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.     

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.     

Simulation of transient flow in pipeline systems due to load rejection and load acceptance by hydroelectric power plants

In this paper, the implicit method of characteristics (IMOC) is extended to simulate the transient flow caused by load variation of hydroelectric power plants. The IMOC was proposed recently by the authors to remove the limitations of the commonly used conventional method of characteristics (MOC) and subsequently used for the simulation of transient flow caused by valve closure and pump failure. In the IMOC, an element-wise definition is used for all devices that may be used in a pipeline system and the corresponding equations are then assembled to form a system of equations that is solved for the unknown nodal heads and flows at each time step. In this paper, the IMOC is extended for the simulation of transient flow in the pipeline system upstream of a hydroelectric power plant. For this, an element-wise definition of the hydroelectric power plants and simple surge tank is proposed and their governing equations are derived in a matrix form required by the IMOC. Then these equations are assembled along with those of other devices of the pipeline system to construct a system of equations that is solved for the unknown head and flow. To further improve the convergence characteristics of the method, an improved formulation of the turbine is also proposed. In the proposed formulation, one of the turbine characteristic parameters, namely turbine net head, is used in addition to turbine head and discharges, to formulate the behavior of the turbine element. The method is applied to a problem of transient flow caused by load rejection and acceptance and the results are presented and compared with those of the explicit MOC. The results show the ability of the proposed method to accurately predict the variations of head and flow in the cases considered. It is also shown that the improved formulation of the turbine is computationally more efficient than the original formulation while producing the same results.     

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.     

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.     

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.     

Improved dynamic cutting force model in ball-end milling. Part I: theoretical modelling and experimental calibration

An accurate cutting force model of ball-end milling is essential for precision prediction and compensation of tool deflection that dominantly determines the dimensional accuracy of the machined surface. This paper presents an improved theoretical dynamic cutting force model for ball-end milling. The three-dimensional instantaneous cutting forces acting on a single flute of a helical ball-end mill are integrated from the differential cutting force components on sliced elements of the flute along the cutter-axis direction. The size effect of undeformed chip thickness and the influence of the effective rake angle are considered in the formulation of the differential cutting forces based on the theory of oblique cutting. A set of half immersion slot milling tests is performed with a one-tooth solid carbide helical ball-end mill for the calibration of the cutting force coefficients. The recorded dynamic cutting forces are averaged to fit the theoretical model and yield the cutting force coefficients. The measured and simulated dynamic cutting forces are compared using the experimental calibrated cutting force coefficients, and there is a reasonable agreement. A further experimental verification of the dynamic cutting force model will be presented in a follow-up paper.     

Free vibration analysis of sandwich beam carrying sprung masses

In this paper, free vibration of three-layered symmetric sandwich beam carrying sprung masses is investigated using the dynamic stiffness method and the finite element formulation. First the governing partial differential equations of motion for one element are derived using Hamilton’s principle. Closed form analytical solution of these equations is determined. Applying the effect of sprung masses by replacing each sprung mass with an effective spring on the boundary condition of the element, the element dynamic stiffness matrix is developed. These matrices are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by the use of numerical techniques and the well known Wittrick–Williams algorithm. Free vibration analysis using the finite element method is carried out by increasing one degree of freedom for each sprung mass. Finally, some numerical examples are discussed using the dynamic stiffness method and the finite element formulation. After verification of the present model, the effect of various parameters such as mass and stiffness of the sprung mass is studied on the natural frequencies.     

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     

Dynamic analysis of a reciprocating compression mechanism considering hydrodynamic forces

In this paper, a dynamic analysis of the reciprocating compression mechanism of a small refrigeration compressor is performed. In the problem formulation of the mechanism dynamics, the viscous frictional force between the piston and the cylinder wall is considered in order to determine the coupled dynamic behaviors of the piston and the crankshaft. Simultaneous solutions are obtained for the equations of motion of the reciprocating mechanism and the time-dependent Reynolds equations for the lubricating film between the piston and the cylinder wall and for the oil films on the journal bearings. The hydrodynamic forces of the journal bearings are calculated by using a finite bearing model along with the Gümbel boundary condition. A Newton-Raphson procedure is employed in solving the nonlinear equations for the piston and crankshaft. The developed computer program can be used to calculate the complete trajectories of the piston and the crankshaft as functions of the crank angle under compressor-running conditions. The results explored the effects of the radial clearance of the piston, oil viscosity, and mass and mass moment of inertia of the piston and connecting rod on the stability of the compression mechanism.     

Measuring organisational agility before and after implementation of TADS

The ever increasing competition compels the modern organisations to react quickly in accordance with this kind of dynamic demands of the customers, which is referred to as agility, and currently researchers are addressing these capabilities under the field agile manufacturing. The success of achieving agility lies in designing agile-friendly products. In this direction, very little researches have been pursued. In order to fill this gap, a model called total agile design system (TADS) is proposed. The implementation study conducted to examine this model in a traditional manufacturing company is briefly appraised. A scoring model has been used for measuring agility before and after implementation of TADS. The implementation study revealed the improvement of agility by 10%. This improvement is appreciable in traditional manufacturing organisation where only the mass production-based practices are only currently practiced.     

Pull-in analysis of an electrostatically actuated nano-cantilever beam with nonlinearity in curvature and inertia

In this paper, the nonlinear behavior of electrostatically actuated carbon nanotubes (CNTs) is investigated based on a comprehensive model with nonlinearity in curvature, inertia and electrostatic force. The aim of this study is to show when the nonlinear formulation needs to be taken into account and when the linear formulation can simulate the system behavior accurately. The model comprises a cantilevered CNT suspended over a fixed electrode plate from which a DC potential difference is imposed. A relatively large gap between the CNT and the ground plate is considered. The versatile Galerkin method is employed to reduce the nonlinear integro-differential equations of motion to a set of nonlinear ordinary differential equations in time, and then, the reduced equations are solved by direct numerical integration. Dynamic response of the system before and beyond the pull-in voltages and effect of gap to length ratio of the CNT are studied. It is shown that in a large gap to length ratio, when the applied voltage is close to the corresponding pull-in voltage the nonlinear terms have a profound role in the dynamic behavior of the system. Eventually, the contribution of nonlinear terms are examined and it is found that the nonlinear inertia and curvature terms have softening and hardening effects, respectively, whereas the hardening effect of the nonlinear curvature has a major contribution.     

Counterweight optimization for reducing dynamic effects of clearance at a revolute joint

An optimal design formulation is developed to reduce undesirable dynamic effects due to clearance at a joint. The objective function to be minimized is the maximum ratio of the rate of change of the joint force direction ( ) to the magnitude of the joint force (R), i.e. max ( ) as calculated from the nominal mechanism without clearances. Design variables are the magnitude and the location of an added mass attached to each link.

Numerical examples for an offset slider crank mechanisms are considered. To check suitability of the objective function, the initial and optimized systems are simulated dynamically by integrating the system model equations and the phenomena of contact loss compared. It is found that although max ( ) is not a function of the magnitude of clearances and not of dimensionless form, it is a reasonable indicator of the contact loss phenomena for the single clearance system considered. The input torques have also been obtained and compared, shown to be more uniform in the optimized system.     

INVERSE DYNAMIC FORMULATION OF A NOVEL HYBRID MACHINE TOOL

In recent years, hybrid devices have increasingly received more research. However, few of researchers studied the dynamic analysis. The inverse dynamic analysis of a novel hybrid machine tool designed in Tsinghua University is presented. The hybrid machine tool under consideration consists of parallel and serial structures, which is based on a new 2-DOF parallel platform and serial orientations. The kinematics and the dynamic equations are studied first for the parallel structure through Newton-Euler approach. And then, the dynamic analysis for serial structures is conducted. Finally, a closed-form inverse dynamic formulation is derived by using some elimination techniques. Some simulation results are also given.     

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.     

A multi-field approach to modeling the dynamic response of cellular materials

A multi-field approach is developed for simulating the continuum-scale mechanical response of cellular materials. This approach departs from traditional methods used to model cellular materials, which focus almost exclusively on the mechanical response of the cellular solid, while essentially ignoring the fluids permeating these material systems. In the present work, conservation equations are derived in multi-field form, producing a coupled set of governing equations with source terms depending on gradients in the cellular solid stress, but also on gradients in the permeating fluid pressure and momentum exchange resulting from relative motion between the cellular solid and permeating fluid fields. The multi-field equations of motion are implemented in a standard finite-volume computational test bed and used to study the dynamic response of cellular material systems. The influence of various permeating fluids, along with the effects of aperture size, loading rate, and boundary conditions, also are examined. By incorporating an advanced constitutive model for cellular solids into a multi-field response formulation, a promising new approach for simulating the finite-strain dynamic response of cellular materials is offered. Results demonstrate that the permeating fluid can play a major role in the general response of cellular material systems, contributing to the overall load-carrying capacity of the materials and affecting rate dependence and signal propagation speeds. Furthermore, the results point to the usefulness of the multi-field formulation and provide evidence to suggest that any modeling approach developed for cellular materials gives a proper accounting of the pressure evolution and flow behavior of the fluids present in these material systems.     

考虑变形耦合的受冲击柔性机械臂动力学建模

将刚-柔耦合体动力学的新建模理论应用于受冲击柔性机械臂的研究。将柔性机械臂简化为弹性梁,在梁的纵向变形中考虑了变形耦合量,计及了这种耦合对大范围运动的影响。利用Lagrange方程建立了机械臂的动力学方程。将受碰撞冲击后柔性机械臂的瞬态响应,作为求解动力学方程组的初始条件。针对刚体模型和柔性耦合模型进行了数值仿真计算,表明柔性耦合模型更加符合实际情况,为柔性机械臂的动力学分析与控制提供了依据。     

柔索驱动并联机器人动力学建模与数值仿真

柔索驱动并联机器人采用柔索代替连杆作为驱动元件,并结合了并联机构和柔索驱动的优点。500 m口径大射电望远镜(Five-hundred meter aperture spherical radio telescope, FAST)粗调系统通过6根索长的协调变化使馈源舱作跟踪射电源的6自由度运动,其工作特点与并联机器人类似,因此可被看作柔索驱动并联机器人。基于此,根据FAST 5 m缩比试验模型,首先应用悬链线解析表达式推导出柔索两端固定时索端拉力与索长之间的关系,用于求解特定长度的驱动柔索对处于某一位姿的馈源舱的作用力。其次,对该舱索系统进行逆运动学分析,采用拉格朗日方程建立柔索驱动并联机器人的逆动力学模型。最后,针对FAST 5 m缩比模型的设计方案进行动力学仿真,数值结果表明该动力学建模是合理的。     

6-PSS并联机器人动力学模型的牛顿-欧拉方法

给出了动平台的速度、加速度与各个构件的速度、加速度之间的关系,以牛顿-欧拉法为基础,对6-PSS并联机构各个杆件和动平台分别建立牛顿-欧拉方程,并消去其内力,以动平台为研究对象导出6-PSS并联机构动力学方程的开式和闭式形式,最后进行了数值仿真,给出了动平台做一定曲线运动时各驱动力的变化曲线.为基于动力学的控制奠定了基础.