Dynamics of Flexible Multibody Systems with Non-Holonomic Constraints: A Finite Element Approach

In this article it is shown how non-holonomic constraints can beincluded in the formulation of the dynamic equations of flexiblemultibody systems. The equations are given in state space formwith the degrees of freedom, their derivatives and the kinematiccoordinates as state variables, which circumvents the use ofLagrangian multipliers. With these independent state variables forthe system the derivation of the linearized equations of motion isstraightforward. The incorporation of the method in a finiteelement based program for flexible multibody systems is discussed.The method is illustrated by three examples, which show, amongother things, how the linearized equations can be used to analysethe stability of a nominal steady motion.     

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.     

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.     

General Properties and Termination Conditions for Soft Constraint Propagation

Soft constraints based on semirings are a generalization of classical constraints, where tuples of variables' values in each soft constraint are associated to elements from an algebraic structure called semiring. This framework is able to express, for example, fuzzy, classical, weighted, valued and over-constrained constraint problems.Classical constraint propagation has been extended and adapted to soft constraints by defining a schema for soft constraint propagation [8]. On the other hand, in [1–3] it has been proven that most of the well known constraint propagation algorithms for classical constraints can be cast within a single schema.In this paper we combine these two schemas and we provide a more general framework where the schema of [3] can be used for soft constraints. In doing so, we generalize the concept of soft constraint propagation, and we provide new sufficient and independent conditions for its termination.     

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.     

Fuzzy Control for Equilibrium and Roll-Angle Tracking of an Unmanned Bicycle

This study presents steady turning motion and roll-angle tracking controls for an unmanned bicycle. The equations of motion describing the dynamics of a bicycle are developed using Lagrange's equations for quasi-coordinates. Pure rolling without slipping constraints between the ground and two wheels are also considered in this model. These constraints introduce four holonomic and four non-holonomic constraint equations to the model. For the developed bicycle dynamics, one PID and one fuzzy controller that create steering torque are derived to recover the balance of the bicycle from a near-fall state. Furthermore, another fuzzy controller is added for controlling the bicycle to a desired roll angle which leads to its steady circular motion. The bicycle can track a given roll angle while maintaining its balance. The effectiveness of the control schemes is proved by simulation results.     

Periodic motion planning for virtually constrained Euler–Lagrange systems

The paper suggests an explicit form of a general integral of motion for some classes of dynamical systems including n-degrees of freedom Euler–Lagrange systems subject to (n-1) virtual holonomic constraints. The knowledge of this integral allows to extend the classical results due to Lyapunov for detecting a presence of periodic solutions for a family of second order systems, and allows to solve the periodic motion planning task for underactuated Euler–Lagrange systems, when there is only one not directly actuated generalized coordinate. As an illustrative example, we have shown how to create a periodic oscillation of the pendulum for a cart–pendulum system and how then to make them orbitally exponentially stable following the machinery developed in [A. Shiriaev, J. Perram, C. Canudas-de-Wit, Constructive tool for an orbital stabilization of underactuated nonlinear systems: virtual constraint approach, IEEE Trans. Automat. Control 50 (8) (2005) 1164–1176]. The extension here also considers time-varying virtual constraints.     

Dynamics and Stability of Blind Grasping of a 3-Dimensional Object under Non-holonomic Constraints

A mathematical model expressing the motion of a pair of multi-DOF robot fingers with hemi-spherical ends, grasping a 3-D rigid object with parallel flat surfaces, is derived, together with non-holonomic constraints. By referring to the fact that humans grasp an object in the form of precision prehension, dynamically and stably by opposable forces, between the thumb and another finger (index or middle finger), a simple control signal constructed from finger-thumb opposition is proposed, and shown to realize stable grasping in a dynamic sense without using object information or external sensing (this is called "blind grasp" in this paper). The stability of grasping with force/torque balance under non-holonomic constraints is analyzed on the basis of a new concept named "stability on a manifold". Preliminary simulation results are shown to verify the validity of the theoretical results.     

Motion planning for mobile manipulators with a non-holonomic constraint using the FSP (full space parameterization) method

The efficient utilization of the motion capabilities of mobile manipulators, i.e., manipulators mounted on mobile platforms, requires the resolution of the kinematically redundant system formed by the addition of the degrees of freedom (DOF) of the platform to those of the manipulator. At the velocity level, the linearized Jacobian equation for such a redundant system represents an underspecified system of algebraic equations, which can be subject to a varying set of contraints such as a non-holonomic constraint on the platform motion, obstacles in the workspace, and various limits on the joint motions. A method, which we named the Full Space Parameterization (FSP), has recently been developed to resolve such underspecified systems with constraints that may vary in time and in number during a single trajectory. In this article, we first review the principles of the FSP and give analytical solutions for constrained motion cases with a general optimization criterion for resolving the redundancy. We then focus on the solutions to (1) the problem introduced by the combined use of prismatic and revolute joints (a common occurrence in practical mobile manipulators), which makes the dimensions of the joint displacement vector components non-homogeneous, and (2) the treatment of a non-holonomic constraint on the platform motion. Sample implementations on several large-payload mobile manipulators with up to 11 DOF are discussed. Comparative trajectories involving combined motions of the platform and manipulator for problems with obstacle and joint limit constraints, and with non-holonomic contraints on the platform motions, are presented to illustrate the use and efficiency of the FSP approach in complex motion planning problems. © 1996 John Wiley & Sons, Inc.     

一种基于知识树和约束的柔性活动动态细化方法

柔性工作流在应对业务建模过程中的动态不确定因素、提高工作流系统的柔性具有巨大的优势,然而,柔性活动的动态细化一直是柔性工作流建模和应用的一个难点。因此,提出一种基于知识树和约束的柔性活动动态细化方法。该方法以知识树的包含和泛化关系作为启发信息,以活动选取约束和时序约束作为指导和校验,实现柔性活动的动态细化。在介绍了知识树及其蕴含关系以及活动选取约束和时序约束规则的基础上,给出了柔性活动的动态细化算法,描述了活动选取校验和时序约束校验算法。最后给出了算法的实现和实例分析,其结果表明了所提方法的有效性,能够很好地解决柔性活动的细化问题。     

Nicolog: A simple yet powerful cc(FD) language

In this paper, we describe Nicolog, a language with capabilities similar to recently developed constraint logic programming (CLP) languages such as CLP(BNR), clp(FD), and cc(FD). Central to Nicolog are projection constraints (PCs), a sublanguage for compiling and optimizing constraint propagation in numeric and Boolean domains. PCs are an interesting generalization of the indexical constraints introduced in cc(FD) and also found in clp(FD). Nicolog compiles a very general class of built-in constraints into equivalent sets of PCs, allowing an arbitrary mixture of integer (easily extensible to real) and Boolean operations. Nicolog also lets the user program PCs directly, making it possible to implement new sophisticated propagation procedures. We show that PCs are a simple, efficient, and flexible way to implement most of the propagation procedures possible in other FD CLP systems. These include procedures for cardinality, constructive disjunction, implication, and mixed Boolean/numeric constraints. Empirical results with a simple prototype Nicolog implementation based on the WAM architecture show it can solve hard problems with speed comparable to the fastest existing CLP systems.     

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.     

Efficient Symbolic Analysis for Parallelizing Compilers and Performance Estimators

Symbolic analysis is of paramount importance for parallelizing compilers and performance estimators to examine symbolic expressions with program unknowns such as machine and problem sizes and to solve queries based on systems of constraints (equalities and inequalities). This paper describes novel techniques for counting the number of solutions to a system of constraints, simplifying systems of constraints, computing lower and upper bounds of symbolic expressions, and determining the relationship between symbolic expressions. All techniques target wide classes of linear and non-linearsymbolic expressions and systems of constraints. Our techniques have been implemented and are used as part of a parallelizing compiler and a performance estimator to support analysis and optimization of parallel programs. Various examples and experiments demonstrate the effectiveness of our symbolic analysis techniques.     

Nonminimal Kane's Equations of Motion for Multibody Dynamical Systems Subject to Nonlinear Nonholonomic Constraints

Nonholonomic constraint equations that are nonlinear in velocities are incorporated with Kane's dynamical equations by utilizing the acceleration form of constraints, resulting in Kane's nonminimal equations of motion, i.e. the equations that involve the full set of generalized accelerations. Together with the kinematical differential equations, these equations form a state-space model that is full-order, separated in the derivatives of the states, and involves no Lagrange multipliers. The method is illustrated by using it to obtain nonminimal equations of motion for the classical Appell–Hamel problem when the constraints are modeled as nonlinear in the velocities. It is shown that this fictitious nonlinearity has a predominant effect on the numerical stability of the dynamical equations, and hence it is possible to use it for improving the accuracy of simulations. Another issue is the dynamics of constraint violations caused by integration errors due to enforcing a differentiated form of the constraint equations. To solve this problem, the acceleration form of the constraint equations is augmented with constraint stabilization terms before using it with the dynamical equations. The procedure is illustrated by stabilizing the constraint equations for a holonomically constrained particle in the gravitational field.     

Contouring Control for Multi‐Axis Motion Systems With Holonomic Constraints: Theory and Application to a Parallel System

In this paper, the contouring control problem for the constrained multi‐axis motion system is studied. The method of equivalent errors, previously proposed for unconstrained motion systems, is generalized to the system with holonomic constraints. It is shown that the method can be applied to the constrained system provided that the constraints satisfy a proper condition. Because of the constraints, the states in the control law are not completely independent. The unavailable states can be estimated using linear approximation from the constraint equations. As an illustrative example, the proposed method is applied to a parallel motion system with complicated dynamics. A contouring controller is designed using the method of equivalent errors incorporated with integral sliding mode control. Simulation results for contouring circular, elliptic, and square paths verify the effectiveness of the proposed method.     

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.     

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.     

Identifying redundancy in multi-dimensional knapsack constraints based on surrogate constraints

Redundancy identification techniques play an important role in improving the solvability of a linear program. In this paper, we address the redundancy in multi-dimensional knapsack constraints by proposing a new redundancy identification method. The proposed method is based on the constraint intercepts of Paulraj, Chellappan, and Natesan [A heuristic approach for identification of redundant constraints in linear programming models, Int. J. Comput. Math. 83 (2006), pp. 675–683] and surrogate constraints. In it, feasibility problems are constructed in order to determine the redundancy of the constraints, and are solved by a heuristic algorithm, which is developed to check the redundancy fast. The results of computational experiments show that the proposed method may be used in a preprocessing stage in order to reduce the number of knapsack constraints.     

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.