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On Finite Rotations and the Noncommutativity Rate Vector

机译：关于有限旋转和非对易率向量

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摘要

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.
机译：方向矢量微分方程首先由John Bortz推导，以提高捷联惯性导航姿态算法的精度。这些算法以前依赖于方向余弦矩阵微分方程的直接积分。提出了使用几何代数的Bortz方程的紧致推导。除了与文献中的任何推导一样简单和直接之外，这种推导也具有普遍意义，因为它产生了可在任何维度上应用的Bortz方程形式，而不仅仅是常规3D情况。提出的推导具有进一步的优势，即它不依赖于表示旋转的多种方法，并且以单个代数框架表示。除了新的推论之外，人们还质疑是否有必要在捷联惯性导航系统（SDINS）中使用这种方程的原因是有限旋转的非对易性的影响，以及使用Bortz方程的其他理由被争论。

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