Recursive Newton–Euler formulation for flexible dynamic manufacturing analysis of open-loop robotic systems

The main objective of this paper is to develop a recursive formulation for the flexible dynamic manufacturing analysis of open-loop robotic systems. The nonlinear generalized Newton–Euler equations are used for flexible bodies that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars, vectors and matrices that depend on the spatial coordinates as well as the assumed displacement fields. These time invariant quantities represent the dynamic manufacturing couplings between the rigid body motion and elastic deformation. This formulation applies recursive procedures with the generalized Newton–Euler equations for flexible bodies to obtain a large, loosely coupled system equation describing motion in flexible manufacturing systems. The techniques used to solve the system equations can be implemented in any computer system. The algorithms presented in this investigation are illustrated using cylindrical joints for open-loop robotic systems, which can be easily extended to revolute, slider and rigid joints. The recursive Newton–Euler formulation developed in this paper is demonstrated with a robotic system using cylindrical mechanical joints.     

A recursive algorithm for generating the equations of motion of spatial mechanical systems with application to the five-point suspension

In this paper, a recursive formulation for generating the equations of motion of spatial mechanical systems is presented. The rigid bodies are replaced by a dynamically equivalent constrained system of particles which avoids introducing any rotational coordinates. For the open-chain system, the equations of motion are generated recursively along the serial chains using the concepts of linear and angular momenta. Closed-chain systems are transformed to open-chain systems by cutting suitable kinematic joints and introducing cut-joint constraints. The formulation is used to carry out the dynamic analysis of multi-link five-point suspension. The results of the simulation demonstrate the generality and simplicity of the proposed dynamic formulation.     

Simulation and experimental methods for media transport system: Part I,Three-dimensional sheet modeling using relative coordinate

This research presents a three-dimensional modeling technique for a flexible sheet A relative cooidinate formulation is used to represent the kinematics of the sheet The three-dimensional flexible sheet is modeled by multi-rigid bodies interconnected by out-of-plane joints and plate force elements A parent node is designated as a master body and is connected to the ground by a floating joint to cover the rigid motion of the flexible sheet in space Since the in-plane deformation of a sheet such as a paper and a film is relatively small, compared to out-of-plane deformation, only the out-of-plane deformation is accounted for in this research The recursive formulation has been adopted to solve the equations of motion efficiently An example is presented to show the validity of the proposed method     

A computer method for the dynamic analysis of a system of rigid bodies in plane motion

This paper presents a computer method for the dynamic analysis of a system of rigid bodies in plane motion. The formulation rests upon the idea of replacing a rigid body by a dynamically equivalent constrained system of particles. Newton’s second law is applied to study the motion of the resulting system of particles without introducing any rotational coordinates. A velocity transformation is used to transform the equations of motion to a reduced set. For an open-chain, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For a closed-chain, suitable joints should be cut and few cut-joints constraint equations should be included. An example of a closed-chain is used to demonstrate the generality and efficiency of the proposed method.     

Dynamics of the macpherson strut motor-vehicle suspension system in point and joint coordinates

In this paper the dynamic analysis of the Macpherson strut motor-vehicle suspension system is presented. The equations of motion are formulated using a two-step transformation. Initially, the equations of motion are derived for a dynamically equivalent constrained system of particles that replaces the rigid bodies by applying Newton’s second law. The equations of motion are then transformed to a reduced set in terms of the relative joint variables. Use of both Cartesian and joint variables produces an efficient set of equations without loss of generality. For open chains, this process automatically eliminates all of the non-working constraint forces and leads to an efficient solution and integration of the equations of motion. For closed loops, suitable joints should be cut and few cut-joints constraint equations should be included for each closed chain. The chosen suspension includes open and closed loops with quarter-car model. The results of the simulation indicate the simplicity and generality of the dynamic formulation.     

三维大变形梁系统的动力学建模与仿真*

对三维大变形柔性梁系统的动力学建模和仿真进行了研究。采用绝对节点坐标法描述柔性体的大变形和大位移运动,并由此建立三维大变形柔性梁系统的动力学模型。在此动力学模型基础上,编制动力学仿真软件,实现了对三维大变形柔性梁系统的动力学仿真。给出了两个动力学仿真算例。第一个对柔性单摆自由下落进行了动力学仿真,并与现有文献结果相比较,验证了模型的正确性。第二个对空间柔性双摆的自由下落过程进行了动力学仿真,并将模型计算的结果与使用ADAMS软件计算的结果进行比较。研究结果表明,ADAMS在计算大变形物体动力学时具有局限性,而所得的模型能够有效地对三维大变形柔性梁系统的动力学进行仿真解决这类动力学问题。     

5R柔性并联机器人系统运动微分方程

为了描述平面5R柔性并联机器人的运动学和动力学特性,需要建立机器人的运动微分方程。针对刚性活动平台和柔性杆件的运动学耦合特点,改进了一套适用于刚体、柔性体耦合的有限元建模方法,推导出单元弹性广义坐标相对于系统弹性广义坐标的转换矩阵,综合考虑了科氏阻尼、离心刚度和几何非线性的影响,利用运动弹性动力学理论,建立了平面5R柔性并联机器人的运动微分方程,避免了采用运动学和动力学约束方程的弊端,提高了建模精度。计算实例表明,该方程反映了机器人的弹性振动特性,杆件的弹性变形对机器人的运动误差具有重要影响。     

卫星太阳阵展开锁紧过程冲击振动

为了研究铰链副对卫星太阳阵展开过程接触碰撞的扰动影响,针对其链式拓扑结构漂浮基无根树特点,描述其运动学关系,并利用Largrange和Newton方法推导太阳阵系统递推动力学模型,克服了非线性多自由度系统获得解析解的困难;进一步通过动量守恒,并采用非线性弹簧阻尼及摩擦的接触碰撞约束力,构建太阳阵展开过程的广义动力学模型。通过太阳阵展开过程的接触碰撞数值仿真,研究铰链副接触碰撞对卫星太阳阵多体系统的影响,结果较好地预测了太阳阵展开历程及卫星姿态的动态行为,结论对卫星太阳阵展开过程分析与控制具有重要的理论价值及工程实际意义。     

基于广义坐标形式牛顿－欧拉方法的空间并联机构动力学正问题分析

基于具有广义坐标形式的牛顿－欧拉方法建立一般空间并联机构动力学正问题建模与仿真的通用方法。在动力学建模时,选取空间并联机构动平台的工作空间变量作为系统广义坐标,从各构件的牛顿－欧拉方程出发,结合运动学逆解和虚功原理,得到与广义坐标相对应的系统动力学方程－运动微分方程和系统理想铰约束力方程。对6-UPS型空间并联机构进行动力学建模,计算外力作用下的动态响应,验证了该方法的有效性。理论分析和仿真案例表明,该方法可以简化并联机构动力学正问题的建模步骤,提高计算效率。     

Kinematic and dynamic modeling of viscoelastic robotic manipulators using Timoshenko beam theory: theory and experiment

This paper presents an investigation into the development of modeling of n-viscoelastic robotic manipulators. The dynamic model of the system is derived using Gibbs-Appell formulation and assumed mode method. When the beam is short in length direction, shear deformation is a factor that may have significant effects on system dynamic. So, in modeling, the assumption of Timoshenko beam theory and associated mode shapes has been considered. Although including the effect of damping in continuous systems makes the formulations more complicated, two important damping mechanisms, namely, Kelvin-Voigt damping as internal damping and the viscous air damping as external damping have been considered. Based on derived formulation, a non-linear recursive algorithm is developed for deriving the inverse dynamic equation of motion, systematically. The performance of the proposed algorithm was assessed in terms of the required mathematical operations for deriving the kinematic and dynamic equations of the mechanical system. Finally, to validate the proposed formulation, a comparative assessment between the results achieved from experiment and simulation is presented in time and frequency domains.     

Comparative study of tracking control for a mobile manipulator: Nonholonomic and dynamic cases

This paper gives an in-depth treatment of the modeling and control of a mobile manipulator which consists of a robotic manipulator mounted upon a mobile robot. By neglecting slip of the platform's tires, nonholonomic constraints are introduced into the equations of motion. By considering wheel slip, the assumption of nonholonomic motion is violated. Nonholonomic and dynamic models of a mobile manipulator are developed and compared using the Lagrange-d'Alembert formulation and the Newton-Euler method, respectively. The dynamic model which considers wheel slip incorporates a nonlinear tire friction model. The tracking problem is investigated by using imput-output linearization for the nonholonomic model. For the dynamic model, a robust control method based on a matching condition is developed to eliminate the harmful effects of wheel slip, which acts as a disturbance to the system. Then, the effect of wheel slip on the tracking of commanded motion is identified via simulation. The effectiveness of the proposed control algorithm is demonstrated through computer simulation.     

大位移、变拓扑空间机构的动力分析

给出了一种大位移、变拓扑空间机构的动力分析方法。由物体相对坐标和有限元节点坐标描述系统的运动与变形，通过运动相容条件确定系统空间构形的连续性和唯一性，建立了变结构、刚柔多体系统动力学方程。并由算例表明了该方法的精度与效率     

Dynamic modelling of planar mechanisms using point coordinates

In the present study, the dynamic modelling of planar mechanisms that consist of a system of rigid bodies is carried out using point coordiantes. The system of rigid bodies is replaced by a dynamically equivalent constrained system of particles. Then for the resulting equivalent system of particles, the concepts of linear and angular momentums are used to generate the equations of motion without either introducing any rotational coordinates or distributing the external forces and force couples over the particles. For the open loop case, the equations of motion are generated recursively along the open chains. For the closed loop case, the system is transformed to open loops by cutting suitable kinematic joints with the addition of cut-joints kinematic constraints. An example of a multi-branch closed-loop system is chosen to demonstrate the generality and simplicity of the proposed method.     

计及动力刚化的柔体动力学

提出两种计及动力刚化影响的动力学建模方法；有限段方法和一致线性化动力学方法。分析了这两种动力不主动力刚化的机理，有限段方法将柔体动力学问题转化为带有柔性的多刚体体系统动力学问题，计及了几何非线性的影响，适合于解决梁式结构的动力学问题；一致线性化动力学方法将变形场描述成为变形广义坐标的非线性形式，在适当的阶段线性化，可得到一致线性化动力学方程，自然计及了动力刚化项，适合于柔体的小变形问题。     

Synthetic analysis of flexible multibody system including a very flexible body

The progress in developing a dynamic analysis solver has different aspects of improvement in the sense of simulating the behavior of the parts. Among them, dynamics in flexible body and large deformable body have been an issue in recent decades. A modal coordinate formulation has been developed and used for analyzing the flexible body dynamics with a commercial dynamic solver, like in ADAMS. Flexible body dynamics using modal coordinates are reliable when the system’s deflection is relatively small, and generally its accuracy depends on how many relevant modes are used for the system. Conversely, to simulate the behavior of the large deflected body, absolute nodal coordinate formulation is derived and developed. The theory presents the mixed equations of motion, which consider both the absolute nodal coordinates and absolute cartesian orientation coordinates to simulate the large deflection. Its reliability is proved by many researches and experimental data. In this study, a dynamic solver which can handle the flexible bodies is developed. Three kinds of bodies, rigid, flexible and large deformable body, can be simulated. Its validity is verified by comparison with a commercial analysis program. For further studies, the constraints and force elements between different coordinates will be developed. Solving efficiency would be another major concern to be improved. This paper was presented at the 4th Asian Conference on Multibody Dynamics(ACMD2008), Jeju, Korea, August 20–23, 2008. Ji Won Yoon received B.S. and M.S. degrees in Mechanical Engineering from Ajou University in 2004 and 2006, respectively. Mr. Yoon is currently a Ph.D student at the School of Mechanical Engineering at Ajou University in Suwon, Korea. He is serving as an instructor for undergraduate students. Mr. Yoon’s research interests are in the area of multibody dynamics, flexible body dynamics, and fatigue analysis.     

COMPUTER AIDED ANALYSIS OF FLEXI－BLE MEMBER IN CONSIDERATION OF THE EFFECTS OF DYNAMIC STI－FFEN－ING

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Complete dynamic modeling and approximate state space equations of the flexible link manipulator

This work treats the problem of dynamic modeling and state space approximation for robotic manipulators with flexibility. Case studies are planar manipulators with a single flexible link together with clamped-free ends and tip mass conditions. In this paper, complete dynamic modeling of the flexible beam without premature linearization in the formulation of the dynamics equations is developed, whereby this model is capable of reproducing nonlinear dynamic effects, such as the beam stiffening due to the centrifugal and the Coriolis forces induced by rotation of the joints, giving it the capability to predict reliable dynamic behavior. On the other hand, in order to show the joint flexibility effects on the model dynamic behaviors, manipulator with structural and joint flexibility is considered. Thus, a reliable model for flexible beam is then presented. The model is founded on two basic assumptions: inextensibility of the neutral fiber and moderate rotations of the cross sections in order to account for the foreshortening of the beam due to bending. To achieve flexible manipulator control, the standard form of state space equations for a flexible manipulator system (flexible link and actuator) is very important. In this study, finite difference method for discretization of the dynamic equations is used and the state space equations of the flexible link with tip mass considering complete dynamic of the system are obtained. Simulation results indicated substantial improvements on dynamic behavior and it is shown that the joint flexibility has a considerable effect on the dynamic behavior of rotating flexible arm that should not be simply neglected. The effects of tip mass is proved to be increasing the elastic deformations?? amplitudes and increasing stability.     

滚动轴承刚柔多体接触动力学分析

以柔性多体动力学理论为基础,提出了以相对坐标描述刚柔多体接触滚动轴承的递推数值分析方法。以球轴承为例,研究了滚动轴承刚柔多体接触输出响应的动态特性。综合考虑滚动体与轴承内外圈滚道的时变接触刚度、柔性接触、摩擦、离心力等非线性因素,分析出转速、摩擦力和预紧力对滚动轴承的时变接触力和时变振动位移的影响。结果表明,以相对坐标描述刚柔多体接触滚动轴承的动力学模型,可以有效地解决滚动体与滚道间的时变接触振动响应和柔性接触等非线性接触动力学问题。     

无根欠驱动冗余机器人动力耦合特性研究

无根多刚体系统和欠驱动冗余机器人系统实质上都属二阶非完整动力系统,其位姿空间约束方程不能满足控制要求,一般基于动力学方程对系统进行控制,即通过关节间动力耦合作用约束被动关节运动,因此此类机器人可控性分析的重点在于系统耦合运动特性研究。基于动力学虚设机构法及非完整系统微分变分原理,建立了无根欠驱动冗余机器人的动力学模型;针对虚设关节、主、被动关节进行动力学模型解耦,推导出了系统的二阶非完整约束方程及被动关节的加速度表达式;在此基础上,通过定义表征被动关节耦合运动的性能指标,针对不同位置主动关节输入参数对被动关节可控性的影响进行了仿真分析,得到了提高无根欠驱动冗余机器人可控性的有益结论,为实际欠驱动冗余机器人输入控制提供了参考。     

Subsystem synthesis methods with independent coordinates for real-time multibody dynamics

For real time dynamic simulation, two different subsystem synthesis methods with independent generalized coordinates have been developed and compared In each formulation, the subsystem equations of motion are generated in terms of independent generalized coordinates The first formulation is based on the relative Cartesian coordinates with respect to moving subsystem base body The second foimulation is based on the relative joint coordinates using recursive formulation Computational efficiency of the formulations has been compared theoretically by the arithmetic operational counts In order to verify real-time capability of the formulations, bump run simulations of a quarter car model with SLA suspension subsystem have been carried out to measure the actual CPU time