Constraint Drift Research Articles

A Lagrangian method for constrained dynamics in tensegrity systems with compressible bars

This paper presents a Lagrangian approach to simulating multibody dynamics in a tensegrity framework with an ability to tackle holonomic constraint violations in an energy-preserving scheme. Governing equations are described using non-minimum coordinates to simplify descriptions of the structure’s kinematics. To minimize constraint drift arising from this redundant system, the direct correction method has been employed in conjunction with a novel energy-correcting scheme that treats the total mechanical energy of the system as a supplementary constraint. The formulation has been extended to allow tensegrity structures with compressible bars, allowing for further discussion on potential choices for softer bar materials. The benchmark example involving a common tensegrity structure demonstrates the superiority of the presented formulation over Simscape Multibody in terms of motion accuracy as well as energy conservation. The effectiveness of the energy correction scheme is found to be increasing with the extent of deformations in the structure.

A robust nonsmooth generalized-$\alpha $ scheme for flexible systems with impacts

The aim of this work is the development of a robust and accurate time integrator for the simulation of the dynamics of multibody systems composed of rigid and/or flexible bodies subject to frictionless contacts and impacts. The integrator is built upon a previously developed nonsmooth generalized-$\alpha $ scheme time integrator which was able to deal well with nonsmooth dynamical problems avoiding any constraint drift phenomena and capturing vibration effects without introducing too much numerical dissipation. However, when dealing with problems involving nonlinear bilateral constraints and/or flexible elements, it is necessary to adopt small time-step sizes to ensure the convergence of the numerical scheme. In order to tackle these problems more efficiently, a fully decoupled version of the nonsmooth generalized-$\alpha $ method is proposed in this work, avoiding these inconveniences. Several examples are considered to assess its accuracy and robustness.

Modeling and simulation of longitudinal dynamics coupled with clutch engagement dynamics for ground vehicles

During the engagement of the dry clutch in automotive transmissions, clutch judder may occur. Vehicle suspension and engine mounts couple the torsional and longitudinal models, leading to oscillations of the vehicle body that are perceived by the driver as poor driving quality. This paper presents an effective formulation for the modeling and simulation of longitudinal dynamics and powertrain torsional dynamics of the vehicle based on non-smooth dynamics of multibody systems. In doing so friction forces between wheels and the road surface are modeled along with friction torque in the clutch using Coulomb’s friction law. First, bilateral constraint equations of the system are derived in Cartesian coordinates and the dynamical equations of the system are developed using the Lagrange multiplier technique. Complementary formulations are proposed to determine the state transitions from stick to slip between wheels and road surface and from the clutch. An event-driven scheme is used to represent state transition problem, which is solved as a linear complementarity problem (LCP), with Baumgarte’s stabilization method applied to reduce constraint drift. Finally, the numerical results demonstrate that the modeling technique is effective in simulating the vehicle dynamics. Using this method stick-slip transitions between driving wheel and the road surface and from the clutch, as a form of clutch judder, are demonstrated to occur periodically for certain values of the parameters of input torque from engine, and static and dynamic friction characteristics of tire/ground contact patch and clutch discs.

A systematic approach for designing analytical dynamics and servo control of constrained mechanical systems

A systematic approach for designing analytical dynamics and servo control of constrained mechanical systems is proposed. Fundamental equation of constrained mechanical systems is first obtained according to Udwadia-Kalaba approach which is applicable to holonomic and nonholonomic constrained systems no matter whether they satisfy the D'Alember's principle. The performance specifications are modeled as servo constraints. Constraint-following servo control is used to realize the servo constraints. For this inverse dynamics control problem, the determination of control inputs is based on the Moore-Penrose generalized inverse to complete the specified motion. Secondorder constraints are used in the dynamics and servo control. Constraint violation suppression methods can be adopted to eliminate constraint drift in the numerical simulation. Furthermore, this proposed approach is applicable to not only fully actuated but also underactuated and redundantly actuated mechanical systems. Two-mass spring system and coordinated robot system are presented as examples for illustration.

Method for modeling and simulation of aircraft taxiing with unilateral and non-holonomic constraints

The motion of the aircraft taxiing along the ground was related to wind load and asymmetric braking torque. The method for modeling and simulation of the aircraft was presented on the basis of the dynamics of the non-smooth multibody system with non-holonomic constraints in order to analyze the dynamic behavior of the aircraft taxiing along the ground. The aircraft was treated as the multibody system which consisted of a rigid fuselage,two wheels of main landing gears and a wheel of nose landing gear. Two wheels of main landing gears rolled on the ground without slip. The lateral slip of the wheel on the nose landing gear was taken into account when the aircraft taxied along the ground. The dynamic equations of the system were obtained by Routh 's equations and the constraint-stabilized method to reduce constraint drift. The frictional model between the wheels and the ground was the model of Coulomb friction which was described by the set value of function to determine whether the wheel of the nose landing gear slipped in lateral direction or not. Finally,numerical examples were presented to analyze the dynamic behavior of the aircraft undertook the wind load and the asymmetric braking torque.

Modeling and analysis of rigid multibody systems with driving constraints and frictional translation joints

An approach is proposed for modeling and analyses of rigid multibody systems with frictional translation joints and driving constraints. The geometric constraints of translational joints with small clearance are treated as bilateral constraints by neglecting the impact between sliders and guides. Firstly, the normal forces acting on sliders, the driving constraint forces (or moments) and the constraint forces of smooth revolute joints are all described by complementary conditions. The frictional contacts are characterized by a setvalued force law of Coulomb’s dry friction. Combined with the theory of the horizontal linear complementarity problem (HLCP), an event-driven scheme is used to detect the transitions of the contact situation between sliders and guides, and the stick-slip transitions of sliders, respectively. And then, all constraint forces in the system can be computed easily. Secondly, the dynamic equations of multibody systems are written at the acceleration-force level by the Lagrange multiplier technique, and the Baumgarte stabilization method is used to reduce the constraint drift. Finally, a numerical example is given to show some non-smooth dynamical behaviors of the studied system. The obtained results validate the feasibility of algorithm and the effect of constraint stabilization.

Modeling and simulation of the nonsmooth planar rigid multibody systems with frictional translational joints

The main purpose of this paper is to present a modeling and simulation method for the rigid multibody system with frictional translational joints. The small clearance between a slider and guide is considered. The geometric constraints of the translational joints are treated as bilateral constraints and the impacts between sliders and guides are neglected when the clearance sizes of the translational joints are very small. The contact situations of the normal forces acting on the sliders are described by inequalities and complementarity conditions, while the frictional contacts are characterized by a set-valued force law of the type of Coulomb’s law for dry friction. The dynamic equations of the multibody systems with normal and tangential contact forces are written on the acceleration-force level using the Lagrange multiplier technique. The problem of the transitions of the contact situation of the normal forces acting on sliders and the transitions of the stick-slip of the sliders in the system is formulated as a horizontal linear complementarity problem (HLCP), which is solved by event-driven method. Baumgarte’s stabilization method is used to decrease the constraint drift. Finally, two typical mechanisms are considered as demonstrative application examples. The numerical results obtained show some dynamical behaviors of the systems with frictional translational joints and constraint stabilization effect.

A PID Type Constraint Stabilization Method for Numerical Integration of Multibody Systems

The dynamic equations of motion of the constrained multibody mechanical system are mixed differential-algebraic equations (DAEs). The numerical solution of the DAE systems solved using ordinary-differential equation (ODE) solvers may suffer from constraint drift phenomenon. To solve this problem, Baumgarte proposed a constraint stabilization method in which a position and velocity terms were added in the second derivative of the constraint equation. Baumgarte’s method is a proportional-derivative (PD) type controller design. In this paper, an Iintegrator controller is included to form a proportional-integral-derivative (PID) controller so that the steady state error of the numerical integration can be reduced. Stability analysis methods in the digital control theory are used to find out the correct choice of the coefficients for the PID controller.

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.

Eliminating constraint drift in the numerical simulation of constrained dynamical systems

By means of the Udwadia–Kalaba approach we propose an explicit equation of constrained motion developed to simulate constrained dynamical systems without error accumulation due to constraint drift. The basic idea is to embed a small virtual force and a small virtual impulse to the equation of motion, in order to avoid the drift typically experienced in constrained multibody simulations. The embedded correction terms are selected to minimally alter the dynamics in an acceleration and kinetic energy norm sense. The formulation allows one to use a standard ODE solver, avoiding the need for iterative constraint stabilization. The equation is based on the pseudoinverse of a constraint matrix such that it can be used under redundant constraints and kinematic singularities. The proposed method takes into account the finite word-length of the computational environment, and also accommodates possibly inconsistent initial conditions.

A Discussion of Low-Order Numerical Integration Formulas for Rigid and Flexible Multibody Dynamics

The premise of this work is that the presence of high stiffness and/or frictional contact/impact phenomena limits the effective use of high order integration formulas when numerically investigating the time evolution of real-life mechanical systems. Producing a numerical solution relies most often on low-order integration formulas of which the paper investigates three alternatives: Newmark, HHT, and order 2 BDFs. Using these methods, a first set of three algorithms is obtained as the outcome of a direct index-3 discretization approach that considers the equations of motion of a multibody system along with the position kinematic constraints. The second batch of three algorithms draws on the HHT and BDF integration formulas and considers, in addition to the equations of motion, both the position and velocity kinematic constraint equations. Numerical experiments are carried out to compare the algorithms in terms of several metrics: (a) order of convergence, (b) energy preservation, (c) velocity kinematic constraint drift, and (d) efficiency. The numerical experiments draw on a set of three mechanical systems: a rigid slider-crank, a slider-crank with a flexible body, and a seven body mechanism. The algorithms investigated show good performance in relation to the asymptotic behavior of the integration error and, with one exception, result in comparable CPU simulation times with a small premium being paid for enforcing the velocity kinematic constraints.

Review of Contemporary Approaches for Constraint Enforcement in Multibody Systems

A hallmark of multibody dynamics is that most formulations involve a number of constraints. Typically, when redundant generalized coordinates are used, equations of motion are simpler to derive but constraint equations are present. Approaches to dealing with high index differential algebraic equations, based on index reduction techniques, are reviewed and discussed. Constraint violation stabilization techniques that have been developed to control constraint drift are also reviewed. These techniques are used in conjunction with algorithms that do not exactly enforce the constraints. Control theory forms the basis for a number of these methods. Penalty based techniques have also been developed, but the augmented Lagrangian formulation presents a more solid theoretical foundation. In contrast to constraint violation stabilization techniques, constraint violation elimination techniques enforce exact satisfaction of the constraints, at least to machine accuracy. Finally, as the finite element method has gained popularity for the solution of multibody systems, new techniques for the enforcement of constraints have been developed in that framework. The goal of this paper is to review the features of these methods, assess their accuracy and efficiency, underline the relationship among the methods, and recommend approaches that seem to perform better than others.

Adaptive realization of desired constraint stabilization dynamics in the control of multibody systems

This paper presents a novel way of stabilizing the constraint drift dynamics in the numerical simulation of multibody systems. This formulation is applicable for a large class of uncertain mechanical systems described by nonlinear differential algebraic equations that are subject to holonomic constraints. In the absence of uncertainty in the system inertia parameters, it is possible to develop stabilization relationships that suppress the accumulation of error in the constraint equations during the time integration process. In order to account for ignorance in the parameters, we propose a model reference adaptive control scheme that ensures the asymptotic realization of the desired (reference) constraint violation dynamics. Special attention is given to the case of redundantly actuated systems, as is typical for robotics. For this class of problems, we direct special attention to optimization criteria that achieve any desired manoeuvre using minimum control effort and coordination between the redundant set of actuators. An example application demonstrates the effectiveness and practicality of the proposed adaptive control formulation.

Smallest g-supersolution with constraint

In this paper the existence and uniqueness of the smallest g-supersolution for BSDE is discussed in the case without Lipschitz condition imposing on both constraint function and drift coefficient in the different method from the one with Lipschitz condition. Then by considering (ξ, g) as a parameter of BSDE, and (ξα, gα) as a class of parameters for BSDE, where α belongs to a set Open image in new window , for every Open image in new window there exists a pair of solution {Ya, Za} for the BSDE, the properties of Open image in new window which is also a solution for some BSDE is studied. This result may be used to discuss optimal problems with recursive utility.