A matrix formulation for the dynamic analysis of planar mechanisms using point coordinates and velocity transformation

This paper presents a matrix formulation for the dynamic analysis of planar mechanisms consisting of interconnected rigid bodies. The formulation initially uses the rectangular Cartesian coordinates of an equivalent constrained system of particles to define the configuration of the mechanical system. This results in a simple and straightforward procedure for generating the equations of motion. The equations of motion are then derived in terms of relative joint coordinates through the use of a velocity transformation matrix. The velocity transformation matrix relates the relative joint velocities to the Cartesian velocities. For the open-loop case, this process automatically eliminates all of the non-working constraint forces and leads to an efficient integration of the equations of motion. For the closed-loop case, suitable joints should be cut and few cut-joints constraint equations should be included for each closed loop. Two examples are used to demonstrate the generality and efficiency of the proposed method.     

Dynamic Analysis of Planar Linkages: A Recursive Approach

In the present study, the equations of motion for generalized planar linkages that consist of a system of rigid bodies with all common types of kinematic joints are derived using a recursive approach. 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 and introducing 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.     

Dynamic Analysis of Spatial Linkages: A Recursive Approach

In this paper, recursive equations of motion of spatial linkages are presented. The method uses the concepts of linear and angular momentums to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotational transformation matrix. For the open-chain system, the equations of motion are generated recursively along the serial chains. Closed-chain system is transformed to open-chain by cutting suitable kinematic joints and introducing cut–joint constraints. An example is chosen to demonstrate the generality and simplicity of the developed formulation.     

Stability case study of the ACROBOTER underactuated service robot

The dynamics of classical robotic systems are usually described by ordinary differential equations via selecting a minimum set of independent generalized coordinates. However, different parameterizations and the use of a nonminimum set of (dependent) generalized coordinates can be advantageous in such cases when the modeled device contains closed kinematic loops and/or it has a complex structure. On one hand, the use of dependent coordinates, like natural coordinates, leads to a different mathematical representation where the equations of motion are given in the form of differential algebraic equations. On the other hand, the control design of underactuated robots usually relies on partial feedback linearization based techniques which are exclusively developed for systems modeled by independent coordinates. In this paper, we propose a different control algorithm formulated by using dependent coordinates. The applied computed torque controller is realized via introducing actuator constraints that complement the kinematic constraints which are used to describe the dynamics of the investigated service robotic system in relatively simple and compact form. The proposed controller is applied to the computed torque control of the planar model of the ACROBOTER service robot. The stability analysis of the digitally controlled underactuated service robot is provided as a real parameter case study for selecting the optimal control gains.     

The Kinematics and the Full Minimal Dynamic Model of a 6-DOF Parallel Robot Manipulator

In this paper we present a particular architecture of parallel robots which has six-degrees-of-freedom (6-DOF) with only three limbs. The particular properties of the geometric and kinematic models with respect to that of a classical parallel robot are presented. We show that inverse problems have an analytical solution. However, to solve the direct problems, an efficient numerical procedure which needs to inverse only a 3 × 3 passive Jacobian matrix is proposed. In a second step, dynamic equations are derived using the Lagrangian formalism where the joint variables are passive and active joint coordinates. Based on the geometrical properties of the robot, the equations of motion are derived in terms of only nine coordinates related by three kinematic constraints instead of 18 joint coordinates. The computational cost of the dynamic model obtained is reduced by using a minimum set of base inertial parameters.     

A Recursive Formulation for Constrained Mechanical System Dynamics: Part II. Closed Loop Systems

ABSTRACT

ABSTRACT A recursive formulation of the equations of motion of constrained mechanical systems with closed loops is derived, using tools of variational and vector calculus. Kinematic couplings between pairs of contiguous bodies presented in Part 1 of this paper are generalized. Lagrange multipliers are introduced to account for the effects of joints that are cut to define a tree structure. Constraint Jacobian terms are added to the reduced variational equations derived in Part I. Cut-joint constraint acceleration equations are derived, to complete the reduced equations of motion. Lagrange multipliers associated with each cut-joint are eliminated at the first junction body encountered that permits closing the loop that constraints in cut joint. The inductive algorithm developed in Part I is used to calculate accelerations for the system. A multi-loop compressor is analyzed to illustrate use of the method.     

A velocity transformation method for the nonlinear dynamic simulation of railroad vehicle systems

The solution of the constrained multibody system equations of motion using the generalized coordinate partitioning method requires the identification of the dependent and independent coordinates. Using this approach, only the independent accelerations are integrated forward in time in order to determine the independent coordinates and velocities. Dependent coordinates are determined by solving the nonlinear constraint equations at the position level. If the constraint equations are highly nonlinear, numerical difficulties can be encountered or more Newton–Raphson iterations may be required in order to achieve convergence for the dependent variables. In this paper, a velocity transformation method is proposed for railroad vehicle systems in order to deal with the nonlinearity of the constraint equations when the vehicles negotiate curved tracks. In this formulation, two different sets of coordinates are simultaneously used. The first set is the absolute Cartesian coordinates which are widely used in general multibody system computer formulations. These coordinates lead to a simple form of the equations of motion which has a sparse matrix structure. The second set is the trajectory coordinates which are widely used in specialized railroad vehicle system formulations. The trajectory coordinates can be used to obtain simple formulations of the specified motion trajectory constraint equations in the case of railroad vehicle systems. While the equations of motion are formulated in terms of the absolute Cartesian coordinates, the trajectory accelerations are the ones which are integrated forward in time. The problems associated with the higher degree of differentiability required when the trajectory coordinates are used are discussed. Numerical examples are presented in order to examine the performance of the hybrid coordinate formulation proposed in this paper in the analysis of multibody railroad vehicle systems.     

Application of joint coordinates and homogeneous transformations to modeling of vehicle dynamics

The paper presents a method of modeling dynamics of multibody systems with open and closed kinematic chains. The joint coordinates and homogeneous transformations are applied in order to formulate the equations of motion of a rigid body. In this method, constraint equations are introduced only in the case when closed subchains are considered or when the joint reactions have to be calculated. This allows the number of generalized coordinates in the system to be reduced in comparison to the case when absolute coordinates are applied. It is shown how the method can be applied to modeling of vehicle dynamics. The calculation results are compared with those obtained when the ADAMS/Car package is used. Experimental verification has been performed and is reported in the paper, as well. In both cases, a good correspondence of results has been achieved. Final remarks concerning advantages and disadvantages of the method are formulated at the end of the paper.     

Numerical study of the noninertial systems: application to train coupler systems

Car coupler forces have a significant effect on the longitudinal train dynamics and stability. Because the coupler inertia is relatively small in comparison with the car inertia; the high stiffness associated with the coupler components can lead to high frequencies that adversely impact the computational efficiency of train models. The objective of this investigation is to study the effect of the coupler inertia on the train dynamics and on the computational efficiency as measured by the simulation time. To this end, two different models are developed for the car couplers; one model, called the inertial coupler model, includes the effect of the coupler inertia, while in the other model, called the noninertial model, the effect of the coupler inertia is neglected. Both inertial and noninertial coupler models used in this investigation are assumed to have the same coupler kinematic degrees of freedom that capture geometric nonlinearities and allow for the relative translation of the draft gears and end of car cushioning (EOC) devices as well as the relative rotation of the coupler shank. In both models, the coupler kinematic equations are expressed in terms of the car body and coupler coordinates. Both the inertial and noninertial models used in this study lead to a system of differential and algebraic equations that are solved simultaneously in order to determine the coordinates of the cars and couplers. In the case of the inertial model, the coupler kinematics is described using the absolute Cartesian coordinates, and the algebraic equations describe the kinematic constraints imposed on the motion of the system. In this case of the inertial model, the constraint equations are satisfied at the position, velocity, and acceleration levels. In the case of the noninertial model, the equations of motion are developed using the relative joint coordinates, thereby eliminating systematically the algebraic equations that represent the kinematic constraints. A quasistatic force analysis is used to determine a set of coupler nonlinear force algebraic equations for a given car configuration. These nonlinear force algebraic equations are solved iteratively to determine the coupler noninertial coordinates which enter into the formulation of the equations of motion of the train cars. The results obtained in this study showed that the neglect of the coupler inertia eliminates high frequency oscillations that can negatively impact the computational efficiency. The effect of these high frequencies that are attributed to the coupler inertia on the simulation time is examined using frequency and eigenvalue analyses. While the neglect of the coupler inertia leads, as demonstrated in this investigation, to a much more efficient model, the results obtained using the inertial and noninertial coupler models show good agreement, demonstrating that the coupler inertia can be neglected without having an adverse effect on the accuracy of the solution.     

有多余坐标完整系统的自由运动

对于完整力学系统,若选取的参数不是完全独立的,则称为有多余坐标的完整系统. 由于完整力学系统的第二类Lagrange 方程中没有约束力,故为研究完整力学系统的约束力,需采用有多余坐标的带乘子的Lagrange方程或第一类Lagrange 方程. 一些动力学问题要求约束力不能为零,而另一些问题要求约束力很小. 如果约束力为零,则称为系统的自由运动问题. 本文提出并研究了有多余坐标完整系统的自由运动问题. 为研究系统的自由运动,首先,由d'Alembert-Lagrange 原理, 利用Lagrange 乘子法建立有多余坐标完整系统的运动微分方程;其次,由多余坐标完整系统的运动方程和约束方程建立乘子满足的代数方程并得到约束力的表达式;最后,由约束系统自由运动的定义,令所有乘子为零,得到系统实现自由运动的条件. 这些条件的个数等于约束方程的个数,它们依赖于系统的动能、广义力和约束方程,给出其中任意两个条件,均可以得到实现自由运动时对另一个条件的限制. 即当给定动能和约束方程,这些条件会给出实现自由运动时广义力之间的关系. 当给定动能和广义力,这些条件会给出实现自由运动时对约束方程的限制. 当给定广义力和约束方程,这些条件会给出实现自由运动时对动能的限制. 文末,举例并说明方法和结果的应用.      

Forces associated with non-linear non-holonomic constraint equations

A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by non-holonomic equations that are inherently non-linear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known non-holonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as non-holonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor first to form the former, larger set and subsequently perform matrix multiplications.     

Computer Implementation of the Absolute Nodal Coordinate Formulation for Flexible Multibody Dynamics

Deformable components in multibody systems are subject to kinematic constraints that represent mechanical joints and specified motion trajectories. These constraints can, in general, be described using a set of nonlinear algebraic equations that depend on the system generalized coordinates and time. When the kinematic constraints are augmented to the differential equations of motion of the system, it is desirable to have a formulation that leads to a minimum number of non-zero coefficients for the unknown accelerations and constraint forces in order to be able to exploit efficient sparse matrix algorithms. This paper describes procedures for the computer implementation of the absolute nodal coordinate formulation' for flexible multibody applications. In the absolute nodal coordinate formulation, no infinitesimal or finite rotations are used as nodal coordinates. The configuration of the finite element is defined using global displacement coordinates and slopes. By using this mixed set of coordinates, beam and plate elements can be treated as isoparametric elements. As a consequence, the dynamic formulation of these widely used elements using the absolute nodal coordinate formulation leads to a constant mass matrix. It is the objective of this study to develop computational procedures that exploit this feature. In one of these procedures, an optimum sparse matrix structure is obtained for the deformable bodies using the QR decomposition. Using the fact that the element mass matrix is constant, a QR decomposition of a modified constant connectivity Jacobian matrix is obtained for the deformable body. A constant velocity transformation is used to obtain an identity generalized inertia matrix associated with the second derivatives of the generalized coordinates, thereby minimizing the number of non-zero entries of the coefficient matrix that appears in the augmented Lagrangian formulation of the equations of motion of the flexible multibody systems. An alternate computational procedure based on Cholesky decomposition is also presented in this paper. This alternate procedure, which has the same computational advantages as the one based on the QR decomposition, leads to a square velocity transformation matrix. The computational procedures proposed in this investigation can be used for the treatment of large deformation problems in flexible multibody systems. They have also the advantages of the algorithms based on the floating frame of reference formulations since they allow for easy addition of general nonlinear constraint and force functions.     

Dynamic recursive decoupling method for closed-loop flexible mechanical systems

In this paper, a new method for the dynamic analysis of a closed-loop flexible kinematic mechanical system is presented. The kinematic and force models are developed using absolute reference, joint relative, and elastic coordinates as well as joint reaction forces. This recursive formulation leads to a system of loosely coupled equations of motion. In a closed-loop kinematic chain, cuts are made at selected auxiliary joints in order to form spanning tree structures. Compatibility conditions and reaction force relationships at the auxiliary joints are adjoined to the equations of open-loop mechanical systems in order to form closed-loop dynamic equations. Using the sparse matrix structure of these equations and the fact that the joint reaction forces associated with elastic degrees of freedom do not represent independent variables, a method for decoupling the joint and elastic accelerations is developed. Unlike existing recursive formulations, this method does not require inverse or factorization of large non-linear matrices. It leads to small systems of equations whose dimensions are independent of the number of elastic degrees of freedom. The application of dynamic decoupling method in dynamic analysis of closed-loop deformable multibody systems is also discussed in this paper. The use of the numerical algorithm developed in this investigation is illustrated by a closed-loop flexible four-bar mechanism.     

On the use of linear graph theory in multibody system dynamics

Multibody dynamics involves the generation and solution of the equations of motion for a system of connected material bodies. The subject of this paper is the use of graph-theoretical methods to represent multibody system topologies and to formulate the desired set of motion equations; a discussion of the methods available for solving these differential-algebraic equations is beyond the scope of this work. After a brief introduction to the topic, a review of linear graphs and their associated topological arrays is presented, followed in turn by the use of these matrices in generating various graph-theoretic equations. The appearance of linear graph theory in a number of existing multibody formulations is then discussed, distinguishing between approaches that use absolute (Cartesian) coordinates and those that employ relative (joint) coordinates. These formulations are then contrasted with formal graph-theoretic approaches, in which both the kinematic and dynamic equations are automatically generated from a single linear graph representation of the system. The paper concludes with a summary of results and suggestions for further research on the graph-theoretical modelling of mechanical systems.     

Forward and inverse dynamics of nonholonomic mechanical systems

This paper deals with the forward and the inverse dynamic problems of mechanical systems subjected to nonholonomic constraints. The intrinsically dual nature of these two problems is identified and utilised to develop a systematic approach to formulate and solve them according to an unified framework. The proposed methodology is based on the fundamental equations of constrained motion which derive from Gauss’s principle of least constraint. The main advantage arising from using the fundamental equations of constrained motion is that they represent an effective method capable to derive the generalised acceleration of a mechanical system, constrained in general by a set of nonholonomic constraints, together with the generalized constraint forces (forward dynamics). When the constraint equations are used to represent the desired behaviour of the mechanical system under study, the generalised constraint forces deriving from the fundamental equations of constrained motion provide the control actions which reproduce the specified motion for the system (inverse dynamics). This approach is systematically extended to underactuated mechanical systems introducing a new method named underactuation equivalence principle. The underactuation equivalence principle is founded on the key idea that the underactuation property of a mechanical system can be mathematically represented using a particular set of nonholonomic constraint equations. Two simple case-studies are reported to exemplify the proposed methodology. In the first case-study the computation of the generalised constraint forces relative to the revolute joint constraints of a physical pendulum is illustrated. In the second case-study the calculation of the control action which solves the swing-up problem for an inverted pendulum is described.     

Dynamics of a Two-Pendulum Model with Impact Interaction and an Elastic Support

This paper examines the dynamic behavior of a double pendulummodel with impact interaction. One of the masses of the two pendulumsmay experience impacts against absolutely rigid container wallssupported by an elastic system forming an inverted pendulum restrainedby a torsional elastic spring. The system equations of motion arewritten in terms of a non-smooth set of coordinates proposed originallyby Zhuravlev. The advantage of non-smooth coordinates is that theyeliminate impact constraints. In terms of the new coordinates, thepotential energy field takes a cell-wise non-local structure, and theimpact events are treated geometrically as a crossing of boundariesbetween the cells. Based on a geometrical treatment of the problem,essential physical system parameters are established. It is found thatunder resonance parametric conditions of the linear normal modes thesystem's response can be either bounded or unbounded, depending on thesystem's parameters. The ability of the system to absorb energy from anexternal source essentially depends on the modal inclination angle,which is related to the principal coordinates.     

Slow Motion of an Assemblage of Porous Spherical Shells Relative to a Fluid

The body-force-driven motion of a homogeneous distribution of spherically symmetric porous shells in an incompressible Newtonian fluid and the fluid flow through a bed of these shell particles are investigated analytically. The effect of the hydrodynamic interaction among the porous shell particles is taken into account by employing a cell-model representation. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field around a single particle in a unit cell, and the drag force acting on the particle by the fluid is obtained in closed forms. For a suspension of porous spherical shells, the mobility of the particles decreases or the hydrodynamic interaction among the particles increases monotonically with a decrease in the permeability of the porous shells. The effect of particle interactions on the creeping motion of porous spherical shells relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solution describing the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres. The particle-interaction behavior for a suspension of porous spherical shells with a relatively low permeability may be approximated by that of permeable spheres when the porous shells are sufficiently thick.     

多体系统接触碰撞问题的牛顿-欧拉线性互补方法

基于非光滑动力学方法的多体系统接触碰撞分析是目前多体系统动力学的研究热点.本文采用牛顿-欧拉方法建立多体系统接触、碰撞问题的动力学模型,给出一种牛顿-欧拉型线性互补公式.该建模方法与目前一般采用的拉格朗日建模方法的不同之处是约束条件中除了库仑摩擦、单边约束之外还含有光滑等式约束.在建立系统动力学模型时,首先解除摩擦约束和单边约束得到原系统对应的基本系统.牛顿-欧拉方法采用最大数目坐标建立基本系统的动力学方程,由于坐标不相互独立,因此基本系统中带有等式约束,其数学模型为一组微分代数方程.借助约束雅可比矩阵,在基本系统微分代数方程中添加摩擦接触和单边约束对应的拉氏乘子,就可以得到系统全局运动的具有变拓扑结构特征的动力学方程,再结合非光滑约束互补条件便可构成完备的系统动力学模型.完备的动力学模型由动力学微分方程以及等式约束和不等式约束组成.线性互补公式采用分块矩阵形式进行推导,简化了推导过程.数值计算采用基于线性互补的时间步进算法.时间步进算法是目前流行的非光滑数值算法,其突出特点是可以免去数值积分中繁琐的事件检测过程,而数值积分过程中通过对线性互补问题的求解可以确定系统的触-离状态.通过对典型的曲柄滑块间隙机构进行数值分析,验证本文方法的有效性.