Analytical method for constrained mechanical and structural systems

The objective of this study is to present an accurate and simple method to describe the motion of constrained mechanical or structural systems. The proposed method is an elimination method to require less effort in computing Moore-Penrose inverse matrix than the generalized inverse method provided by Udwadia and Kalaba. Considering that the results by numerical integration of the derived second-order differential equation to describe constrained motion veer away the constrained trajectories, this study presents a numerical integration scheme to obtain more accurate results. Applications of holonomically or nonholonomically constrained systems illustrate the validity and effectiveness of the proposed method.     

Explicit motion of dynamic systems with position constraints

Although many methodologies exist for determining the constrained equations of motion, most of these methods depend on numerical approaches such as the Lagrange multiplier’s method expressed in differential/algebraic systems. In 1992, Udwadia and Kalaba proposed explicit equations of motion for constrained systems based on Gauss’s principle and elementary linear algebra without any multipliers or complicated intermediate processes. The generalized inverse method was the first work to present explicit equations of motion for constrained systems. However, numerical integration results of the equation of motion gradually veer away from the constraint equations with time. Thus, an objective of this study is to provide a numerical integration scheme, which modifies the generalized inverse method to reduce the errors. The modified equations of motion for constrained systems include the position constraints of index 3 systems and their first derivatives with respect to time in addition to their second derivatives with respect to time. The effectiveness of the proposed method is illustrated by numerical examples.     

Numerical integration scheme to reduce the errors in the satisfaction of constrained dynamic equation

Assuming that the existence of the constraints yields the change of the inertia force, this study derives the time-varying mass matrix for describing the constrained dynamic equation. It is displayed that the results corresponds with the ones by Udwadia and Kalaba. The numerical results obtained by integrating the constrained dynamic equation of second-order differential equations yield the errors in the satisfaction of the constraints. Modifying the derived dynamic equation this study presents a numerical algorithm to reduce the errors and to compute more precise motion. It is illustrated that the proposed method can be more precisely utilized in constrained mechanical systems through two applications of constrained nonlinear robotic systems.     

Structural and mechanical systems subjected to constraints

The characteristics of dynamic systems subjected to multiple linear constraints are determined by considering the constrained effects. Although there have been many researches to investigate the dynamic characteristics of constrained systems, most of them depend on numerical analysis like Lagrange multipliers method. In 1992, Udwadia and Kalaba presented an explicit form to describe the motion for constrained discrete systems. Starting from the method, this study determines the dynamic characteristics of the systems to have positive semidefinite mass matrix and the continuous systems. And this study presents a closed form to calculate frequency response matrix for constrained systems subjected to harmonic forces. The proposed methods that do not depend on any numerical schemes take more generalized forms than other research results.     

Computational method for dynamic analysis of constrained mechanical systems using partial velocity matrix transformation

A computational method for the dynamic analysis of a constrained mechanical system is presented in this paper. The partial velocity matrix, which is the null space of the Jacobian of the constraint equations, is used as the key ingredient for the derivation of reduced equations of motion. The acceleration constraint equations are solved simultaneously with the equations of motion. Thus, the total number of equations to be integrated is equivalent to that of the pseudo generalized coordinates, which denote all the variables employed to describe the configuration of the system of concern. Two well-known conventional methods are briefly introduced and compared with the present method. Three numerical examples are solved to demonstrate the solution accuracy, the computational efficiency, and the numerical stability of the present method.     

基于加权广义逆求解被动过约束刚-柔耦合并联机构的受力问题*

提出一种直接应用力映射矩阵的加权广义逆求解一般被动过约束并联机构(指分支提供给动平台的约束力/力偶数目大于等于1的被动过约束并联机构)的超静定受力问题的方法。基于小变形叠加原理和螺旋理论得到了分支约束力螺旋系刚度矩阵的一般表达式。推导一般被动过约束并联机构的各分支约束力螺旋系幅值与六维外力之间的映射关系式,进而证明了该关系式恰好是力映射矩阵的加权广义逆,其中加权矩阵为各分支约束力螺旋系刚度矩阵组成的分块对角阵的逆。以3-RRC空间三自由度被动过约束并联机构和一个并联振动平台为例,直接应用加权广义逆求解了机构各分支约束力螺旋系的幅值,并进行仿真验证。基于加权广义逆求解被动过约束并联机构的超静定受力问题大大简化了求解过程且解的形式统一、简单。     

Calculating Jacobian coefficients of primitive constraints with respect to Euler parameters

It is a fundamental problem to calculate Jacobian coefficients of constraint equations in assembly constraint solving because most approaches to solving an assembly constraint system will finally resort to a numerical iterative method that requires the first-order derivatives of the constraint equations. The most-used method of deriving the Jacobian coefficients is to use virtual rotation which is originally presented to derive the equations of motion of constrained mechanical systems. However, when Euler parameters are adopted as the state variables to represent the transformation matrix, using the virtual rotation will yield erroneous formulae of Jacobian coefficients. The reason is that Euler parameters are incompatible with virtual rotation. In this paper, correct formulae of Jacobian coefficients of geometric constraints with respect to Euler parameters are presented in both Cartesian coordinates and relative generalized coordinates. Experimental results show that our proposed formulae make Newton–Raphson iterative method converge faster and more stable.     

Steady-state equilibrium analysis of a multibody system driven by constant generalized speeds

A formulation which seeks steady-state equilibrium positions of constrained multibody systems driven by constant generalized speeds is presented in this paper. Since the relative coordinates are employed, constraint equations at cut joints are incorporated into the formulation. To obtain the steady-state equilibrium position of a multibody system, nonlinear equations are derived and solved iteratively. The nonlinear equations consist of the force equilibrium equations and the kinematic constraint equations. To verify the effectiveness of the proposed formulation, two numerical examples are solved and the results are compared with those of a commercial program.     

一种并联机构天线座的运动学研究

以3UPS-U型并联机构天线座为研究对象,计算此机构的自由度,建立并联天线座静平台和动平台的坐标系,确定位姿变换矩阵,进而推导出位姿逆解。针对位姿正解的难题,利用方向余弦矩阵约束条件和推杆长度约束方程构造出方程组,求解出姿态角的封闭解。根据此并联天线座的几何特殊性,推导出雅可比矩阵和速度逆解、正解表达式。以理论建模为基础,结合具体参数,进行了数值仿真,求解出位姿逆解和速度逆解。该研究工作为更深一步的理论研究打下基础,为进一步的工程应用提供了理论依据。     

多指抓取的实时力优化算法

根据非线性摩擦约束与特定结构矩阵正定性之间的等价性,将多指抓取力规划问题描述为线性约束正定矩阵对应的平滑流形最优化问题,并且采用线性约束梯度流方法计算得到最优的抓取力。当手指数目较多时,高维描述矩阵限制了传统线性约束梯度流表达式的计算速度,为了解决该问题,基于力优化过程中描述矩阵的仿射约束特性,提出一种适合实时应用的基于抓取力矢量的线性约束梯度流算法。该算法取代描述矩阵,采用抓取力矢量表示线性约束梯度流,大大降低了线性约束梯度流表达式的维数和计算量。以摩擦点接触情况下的四指灵巧手为对象, 采用该算法进行抓取力计算,分析权重因子和步距因子对于计算结果和收敛速度的影响,证明该算法的正确性和实时性。     

Dynamic modeling and simulation of multi-body systems using the Udwadia-Kalaba theory

Laboratory experiments were conducted for falling U-chain,but explicit analytic form of the general equations of motion was not presented.Several modeling methods were developed for fish robots,however they just focused on the whole fish’s locomotion which does little favor to understand the detailed swimming behavior of fish.Udwadia-Kalaba theory is used to model these two multi-body systems and obtain explicit analytic equations of motion.For falling U-chain,the mass matrix is non-singular.Second-order constraints are used to get the constraint force and equations of motion and the numerical simulation is conducted.Simulation results show that the chain tip falls faster than the freely falling body.For fish robot,two-joint Carangiform fish robot is focused on.Quasi-steady wing theory is used to approximately calculate fluid lift force acting on the caudal fin.Based on the obtained explicit analytic equations of motion(the mass matrix is singular),propulsive characteristics of each part of the fish robot are obtained.Through these two cases of U chain and fish robot,how to use Udwadia-Kalaba equation to obtain the dynamical model is shown and the modeling methodology for multi-body systems is presented.It is also shown that Udwadia-Kalaba theory is applicable to systems whether or not their mass matrices are singular.In the whole process of applying Udwadia-Kalaba equation,Lagrangian multipliers and quasi-coordinates are not used.Udwadia-Kalaba theory is creatively applied to dynamical modeling of falling U-chain and fish robot problems and explicit analytic equations of motion are obtained.     

光滑平面约束下的活动线缆物性建模与运动仿真技术

针对活动线缆在安装和服役过程中受到单面约束时的运动过程仿真问题,提出一种基于弹性细杆静力学理论的光滑平面约束下的活动线缆物性建模与仿真方法。该方法通过分析活动线缆在光滑平面约束下的受力情况,将活动线缆所受到的约束力用拉格朗日乘子表达,将节点与平面的相对空间位置关系作为限制活动线缆运动的约束方程,得到分布力作用下的活动线缆Kirchhoff平衡方程,并建立活动线缆的物理模型。进而采用非线性约束优化方法求解模型。最后根据不同假设条件进行算例测试,通过比较活动线缆的仿真形态和实际形态,验证模型和数值解法的正确性。结果表明,该模型和解法能充分真实地表达活动线缆的基本物理特性和活动线缆受到光滑平面约束时的形变特点。     

环状可展机构运动学分析方法及应用研究

将Moore_Penrose广义逆理论与空间连杆机构的运动传递矩阵相结合,提出了一种多杆可展机构的运动学分析方法,利用Gan和Pellegrino方法得到了机构的运动约束方程以及约束方程的雅可比矩阵,依据雅可比矩阵的奇异性和欠秩特性,获得了机构运动参数的最小二乘解。以四杆可展机构为算例,揭示了构件的截面夹角对机构收拢角和展开角的影响规律,并获得了展开运动过程中构件间夹角的变化曲线,为环状可展机构的运动控制奠定了基础。     

金属橡胶单自由度干摩擦隔振系统的理论分析

提出利用静态实验数据求解金属橡胶隔振系统的运动微分方程，并根据方程的数值解来分析该隔振系统的动态特性的新方法，在不同外力激励的条件下，应用该方法对一种金属橡胶隔振器振系统进行动态性的理论分析。     

新型基于6—PSS三维平台机构的并联微动机器人

提出一种新颖的基于6－PSS并联三维平台机的微动机器人并介绍其结构布局特点,应用Jacobian矩阵和力Jacobian矩阵的子阵,分别对其线速度与角速度和力与力矩的各项同性进行了分析计算,并建立显式的微位移正解和反解方程,为其设计和使用提供理论依据。分析计算结果表明,该微动机器人算法与控制简单,微位移解耦和速度与力向向同性,具有最佳的运动和力传递性能。     

Dimensional Synthesis of a 3-DOF Parallel Manipulator with Full Circle Rotation

Parallel robots are widely used in the academic and industrial fields. In spite of the numerous achievements in the design and dimensional synthesis of the low-mobility parallel robots, few research ef...     

六杆机构的运动仿真及方法比较

简要介绍了Simulink与Simmechanics在六杆机构运动分析中的应用。以矢量法建立运动学矩阵方程,分别采用Simulink、Simmechanics进行运动仿真。这两种方法求解效率高,在机构性能分析中具有一定的应用价值。     

基于速度变换的Stewart平台机械手动力学分析

提出了一种新的Ｓｔｅｗａｒｔ平台机械手动力学分析方法─—基于速度变换的坐标缩并方法,该方法根据Ｓｔｅｗａｒｔ平台的道运动学特点,直接导出独立广义速度与非独立广义速度之间的解析变换关系,无需构造约束方程的Ｊａｃｏｂｉａｎ矩阵,也不必进行各种复杂的矩阵分解运算,大大减少了计算量,同时也避免了由矩阵分解引入的数值稳定问题。通过分析证明该方法具有很高的计算效率,而且非常适合并行计算。该方法对其他形式的并联机器人机构同样适用。     

6-PPPS正交六自由度并联机构的位置正反解

针对一种新型6-PPPS并联机构,提出方向余弦阵法进行位置正反解的解析求解。利用机构运动副的运动参数构建并联动平台的方向余弦阵,通过添加约束条件,得到唯一的反解,同样方法可得出所有从动移动副的运动参数。根据方向余弦阵的性质构建约束方程,利用Maple软件辅助求得8组解析正解。根据球铰的约束,发现该机构实际正解唯一,给出了唯一正解的求取方法。     

Modal analysis of constrained multibody systems undergoing constant accelerated motions

The modal characteristics of constrained multibody systems undergoing constant accelerated motions are investigated in this paper. Relative coordinates are employed to derive the equations of motion, which are generally nonlinear in terms of the coordinates. The dynamic equilibrium position of a constrained multibody system needs to be obtained from the nonlinear equations of motion, which are then linearized at the dynamic equilibrium position. The mass and the stiffness matrices for the modal analysis can be obtained from the linearized equations of motion. To verify the effectiveness and the accuracy of the proposed method, two numerical examples are solved and the results obtained by using the proposed method are compared with those obtained by analytical and other numerical methods. The proposed method is found to be accurate as well as effective in predicting the modal characteristics of constrained multibody systems undergoing constant accelerated motions.