# Analytical Formulation of Relationship Between Sensors and Euler Angle Errors for Arbitrary Stationary Alignment Based on Accelerometer and Magnetometer

## TL;DR

This paper develops a mathematical model to understand how sensor errors affect orientation measurements in devices like AHRS, regardless of sensor alignment.

## Contribution

The paper introduces an analytical formulation linking sensor errors to Euler angle errors for arbitrary alignments using accelerometer and magnetometer data.

## Key findings

- The error formulation accurately describes total and partial Euler angle errors for aligned and misaligned sensor configurations.
- Sensor error factors like scale, installation, and offset are shown to influence Euler angle errors through linear relationships.
- Simulations confirmed the formulation's effectiveness in analyzing error contributions under arbitrary Euler angles.

## Abstract

An attitude and heading reference system (AHRS) based on the inertial measurement unit is crucial for various applications. In an AHRS, stationary alignments are performed to determine the initial orientation of the sensor frame with respect to the navigation frame. However, the stationary alignment accuracy is affected by sensor error factors. Therefore, several studies have attempted to analyze and minimize the effects of these errors. However, there have been no studies describing and analyzing the Euler angle errors for various sensor orientations. This paper presents the analytical formulation of the relationship between the sensor and the Euler angle errors based on accelerometer and magnetometer signals, regardless of alignment between the sensor and the navigation frames. We selected three-axis attitude determination (TRIAD) as the stationary alignment method and considered the scale, installation, and the offset errors, including noise and constant bias, as sensor error factors. The presented formulation describes the relationship between the sensor error factors and the Euler angle errors as a linear equation. To analyze the Euler angle errors, we performed both sensor-aligned and sensor-misaligned simulations in which the Euler angles were 0° and arbitrary, respectively. The results showed that the presented error formulation could describe the total Euler angle errors and the partial errors induced by each sensor error factor for both the sensor-aligned conditions and the arbitrary Euler angle configurations. Thus, the effects of each sensor error factor on the Euler angle errors can be analytically investigated using the presented formulations for random alignment.