New algorithms for determining the inertial orientation of an object

Kinematic equations and algorithms for the operation of strapdown inertial navigation systems intended for the high-accuracy determination of the inertial orientation parameters (the Euler (Rodrigues–Hamilton) parameters) of a moving object are considered. Together with classical orientation equations, Hamilton's quaternions and new kinematic differential equations in four-dimensional (quaternion) skew-symmetric operators are used that are matched with the classical rotation quaternion and the quaternion rotation matrix using Cayley's formulae. New methods for solving the synthesized kinematic equations are considered: a one-step quaternion orientation algorithm of third-order accuracy and two-step algorithms of third- and fourth-order accuracy in four-dimensional skew-symmetric operators for calculating the parameters of the spatial position of an object. The algorithms were constructed using the Picard method of successive approximations and employ primary integral information from measurements of the absolute angular velocity of the object as the input information, and have advantages over existing algorithms of a similar order with respect to their accuracy and simplicity.