One case of analytic integrability of a nonstationary matrix ordinary differential equation

Abstract

We find the general solution of the equation for the error in the matrix of direction cosines of the form ẋ = αx + β, where α(t) is a skew-symmetric matrix with zero main diagonal, β(t) is the product of a skew-symmetric matrix by a matrix exponential, and α(t) and β(t) are functions of direction cosines, the angular velocity of the moving trihedral, and the error in the determination of the angular velocity. We point out the possibility of using the general solution to construct approximate formulas for the analysis of the orientation error. © 2008 MAIK Nauka.