This paper proposes a novel formulation to the problem of the five-precision points path synthesis for planar four-bar linkages with given fixed pivots. Firstly, the coupler-curve equation of planar four-bar linkages is revisited and derived based on the conformal geometric algebra (CGA), whose derivation is operated geometrically and free of coordinate. Secondly, based on the derived coupler-curve equation and the variable substitution, the seven constraint equations in seven variables for the problem are formulated. Thirdly, the Gröbner bases under the group degree reverse lexicographic ordering for the problem are reduced using computer algebra. Then, a 19 by 19 Sylvester resultant is constructed by selecting 19 Gröbner bases from 48 ones. Finally, a 36th degree univariate equation is directly obtained from the determinant of the resultant. At last, two numerical examples are provided to demonstrate the proposed method. The advantages of the novel method lie in that seven constraint equations are derived based on the coupler curve, and the novel formulation is concise and the computing speed is faster than before.

This paper presents a novel geometric modeling method for direct displacement analysis of 6-4 Stewart platforms based on conformal geometric algebra (CGA). First, a geometric constraint relationship of four lines and a plane intersecting at a point is published. Second, a new coordinate-invariant geometric constraint equation of 6-4 Stewart platforms is deduced by CGA operation. Third, five polynomial equations are established by CGA theory. Fourth, based on the above six equations, a 5 x 5 Sylvester's matrix is formulated by using Sylvester's Dialytic elimination method and Gr & ouml;bner bases method under the graded reverse lexicographical order. Finally, the coordinates of four points on the moving platform are revealed. Besides, a numerical example is used to prove the validity of the proposed method. The novelty of this study is that a whole geometric modeling method by geometric constraint relationship of four lines and a plane intersecting at a point is put forward under the CGA framework, which has good intuition and offers a novel idea for solving the other complex mechanisms. At the same time, Sylvester's matrix constructed by this method is the smallest one in the known literature for forward displacement analysis of 6-4 Stewart platforms.

This paper proposes a geometric modeling and algebraic solution method for the displacement analysis of the three seven-link Baranov trusses based on conformal geometric algebra (CGA). Under the frame of CGA, a coordinate-free formulation for the basic four-link kinematic chain is derived in terms with the area sign of the triangle and CGA operation. Then, based on the aforementioned formulation and the distance relationship, two constraint equations for the displacement analysis problem are formulated. Finally, a high-degree univariate equation is derived by one-step resultant elimination. The contribution of the paper is that the derivation of two constraint equations is free of coordinate and the elimination procedure is greatly simplified to only one-step due to the reduction of the number of equations using CGA. At last, numerical examples are given to confirm the correctness of the method. The proposed method offers a new sight for solving the displacement analysis problem of other Baranov trusses.

摘要： 基于共形几何代数(Conformal geometric algebra, CGA)，提出了一种求解SRS(S：球副，R：转动副)7自由度机械臂逆运动分析的臂型角参数化法。首先，根据臂型角参数化，在共形几何代数框架下，使用点、线、面、球和点对等基本几何体的相交、分离和对偶等运算，得到机械臂5个关节点的位置；接着，通过构造过关节点的直线与平面，使用构造的直线间或构造的平面间的内积，得出机械臂7个关节转角的余弦表达，从而获得了机械臂逆运动学分析的所有16组解析解；最后，以Kinova Jaco2 7自由度机械臂为数值实例求解，并通过SolidWorks仿真验证了所提方法的正确性。结果表明提出的方法具有几何直观性强和计算简明的特点，为其他7自由度机械臂的逆运动学求解提供了一种新思路。

基于共形几何代数,提出了一种9 杆巴氏桁架位移分析的新几何建模和计算方法.在共形几何代数框架下,根据点、球、面、点对等基本几何元素的表示方法以及三角形面积的有向性,通过几何体的相交、分离和对偶运算,建立2 个位移约束方程式;然后,通过一步结式消元得到该问题的一元 54 次方程,无增根无漏根.所提方法的优势在于2 个位移约束方程的推导脱离了坐标系,且约束方程数的减少简化了方程组的消元过程.数值实例表明了所提方法的正确性,为其他9 杆巴氏桁架的位移分析求解理论提供了一种新思路.

将倍矩阵引入到串联机械手运动学分析的建模中,结合四维旋转矩阵,提出了一种空间1P5R串联机械手逆运动学分析的新方法.基于倍矩阵和四维旋转矩阵对空间1P5R串联机械手进行运动学建模,直接得到14个逆运动学约束方程;将14个方程通过线性消元和Sylvester结式消元,得到只含1个关节变量的一元16次方程.通过数字实例求解和SolidWorks软件仿真验证了所提方法的正确性.所提方法最大的优势在于可直接获得14个逆运动学约束方程,不需要进行矢量运算或者投影等,为空间其他串联机械手的逆运动学建模提供了一种新思路.

空间刚体变换的描述方法是机器人机构运动学分析的一个重要内容,描述方法对机构性能分析起着重要作用,不同的描述方法会导致机构运动学建模和求解的难易不同.基于对偶矩阵理论,类比倍四元数与对偶四元数的映射关系,提出并推导了空间刚体位移和变换的倍矩阵描述新方法.该方法对空间线变换进行描述,可以描述空间刚体的全位姿变换.详细推导并证明了四维旋转矩阵、倍四元数和倍矩阵这三种空间刚体变换的近似描述方法之间的相互转换关系.这三种描述方法将三维空间刚体的平移转换为4维空间的旋转,进而实现了刚体平移和旋转量纲的统一,为其比较大小提供了一种尺度.

为了解决空间6R串联机械手的逆运动学问题使用矩阵方法建模时,需要进行矢量运算或投影运算的问题,基于四维旋转矩阵和倍矩阵,提出了一种建模新方法.根据三维空间刚体变换的四维旋转矩阵和倍矩阵表示,建立空间6R串联机械手的正运动学方程.通过变量分离,直接得到14个逆运动学基本约束方程;通过线性消元和Sylvester结式消元,将其转化为求解一个16阶矩阵的特征值问题,得到该问题的16组解.采用数值实例和SolidWorks仿真验证了新方法的正确性.新方法的优势在于可以直接得到14个逆运动学约束方程,不需要进行矢量运算或者投影等,并且由于新方法将三维空间中的平移变换近似为四维空间中的旋转变换,故而可以统一求解含有R、P和C副的空间串联机械手逆运动学问题.

This paper put forward a new geometric modeling method for forward displacement analysis of 3-RPR planar parallel mechanisms (PPMs) under the conformal geometric algebra (CGA) framework. A geometric constraint relationship of four lines intersecting at a point is first given. Then, a new coordinate-invariant geometric constraint equation of 3-RPR PPMs is elaborated via CGA operation. Next, an univariate polynomial equation without algebraic elimination is published. At last, the coordinates of three points on the mobile platform are figured out. The best part of this research is that a complete geometric modeling method by geometric constraint relationship of four lines intersecting at a point is formulated under the CGA framework, which has good intuition and offers a novel idea for solving the other complex mechanisms. Three numerical examples are utilized to confirm the effectiveness of the proposed method.

This paper demonstrates a novel geometric modeling and computational method of the family of spatial parallel mechanisms (PMs) with 3-R(P)S structure for direct kinematic analysis based on the point pair relationship. The point pair relationship, which is derived from the framework of conformal geometric algebra (CGA), consists of the relationship between the point and the point pair and two point pairs. The first research is on the distance relationship between the point and the point pair. Second, the derivation of the distance relationship between two point pairs is based on the aforementioned result, which shows the mathematical homogeneity. Third, two formulations for a point of the point pairs that satisfy the distance relationship between two point pairs are reduced. Fourth, the point pair relationship is applied to solve the direct kinematic analysis of the spatial parallel mechanism with 3-R(P)S structure. Finally, four numerical examples are provided to verify the validity of the proposed algorithm. Overall, the proposed method can be generalized for the direct kinematics of a series of spatial parallel mechanisms with 3-R(P)S structure.

This paper presents a novel analytical approach for synthesizing a path-generation mechanism without any limitation on the number of precision points. Compared with classical analytical methods, the novelty of the proposed method is that the synthesis equations were established based on the relationship between the design variables and the Fourier coefficients of the path instead of the displacement matrix. Based on the presented synthesis equations, the path generation problem of planar four-bar linkages was reduced to solving two polynomial equations of low complexity. Moreover, a general formula was derived from the analytical solutions of the new synthesis equations, whereby the design variables could be directly calculated with the Fourier coefficients of the prescribed path. Five examples are provided to evaluate the efficiency and accuracy of the proposed method. The findings indicate that the proposed method is simple, efficient, and readily programmed. (C) 2019 Elsevier Ltd. All rights reserved.

In order to cope with the requirements of the coordinate establishment and elimination process, in the process of the modelling and computing for the forward kinematic analysis of general planar parallel manipulators, a geometric modeling and free-elimination computing method for the forward kinematics of planar parallel manipulators is proposed using conformal geometric algebra (CGA). Under the frame of CGA, two of the three coordinates in the moving platform are formulated by the intersecting, dissecting and dual of the basic geometric entities; in terms with the area sign of the moving triangular platform, the characteristic polynomial equation is derived by a sequence of geometric algebra operation and simplification; a high-degree polynomial equation for planar parallel manipulators of any link parameters is deduced by tangent-half-angle substitution, Euler-angle substitution or no substitution. The derivation of the characteristic polynomial is free of coordinate and no elimination process and no assumption are required. Numerical examples are given to validate the correctness of the procedure and that the proposed algorithm is feasible to all cases of planar parallel manipulators including the special and general structures. At last, the results show that the proposed algorithm has a readily numerical robustness and provides a new sight for the theoretical solution to the kinematics of planar parallel manipulators.

In order to overcome the disadvantage of conventional precise point methods,optimization approaches and numerical atlas,and further improve the accuracy and efficiency of rigid-body guidance synthesis of planar four-bar linkages with prescribes timing,a novel analytical approach with Fourier series was presented to solve the rigid-body guidance synthesis problem of planar four-bar linkages.Firstly,according to the periodicity of the linkage rotation angle function,the series mathematical formula of the linkage rotation angle function was established by using Fourier series theory.The harmonic parameters of the linkage rotation angle function were obtained by DFT.As a result,the linkage rotation angle function was expressed as the summation of a Fourier series with the input angle as the variable.The harmonic parameters relationship between the linkage rotation angle function and the rigid-body guidance rotation angle function was obtained by analysing the internal connection between them in the planar motion.Then,according to the theory of complex vector,the vector loop equation of planar four-bar linkages was established.The linkage rotation angle function was formulated according to the Fourier series,and was substituted into a vector loop equation.By using eliminated element method,the vector loop equation was changed into a complex number equation that contained only the fundamental dimensions of mechanism, the harmonic parameters of the linkage rotation angle function and the input angle.With regard to the properties of the complex exponent, the mathematic expression containing the fundamental dimensions of mechanism and the harmonic parameters of the linkage rotation angle function was obtained.Accordingly,the relationship between the fundamental dimensions of mechanism and the harmonic parameters of the rigid-body guidance rotation angle function was obtained by variable substitution in terms of the previous harmonic parameters relationship.Based on this relationship,the new design equations for rigid-body guidance synthesis of planar four-bar linkages were established.A cubic equation that contained only the fundamental dimensions of mechanism and the harmonic parameters of the rigid-body guidance rotation angle function was obtained by dialytic elimination.Through solving the cubic equation,a general formula was derived for the rigid-body guidance synthesis problem of planar four-bar linkages using the harmonic parameters of the rigid -body guidance rotation angle function.After the fundamental dimensions of mechanism are determined,the harmonic parameters of position of rigid-body guidance were obtained by DFT.According to the harmonic parameters relationship between the linkage rotation angle function and the position of rigid-body guidance,the formula,which can compute the real size and installing dimensions of the linkage mechanism,was obtained.Based on the aforementioned theory,the procedure of solving rigid-body guidance synthesis problem by the proposed method can be obtained.As a result,a new analytical method for rigid-body guidance synthesis of planar four-bar linkages with prescribes timing was established.Based on the synthesis procedure,the computer programs have been developed for the proposed method by MATLAB.An example was provided to verify the validity and feasibility of the proposed method.Verification results showed that the proposed approach can overcome the shortage of precise point method and directly to solve rigid-body guidance synthesis problems for planar four-bar linkages with no limitations on the number of precision points.Compared with numerical atlas and optimization approaches,the proposed method avoided the use of extensive numerical atlas databases and optimal initial solutions,and obtained the results by solving the equation.Therefore,this approach has the characteristics of high accuracy,fast solution velocity and high repeatability,and is suitable for computer programming.The research of this approach provided the theory basis for development of convenient synthesis software.

This paper presents a unified geometric modeling and solution procedure for direct kinematic analysis of a class of parallel mechanisms based on conformal geometric algebra (CGA). After locking the actuated joints, such parallel mechanisms will be turned into a 3-RS structure, which is composed of two triangular platforms connected by three RS serial chains in parallel. Using the proposed approach, the univariate polynomial equation for the direct kinematic analysis of these parallel mechanisms can be derived in three steps. Firstly, the positions of two of the three spherical joints on the moving platform are formulated by the intersection, dissection and dual of the basic geometric entities under the frame of CGA. Secondly, a coordinate-invariant equation expressed in terms of geometric entities is derived via CGA operation. Thirdly, a univariate polynomial equation is obtained directly from the aforementioned coordinate-invariant equation by using tangent-half-angle substitution. Several case studies are then presented to verify the solution procedure. The novelties of this approach lie in that: (1) The formulation is concise and coordinate-invariant and has intrinsic geometric intuition due to the use of CGA; (2) No algebraic elimination procedure is required to derive the univariate polynomial equation; and (3) The proposed approach is applicable to the direct kinematics of this family of parallel mechanisms with any link parameters. (c) 2018 Elsevier Ltd. All rights reserved.

In this paper, a new algorithm for the forward displacement analysis of a general 6-3 Stewart platform(6-3SPS) based on conformal geometric algebra (CGA) is presented. First, a 6-3SPS structure is changed into an equivalent 2RPS-2SPS structure. Then, two kinematic constraint equations are established based on the geometric characteristics, one of which is built according to the point characteristic four-ball intersection in CGA. A 16th-degree univariate polynomial equation is derived from the aforementioned two equations by the Sylvester resultant elimination. Finally, a numerical example is given to verify the algorithm.

In this paper, we propose a novel solution process for the forward kinematics of general 3-RPS parallel mechanisms based on conformal geometric algebra (CGA). First of all, the position of one of the three spherical joints is expressed in the rotational angle about the axis of one kinematic chain RPS. Secondly, the other two spherical joints can be determined by this angle via CGA operation. Thirdly, an explicit 16th-degree univariate polynomial equation is reduced from two geometric constraint equations. At last, one numerical example is employed to verify the solution procedure. The novelties of this paper lie in that (1) the modeling formulation and the elimination procedure have intrinsic geometric intuition due to the use of CGA and (2) the 16th-degree polynomial equation without extraneous roots is explicitly formulated and suitable for implementing the mathematics mechanization of this problem.

A novel analytical approach is presented to solve path synthesis problems for planar five-bar linkages with timing.Firstly,the planar five-bar linkage is decomposed into two dyads for the purpose of decoupling the design parameters of the linkage.The relationship between the design parameters of the linkage and the harmonic parameters of the path is obtained,based on which,the new synthesis equations with different transmission ration for planar five-bar linkages are established.A quartic equation is obtained by dialytic elimination.As a result,a general formula is derived for calculation of the design parameters of a planar five-bar linkage.The design parameters of two dyads are obtained with the general formula respectively.At last,all design parameters of the planar five-bar linkage are obtained.Two numerical examples are provided to verify the effectiveness of the proposed algorithm.

The problem of five precision points path synthesis of planar four-bar linkage can be divided into four types in term of the input parameters. A unified formulation for the four types is built based on the planar displacement matrix. Next, the corresponding resultant matrix is constructed based on Groebner bases generated by applying the new term ordering (the groups graded reverse lexicographic ordering, <(ggrevlex)) for four types. Then, a high-degree univariate polynomial equation is accordingly obtained. At last, several examples are provided to validate the algorithm and the solutions are verified in the software SAM. And it is concluded that type I has 36 solutions, type II has 64 including 16 degenerate solutions, type III has 92 solutions and,type IV has 82 solutions including 16 degenerate solutions.

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 Gröbner 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 Gröbner 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.

In this paper, we present a new algebraic elimination algorithm for the inverse static force analysis of a special planar three-spring system. The system consists of three linear springs joined to the ground at the two fixed pivots and connected to the two moving pivots at the platform. When exerted by specified static force, the goal of inverse static analysis is to determine all the equilibrium configurations. First of all, a system of seven polynomial equations in seven variables is established based on the geometric constraint and static force balancing. Then, four basic constraint equations in four variables are obtained by variable substitution. Next, a 20 by 20 resultant matrix is reduced by means of three consecutive Sylvester elimination process. Finally, a 54th-degree univariate polynomial equation is directly derived without extraneous roots in the computer algebra system Mathematica 9.0. At last, a numerical example is given to verify the elimination procedure.