On-Line Symbolic Constraint Embedding for Simulation of Hybrid Dynamical Systems

In this paper we present a simulator designed to handle multibody systems with changing constraints, wherein the equations of motion for each of its constraint configurations are formulated in minimal ODE form with constraints embedded before they are passed to an ODE solver. The constraint-embedded equations are formulated symbolically according to a re-combination of terms of the unconstrained equations, and this symbolic process is undertaken on-line by the simulator. Constraint-embedding undertaken on-the-fly enables the simulation of systems with an ODE solver for which constraints are not known prior to simulation start or for which the enumeration of all constraint conditions would be unwieldy because of their complexity or number. Issues of drift associated with DAE solvers that usually require stabilization are sidestepped with the constraint-embedding approach. We apply nomenclature developed for hybrid dynamical systems to describe the system with changing constraints and to distinguish the roles of the forward dynamics solver, a collision detector, and an impact resolver. We have prototyped the simulator in MATLAB and demonstrate the design using three representative examples.     

Dynamics of Multibody Systems Subjected to Impulsive Constraints

This paper presents a procedure for studying dynamics of multibodysystems subjected to impulsive constraints which may be either holonomicor nonholonomic. The procedure automatically incorporates the effects ofimpulsive constraints through its analysis. The governing equationsthemselves are developed from Kane's equations, using partial velocityvectors and partial angular velocity vectors. Explicit expressions forthe coefficients and terms of the governing equations are presented. Twosubcases are studied: (1) the constraints are instantaneously applied andcontinue to act; and (2) the constraints are instantaneously applied andlifted immediately after completion of impact. The internal impulses ateach joint and constraint impulses associated with the impulsiveconstraints are calculated. The procedure is checked with two examples,whose solutions are established.     

具有环状运动约束的悬臂输流管道的非线性振动特征

本文对具有环状运动约束的悬臂输流管道的空间弯曲振动进行研究,目的在于考察约束刚度系数、约束放置位置对管道的两类周期运动(包括平面周期运动和空间周期运动)及其稳定性的影响规律.首先,在已有文献的基础上,将运动约束对管道的作用模拟成非线性立方弹簧模,得出振动方程.其次,运用Galerkin方法将振动方程离散成常微分方程组,结合基于中心流形—范式理论的投影法与平均法,给出了决定系统定性动力学性质的相关系数(包括临界特征值随流速的变化率及非线性共振项),取模态截断数为6,在几组约束刚度值和约束位置处计算了上述系数,据此考察了运动约束对管道的周期运动的影响,总结出了如下结论：在约束位置取定时增加约束刚度,或在约束刚度取定时增大约束位置至管的固定端的距离,均会使得管道的稳定平面周期运动对应的质量比区间减小,稳定空间周期运动对应的质量比区间增大;约束位置距管的固定端越远,约束刚度的变化对管道动力学行为的影响越明显.最后,对上述通过投影法和平均法得出的结论,本文在特定的质量比处数值求解了原振动方程的6模态Galerkin离散化方程,绘制了位形图、相图和Poincaré映射图,计算了频率,从而验证了相关的分析.     

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.     

On Derivation of Motion Equations for Systems with Non-Holonomic High-Order Program Constraints

The paper develops and discusses the generalization of modeling methods for systems with non-holonomic constraints. The classification of constraints has been revisited and a concept of program constraints introduced. High-order non-holonomic constraints (HONC), as presented in examples, are the generalization of the constraint concept and may, as a constraint class, include many of motion requirements that are put upon mechanical systems. Generalized program motion equations (GPME) that have been derived in the paper can be applied to systems with HONC. Concepts of virtual displacements and a generalized variational principle for high-order constraints are presented. Classical modeling methods for non-holonomic systems based on Lagrange equations with multipliers, Maggi, Appell–Gibbs, Boltzman–Hamel, Chaplygin and others are peculiar cases of GPME. The theory has been illustrated with examples of high-order constraints. Motion equations have been derived for a system subjected to a constraint that programmed a trajectory curvature profile. Efficiency, advantages and disadvantages of GPME have been discussed.     

含摩擦滑移铰及驱动约束多刚体系统数值算法

研究了具有驱动约束及非光滑滑移铰多体系统动力学方程的建模与数值计算方法．将驱动约束视为非定常约束，非光滑滑移铰视为双边定常约束，滑移铰的摩擦模型采用库仑摩擦模型；应用第一类Lagrange方程建立系统的动力学方程，应用距离函数建立滑移铰的约束方程；将线性互补方法和Baumgarte约束稳定化方法引入，以解决滑移铰法向约束力的计算以及约束方程违约问题．最后应用曲柄摇杆机构作为算例，说明该方法的有效性．     

On Nonlinear Yaw Model for Directional Responses of Log Hauling Trucks

A nonlinear yaw model for dynamics of log hauling trucks is developed based on Kane's equations. The relationship between the articulation yaw angles, between the vehicle units, and the sliding length of the drawbar has been investigated using a new procedure and introduced into the governing equations of motion. Analytical expressions of equations of motion for a tractor/jeep/pole-trailer system are obtained. The numerical results for directional responses of the system, under some maneuvers, are checked against measured responses in field tests. Also a comparison of the results for the nonlinear and linear yaw models is carried out and the modifying effects of the nonlinear terms are noted.     

On the First-Order Decoupling of Dynamical Equations of Motion for Elastic Multibody Systems as Applied to a Two-Link Flexible Manipulator

An improved method for deriving elastic generalized coordinatesis considered. Then Kane's equations of motion for multibody systemsconsisting of an arbitrary number of rigid and elastic bodies ispresented. The equations are in general form and are applicable for anydesired holonomic system. Flexibility in choosing generalized speeds interms of generalized coordinate derivatives in Kane's method is used. Itis shown that proper choice of a congruency transformation betweengeneralized coordinate derivatives and generalized speeds leads toequations of motion for holonomic multibody systems consisting of anarbitrary number of rigid and elastic bodies. These equations aredecoupled in first-order terms. In order to show the use of this method,a simple system consisting of a lumped mass, a spring and a clamped-freeelastic beam is modeled. Finally, the numerical implementation ofdecoupling using congruency transformation is discussed and shown viasimulation of a two-degrees-of-freedom flexible robot.     

Simulation of Motion of Constrained Multibody Systems Based on Projection Operator

This paper presents an efficient dynamic formulation for solvingDifferential Algebraic Equations (DAE) by using the notion of orthogonalprojection. Firstly, the constraint equations are expressed explicitlyat acceleration level by using the notion of the orthogonal projection.Secondly, the Lagrangian multiplier is eliminated from the dynamicsequation by the projection operator. Then, the resultant equations areconsolidated into one equation which explicitly correlates theacceleration to the generalized force through a so-called constraint mass matrix. It is proved that the constraint mass matrix isalways invertible and hence the acceleration can be computed in aclosed-form manner even with the presence of redundant constraints or asingular configuration. The equation of motion is given explicitly in arelatively compact form, which can lead to computational efficiency. Italso has a useful physical interpretation, as the component of thegeneralized force contributing to motion dynamics is readily derivedform the formulation. Finally, results obtained from numericalsimulation of motion of a five-bar mechanism is documented.     

带非线性等式约束无迹卡尔曼滤波方法

针对带非线性等式约束的非线性系统的状态估计问题,给出了一种新形式的基于无迹卡尔曼滤波及伪观测手段的处理约束的状态估计方法（SPUKF）。在该方法中原动态系统被虚拟地分离成两个并行的子系统,各时刻的状态估计由基于这两个子系统构建的两套滤波链交替得到。相对于伪观测法中的序贯形式估计器,SPUKF无需事先确定观测及约束的处理次序且能获得更好的估计结果,故可以用来解决序贯方法中观测与约束的处理次序问题。由钟摆运动的实例仿真结果看到,SPUKF不仅有好于序贯形式无迹卡尔曼滤波的估计效果,误差改善比达到22%左右,而且算法运行时间与序贯形式估计器相近。此外,其估计效果还与批处理无迹卡尔曼滤波相当。     

Mathematical programs with complementarity constraints in the context of inverse optimal control for locomotion

In this paper an inverse optimal control problem in the form of a mathematical program with complementarity constraints (MPCC) is considered and numerical experiences are discussed. The inverse optimal control problem arises in the context of human navigation where the body is modelled as a dynamical system and it is assumed that the motions are optimally controlled with respect to an unknown cost function. The goal of the inversion is now to find a cost function within a given parametrized family of candidate cost functions such that the corresponding optimal motion minimizes the deviation from given data. MPCCs are known to be a challenging class of optimization problems typically violating all standard constraint qualifications (CQs). We show that under certain assumptions the resulting MPCC fulfills CQs for MPCCs being the basis for theory on MPCC optimality conditions and consequently for numerical solution techniques. Finally, numerical results are presented for the discretized inverse optimal control problem of locomotion using different solution techniques based on relaxation and lifting.     

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.     

A New Linear Method for Camera Self-Calibration with Planar Motion

We consider the self-calibration (affine and metric reconstruction) problem from images acquired with a camera with unchanging internal parameters undergoing planar motion. The general self-calibration methods (modulus constraint, Kruppa equations) are known to fail with this camera motion. In this paper we give two novel linear constraints on the coordinates of the plane at infinity in a projective reconstruction for any camera motion. In the planar case, we show that the two constraints are equivalent and easy to compute, giving us a linear version of the quartic modulus constraint. Using this fact, we present a new linear method to solve the self-calibration problem with planar motion of the camera from three or more images. This work was partly supported by project BFM2003-02914 from the Ministerio de Ciencia y Tecnología (Spain). Ferran Espuny received the MSc in Mathematics in 2002 from the Universitat de Barcelona, Spain. He is currently a PhD student and associate professor in the Departament d’àlgebra i Geometria at Universitat de Barcelona, Spain. His research, supervised by Dr. José Ignacio Burgos Gil, is focussed on self-calibration and critical motions for both pinhole and generic camera models.     

Principle of Dynamical Balance for Multibody Systems

A new formulation for multibody system dynamics is developed based on the concept of dynamical balance. In particular, we address the problem how to compose two known subsystem dynamics to generate the equations of motion for a composite system. The principle states that dynamical balance should hold between two subsystems, or the so-called d'Alembertian wrenches and torques of two subsystems should balance each other, for composite systems. The notion of body twists and wrenches is utilized to describe the principle. According to the principle, the dynamical balance condition is obtained just by taking the dual expression of the kinematical constraint in terms of the d'Alembertian wrenches and torques of subsystem dynamics. This work was supported by the Korea Research Foundation Grant (KRF-2003-003-D00015).     

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.     

Time‐optimal constant speed motion program for multiple cooperating manipulators

This article presents a systematic approach for synthesizing the time‐optimal constant speed motion program for multiple manipulators moving a commonly held object along a specified Cartesian trajectory. In this approach, the motion program is constructed by using piecewise polynomials to blend the acceleration, constant speed, and deceleration periods. The polynomials are interpolated according to the boundary and continuity conditions to obtain a smooth and continuous profile. With this formulation, it is shown that the final form of the motion program can be established in terms of the initial acceleration, the constant operation speed, and the finial deceleration. The optimum values of these terms to allow the given trajectory to be executed in minimum time are determined based on the parametric dynamic equations of the system and the torque constraints of the actuator. This approach is conceptually straightforward and can be applied to various multirobot systems with nonlinear actuator constraints. ©1999 John Wiley & Sons, Inc.     

On the Equations of Kinematics and Dynamics of Constrained Mechanical Systems

The method of constructing of kinematical and dynamicalequations of mechanical systems, applied to numerical realization, isproposed in this paper. The corresponding difference equations, whichare obtained, give a guarantee of computations with given precision. Theequations of programmed constraints and those of constraintperturbations are defined. The stability of the programmed manifold fornumerical solutions of the kinematical and dynamical equations isobtained by means of corresponding construction of the constraintperturbation equations. The dynamical equations of system withprogrammed constraints are set up in the form of Lagrange equations ingeneralized coordinates. Certain inverse problems of rigid body dynamicsare considered.     

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.     

State-constrained optimal control problems governed by coupled nonlinear wave equations with memory

This work considers the optimal control problems governed by coupled nonlinear wave equations with memory, where the state constraint is a type of integral. First, we investigate the existence of the optimal solution for the optimal control problem, and then we deduce the maximum principle for the optimal solution with state constraint. Moreover, we give a concrete example which is covered by the main result.