Equivalency Between Strapdown Inertial Navigation Coning and Sculling Integrals/Algorithms

Abstract

This paper develops a generic equivalency between strapdown inertial navigation coning and sculling integrals and algorithms. The equivalency allowsa previously derived coning algorithm to be converted to itscorresponding sculling algorithm using a simple mathematical formula. Two examples are provided illustrating the coning-to- sculling algorithm conversion process. The results are verie ed by comparing them against previously derived coning and sculling algorithms. WO key calculations performed in strapdown inertial naviga- tion systems are updating the body frame (inertial sensor axes ) attitudeandupdating thevehiclevelocity.The attitudeupdatecalcu- lation includes an integral term (denotedas coning) that is nonzero when the vehicle' s angular rate vector is rotating. The velocity up- date calculation includes an integral term (denotedas sculling) that is nonzero when the vehicle' s angular rate or specie c force accel- eration vector is rotating, or when the ratio of the angular rate to specie c force acceleration magnitude is not constant. To improvetheaccuracy ofthe attitudeand velocityupdatecalcu- lations, particularly in environments where the angular rate vector or specie c force acceleration vector rotation rate is large, high-rate algorithms have been developed for the coning and sculling inte- grals. The e rst detailed optimization of algorithms for the coning integral appeared in a paper by R. Miller. 1 Miller' s procedure was then applied and extended in a variety of papers, two of which are Refs. 2 and 3. A detailed description of the coning integral algo- rithm design process is provided in Ref. 4. Work on the design of sculling algorithms has not been as extensive as that of coning algo- rithms.Somerecentworkdetailingthedesignofscullingalgorithms is provided in Refs. 5 -7. In Ref. 5 Savage provides an analytical de- scription of sculling in two forms; the e rst having only one term, denoted as the composite sculling/velocity rotation compensation integral, and the second having two terms, denoted as velocity rota- tioncompensationandthescullingintegral.Anexampleisprovided that develops a digital algorithm for calculating the sculling inte- gral term. In Ref. 6 Ignagni derives a class of optimized sculling algorithms for the composite sculling/velocity rotation compensa- tionintegral and demonstrates aduality between the derived classof sculling algorithms and a previously derived class of coning algo- rithms. In addition, Ref. 6 provides a detailed example illustrating the derivation of one sculling algorithm solution and compares it to a previously derived coning algorithm solution (Ref. 2, algorithm 3). In Ref. 7 Mark and Tazartes develop a sculling algorithm using a different approach than that in Refs. 5 and 6. Both approaches are valid and have been successfully applied in strapdown systems. Thispaperonlydeals with sculling algorithmforms found inRefs.5 and 6. This paper develops a generic equivalency between the coning andsculling integrals and extends it to algorithms that take the same form as those in Refs. 5 and 6. The equivalency allows one to con- vert an already derived coning algorithm to its sculling algorithm counterpart using a simple mathematical formula. The paper e rst introduces the coning and sculling integral equations. Generic inte- gral/algorithm equivalencies are then developed showing how con-