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Masaki Kobayashi - One of the best experts on this subject based on the ideXlab platform.
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Split Quaternion-Valued Twin-Multistate Hopfield Neural Networks
Advances in Applied Clifford Algebras, 2020Co-Authors: Masaki KobayashiAbstract:Complex-valued Hopfield neural networks have been extended to 4-dimensional models using Quaternions and commutative Quaternions. The algebra of split Quaternions is another 4-dimensional hypercomplex number system. In this work, a split quaternion-valued Hopfield neural network is defined. By the computer simulations using the twin-multistate activation function and projection rule, we evaluate the noise tolerance.
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Twin-multistate commutative quaternion Hopfield neural networks
Neurocomputing, 2018Co-Authors: Masaki KobayashiAbstract:Abstract The complex-valued Hopfield neural network (CHNN) can deal with multi-level information, and has often been applied to the storage of image data. The quaternion Hopfield neural network (QHNN) is a multistate model of a Hopfield neural network, and requires half the connection weight parameters of CHNN. In this study, we propose a commutative quaternion Hopfield neural network (CQHNN) as the analogy of QHNN. The multiplication of commutative Quaternions is commutative and convenient, unlike that of Quaternions. We compared the noise tolerance of CQHNNs and QHNNs by computer simulation, and discuss the simulation results from the perspective of rotational invariance and self loops.
Panagiotis Tsiotras - One of the best experts on this subject based on the ideXlab platform.
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Extended Kalman Filter for spacecraft pose estimation using dual Quaternions
2015 American Control Conference (ACC), 2015Co-Authors: Nuno Filipe, Michail Kontitsis, Panagiotis TsiotrasAbstract:Based on the highly successful Quaternion Multiplicative Extended Kalman Filter (Q-MEKF) for spacecraft attitude estimation using unit Quaternions, this paper proposes a Dual Quaternion Multiplicative Extended Kalman Filter (DQ-MEKF) for spacecraft pose (i.e., attitude and position) and linear and angular velocity estimation using unit dual Quaternions. By using the concept of error unit dual quaternion, defined analogously to the concept of error unit quaternion in the Q-MEKF, this paper proposes, as far as the authors know, the first multiplicative EKF for pose estimation. Compared to existing literature, the state of the DQ-MEKF only includes six elements of a unit dual quaternion, instead of eight, resulting in obvious computational savings. A version of the DQ-MEKF is presented that takes only discrete-time pose measurements with noise and, hence, is suitable for uncooperative satellite proximity operation scenarios where the chaser satellite has only access to measurements of the relative pose, but requires the relative velocities for control. Finally, the DQ-MEKF is experimentally validated and compared with two alternative EKF formulations on a 5-DOF air-bearing platform.
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Extended Kalman Filter for Spacecraft Pose Estimation Using Dual Quaternions
Journal of Guidance Control and Dynamics, 2015Co-Authors: Nuno Filipe, Michail Kontitsis, Panagiotis TsiotrasAbstract:Based on the highly successful quaternion multiplicative extended Kalman filter for spacecraft attitude estimation using unit Quaternions, this paper proposes a dual quaternion multiplicative extended Kalman filter for spacecraft pose (i.e., attitude and position) and linear and angular velocity estimation using unit dual Quaternions. By using the concept of error unit dual quaternion, defined analogously to the concept of error unit quaternion in the quaternion multiplicative extended Kalman filter, this paper proposes, as far as the authors know, the first multiplicative extended Kalman filter for pose estimation. The state estimate of the dual quaternion multiplicative extended Kalman filter can directly be used by recently proposed pose controllers based on dual Quaternions, without any additional conversions, thus providing an elegant solution to the output dynamic compensation problem of the full six degree-of-freedom motion of a rigid body. Three formulations of the dual quaternion multiplicative e...
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ACC - Extended Kalman Filter for spacecraft pose estimation using dual Quaternions
2015 American Control Conference (ACC), 2015Co-Authors: Nuno Filipe, Michail Kontitsis, Panagiotis TsiotrasAbstract:Based on the highly successful Quaternion Multiplicative Extended Kalman Filter (Q-MEKF) for spacecraft attitude estimation using unit Quaternions, this paper proposes a Dual Quaternion Multiplicative Extended Kalman Filter (DQ-MEKF) for spacecraft pose (i.e., attitude and position) and linear and angular velocity estimation using unit dual Quaternions. By using the concept of error unit dual quaternion, defined analogously to the concept of error unit quaternion in the Q-MEKF, this paper proposes, as far as the authors know, the first multiplicative EKF for pose estimation. Compared to existing literature, the state of the DQ-MEKF only includes six elements of a unit dual quaternion, instead of eight, resulting in obvious computational savings. A version of the DQ-MEKF is presented that takes only discrete-time pose measurements with noise and, hence, is suitable for uncooperative satellite proximity operation scenarios where the chaser satellite has only access to measurements of the relative pose, but requires the relative velocities for control. Finally, the DQ-MEKF is experimentally validated and compared with two alternative EKF formulations on a 5-DOF air-bearing platform.
Eckhard Hitzer - One of the best experts on this subject based on the ideXlab platform.
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The orthogonal planes split of Quaternions and its relation to quaternion geometry of rotations
Journal of Physics: Conference Series, 2015Co-Authors: Eckhard HitzerAbstract:Recently the general orthogonal planes split with respect to any two pure unit Quaternions f,g ∈ H, f2 = g2 = -1, including the case f = g, has proved extremely useful for the construction and geometric interpretation of general classes of double-kernel quaternion Fourier transformations (QFT) [7]. Applications include color image processing, where the orthogonal planes split with f = g = the grayline, naturally splits a pure quaternionic three-dimensional color signal into luminance and chrominance components. Yet it is found independently in the quaternion geometry of rotations [3], that the pure quaternion units f, g and the analysis planes, which they define, play a key role in the geometry of rotations, and the geometrical interpretation of integrals related to the spherical Radon transform of probability density functions of unit Quaternions, as relevant for texture analysis in crystallography. In our contribution we further investigate these connections.
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The Orthogonal Planes Split of Quaternions and Its Relation to Quaternion Geometry of Rotations
viXra, 2014Co-Authors: Eckhard HitzerAbstract:Recently the general orthogonal planes split with respect to any two pure unit Quaternions $f,g \in \mathbb{H}$, $f^2=g^2=-1$, including the case $f=g$, has proved extremely useful for the construction and geometric interpretation of general classes of double-kernel quaternion Fourier transformations (QFT) [E.Hitzer, S.J. Sangwine, The orthogonal 2D planes split of Quaternions and steerable quaternion Fourier Transforms, in E. Hitzer, S.J. Sangwine (eds.), "Quaternion and Clifford Fourier Transforms and Wavelets", TIM \textbf{27}, Birkhauser, Basel, 2013, 15--39.]. Applications include color image processing, where the orthogonal planes split with $f=g=$ the grayline, naturally splits a pure quaternionic three-dimensional color signal into luminance and chrominance components. Yet it is found independently in the quaternion geometry of rotations [L. Meister, H. Schaeben, A concise quaternon geometry of rotations, MMAS 2005; \textbf{28}: 101--126], that the pure quaternion units $f,g$ and the analysis planes, which they define, play a key role in the spherical geometry of rotations, and the geometrical interpretation of integrals related to the spherical Radon transform of probability density functions of unit Quaternions, as relevant for texture analysis in crystallography. In our contribution we further investigate these connections.
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The orthogonal planes split of Quaternions and its relation to quaternion geometry of
2014Co-Authors: Eckhard HitzerAbstract:Recently the general orthogonal planes split with respect to any two pure unit Quaternions f;g 2 H, f 2 = g 2 = 1, including the case f = g, has proved extremely useful for the construction and geometric interpretation of general classes of double-kernel quaternion Fourier transformations (QFT) (7). Applications include color image processing, where the orthogonal planes split with f = g = the grayline, naturally splits a pure quaternionic three-dimensional color signal into luminance and chrominance components. Yet it is found independently in the quaternion geometry of rotations (3), that the pure quaternion units f;g and the analysis planes, which they define, play a key role in the spherical geometry of rotations, and the geometrical interpretation of integrals related to the spherical Radon transform of probability density functions of unit Quaternions, as relevant for texture analysis in crystallography. In our contribution we further investigate these connections.
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ops qfts a new type of quaternion fourier transforms based on the orthogonal planes split with one or two general pure Quaternions
arXiv: Rings and Algebras, 2013Co-Authors: Eckhard HitzerAbstract:We explain the orthogonal planes split (OPS) of Quaternions based on the arbitrary choice of one or two linearly independent pure unit Quaternions $f,g$. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by $f,g$, or by only one pure unit quaternion $f$, comment on their geometric meaning, and establish inverse transformations. Keywords: Clifford geometric algebra, quaternion geometry, quaternion Fourier transform, inverse Fourier transform, orthogonal planes split
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OPS‐QFTs: A New Type of Quaternion Fourier Transforms Based on the Orthogonal Planes Split with One or Two General Pure Quaternions
arXiv: Rings and Algebras, 2011Co-Authors: Eckhard HitzerAbstract:We explain the orthogonal planes split (OPS) of Quaternions based on the arbitrary choice of one or two linearly independent pure unit Quaternions f,g. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by f,g, or by only one pure unit quaternion f, comment on their geometric meaning, and establish inverse transformations.
Nuno Filipe - One of the best experts on this subject based on the ideXlab platform.
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Extended Kalman Filter for spacecraft pose estimation using dual Quaternions
2015 American Control Conference (ACC), 2015Co-Authors: Nuno Filipe, Michail Kontitsis, Panagiotis TsiotrasAbstract:Based on the highly successful Quaternion Multiplicative Extended Kalman Filter (Q-MEKF) for spacecraft attitude estimation using unit Quaternions, this paper proposes a Dual Quaternion Multiplicative Extended Kalman Filter (DQ-MEKF) for spacecraft pose (i.e., attitude and position) and linear and angular velocity estimation using unit dual Quaternions. By using the concept of error unit dual quaternion, defined analogously to the concept of error unit quaternion in the Q-MEKF, this paper proposes, as far as the authors know, the first multiplicative EKF for pose estimation. Compared to existing literature, the state of the DQ-MEKF only includes six elements of a unit dual quaternion, instead of eight, resulting in obvious computational savings. A version of the DQ-MEKF is presented that takes only discrete-time pose measurements with noise and, hence, is suitable for uncooperative satellite proximity operation scenarios where the chaser satellite has only access to measurements of the relative pose, but requires the relative velocities for control. Finally, the DQ-MEKF is experimentally validated and compared with two alternative EKF formulations on a 5-DOF air-bearing platform.
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Extended Kalman Filter for Spacecraft Pose Estimation Using Dual Quaternions
Journal of Guidance Control and Dynamics, 2015Co-Authors: Nuno Filipe, Michail Kontitsis, Panagiotis TsiotrasAbstract:Based on the highly successful quaternion multiplicative extended Kalman filter for spacecraft attitude estimation using unit Quaternions, this paper proposes a dual quaternion multiplicative extended Kalman filter for spacecraft pose (i.e., attitude and position) and linear and angular velocity estimation using unit dual Quaternions. By using the concept of error unit dual quaternion, defined analogously to the concept of error unit quaternion in the quaternion multiplicative extended Kalman filter, this paper proposes, as far as the authors know, the first multiplicative extended Kalman filter for pose estimation. The state estimate of the dual quaternion multiplicative extended Kalman filter can directly be used by recently proposed pose controllers based on dual Quaternions, without any additional conversions, thus providing an elegant solution to the output dynamic compensation problem of the full six degree-of-freedom motion of a rigid body. Three formulations of the dual quaternion multiplicative e...
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ACC - Extended Kalman Filter for spacecraft pose estimation using dual Quaternions
2015 American Control Conference (ACC), 2015Co-Authors: Nuno Filipe, Michail Kontitsis, Panagiotis TsiotrasAbstract:Based on the highly successful Quaternion Multiplicative Extended Kalman Filter (Q-MEKF) for spacecraft attitude estimation using unit Quaternions, this paper proposes a Dual Quaternion Multiplicative Extended Kalman Filter (DQ-MEKF) for spacecraft pose (i.e., attitude and position) and linear and angular velocity estimation using unit dual Quaternions. By using the concept of error unit dual quaternion, defined analogously to the concept of error unit quaternion in the Q-MEKF, this paper proposes, as far as the authors know, the first multiplicative EKF for pose estimation. Compared to existing literature, the state of the DQ-MEKF only includes six elements of a unit dual quaternion, instead of eight, resulting in obvious computational savings. A version of the DQ-MEKF is presented that takes only discrete-time pose measurements with noise and, hence, is suitable for uncooperative satellite proximity operation scenarios where the chaser satellite has only access to measurements of the relative pose, but requires the relative velocities for control. Finally, the DQ-MEKF is experimentally validated and compared with two alternative EKF formulations on a 5-DOF air-bearing platform.
Yusuf Yayli - One of the best experts on this subject based on the ideXlab platform.
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Lie Algebra of Unit Tangent Bundle
Advances in Applied Clifford Algebras, 2016Co-Authors: Murat Bekar, Yusuf YayliAbstract:In this paper, semi-Quaternions are studied with their basic properties. Unit tangent bundle of \({{\mathbb {R}}^2}\) is also obtained by using unit semi-Quaternions and it is shown that the set \({{T {\mathbb {R}}^2}}\) of all unit semi-Quaternions based on the group operation of semi-quaternion multiplication is a Lie group. Furthermore, the vector space matrix of angular velocity vectors forming the Lie algebra \({{T_{1} {\mathbb {R}}^2}}\) of the group \({{T {\mathbb {R}}^2}}\) is obtained. Finally, it is shown that the rigid body displacements obtained by using semi-Quaternions correspond to planar displacements in \({{\mathbb {R}^3}}\).
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Canal Surfaces with Quaternions
Advances in Applied Clifford Algebras, 2016Co-Authors: Selahattin Aslan, Yusuf YayliAbstract:Quaternions are more usable than three Euler angles in the three dimensional Euclidean space. Thus, many laws in different fields can be given by the Quaternions. In this study, we show that canal surfaces and tube surfaces can be obtained by the quaternion product and by the matrix representation. Also, we show that the equation of canal surface given by the different frames of its spine curve can be obtained by the same unit quaternion. In addition, these surfaces are obtained by the homothetic motion. Then, we give some results.
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Spline Split Quaternion Interpolation in Minkowski Space
Advances in Applied Clifford Algebras, 2013Co-Authors: Raheleh Ghadami, Javad Rahebi, Yusuf YayliAbstract:Spherical spline quaternion interpolation has been done on sphere in Euclidean space using Quaternions. In this paper, we have done the spline split quaternion interpolation on hyperbolic sphere in Minkowski space using split Quaternions and metric Lorentz. This interpolation curve is called spherical spline split quaternion interpolation in Minkowski space (MSquad).
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Involutions of Complexified Quaternions and Split Quaternions
Advances in Applied Clifford Algebras, 2013Co-Authors: Murat Bekar, Yusuf YayliAbstract:An involution or anti-involution is a self-inverse linear mapping. Involutions and anti-involutions of real Quaternions were studied by Ell and Sangwine [ 15 ]. In this paper we present involutions and antiinvolutions of biQuaternions (complexified Quaternions) and split Quaternions. In addition, while only quaternion conjugate can be defined for a real quaternion and split quaternion, also complex conjugate can be defined for a biquaternion. Therefore, complex conjugate of a biquaternion is used in some transformations beside quaternion conjugate in order to check whether involution or anti-involution axioms are being satisfied or not by these transformations. Finally, geometric interpretations of real quaternion, biquaternion and split quaternion involutions and anti-involutions are given.
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Matrix Representation of Dual Quaternions
gazi university journal of science, 2013Co-Authors: Mehdi Jafari, Mücahit Meral, Yusuf YayliAbstract:After a review of some properties of dual Quaternions, De Moivre's and Euler's formulas for the matrices associated with these Quaternions are studied. In special case, De Moivre's formula implies that there are uncountably many matrices of unit dual Quaternions satisfying 4 A n= I for n≥3. Also; we give the relation between the powers of matrices of dual Quaternions. Key words:De-Moiver's formula, Hamilton operator, Dual quaternion 2010 Mathematics Subject Classification: 11R52; 15A99
