Design and Evaluation of High-order Non-commutativity Error Compensation Algorithm in Dynamics
Maosong Wang, National University of Defense Technology, China & The University of Calgary, Canada; Wenqi Wu, Xiaofeng He, National University of Defense Technology, China
Location: Big Sur
Attitude updating is an essential part for strapdown inertial navigation systems, and the corresponding algorithms are widely used and studied in modern days. For example, the rotation vector based algorithms [1, 2], the angular rate based algorithm , the Rodrigues vector iteration based algorithm , the Picard iteration based algorithm . But the most widely used is the rotation vector based angular increment coning algorithm. Miller first introduced the concept of improving the accuracy of the coning correction by using coning vibration environment . Ignagni  further proved that the coefficients of the algorithms which were optimized under pure coning environments were also optimal under generalized vibrational environments. Then, Ignagni  developed a compressed form coning algorithm based on Taylor series expansion in powers of coning frequency, and proved that the uncompressed form was equivalent to the compressed form in pure coning environments. The compressed form assumes that the cross products between time displaced integrated rate samples are a function of time only, while the uncompressed form assumes that all the possible cross products of the integrated angular rate samples are taken into account . The same frequency-series expansion based method was then used in later in [9, 10]. Savage  presented an approach called the explicit frequency shaping (EFS) for the coning algorithm design, in which the coefficients are designed by using least-squares error minimization rather than the frequency-series expansion to achieve optimal coning performance. These coning algorithms are optimal over different coning frequency ranges, but they are not optimal under general motion conditions . Song’s paper  represents a major advancement in coning algorithm design for optimal performance in both high-frequency vibration and low-frequency maneuver environments. Using a Taylor series expansion of the maneuver profile representation, Song’s approach expanded a previously designed compressed coning algorithm into an uncompressed form that optimizes accuracy under maneuver environment while maintaining the optimality of the original compressed algorithm. Though Song’s paper considered both high frequency and low frequency coning environments, the triple-cross-product terms of the non-commutativity error were not taken into consideration.
In 2015, the third-order terms of non-commutativity error were considered for coning algorithms design, and the algorithms showed higher accuracy than the traditional algorithms in coning and rotation coexisting environments . In contrast to the design process presented in , the two third-order terms of the non-commutativity error are combined in this paper, and the third-order algorithms are derived analytically based on the third-order Picard component solutions of rotation vector. Additionally, the fourth-order algorithm is also developed based on the fourth-order Picard component solutions of the rotation vector, which can further improve the rotation vector compensation accuracy under maneuver environments. A new general rotation vector computational structure is proposed, which allows the use of previous high frequency operated second-order coning algorithms to be combined with the high-order rotation vector correction algorithms (third-order and fourth-order) at a slower attitude update rate. The new algorithms can improve the accuracy of second-order coning algorithms under maneuvers and vibration environments.
The performance of the new algorithms are evaluated based on real experiment data, which are from pure coning environment, pure large angular rate maneuver environment, and large angular rate maneuvering with angular rate vibration coexisting environments. Twenty minutes' simulation show that, the accuracies of the new high-order algorithms have no advantages over traditional algorithms in the small amplitude pure coning motion and pure large angular rate maneuvering environments. However, in both the large amplitude coning and the large angular rate maneuvering with angular rate vibration coexisting environments, the new high-order algorithms perform better adaptability and higher accuracy. Especially when the platform is in severe coning motion and large angular rate coexisting environments, the advantage of the new high-order algorithms is more distinct.
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