Baumgarte Stabilization Research Articles

Influence of two kinds of clearance joints on the dynamics of planar mechanical system based on a modified contact force model

This study takes the slider-crank mechanism with revolute joint and translational joint as the research object and studies the contact force model of the clearance joint and the influence of the hybrid clearance joints on the nonlinear dynamic behavior of the mechanism. A modified contact force model is established based on the simplified elastic oscillator model, which can be used as a normal force in clearance joint. In the new contact force model, the component n of the indentation depth can be arbitrarily selected and it can support the calculation of contact force for both fully elastic recovery, non-elastic recovery and fully inelastic recovery. Based on the LuGre friction model, the tangential friction model of the clearance joint is given. Thus, the normal force and tangential force during the dynamic contact of the clearance joint are formed. Combining Lagrange’s equations of the first kind with the modified normal force and tangential friction force, the dynamic equations of the multi-body system with clearance joints are established. The Baumgarte stabilization method is used to improve the numerical stability. The correctness of the dynamic prediction model in the mechanism with clearance joint is verified by experiment. The dynamic analysis of the slider-crank mechanism with mixed clearance joints shows that the revolute clearance joint has a greater influence on the mechanism than the translational clearance, and the revolute clearance joint plays a leading role in the dynamic response.

Application of the Bezier integration technique with enhanced stability in forward dynamics of constrained multibody systems with Baumgarte stabilization method

Application of the Bezier integration technique with enhanced stability in forward dynamics of constrained multibody systems with Baumgarte stabilization method

Non-smooth dynamic modeling and simulation of an unmanned bicycle on a curved pavement

The non-smooth dynamic model of an unmanned bicycle is established to study the contact-separate and stick-slip non-smooth phenomena between wheels and the ground. According to the Carvallo-Whipple configuration, the unmanned bicycle is reduced to four rigid bodies, namely, rear wheel, rear frame, front fork, and front wheel, which are connected by perfect revolute joints. The interaction between each wheel and the ground is simplified as the normal contact force and the friction force at the contact point, and these forces are described by the Hunt-Crossley contact force model and the LuGre friction force model, respectively. According to the characteristics of flat and curved pavements, calculation methods for contact forces and their generalized forces are presented. The dynamics of the system is modeled by the Lagrange equations of the first kind, a numerical solution algorithm of the dynamic equations is presented, and the Baumgarte stabilization method is used to restrict the drift of the constraints. The correctness of the dynamic model and the numerical algorithm is verified in comparison with the previous studies. The feasibility of the proposed model is demonstrated by simulations under different motion states.

Inverse dynamics‐based formulation of finite horizon optimal control problems for rigid‐body systems

Inverse dynamics‐based formulation of finite horizon optimal control problems for rigid‐body systems

Forward dynamics analysis of origami-folded deployable spacecraft structures

A fast dynamics model that captures the deployment dynamics of self-actuated, origami-inspired, folded planar spacecraft structures is desired for design and verification applications. In this paper, a general simulation framework for numerically generating the equations of motion of any structure that complies with a set of pattern assumptions is presented. The framework is built through application of the articulated body forward dynamics algorithm and the tree-augmented approach for closed-chain forward dynamics. These are multi-body dynamics approaches developed in the literature for complex robotic manipulator systems. Unique adaptations are required to address the highly constrained nature of a folding structure. This solution is desirable due to the computational efficiency of the base algorithms and the ability to analyze multiple systems without reformulation of the core dynamics algorithm. Baumgarte stabilization techniques are applied to address constraint violations, but are found to be a continuing challenge for folding structures with several closed-chains.

Lumped, Constrained Cable Modeling With Explicit State-Space Formulation Using An Elastic Version of Baumgarte Stabilization

Abstract This paper presents a modeling approach for efficient simulation of slender structures, such as wires, cables and ropes. Lumped structural elements are connected using constraints. These are solved explicitly, using an elastic version of Baumgarte stabilization. This avoids singularities in the matrix inversions. The resulting explicit state-space formulation filters the higher order dynamics and can be solved using simple numerical integration methods. Constraints are demonstrated for modeling different aspects: Internal cable forces, one cable sliding along another cable and contact between cable and seabed. Also, a cable initialization routine is presented for rapid building of different interconnected cable geometries, ranging from cases in offshore crane operations to in-sea equipment such as seismic cables. Two case studies are presented to illustrate the effectiveness and the robustness of the proposed modeling approach; the first one being a test of two connected, sinking cables, and the last one being a larger case demonstrating the use of the cable library in an offshore seismic survey case.

Quasi-time-optimal steering law for low-thrust orbit transfer considering angular momentum and torque constraints

In this study, a quasi-time-optimal steering law for low-thrust orbit transfer is investigated. In the proposed steering law, the satellite attitude is controlled to suppress its error with respect to the ideal thruster direction under the orthogonality constraint between the Sun direction and the solar array driving mechanism's rotational axis as well as the angular momentum and torque constraints. All these constraints are satisfied by introducing a structure similar to Baumgarte's stabilization method and by formulating a quadratically constrained quadratic programming problem which can be analytically solved to obtain the control torque. Several numerical simulations show that the transfer time difference from the ideal unconstrained transfer is less than a few percent.

Mathematical modeling and dynamic characteristic analysis of a novel parallel tracking mechanism for inter-satellite link antenna

Inter-satellite link antenna tracking mechanism is an important enabling component of relay satellite for continuous real-time tracking of medium and low orbit target spacecraft. Facing the major demand of advanced technology development in aerospace engineering, this paper mainly tackles the problem on the rigid-flexible coupling dynamic modeling and dynamic characteristics analysis of a novel parallel tracking mechanism. Firstly, adopting a novel shape function of three-dimensional isoparametric beam element with curvature, the elasto-dynamic model is derived with floating coordinate method and Lagrangian equation. Then the natural frequencies of the system are investigated and ANSYS software is utilized to verify the model. An efficient numerical solution strategy consisting of the implicit integration algorithm and Baumgarte stabilization technology is employed to precisely evaluate the dynamic response of the system. Meanwhile, trajectory planning is carried out for high tracking precision. Furthermore, the validity of the numerical results of dynamic response is illustrated by simulations with the ADAMS software. Finally, the natural frequency sensitivity analysis is implemented by direct differential method with a novel sensitivity index to identify the structure weakness of the mechanism. A novel design idea for the cross-sectional parameters of each flexible beam is proposed with the sensitivity index. Above mentioned research results enrich the basic theoretical researches of parallel mechanism and lay a theoretical foundation for the practical application of tracking mechanism.

Determination of constraint stabilization parameters with multiple roots of characteristic equation

During a numerical integration of motion equations of a constrained mechanical system effect of deviations from initial data can be neglected with the help of Baumgarte stabilization method. According to this method first time derivatives of constraint equations are not considered as first integrals and are equated to an arbitrarily linear form of constraints themselves. The coefficients of these forms are called perturbation parameters. So, the problem of constraint stabilization is reduced to a problem of finding an appropriate range of perturbation parameters’ values. This problem can be solved by obtaining these values with the help of stability conditions. The procedure of evaluation is getting even more easier if the perturbation coefficients correspond to the multiple root case of its characteristic equation.

Effect of friction on the dynamic analysis of slider-crank mechanism with clearance joint

In this paper, we are interested in the effect of friction on dynamic behavior of a crank-slider mechanism with clearance joint. Due to the clearance existence in the revolute joint, it is important to choose an appropriate contact force model to analyze the dynamic response, the dynamic equations are established by combining Lagrange’s equation of the first kind with modified contact force model and LuGre friction model, the Baumgarte stabilization approach was used to improve the numerical stability. Simulation and experimental tests were carried out to verify this model. The system dynamic response hysteresis, and the more energy consumption when friction is considered in the mechanism. The influence of friction coefficient on the nonlinear dynamic characteristics of the mechanism with clearance joint is analyzed. The dynamic performance of the mechanism considering friction presents significant differences at the accelerations level. Friction plays a positive role in the stability of the system.

Modeling and simulation of planar multibody systems with revolute clearance joints considering stiction based on an LCP method

The main objective of this study is to present a general modeling and simulation methodology for planar rigid multibody systems with revolute clearance joints considering the effect of stiction. The normal and tangential contact forces acting at the contact points between the components of the clearance joints are described by a Hertzian-based contact force model and the Coulomb dry friction model, respectively. The equations of motion of the systems are derived on the basis of Lagrange equations of the first kind and the Baumgarte stabilization method. Based on the non-smooth dynamics method, the determination of stick-slip transitions and the calculation of static frictional forces of the contact points are transformed as a linear complementarity problem (LCP), which is solved with an event-driven scheme. Finally, two numerical examples are given as demonstrative applications. The results show that the stiction in the revolute clearance joints influences the dynamic response of the multibody systems.

Comparison and analysis of two Coulomb friction models on the dynamic behavior of slider-crank mechanism with a revolute clearance joint

The objective of this study is to investigate the effects of the Coulomb dry friction model versus the modified Coulomb friction model on the dynamic behavior of the slider-crank mechanism with a revolute clearance joint. The normal and tangential forces acting on the contact points between the journal and the bearing are described by using a Hertzian-based contact force model and the Coulomb friction models, respectively. The dynamic equations of the mechanism are derived based on the Lagrange equations of the first kind and the Baumgarte stabilization method. The frictional force is solved via the linear complementarity problem (LCP) algorithm and the trial-and-error algorithm. Finally, three numerical examples are given to show the influence of the two Coulomb friction models on the dynamic behavior of the mechanism. Numerical results show that due to the stick friction, the slider-crank mechanism may exhibit stick-slip motion and can balance at some special positions, while the mechanism with ideal joints cannot.

Dynamic modeling of constrained planar multibody systems: A case of lower limbs rehabilitative robot

An explicit equation of motion for constrained systems, called the Udwadia-Kalaba equation, opens up a new way of dynamic modeling of constrained multibody systems, which is accomplished in a straightforward and novel three-step process. However, constraint violation arises from violation of the lower-order constraint equations because the aforementioned equations incorporate only the acceleration level constraints. There are two methods, Baumgarte stabilization method (BSM) and modified method, are implemented to realize the constraint violation suppression. The comparison of the BSM and the modified method is provided, based on numerical simulations for a sample constrained system that represents a lower limbs rehabilitative robot with continuous passive motion for therapy and rehabilitation. The results of numerical simulation obtained by the modified method are superior to that obtained by BSM, which confirms the numerical effectiveness and accuracy of the explicit solution based on the modified method to the constrained planar multibody systems.

Chaos phenomenon and stability analysis of RU-RPR parallel mechanism with clearance and friction

The chaos phenomenon often exists in the dynamics system of the mechanism with clearance and friction, which has obvious effect on the stability of the mechanism, then it is worthy of attention for identifying the relationship between the friction coefficient and the stability of the mechanism. Two rotational degrees of freedom decoupled parallel mechanism RU-RPR is taken as the research object. Considering the clearance existing in the revolute pair, Lankarani–Nikravesh contact force model is used to calculate the normal contact force, and the Coulomb friction force model is used to calculate the tangential contact force. The dynamics model is established using Newton–Euler equations, and the Baumgarte stabilization method is used to keep the stability of the numerical analysis. Then, the equations are solved using the fourth adaptive Runge–Kutta method, and the effect of the revolute pair’s clearance on the dynamic behavior is analyzed. Poincare mapping is plotted, and the bifurcation diagrams are analyzed with varying the friction coefficient corresponding to different values of clearance size. The research contents possess a certain theoretical guidance significance and practical application value on the analysis of the chaotic motion and its stability in the dynamics of the parallel mechanism.

Nonlinear dynamic modeling and performance analysis of a redundantly actuated parallel manipulator with multiple actuation modes based on FMD theory

A novel redundantly actuated parallel manipulator (PM) with multiple actuation modes was developed recently by the author. In this paper, the further research is implemented and a systematic methodology is proposed to develop the rigid–flexible coupling dynamic model (RFDM) of the novel PM based on the flexible multi-body dynamics theory. Firstly, under the floating frame of reference, an arbitrary flexible link of the novel PM is regarded as an Euler–Bernoulli beam and the finite element approach is employed to discretize the flexible beam. Then, one kind of planar beam element with lumped masses and moments of inertia at both ends is presented and a corresponding dynamic model is deduced based on the Lagrangian formulation. On this basis, the RFDM of system is formulated by virtue of the augmented Lagrangian multipliers approach. Given the stiff characteristic of system dynamic model, the hybrid TR-BDF2 numerical algorithm combined with Baumgarte stabilization approach is employed to address the nonlinear RFDM so as to balance the solution efficiency and precision. Based on the RFDM and its degradation model, i.e., the rigid dynamic model, one dynamic simulation experiment is designed to investigate the dynamic performance of the PM. Numerical results indicate that the practical motion of the PM manifests rigid–flexible coupling characteristic, and the redundant actuation modes can attenuate the effect of link flexibility and improve the trajectory tracking precision of end-effector in comparison with the non-redundant actuation modes. Finally, to validate the presented methodology, the obtained numerical results are compared with a virtual prototype model developed based on SimMechanics. The results will be helpful for structure optimization and efficient controller design of the PM with multiple actuation modes.

Application of Hilber-Hughes-Taylor- method to dynamics of flexible multibody system with contact and constraint

Numerical characteristics of the Hilber-Hughes-Taylor- (HHT-) method for the differential-algebraic equations (DAEs) in impact dynamics of flexible multibody systems are investigated. The research is based on a dynamic process of a flexible beam rotating about a fixed axis, whichis under the action of gravity and collides with a rigid plane. Therefore, the dynamic transformation and solution of flexible multibody system are divided into two parts. The Lagrange's equations of the second kind are used to derive the dynamic equations before and after impact, whereas the contact constraint method (CCM) is adopted to simulate the contact process. Compared with other methods, the CCM can describe the contact process accurately and avoid choosing the additional parameters. A set of the differential equations are transformed into a set of the DAEs due to the added constraint equations into impact process. Normally the dynamic equations of the flexible multibody system are index-3 DAEs. Solving a system of the index-3 DAEs directly by an integration algorithm would be subject to ill-conditioning and poor global convergence properties, so it is reasonable to find the methods that avoid both drawbacks and dependence on the constraint information. In order to solve this complex process, the HHT- method is used in the impact dynamic simulation by introducing the Gear-Gupta-Leimkuhler formulation. The coefficient of the HHT- method can be used to control the numerical dissipation, and it also represents asymptotic annihilation of the high frequency response. The smaller the value of , the more the damping is induced in the numerical solution. The Baumgarte's stabilization method is the most famous one for index-3 DAEs. Unfortunately, no general way can be adopted to determine the coefficients of the Baumgarte's stabilization method. It is the main reason for the numerical stability problems. It is necessary to study the influences of coefficients of the former two methods. Simultaneously, the simulation results from the HHT- method are compared with those from the Baumgarte's stabilization method to calculate the CCM model, and the Newmark method is used to solve the ODEs by using the continuous contact force model. The influence of the modal truncation N on the numerical method is also taken into account. Furthermore, the influences of N and the coefficient of HHT- method on the velocity and acceleration constraints in the multibody system are analyzed. Results have shown that the choice of the stabilization coefficients exerts a greater influence on the simulation results, such as the dynamic responses and the constraints, than that of the coefficient . Meanwhile, the HHT- method has an influence on the choice of coefficient and numerical damping properties. This numerical damping property can reduce the effect of high order modes induced by impact. Finally, the increase of N causes the sharpening default of both velocity and acceleration constraints.

A continuous analysis method of planar rigid-body mechanical systems with two revolute clearance joints

In this paper we are interested in the dynamic behavior of a slider-crank mechanism with single and two revolute clearance joints. Due to the clearance existence in the revolute joints, it is important to choose an appropriate contact force model in analyzing the dynamic response of a slider-crank mechanism with clearances. The dynamic equations are established by combining the Newton–Euler equations with modified contact force model and improved Coulomb friction force model, and the Baumgarte stabilization approach is used to improve the numerical stability. According to numerical and experimental results, the method of continuous contact can be verified to be reasonable. Comparing dynamic the response between one clearance joint and two clearance joints in a crank-slider mechanism, it is easy to find a significant mutual coupling region due to the presence of two clearance joints by simply contact figures. The dynamic response in a mechanism with two clearance joints is not a simple superposition of that in mechanism with one clearance joint. Therefore, all the joints in a multibody system should be modeled as clearance joints.

Modeling and simulation of structural components in recursive closed-loop multibody systems

Passenger cars, transit buses, railroad vehicles, off-highway trucks, earth moving equipment and construction machinery contain structural and light-fabrications (SALF) components that are prone to excessive vibration due to rough terrains and work-cycle loads’ excitations. SALF components are typically modeled as flexible components in the multibody system allowing the analysts to predict elastic deformation and hence the stress levels under different loading conditions. Including SALF component in the multibody system typically generates closed-kinematic loops. This paper presents an approach for integrating SALF modeling capabilities as a flexible body in a general-purpose multibody dynamics solver that is based on joint-coordinates formulation with the ability to handle closed-kinematic loops. The spatial algebra notation is employed in deriving the spatial multibody dynamics equations of motion. The system kinematic topology matrix is used to project the Cartesian quantities into the joint subspace, leading to a condensed set of nonlinear equations with minimum number of generalized coordinates. The proposed flexible body formulation utilizes the component mode synthesis approach to reduce the large number of finite element degrees of freedom to a small set of generalized modal coordinates. The resulting reduced flexible body model has two main characteristics: the stiffness matrix is constant while the mass matrix depends on the elastic modal coordinates. A consistent set of pre-computed inertia shape integrals are identified and used to update the modal mass matrix at each time step. The implementation of the component mode synthesis approach in a closed-loop recursive multibody formulation is presented. The kinematic equations are modified to include the effect of the flexible body modal elastic coordinates. Also, modified constraint equations that include the effect of flexibility at the joint connections and the necessary details of the Jacobian matrix are presented. Baumgarte stabilization approach is used to stabilize the constraint equations without using iterative schemes. A sample results for flexible body impeded in a closed system will be presented to demonstrate the above mentioned approach.

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.

A new representation of systems with frictional unilateral constraints and its Baumgarte-like relaxation

This paper proposes a new representation of multibody mechanical systems involving three-dimensional frictional unilateral constraints. The new representation is of the form of a differential algebraic inclusion (DAI) employing a normal cone with a non-Euclidean, singular norm metric. It can be seen as a generalization of a differential algebraic equation (DAE) using Lagrange multipliers, which has been used to represent mechanical systems with equality constraints. The paper also presents an approach to approximate the aforementioned DAI by another form of DAI, which can be equivalently converted into an ordinary differential equation (ODE). The approach can be seen as a generalization of the Baumgarte stabilization, which was originally developed for DAEs. The new DAI representation and its ODE approximation are illustrated with some simple examples.