On Finite Rotations and the Noncommutativity Rate Vector

The orientation vector differential equation was first derived by John Bortz to improve the accuracy of strapdown inertial navigation attitude algorithms. These algorithms previously relied on the direct integration of the direction cosine matrix differential equation. A compact derivation of the Bortz equation using geometric algebra is presented. Aside from being as simple and direct as any derivation in the literature, this derivation is also entirely general in that it yields a form of the Bortz equation that is applicable in any dimension, not just the conventional 3D case. The derivation presented has the further advantage that it does not rely on multiple methods of representing rotations and is expressed in a single algebraic framework. In addition to the new derivation, the validity of the notion that it is the effect of the noncommutativity of finite rotations that necessitates the use of such an equation in strapdown inertial navigation systems (SDINS) is questioned, and alternative justification for using the Bortz equation is argued.