Coupling characteristics of rigid body motion and elastic deformation of a 3-PRR parallel manipulator with flexible links

Modeling of multibody dynamics with flexible links is a challenging task, which not only involves the effect of rigid body motion on elastic deformations, but also includes the influence of elastic deformations on rigid body motion. This paper presents coupling characteristics of rigid body motions and elastic motions of a 3-PRR parallel manipulator with three flexible intermediate links. The intermediate links are modeled as Euler–Bernoulli beams with pinned-pinned boundary conditions based on the assumed mode method (AMM). Using Lagrange multipliers, the fully coupled equations of motions of the flexible parallel manipulator are developed by incorporating the rigid body motions with elastic motions. The mutual dependence of elastic deformations and rigid body motions are investigated from the analysis of the derived equations of motion. Open-loop simulation without joint motion controls and closed-loop simulation with joint motion controls are performed to illustrate the effect of elastic motion on rigid body motions and the coupling effect amongst flexible links. These analyses and results provide valuable insight to the design and control of the parallel manipulator with flexible intermediate links.     

Dynamic Simulation of Planar Flexible Link Manipulators using Adaptive Modal Integration

In this paper a modal integration method is proposed for analyzingthe dynamic behavior of multi-link planar flexible manipulators. Anon-linear finite element method is employed to derive theequations of motion in terms of a mixed set of generalizedcoordinates of the manipulator with rigid links and deformationparameters that characterize flexible deformations of the links.Using a perturbation method the vibrational motion of themanipulator is modeled as a first-order perturbation of thenon-linear nominal rigid link motion. For that purpose theflexible dynamic manipulator model is split into two parts. Arigidified model describes the nominal rigid link motion. Alinear system linearized about the nominal trajectory describesthe vibrational motion. In order to reduce the dimension of thelinearized system, a modal reduction technique is proposed. Thenmodal integration can be applied using only a small number of lowfrequency modes. The mode-acceleration concept is used to accountfor the pseudo static contribution of the high frequency modes.Applied to the motion of a manipulator mechanism the method isreferred to as adaptive modal integration since thetime-varying nature of the mode shape functions is taken intoaccount.A flexible two-link manipulator is analyzed to illustrate theperformance of the solution method. Comparisons between solutionsutilizing non-linear and perturbation analyzes with and withoutmodal integration show a good agreement. In a simulation with onlya few modes the accuracy is kept, whereas the computation time isdrastically reduced.     

柔性臂漂浮基空间机器人建模与轨迹跟踪控制

利用拉格朗日法和假设模态方法建立了末端柔性的两臂漂浮基空间机器人的非线性动力学方程．通过坐标变换,推导出一种新的以可测关节角为变量的全局动态模型，并在此基础上运用基于模型的非线性解耦反馈控制方法得到关节相对转角与柔性臂的弹性变形部分解耦形式控制方程．最后，讨论了柔性臂漂浮基空间机器人的轨迹跟踪问题，并通过仿真实例计算,表明该模型转换及控制方法对于柔性臂漂浮基空间机器人末端轨迹跟踪控制的有效性．     

A systematic formulation of dynamic equations for robot manipulators with elastic links

Although a variety of formulation schemes for the dynamic equations of robot manipulators with rigid links can be found in the literature, an efficient method of the formulation for robot manipulators with elastic links is not well known. Accordingly, this work presents the derivation of the equations of motion for application to mechanical manipulators with elastic links. The formulation is conducted analytically using Hamilton's principle. The resultant equations consist of the terms of inertial, Coriolis, centrifugal, gravitational, and exerted forces. They are expressed in terms of a set of independent generalized coordinates. In contrast to conventional variational approaches, the present method provides an efficient and systematic way for obtaining the compact symbolic equations of flexible manipulator systems. Two examples are presented to illustrate the proposed methodology. Firstly, a three-link flexible manipulator with three revolute joints is studied. A flexible manipulator consisting of a prismatic joint and a discrete mass is the second model.     

Complement of Source Equation of Elastic Line

A new notion of joint, defined in terms of the state of motor (active or locked) and type of the elastic or rigid element, gear and/or link that follows after the motor, is introduced. Special attention is paid to the motion of the flexible links in the robotic configuration. The paper deals with the relationship between the equation of elastic line equilibrium, the “Euler–Bernoulli approach” (EBA), and equation of motion at the point of elastic line tip, the “Lumped-mass approach” (LMA). The Euler–Bernoulli equations (which have for a long time been used in the literature) should be expanded according to the requirements of the motion complexity of elastic robotic systems. The Euler–Bernoulli equation (based on the known laws of dynamics) should be supplemented with all the forces that are participating in the formation of the elasticity moment of the considered mode. This yields the difference in the structure of Euler–Bernoulli equations for each mode. The stiffness matrix is a full matrix. Mathematical model of the actuators also comprises coupling between elasticity forces. Particular integral of Daniel Bernoulli should be supplemented with the stationary character of elastic deformation of any point of the considered mode, caused by the present forces. General form of the elastic line is a direct outcome of the system motion dynamics, and cannot be described by one scalar equation but by three equations for position and three equations for orientation of every point on that elastic line. The choice of reference trajectory is analyzed. Simulation results are shown for a selected robotic example involving the simultaneous presence of elasticity of the gear and of the link (two modes), as well as the environment force dynamics.     

A systematic method of dynamics for flexible robot manipulators

In this article, a systematic method to derive dynamic equations of motion for flexible robot manipulators is developed by using the Lagrangian assumed modes method. The proposed method can be applied to dynamic simulation and control system design for flexible robot manipulators. In the proposed method, the link deflection is described by a truncated modal expansion. The operations of only 3x3 matrices and/or 3 × 1 vectors exist in the method. All the dynamics computations are performed in the link coordinate systems, where the kinematics informations are computed with the forward recursion from the base to the hand tip and the dynamics informations are computed with the return recursion. As generally compared with other existing methods, the method proposed in this article is, computationally, more simple, systematic, and efficient. A computational simulation for a single-link flexible robot manipulator is presented to verify the proposed method. © 1992 John Wiley & Sons, Inc.     

A constrained assumed modes method for dynamics of a flexible planar serial robot with prismatic joints

In the present study, for the first time, flexible multibody dynamics for a three-link serial robot with two flexible links having active prismatic joints is presented using an approximate analytical method. Transverse vibrations of flexible links/beams with prismatic joints have complicated differential equations. This complexity is mostly due to axial motion of the links. In this study, first, vibration analysis of a flexible link sliding through an active prismatic joint having translational motion is considered. A rigid-body coordinate system is used, which aids in obtaining a new and rather simple form of the kinematic differential equation without the loss of generality. Next, the analysis is extended to include dynamic forces for a three-link planar serial robot called PPP (Prismatic, Prismatic, Prismatic), in which all joints are prismatic and active. The robot has a rigid first link but flexible second and third links. To model the prismatic joint, time-variant constraints are written, and a motion equation in a form of virtual displacement and virtual work of forces/moments is obtained. Finally, an approximate analytical method called the “constrained assumed modes method” is presented for solving the motion equations. For a numerical case study, approximate analytical results are compared with finite element results, which show that the two solutions closely follow each other.     

Symbolic modeling and dynamic simulation of robotic manipulators with compliant links and joints

The explicit, non-recursive symbolic form of the dynamic model of robotic manipulators with compliant links and joints are developed based on a Lagrangian-assumed mode of formulation. This form of dynamic model is suitable for controller synthesis, as well as accurate simulations of robotic applications. The final form of the equations is organized in a form similar to rigid manipulator equations. This allows one to identify the differences between rigid and flexible manipulator dynamics explixitly. Therefore, current knowledge on control of rigid manipulators is likely to be utilized in a maximum way in developing new control algorithms for flexible manipulators.

Computer automated symbolic expansion of the dynamic model equations for any desired manipulator is accomplished with programs written based on commercial symbolic manipulation programs (SMP, MACSYMA, REDUCE). A two-link manipulator is used as an example. Computational complexity involved in real-time control, using the explicit, non-recursive form of equations, is studied on single CPU and multi-CPU parallel computation processors.     

A recursive, numerically stable, and efficient simulation algorithm for serial robots with flexible links

A methodology for the formulation of dynamic equations of motion of a serial flexible-link manipulator using the decoupled natural orthogonal complement (DeNOC) matrices, introduced elsewhere for rigid bodies, is presented in this paper. First, the Euler Lagrange (EL) equations of motion of the system are written. Then using the equivalence of EL and Newton–Euler (NE) equations, and the DeNOC matrices associated with the velocity constraints of the connecting bodies, the analytical and recursive expressions for the matrices and vectors appearing in the independent dynamic equations of motion are obtained. The analytical expressions allow one to obtain a recursive forward dynamics algorithm not only for rigid body manipulators, as reported earlier, but also for the flexible body manipulators. The proposed simulation algorithm for the flexible link robots is shown to be computationally more efficient and numerically more stable than other algorithms present in the literature. Simulations, using the proposed algorithm, for a two link arm with each link flexible and a Space Shuttle Remote Manipulator System (SSRMS) are presented. Numerical stability aspects of the algorithms are investigated using various criteria, namely, the zero eigenvalue phenomenon, energy drift method, etc. Numerical example of a SSRMS is taken up to show the efficiency and stability of the proposed algorithm. Physical interpretations of many terms associated with dynamic equations of flexible links, namely, the mass matrix of a composite flexible body, inertia wrench of a flexible link, etc. are also presented. The work has been carried out in the Dept. of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India.     

Analytical development of dynamic equations of motion for a three-dimensional flexible link manipulator with revolute and prismatic joints

In this paper, a mathematical model capable of handling a three-dimensional (3D) flexible n-degree of freedom manipulator having both revolute and prismatic joints is considered. This model is used to study the longitudinal, transversal, and torsional vibration characteristics of the robot manipulator and obtain kinematic and dynamic equations of motion. The presence of prismatic joints makes the mathematical derivation complex. In this paper, for the first time, prismatic joints as well as revolute joints have been considered in the structure of a 3D flexible n-degree of freedom manipulator. The kinematic and dynamic equations of motion representing longitudinal, transversal, and torsional vibration characteristics have been solved in parametric form with no discretization. In this investigation, in order to obtain an analytical solution of the vibrational equations, a novel approach is presented using the perturbation method. By solving the equations of motion, it is shown that mode shapes of the link with prismatic joints can be modeled as the equivalent clamped beam at each time instant. As an example, this method is applied to a three degrees of freedom robot with revolute and prismatic joints. The obtained equations are solved using the perturbation method and the results are used to simulate vibrational behavior of the manipulator.     

多移动机械臂系统动力学建模以及稳定性分析

随着社会生产力的发展和发展需求的提高,移动机械臂凭借着自身优势,受到学术界和工业界的广泛关注.但在许多工作场景下,单个移动机械臂有着自由度数以及载荷的限制,无法顺利完成任务.为了更好地满足任务需求,多移动机械臂系统应运而生.在上述工业背景下,本文建立了多移动机械臂系统的动力学模型,并针对该动力学方程进行了稳定性分析.首先通过拉格朗日方程建立单个移动机械臂的动力学方程,将多体动力学软件仿真结果同动力学模型数值计算结果进行对比,验证了模型的正确性.随后联立多个移动机械臂的动力学方程和操作对象的动力学方程,得到封闭形式的多移动机械臂系统的动力学方程.再利用关节位置误差和速度误差设计李雅普诺夫函数,通过反步法获得了关节力矩的控制律.最后在多体动力学软件仿真中,察看轨迹是否能跟踪上期望信号来检验控制律的有效性.     

Dynamics and elasto-dynamics optimization of a 2-DOF planar parallel pick-and-place robot with flexible links

Dynamic modeling and analysis of a 2-DOF translational parallel robot with flexible links for high-speed pick-and-place operation is presented in this paper. Optimization is implemented with the goal to improve the dynamic accuracy of the end-effector at high speed. The governing equations of flexible links within the robot are formulated in the floating reference frame using Euler–Lagrange method, leading to a global FEM model being generated using the KED (Kineto-Elasto-Dynamics) technique. The dynamic characteristics of the robot are then investigated by model analysis. A numerical dynamic index is proposed to identity the range of natural frequency when the robot reaches different configurations. The comparisons are made between the optimized and original designs in terms of dynamic stress and response.     

Force control of a two-linke planar manipulator with one flexible link

Force control of a two-link planar manipulator with one flexible link is considered in this study. The equations of motion are derived using the extended Hamilton's principle with only structural flexibility effects included in the dynamic model. The linear quadratic Gaussian/loop transfer recovery (LQG/LTR) design methodology is exploited to design a robust feedback control system that can handle modelling errors and sensor noise, and operate on Cartesian space trajectory errors. The LQG/LTR compensator together with a feedforward loop is used to simultaneously control the force exerted by the flexible manipulator normal to the environment and the position of the end-point in a direction tangent to the environment. Simulated results are presented for a numerical example.     

大范围运动柔性机械臂斜碰撞动力学分析

对作大范围运动柔性机械臂系统,进行斜碰撞动力学分析.基于柔性多体系统刚柔耦合动力学理论,计入耦合变形项,全面考虑大范围刚体运动与弹性小变形运动的耦合,建立系统连续动力学方程.引入斜碰撞力学模型,将法向和切向碰撞力以广义力的形式加入动力学方程中,对系统进行斜碰撞动力学建模分析.法向碰撞模型选取基于连续接触力法的非线性弹簧阻尼模型,切向碰撞模型选取一种修正Coulomb摩擦模型,对切向摩擦力进行统一描述.给出接触、分离判据,实现不同状态的动力学模型转换与求解.对斜碰撞全局动力学进行了仿真验证,分析了柔性机械臂全局过程的动力学特性变化以及碰撞对大范围运动和小变形运动的作用,并对比了不同碰撞方向对大范围运动、变形、机械能、碰撞力等动力学参数的影响.     

漂浮基柔性空间机械臂姿态与关节协调运动的Terminal滑模控制

讨论了载体位置无控、姿态受控情况下,具有外部扰动的漂浮基柔性空间机械臂载体姿态与各关节协调运动的控制问题．基于假想模态法、系统动量守恒关系及拉格朗日方法,建立了漂浮基柔性空间机械臂系统的动力学方程,并将其转化为系统控制状态方程．以此为基础,根据Terminal滑模控制技术,给出了系统相关Terminal滑模面的数学表达式,在此基础上提出了具有外部扰动情况下漂浮基柔性空间机械臂载体姿态与各关节协调运动的Terminal滑模控制方案．提出的控制方案不但确保了闭环系统滑模阶段的存在性,同时通过Terminal滑模函数的适当选取,还保证了输出误差在有限时间内的收敛性．此外,由于确保了无论何种情况下系统初始状态均在Terminal滑模面上,从而消除了其它滑模控制方法常有的到达阶段,使得闭环系统具有全局鲁棒和稳定性．一个平面两杆漂浮基柔性空间机械臂的系统数值仿真,证实了方法的有效性．     

Dynamic modeling and design optimization of a 3-DOF spherical parallel manipulator

This paper deals with the dynamic modeling and design optimization of a three Degree-of-Freedom spherical parallel manipulator. Using the method of Lagrange multipliers, the equations of motion of the manipulator are derived by considering its motion characteristics, namely, all the components rotating about the center of rotation. Using the derived dynamic model, a multiobjective optimization problem is formulated to optimize the structural and geometric parameters of the spherical parallel manipulator. The proposed approach is illustrated with the design optimization of an unlimited-roll spherical parallel manipulator with a main objective to minimize the mechanism mass in order to enhance both kinematic and dynamic performances.     

Speed control strategy of dual flexible servo system considering time-varying parameters for flexible manipulator with an axially translating arm

Many factors lead to fluctuations of output speed in the motor side during the movement processes in a flexible manipulator with an axially translating arm. These factors include the time-varying characteristics of manipulator parameters, external disturbance, and flexibility in the manipulator. The vibration of flexible manipulators is strengthened by fluctuations of output speed, which seriously affects the motion accuracy. Therefore, a variable parameters proportional–integral (PI) control strategy based on disturbance observer is adopted to reduce fluctuations of output speed in a dual flexible servo system. First, the dynamic model of the dual flexible servo system is established by the assumed mode method. Then, the variable parameters PI control strategy is applied to the servo system, and the controller parameters in different conditions are selected by pole placement strategies. Finally, numerical simulations and control experiments of the servo system are carried out. The results show that the control strategy using the variable parameters and disturbance observer could reduce fluctuations of output speed and improve the motion accuracy of flexible manipulators.     

Chain Vibration and Dynamic Stress in Three-Dimensional Multibody Tracked Vehicles

A three-dimensional computational finite element procedure for the vibration and dynamic stress analysis of the track link chains of off-road vehicles is presented in this paper. The numerical procedure developed in this investigation integrates classical constrained multibody dynamics methods with finite element capabilities. The nonlinear equations of motion of the three-dimensional tracked vehicle model in which the track link s are considered flexible bodies, are obtained using the floating frame of reference formulation. Three-dimensional contact force models are used to describe the interaction of the track chain links with the vehicle components and the ground. The dynamic equations of motion are first presented in terms of a coupled set of reference and elastic coordinates of the track links. Assuming that the structural flexibility of the track links does not have a significant effect on their overall rigid body motion as well as the vehicle dynamics, a partially linearized set of differential equations of motion of the track links is obtained. The equations associated with the rigid body motion are used to predict the generalized contact, inertia, and constraint forces associated with the deformation degrees of freedom of the track links. These forces are introduced to the track link flexibility equations which are used to calculate the deformations of the links resulting from the vehicle motion. A detailed three-dimensional finite element model of the track link is developed and utilized to predict the natural frequencies and mode shapes. The terms that represent the rigid body inertia, centrifugal and Coriolis forces in the equations of motion associated with the elastic coordinates of the track link are described in detail. A computational procedure for determining the generalized constraint forces associated with the elastic coordinates of the deformable chain links is presented. The finite element model is then used to determine the deformations of the track links resulting from the contact, inertia, and constraint forces. The results of the dynamic stress analysis of the track links are presented and the differences between these results and the results obtained by using the static stress analysis are demonstrated.     

Hybrid force and motion control of flexible joint parallel manipulators using inverse dynamics approach

An inverse dynamics control algorithm is developed for hybrid motion and contact force trajectory tracking control of flexible joint parallel manipulators. First, an open-tree structure is considered by the disconnection of adequate number of unactuated joints. The loop closure constraint equations are then included. Elimination of the joint reaction forces and the other intermediate variables yield a fourth-order relation between the actuator torques and the end-effector position and contact force variables, showing that the control torques do not have an instantaneous effect on the end-effector contact forces and accelerations because of the flexibility. The proposed control law provides simultaneous and asymptotically stable control of the end-effector contact forces and the motion along the constraint surfaces by utilizing the feedback of positions and velocities of the actuated joints and rotors. A two degree of freedom planar parallel manipulator is considered as an example to illustrate the effectiveness of the method.     

Design and kinetostatic analysis of a new parallel manipulator

This paper proposes an innovative design for a parallel manipulator that can be applied to a machine tool. The proposed parallel manipulator has three degrees of freedom (DOFs), including the rotations of a moving platform about the x and y axes and a translation of this platform along the z-axis. A passive link is introduced into this new parallel manipulator in order to increase the stiffness of the system and eliminate any unexpected motion. Both direct and inverse kinematic problems are investigated, and a dynamic model using a Newton–Euler approach is implemented. The global system stiffness of the proposed parallel manipulator, which considers the compliance of links and joints, is formulated and the kinetostatic analysis is conducted. Finally, a case study is presented to demonstrate the applications of the kinematic and dynamic models and to verify the concept of the new design.