状态空间中约束系统的运动方程

引入状态变量表示力学系统的约束方程;建立状态空间中运动约束系统的新型变分原理;导出运动约束系统的带乘子的运动微分方程和广义状态变量运动微分方程;证明状态空间中运动约束系统的运动方程是奇异的;举例说明所得结果的应用.     

Augmented Lagrangian Formulation: Geometrical Interpretation and Application to Systems with Singularities and Redundancy

A geometric interpretation of the augmented Lagrangian formulation ofBayo et al. (Comput. Methods Appl. Mech. Engrg. 71, 1988, 183–195), appliedto equations of motion in relative and Cartesian coordinates, ispresented. Instead of imposing constraints on a system in thetraditional sense, large artificial masses resisting in the constraineddirections are added, and the system motion is enforced to evolveprimarily in the directions with smaller masses (in the unconstraineddirections). Then, the residual motion in the constrained directions isremoved by applying the constraint reactions to the system, estimatedeffectively in few iterations. The formulation is comparatively simpleand leads to computationally efficient numerical codes. Usefulapplications of the formulation to the dynamic analysis of constrainedmultibody systems with possible singular configurations, massless linksand redundant constraints are shown. The theoretical background isfollowed by some remarks on the modeling precautions and assistedcomputational peculiarities of the method. The results of numericalsimulation of motion of a parallel five-bar and a parallel four-barlinkage are reported.     

Equations of Motion of Planar Serial Chains of Rigid Bodies

In the present study, the equations of motion of planar serialchains that consist of a system of rigid bodies with all commontypes of kinematic joints are presented. The system of rigid bodiesis replaced by a dynamically equivalent constrained system ofparticles. The concepts of linear and angular momentums are used togenerate the equations of motion for the resulting equivalentconstrained system of particles without either introducing anyrotational coordinates or distributing the external forces and forcecouples over the particles. The method can be applied for open orclosed chains. An example of a closed chain is chosen to demonstratethe generality and simplicity of the proposed method.     

Stabilization Methods for the Integration of DAE in the Presence of Redundant Constraints

The use of multibody formulations based on Cartesian or naturalcoordinates lead to sets of differential-algebraic equations that haveto be solved. The difficulty in providing compatible initial positionsand velocities for a general spatial multibody model and the finiteprecision of such data result in initial errors that must be correctedduring the forward dynamic solution of the system equations of motion.As the position and velocity constraint equations are not explicitlyinvolved in the solution procedure, any integration error leads to theviolation of these equations in the long run. Another problem that isvery often impossible to avoid is the presence of redundant constraints.Even with no initial redundancy it is possible for some systems toachieve singular configurations in which kinematic constraints becometemporarily redundant. In this work several procedures to stabilize thesolution of the equations of motion and to handle redundant constraintsare revisited. The Baumgarte stabilization, augmented Lagrangian andcoordinate partitioning methods are discussed in terms of theirefficiency and computational costs. The LU factorization with fullpivoting of the Jacobian matrix directs the choice of the set ofindependent coordinates, required by the coordinate partitioning method.Even when no particular stabilization method is used, a Newton–Raphsoniterative procedure is still required in the initial time step tocorrect the initial positions and velocities, thus requiring theselection of the independent coordinates. However, this initialselection does not guarantee that during the motion of the system otherconstraints do not become redundant. Two procedures based on the singlevalue decomposition and Gram–Schmidt orthogonalization are revisited forthe purpose. The advantages and drawbacks of the different procedures,used separately or in conjunction with each other and theircomputational costs are finally discussed.     

An Efficient Dynamic Formulation for Multibody Systems

The objective of this article is to present an efficient extension ofRosenthal's order-n algorithm to multibody systems containing closedloops. The equations of motion are created by using relative coordinatesand partial velocity theory. Closed topological loops are handled by cutjoint technique. The set of constraint equations of cut joints isadjoined to the system's equation of motion by using Lagrangemultipliers. This results in the equation of motion as adifferential-algebraic equation (DAE) rather than an ordinarydifferential equation. This DAE is then solved by applying the extendedRosenthal's order-n algorithm proposed in this article. While solvingDAE, violation of the kinematic constraint equations of cut joints iscorrected by coordinate projection method. Some numerical simulationsare carried out to demonstrate efficiency of the proposed method.     

Modeling and control of two constrained manipulators

The coordinated control of two manipulators in the presence of environment constraints is studied in this paper. Such a control method is needed in applications in which the two manipulators grasp a common object whose motion is constrained by environments. The two manipulators are not only constrained with each other, but also constrained by the environment in their workspace. It is realized that the motion and constraint equations obtained directly from mechanics are not suitable for the control purpose. A set of equivalent equations are derived, which are in the standard form of the nonlinear system representation with clear state equations and output equations. A nonlinear feedback is found which exactly linearizes and decouples the dynamic nonlinear system of the two constrained manipulators. The coordinated controller design is then carried out based on the linearized system by using linear system theory.     

Small Vibrations Superimposed on a Prescribed Rigid Body Motion

A method for analysing flexible multibody systems in which theelastic deformations are small is presented. The motion isconsidered a gross non-linear rigid body motion with small linearvibrations superimposed on it. For periodic gross motion, thisresults in a system of rheolinear differential equations for thedeformations with periodic coefficients. The determination of therequired equations with a program for flexible multibody systemsis discussed which calculates, besides the periodic gross motion,the linearized, or variational, equations of motion. Periodicsolutions are determined with a harmonic balance method, whiletransient solutions are obtained by an averaging method. Thestability of the periodic solutions is considered. The procedurehas a high computational efficiency and leads to more insight intothe structure of solutions. The method is applied to a pendulumwith an elliptical motion of its support point, a slider-crankmechanism with flexible connecting rod, a rotor system, and aCardan drive shaft with misalignment.     

考虑刚-柔-热耦合的板结构多体系统的动力学建模

对热载荷作用下中心刚体与大变形薄板多体系统的动力学建模问题进行研究.基于Kirchhoff假设,从格林应变和曲率与绝对位移的非线性关系式出发,推导了非线性广义弹性力阵,用绝对节点坐标法建立了大变形矩形薄板的有限元离散的动力学变分方程.为了考虑刚体姿态运动、弹性变形和温度变化的相互耦合作用,推导了热流密度与绝对节点坐标之间的关系式.引入系统的运动学约束方程,建立了中心刚体-矩形板多体系统的考虑刚-柔-热耦合的热传导方程和带拉格朗日乘子的第一类拉格朗日动力学方程.为了有效地提高计算效率,将改进的中心差分法和广义-α法相结合,求解热传导方程和动力学方程,差分后的方程通过牛顿迭代法耦合求解.对刚-柔耦合和刚-柔-热三者耦合两种模型的仿真结果进行比较表明,刚体运动对温度梯度和热变形的影响显著.此外,本文建模方法考虑了几何非线性项,因此也考虑了热膨胀引起的轴向变形对横向变形的影响.     

An optimization method for overdetermined kinematic problems formulated with natural coordinates

In this paper, we present an optimization method for solving the nonlinear constrained optimization problem arising from a motion reconstruction problem formulated with natural coordinates. A motion reconstruction problem consists in a kinematic analysis of a rigid multibody system whose motion is usually overdetermined by an excess of data. The method has been applied to the analysis of human motion which is a typical case of an overdetermined kinematic problem as a large number of markers are usually placed on a subject to capture its movement. The efficiency of the method has been tested both with computer-simulated and real experimental data using models that include open and closed kinematic loops.     

Elimination of Constraint Violation and Accuracy Aspects in Numerical Simulation of Multibody Systems

Multibody systems are often modeled as constrained systems, and theconstraint equations are involved in the dynamics formulations. To makethe arising governing equations more tractable, the constraint equationsare differentiated with respect to time, and this results in unstablenumerical solutions which may violate the lower-order constraintequations. In this paper we develop a methodology for numerically exactelimination of the constraint violations, based on appropriatecorrections of the state variables (after each integration step) withoutany modification in the motion equations. While the elimination ofviolation of position constraints may require few iterations, theviolation of velocity constraints is removed in one step. The totalenergy of the system is sometimes treated as another measure of theintegration process inaccuracy. An improved scheme for one-stepelimination of the energy constraint violation is proposed as well. Theconclusion of this paper is, however, that the energy conservation is ofminor importance as concerns the improvement of accuracy of numericalsimulations. Some test calculations are reported.     

A geometrical approach to the motion planning problem for a submerged rigid body

The main focus of this article is the motion planning problem for a deeply submerged rigid body. The equations of motion are formulated and presented by use of the framework of differential geometry and these equations incorporate external dissipative and restoring forces. We consider a kinematic reduction of the affine connection control system for the rigid body submerged in an ideal fluid, and present an extension of this reduction to the forced affine connection control system for the rigid body submerged in a viscous fluid. The motion planning strategy is based on kinematic motions; the integral curves of rank one kinematic reductions. This method is of particular interest to autonomous underwater vehicles which cannot directly control all six degrees of freedom (such as torpedo-shaped autonomous underwater vehicles) or in case of actuator failure (i.e. under-actuated scenario). A practical example is included to illustrate our technique.     

On the constraints violation in forward dynamics of multibody systems

It is known that the dynamic equations of motion for constrained mechanical multibody systems are frequently formulated using the Newton–Euler’s approach, which is augmented with the acceleration constraint equations. This formulation results in the establishment of a mixed set of partial differential and algebraic equations, which are solved in order to predict the dynamic behavior of general multibody systems. The classical solution of the equations of motion is highly prone to constraints violation because the position and velocity constraint equations are not fulfilled. In this work, a general and comprehensive methodology to eliminate the constraints violation at the position and velocity levels is offered. The basic idea of the described approach is to add corrective terms to the position and velocity vectors with the intent to satisfy the corresponding kinematic constraint equations. These corrective terms are evaluated as a function of the Moore–Penrose generalized inverse of the Jacobian matrix and of the kinematic constraint equations. The described methodology is embedded in the standard method to solve the equations of motion based on the technique of Lagrange multipliers. Finally, the effectiveness of the described methodology is demonstrated through the dynamic modeling and simulation of different planar and spatial multibody systems. The outcomes in terms of constraints violation at the position and velocity levels, conservation of the total energy and computational efficiency are analyzed and compared with those obtained with the standard Lagrange multipliers method, the Baumgarte stabilization method, the augmented Lagrangian formulation, the index-1 augmented Lagrangian, and the coordinate partitioning method.     

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.     

Dynamic behaviour of a mechanical system of which a part performs a prescribed relative motion

A method is presented to solve by means of a finite element analysis the problem of the dynamic behaviour of a mechanical system, a part of which performs a prescribed relative motion. The kinematic coupling of this part to the rest of the system and the derivation of the matrix equations of motion are discussed, as well as the applied condensation method. Moreover the results of a computation based on this method are compared to those obtained by neglecting the dynamic interactions of both the parts of such a mechanical system.     

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.     

A compact smoothing-differentiation and projection approach for the kinematic data consistency of biomechanical systems

The estimation of the skeletal motion obtained from marker-based motion capture systems affects the results of the kinematic and dynamic analysis of biomechanical systems. The main source of error is the inaccuracy of velocities and accelerations derived from experimentally measured displacements of markers placed on the skin of joints. This error is mainly due to the amplification of high-frequency low-amplitude noise introduced by the motion capture system when the raw displacement signals are differentiated. Another source of error is the skin motion artifact that produces violations of the kinematic constraint equations of the multibody system. An integrated smoothing-differentiation-projection approach to ensure the kinematic data consistency in the context of the analysis of biomechanical systems is presented. The raw data differentiation problem is solved by applying a single-step smoothing-differentiation technique based on the Newmark integration scheme. A systematic multibody procedure is proposed based on the projection of the positions and its smoothed derivatives into their corresponding constraint manifolds to ensure the kinematic data consistency. Several benchmark kinematic signals that include an acquired nonstationary mono-dimensional motion of biomechanical origin and computer generated data of a four-bar mechanism were processed using the proposed method to study its performance.     

A DAE Approach to Flexible Multibody Dynamics

The present work deals with the dynamics of multibody systems consisting ofrigid bodies and beams. Nonlinear finite element methods are used to devise a frame-indifferent spacediscretization of the underlying geometrically exact beam theory. Both rigid bodies and semi-discrete beams are viewed as finite-dimensional dynamical systems with holonomic constraints. The equations of motion pertaining to the constrained mechanical systems under considerationtake the form of Differential Algebraic Equations (DAEs).The DAEs are discretized directly by applying a Galerkin-based method.It is shown that the proposed DAE approach provides a unified framework for the integration of flexible multibody dynamics.     

Quadruped free gait generation for straight-line and circular trajectories

A method of free gait generation is proposed utilizing the primary/secondary gait for both straight line and circular body trajectories. The primary gait is a fixed sequence of leg transfers with modified leg-ends kinematic limits according to the presence of obstacles, while the secondary gait is a flexible gait which is generated to adjust the leg-end position. The primary gait is generated considering the following four constraints: stability constraint, kinematic constraint, sequential constraint and neighboring constraints. A generalized reference coordinate (GRC) system is introduced to describe the vehicle motion. Using the GRC system, all constraints and obstacle influences are expressed by only one set of equations despite the difference of motion mode. The efficiency of free gait generation is improved with the proposed method, and the trajectory of the vehicle body can be designed more naturally. Simulation results are given to demonstrate the efficiency of the proposed methodology.