Quaternion Method Research Articles

Flexible method for accurate calibration of large-scale vision metrology system based on virtual 3-D targets and laser tracker

For the purpose of obtaining high-precision in stereo vision calibration, a large-size precise calibration target, which can cover more than half of the field of view is vital. However, large-scale calibration targets are very difficult to fabricate. Based on the idea of error tracing, a high-precision calibration method for vision system with large field of view by constructing a virtual 3-D calibration target with a laser tracker was proposed in this article. A virtual 3-D calibration target that covers the whole measurement space can be established flexibly and the measurement precision of the vision system can be traceable to the laser tracker. First, virtual 3-D targets by calculating rigid body transformation with unit quaternion method were constructed. Then, the high-order distortion camera model was taken into consideration. Besides, the calibration parameters were solved with Levenberg–Marquardt optimization algorithm. In the experiment, a binocular stereo vision system with the field of view of 4 × 3 × 2 m3 was built for verifying the validity and precision of the proposed calibration method. It is measured that the accuracy with the proposed method can be greatly improved comparing with traditional plane calibration method. The method can be widely used in industrial applications, such as in the field of calibrating large-scale vision-based coordinate metrology, and six-degrees of freedom pose tracking system for dimensional measurement of workpiece, as well as robotics geometrical accuracy detection and compensation.

Comparative Study on Two Solutions of 3-RPS Parallel Mechanism Pose

In this paper, the performance of the different methods for solving the pose of parallel mechanism is different in the amount and precision of calculation. Taking typical 3-RPS parallel mechanism as the analysis object, kinematics models of the mechanism were established with quaternion and Rodriguez parameter method respectively, and the bar length of the maximum reachable space was solved by genetic algorithm, so as to compare the advantages and disadvantages of the solution of the model. Finally, numerical simulation is used to compare the amplitude and fluctuation degree of the curve according to the smooth spline interpolation in both calculation and precision. The results show that the Rodriguez parameter method is superior to the quaternion method in the accuracy of pose resolution of 3-RPS parallel mechanism, while the quaternion method is superior in the real-time performance of the solution. This result provides a theoretical basis for the pose control of parallel mechanism.

Research on Action Recognition Method Based on Weighted DTW Algorithm

Action recognition technology is an important part of artificial intelligence. In order to improve the rate of action recognition, this paper designs an action recognition method based on Weighted DTW algorithm. Firstly, the Kinect2.0 is used to obtain the three-dimensional data of the human joint points. Then the quaternion method is used to define the action sequences. The weight of joint is calculated according to the participation in different types of action. Then the improved DTW algorithm is used to design the action recognition experiment. The experimental results show that the action recognition algorithm designed in this paper has better recognition rate and timeliness than traditional DTW algorithm and F-DTW algorithm.

Point Cloud Registration Method for Pipeline Workpieces Based on RANSAC and Improved ICP Algorithms

Aiming at the registration problem of laser-scanned workpiece point cloud data, a point cloud registration method based on RANSAC algorithm and improved ICP algorithm is proposed. Firstly, feature points are selected according to the variation law of the normal vector in the original point cloud data, and the initial matching point set is obtained by establishing the histogram (FPFH) of feature points. Then a random sampling consensus (RANSAC) algorithm is applied to the initial data matching. At last, the nearest point iterative algorithm (ICP) accelerated by k-d tree is used for accurate matching, and the quaternion method is used to obtain the registration parameters. The new algorithm and PCA+ICP algorithm are tested and the experimental results are compared. The results show that the new algorithm can achieve registration, and improve the speed and accuracy of registration, which provide a reference for similar problems.

Point Cloud Registration Method for Pipeline Workpieces Based on PCA and Improved ICP Algorithms

Aiming at the registration problem of point cloud data obtained by laser scanning workpiece, a new automatic registration algorithm of point cloud based on PCA algorithm and improved ICP algorithm is proposed. Firstly, the feature points are selected according to the change rule of normal vectors in the original point cloud data, and the initial matching point set is obtained by establishing the histogram of feature points (FPFH); then the principal component analysis (PCA) is used to match the initial data; Finally using the k-d tree to accelerate the improved iterative method is closest point (ICP) precise matching, and using quaternion method of registration parameters are obtained. Experiments are carried out on the proposed new algorithm and PCA+ICP algorithm, and the experimental results are compared. The results show that the new algorithm can achieve registration, and improve the speed and accuracy of registration, which provide a reference for similar problems.

A point cloud registration algorithm based on normal vector and particle swarm optimization

Based on normal vector and particle swarm optimization (NVP), a point cloud registration algorithm is proposed by searching the corresponding points. It provides a new method for point cloud registration using feature point registration. First, in order to find the nearest eight neighbor nodes, the k-d tree is employed to build the relationship between points. Then, the normal vector and the distance between the point and the center gravity of eight neighbor points can be calculated. Second, the particle swarm optimization is used to search the corresponding points. There are two conditions to terminate the search in particle swarm optimization: one is that the normal vector of node in the original point cloud is the most similar to that in the target point cloud, and the other is that the distance between the point and the center gravity of eight neighbor points of node is the most similar to that in the target point cloud. Third, after obtaining the corresponding points, they are tested by random sample consensus in order to obtain the right corresponding points. Fourth, the right corresponding points are registered by the quaternion method. The experiments demonstrate that this algorithm is effective. Even in the case of point cloud data lost, it also has high registration accuracy.

Stability and stabilization of quaternion-valued neural networks with uncertain time-delayed impulses: Direct quaternion method

In this paper, the exponential stability of impulsive quaternion-valued neural networks (IQVNNs) with time delays is investigated. Both the differential equation and impulses are affected by time-varying delays which may be nonidentical. Uncertain parameters in impulses are also considered. Firstly, the existence and uniqueness of the equilibrium of the IQVNNs are discussed with the help of homeomorphic mapping theorem. Then, by utilizing Lyapunov functions and a newly developed inequality, several sufficient criteria are established in the form of quaternion-valued linear matrix inequalities (LMIs). It should be noted that, compared to most of existing results which are obtained by decomposing quaternion-valued neural networks (QVNNs) into four real-valued neural networks (RVNNs) or two complex-valued neural networks (CVNNs), our results are easier to be verified since the quaternion-valued LMIs exhibit lower dimensions. Then, based on obtained results, one impulsive controller is designed to realize the stabilization of delayed QVNNs. Finally, two numerical examples are presented to verify the effectiveness and merits of the theoretical analysis.

Three-dimensional indoor location estimation using single inertial navigation system with linear regression

This paper presents a 3D pedestrian position estimation algorithm using a waist-mounted inertial measurement unit (IMU) sensor. When using the IMU for position estimation, the measurement accuracy is degraded by the influence of the gyroscope’s drift error. This study proposes reducing the accumulated drift error by using the quaternion method to obtain an accurate Euler angle and combining this with step estimation to achieve 2D location estimation. For an indoor pedestrian positioning system, a pedestrian’s height accuracy is related to the reliability of the entire positioning system. In order to enhance 3D location estimation, an estimation algorithm for realizing the indoor height of pedestrians is proposed. This algorithm, which is based on the acceleration value of the accelerometer without using any external equipment, is combined with an algorithm which is additionally proposed herein. Moreover, to obtain corresponding decision points to distinguish between the behaviors of upstairs and downstairs movement, and to solve the problem of different decision points in different experiments, a linear regression algorithm is adopted, which improves the generalization ability of the algorithm for distinguishing between upstairs and downstairs movement. The effectiveness and validity of the proposed algorithm in 2D and 3D space is demonstrated by practical experiments.

Accurate attitude estimation of HB2 standard model based on QNCF in hypersonic wind tunnel test

The accuracy of the HB2 standard model attitude measurement has an important impact on the hypersonic wind tunnel data assessment. The limited size of the model and the existence of external vibrations make it challenging to obtain real-time reliable attitude measurement. To reduce the influence of attitude errors on test results, this paper proposes a Quaternion Nonlinear Complementary Filter (QNCF) attitude determination algorithm based on Microelectromechanical Inertial Measurement Unit (MEMS-IMU). Firstly, the threshold-based PI control strategy is adopted to eliminate noise effect according to the Acceleration Magnitude Detector (AMD). Then, the flexible quaternion method is updated to carry out attitude estimation which is operational and easy to be embedded in the Field Programmable Gate Array (FPGA). Finally, a high-precision three-axis turntable test and a hypersonic wind tunnel test are performed. The results show that the pitch-roll attitude errors are within 0.05° and 0.08° in the high-precision three-axis turntable test in a calculation time of 100 s respectively, and the attitude error is within 0.3° after the elastic angle correction in the hypersonic wind tunnel test. The proposed method can provide accurate real-time attitude reference for the analysis of the actual movement of the model, exhibiting certain engineering application value with robustness and simplicity.

Hypercomplex Widely Linear Estimation Through the Lens of Underpinning Geometry

We provide a rigorous account of the equivalence between the complex-valued widely linear estimation method and the quaternion involution widely linear estimation method with their vector-valued real linear estimation counterparts. This is achieved by an account of degrees of freedom and by providing matrix mappings between a complex variable and an isomorphic bivariate real vector, and a quaternion variable versus a quadri-variate real vector. Furthermore, we show that the parameters in the complex-valued linear estimation method, the complex-valued widely linear estimation method, the quaternion linear estimation method, the quaternion semi-widely linear estimation method, and the quaternion involution widely linear estimation method include distinct geometric structures imposed on complex numbers and quaternions, respectively, whereas the real-valued linear estimation methods do not exhibit any structure. This key difference explains, both in theoretical and practical terms, the advantage of estimation in division algebras (complex, quaternion) over their multivariate real vector counterparts. In addition, we discuss the computational complexities of the estimators of the hypercomplex widely linear estimation methods.

Quaternion and octonion-based finite element analysis methods for computing multiple first order derivatives

The complex Taylor series expansion method for computing accurate first order derivatives is extended in this work to quaternion, octonion and any order Cayley-Dickson algebra. The advantage of this new approach is that highly accurate multiple first order derivatives can be obtained in a single analysis. Quaternion and octonion-based finite element analysis methods were developed in order to compute up to three (quaternion) and up to seven (octonion) first order derivatives of shape, material properties, and/or loading conditions in a single analysis. The traditional finite element formulation was modified such that each degree-of-freedom was augmented with three or seven additional imaginary nodes. The quaternion and octonion-based methods were integrated within the Abaqus commercial finite element code through a user element subroutine. Numerical examples are presented for thermal conductivity and linear elasticity; however, the methodology is general. The results indicate that the quaternion and octonion-based methods provide derivatives of the same high accuracy as the complex finite element method but are significantly more efficient. A Fortran code to solve a simple seven variable quaternion example is given in the Appendix.

Hole position quick modification method for automatic drilling and riveting system considering workpiece pose deviation

Workpiece pose deviation is inevitable during automatic drilling and riveting in aircraft assembly, and it either leads to increased time cost or decreased position accuracy with the present methods. This study aims to propose a quick modification method to modify the coordinates of assembly holes based on online detection of the position of pre-assembly holes considering the effect of abnormal data caused by workpiece pose deviation. In the case of hole-making in the double curvature skin region of three pre-assembly holes, the space transformation matrix is determined using the unit quaternion method in accordance with the on-line detection of the hole position coordinates. In the case of more than three assembly holes with abnormal data, a new modification method using a random sampling consistency RANSAC algorithm is used to optimize the spatial transformation matrix and eliminate the abnormal data effectively. This method has been integrated into automatic drilling and riveting systems. Numerical simulation and experiment tests were conducted, and the result comparison showed that the method has obvious positive effect dealing with the abnormal data considering component pose deviation, which proves the effectiveness of the method.

Image descriptor based on local color directional quaternionic pattern

Most of the existing local binary pattern methods discard local color differences by holding their binary information. And local spatial information is also neglected. To address these problems, a robust color image descriptor local color directional quaternionic pattern (LCDQP) is proposed. In the descriptor, the color distance map (CDM) is generated in the RGB color space to capture the color distribution among three channels. According to the distribution of CDM elements and their neighbors, four edge models are defined to describe the change trend of the original image. Then, based on this, the LCDQP strings are gained according to the distribution of four models in the four directions of 0 deg, 45 deg, 90 deg, and 135 deg. Finally, an effective quaternionic code method is adopted to construct the LCDQP descriptor. The proposed descriptor not only captures the local color features but also reflects the spatial structure information. Experiments on four representative databases demonstrate that the proposed descriptor is superior to other state-of-the-art approaches.

Relative sensor registration with two‐step method for state estimation

State estimation suffers from some new challenging problems with a multi-platform multi-sensor observation system. An important problem for multisensor integration is that the data from the local sensors needs to be transformed into a common reference frame free of systematic bias or registration. In this study, the relative sensor registration problem is discussed. It aligns measurement from the global sensor with the local sensor under the assumptions that the global sensor is bias free and all biases reside with the local sensor. The traditional methods failed in the condition when attitude bias becomes large because the error caused by linearisation of rotation matrix increases with growing attitude bias. Motivated by this, a two-step method is established. By estimating the measurement bias through augmented extended Kalman filter in local sensor coordinate independent of attitude and location bias, and by introducing the unit quaternion method compute the attitude and location bias, the proposed method can avoid the problem the traditional methods faced. Simulation examples are provided to verify the proposed method by comparing with the existing linear least square algorithm.

Quaternionic rank-reduction methods for vector-field seismic data processing

The quaternion signal model can naturally represent vector-field data in a physically consistent approach, plausibly preserving and exploiting its vector relationships. The Singular Spectrum Analysis (SSA) filter can be extended to the case of multicomponent seismic data via the non-commutative quaternion algebra. Adoption of quaternions requires a shift from real or complex signal processing to quaternion-based signal processing. In this paper, the quaternion-defined seismic data is transformed to the frequency–space (f−x) domain via the quaternion Fourier transform (QFT), and rank-reduction can be performed via the quaternion SVD (QSVD) following the SSA methodology. Efficient rank-reduction alternatives, such as Lanczos bidiagonalization via convolutions, are used to allow significant gains in processing time. Simultaneous denoising and reconstruction of synthetic multicomponent seismic records are used to illustrate and compare the quaternionic versions of the SSA against its scalar version.

Quadrotor aircraft attitude control algorithm based on improved UKF

The quadrotor aircraft attitude estimation algorithm which is based on the adaptive attenuation memory unscented Kalman filter algorithm is proposed. The UKF algorithm uses the quaternion method to solve the attitude. The drift output of the gyroscope amplified as part of the state variable, and the attitude angle is used as the measurement variable to establish a filtering model, which effectively suppresses the attitude error caused by the gyro drift. The attenuation memory filter and adaptive algorithm are introduced based on the UKF algorithm to solve the problem of the accuracy degradation of the system model when it is abnormally interfered. The filter divergence is suppressed and the robustness and stability of the system are improved. The simulation results show that the algorithm can effectively improve the attitude estimation accuracy of the aircraft compared with the traditional UKF algorithm.

The dual spatial quaternionic expression of ruled surfaces

In this paper, the ruled surface which corresponds to a curve on dual unit sphere is rederived with the help of dual spatial quaternions. We extend the term of dual expression of ruled surface using dual spatial quaternionic method. The correspondences in dual space of closed ruled surfaces are quaternionically expressed. As a consequence, the integral invariants of these surfaces and the relationships between these invariants are shown

Quaternion-based weighted nuclear norm minimization for color image denoising

The quaternion method plays an important role in color image processing, because it represents the color image as a whole rather than as a separate color space component, thus naturally handling the coupling among color channels. The weighted nuclear norm minimization (WNNM) scheme assigns different weights to different singular values, leading to more reasonable image representation method. In this paper, we propose a novel quaternion weighted nuclear norm minimization (QWNNM) model and algorithm under the low rank sparse framework. The proposed model represents the color image as a low rank quaternion matrix, where quaternion singular value decomposition can be calculated by its equivalent complex matrix. We solve the QWNNM by adaptively assigning different singular values with different weights. Color image denoising is implemented by QWNNM based on non-local similarity priors. In this new color space, the inherent color structure can be well preserved during image reconstruction. For high noise levels, we apply a Gaussian low pass filter (LPF) to the noisy image as a preprocessing before QWNNM, which reduces the iteration numbers and improves the denoised results. The experimental results clearly show that the proposed method outperforms K-SVD, QKSVD and WNNM in terms of both quantitative criteria and visual perceptual.

Relative pose measurement of moving rigid bodies based on binocular vision

To solve the shortcomings of the traditional monocular vision measurement algorithm, which needs to be solved iteratively, this paper proposes a method based on binocular vision for measuring the relative pose of moving rigid bodies. In this paper, the coordinates of the feature points of the follower rigid body coordinate system are obtained by using the method of least squares. At the same time, combined with the dual quaternion algorithm, the leader and follower rigid bodies’ coordinate systems are uniformly described, and the model takes the minimum sum of squared errors as the objective function and solves the pose parameters by the dual quaternion method. The simulation results show that the dual quaternion algorithm not only overcomes the shortcomings of the traditional quaternion and translation vector split description coordinate system transformation but also has strong stability; the relative position error is less than 0.05 m, and the relative attitude error is less than 0.2°, which can meet the measurement accuracy requirements of the relative pose of the moving rigid body.

Optimization method of tool axis vector based on kinematical characteristics of rotary feed axis for curved surface machining

Complex surface parts are widely used in the industrial applications, and 5-axis NC machining with ball-end cutter is the commonly adopted method for curved surface parts. Due to the additional rotary feed axis comparing with 3-axis NC machining, the tool orientation control is complex for curved surface machining. With the more complexity of curved surface parts, it is a known problem that the large incoherent movement of the rotary feed axis will easily appear in curved surface machining, which may even be beyond the kinematical performances of the rotary feed axis in machine tool, so as to affect the machining quality of curved surface parts. In order to overcome this issue, an optimization method of tool axis vector based on the kinematical characteristics of the rotary feed axis for curved surface machining is proposed. Firstly, the optimizing interval of the toolpath for tool axis vector is selected based on the relationship between the rotation angle of rotary feed axis and the accumulation arc length of toolpath. Then, the equalization method of tool axis vector based on the quaternion method is used to optimize the tool axis vector with the kinematical characteristics of rotary feed axis. Finally, the angular change curve of rotary feed axis with the optimized angular value and cumulative arc length is adjusted by the principle of least-squares fitting after the local optimization of tool axis vector. Simulation and experiment on test parts are carried out to verify the validity of the proposed method, and the achievements are significant to improve the processing quality of complex curved surface parts.