• Communications of the Korean Mathematical Society
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• v.17 no.4
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• pp.625-633
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• 2002

Recently, Chen establishes sharp relationship between the k-Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. In this paper, we establish sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in quaternion projective spaces.

• Bulletin of the Korean Mathematical Society
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• v.48 no.5
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• pp.969-990
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• 2011

Assume that X, partitioned into $2{\times}2$ block form, is a solution of the system of quaternion matrix equations $A_1XB_1$ = $C_1,A_2XB_2=C_2$. We in this paper give the maximal and minimal ranks of the submatrices in X, and establish necessary and sufficient conditions for the submatrices to be zero, unique as well as independent. As applications, we consider the common inner inverse G, partitioned into $2{\times}2$ block form, of two quaternion matrices M and N. We present the formulas of the maximal and minimal ranks of the submatrices of G, and describe the properties of the submatrices of G as well. The findings of this paper generalize some known results in the literature.

• Proceedings of the KIEE Conference
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• 2001.07d
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• pp.2092-2095
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• 2001

In this paper, the performance of a new alternative form of three-axis attitude estimation algorithm for a rigid body is evaluated via simulation for the situation where the observed vectors are the estimated baselines of a GPS antenna array. This method is derived based on a simple iterative nonlinear least-squares with four elements of quaternion parameter. The representation of quaternion parameters for three-axis attitude of a rigid body is free from singularity problem. The performance of the proposed algorithm is compared with other eight existing methods, such as, Transformation Method (TM), Vector Observation Method (VOM), TRIAD algorithm, two versions of QUaternion ESTimator (QUEST), Singular Value Decomposition (SVD) method, Fast Optimal Attitude Matrix (FOAM), Slower Optimal Matrix Algorithm (SOMA).

• International Journal of Aeronautical and Space Sciences
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• v.12 no.1
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• pp.24-36
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• 2011

A new approach for spacecraft absolute attitude estimation based on the unscented Kalman filter (UKF) is extended to relative attitude estimation and navigation. This approach for nonlinear systems has faster convergence than the approach based on the standard extended Kalman filter (EKF) even with inaccurate initial conditions in attitude estimation and navigation problems. The filter formulation employs measurements obtained from a vision sensor to provide multiple line(-) of(-) sight vectors from the spacecraft to another spacecraft. The line-of-sight measurements are coupled with gyro measurements and dynamic models in an UKF to determine relative attitude, position and gyro biases. A vector of generalized Rodrigues parameters is used to represent the local error-quaternion between two spacecraft. A multiplicative quaternion-error approach is derived from the local error-quaternion, which guarantees the maintenance of quaternion unit constraint in the filter. The scenario for bounded relative motion is selected to verify this extended application of the UKF. Simulation results show that the UKF is more robust than the EKF under realistic initial attitude and navigation error conditions.

• Journal of Mechanical Science and Technology
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• v.20 no.3
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• pp.376-381
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• 2006

In this paper, the quaternion, the dual Euler, and the direction cosine methods are numerically compared using a non-aerodynamic 6 degree-of-freedom rigid model at all-attitude angles of an aircraft. The dual Euler method turns out to be superior to the others in the applications because it shows better numerical accuracy, stability, and robustness in integration step sizes. The dual Euler method is affordably less efficient than the quaternion method in terms of computational cost. Numerical accuracy and stability, which allow larger integration step sizes, are more critical in modern real-time applications than computational efficiency because of today's increased computational power. If the quaternion method is required because of constraints in computation time, then a suppression mechanism should be provided for algebraic constraint errors which will eventually add computational burden.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1285-1301
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• 2018

In this paper, we mainly consider the determinantal representations of the unique solution and the general solution to the restricted system of quaternion matrix equations $$\{{A_1X=C_1\\XB_2=C_2,}\;{{\mathcal{R}}_r(X){\subseteq}T_1,\;{\mathcal{N}}_r(X){\supseteq}S_1$$, respectively. As an application, we show the determinantal representations of the general solution to the restricted quaternion matrix equation $$AX+Y B=E,\;{\mathcal{R}}_r(X){\subseteq}T_1,\;{\mathcal{N}}_(X){\supseteq}S_1,\;{\mathcal{R}}_l(Y){\subseteq}T_2,\;{\mathcal{N}}_l (Y){\supseteq}S_2$$. The findings of this paper extend some known results in the literature.

• Journal of the Korean Mathematical Society
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• v.34 no.2
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• pp.321-336
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• 1997

Let S be the Ricci curvature of an n-dimensional compact minimal totally real submanifold M of a quaternion projective space $HP^n (c)$ of quaternion sectional curvature c. We proved that if $S \leq \frac{16}{3(n -2)}c$, then either $S \equiv \frac{4}{n - 1}c$ (i.e. M is totally geodesic or $S \equiv \frac{16}{3(n - 2)}c$. All compact minimal totally real submanifolds of $HP^n (c)$ satisfy in $S \equiv \frac{16}{3(n - 2)}c$ are determined.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.81-96
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• 2004

The level-m scaled circulant factor matrix over the complex number field is introduced. Its diagonalization and spectral decomposition and representation are discussed. An explicit formula for the entries of the inverse of a level-m scaled circulant factor matrix is presented. Finally, an algorithm for finding the inverse of such matrices over the quaternion division algebra is given.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.791-807
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• 1997

There are several kinds of orders in a quaternion algebra. In this article, the relation between the orders is studied.

• Journal of the Korean Society for Aeronautical & Space Sciences
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• v.47 no.11
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• pp.795-802
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• 2019

This paper deals with position-attitude coupling motion during spacecraft relative operation, and suggests dual quaternion-based kinematics for the problem. The position-attitude coupling motion can occur when the target point is located at an arbitrary point on the satellite body, not the center of mass. This is especially apparent in close proximity operation case. The dual quaternion-based kinematics directly reflects the angular velocity state, so that the coupling motion in which the change of attitude affects the position can be concisely defined. In this study, a new dual quaternion-based kinematics is presented along with a conventional approach to solve the coupling problem. Numerical simulations show that the position error for the target point is generated by the coupling motion, and verify that the dual quaternion-based kinematics can solve this problem.