Improving the Precision and Speed of Euler Angles Computation from Low-Cost Rotation Sensor Data

Abstract

This article compares three different algorithms used to compute Euler angles from data obtained by the angular rate sensor (e.g., MEMS gyroscope)—the algorithms based on a rotational matrix, on transforming angular velocity to time derivations of the Euler angles and on unit quaternion expressing rotation. Algorithms are compared by their computational efficiency and accuracy of Euler angles estimation. If attitude of the object is computed only from data obtained by the gyroscope, the quaternion-based algorithm seems to be most suitable (having similar accuracy as the matrix-based algorithm, but taking approx. 30% less clock cycles on the 8-bit microcomputer). Integration of the Euler angles’ time derivations has a singularity, therefore is not accurate at full range of object’s attitude. Since the error in every real gyroscope system tends to increase with time due to its offset and thermal drift, we also propose some measures based on compensation by additional sensors (a magnetic compass and accelerometer). Vector data of mentioned secondary sensors has to be transformed into the inertial frame of reference. While transformation of the vector by the matrix is slightly faster than doing the same by quaternion, the compensated sensor system utilizing a matrix-based algorithm can be approximately 10% faster than the system utilizing quaternions (depending on implementation and hardware).

Indoor mobile robot attitude estimation with MEMS gyroscope

“…Rotation vector at the moment t can be obtained from gyroscope readings (scaled in rad/s), using quaternion mathematics, commonly applied to rotation, orientation and tracking problems in navigational systems [9][10][11][12]. For any sufficiently small dt, a rotation vector can be derived directly from 3-axis gyroscope reading: (2) where: rv -rotation vector, gx, gy, gz -gyroscope readings (axes X, Y and Z) scaled to rad/s.…”

Low-Cost Mini Radar: Design Prototyping and Tests

“…Different strategies can be implemented to compute the attitude of a moving object [14][15][16]; moreover, several methods can be used to describe the orientation of a moving object: rotation vectors [13] which describe a rotation by an axis of rotation and an angle around it [17]; Euler angles, which represent rotation of a rigid body by decomposing it into consecutive rotations around three different axes [18]; and quaternions [13]. Although Euler angles have singularities and are less accurate than quaternions when used to integrate incremental changes in attitude over time, they are the most widely used.…”

Intelligent Real-Time MEMS Sensor Fusion and Calibration

“…There are several methods of angular velocity integration into Euler angles; we usually use the matrix-based algorithm. Rotation is expressed as the 3D transformation matrix R updated by the infinitesimal update matrix computed from each sample of the angular velocity [14]: …”