. In Fig. \ref{['linear']}, Xsens is attached to a carrier that is connected to a rail. The movement of Xsens is controlled manually. The external acceleration $A_d$ is generated by the movement of Xsens, while the magnetic disturbance $M_d$ is caused by the approaching of pliers. In Fig. \ref{['rotation']}, Xsens is attached to the shank of an exoskeleton, and the exoskeleton is commanded to imitate the walking of a human. The magnitude of $A_d$ depends on the rotation frequency $f$, while $M_d$ is determined by the distance between the pliers and Xsens. The plastic carrier has a height of 12 cm which is sufficient to isolate the magnetic effects caused by rails or exoskeletons.

Theorems & Definitions (3)

• Definition 1
• proof
• proof