Comparative study of attitude control methods based on Euler angles, quaternions, angle–axis pairs and orientation matrices

Abstract

This paper presents a comparative study about the attitude control methods based on four commonly used error indicators, namely the triad of 3-2-1 deviational Euler angles, the error quaternion, the deviational angle–axis pair and the orientation error matrix. These error indicators are used here with the same backstepping control law to have a common basis of comparison. This control law makes the controller track a restoring angular velocity generated here specifically for each error indicator. This comparative study shows that all these error indicators can be used satisfactorily even in the critical orientations associated with them by taking special measures. For the deviational Euler angle triad, a critical orientation is a singularity, at which the angles become indefinite. Unless the vehicle is stationary, this indefiniteness is resolved here by applying L’Hopital’s Rule on the angular velocity information. The Euler angle triad has also a multiplicity problem. It is solved here by using the novel criterion of minimal deviation angles. For the other error indicators, a critical orientation is an antipodal orientation, which is opposite to the desired one. In an antipodal orientation, the error quaternion and the deviational angle–axis pair cannot be determined through the customary formulas. They are determined here by using the novel specially introduced formulas. Besides, they may suffer from the unwinding phenomenon in an ordinary orientation. This phenomenon is prevented here by keeping the scalar part of the error quaternion non-negative and the deviation angle between 0° and 180°. For the orientation error matrix, a stationary and undisturbed antipodal orientation is an unstable equilibrium, in which the ordinary backstepping control law becomes ineffective for driving the system into action. This problem is solved here by adding an extra term to the ordinary backstepping control law.

State Estimation of Axisymmetric Target Based on Beacon Linear Features and View Relation

“…Each experiment fixes the total effective frames = 100, under the condition of three kinds of detection diagram, as shown in Figure 8; blue box, blue dot, green line, and red dot represent target detection and its confidence, beacon corner, straight line feature, and center point detection results, respectively. Regarding the description of the rotation state of the target object, there are various methods to describe the rotation state, and the commonly used methods are Euler angles [30] and quaternion [31]. Among them, the quaternion state description method tends to use numbers with certain calculation rules to describe the rotation state of the target object, which can be displayed intuitively through the certain calculation.…”

“…The measurement results expressed by the RPY angle are shown in Tables 2-4, respectively. Regarding the description of the rotation state of the target object, there are various methods to describe the rotation state, and the commonly used methods are Euler angles [30] and quaternion [31]. Among them, the quaternion state description method tends to use numbers with certain calculation rules to describe the rotation state of the target object, which can be displayed intuitively through the certain calculation.…”

Linear Continuous Sliding Mode-based Attitude Controller with Modified Rodrigues Parameters Feedback

“…The rigid body satellite dynamics with a 3 × 3 symmetric matrix of inertia calculated about its center of mass J > 0 Kg • m 2 , J ∈ R 3×3 , and the control torque expressed in the body frame τ (N • m), τ ∈ R 3 , is defined in (1). The attitude kinematics represented by modified Rodrigues parameters (MRP) is shown in ( 2)-( 3) [5], where [σ db ]…”

“…Research efforts have been conducted to design the attitude control systems using various kinematics representations. For instance, Ozgoren [1] reported a comparison of the attitude control systems design based on Euler angles, quaternion, Euler angleaxis pairs, and orientation matrices (or rotation matrices [2]). Another kinematics representation that has been also widely studied is modified Rodrigues parameters (MRP), e.g., [3], [4], [5], to name a few.…”

GNSS attitude estimation based on adaptive Kalman filtering using phase measurement

“…These methods are introduced as follows. The former is to first calculate the baseline coordinate vectors in the reference coordinate system which can be the earth‐centred‐earth‐fixed (ECEF) frame or local level system (LLS), and then the attitude parameters expressed as Euler angles, Gibbs vectors, quaternions or attitude matrix [25–27] are calculated according to the transformation between the reference coordinate system and the BF. The direct attitude determination (Direct) method and the LS attitude estimation method are carried out in the position domain, the accuracy of those two methods is compared and analysed in [14].…”