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.     

An extended divide-and-conquer algorithm for a generalized class of multibody constraints

An extension to the divide-and-conquer algorithm (DCA) is presented in this paper to model constrained multibody systems. The constraints of interest are those applied to the system due to the inverse dynamics or control laws rather than the kinematically closed loops which have been studied in the literature. These imposed constraints are often expressed in terms of the generalized coordinates and speeds. A set of unknown generalized constraint forces must be considered in the equations of motion to enforce these algebraic constraints. In this paper dynamics of this class of multibody constrained systems is formulated using a Generalized-DCA. In this scheme, introducing dynamically equivalent forcing systems, each generalized constraint force is replaced by its dynamically equivalent spatial constraint force applied from the appropriate parent body to the associated child body at the connecting joint without violating the dynamics of the original system. The handle equations of motion are then formulated considering these dynamically equivalent spatial constraint forces. These equations in the GDCA scheme are used in the assembly and disassembly processes to solve for the states of the system, as well as the generalized constraint forces and/or Lagrange multipliers.     

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.     

Design sensitivity analysis and optimization of dynamic response

The paper presents methods of design sensitivity analysis and optimization of dynamic response of mechanical and structural systems. A key feature of the paper is the development of procedures to handle point-wise state variable constraints. Difficulties with a previous treatment where such constraints were transformed to equivalent integral constraints are noted and explained from theoretical as well as engineering standpoints. An alternate treatment of such constraints is proposed, developed and evaluated. In this treatment each point-wise state variable constraint is replaced by several constraints that are imposed at all the local max-points for the original constraint function. The differential equations of motion are formulated in the first-order form so as to handle more general problems. The direct differentiation and adjoint variable methods of design sensitivity analysis to deal with the point-wise constraints are presented. With the adjoint variable methods, there are two ways of calculating design sensitivity coefficients. The first approach uses an impulse load and the second approach uses a step load for the corresponding adjoint equation. Since the adjoint variable methods are better for a large class of problems, an efficient computational algorithm with these methods is presented in detail. Optimum results for several problems are obtained and compared with those available in the literature. The new formulation works extremely well as precise optimum designs are obtained.     

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.     

A parallel Hamiltonian formulation for forward dynamics of closed-loop multibody systems

This paper presents a novel recursive divide-and-conquer formulation for the simulation of complex constrained multibody system dynamics based on Hamilton’s canonical equations (HDCA). The systems under consideration are subjected to holonomic, independent constraints and may include serial chains, tree chains, or closed-loop topologies. Although Hamilton’s canonical equations exhibit many advantageous features compared to their acceleration based counterparts, it appears that there is a lack of dedicated parallel algorithms for multi-rigid-body system dynamics based on the Hamiltonian formulation. The developed HDCA formulation leads to a two-stage procedure. In the first phase, the approach utilizes the divide and conquer scheme, i.e., a hierarchic assembly–disassembly process to traverse the multibody system topology in a binary tree manner. The purpose of this step is to evaluate the joint velocities and constraint force impulses. The process exhibits linear \(O(n)\) (\(n\) – number of bodies) and logarithmic \(O(\log_{2}{n})\) numerical cost, in serial and parallel implementations, respectively. The time derivatives of the total momenta are directly evaluated in the second parallelizable step of the algorithm. Sample closed-loop test cases indicate very small constraint violation errors at the position and velocity level as well as marginal energy drift without any additional form of constraint stabilization techniques involved in the solution process. The results are comparatively set against more standard acceleration based Featherstone’s DCA approach to indicate the performance of the HDCA algorithm.     

Solvability of reactions in rigid multibody systems with redundant nonholonomic constraints

The problem of calculating joint reaction forces in rigid body mechanisms with redundant constraints, both geometric and nonholonomic, is discussed. When constraint equations are dependent, some of the constraint reactions are unsolvable, i.e., cannot be uniquely determined using a rigid body model, whereas some others may be solvable. In this paper, analytic conditions, which must be fulfilled to obtain unique values of selected reaction forces in the presence of dependent nonholonomic constraints, are presented and proven. The concept of direct sum, known from linear algebra, is exploited. These purely mathematical conditions are followed by numerical methods that enable detection of constraints with uniquely solvable reactions. Similar conditions and methods were proposed earlier for holonomic systems. In this contribution, they are generalized to the case of linear nonholonomic constraints. An example of constraint reactions solvability analysis, for a mechanism subjected to redundant nonholonomic constraints, is presented.     

Control of discrete event systems with respect to strict duration: Supervision of an industrial manufacturing plant

In this paper, we propose a (max, +)-based method for the supervision of discrete event systems subject to tight time constraints. Systems under consideration are those modeled as timed event graphs and represented with linear (max, +) state equations. The supervision is addressed by looking for solutions of constrained state equations associated with timed event graph models. These constrained state equations are derived by reducing duration constraints to elementary constraints whose contributions are injected in the system’s state equations. An example for supervisor synthesis is given for an industrial manufacturing plant subject to a strict temporal constraint, the thermal treatment of rubber parts for the automotive industries. Supervisors are calculated and classified according to their performance, considering their impact on the production throughput.     

Singularity-free simulation of closed loop multibody systems by using null space of Jacobian matrix

Numerical simulation of closed loop multibody systems is associated with numerical solution of equations of motion which are, in general, in the form of DAE’s index-3 systems. For assuring continuous simulation, one should overcome some difficulties such as stabilization of the constraint equations, singular configuration of the system. In this paper, the system equations of motion with the Lagrange multipliers is rewritten by introducing generalized reaction forces. The combination with the condition of ideality of constraints leads to the system of equations which can be solved by numerical techniques smoothly, even over singular positions. Based on the new criterion of ideality of constraints, which relates generalized reaction forces and the null space matrix of Jacobian matrix, it is possible also to remove reaction forces and use only the reduced system of equations with null space matrix for passing singular positions. In order to prevent the constraint equations from the accumulated errors of integral time, the method of position and velocity projection has been exploited. Some numerical experiments are carried out to verify the proposed approach.     

Impulsive motion of multibody systems

This article uses the piecewise model and Kane’s method to present a procedure for studying impulsive motion of multibody systems. Impulsive motion occurs when the system is subject to either impulsive forces or impulsive constraints, or when subjected to both simultaneously. The Appellian classification of impulsive constraints and the corresponding equations of impulsive motion of the multibody system are discussed. The governing equations are derived based upon multibody formulation procedures developed by Huston. Constraint impulses associated with finite and impulsive constraints are incorporated into impact dynamical equations through the impulsive Lagrange multipliers. The kinetic energy change of the scleronomic multibody system due to the impact is derived. Newton’s impact law is treated as an impulsive constraint equation to study single-point frictionless collision between two multibody systems. Several examples are used to demonstrate and validate the procedure.     

Simplifying Subtyping Constraints: A Theory

This paper offers a theoretical study of constraint simplification, a fundamental issue for the designer of a practical type inference system with subtyping. In the simpler case where constraints are equations, a simple isomorphism between constrained type schemes and finite state automata yields a complete constraint simplification method. Using it as a guide for the intuition, we move on to the case of subtyping, and describe several simplification algorithms. Although no longer complete, they are conceptually simple, efficient, and very effective in practice. Overall, this paper gives a concise theoretical account of the techniques found at the core of our type inference system. Our study is restricted to the case where constraints are interpreted in a non-structural lattice of regular terms. Nevertheless, we highlight a small number of general ideas, which explain our algorithms at a high level and may be applicable to a variety of other systems.     

Discussion of the Gear–Gupta–Leimkuhler method for impacting mechanical systems

In multibody simulation, the Gear–Gupta–Leimkuhler method or stabilized index 2 formulation for persistent contacts enforces constraints on position and velocity level at the same time. It yields a robust numerical discretization of differential algebraic equations avoiding the drift-off effect and is often more effective than decreasing the time step size to preserve geometric characteristics. In this work, we carry over these benefits to impacting mechanical systems with unilateral constraints. For this kind of a mechanical system, adding the position level constraint to an (event-capturing) timestepping scheme on velocity level even maintains physical consistency of the impulsive discretization. Hence, we propose a timestepping scheme based on Moreau’s midpoint rule, which enables to achieve not only compliance of the impact law, but also of the nonpenetration constraint. The choice of a decoupled and consecutive evaluation of the respective constraints can be interpreted as a not energy-consistent coordinate projection to the nonpenetration constraint at the end of each time step. It is the implicit coupling of position and velocity level, which yields satisfactory results. An implicit evaluation of the right hand side improves stability properties without additional cost. With the prox function formulation, the overall set of nonsmooth equations is solved by a nonsmooth Newton method. Results from simulations of a slider-crank mechanism with unilateral constraints demonstrate the capability of our approach.     

Optimal approximation of high-order systems subject to polynomial inputs

In this paper the optimal approximation of a high-order system, subject to polynomial inputs, is investigated.

It is shown that the constraints on the asymptotic behaviour are easily taken into account as structure constraints. Thus the problem is reduced to a minimization without constraint. Expressions for the gradient of the criterion are then given in the case of a unit step response.

An extension to the case of unstable systems is also proposed.     

装配位置约束建模及求解

以基本几何约束组合统一表达装配约束,为提高求解效率,研究了姿态约束和位置约束的可解耦情况下位置约束的解析求解.将基本位置约束映射为移动空间并以参数方程表达,通过移动空间的增量解析求交,满足约束;在姿态约束和位置约束的不可解耦情况,联立基本约束进行整体数值法求解.文中方法保持了基本约束表达的独立性,适合于欠约束系统和完整约束系统.     

一种能量收集嵌入式系统自适应调度算法

能量收集嵌入式系统(energy harvesting embedded system,简称EHES)的任务调度算法需要考虑能量收集单元的能量输出、能量存储单元的能量水平和能量消耗单元的能耗.实时任务在满足能量约束的条件下,才可能满足时间约束.在这个背景下,传统固定优先级调度算法不再适用于EHES.提出一种基于分组的自适应任务调度算法,它能根据能量收集单元由于能量输出的不确定性而造成的非能量约束情况和能量约束情况,自适应地选择任务调度算法.在非能量约束的情况下,减少任务抢占次数,增强任务的可调度性;在能量约束情况下,减少电池模式切换次数,提高能量存储单元的平均能量水平,从而降低系统能量约束.在一个可进行大范围任务集合仿真的实验环境下对提出的算法进行验证,并将基于分组的自适应调度算法与现有的两个经典算法进行了对比.     

Multibody dynamics with redundant constraints and singular mass matrix: existence, uniqueness, and determination of solutions for accelerations and constraint forces

This paper addresses some important issues for multibody dynamics; issues that are basic and really not too difficult to solve, but rarely considered in the literature. The aim of this paper is to contribute to the resolution and clarification of these topics in multibody dynamics. There are many formulations for determining the equations of motion in constrained multibody systems. This paper will focus on three of the most important methods: the Lagrange equations of the first kind, the null space method and the Maggi equations. In all cases we consider singular inertia matrices and redundant constraint equations. We assume that the inertia matrix is positive-semidefinite (symmetric) and that the constraint equations may be redundant but always consistent. It is demonstrated that the aforementioned dynamic formulations lead to the same three mathematical conditions of existence and uniqueness of solutions, conditions that have at the same time a clear physical meaning. We conclude that the mathematical problem always has a solution if the physical problem is well conditioned. This paper also addresses the problem of determining the constraint forces in the case of redundant constraints. This problem is examined from a broad perspective. We will present several examples and a simple method to find practical solutions in cases where the forces of constraint are undetermined. The method is based on the weighted minimum norm condition. A physical interpretation of this minimum norm condition is provided in detail for all examples. In some cases a comparison with the results obtained by considering flexibility is included.     

Methods for constraint violation suppression in the numerical simulation of constrained multibody systems – A comparative study

Multibody systems are frequently modeled as constrained systems, and the arising governing equations incorporate the closing constraint equations at the acceleration level. One consequence of accumulation of integration truncation errors is the phenomenon of violation of the lower-order constraint equations by the numerical solutions to the governing equations. The constraint drift usually tends to increase in time and may spoil reliability of the simulation results. In this paper a comparative study of three methods for constraint violation suppression is presented: the popular Baumgarte’s constraint violation stabilization method, a projective scheme for constraint violation elimination, and a novel scheme patterned after that proposed recently by Braun and Goldfarb [D.J. Braun, M. Goldfarb, Eliminating constraint drift in the numerical simulation of constrained dynamical systems, Comput. Meth. Appl. Mech. Engrg., 198 (2009) 3151–3160]. The methods are confronted with respect to simplicity in applications, numerical effectiveness and influence on accuracy of the constraint-consistent motion.     

Optimal feedback controls in deterministic two-machine flowshopswith finite buffers

This paper deals with an optimal control problem of deterministic two-machine flowshops. Since the sizes of both internal and external buffers are practically finite, the problem is one with state constraints. The Hamilton-Jacobi-Bellman (HJB) equations of the problem involve complicated boundary conditions due to the presence of the state constraints, and as a consequence the usual “verification theorem” may not work for the problem. To overcome this difficulty, it is shown that any function satisfying the HJB equations in the interior of the state constraint domain must be majorized by the value function. The main techniques employed are the “constraint domain approximation” approach and the “weak-Lipschitz” property of the value functions developed in preceding papers. Based on this, an explicit optimal feedback control for the problem is obtained     

Linear constraint equations for continuous support conditions in finite element analysis

Discretised structural models such as by finite elements imply discretised support conditions. In some cases such as plates on elastic foundation or slabs on large interacting columns an improved formulation of the continuous support conditions is desirable. This can be achieved by means of linear constraint equations. The numerical treatment of linear constraints is discussed for the method of elimination of variables as well as for the method of Lagrange multipliers. Then specific constraint equations for different accuracy requirements are derived, which can be used to constrain rectangular flat shell elements of arbitrary shape functions. These constraints introduce six generalized displacements according to the rigid body motions of the element and transmit the corresponding generalized reactions on the nodal degrees of freedom in a way consistent with distributed reactions. The effect on the strain energy of a square shell element is shown for the different constraint equations. As an application, the linear constraints are used to represent the continuous interaction of columns with the plate in a flat slab structure. Comparison of the finite element solutions with analytical results shows that the derived constraint equations allow a considerably improved formulation of continuous support conditions.     

A Logical Approach to Quality of Service Specification in Video Databases

Quality of Service (QoS) is defined as a set of perceivable attributes expressed in a user-friendly language with parameters that may be objective or subjective. Objective parameters are those related to a particular service and are measurable and verifiable. Subjective parameters are those based on the opinions of the end-users. We believe that quality of service should become an integral part of multimedia database systems and users should be able to query by requiring a quality of service from the system. The specification and enforcement of QoS presents an interesting challenge in multimedia systems development. A deal of effort has been done on QoS specification and control at the system and the network levels, but less work has been done at the application/user level. In this paper, we propose a language, in the style of constraint database languages, for formal specification of QoS constraints. The satisfaction by the system of the user quality requirements can be viewed as a constraint satisfaction problem, and the negotiation can be viewed as constraint optimization. We believe this paper represents a first step towards the development of a database framework for quality of service management in video databases. The contribution of this paper lies in providing a logical framework for specifying and enforcing quality of service in video databases. To our knowledge, this work is the first from a database perspective on quality of service management.