In this paper,we shall prove that for any positive interger n,there exists non-trivialcommutative finite semigroup of idempotent consisting of some n×n real quaternion matri-ces which is lower semilattice.In the ...In this paper,we shall prove that for any positive interger n,there exists non-trivialcommutative finite semigroup of idempotent consisting of some n×n real quaternion matri-ces which is lower semilattice.In the process of solving this problem we shall see thatmany properties of generalized inverses for complex matrices still hold for quaternions ma-展开更多
Abstract There exists rare integration formulas over quaternions due to the noncommutativity of quarternions multplication. Based on the class operator＇s formula of rotation group we derive a Gaussian integral formul...Abstract There exists rare integration formulas over quaternions due to the noncommutativity of quarternions multplication. Based on the class operator＇s formula of rotation group we derive a Gaussian integral formula for quaternions, which is similar in form to the integration for coherent state＇s completeness relation.展开更多
Let H be the real quaternion field,C and R be the complex and real field respectively.Clearly R（?）C（?）H. Let H<sup>m×n</sup> denote the set of all m×n matrices over H.If A=(a<sub>rs<...Let H be the real quaternion field,C and R be the complex and real field respectively.Clearly R（?）C（?）H. Let H<sup>m×n</sup> denote the set of all m×n matrices over H.If A=(a<sub>rs</sub>)∈H<sup>m×n</sup>,then there exist A<sub>1</sub> and A<sub>2</sub>∈C<sup>m×n</sup> such that A=A<sub>1</sub>+A<sub>2</sub>j.Let A<sub>C</sub> denote the complexrepresentation of A,that is the 2m×2n complex matrix Ac=（（A<sub>1</sub>/A<sub>2</sub>）（-A<sub>2</sub>/A<sub>1</sub>））（see[1,2]）.We denote by A<sup>D</sup> the Drazin inverse of A∈H<sup>m×n</sup> which is the unique solution of the e-展开更多
Quatemions complementary filter attitude algorithm was conducted on the unmanned aerial vehicle （UAV） platform. This introduces traditional attitude algorithm and attitude quaternion complementary filter algorithm d...Quatemions complementary filter attitude algorithm was conducted on the unmanned aerial vehicle （UAV） platform. This introduces traditional attitude algorithm and attitude quaternion complementary filter algorithm difference, and the attitude quatemion complementary filter algorithm realization are introduced in details展开更多
This paper proposes a new type of control laws for free rigid bodies. The start point is the dual quaternion and its characteristics. The logarithm of a dual quaternion is defined, based on which kinematic control law...This paper proposes a new type of control laws for free rigid bodies. The start point is the dual quaternion and its characteristics. The logarithm of a dual quaternion is defined, based on which kinematic control laws can be developed. Global exponential convergence is achieved using logarithmic feedback via a generalized proportional control law, and an appropriate Lyapunov function is constructed to prove the stability. Both the regulation and tracking problems are tackled. Omnidirectional control is discussed as a case study. As the control laws can handle the interconnection between the rotation and translation of a rigid body, they are shown to be more applicable than the conventional method.展开更多
The multi axis coupling attitude control of a spacecraft with thrusters for attitude tracking is investigated. The attitude kinematics and dynamics are both described by error quaternions. The four error quaternion dy...The multi axis coupling attitude control of a spacecraft with thrusters for attitude tracking is investigated. The attitude kinematics and dynamics are both described by error quaternions. The four error quaternion dynamic equations are then transformed into four perturbed double integrators via linear transformations. An on off controller is designed based on the perturbed double integrators. The controller is determined by parabolic switching functions of the scalar error quaternion and the transfor...展开更多
We introduce a total order and an absolute value function for dual numbers.The absolute value function of dual numbers takes dual number values,and has properties similar to those of the absolute value function of rea...We introduce a total order and an absolute value function for dual numbers.The absolute value function of dual numbers takes dual number values,and has properties similar to those of the absolute value function of real numbers.We define the magnitude of a dual quaternion,as a dual number.Based upon these,we extend 1-norm,co-norm,and 2-norm to dual quaternion vectors.展开更多
This paper aims to present, in a unified manner, algebraic techniques for linear equations which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, s...This paper aims to present, in a unified manner, algebraic techniques for linear equations which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies the problem of v-quaternionic linear equations by means of a complex representation and a real representation of v-quaternion matrices, and gives two algebraic methods for solving v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for quaternionic and split quaternionic linear equations in quaternionic and split quaternionic mechanics.展开更多
Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. A...Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.展开更多
Recently, there has been considerable effort to bring together quaternion-based representations of spatial displacements with curve design techniques in Computer Aided Geometric Design (CAGD) to develop methods for sy...Recently, there has been considerable effort to bring together quaternion-based representations of spatial displacements with curve design techniques in Computer Aided Geometric Design (CAGD) to develop methods for synthesizing freeform Cartesian motions. These methods have a wide range of applications from computer graphics,Cartesian motion planning for robot manipulators to task specification and motion approximation for spatial mechanism design. This paper compares the use of quaternions, dual quaternions, and double quaternions for freeform motion synthesis in a CAD environment.展开更多
In this paper,Schwarz-type lemmas for different classes of quaternion functions are obtained.Firstly,some properties of symmetric points are given.Secondly,the Schwarz-type lemma and the Schwarz-Pick-type theorem for ...In this paper,Schwarz-type lemmas for different classes of quaternion functions are obtained.Firstly,some properties of symmetric points are given.Secondly,the Schwarz-type lemma and the Schwarz-Pick-type theorem for quaternion regular functions are obtained.Finally,the Schwarz-type lemma for quaternion k-regular functions is derived.展开更多
This paper investigates the modeling and the practical predefined-time(PdT)tracking control problems for a fully actuated disk-shaped autonomous underwater vehicle(AUV)with six degrees of freedom.To overcome the gimba...This paper investigates the modeling and the practical predefined-time(PdT)tracking control problems for a fully actuated disk-shaped autonomous underwater vehicle(AUV)with six degrees of freedom.To overcome the gimbal lock problem inherent in Euler angle representation,unit quaternions are adopted to model the AUV,accounting for internal uncertainties and external disturbances.Then,an improved time-varying function is introduced,which serves as the basis for designing a nonsingular sliding surface and sliding mode controller with PdT stability.This approach ensures that the tracking errors converge within a predefined time,independent of initial conditions and design parameters.Compared with traditional PdT controllers,the proposed method eliminates singularities,enhances the precision of convergence time estimation,and typically yields smaller,smoother initial control inputs,thus improving its potential for engineering applications.Numerical simulations validate the effectiveness and performance of the proposed controller.展开更多
In this paper,an algorithm on measurement noise with adaptive strong tracking unscented Kalman filter(ASTUKF)is advanced to improve the precision of pose estimation and the stability for data computation.To suppress h...In this paper,an algorithm on measurement noise with adaptive strong tracking unscented Kalman filter(ASTUKF)is advanced to improve the precision of pose estimation and the stability for data computation.To suppress high-frequency noise,an infinite impulse response filter(IIRF)is introduced at the front end of ASTUKF to preprocess the original data.Then the covariance matrix of the error is corrected and the measurement noise is estimated in the process of filtering.After that,the data from the experiment were tested on the hardware experiment platform.The experimental results show that compared to the traditional extended Kalman filter(EKF)and unscented Kalman filter(UKF)algorithms,the root mean square error(RMSE)of the roll axis results from the algorithm proposed in this paper is respectively reduced by approximately 57.5%and 36.1%;the RMSE of the pitch axis results decreases by nearly 58.4%and 51.5%,respectively;and the RMSE of the yaw axis results decreases almost 62.8%and 50.9%,correspondingly.The above results indicate that the algorithm enhances the ability of resisting high-frequency vibration interference and improves the accuracy of attitude solution.展开更多
On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial ...On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial order,give the explicit formulations of the monotonicity solution sets A{1;≥,T_1;≤B^(1)}and B{1;≥,T_2≥A^(1)}for the(1)-inverse,and two results of the monotonicity charac teriaztion for the(1,2)-inverse.展开更多
In this paper, by means of an isomorphism, we express the Clifford algebra Cl<sub>5,3</sub> as hyperquaternion algebra H ⊗H ⊗H ⊗H (a four-fold tensor product of quaternion alg...In this paper, by means of an isomorphism, we express the Clifford algebra Cl<sub>5,3</sub> as hyperquaternion algebra H ⊗H ⊗H ⊗H (a four-fold tensor product of quaternion algebras) and we provide the hyperquaternionic approach to the inner product null space (IPNS) representation of conic sections.展开更多
The laser gyro is most su it able for building the strap down inertial navigation system (SINS), and its acc uracy of attitude algorithm can enormously affect that of the laser SINS. This p aper develops three improv...The laser gyro is most su it able for building the strap down inertial navigation system (SINS), and its acc uracy of attitude algorithm can enormously affect that of the laser SINS. This p aper develops three improved algorithmal expressions for strap down attitude ut ilizing the angular increment output by the laser gyro from the last two and cur rent updating periods according to the number of gyro samples, and analyses the algorithm error in the classical coning motion. Compared with the conventional algorithms, simulational results show that this improved algorithm has higher precision. A new way to improve the rotation vector algorithms is provided.展开更多
In order to effectively restore color noisy images with the mixture of Gaussian noise and impulse noise,a new algorithm is proposed using the quaternion-based holistic processing idea for color images.First,a color im...In order to effectively restore color noisy images with the mixture of Gaussian noise and impulse noise,a new algorithm is proposed using the quaternion-based holistic processing idea for color images.First,a color image is represented by a pure quaternion matrix.Secondly,according to the different characteristics of the Gaussian noise and the impulse noise,an algorithm based on quaternion directional vector order statistics is used to detect the impulse noise. Finally,the quaternion optimal weights non-local means filter （QOWNLMF）for Gaussian noise removal is improved for the mixed noise removal.The detected impulse noise pixels are not considered in the calculation of weights.Experimental results on five standard images demonstrate that the proposed algorithm performs better than the commonly used robust outlyingness ratio-nonlocal means （ROR-NLM）algorithm and the optimal weights mixed filter （OWMF）.展开更多
The traditional algorithms for formation flying satellites treat the satellite position and attitude sepa- rately. A novel algorithm combining satellite attitude with position is proposed. The principal satellite traj...The traditional algorithms for formation flying satellites treat the satellite position and attitude sepa- rately. A novel algorithm combining satellite attitude with position is proposed. The principal satellite trajectory is obtained by dual quaternion interpolation, then the relative position and attitude of the deputy satellite are ob- tained by dual quaternion modeling on the principal satellite. Through above process, relative position and atti- tude are unified. Compared with the orbital parameter and the quaternion methods, the simulation result proves that the algorithm can unify position and attitude, and satisfy the precision requirement of formation flying satel- lites.展开更多
A new method is illustrated for processing the output of a set of triad orthogonal rate gyros and accelerometers to reconstruct vehicle navigation parameters(attitude, velocity, and position). The paper introduces two...A new method is illustrated for processing the output of a set of triad orthogonal rate gyros and accelerometers to reconstruct vehicle navigation parameters(attitude, velocity, and position). The paper introduces two vectors with dimensions 4×1 as velocity and position quaternions.The navigation equations for strapdown systems are nonlinear but after using these parameters, the navigation equations are converted into a pseudo-linear system. The new set of navigation equations has an analytical solution and the state transition matrix is used to solve the linear timevarying differential equations through time series. The navigation parameters are updated using the new formulation for strapdown navigation equations. Finally, the quaternions of velocity and position are converted into the original position and velocity vectors. The combination of the coning motion and a translational oscillatory trajectory is used to evaluate the accuracy of the proposed algorithm. The simulations show significant improvement in the accuracy of the inertial navigation system, which is achieved through the mentioned algorithm.展开更多
文摘In this paper,we shall prove that for any positive interger n,there exists non-trivialcommutative finite semigroup of idempotent consisting of some n×n real quaternion matri-ces which is lower semilattice.In the process of solving this problem we shall see thatmany properties of generalized inverses for complex matrices still hold for quaternions ma-
基金National Natural Science Foundation of China under Grant No.10475056
文摘Abstract There exists rare integration formulas over quaternions due to the noncommutativity of quarternions multplication. Based on the class operator＇s formula of rotation group we derive a Gaussian integral formula for quaternions, which is similar in form to the integration for coherent state＇s completeness relation.
基金Supported by the Natural Science Foundation of jiangxi
文摘Let H be the real quaternion field,C and R be the complex and real field respectively.Clearly R（?）C（?）H. Let H<sup>m×n</sup> denote the set of all m×n matrices over H.If A=(a<sub>rs</sub>)∈H<sup>m×n</sup>,then there exist A<sub>1</sub> and A<sub>2</sub>∈C<sup>m×n</sup> such that A=A<sub>1</sub>+A<sub>2</sub>j.Let A<sub>C</sub> denote the complexrepresentation of A,that is the 2m×2n complex matrix Ac=（（A<sub>1</sub>/A<sub>2</sub>）（-A<sub>2</sub>/A<sub>1</sub>））（see[1,2]）.We denote by A<sup>D</sup> the Drazin inverse of A∈H<sup>m×n</sup> which is the unique solution of the e-
基金supported by the National Science and Technology(2015BAK06B04)
文摘Quatemions complementary filter attitude algorithm was conducted on the unmanned aerial vehicle （UAV） platform. This introduces traditional attitude algorithm and attitude quaternion complementary filter algorithm difference, and the attitude quatemion complementary filter algorithm realization are introduced in details
文摘This paper proposes a new type of control laws for free rigid bodies. The start point is the dual quaternion and its characteristics. The logarithm of a dual quaternion is defined, based on which kinematic control laws can be developed. Global exponential convergence is achieved using logarithmic feedback via a generalized proportional control law, and an appropriate Lyapunov function is constructed to prove the stability. Both the regulation and tracking problems are tackled. Omnidirectional control is discussed as a case study. As the control laws can handle the interconnection between the rotation and translation of a rigid body, they are shown to be more applicable than the conventional method.
基金National Natural Science F oundation of China(No.10 172 0 12 )
文摘The multi axis coupling attitude control of a spacecraft with thrusters for attitude tracking is investigated. The attitude kinematics and dynamics are both described by error quaternions. The four error quaternion dynamic equations are then transformed into four perturbed double integrators via linear transformations. An on off controller is designed based on the perturbed double integrators. The controller is determined by parabolic switching functions of the scalar error quaternion and the transfor...
基金supported by Hong Kong Innovation and Technology Commission(InnoHK Project CIMDA)supported by the National Natural Science Foundation of China(No.11971138)+3 种基金the Natural Science Foundation of Zhejiang Province of China(Nos.LY19A010019,LD19A010002)supported by Hong Kong Research Grants Council(Project 11204821)Hong Kong Innovation and Technology Commission(InnoHK Project CIMDA)City University of Hong Kong(Project 9610034).
文摘We introduce a total order and an absolute value function for dual numbers.The absolute value function of dual numbers takes dual number values,and has properties similar to those of the absolute value function of real numbers.We define the magnitude of a dual quaternion,as a dual number.Based upon these,we extend 1-norm,co-norm,and 2-norm to dual quaternion vectors.
文摘This paper aims to present, in a unified manner, algebraic techniques for linear equations which are valid on both the algebras of quaternions and split quaternions. This paper, introduces a concept of v-quaternion, studies the problem of v-quaternionic linear equations by means of a complex representation and a real representation of v-quaternion matrices, and gives two algebraic methods for solving v-quaternionic linear equations. This paper also gives a unification of algebraic techniques for quaternionic and split quaternionic linear equations in quaternionic and split quaternionic mechanics.
文摘Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial differential equations of higher order with real and complex coefficients and with variable coefficients with or without boundary conditions are considered.
文摘Recently, there has been considerable effort to bring together quaternion-based representations of spatial displacements with curve design techniques in Computer Aided Geometric Design (CAGD) to develop methods for synthesizing freeform Cartesian motions. These methods have a wide range of applications from computer graphics,Cartesian motion planning for robot manipulators to task specification and motion approximation for spatial mechanism design. This paper compares the use of quaternions, dual quaternions, and double quaternions for freeform motion synthesis in a CAD environment.
基金supported by the Innovation Foundation of the School of Mathematical Sciences in Hebei Normal University in 2025(ycxzzbs202503)the NSF of Hebei Province(A2023205006,A2022208007,A2023205045,A2024208005)+2 种基金the Hebei Research Center of the Basic Discipline Pure Mathematics,the Key Development Foundation of Hebei Normal University(L2024ZD08)the NSFC(12431005)the Funding Project of Central Guidance for Local Scientific and Technological Development(246Z7608G).
文摘In this paper,Schwarz-type lemmas for different classes of quaternion functions are obtained.Firstly,some properties of symmetric points are given.Secondly,the Schwarz-type lemma and the Schwarz-Pick-type theorem for quaternion regular functions are obtained.Finally,the Schwarz-type lemma for quaternion k-regular functions is derived.
基金supported in part by the National Natural Science Foundation of China(62373107)the“Zhishan”Scholars Programs of Southeast University(2242023R40011).
文摘This paper investigates the modeling and the practical predefined-time(PdT)tracking control problems for a fully actuated disk-shaped autonomous underwater vehicle(AUV)with six degrees of freedom.To overcome the gimbal lock problem inherent in Euler angle representation,unit quaternions are adopted to model the AUV,accounting for internal uncertainties and external disturbances.Then,an improved time-varying function is introduced,which serves as the basis for designing a nonsingular sliding surface and sliding mode controller with PdT stability.This approach ensures that the tracking errors converge within a predefined time,independent of initial conditions and design parameters.Compared with traditional PdT controllers,the proposed method eliminates singularities,enhances the precision of convergence time estimation,and typically yields smaller,smoother initial control inputs,thus improving its potential for engineering applications.Numerical simulations validate the effectiveness and performance of the proposed controller.
基金supported by the Key Research and Development Program of Shaanxi Province(No.2024NC-YBXM-246)the Shaanxi Provincial Science and Technology Department(No.2024JC-YBQN-0725)+1 种基金the Education Department of Shaanxi Province(No.23JK0371)the Shaanxi University of Technology(No.SLGRCQD2318).
文摘In this paper,an algorithm on measurement noise with adaptive strong tracking unscented Kalman filter(ASTUKF)is advanced to improve the precision of pose estimation and the stability for data computation.To suppress high-frequency noise,an infinite impulse response filter(IIRF)is introduced at the front end of ASTUKF to preprocess the original data.Then the covariance matrix of the error is corrected and the measurement noise is estimated in the process of filtering.After that,the data from the experiment were tested on the hardware experiment platform.The experimental results show that compared to the traditional extended Kalman filter(EKF)and unscented Kalman filter(UKF)algorithms,the root mean square error(RMSE)of the roll axis results from the algorithm proposed in this paper is respectively reduced by approximately 57.5%and 36.1%;the RMSE of the pitch axis results decreases by nearly 58.4%and 51.5%,respectively;and the RMSE of the yaw axis results decreases almost 62.8%and 50.9%,correspondingly.The above results indicate that the algorithm enhances the ability of resisting high-frequency vibration interference and improves the accuracy of attitude solution.
文摘On the basis of the paoers[3—7],this paper study the monotonicity problems for the positive semidefinite generalized inverses of the positive semidefinite self-conjugate matrices of quaternions in the Lowner partial order,give the explicit formulations of the monotonicity solution sets A{1;≥,T_1;≤B^(1)}and B{1;≥,T_2≥A^(1)}for the(1)-inverse,and two results of the monotonicity charac teriaztion for the(1,2)-inverse.
文摘In this paper, by means of an isomorphism, we express the Clifford algebra Cl<sub>5,3</sub> as hyperquaternion algebra H ⊗H ⊗H ⊗H (a four-fold tensor product of quaternion algebras) and we provide the hyperquaternionic approach to the inner product null space (IPNS) representation of conic sections.
文摘The laser gyro is most su it able for building the strap down inertial navigation system (SINS), and its acc uracy of attitude algorithm can enormously affect that of the laser SINS. This p aper develops three improved algorithmal expressions for strap down attitude ut ilizing the angular increment output by the laser gyro from the last two and cur rent updating periods according to the number of gyro samples, and analyses the algorithm error in the classical coning motion. Compared with the conventional algorithms, simulational results show that this improved algorithm has higher precision. A new way to improve the rotation vector algorithms is provided.
基金The National Natural Science Foundation of China(No.61572258,61173141,61271312,61232016,61272421)the Natural Science Foundation of Jiangsu Province(No.BK2012858,BK20151530)+1 种基金the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(No.13KJB520015)Open Fund of Jiangsu Engineering Center of Network Monitoring(No.KJR1404)
文摘In order to effectively restore color noisy images with the mixture of Gaussian noise and impulse noise,a new algorithm is proposed using the quaternion-based holistic processing idea for color images.First,a color image is represented by a pure quaternion matrix.Secondly,according to the different characteristics of the Gaussian noise and the impulse noise,an algorithm based on quaternion directional vector order statistics is used to detect the impulse noise. Finally,the quaternion optimal weights non-local means filter （QOWNLMF）for Gaussian noise removal is improved for the mixed noise removal.The detected impulse noise pixels are not considered in the calculation of weights.Experimental results on five standard images demonstrate that the proposed algorithm performs better than the commonly used robust outlyingness ratio-nonlocal means （ROR-NLM）algorithm and the optimal weights mixed filter （OWMF）.
基金Supported by the National Natural Science Foundation of China(60974107)the Research Foundation of Nanjing University of Aeronautics and Astronautics(2010219)~~
文摘The traditional algorithms for formation flying satellites treat the satellite position and attitude sepa- rately. A novel algorithm combining satellite attitude with position is proposed. The principal satellite trajectory is obtained by dual quaternion interpolation, then the relative position and attitude of the deputy satellite are ob- tained by dual quaternion modeling on the principal satellite. Through above process, relative position and atti- tude are unified. Compared with the orbital parameter and the quaternion methods, the simulation result proves that the algorithm can unify position and attitude, and satisfy the precision requirement of formation flying satel- lites.
文摘A new method is illustrated for processing the output of a set of triad orthogonal rate gyros and accelerometers to reconstruct vehicle navigation parameters(attitude, velocity, and position). The paper introduces two vectors with dimensions 4×1 as velocity and position quaternions.The navigation equations for strapdown systems are nonlinear but after using these parameters, the navigation equations are converted into a pseudo-linear system. The new set of navigation equations has an analytical solution and the state transition matrix is used to solve the linear timevarying differential equations through time series. The navigation parameters are updated using the new formulation for strapdown navigation equations. Finally, the quaternions of velocity and position are converted into the original position and velocity vectors. The combination of the coning motion and a translational oscillatory trajectory is used to evaluate the accuracy of the proposed algorithm. The simulations show significant improvement in the accuracy of the inertial navigation system, which is achieved through the mentioned algorithm.
