Orientation and Kinematics of Rotation: Quaternionic and Four-Dimensional Skew-Symmetric Operators, Equations, and Algorithms
Tóm tắt
This paper develops a theory of three- and four-dimensional skew-symmetric rotation operators generated by exponential representations of orthogonal operators or their representations using Cayley’s formulas. The generating orthogonal operators involve the direction angle cosine matrix, the quaternionic matrix of Euler’s parameters, and the Hamilton rotation quaternion. Novel matrix and quaternion kinematic equations for the rotation of a rigid body based on four-dimensional skew-symmetric matrices and on quaternions with zero scalar parts (in associated quaternions) are proposed. It is shown that they are advantageous compared to the known kinematic equations of rotation based on three-dimensional skew-symmetric matrices and vector kinematic equations. As a topical application of the proposed equations, the paper considers the construction of high-precision algorithms for determining the orientation of a moving object in an inertial coordinate system using a strapless inertial navigation system. Fourth-order (even-order) skew-symmetric matrices and associated quaternions have qualitative advantages over third-order (odd-order) skew-symmetric matrices and vectors. This makes the use of the proposed kinematic equations of rotation in orientation and navigation problems much more efficient compared to traditionally used equations in three-dimensional skew-symmetric operators.
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