The superiority of hypothetical quantum computers is not due to faster calculations but due to different scheme of calculations running on special hardware. At the same time, one should realize that quantum computers ...The superiority of hypothetical quantum computers is not due to faster calculations but due to different scheme of calculations running on special hardware. At the same time, one should realize that quantum computers would only provide dramatic speedups for a few specific problems, for example, factoring integers and breaking cryptographic codes in the conventional quantum computing approach. The core of quantum computing follows the way a state of a quantum system is defined when basic things interact with each other. In the conventional approach, it is implemented through the tensor product of qubits. In the suggested geometric algebra formalism simultaneous availability of all the results for non-measured observables is based on the definition of states as points on a three-dimensional sphere, which is very different from the usual Hilbert space scheme.展开更多
In this paper, the source localization by utilizing the measurements of a single electromagnetic （EM） vector-sensor is investigated in the framework of the geometric algebra of Euclidean 3-space. In order to describ...In this paper, the source localization by utilizing the measurements of a single electromagnetic （EM） vector-sensor is investigated in the framework of the geometric algebra of Euclidean 3-space. In order to describe the orthogonality among the electric and magnetic measurements, two multivectors of the geometric algebra of Euclidean 3-space （G3） are used to model the outputs of a spatially collocated EM vector-sensor. Two estimators for the wave propagation vector estimation are then formulated by the inner product between a vector and a bivector in the G3. Since the information used by the two estimators is different, a weighted inner product estimator is then proposed to fuse the two estimators together in the sense of the minimum mean square error （MMSE）. Analytical results show that the statistical performances of the weighted inner product estimator are always better than its traditional cross product counterpart. The efficacy of the weighted inner product estimator and the correctness of the analytical predictions are demonstrated by simulation results.展开更多
Singular configurations must be avoided in path planning and control of a parallel manipulator. However, most studies rarely focus on an overall singularity loci distribution of lower-mobility parallel mechanisms. Geo...Singular configurations must be avoided in path planning and control of a parallel manipulator. However, most studies rarely focus on an overall singularity loci distribution of lower-mobility parallel mechanisms. Geometric algebra is employed in analysis of singularity of a 3-RPS parallel manipulator. Twist and wrench in screw theory are represented in geometric algebra. Linear dependency of twists and wrenches are described by outer product in geometric algebra. Reciprocity between twists and constraint wrenches are reflected by duality. To compute the positions of the three spherical joints of the 3-RPS parallel manipulator, Tilt-and-Torsion angles are used to describe the orientation of the moving platform. The outer product of twists and constraint wrenches is used as an index for closeness to singularity（ICS） of the 3-RPS parallel manipulator. An overall and thorough perspective of the singularity loci distribution of the 3-RPS parallel manipulator is disclosed, which is helpful to design, trajectory planning and control of this kind of parallel manipulator.展开更多
The research considers wavelike objects that are elements of even subalgebra of geometric algebra in three dimensions. The used formalism particularly eliminates long existing confusion about the reasons behind the ap...The research considers wavelike objects that are elements of even subalgebra of geometric algebra in three dimensions. The used formalism particularly eliminates long existing confusion about the reasons behind the appearance of the imaginary unit in quantum mechanics and introduces clear definition of wave functions. When a wave function acts through the Hopf fibration on a localized geometric algebra element, that is executing a measurement, the result can be named as “collapse” of the wave function.展开更多
In this paper,it is created a one-to-one map between the Dirac Equation in Matrix Algebra and the Dirac Equation in Geometric Algebra Cl3,0.The eight free parameters of the spinors in matrix representation correspond ...In this paper,it is created a one-to-one map between the Dirac Equation in Matrix Algebra and the Dirac Equation in Geometric Algebra Cl3,0.The eight free parameters of the spinors in matrix representation correspond to the coefficients of the eight elements of the Geometric Algebra Cl3,0(scalars,three vectors,three bivectors and the trivector).This way,a one-to-one mapping between matrix representation and Geometric Algebra Cl3,0 can be obtained.展开更多
In this paper,we will show how Geometric Algebra expands the three spatial dimensions into entities of 8 degrees of freedom.It is also explained that one of these degrees of freedom(the trivector)can be considered to ...In this paper,we will show how Geometric Algebra expands the three spatial dimensions into entities of 8 degrees of freedom.It is also explained that one of these degrees of freedom(the trivector)can be considered to be the time(so no ad-hoc extra dimension is necessary).The square of the trivector is negative,solving this way the issue of the negative signature of the time(not necessary any ad-hoc metric indicating this,it is a property of time that appears naturally).Also,we will show that we can try to prove this experimentally looking for the electromagnetic trivector,an entity that should exist according to GA.Also,some comments regarding the similarities with E8 theory are given.Mainly that E8 theory considers 8 dimensions,exactly the same,emerging naturally in this paper.But not only that,also some similarities regarding how gravity can be understood,and others are presented.展开更多
In this paper,we will use Geometric Algebra to be able to embed the Klein-Gordon equation for a particle in a non-Euclidean field(gravitational field).This way,we will obtain an expression similar to the Dirac equatio...In this paper,we will use Geometric Algebra to be able to embed the Klein-Gordon equation for a particle in a non-Euclidean field(gravitational field).This way,we will obtain an expression similar to the Dirac equation,but with a slight change in one of the terms.This variation is produced and depends on the curvature of the space where the particle lies in(the Ricci scalar).In a similar manner,we will find variations in the equation for the energy of a particle and in the Einstein gravitational equation that will depend again on the value of the Ricci scalar(the curvature of the space where the particle lies in).An important outcome will be an equation that limits the value of the Ricci scalar depending on the value of the mass that provokes it(the value of the mass,not the mass density),highly reducing the possibilities of arriving at singularities.In fact,the value of this R has been found to be equal to the cosmological constant(both in the order of 1E-52),making it a perfect candidate for the dark energy.Also,the magnetic-like effects of gravitation coming from the equations are sufficient to explain the rotation of the galaxies(NGC 1560,NGC 3198 and NGC 3115)without the need for dark matter.展开更多
In this paper,it will be calculated the electromagnetic field strength and the Lorentz force in Geometric Algebra Cl_(3,0).And it will be compared with their equivalent in the tensor covariant formalism.Also,four extr...In this paper,it will be calculated the electromagnetic field strength and the Lorentz force in Geometric Algebra Cl_(3,0).And it will be compared with their equivalent in the tensor covariant formalism.Also,four extra equations not appearing in the classical formalism will be obtained and its meaning will be explained.In the same way,the electromagnetic Field strength elements and the velocity multivector of the particle will be expanded.New equations and new elements appearing will be explained,being the most important one,the Electromagnetic trivector B_(xyz).A field to be added to the magnetic(bivector-like)and electric(vector-like)fields.Lastly,an insight of the possible implications of these learnings in the Dirac Equation will be commented.展开更多
Parallel mechanisms with fewer degrees of freedom that incorporate two or more SPR limbs have been widely adopted in industrial applications in recent years.However,notable theoretical gaps persist,particularly in the...Parallel mechanisms with fewer degrees of freedom that incorporate two or more SPR limbs have been widely adopted in industrial applications in recent years.However,notable theoretical gaps persist,particularly in the field of analytical solutions for forward kinematics.To address this,this paper proposes an innovative forward kinematics analysis method based on Conformal Geometric Algebra(CGA)for complex hybrid mechanisms formed by serial concatenation of such parallel mechanisms.The method efficiently represents geometric elements and their operational relationships by defining appropriate unknown parameters.It constructs fundamental geometric objects such as spheres and planes,derives vertex expressions through intersection and dual operations,and establishes univariate high-order equations via inner product operations,ultimately obtaining complete analytical solutions for the forward kinematics of hybrid mechanisms.Using the(2-SPR+RPS)+(3-SPR)serial-parallel hybrid mechanism as a validation case,three configuration tests implemented in Mathematica demonstrate that:for each configuration,the upper 3-SPR mechanism yields 15 mathematical solutions,while the lower 2-SPR+RPS mechanism yields 4 mathematical solutions.After geometric constraint filtering,a unique physically valid solution is obtained for each mechanism.SolidWorks simulations further verify the correctness and reliability of the model.This research provides a reliable analytical method for forward kinematics of hybrid mechanisms,holding significant implications for advancing their applications in high-precision scenarios.展开更多
We propose a new Geographic Information System (GIS) three-dimensional (3D) data model based on conformal geometric algebra (CGA). In this approach, geographic objects of different dimensions are mapped to the corresp...We propose a new Geographic Information System (GIS) three-dimensional (3D) data model based on conformal geometric algebra (CGA). In this approach, geographic objects of different dimensions are mapped to the corresponding basic elements (blades) in Clifford algebra, and the expressions of multi-dimensional objects are unified without losing their geometric meaning. Geometric and topologic computations are also processed in a clear and coordinates-free way. Under the CGA framework, basic geometrics are constructed and expressed by the inner and outer operators. This expression combined geometrics of different dimensions and metric relations. We present the structure of the framework, data structure design, and the data storage, editing and updating mechanisms of the proposed 3D GIS data model. 3D GIS geometric and topological analyses are performed by CGA metric, geometric and topological operators using an object-oriented approach. Demonstrations with 3D residence district data suggest that our 3D data model expresses effectively geometric objects in different dimensions, which supports computation of both geometric and topological relations. The clear and effective expression and computation structure has the potential to support complex 3D GIS analysis, and spatio-temporal analysis.展开更多
WE use the geometric algebra in refs. [1, 2] to study hyperbolic geometry. The n-dimensional hyperbolic space H<sup>n</sup> is taken to be the unit sphere of G<sub>1</sub> (I<sub>-n,1<...WE use the geometric algebra in refs. [1, 2] to study hyperbolic geometry. The n-dimensional hyperbolic space H<sup>n</sup> is taken to be the unit sphere of G<sub>1</sub> (I<sub>-n,1</sub>) with antipodal points identified. We study typically H<sup>2</sup>: the dualities between generalized point and generalized line, between generalized triangle and imaginary triangle; convex generalized triangles; Lorentz transformations; generalized circles and double-cycles, etc. Below we list some of the results.展开更多
To investigate the forward kinematics problem of parallel mechanisms with complex limbs and to expand the applicability of the powerful tool of Conformal Geometric Algebra(CGA),a CGA-based modeling and solution method...To investigate the forward kinematics problem of parallel mechanisms with complex limbs and to expand the applicability of the powerful tool of Conformal Geometric Algebra(CGA),a CGA-based modeling and solution method for a class of parallel platforms with 3-RE structure after locking the actuated joints is proposed in this paper.Given that the angle between specific joint axes of limbs remains constant,a set of geometric constraints for the forward kinematics of parallel mechanisms(PM)are determined.After translating unit direction vectors of these joint axes to the common starting point,the geometric constraints of the angle between the vectors are transformed into the distances between the endpoints of the vectors,making them easier to handle.Under the framework of CGA,the positions of key points that determine the position and orientation of the moving platform can be intuitively determined by the intersection,division,and duality of basic geometric entities.By employing the tangent half-angle substitution,the forward kinematic analysis of the parallel mechanisms leads to a high-order univariate polynomial equation without the need for any complex algebraic elimination operations.After solving this equation and back substitution,the position and pose of the MP can be obtained indirectly.A numerical case is utilized to confirm the effectiveness of the proposed method.展开更多
The solution for the forward displacement analysis（FDA） of the general 6-6 Stewart mechanism（i.e., the connection points of the moving and fixed platforms are not restricted to lying in a plane） has been extensive...The solution for the forward displacement analysis（FDA） of the general 6-6 Stewart mechanism（i.e., the connection points of the moving and fixed platforms are not restricted to lying in a plane） has been extensively studied, but the efficiency of the solution remains to be effectively addressed. To this end, an algebraic elimination method is proposed for the FDA of the general 6-6 Stewart mechanism. The kinematic constraint equations are built using conformal geometric algebra（CGA）. The kinematic constraint equations are transformed by a substitution of variables into seven equations with seven unknown variables. According to the characteristic of anti-symmetric matrices, the aforementioned seven equations can be further transformed into seven equations with four unknown variables by a substitution of variables using the Grobner basis. Its elimination weight is increased through changing the degree of one variable, and sixteen equations with four unknown variables can be obtained using the Grobner basis. A 40th-degree univariate polynomial equation is derived by constructing a relatively small-sized 9 × 9 Sylvester resultant matrix. Finally, two numerical examples are employed to verify the proposed method. The results indicate that the proposed method can effectively improve the efficiency of solution and reduce the computational burden because of the small-sized resultant matrix.展开更多
The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing ...The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. So it is worth popularizing in the teaching of undergraduate physics and mathematics. Clifford algebras can be directly generalized to 2<sup>n</sup>-ary associative algebras. In this generalization, the matrix representation of the orthonormal basis of space-time plays an important role. The matrix representation carries more information than the abstract definition, such as determinant and the definition of inverse elements. Without this matrix representation, the discussion of hypercomplex numbers will be difficult. The zero norm set of hypercomplex numbers is a closed set of special geometric meanings, like the light-cone in the realistic space-time, which has no substantial effect on the algebraic calculus. The physical equations expressed in Clifford algebra have a simple formalism, symmetrical structure, standard derivation, complete content. Therefore, we can hope that this magical algebra can complete a new large synthesis of modern science.展开更多
The superiority of hypothetical quantum computers is not due to faster calculations but due to different schemes of calculations running on special hardware. The core of quantum computing follows the way a state of a ...The superiority of hypothetical quantum computers is not due to faster calculations but due to different schemes of calculations running on special hardware. The core of quantum computing follows the way a state of a quantum system is defined when basic things interact with each other. In conventional approach it is implemented through tensor product of qubits. In the geometric algebra formalism simultaneous availability of all the results for non-measured observables is based on the definition of states as points on three-dimensional sphere.展开更多
The current work considers the sprefield wave functions received as special g-qubit solutions of Maxwell equations in the terms of geometric algebra. I will call such g-qubits spreons or sprefields. The purpose of thi...The current work considers the sprefield wave functions received as special g-qubit solutions of Maxwell equations in the terms of geometric algebra. I will call such g-qubits spreons or sprefields. The purpose of this article is to analyze behavior of such wave functions in scattering and measurements. It is shown that sprefields are defined through the whole three-dimensional space at all values of the time parameter. They instantly change all their values when get scattered, that is subjected to Clifford translation. In “measurements”, when a sprefield acts on a static geometric algebra element through the Hopf fibration, sprefield collapses and new geometric algebra non static, rotating element is thereby created.展开更多
Geometric Algebra formalism opens the door to developing a theory deeper than conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three...Geometric Algebra formalism opens the door to developing a theory deeper than conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions, unambiguous definition of states, observables, measurements, Maxwell equations solution in those terms, bring into reality a kind of physical fields spreading through the whole three-dimensional space and values of the time parameter. The fields can be modified instantly in all points of space and time values, thus eliminating the concept of cause and effect, and perceiving of one-directional time. In the suggested theory all measured observable values get available all together, not through looking one by one. In this way quantum computer appeared to be a kind of analog computer keeping and instantly processing information by and on sets of objects possessing an infinite number of degrees of freedom. As practical implementation, the multithread GPUs bearing the CUDA language functionality allow to simultaneously calculate observable measurement values at a number of space/time discrete points only restricted by the GPU threads capacity.展开更多
A primordial field Self-interaction Principle, analyzed in Hestenes’ Geometric Calculus, leads to Heaviside’s equations of the gravitomagnetic field. When derived from Einstein’s nonlinear field equations Heaviside...A primordial field Self-interaction Principle, analyzed in Hestenes’ Geometric Calculus, leads to Heaviside’s equations of the gravitomagnetic field. When derived from Einstein’s nonlinear field equations Heaviside’s “linearized” equations are known as the “weak field approximation”. When derived from the primordial field equation, there is no mention of field strength;the assumption that the primordial field was predominant at the big bang rather suggests that ultra-strong fields are governed by the equations. This aspect has physical significance, so we explore the assumption by formulating the gauge field version of Heaviside’s theory. We compare with recent linearized gravity formulations and discuss the significance of differences.展开更多
The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from changing of complex numbers by geometrically feasible objects in three dim...The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from changing of complex numbers by geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of weird quantum mechanical features, for example primitively considering atoms as a kind of solar system. The three-sphere S<sup>3</sup> becomes the playground of the torsion kind states eliminating abstract Hilbert space vectors. The S<sup>3</sup> points evolve, governed by updated Schrodinger equation, and act in measurements on observable as operators.展开更多
Superposition and entanglement are two theoretical pillars quantum computing rests upon. In the g-qubit theory quantum wave functions are identified by points on the surface of three-dimensional sphere S<sup>3&l...Superposition and entanglement are two theoretical pillars quantum computing rests upon. In the g-qubit theory quantum wave functions are identified by points on the surface of three-dimensional sphere S<sup>3</sup>. That gives different, more physically feasible explanation of what superposition and entanglement are. The core of quantum computing scheme should be in manipulation and transferring of wave functions on S<sup>3</sup> as operators acting on observables and formulated in terms of geometrical algebra. In this way quantum computer will be a kind of analog computer keeping and processing information by sets of objects possessing infinite number of degrees of freedom, contrary to the two value bits or two-dimensional Hilbert space elements, qubits.展开更多
文摘The superiority of hypothetical quantum computers is not due to faster calculations but due to different scheme of calculations running on special hardware. At the same time, one should realize that quantum computers would only provide dramatic speedups for a few specific problems, for example, factoring integers and breaking cryptographic codes in the conventional quantum computing approach. The core of quantum computing follows the way a state of a quantum system is defined when basic things interact with each other. In the conventional approach, it is implemented through the tensor product of qubits. In the suggested geometric algebra formalism simultaneous availability of all the results for non-measured observables is based on the definition of states as points on a three-dimensional sphere, which is very different from the usual Hilbert space scheme.
基金National Natural Science Foundation of China(61171127)National Basic Research Program of China(2011CB302903)
文摘In this paper, the source localization by utilizing the measurements of a single electromagnetic （EM） vector-sensor is investigated in the framework of the geometric algebra of Euclidean 3-space. In order to describe the orthogonality among the electric and magnetic measurements, two multivectors of the geometric algebra of Euclidean 3-space （G3） are used to model the outputs of a spatially collocated EM vector-sensor. Two estimators for the wave propagation vector estimation are then formulated by the inner product between a vector and a bivector in the G3. Since the information used by the two estimators is different, a weighted inner product estimator is then proposed to fuse the two estimators together in the sense of the minimum mean square error （MMSE）. Analytical results show that the statistical performances of the weighted inner product estimator are always better than its traditional cross product counterpart. The efficacy of the weighted inner product estimator and the correctness of the analytical predictions are demonstrated by simulation results.
基金Supported by National Natural Science Foundation of China(Grant No.51135008)Zhejiang Provincial Natural Science Foundation of China(Grant No.LZ14E050005)
文摘Singular configurations must be avoided in path planning and control of a parallel manipulator. However, most studies rarely focus on an overall singularity loci distribution of lower-mobility parallel mechanisms. Geometric algebra is employed in analysis of singularity of a 3-RPS parallel manipulator. Twist and wrench in screw theory are represented in geometric algebra. Linear dependency of twists and wrenches are described by outer product in geometric algebra. Reciprocity between twists and constraint wrenches are reflected by duality. To compute the positions of the three spherical joints of the 3-RPS parallel manipulator, Tilt-and-Torsion angles are used to describe the orientation of the moving platform. The outer product of twists and constraint wrenches is used as an index for closeness to singularity（ICS） of the 3-RPS parallel manipulator. An overall and thorough perspective of the singularity loci distribution of the 3-RPS parallel manipulator is disclosed, which is helpful to design, trajectory planning and control of this kind of parallel manipulator.
文摘The research considers wavelike objects that are elements of even subalgebra of geometric algebra in three dimensions. The used formalism particularly eliminates long existing confusion about the reasons behind the appearance of the imaginary unit in quantum mechanics and introduces clear definition of wave functions. When a wave function acts through the Hopf fibration on a localized geometric algebra element, that is executing a measurement, the result can be named as “collapse” of the wave function.
文摘In this paper,it is created a one-to-one map between the Dirac Equation in Matrix Algebra and the Dirac Equation in Geometric Algebra Cl3,0.The eight free parameters of the spinors in matrix representation correspond to the coefficients of the eight elements of the Geometric Algebra Cl3,0(scalars,three vectors,three bivectors and the trivector).This way,a one-to-one mapping between matrix representation and Geometric Algebra Cl3,0 can be obtained.
文摘In this paper,we will show how Geometric Algebra expands the three spatial dimensions into entities of 8 degrees of freedom.It is also explained that one of these degrees of freedom(the trivector)can be considered to be the time(so no ad-hoc extra dimension is necessary).The square of the trivector is negative,solving this way the issue of the negative signature of the time(not necessary any ad-hoc metric indicating this,it is a property of time that appears naturally).Also,we will show that we can try to prove this experimentally looking for the electromagnetic trivector,an entity that should exist according to GA.Also,some comments regarding the similarities with E8 theory are given.Mainly that E8 theory considers 8 dimensions,exactly the same,emerging naturally in this paper.But not only that,also some similarities regarding how gravity can be understood,and others are presented.
文摘In this paper,we will use Geometric Algebra to be able to embed the Klein-Gordon equation for a particle in a non-Euclidean field(gravitational field).This way,we will obtain an expression similar to the Dirac equation,but with a slight change in one of the terms.This variation is produced and depends on the curvature of the space where the particle lies in(the Ricci scalar).In a similar manner,we will find variations in the equation for the energy of a particle and in the Einstein gravitational equation that will depend again on the value of the Ricci scalar(the curvature of the space where the particle lies in).An important outcome will be an equation that limits the value of the Ricci scalar depending on the value of the mass that provokes it(the value of the mass,not the mass density),highly reducing the possibilities of arriving at singularities.In fact,the value of this R has been found to be equal to the cosmological constant(both in the order of 1E-52),making it a perfect candidate for the dark energy.Also,the magnetic-like effects of gravitation coming from the equations are sufficient to explain the rotation of the galaxies(NGC 1560,NGC 3198 and NGC 3115)without the need for dark matter.
文摘In this paper,it will be calculated the electromagnetic field strength and the Lorentz force in Geometric Algebra Cl_(3,0).And it will be compared with their equivalent in the tensor covariant formalism.Also,four extra equations not appearing in the classical formalism will be obtained and its meaning will be explained.In the same way,the electromagnetic Field strength elements and the velocity multivector of the particle will be expanded.New equations and new elements appearing will be explained,being the most important one,the Electromagnetic trivector B_(xyz).A field to be added to the magnetic(bivector-like)and electric(vector-like)fields.Lastly,an insight of the possible implications of these learnings in the Dirac Equation will be commented.
基金Supported by Hebei Provincial Natural Science Foundation(Grant No.F2024202052)National Natural Science Foundation of China(Grant No.52175019)+3 种基金Beijing Municipal Natural Science Foundation(Grant No.L222038)Beijing Nova Programme Interdisciplinary Cooperation Project(Grant No.20240484699)Joint Funds of Industry-University-Research of Shanghai Academy of Spaceflight Technology(Grant No.SAST2022-017)Beijing Municipal Key Laboratory of Space-ground Interconnection and Convergence of China and Key Laboratory of IoT Monitoring and Early Warning,Ministry of Emergency Management。
文摘Parallel mechanisms with fewer degrees of freedom that incorporate two or more SPR limbs have been widely adopted in industrial applications in recent years.However,notable theoretical gaps persist,particularly in the field of analytical solutions for forward kinematics.To address this,this paper proposes an innovative forward kinematics analysis method based on Conformal Geometric Algebra(CGA)for complex hybrid mechanisms formed by serial concatenation of such parallel mechanisms.The method efficiently represents geometric elements and their operational relationships by defining appropriate unknown parameters.It constructs fundamental geometric objects such as spheres and planes,derives vertex expressions through intersection and dual operations,and establishes univariate high-order equations via inner product operations,ultimately obtaining complete analytical solutions for the forward kinematics of hybrid mechanisms.Using the(2-SPR+RPS)+(3-SPR)serial-parallel hybrid mechanism as a validation case,three configuration tests implemented in Mathematica demonstrate that:for each configuration,the upper 3-SPR mechanism yields 15 mathematical solutions,while the lower 2-SPR+RPS mechanism yields 4 mathematical solutions.After geometric constraint filtering,a unique physically valid solution is obtained for each mechanism.SolidWorks simulations further verify the correctness and reliability of the model.This research provides a reliable analytical method for forward kinematics of hybrid mechanisms,holding significant implications for advancing their applications in high-precision scenarios.
基金supported by National High Technology R & D Program of China (Grant No. 2009AA12Z205)Key Project of National Natural Science Foundation of China (Grant No. 40730527)National Natural Science Foundation of China (Grant No. 41001224)
文摘We propose a new Geographic Information System (GIS) three-dimensional (3D) data model based on conformal geometric algebra (CGA). In this approach, geographic objects of different dimensions are mapped to the corresponding basic elements (blades) in Clifford algebra, and the expressions of multi-dimensional objects are unified without losing their geometric meaning. Geometric and topologic computations are also processed in a clear and coordinates-free way. Under the CGA framework, basic geometrics are constructed and expressed by the inner and outer operators. This expression combined geometrics of different dimensions and metric relations. We present the structure of the framework, data structure design, and the data storage, editing and updating mechanisms of the proposed 3D GIS data model. 3D GIS geometric and topological analyses are performed by CGA metric, geometric and topological operators using an object-oriented approach. Demonstrations with 3D residence district data suggest that our 3D data model expresses effectively geometric objects in different dimensions, which supports computation of both geometric and topological relations. The clear and effective expression and computation structure has the potential to support complex 3D GIS analysis, and spatio-temporal analysis.
文摘WE use the geometric algebra in refs. [1, 2] to study hyperbolic geometry. The n-dimensional hyperbolic space H<sup>n</sup> is taken to be the unit sphere of G<sub>1</sub> (I<sub>-n,1</sub>) with antipodal points identified. We study typically H<sup>2</sup>: the dualities between generalized point and generalized line, between generalized triangle and imaginary triangle; convex generalized triangles; Lorentz transformations; generalized circles and double-cycles, etc. Below we list some of the results.
基金Supported by National Natural Science Foundation of China (Grant No. 52175019)Beijing Municipal Natural Science Foundation of China (Grant No. L222038)+3 种基金Beijing Nova Programme Interdisciplinary Cooperation Project of China (Grant No. 20240484699)Joint Funds of Industry-University-Research of Shanghai Academy of Spaceflight Technology of China (Grant No. SAST2022-017)Beijing Municipal Key Laboratory of Space-ground Interconnection and Convergence of ChinaKey Laboratory of IoT Monitoring and Early Warning,Ministry of Emergency Management of China
文摘To investigate the forward kinematics problem of parallel mechanisms with complex limbs and to expand the applicability of the powerful tool of Conformal Geometric Algebra(CGA),a CGA-based modeling and solution method for a class of parallel platforms with 3-RE structure after locking the actuated joints is proposed in this paper.Given that the angle between specific joint axes of limbs remains constant,a set of geometric constraints for the forward kinematics of parallel mechanisms(PM)are determined.After translating unit direction vectors of these joint axes to the common starting point,the geometric constraints of the angle between the vectors are transformed into the distances between the endpoints of the vectors,making them easier to handle.Under the framework of CGA,the positions of key points that determine the position and orientation of the moving platform can be intuitively determined by the intersection,division,and duality of basic geometric entities.By employing the tangent half-angle substitution,the forward kinematic analysis of the parallel mechanisms leads to a high-order univariate polynomial equation without the need for any complex algebraic elimination operations.After solving this equation and back substitution,the position and pose of the MP can be obtained indirectly.A numerical case is utilized to confirm the effectiveness of the proposed method.
基金Supported by National Natural Science Foundation of China(Grant No.51375059)National Hi-tech Research and Development Program of China(863 Program,Grant No.2011AA040203)+1 种基金Special Fund for Agro-scientific Research in the Public Interest of China(Grant No.201313009-06)National Key Technology R&D Program of the Ministry of Science and Technology of China(Grant No.2013BAD17B06)
文摘The solution for the forward displacement analysis（FDA） of the general 6-6 Stewart mechanism（i.e., the connection points of the moving and fixed platforms are not restricted to lying in a plane） has been extensively studied, but the efficiency of the solution remains to be effectively addressed. To this end, an algebraic elimination method is proposed for the FDA of the general 6-6 Stewart mechanism. The kinematic constraint equations are built using conformal geometric algebra（CGA）. The kinematic constraint equations are transformed by a substitution of variables into seven equations with seven unknown variables. According to the characteristic of anti-symmetric matrices, the aforementioned seven equations can be further transformed into seven equations with four unknown variables by a substitution of variables using the Grobner basis. Its elimination weight is increased through changing the degree of one variable, and sixteen equations with four unknown variables can be obtained using the Grobner basis. A 40th-degree univariate polynomial equation is derived by constructing a relatively small-sized 9 × 9 Sylvester resultant matrix. Finally, two numerical examples are employed to verify the proposed method. The results indicate that the proposed method can effectively improve the efficiency of solution and reduce the computational burden because of the small-sized resultant matrix.
文摘The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. So it is worth popularizing in the teaching of undergraduate physics and mathematics. Clifford algebras can be directly generalized to 2<sup>n</sup>-ary associative algebras. In this generalization, the matrix representation of the orthonormal basis of space-time plays an important role. The matrix representation carries more information than the abstract definition, such as determinant and the definition of inverse elements. Without this matrix representation, the discussion of hypercomplex numbers will be difficult. The zero norm set of hypercomplex numbers is a closed set of special geometric meanings, like the light-cone in the realistic space-time, which has no substantial effect on the algebraic calculus. The physical equations expressed in Clifford algebra have a simple formalism, symmetrical structure, standard derivation, complete content. Therefore, we can hope that this magical algebra can complete a new large synthesis of modern science.
文摘The superiority of hypothetical quantum computers is not due to faster calculations but due to different schemes of calculations running on special hardware. The core of quantum computing follows the way a state of a quantum system is defined when basic things interact with each other. In conventional approach it is implemented through tensor product of qubits. In the geometric algebra formalism simultaneous availability of all the results for non-measured observables is based on the definition of states as points on three-dimensional sphere.
文摘The current work considers the sprefield wave functions received as special g-qubit solutions of Maxwell equations in the terms of geometric algebra. I will call such g-qubits spreons or sprefields. The purpose of this article is to analyze behavior of such wave functions in scattering and measurements. It is shown that sprefields are defined through the whole three-dimensional space at all values of the time parameter. They instantly change all their values when get scattered, that is subjected to Clifford translation. In “measurements”, when a sprefield acts on a static geometric algebra element through the Hopf fibration, sprefield collapses and new geometric algebra non static, rotating element is thereby created.
文摘Geometric Algebra formalism opens the door to developing a theory deeper than conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions, unambiguous definition of states, observables, measurements, Maxwell equations solution in those terms, bring into reality a kind of physical fields spreading through the whole three-dimensional space and values of the time parameter. The fields can be modified instantly in all points of space and time values, thus eliminating the concept of cause and effect, and perceiving of one-directional time. In the suggested theory all measured observable values get available all together, not through looking one by one. In this way quantum computer appeared to be a kind of analog computer keeping and instantly processing information by and on sets of objects possessing an infinite number of degrees of freedom. As practical implementation, the multithread GPUs bearing the CUDA language functionality allow to simultaneously calculate observable measurement values at a number of space/time discrete points only restricted by the GPU threads capacity.
文摘A primordial field Self-interaction Principle, analyzed in Hestenes’ Geometric Calculus, leads to Heaviside’s equations of the gravitomagnetic field. When derived from Einstein’s nonlinear field equations Heaviside’s “linearized” equations are known as the “weak field approximation”. When derived from the primordial field equation, there is no mention of field strength;the assumption that the primordial field was predominant at the big bang rather suggests that ultra-strong fields are governed by the equations. This aspect has physical significance, so we explore the assumption by formulating the gauge field version of Heaviside’s theory. We compare with recent linearized gravity formulations and discuss the significance of differences.
文摘The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from changing of complex numbers by geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of weird quantum mechanical features, for example primitively considering atoms as a kind of solar system. The three-sphere S<sup>3</sup> becomes the playground of the torsion kind states eliminating abstract Hilbert space vectors. The S<sup>3</sup> points evolve, governed by updated Schrodinger equation, and act in measurements on observable as operators.
文摘Superposition and entanglement are two theoretical pillars quantum computing rests upon. In the g-qubit theory quantum wave functions are identified by points on the surface of three-dimensional sphere S<sup>3</sup>. That gives different, more physically feasible explanation of what superposition and entanglement are. The core of quantum computing scheme should be in manipulation and transferring of wave functions on S<sup>3</sup> as operators acting on observables and formulated in terms of geometrical algebra. In this way quantum computer will be a kind of analog computer keeping and processing information by sets of objects possessing infinite number of degrees of freedom, contrary to the two value bits or two-dimensional Hilbert space elements, qubits.
