Quaternion Research Articles

SPIRAL: An efficient algorithm for the integration of the equation of rotational motion

We introduce Spiral, a third-order integration algorithm for the rotational motion of extended bodies. It requires only one force calculation per time step, does not require quaternion normalization at each time step, and can be formulated for both leapfrog and synchronous integration schemes, making it compatible with many particle simulation codes. The stability and precision of Spiral exceed those of state-of-the-art algorithms currently used in popular DEM codes such as Yade, MercuryDPM, LIGGGHTS, PFC, and more, at only slightly higher computational cost. Also, beyond DEM, we see potential applications in all numerical simulations that involve the 3D rotation of extended bodies.

Orientation Quaternion Norm Correction in SINS Algorithms: Calculation Normalization Schemes and Their Efficiency

The problem of correcting the norm of the computed orientation quaternion in the algorithms for the operation of strapdown inertial navigation systems is considered. Two existing approaches to the correction process are considered, the first approach is to normalize the rotation quaternion at the computation cycle, the second approach is to normalize the resulting quaternion. 5 well-known calculation schemes for norm correction are given. To simulate the test motion, we used an analytical quaternion kinematic rotation model based on a sequence of three rotations corresponding to the Krylov angles. The case of linear dependence of elementary rotation angles on time is considered. The model provides analytical representation of the projections of the angular velocity vector of the rigid body on the associated axes and the corresponding quasi-coordinates on the calculation cycle. The results of numerical simulation of the reference motion for a given set of frequencies are presented as dependences of the projections of the angular velocity vector of the rigid body on time and the constructed trajectories in the configuration space of the orientation parameters.
 To determine the rotation quaternion on a cycle, the Miller algorithm was used, which makes it possible to obtain an increase in the orientation vector based on ideal information from the angular velocity sensors in the form of quasi-coordinates. The transformation into a rotation quaternion occurs with the help of the corresponding expansions of the trigonometric functions of the true rotation angle (modulus of the orientation vector) in a series.
 Based on a numerical experiment, it is shown that the best result of correcting the norm of the calculated quaternion in the sense of the minimum error of the norm is given by one of the finite normalization schemes, for which there is no division operation and ensures the stability of the norm correction in time. The results of numerical simulation of the model rotational motion of a rigid body and the development of schemes for correcting the norm of the calculated orientation quaternion are presented.
 Keywords: orientation quaternion, Miller's orientation algorithm, SINS, norm error, computational drift, analytical reference model, quasi-coordinates, numerical simulation.

Calculation of the results of basic arithmetic operations with quaternions defined in the interval form

The goal of the work. Creation of an EXCEL-oriented calculator for determining the results of basic arithmetic operations with quaternions, which are given by hyperbolic interval numbers. The subject of research is the set of quaternions and arithmetic operations defined on this set. Research method: Algorithmic and numerical analysis of procedures for performing basic arithmetic operations with quaternions given by hyperbolic interval numbers. The obtained results. The definition of a quaternion, the coefficients of which are interval numbers, is given. It was obtained the ratios for determining in the interval form the following characteristics: quaternion norm, quaternion modulus, quaternion vector part modulus, quaternion argument, quaternion sign was obtained. It was obtained the ratios for determining in the interval form the following basic arithmetic operations: sum of quaternions, difference of quaternions, multiplication of a constant value by a quaternion, scalar product of quaternions, product of vector parts of quaternions, product of quaternions, vector product of quaternions, left and right division of quaternions. The ratio for calculating the determinants of the second order, the elements of which are defined in the interval form, is obtained. The ratio for the approximate calculation of determinants of arbitrary order, the elements of which are defined in the interval form, is obtained. It is shown that the operations of multiplication (division) of interval numbers and raising them to an integer power are expedient to perform when they have a hyperbolic form. It is advisable to perform the addition (subtraction) operation with interval numbers given in the classic form or in the CENTER-RADIUS system. It is better to use the last form in the case of determining the coefficients of the quaternions based on the results of technological calculations. Screenshots of the formulas for determining the vector product of quaternions are given, provided that the coefficients at their orths are represented by interval numbers.

Adaptive robust maximum correntropy cubature Kalman filter for spacecraft attitude estimation

In this paper, an adaptive robust maximum correntropy cubature Kalman filter (AMCCKF) for spacecraft attitude estimation is proposed. The main aim is to enable the algorithm to resist the influence of noise non-Gaussian and insufficient statistical information on process noise in the system. Based on the maximum correntropy criterion in this study, the robust filtering algorithm is derived by establishing the regression model. In addition, the Cauchy kernel function is used to replace the Gaussian kernel function in the cost function to prevent a singular matrix from appearing in algorithm operation. Furthermore, the process noise is corrected by an adaptive fading factor. The quaternion normalization problem is solved by constructing a minimum constraint cost function using additive quaternion property. The simulation results show that the proposed AMCCKF algorithm has better estimation accuracy and robustness under the conditions of non-Gaussian noise and insufficient statistical information of process noise.

Quaternion Scalar and Vector Norm Decomposition: Quaternion PCA for Color Face Recognition.

This paper proposes a decomposition called quaternion scalar and vector norm decomposition (QSVND) for approximation problems in color image processing. Different from traditional quaternion norm approximations that are always the single objective models (SOM), QSVND is adopted to transform the SOM into the bi-objective model (BOM). Furthermore, regularization is used to solve the BOM problem as a common scalarization method, which converts the BOM into a more reasonable SOM. This can handle over-fitting or under-fitting problems neglected in this kind of research for quaternion representation (QR) in color image processing. That is how to treat redundancy caused by the extra scalar part when the vector part of a quaternion is used to represent a color pixel. We apply QSVND to quaternion principal component analysis (QPCA) for color face recognition (FR), which can deal with the phenomenon of under-fitting of vector part norm approximation. Comparisons with the competing approaches on AR, FERET, FEI, and KDEF&AKDEF databases consistently show the superiority of the proposed approach for color FR.

Adaptive quaternion particle filter using generalized likelihood ratio test for aircraft attitude estimation in the presence of anomalous measurement

An adaptive quaternion particle filter (QPF) based on the generalized likelihood ratio test (GLRT) is proposed for aircraft attitude estimation in the presence of an anomalous measurement. The framework of the QPF is employed to guarantee quaternion normalization. To cope with an anomalous measurement when interference occurs, the proposed algorithm uses the GLRT to detect an anomalous measurement and determine the interference source. An adaptation scheme is used to tune the measurement noise covariance matrix after the interference source has been determined so that the algorithm can remain robust against interference. Numerical simulations are carried out to evaluate the performance of the proposed algorithm and compare it with the QPF algorithm. The results show that the proposed algorithm outperforms the standard one in attitude estimation accuracy under anomalous measurement conditions.

Quaternions: what are they, and why do we need to know?

Quaternions: what are they, and why do we need to know?

Registration of Three-Dimensional Point Clouds Based on Quaternions Using Linear Features

Registration of Three-Dimensional Point Clouds Based on Quaternions Using Linear Features

The polar form of hyperbolic commutative quaternions

The polar form of hyperbolic commutative quaternions

Solution of an inverse kinematics problem using dual quaternions

Solution of an inverse kinematics problem using dual quaternions

Hopf Hypersurfaces in Complex Two-plane Grassmannians with Generalized Tanaka-Webster Reeb-parallel Structure Jacobi Operator

Regarding the generalized Tanaka-Webster connection, we considered a new notion of \(\mathfrak{D}^ \bot\)-parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G2(ℂm+2) and proved that a real hypersurface in G2(ℂm+2) with generalized Tanaka-Webster \(\mathfrak{D}^ \bot\)-parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍPn in G2(ℂm+2), where m = 2n.

Attitude quaternion continuous self-calibration model of hybrid inertial navigation system

Attitude quaternion continuous self-calibration model of hybrid inertial navigation system

Convolutional neural network-based method for real-time orientation indexing of measured electron backscatter diffraction patterns

Electron backscatter diffraction (EBSD) is the most commonly used technique for obtaining spatially resolved microstructural information from polycrystalline materials. We have developed two convolutional neural network approaches based on domain transform and transfer learning to reconstruct crystal orientations from electron backscatter diffraction patterns. Our models are robust to experimentally measured image noise and index orientations as fast as the highest EBSD scanning rates. We demonstrate that the quaternion norm metric is a strong indicator for assessing the reliability of the reconstructions in the absence of the ground truth. We demonstrate the applicability of the current methods on a tantalum sample.

Adaptive Robust Nonlinear Filtering for Spacecraft Attitude Estimation Based on Additive Quaternion

This paper proposes a new Huber-based and covariance matching-based adaptive robust spacecraft attitude estimation algorithm. By constructing a nonlinear regression model, the measurement noise based on robust filtering is derived. The error function is used to improve the Huber density function, which can enhance the estimation accuracy of the robust algorithm. In addition, the process noise is corrected by introducing a fading factor. The Kalman filter gain derived from the additive quaternion is employed to satisfy the quaternion normalization constraint and avoid the covariance singularity. Simulation results show that the proposed adaptive robust attitude estimation algorithms are more robust than the existing methods under large initial attitude error conditions.

The Conjugate Frame Method Relativity

Researchers typically use four dimensional vectors in Minkowski space-time to derive Albert Einstein's Special Theory of Relativity equations. In this work, I have created an original method that uses only scalar quantities in the derivation of relativity equations. The Conjugate Frame Method reproduces the time, mass, and length equations of Special Relativity. Equations for the relativistic electric charge and relativistic temperature are also derived using this method. Unlike the equations of Special Relativity, the Conjugate Frame Method Relativity equations are applicable to both inertial and non-inertial reference frames. The creation of the Conjugate Frame Method was motivated by work done in real quaternion relativity [1,2].

Determinantal Representations of Solutions to Systems of Quaternion Matrix Equations

Within the framework of the theory of quaternion column–row determinants and using determinantal representations of the Moore–Penrose inverse previously obtained by the author, we get explicit determinantal representation formulas of solutions (analogs of Cramer’s rule) to the systems of quaternion matrix equations $$ \mathbf{A}_{1}{} \mathbf{X}=\mathbf{C}_{1}$$ , $$ \mathbf{X}\mathbf{B}_{2}=\mathbf{C}_{2} $$ , and $$ \mathbf{A}_{1}{} \mathbf{X}=\mathbf{C}_{1}$$ , $$\mathbf{A}_{2} \mathbf{X}=\mathbf{C}_{2} $$ .

Adaptive robust cubature Kalman filtering for satellite attitude estimation

This paper is concerned with the adaptive robust cubature Kalman filtering problem for the case that the dynamics model error and the measurement model error exist simultaneously in the satellite attitude estimation system. By using Hubel-based robust filtering methodology to correct the measurement covariance formulation of cubature Kalman filter, the proposed filtering algorithm could effectively suppress the measurement model error. To further enhance this effect and reduce the impact of the dynamics model error, two different adaptively robust filtering algorithms, one with the optimal adaptive factor based on the estimated covariance matrix of the predicted residuals and the other with multiple fading factors based on strong tracking algorithm, are developed and applied for the satellite attitude estimation. The quaternion is employed to represent the global attitude parameter, and three-dimensional generalized Rodrigues parameters are introduced to define the local attitude error. A multiplicative quaternion error is derived from the local attitude error to maintain quaternion normalization constraint in the filter. Simulation results indicate that the proposed novel algorithm could exhibit higher accuracy and faster convergence compared with the multiplicative extended Kalman filter, the unscented quaternion estimator, and the adaptive robust unscented Kalman filter.

Semi-automatic Camera Calibration Using Quaternions

Semi-automatic Camera Calibration Using Quaternions

Robust and Accurate Capture of Human Joint Pose Using an Inertial Sensor.

Wearable inertial measurement units (IMU) measuring acceleration, earth magnetic field, and gyroscopic measurements can be considered for capturing human skeletal postures in real time. Number of movement disorders require accurate and robust estimation of the human joint pose. Though these movements are inherently slow, the accuracy of estimation is vital as many subtle moment patterns, such as tremor are useful to capture under many assessments scenarios. Also, as the end user is a patient with movement disabilities, the practical wearability aspects impose stringent requirements such as the use of minimal number of sensors as well as positioning them in conformable areas of the human body; particularly for longer term monitoring. Estimating skeletal and limb orientations to describe human posture dynamically via model-based approaches poses numerous challenges. In this paper, we convey that the use of measurement conversion ideas-a representation signifying a linear characterization of an inherently non linear estimation problem, pragmatically improves the overall estimation of the limb orientation. A quaternion, as opposed to the Euler angle-based approach is adopted to avoid Gimbal lock scenarios. We also lay a systematic basis for quaternion normalization, typically performed in the pre-filtering stage, by introducing an optimization-based mathematical justification. A robust version of the extended Kalman filter is configured to amalgamate the underlying ideas in enhancing the overall system performance while providing a structured and a comprehensive approach to IMU-based real time human pose estimation problem, particularly in a movement disability capture context.

Enforcing an Algebraic Constraint in Extended Kalman Filter Design

A new method is presented for optimally enforcing an algebraic constraint in Kalman filtering. The method is illustrated in the setting of inertial navigation using extended Kalman filtering. It is shown that to first order in the estimation error the constrained optimal update of the quaternion portion of the state vector is a projection of the unconstrained update onto the constraint space, and the update increment for the quaternion is perpendicular to the a priori estimate. The procedure is easy to implement and appears to provide a nearly optimal solution to the problem of quaternion normalization in the setting of inertial navigation. More importantly, it provides a correction for the covariance update that significantly enhances filter stability. Examples involving initial alignment and inertial navigation illustrate the efficacy of the approach, and the method is compared to several other methods in the literature.