Analytical Solutions in the Darboux Problem and the Bortz Equation and the Approach to Orientation Algorithms Based on Them

We consider the problem of determining the angular position of a rigid body in space from its known angular velocity and initial position (the Darboux problem) in the quaternion setting. Based on the exact solution of the Bortz approximate differential equation with respect to the orientation vector of the rigid body, we analytically solve the problem to determine the quaternion of the orientation of the rigid body for each arbitrary angular velocity and small rotation angle of the rigid body. Based on this solution, we propose an approach to design a new algorithm to compute the orientation of moving objects by means of strapdown inertial navigation systems.