Constrained Multibody Systems Research Articles

Simulation of Multibody Systems with Nonholonomic Constraints

AbstractThe paper deals with the nonholonomic multibody system dynamics from a point of view resulting from some present applications in high‐tec areas like high‐speed train technology or biomechanics of some disciplines in high‐performance sports. A formulation of nonholonomic constraints which are linear related to generalized velocities is based on a derivative‐free approach for generating Lagrangian motion equations of multibody systems with kinematical tree structure as well as for constrained multibody systems. This has been done by using di.erential‐geometric concepts in a Riemannian space. The ideas are illustrated by the classical edge condition on double‐curved surfaces. The surfaces are described by C2‐vector functions, for example by NURBS‐approximation. As an example a bobsleigh is regarded moving on a double‐curved surface.

Equivalent Control of Constrained Multibody Systems

This paper investigates the title problem from the perspective ofdetermining which forcesystems, although different, will have the same dynamic effectwhen applied to a mechanical system.A procedure is presented for determining how differences incontrol force/moment input are nullifiedby constraints on the system. This in turn provides a basis foroptimal control. The theoreticaldevelopment is illustrated by a series of examples with a model ofa redundant manipulator – a constrained/controlled triple pendulum.

Simulation of Motion of Constrained Multibody Systems Based on Projection Operator

This paper presents an efficient dynamic formulation for solvingDifferential Algebraic Equations (DAE) by using the notion of orthogonalprojection. Firstly, the constraint equations are expressed explicitlyat acceleration level by using the notion of the orthogonal projection.Secondly, the Lagrangian multiplier is eliminated from the dynamicsequation by the projection operator. Then, the resultant equations areconsolidated into one equation which explicitly correlates theacceleration to the generalized force through a so-called constraint mass matrix. It is proved that the constraint mass matrix isalways invertible and hence the acceleration can be computed in aclosed-form manner even with the presence of redundant constraints or asingular configuration. The equation of motion is given explicitly in arelatively compact form, which can lead to computational efficiency. Italso has a useful physical interpretation, as the component of thegeneralized force contributing to motion dynamics is readily derivedform the formulation. Finally, results obtained from numericalsimulation of motion of a five-bar mechanism is documented.

A Lie-Group Formulation of Kinematics and Dynamics of Constrained MBS and Its Application to Analytical Mechanics

A Lie-group formulation for the kinematics and dynamics ofholonomic constrained mechanical systems (CMS) is presented. The kinematics ofrigid multibody systems (MBS) is described in terms of the screw system of theMBS. Using Lie-algebraic properties of screw algebra, isomorphicto se(3), allows a purely algebraic derivation of the Lagrangian motion equations. As such the Lie-group SE(3) ⊗... ⊗ SE(3) (n copies) is theambient space of a MBS consisting of n rigid bodies. Any parameterizationof the ambient space corresponds to a chart on the MBS configuration space ℝn. The key to combine differential geometric and Lie-algebraic approaches is the existence of kinematic basic functions whichare push forward maps from the tangent bundle Tℝn to the Lie-algebra of the ambient space.

Steady-state equilibrium analysis of a multibody system driven by constant generalized speeds

A formulation which seeks steady-state equilibrium positions of constrained multibody systems driven by constant generalized speeds is presented in this paper. Since the relative coordinates are employed, constraint equations at cut joints are incorporated into the formulation. To obtain the steady-state equilibrium position of a multibody system, nonlinear equations are derived and solved iteratively. The nonlinear equations consist of the force equilibrium equations and the kinematic constraint equations. To verify the effectiveness of the proposed formulation, two numerical examples are solved and the results are compared with those of a commercial program.

A Constrained Multibody System Dynamics Avoiding Kinematic Singularities.

In the analysis of constrained multibody systems, the constraint reaction forces are normally expressed in terms of the constraint equations and a vector of Lagrange multipliers. Because it fails to incorporate conservation of momentum, the Lagrange multiplier method is deficient when the constraint Jacobian matrix is singular. This paper presents an improved dynamic formulation for the constrained multibody system. In our formulation, the kinematic constraints are still formulated in terms of the joint constraint reaction forces and moments; however, the formulations are based on a second-order Taylor expansion so as to incorporate the rigid body velocities. Conservation of momentum is included explicitly in this method; hence the problems caused by kinematic singularities can be avoided. In addition, the dynamic formulation is general and applicable to most dynamic analyses. Finally the 3-leg Stewart platform is used for the example of analysis.

F-6-4-2 Linearization and Vibration Analysis of Constrained Multibody Systems with Constant Generalized Speeds

F-6-4-2 Linearization and Vibration Analysis of Constrained Multibody Systems with Constant Generalized Speeds

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.

Stabilization Method for the Numerical Integration of Controlled Multibody Mechanical System. A Hybrid Integration Approach.

Simulation of a controlled multibody mechanical system, with a nonstiff mechanical subsystem and a stiff control subsystem, usually suffers from long run time. In this paper, a hybrid integration scheme combined with the constraint stabilization method is proposed to overcome this difficulty. The dynamic equations of motion of a constrained multibody mechanical system is a mixed differential-algebraic equation (DAE). The constraint stabilization method proposed by Baumgarte is used to integrate this equation. Correct choice of the α and β coefficients used in the constraint stabilization method is found. The hybrid integration scheme consists of two parts: a nonstiff integration algorithm for the mechanical subsystem and a stiff integration algorithm for the control subsystem. Simulation results show the computational efficiency of the proposed integration scheme.

An Augmented Formulation for Mechanical Systems with Non-Generalized Coordinates: Application to Rigid Body Contact Problems

In computational multibody algorithms, the kinematic constraintequations that describe mechanical joints and specified motiontrajectories must be satisfied at the position, velocity andacceleration levels. For most commonly used constraint equations, onlyfirst and second partial derivatives of position vectors with respect tothe generalized coordinates are required in order to define theconstraint Jacobian matrix and the first and second derivatives of theconstraints with respect to time. When the kinematic and dynamicequations of the multibody systems are formulated in terms of a mixedset of generalized and non-generalized coordinates, higher partialderivatives with respect to these non-generalized coordinates arerequired, and the neglect of these derivatives can lead to significanterrors. In this paper, the implementation of a contact model in generalmultibody algorithms is presented as an example of mechanical systemswith non-generalized coordinates. The kinematic equations that describethe contact between two surfaces of two bodies in the multibody systemare formulated in terms of the system generalized coordinates and thesurface parameters. Each contact surface is defined using twoindependent parameters that completely define the tangent and normalvectors at an arbitrary point on the body surface. In the contact modeldeveloped in this study, the points of contact are searched for on lineduring the dynamic simulation by solving the nonlinear differential andalgebraic equations of the constrained multibody system. It isdemonstrated in this paper that in the case of a point contact andregular surfaces, there is only one independent generalized contactconstraint force despite the fact that five constraint equations areused to enforce the contact conditions.

Null Space Integration Method for Constrained Multibody Systems with No Constraint Violation

A method for integrating equations of motion of constrained multibodysystems with no constraint violation is presented. A mathematical model,shaped as a differential-algebraic system of index 1, is transformedinto a system of ordinary differential equations using the null-spaceprojection method. Equations of motion are set in a non-minimal form.During integration, violations of constraints are corrected by solvingconstraint equations at the position and velocity level, utilising themetric of the system's configuration space, and projective criterion to thecoordinate partitioning method. The method is applied to dynamicsimulation of 3D constrained biomechanical system. The simulation resultsare evaluated by comparing them to the values of characteristicparameters obtained by kinematic analysis of analyzed motion based onmeasured kinematic data.

Collocation Methods for the Investigation of Periodic Motions of Constrained Multibody Systems

The investigation of periodic motions of constrained multibody systems requires the numerical solution of differential-algebraic boundary value problems. After briefly surveying the basics of periodic motion analysis the paper presents an extension of projected collocation methods [6] to a special class of boundary Value problems for multibody system equations with position and velocity constraints. These methods can be applied for computing stable as well as unstable periodic motions. Furthermore they provide stability information, which can be used to detect bifurcations on periodic branches. The special class of equations stemming from contact problems like in railroad systems [22] can be handled as well. Numerical experiments with a wheelset model demonstrate the performance of the algorithms (Less)

A direct violation correction method in numerical simulation of constrained multibody systems

A new direct violation correction method for constrained multibody systems is presented. It can correct the value of state variables of the systems directly so as to satisfy the constraint equations of motion. During the integration of the dynamic equations of constrained multibody systems, this method can efficiently control the violations of constraint equations within any given accuracy at each time-step. Compared to conventional indirect methods, especially Baumgarte's Constraint Violation Stabilization Method, this method has clear physical meaning, less calculation and obvious correction effect. Besides, this method has minor effect on the form of the dynamic equations of systems, so it is stable and highly accurate. A numerical example is provided to demonstrate the effectiveness of this method.

Combined DAE and Sliding Mode Control Methods for Simulation of Constrained Mechanical Systems

In dynamic analysis of constrained multibody systems (MBS), the computer simulation problem essentially reduces to finding a numerical solution to higher-index differential-algebraic equations (DAE). This paper presents a hybrid method composed of multi-input multi-output (MIMO), nonlinear, variable-structure control (VSC) theory and post-stabilization from DAE solution theory for the computer solution of constrained MBS equations. The primary contributions of this paper are: (1) explicit transformation of constrained MBS DAE into a general nonlinear MIMO control problem in canonical form; (2) development of a hybrid numerical method that incorporates benefits of both Sliding Mode Control (SMC) and DAE stabilization methods for the solution of index-2 or index-3 MBS DAE; (3) development of an acceleration-level stabilization method that draws from SMC’s boundary layer dynamics and the DAE literature’s post-stabilization; and (4) presentation of the hybrid numerical method as one way to eliminate chattering commonly found in simulation of SMC systems. The hybrid method presented can be used to simulate constrained MBS systems with either holonomic, nonholonomic, or both types of constraints. In addition, the initial conditions (ICs) may either be consistent or inconsistent. In this paper, MIMO SMC is used to find the control law that will provide two guarantees. First, if the constraints are initially not satisfied (i.e., for inconsistent ICs) the constraints will be driven to satisfaction within finite time using SMC’s stabilization method, urobust,i=−ηisgnsi. Second, once the constraints have been satisfied, the control law, ueq and hybrid stabilization techniques guarantee surface attractiveness and satisfaction for all time. For inconsistent ICs, Hermite-Birkhoff interpolants accurately locate when each surface reaches zero, indicating the transition time from SMC’s stabilization method to those in the DAE literature. [S0022-0434(00)02404-7]

Force Coupling Versus Differential Algebraic Description of Constrained Multibody Systems

The force coupling and the differential algebraic descriptionof constrained multibody systems are compared and applied to two testmodels used in vehicle dynamics. Based on the Newtonian and Eulerianequations the equations of motion for local subsystems are presentedusing explicitly and implicitly formulated constraints. In the nextstage the coupling of local subsystems using implicit algebraicconstraints and an approximative description of the constraints byapplied coupling forces is explained. The two coupling methods areapplied to test models used in vehicle dynamics to examine theircharacteristics by numerical integration with different methods. Themost efficient among the tested DAE solvers is identified.

Inverse Dynamic Analysis of Constrained Multibody Systems Considering Friction Forces Acting on Kinematic Joints.

A method for the inverse dynamic analysis of constrained multibody systems considering friction forces acting on kinematic joints is proposed in this paper. The stiction and the sliding which represent zero and non-zero relative motions are considered during the inverse dynamic analysis. Actuating forces to control the position or the orientation of constrained multibody systems are usually calculated in the inverse dynamic analysis. When the friction is considered, an iterative procedure need to be developed to calculate the actuating forces, which are uniquely determined during the sliding but riot uniquely determined during the stiction. Numerical examples are solved to demonstrate the accuracy and the effectiveness of the proposed method.

Inverse Dynamic Analysis of Constrained Multibody Systems Considering Friction Forces on Kinematic Joints

A method for the inverse dynamic analysis of constrained multibody systems considering friction forces acting on kinematic joints is presented in this paper. The stiction and the sliding which represent zero and non-zero relative motions are considered during the inverse dynamic analysis. Actuating forces to control the position or the orientation of constrained multibody systems are usually calculated in the inverse dynamic analysis. An iterative procedure need to be employed to calculate the actuating forces when the friction is considered. Furthermore, the actuating forces are not uniquely determined during the stiction. These difficulties are resolved by the method presented in this paper.

Dynamic Analysis of Constrained Multibody Systems Undergoing Collision with Friction

Dynamic Analysis of Constrained Multibody Systems Undergoing Collision with Friction

Dynamic Analysis of Constrained Multibody Systems Undergoing Collision

This paper presents a method for the dynamic analysis of constrained multibody systems undergoing abrupt collision. The proposed method uses a longer time interval to check collision than that of c onventional method. This reduces the computational effort significantly. To calculate collision points on two colliding rigid bodies, one may introduce constraints of contact. However, this causes reduction of degree of freedom and difficulty of numerical analysis. The proposed method can calculate collision points without above mentioned problems. Three numerical examples are given to demonstrate the computational efficiency and the usefulness of the proposed method

A Mathematical Corrector Method of Constrained Dynamic Systems.

This paper presents a numerical corrector method to find feasible state variables of the constrained multibody systems. For correcting state variables, the Lagrange-Newton method, a nonlinear optimization technique, is used. The iteration formulae derived from the quasi-Newton scheme do not update the Lagrange multipliers in the analysis steps and projects the state variables on the constraint manifold. Therefore, the cost due to the updating of Lagrange multipliers decreases. The method is verified through the convergence theorem denoting a convergence order of numerical solutions and as the corrections are performed along the constraint gradients, the system motion is formed in the null space tangent to the constraint manifold. The simulation example uses a three-dimensional full vehicle model, and the obtained numerical solutions are compared with the ADAMS solutions.