On Position Translation Vector

In [1], Savage described a unified mathematical framework for strapdown inertial navigation algorithm design. The commenting note [2] showed that the " velocity translation vector " (VTV) in the Savage's framework is equivalent to the dual part of an appropriate screw vector in the dual quaternion representation [3] and significantly simplified the VTV rate equation by way of the screw vector rate equation. The current note further derives a new " position translation vector " (PTV) with remarkably simpler rate equation, and proves its connections with Savage's PTV [1]. The developments in this note follow [2, 3] closely for symbol consistency. The " translation " in the context of the dual quaternion approach is a local concept between two considered frames (Footnote 4 in [3]). It could be velocity or position depending on the specific frames under investigation. Where necessary in the sequel, we discriminate the local velocity and position concepts by subscripts v and p , respectively. Define a new thrust position frame that is aligned with the body frame in attitude. The vector from the Earth's origin to the thrust position frame's origin is the integrated thrust velocity [3], namely, the double integrated specific force. When it comes to consider the inertial frame and the thrust position frame, we have the twist between the two frames as , N v     ω ω t  (1) where ω denotes the gyroscope-measured body angular rate and N v t is the body-referenced thrust velocity (see t σ σ t σ σ σ (2) where σ and v  σ are, respectively, the real part and the dual part of the dual quaternion representing the general 2 displacement from the inertial frame to the thrust frame [2, 3]. Then by mimicking the VTV derivation according to Eq. (65) in [3], the double integrated specific force (referenced in the starting frame of the update interval) is expressed as (3) where σ and p  σ are, respectively, the real part and the dual part of the dual quaternion representing the general displacement from the inertial frame to the thrust position frame. Equation (3) is defined differently from the third equation of Eq. (8) in [1], so the dual part p  σ is regarded as a new PTV. The new PTV has an obvious physical meaning in that the dual part of the associated twist in Eq. (1) …