Solution of Approximate Equation for Modified Rodrigues Vector and Attitude Algorithm Designстатья
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Аннотация:DURING operation of strapdown inertial navigation systems(SINS), a vector responsible for the orientation of a rigid body(object) in space can be periodically calculated by the methodof approximate solution of the kinematic equation with respect tothe vector of modified Rodrigues parameters (MRPs) or, in otherwords, the kinematic equation written in three-dimensional skewsymmetricoperators [1] (in the theory and practice of SINS construction,in ultrarapid cycles of algorithms for small angles ofrotation, the nonlinear terms in this equation are neglected). Theangular velocity vector of a rigid body is the input quantity in theequation. Note that the complete nonlinear equation for the MRPsorientation vector of a rigid body is an analog of the full quaternionlinear equation [1,2]; the vector and the quaternion of a rigid bodyorientation are linked by known relations. The approximate lineardifferential equation in skew-symmetric operators is solved byvarious numerical methods, for example, by Picard’s method, andthen, the second iteration of this method in the practice of SINS canbe taken for the final one. This term in the iteration formula ofPicard’s method is called a noncommutative rotation vector, or“coning.” For certain motions of a rigid body, this term makes asignificant contribution to the error of the method. The study ofnoncommutative rotation (or term of the “coning” type) as a kind ofmechanical motion of bodies, separation of numerical algorithmsfor determining the orientation of a rigid body (SINS) for rapid andslow counting cycles are aimed at compensation for the effect of thisphenomenon [3–6]. Meanwhile, for some new angular velocityvector, which is obtained in determining the orientation of a rigidbody (SINS), based on the initial arbitrary angular velocity vectorin unambiguous interchanges of variables in the motion equationsfor a rigid body, the approximate differential equation in skewsymmetricoperators admits of an exact analytic solution. We willshow this.