• The Pure and Applied Mathematics
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• v.22 no.1
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• pp.65-74
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• 2015

In this paper, we provide the notions of dual quaternions and their algebraic properties based on matrices. From quaternion analysis, we give the concept of a derivative of functions and and obtain a dual quaternion Cauchy-Riemann system that are equivalent. Also, we research properties of a regular function with values in dual quaternions and relations derivative with a regular function in dual quaternions.

• Journal of the Korean Society for Aeronautical & Space Sciences
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• v.35 no.7
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• pp.627-632
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• 2007

Dual quaternion is efficient methodology to express rotation and translation of the vehicle's movements in the unified frame work. Recently, a strapdown inertial navigation algorithm based on dual quaternion was introduced. By comparing and analyzing the classical and dual-quaternion algorithms, this paper proposes a new strapdown inertial navigation algorithm that maintains the accuracy benefit of the dual-quaternion algorithm with considerable computational reduction. Simulation results show the efficiency of the proposed hybrid strapdown navigation algorithm.

• East Asian mathematical journal
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• v.29 no.3
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• pp.293-302
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• 2013

We research properties of ternary numbers and hyperholomorphic functions with values in $\mathbb{C}$(2). We represent reduced quaternion numbers and obtain some propertries in dual reduced quaternion systems in view of Clifford analysis. Moreover, we obtain Cauchy theorems with respect to dual reduced quaternions.

• Honam Mathematical Journal
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• v.37 no.4
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• pp.559-567
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• 2015

This paper gives the expression of dual quaternions and provides differential operators in dual quaternions. The paper also represents the derivative of dual quaternion-valued functions by using a corresponding Cauchy-Riemann system in dual quaternions.

• 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.

• 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.

• The Pure and Applied Mathematics
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• v.23 no.1
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• pp.97-104
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• 2016

We give representations of differential operators and rules for addition and multiplication of dual quaternions. Also, we research the notions and properties of a regular function and a corresponding harmonic function with values in dual quaternions of Clifford analysis.

• 한국항공운항학회:학술대회논문집
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• 2005.11a
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• pp.132-135
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• 2005

이체 쿼터니언(dual quaternion)은 선형 미분방정식으로 계산하는 쿼터니언을 회전운동과 병진운동을 동시에 취급하는 이체수(dual numbers)체계로 표현한 형태이다. 성능향상을 위해 이체 쿼터니언을 관성항법 알고리즘에 적용시킨 논문이 새로이 소개되었다. 기존 알고리즘과 새로운 알고리즘을 MATLAB으로 시뮬레이터를 작성하였으며, 정확도와 실행 속도 측면에서 성능을 평가하고자 한다.

• The Pure and Applied Mathematics
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• v.20 no.4
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• pp.251-258
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• 2013

We define an ${\varepsilon}$-regular function in dual quaternions. From the properties of ${\varepsilon}$-regular functions, we represent the Taylor series of ${\varepsilon}$-regular functions with values in dual quaternions.

• ETRI Journal
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• v.43 no.6
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• pp.978-990
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• 2021

This article presents techniques for improved estimation of symbol timing offset (STO) and carrier frequency offset (CFO) for dual-polarization (DP) orthogonal frequency division multiplex (DP-OFDM) systems. Recently, quaternion multiple-input multiple-output OFDM has been proposed for high spectral efficiency communication systems, which can flexibly explore different types of diversities such as space, time, frequency, and polarization. This article focuses on synchronization techniques for DP-OFDM systems using a cyclic prefix, where the application of quaternion algebra leads to new improved estimators. Simulations performed for DP system methods show faster reduction of STO estimator variance with a double-slope line in the logvariance line versus signal-to-noise ratio (SNR) plot compared with singlepolarization (SP) counterparts, and simulations for CFO estimates show a 3-dB gain of DP over SP estimates for same SNR values defined, respectively, for quaternion-valued or complex-valued signals. Cramer-Rao bounds for STO and CFO are derived for the synchronization methods, correlating with the observed gains of DP over SP OFDM systems.