Solution schemes for the system equations of flexible robots

In this work, a method for generating the dynamic equations of flexible robots with open-chain linkage mechanisms is developed. A general transformation matrix associated with the elastic deformation is introduced. In determining the elastic response, a method of separation of variables and the Galerkin's approach are suggested for the boundary-value problem with time-dependent boundary conditions. Besides the formulation scheme, the present work also studies the difficulty of dealing with the inverse kinematic problem, in which the unknowns involve the rigid-body displacements and the elastic deflections. Finally, the ideas presented here have been implemented in a computer simulation, and the formulation of the boundary-value problem has been employed to obtain the equations of motion of a flexible robot. Simulation results are presented.     

A distributed parameter model for the dynamics of flexible-link robots

In this paper a model is developed for kinematic and dynamic analysis of flexible robots undergoing general three-dimensional motion. For modeling robotic links, distributed mass and flexibility are considered without discretization. Some modeling issues are discussed, and parameters characterizing the real design of a robot are introduced into the analysis. The concept of a fictitious rigid link is presented to consider the rigid body motion of a link separately, and to account for possibly complex link shapes. Based on Jourdain's principle, an alternative formulation is proposed to derive the dynamic equations of flexible robots. The equations of motion are developed and analyzed in detail. The vibrations of links are described by linear, inhomogeneous partial differential equations, with homogeneous, nonlinear, time-dependent boundary conditions. © 1998 John Wiley & Sons, Inc.     

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.     

Navigation strategies for multiple autonomous mobile robots moving in formation

The problem of deriving navigation strategies for a fleet of autonomous mobile robots moving in formation is considered. Here each robot is represented by a particle with a spherical effective spatial domain and a specified cone of visibility. The global motion of each robot in the world space is described by the equations of motion of the robot's center of mass. First, methods for formation generation are discussed. Then, simple navigation strategies for robots moving in formation are derived. A sufficient condition for the stability of a desired formation pattern for a fleet of robots each equipped with the navigation strategy based on nearest neighbor tracking is developed. The dynamic behavior of robot fleets consisting of three or more robots moving in formation in a plane is studied by means of computer simulation.     

有环境约束时完成一族跟踪任务的力控制

本文为工业机器人提出了一种极点配置控制法，这种控制方法的优点有：一是它的积分作用消除了机器人的微小扰动和稳态误差；二是能任意设置系统的极点，因此能保证闭环系统的稳定性和规定状态变量的暂态响应；三是加入了加速度反馈，抑制了由电枢电感所引起的机械手的振动，最后，给出了ＰＵＭＡ５６２机器人的计算机仿真和实验结果验证了此控制法的有效性。     

Nonlinear generalized equations of motion for multi-link inverted pendulum systems

In this paper, the equations of motion for a general multi-link inverted pendulum system are derived. Assumptions previously employed to simplify such formulation are removed. The pendulum system is more general and includes nonlinear friction terms to suit various engineering applications. The generalized equations are first developed in the absolute coordinate system using Lagrange's technique, then a simple linear transformation is proposed to obtain the set of nonlinear equations in the DevanitHartenberg coordinate system. The equations of motion for double and triple link inverted pendulum systems are given as examples for such dynamics equations.     

复杂环境下群机器人自组织协同多目标围捕

针对动态多目标围捕,提出了一种复杂环境下协同自组织多目标围捕方法.首先设计了多目标在复杂环境下的运动模型,然后通过对生物群体围捕行为的研究,构建了多目标简化虚拟受力模型.基于此受力模型和提出的动态多目标自组织任务分配算法,提出了群机器人协同自组织动态多目标围捕算法,这两个算法只需多目标和个体两最近邻位置信息以及个体面向多目标中心方向的两最近邻任务信息,计算简单高效,易于实现.接着获得了系统稳定时参数的设置范围.由仿真可知,所提的方法具有较好的灵活性、可扩展性和鲁棒性.最后给出了所提方法相较于其它方法的优势.     

A Partitioned finite element method for dynamical systems

A new analytical method is described for deriving the equations of motion of dynamical systems. The concept is to consider the displacements of the domain to be composed of rigid and elastic components. In contrast to other reduction methods, the domain modeled by finite number of degrees of freedom is discretized into two distinctive types of subdomains. Rigid and elastic subdomains are generated by consistent lumping of the domain properties under unique kinematic constraint relations. Equations of motion of the disjoint subdomains are derived by Lagrange's equations, in conjunction with the shape function matrix represented in partitioned form. This allows reduced sizes of matrices and avoids their possible singularities. Based on the invariance of energies under a compatible partitioned procedure, a simple analytical method is introduced for building the equations of motion of the whole domain from those of the subdomains. The dynamic analysis of a two-node domain with application to a blade-shaft combination is presented to illustrate the method.     

A calibration method for odometry of mobile robots based on the least-squares technique: theory and experimental validation

For a mobile robot, odometry calibration consists of the identification of a set of kinematic parameters that allow reconstructing the vehicle's absolute position and orientation starting from the wheels' encoder measurements. This paper develops a systematic method for odometry calibration of differential-drive mobile robots. As a first step, the kinematic equations are written so as to underline linearity in a suitable set of unknown parameters; thus, the least-squares method can be applied to estimate them. A major advantage of the adopted formulation is that it provides a quantitative measure of the optimality of a test motion; this can be exploited to drive guidelines on the choice of the test trajectories and to evaluate accuracy of a solution. The proposed technique has been experimentally validated on two different mobile robots and, in one case, compared with other existing approaches; the obtained results confirm the effectiveness of the proposed calibration method.     

A finite element formulation for transient analysis of viscoplastic solids with application to stress wave propagation problems

A brief note on a finite element formulation for the transient analysis of viscoplastic solids is presented. Attention is confined to small strains in the present discussion and the emphasis is on stress wave propagation problems. The algorithm is based on straightforward explicit integration of both the equations of motion and the plastic rate equations. The explicit central difference method, which appears suitable for wave propagation problems, is used for integration of the equation of motion. The stress update is accomplished by means of a forward gradient scheme based on an estimate of the plastic flow over a time increment (Peirce, Shih and Needleman, Comput. Struct.18, 875–887, 1984). A number of simple numerical examples are presented to illustrate the method.     

A novel saturated PID-type observer-based controller for wheeled mobile robots with a guaranteed performance considering path curvature

This article proposes a novel saturated PID-type formation controller for a class of nonholonomic wheeled mobile robots (WMRs) of type (m, s) with mobility m and steerability s. Since the path curvature causes a severe deviation of the followers from the leaders' trajectory in the corners, a new definition of output equations is suggested to construct the coordinates of a virtual reference point in the front of each follower based on the concept of the extended look-ahead control that utilizes a corrective angle to compensate the trajectory curvature efficiently. A nonlinear transformation and the robot's kinematic and dynamic equations are heuristically combined to develop a novel second-order Euler–Lagrange formulation of unconstrained errors from constrained errors. Then, a novel saturated PID-type controller is proposed for WMRs to build a desirable formation with a prescribed performance. An innovative nonlinear observer is presented to estimate followers' velocities. An efficient mixture of a radial basis function neural network (RBFNN) and an adaptive mechanism is proposed to compensate for model uncertainties, external disturbances, and network approximation errors. Lyapunov's direct method verifies semi-global uniform ultimate boundedness stability by estimating a gain-dependent region of attraction. Numerical simulations finally show the efficiency of the proposed controller.     

A semi-analytical coupled finite element formulation for shells conveying fluids

A new formulation, based on the semi-analytical finite element method, is proposed for elastic shells conveying fluids. The structural equations are based on the shell element proposed by Ramasamy and Ganesan [Comput Struct 70 (1998) 363] while the fluid model is based on velocity potential. Dynamic pressure acting on the walls is derived from Bernoulli's equation. Imposing the requirement that the normal components of velocity of the solid and fluid be equal, introduces fluid–structure coupling. The proposed technique has been validated using results available in the literature. This study shows that instability occurs at a critical fluid velocity corresponding to the shell circumferential mode with the lowest natural frequency and this phenomenon is also independent of the type of structural boundary conditions imposed.     

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.     

未知动态环境下非完整移动群机器人围捕

针对未知动态障碍物环境下非完整移动群机器人围捕,提出了一种基于简化虚拟受力模型的自组织方法.首先给出了个体机器人的运动方程,然后给出了未知动态环境下目标和动态障碍物的运动模型.通过对复杂环境下围捕行为的分解,抽象出简化虚拟受力模型,基于此受力模型,设计了个体运动控制方法,接着证明了系统的稳定性并给出了参数设置范围.不同情况下的仿真结果表明,本文给出的围捕方法可以使群机器人在未知动态障碍物环境下保持较好的围捕队形,并具有良好的避障性能和灵活性.最后分析了本文与基于松散偏好规则的围捕方法相比的优势.     

Simulation of a viscoelastic flexible multibody system using absolute nodal coordinate and fractional derivative methods

This paper employs a new finite element formulation for dynamics analysis of a viscoelastic flexible multibody system. The viscoelastic constitutive equation used to describe the behavior of the system is a three-parameter fractional derivative model. Based on continuum mechanics, the three-parameter fractional derivative model is modified and the proposed new fractional derivative model can reduce to the widely used elastic constitutive model, which meets the continuum mechanics law strictly for pure elastic materials. The system equations of motion are derived based on the absolute nodal coordinate formulation (ANCF) and the principle of virtual work, which can relax the small deformation assumption in the traditional finite element implementation. In order to implement the viscoelastic model into the absolute nodal coordinate, the Grünwald definition of the fractional derivative is employed. Based on a comparison of the HHT-I3 method and the Newmark method, the HHT-I3 method is used to solve the equations of motion. Another particularity of the proposed method based on the ANCF method lies in the storage of displacement history only during the integration process, reducing the numerical computation considerably. Numerical examples are presented in order to analyze the effects of the truncation number of the Grünwald series (fading memory phenomena) and the value of several fractional model parameters and solution convergence aspects. An erratum to this article can be found at     

Automatic elimination of the undetermined multipliers in Kane's equations using a Pseudo Uptriangular Decomposition (PUTD) method

The formulation of the equations of motion for constrained multibody systems is presented. The procedures employed use Kane's equations as developed by Huston et al. and use a tree-like configuration array as a basis for analysis. A Pseudo Uptriangular Decomposition method along with the Gramm-Schmidt process is developed to modify Kane's equations when the constraint equations are introduced using the undetermined multipliers. This approach is shown to be far reaching in its application and it is tailored for general purpose algorithms used in the development of computer programs for complex structure dynamics.     

Symbolic Finite Element Modeling of Structural Systems

The application of light materials to space structures, aircraft, robots and automobiles has increased the demand for effective algorithms to model and predict the response of structural multibody systems. The understanding of mechanics can assist in developing better design and control strategies. Formulation of mathematical models of a multibody system using manual approaches is a difficult task and prone to errors. For non-linear and/or time-varying systems, numerical formulation provides limited information about physical insight. In this study, a computer-aided symbolic method is used to generate the equations of motion from Lagrange's method. Equations are converted into FORTRAN form ready for simulations and control synthesis. The 4–5th order Runge–Kutta–Fehlberg method (RKF45) was used to numerically solve the system of equations. Two examples, namely a slider–crank mechanism and an aircraft model are presented.     

用小波插值方法实现移动机器人的轨迹追踪控制

基于动力学模型方程，利用小波插值方法，对移动机器人的轨迹追踪控制问题，给出了一种新的方法．该方法在有限时间内，实现了转动速度、前进速度均不为零的期望轨迹追踪．它的突出特点是计算量小、方法简单，期望轨迹可为任意复杂的非线性曲线．在仿真实验里，取得了理想的效果．     

Optimal Trajectory Planning for Flexible Link Manipulators with Large Deflection Using a New Displacements Approach

The main objective of the present paper is to determine the optimal trajectory of very flexible link manipulators in point-to-point motion using a new displacement approach. A new nonlinear finite element model for the dynamic analysis is employed to describe nonlinear modeling for three-dimensional flexible link manipulators, in which both the geometric elastic nonlinearity and the foreshortening effects are considered. In comparison to other large deformation formulations, the motion equations contain constant stiffness matrix because the terms arising from geometric elastic nonlinearity are moved from elastic forces to inertial, reactive and external forces, which are originally nonlinear. This makes the formulation particularly efficient in computational terms and numerically more stable than alternative geometrically nonlinear formulations based on lower-order terms. In this investigation, the computational method to solve the trajectory planning problem is based on the indirect solution of open-loop optimal control problem. The Pontryagin’s minimum principle is used to obtain the optimality conditions, which is lead to a standard form of a two-point boundary value problem. The proposed approach has been implemented and tested on a single-link very flexible arm and optimal paths with minimum effort and minimum vibration are obtained. The results illustrate the power and efficiency of the method to overcome the high nonlinearity nature of the problem.