Baumgarte Stabilization Research Articles

Implicit co‐simulation methods: Stability and convergence analysis for solver coupling approaches with algebraic constraints

The analysis of the numerical stability of co‐simulation methods with algebraic constraints is subject of this manuscript. Three different implicit coupling schemes are investigated. The first method is based on the well‐known Baumgarte stabilization technique. Basis of the second coupling method is a weighted multiplier approach. Within the third method, a classical projection technique is applied. The three methods are discussed for different approximation orders. Concerning the decomposition of the overall system into subsystems, we consider all three possible approaches, i.e. force/force‐, force/displacement‐ and displacement/displacement‐decomposition. The stability analysis of co‐simulation methods with algebraic constraints is inherently related to the definition of a test model. Bearing in mind the stability definition for numerical time integration schemes, i.e. Dahlquist's stability theory based on the linear single‐mass oscillator, a linear two‐mass oscillator is used here for analyzing the stability of co‐simulation methods. The two‐mass co‐simulation test model may be regarded as two Dahlquist equations, coupled by an algebraic constraint equation. By discretizing the co‐simulation test model with a linear co‐simulation approach, a linear system of recurrence equations is obtained. The stability of the recurrence system, which reflects the stability of the underlying coupling method, can simply be determined by an eigenvalue analysis.

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.

Dynamic analysis of a 3-RRR parallel mechanism with multiple clearance joints

The existence of joint clearances causes contact and friction between the joint components, leading to important deviations between the expected behavior and real outcome of the mechanism. A 3-RRR parallel mechanism with six imperfect joints is analyzed in this paper. The dynamic equations are established by combining Newton–Euler equations with Lankarani–Nikravesh contact force model and improved Coulomb friction force model, the Baumgarte stabilization approach is used to improve the numerical stability. The results show that the joint clearances have considerable effects on the displacement and velocity of the moving platform, and have great influence on the acceleration and driving moments. The mechanism with joint clearances also exhibits nonlinear vibration with high frequencies. In order to better understand the different influence parameters of the mechanism, dynamic responses of the system with different loads, moving velocities and moving trajectories are further studied. It indicates that lighter load, moderate velocity and simpler moving trajectory should be adopted when operating the mechanism to ensure the positioning precision and stability.

A Nonlinear Model of a Slack Cable With Bending Stiffness and Moving Ends With Application to Elevator Traveling and Compensation Cables

A nonlinear, planar model of a slack cable with bending stiffness and arbitrarily moving ends is developed. The model uses the slope angle of the centroid line of the cable to describe the motion of the cable, and the resulting integropartial differential equation with constraints is derived using Hamilton’s principle. A new method is developed to obtain the spatially discretized equations, and the Baumgarte stabilization procedure is used to solve the resulting differential-algebraic equations. The model can be used to calculate the equilibria and corresponding free vibration characteristics of the cable, as well as the dynamic response of the cable under arbitrarily moving ends. The results for an equilibrium and free vibration characteristics around the equilibrium are experimentally validated on a laboratory steel band. The methodology is applied to elevator traveling and compensation cables. It is found that a vertical motion of the car can introduce a horizontal vibration of a traveling or compensation cable. The results presented are verified by a commercial finite element software. The current method is shown to be more efficient than the finite element method as it uses a much smaller number of elements to reach the same accuracy. Some other interesting features include the condition for a traveling or compensation cable equilibrium to be closest to a natural loop and a direct proof that the catenary solution is unique.

A Parametric Study on the Baumgarte Stabilization Method for Forward Dynamics of Constrained Multibody Systems

This paper presents and discusses the results obtained from a parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems. The main purpose of this work is to analyze the influence of the variables that affect the violation of constraints, chiefly the values of the Baumgarte parameters, the integration method, the time step, and the quality of the initial conditions for the positions. In the sequel of this process, the formulation of the rigid multibody systems is reviewed. The generalized Cartesian coordinates are selected as the variables to describe the bodies’ degrees of freedom. The formulation of the equations of motion uses the Newton–Euler approach, augmented with the constraint equations that lead to a set of differential algebraic equations. Furthermore, the main issues related to the stabilization of the violation of constraints based on the Baumgarte approach are revised. Special attention is also given to some techniques that help in the selection process of the values of the Baumgarte parameters, namely, those based on the Taylor’s series and the Laplace transform technique. Finally, a slider-crank mechanism with eccentricity is considered as an example of application in order to illustrate how the violation of constraints can be affected by different factors.

Comparison between mathematical modeling and experimental identification of a spatial slider–crank mechanism

In this paper, Hamilton’s principle, Lagrange multiplier, geometric constraints, partitioning method and Baumgarte stabilization method (BSM) are employed to derive the dynamic equations of a spatial slider–crank mechanism that is driven by a servomotor. The formulation considers the effects of links masses, external forces and motor electric inputs. Comparing dynamic responses between the experimental results and numerical simulations, dynamic modeling gives a wonderful interpretation for the spatial slider–crank mechanism. In this paper, a new identification method based on real-coded genetic algorithm (RGA) is presented to identify the parameters of a spatial slider–crank mechanism. The method promotes the calculation efficiency very much, and is calculated by real-code without the operations of encoding and decoding. The results of numerical simulations and experimental results prove that the identification method is feasible. The contributions of this paper are that the comparison of mathematical modeling and identification between numerical simulations and experimental results are all realized.

Elimination of high frequencies in the absolute nodal coordinate formulation

The performance of the coordinate partitioning method, the method of Lagrangian multipliers with Baumgarte's stabilization, and the penalty method to account for non-linear constraints of a linear beam element based on the absolute nodal coordinate formulation is studied. Proper non-linear constraints are selected to avoid high-frequency transverse or shear oscillations of the beam. The results show that harmful oscillations can be successfully eliminated from the beam response and efficiency of the numerical solution improves when these constraints are used in the solution procedure. In case of a swinging pendulum, the calculation speed could be improved, when compared to the non-constrained case. It was found that the overall performance of the Lagrange multiplier method with Baumgarte's stabilization and carefully selected stabilization parameters was the best approach.

On dual Schur domain decomposition method for linear first-order transient problems

This paper addresses some numerical and theoretical aspects of dual Schur domain decomposition methods for linear first-order transient partial differential equations. The spatially discrete system of equations resulting from a dual Schur domain decomposition method can be expressed as a system of differential-algebraic equations (DAEs). In this work, we consider the trapezoidal family of schemes for integrating the ordinary differential equations (ODEs) for each subdomain and present four different coupling methods, corresponding to different algebraic constraints, for enforcing kinematic continuity on the interface between the subdomains. Unlike the continuous formulation, the discretized formulation of the transient problem is unable to enforce simultaneously the continuity of both the primary variable and its rate along the subdomain interface (except for the backward Euler method). Method 1 ( d -continuity) is based on the conventional approach using continuity of the primary variable and we show that this method is unstable for a lot of commonly used time integrators including the mid-point rule. To alleviate this difficulty, we propose a new Method 2 (modified d -continuity) and prove its stability for coupling all time integrators in the trapezoidal family (except the forward Euler). Method 3 ( v -continuity) is based on enforcing the continuity of the time derivative of the primary variable. However, this constraint introduces a drift in the primary variable on the interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte stabilization to limit this drift and we derive bounds for the stabilization parameter to ensure stability. Our stability analysis is based on the “energy” method, and one of the main contributions of this paper is the extension of the energy method (which was previously introduced in the context of numerical methods for ODEs) to assess the stability of numerical formulations for index-2 differential-algebraic equations (DAEs). Finally, we present numerical examples to corroborate our theoretical predictions.

Completions of implicitly defined linear time varying vector fields

The stabilization of constraints through such techniques as Baumgarte stabilization has been used in the simulation community for some time. This and a number of control problems can be viewed as either extending, or modifying, a vector field off of some manifold. Generally these approaches required the equations to have a special structure. Motivated by numerical simulation there has recently been new progress on doing this stabilization in numerically robust ways for larger classes of systems. In this paper we point out how earlier linear time invariant results do not immediately apply to the linear time varying case and then analyze the linear time varying case.

Cluster computing of mechanisms dynamics using recursive formulation

This paper presents the O(n) recursive algorithm for forward dynamics of closed loop kinematic chains adapted to parallel computations on a cluster of workstations. The Newton–Euler equations of motion are formulated in terms of relative coordinates. Closed loop kinematic chains are transformed into open loop chains by cut joint technique. Cut joint constraint and Lagrange multipliers are introduced to complete the equations of motion. Constraint stabilization is performed using the Baumgarte stabilization technique with application to multibody systems with large number of degrees of freedom. Numerical simulations are carried out to study the influence of the degrees of freedom of the multibody system on computational efficiency of the algorithm using the Message Passing Interface (MPI). We also consider the ways of minimization of communication overhead which has significant impact on efficiency in case of cluster computing.

A Comparison between the Stabilization Method and the Post-stabilization Method

A detailed discussion on Baumgarte's stabilization method and Chin's post-stabilization method is carried out together with a numerical comparison. The numerical accuracy can be increased and the numerical stability can be improved by combining the classical numerical method with either of the two stabilization methods. Assuming optimal stabilizing parameter the accuracy is not the same in the two methods and it is not certain which one is the more accurate. While noting the reasons underlying the post-stabilization scheme, it is inadvisable, when applying the latter, to use step lengths greater than are recommended by the original classical integrator with no stabilization.

Dynamic simulation of articulated rigid bodies with contact and collision

We propose a novel approach for dynamically simulating articulated rigid bodies undergoing frequent and unpredictable contact and collision. In order to leverage existing algorithms for nonconvex bodies, multiple collisions, large contact groups, stacking, etc., we use maximal rather than generalized coordinates and take an impulse-based approach that allows us to treat articulation, contact, and collision in a unified manner. Traditional constraint handling methods are subject to drift, and we propose a novel prestabilization method that does not require tunable potentially stiff parameters as does Baumgarte stabilization. This differs from poststabilization in that we compute allowable trajectories before moving the rigid bodies to their new positions, instead of correcting them after the fact when it can be difficult to incorporate the effects of contact and collision. A poststabilization technique is used for momentum and angular momentum. Our approach works with any black box method for specifying valid joint constraints and no special considerations are required for arbitrary closed loops or branching. Moreover, our implementation is linear both in the number of bodies and in the number of auxiliary contact and collision constraints, unlike many other methods that are linear in the number of bodies, but not in the number of auxiliary constraints.

A planar mechanical library in the AMESim simulation software. Part I: Formulation of dynamics equations

This paper presents the mathematical developments of a planar mechanical library implemented in the AMESim simulation tool. Body and joint components are the basic components of this library. Due to the library philosophy requirements, the mathematical models of the components have required a generic vector calculus based formulation of the constraint equations. This formulation uses a set of dependent generalized coordinates. The dynamics equations are obtained from the application of Jourdain’s principle combined with the Lagrange multiplier method. The body component mathematical models consist of differential equations in terms of the dependent generalized coordinates. The joint component mathematical models are based on the Baumgarte stabilization schemes applied to the geometrical, kinematic and acceleration constraint equations. The Lagrange multipliers are the implicit solution of these Baumgarte stabilization schemes. The first main contribution of this paper is the expression of geometrical constraints in terms of vectors and their exploitation in this form. The second important contribution is the adaptation of existing formulations to the AMESim philosophy.

Incorporating illumination constraints in deformable models for shape from shading and light direction estimation

We present a method for the integration of nonlinear holonomic constraints in deformable models and its application to the problems of shape and illuminant direction estimation from shading. Experimental results demonstrate that our method performs better than previous Shape from Shading algorithms applied to images of Lambertian objects under known illumination. It is also more general as it can be applied to non-Lambertian surfaces and it does not require knowledge of the illuminant direction. In this paper, (1) we first develop a theory for the numerically robust integration of nonlinear holonomic constraints within a deformable model framework. In this formulation, we use Lagrange multipliers and a Baumgarte stabilization approach (1972). (2) We also describe a fast new method for the computation of constraint based forces, in the case of high numbers of local parameters. (3) We demonstrate how any type of illumination constraint, from the simple Lambertian model to more complex highly nonlinear models can be incorporated in a deformable model framework. (4) We extend our method to work when the direction of the light source is not known. We couple our shape estimation method with a method for light estimation, in an iterative process, where improved shape estimation results in improved light estimation and vice versa. (5) We perform a series of experiments.

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.

Penalty based Hamiltonian equations for the dynamic analysis of constrained mechanical systems

We propose in this paper a new penalty based Hamiltonian description of the equations of motion of mechanical systems subject ot both holonomic and non-holonomic constraints. This method shows a clear advantage over the previously proposed acceleration based formulation, in terms of numerical efficiency and constraint stabilization. We also show that the new approach is more efficient and robust than both the Lagrange's multiplier method with Baumgarte stabilization, and the use of a minimum set of coordinates. In addition, this new method is robust under singular positions and in the presence of redundant constraints.

Stabilization of DAEs and invariant manifolds

Many methods have been proposed for the stabilization of higher index differential-algebraic equations (DAEs). Such methods often involve constraint differentiation and problem stabilization, thus obtaining a stabilized index reduction. A popular method is Baumgarte stabilization, but the choice of parameters to make it robust is unclear in practice. Here we explain why the Baumgarte method may run into trouble. We then show how to improve it. We further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature. Our approach is to (i) consider stabilization of ODEs with invariants, (ii) discretize the stabilizing term in a simple way, generally different from the ODE discretization, and (iii) use orthogonal projections whenever possible. The best methods thus obtained are related to methods of coordinate projection. We discuss them and make concrete algorithmic suggestions.