Kinematics and Dynamics Hessian Matrices of Manipulators Based on Screw Theory

The complexity of the kinematics and dynamics of a manipulator makes it necessary to simplify the modeling process.However,the traditional representations cannot achieve this because of the absence of coordinate invariance.Therefore,the coordinate invariant method is an important research issue.First,the rigid-body acceleration,the time derivative of the twist,is proved to be a screw,and its physical meaning is explained.Based on the twist and the rigid-body acceleration,the acceleration of the end-effector is expressed as a linear-bilinear form,and the kinematics Hessian matrix of the manipulator（represented by Lie bracket）is deduced.Further,Newton-Euler＇s equation is rewritten as a linear-bilinear form,from which the dynamics Hessian matrix of a rigid body is obtained.The formulae and the dynamics Hessian matrix are proved to be coordinate invariant.Referring to the principle of virtual work,the dynamics Hessian matrix of the parallel manipulator is gotten and the detailed dynamic model is derived.An index of dynamical coupling based on dynamics Hessian matrix is presented.In the end,a foldable parallel manipulator is taken as an example to validate the deduced kinematics and dynamics formulae.The screw theory based method can simplify the kinematics and dynamics of a manipulator,also the corresponding dynamics Hessian matrix can be used to evaluate the dynamical coupling of a manipulator.

Kinematics and Dynamics of Deployable Structures with Scissor-Like-Elements Based on Screw Theory

Because the deployable structures are complex multi-loop structures and methods of derivation which lead to simpler kinematic and dynamic equations of motion are the subject of research effort, the kinematics and dynamics of deployable structures with scissor-like-elements are presented based on screw theory and the principle of virtual work respectively. According to the geometric characteristic of the deployable structure examined, the basic structural unit is the common scissor-like-element（SLE）. First, a spatial deployable structure, comprised of three SLEs, is defined, and the constraint topology graph is obtained. The equations of motion are then derived based on screw theory and the geometric nature of scissor elements. Second, to develop the dynamics of the whole deployable structure, the local coordinates of the SLEs and the Jacobian matrices of the center of mass of the deployable structure are derived. Then, the equivalent forces are assembled and added in the equations of motion based on the principle of virtual work. Finally, dynamic behavior and unfolded process of the deployable structure are simulated. Its figures of velocity, acceleration and input torque are obtained based on the simulate results. Screw theory not only provides an efficient solution formulation and theory guidance for complex multi-closed loop deployable structures, but also extends the method to solve dynamics of deployable structures. As an efficient mathematical tool, the simper equations of motion are derived based on screw theory.

Generalized Kinematics Analysis of Hybrid Mechanisms Based on Screw Theory and Lie Groups Lie Algebras

Advanced mathematical tools are used to conduct research on the kinematics analysis of hybrid mechanisms,and the generalized analysis method and concise kinematics transfer matrix are obtained.In this study,first,according to the kinematics analysis of serial mechanisms,the basic principles of Lie groups and Lie algebras are briefly explained in dealing with the spatial switching and differential operations of screw vectors.Then,based on the standard ideas of Lie operations,the method for kinematics analysis of parallel mechanisms is derived,and Jacobian matrix and Hessian matrix are formulated recursively and in a closed form.Then,according to the mapping relationship between the parallel joints and corresponding equivalent series joints,a forward kinematics analysis method and two inverse kinematics analysis methods of hybrid mechanisms are examined.A case study is performed to verify the calculated matrices wherein a humanoid hybrid robotic arm with a parallel-series-parallel configuration is considered as an example.The results of a simulation experiment indicate that the obtained formulas are exact and the proposed method for kinematics analysis of hybrid mechanisms is practically feasible.

An approach for determination of lateral limit angle in kinematic planar sliding analysis for rock slopes

Planar sliding is one of the frequently observed types of failure in rock slopes.Kinematic analysis is a classic and widely used method to examine the potential failure modes in rock masses.The accuracy of planar sliding kinematic analysis is significantly influenced by the value assigned to the lateral limit angleγlim.However,the assignment ofγlim is currently used generally based on an empirical criterion.This study aims to propose an approach for determining the value ofγlim in deterministic and probabilistic kinematic planar sliding analysis.A new perspective is presented to reveal thatγlim essentially influences the probability of forming a potential planar sliding block.The procedure to calculate this probability is introduced using the block theory method.It is found that the probability is correlated with the number of discontinuity sets presented in rock masses.Thus,different values ofγlim for rock masses with different sets of discontinuities are recommended in both probabilistic and deterministic planar sliding kinematic analyses;whereas a fixed value ofγlim is commonly assigned to different types of rock masses in traditional method.Finally,an engineering case was used to compare the proposed and traditional kinematic analysis methods.The error rates of the traditional method vary from 45%to 119%,while that of the proposed method ranges between 1%and 17%.Therefore,it is likely that the proposed method is superior to the traditional one.

Special Relativity’s “Newtonization” in Complex “Para-Space”: The Two Theories Equivalence Question

Complex model, say C3, of “para-space” as alternative to the real M4 Minkowski space-time for both relativistic and classical mechanics was shortly introduced as reference to our previous works on that subject. The actual aim, however, is an additional analysis of the physical and para-physical phenomena’ behavior as we formally transport observable mechanical phenomena [motion] to non-real interior of the complex domain. As it turns out, such procedure, when properly set, corresponds to transition from relativistic to more classic (or, possibly, just classic) kind of the motion. This procedure, we call the “Newtonization of relativistic physical quantities and phenomena”, first of all, includes the mechanical motion’s characteristics in the C3. The algebraic structure of vector spaces was imposed and analyzed on both: the set of all relativistic velocities and on the set of the corresponding to them “Galilean” velocities. The key point of the analysis is realization that, as a matter of fact, the relativistic theory and the classical are equivalent at least as for the kinematics. This conclusion follows the fact that the two defined structures of topological vector spaces i.e., the structure imposed on sets of all relativistic velocities and the structure on set of all “Galilean” velocities, are both diffeomorphic in their topological parts and are isomorphic as the vector spaces. As for the relativistic theory, the two approaches: the hyperbolic (“classical” SR) with its four-vector formalism and Euclidean, where SR is modeled by the complex para-space C3, were analyzed and compared.

The Kinematics and Field Equations for Porous Media

With a porous medium regarded as an immiscible mixture of multiphase and each phase as a miscible mixture of multi constituent, a systematical research on the kinematics and field equations for porous media is carried out from the point of view of mixture theory. It is shown that the motion of each phase is the mathematical average of the motions of all constituents in the phase, and that the motion of porous media may be described as the motion of the skeleton and the relative motion of each phase with respect to the skeleton. The influence of mass exchange between different constituents in each phase and the influence of mass exchange of same constituent between different phases in porous media are considered in field equations which are self consistent in theory. All the field equations in the references are special cases of the equations proposed in this paper.

具有变泊松运动特性的剪叉式折展机构运动学分析

为提高折展机构的折展率和支撑性能,提出了一种具有变泊松运动特性的剪叉式折展机构并对其进行运动学分析。首先,提出了一种单闭环厚板支撑单元并将其应用于三明治结构,利用ANSYS Workbench软件分析不同形状夹芯层对三明治结构支撑刚度的影响,发现采用厚板支撑单元的三明治结构具有更好的支撑效果和更小的质量,且可通过改变结构设计参数来实现正负泊松比的切换。然后,根据泊松比的定义,设计了一种具有变泊松运动特性的正n边形剪叉式折展机构;基于螺旋理论,通过绘制闭环折展机构的旋量约束拓扑图分析得到其自由度为1,将折展机构分为3种模块并阐述了模块化纵向扩展的原理和过程。最后,建立了m层正n边形剪叉式折展机构的运动学模型并搭建了正四边形剪叉式折展支撑结构实物样机,进一步验证了机构的变泊松运动特性,这可为后续的研究提供理论基础。

一种基于图论理论的大尺度连续变形机构构型综合分析方法

针对航空航天领域大尺度连续变形机构构型问题,提出了一种基于图论理论的大尺度连续变形机构构型综合分析方法。首先,分析了大尺度连续变形机构的组成与基本功能,获得了满足机构拓扑结构、约束条件的要求,对变形机构运动链进行了分析,得到了符合条件的运动链,并对其进行了数学描述;其次,选择并分析了运动链的拓扑对称性,利用特征数组的方法得到了判别矩阵,对运动链进行了同构判别,从而得到了可行的多种运动链形式;再次,建立了各运动链的拓扑图,通过筛选得到了最终的拓扑图,去掉该拓扑图的各个构件,得到了拓扑变换子图和相对应的邻接矩阵,对拓扑图进行了同构判别,再通过改变机架与移动副的位置得到了满足要求的12种构型,画出了其相应的机构拓扑图和机构简图;最后,考虑了大尺度连续变形机构的整体稳定性,避免了尺寸过大等因素,对大尺度变形机构进行了构型优选,得到了符合要求的多种新构型。研究结果表明:该方法能够将大尺度连续变形机构抽象为图形式,获得了22种构型,可根据机构变形要求的不同,改变优选依据,从而得到符合不同要求的构型。

一种新型3-R(4r)(4r)R并联机构及运动学分析

本文提出了一种3-R(4r)(4r)R型全对称三平移并联机构,运用螺旋理论分析了该并联机构实现三平移的机构学原理,进行了驱动输入的选取,分析了机构平台奇异、驱动奇异和支链奇异的特性,建立了机构运动学模型,获得了运动学正、逆解析解,并通过数值仿真验证了正逆解模型的正确性。研究结果表明,该并联机构无平台奇异,运动学模型简单且正解唯一。

2-UPS/UPR/(rT)PS可重构并联机构的运动学分析及应用

目的 针对产品加工包装流水线上的印刷和装箱环节,提出一种具有双构型的2-UPS/UPR/(rT)PS可重构并联机构。方法 基于螺旋理论,分析得出机构在2种构型下自由度的数目及性质;根据修正的Kutzbach-Grübler公式,验证自由度的计算结果;采用封闭矢量法计算2种模式下机构的运动学反解;求解机构在2种模式下的可达工作空间;分析应用实例,研究机构在进行印刷和装箱2个环节作业时驱动支链线位移的变化,使得机构满足喷码和装箱工作需求。结果 2-UPS/UPR/(rT)PS并联机构具有2R1T、2R2T等2种运动模式,工作空间内部连续。结论 2-UPS/UPR/(rT)PS可重构并联机构通过运动副轴线变化,使机构可在2R1T、2R2T这2种运动模式下进行切换,可应用于产品印刷、装箱环节。

铣削加工机器人旋量建模及几何求逆优化

为了提高机器人在铣削加工应用中的性能,将旋量理论应用于铣削加工机器人的建模中。针对仿真过程中逆解计算太慢的问题,提出了一种改进几何法对求逆过程进行优化。该方法以库卡KR60铣削加工机器人为例,考虑铣削加工位置相对固定,忽略了机器人前三个关节引起的逆解变化,从而简化了逆解的求取。仿真结果表明,经过改进后的几何法相对MATLAB软件自带求逆方法能够极大提高运算效率,减少仿真时间。

机械臂运动学和动力学的李群表述

D-H参数法已经被广泛应用到机械手运动建模及分析中,但其所描述的运动都是围绕着x轴和z轴,而不能反映出y轴的运动。为了解决这个问题,采用旋量理论的方法,用旋量来表述出将机械臂的位置和姿态,并进一步分析机械臂的运动学和动力学。相较于传统的D-H法计算运动学正解,旋量理论是从整体上对机械臂的运动进行表述,且不需要中间参考系,几何意义清晰。利用Paden-Kahan子问题分析肘机械臂的逆运动学问题,计算出其运动学逆解,并利用旋量理论和李群-李代数,通过Newton-Euler法建立了高效的递归动力学模型。最后对于用旋量理论求出的肘机械臂运动学逆解进行仿真验证,并计算出其工作空间,结果表明用旋量理论的方法更加简洁、精确、高效。

基于螺旋理论的3自由度并联机构运动学分析

【目的】满足汽车表面加工与车漆喷涂的任务需求,利用机器人进行协同甚至自主完成加工、美化工作,提高工作效率。【方法】设计了一款具备多自由度的汽车车身制造与喷漆机器人,采用3-RPS并联机构作为执行部件支撑件,针对机器人的结构设计特点,主要对机器人末端3-RPS并联机构支撑部分的机构构型和整体运动采用螺旋理论进行分析,利用闭环支链建立机构运动学模型。【结果】利用MATLAB和Admas View软件分别完成了运动学方程求解和运动学仿真,两者结果误差在3%以内。【结论】仿真值与分析值的误差较小,在可忽略范围内,所建立的设计机构运动学方程有效可行,可以利用此结果进行进一步的机器人运动控制设计与相关研究。

油茶果动态抓取Delta机器人正运动学特性研究

针对传统D-H参数法求解机械臂运动学时过程比较复杂的问题,为满足油茶果动态抓取的要求,提出了一种基于旋量理论构建3-R(SS)2Delta机器人正运动学方程的方法。针对3-R(SS)2Delta机器人的结构进行分析;通过旋量理论的指数积公式对该机器人的正运动学特性进行研究,获得机器人的位姿变换矩阵;最后,通过搭建的Delta机器人试验平台运行标准门型轨迹,结果表明:Delta机器人末端执行器能够精准地对传送带上移动的油茶果进行抓取。对比分析正运动学方程的计算值和Delta机器人试验平台试验测量值,得到末端执行器位置值的位置误差为±0.3mm,验证了所提出方法对构建油茶果动态抓取Delta机器人正运动学方程的正确性及可行性,旨在为后续进行油茶果动态抓取的轨迹规划研究奠定基础。

一种3自由度并联抛磨机械臂的运动学分析

针对百叶窗叶片的仿形抛磨要求,研究了一种可为末端执行器提供X向、Y向的移动和Z向转动的3自由度并联机械臂。为了验证机械臂的运动学性能,首先,描述了其机构特征,并基于螺旋理论进行了机构自由度分析;其次,基于闭环矢量法构建了机构运动学方程,对其进行了位置正、逆解分析,并通过具体算例对分析结果进行了初步验证;最后,构建了机构的数值仿真模型和虚拟样机模型,进行了位置、速度的逆运动学仿真分析。仿真结果表明,两种模型的分析结果一致,逆解理论分析结果可信;机械臂各支链可协调运动,运动过程平稳,位移、速度曲线平滑、无突变,可为后续机械臂的尺寸优化和运动控制提供依据。

Time Dilation Cosmology 2

This paper is a further elaboration of the author’s Time Dilation Cosmology (TDC) holographic model that ties gravitation and celestial mechanics and kinematics directly to time dilation, resolving all the major conundrums in astrophysics, and ties astrophysics directly to quantum physics. It begins with a brief summary of the TDC model and contains the new derivation for the time dilation version of the formula for summing relativistic velocities, Einstein’s gravitational constant and the time dilation versions for the Lorentz factor and the Euclidean norm of the 3d velocity vector, the two of which can then be used in the Four-velocity formula. It is demonstrated how orbital curvature is manifested as the resultant of two time dilation-manifested velocities. It also explains why an interferometer cannot distinguish free fall from zero gravity and further elaborates on the author’s previous explanations of how spiral galaxies are formed, and contains mathematical proof that Black Holes are actually Magnetospheric Eternally Collapsing Objects (MECOs) that are massless spacetime vortices.

A Finite and Instantaneous Screw Based Approach for Topology Design and Kinematic Analysis of 5-Axis Parallel Kinematic Machines

Unifying the models for topology design and kinematic analysis has long been a desire for the research of parallel kinematic machines(PKMs). This requires that analytical description, formulation and operation for both finite and instantaneous motions are performed by the same mathematical tool. Based upon finite and instantaneous screw theory, a unified and systematic approach for topology design and kinematic analysis of PKMs is proposed in this paper. Using the derivative mapping between finite and instantaneous screws built in the authors’ previous work, the finite and instantaneous motions of PKMs are analytically described by the simple and non?redundant screws in quasi?vector and vector forms. And topological and parametric models of PKMs are algebraically formulated and related. These related topological and parametric models are ready to do type synthesis and kinematic analysis of PKMs under the unified framework of screw theory. In order to show the validity of the proposed approach, a kind of two?translational and three?rotational(2T3R)5?axis PKMs is taken as example. Numerous new structures of the 2T3R PKMs are synthe?sized as the results of topology design, and their Jacobian matrix is obtained easily for parameter optimization and performance evaluation. Some of the synthesized PKMs have outstanding capabilities in terms of large workspaces and flexible orientations, and have great potential for industrial applications of machining and manufacture. Among them, METROM PKM is a typical example which has attracted a lot of attention from global companies and already been developed as commercial products. The approach is a general and unified approach that can be used in the innovative design of different kinds of PKMs.

Kinematic Solution of Spherical Stephenson-Ⅲ Six-bar Mechanism

A closed-form solution can be obtained for kinematic analysis of spatial mechanisms by using analytical method.However,extra solutions would occur when solving the constraint equations of mechanism kinematics unless the constraint equations are established with a proper method and the solving approach is appropriate.In order to obtain a kinematic solution of the spherical Stephenson-III six-bar mechanism,spherical analytical theory is employed to construct the constraint equations.Firstly,the mechanism is divided into a four-bar loop and a two-bar unit.On the basis of the decomposition,vectors of the mechanism nodes are derived according to spherical analytical theory and the principle of coordinate transformation.Secondly,the structural constraint equations are constructed by applying cosine formula of spherical triangles to the top platform of the mechanism.Thirdly,the constraint equations are solved by using Bezout’ s elimination method for forward analysis and Sylvester’ s resultant elimination method for inverse kinematics respectively.By the aid of computer symbolic systems,Mathematica and Maple,symbolic closed-form solution of forward and inverse displacement analysis of spherical Stephenson-III six-bar mechanism are obtained.Finally,numerical examples of forward and inverse analysis are presented to illustrate the proposed approach.The results indicate that the constraint equations established with the proposed method are much simpler than those reported by previous literature,and can be readily eliminated and solved.

Kinematic and Dynamic Characteristics Analysis of Bennett's Linkage

Bennett's linkage is a spatial fourlink linkage,and has an extensive application prospect in the deployable linkages.Its kinematic and dynamic characteristics analysis has a great significance in its synthesis and application. According to the geometrical conditions of Bennett 's linkage,the motion equations are established,and the expressions of angular displacement,angular velocity and angular acceleration of the followers and the displacement,velocity and acceleration of mass center of link are shown. Based on Lagrange's equation,the multi-rigid-body dynamic model of Bennett's linkage is established. In order to solve the reaction forces and moments of joint,screw theory and reciprocal screw method are combined to establish the computing method.The number of equations and unknown reaction forces and moments of joint are equal through adding link deformation equations. The influence of the included angle of adjacent axes on Bennett 's linkage 's kinematic characteristics,the dynamic characteristics and the reaction forces and moments of joint are analyzed.Results show that the included angle of adjacent axes has a great effect on velocity,acceleration,the reaction forces and moments of Bennett's linkage. The change of reaction forces and moments of joint are apparent near the singularity configuration.

Kinematic Analysis and Trajectory Planning of J-Groove Welding Robot

This paper introduces the complexity and particularity of tube-sphere intersection weld(J-groove weld) and establishes the mathematical model of tube-sphere intersection trajectory.Based on the characteristics of J-groove welds,the computational process of welding gun orientation is first simplified.Then the kinematic algorithm of a welding robot is obtained according to screw theory and exponential product formula.Finally,Solidworks and SimMechanics are employed to simulate the kinematics of the welding robot,which proves the feasibility of the kinematic algorithm.