Error Analysis of Classical Strapdown Velocity Integration Algorithms under maneuvers

Nomenclature SF a , SFx a , SF y a , SFz a = specific force acceleration vector (sensed by accelerometers) and components in frame B B = sensor (body) coordinate frame aligned with strapdown inertial sensor axes 1 m B  = frame B orientation at computer cycle 1 m  () Bt = frame B orientation at time t k r c = combination, k elements selected from a set of r elements without regard to the order of selection I = identity matrix m = computer interval index; as subscript, indicates parameter value at computer cycle m () o = an equivalent infinitesimal to () r = order of derivatives t = time v = velocity vector 1 V , 2 V , 3 V = general vector parameters α ,  = integral of ω in frame B over 1 (,) m tt  time interval and its magnitude 1 SF () m B t  v = exact continuous form of the integrated transformed specific force increment over 1 (,) m tt  iteration time interval 1 () m B arc t  v = velocity integration solution with the approximate rotation compensation term over 1 (,) m tt  iteration time interval 1 () m B erc t  v = velocity integration solution with the exact rotation compensation term over 1 (,) m tt  iteration time interval 1 () m B vtv t  v = velocity integration solution based on the velocity translation vector with inputs from the high-speed algorithm set c c  η = error in c η η = velocity translation vector in the B frame c η = simplified version of η for algorithm usage () t υ = integral of SF a in frame B over 1 (,) m tt  time interval  ,  = rotation vector equivalent to direction cosine matrix 1 () m B Bt C  and its magnitude () c t  , c  = the simplified rotation vector over 1 (,) m tt  and its magnitude ω , x  , y  , z  = angular rate vector and components in the B frame [() ]  = skew symmetric cross-product matrix form of vector () that indicates [() ] ()   . () = first, second, and third time derivatives of ()