Structure of optimized generalized coordinates partitioned vectors for holonomic and non-holonomic systems

The generalized coordinates partitioning is a well-known procedure that can be applied in the framework of a numerical integration of the DAE systems. However, although the procedure proves to be a very useful tool, it is known that an optimization algorithm for the coordinates partitioning is needed to obtain the best performance. In the paper, the optimized partitioning of the generalized coordinates is revisited in the context of a numerical forward dynamics of the holonomic and non-holonomic multibody systems. After a short presentation of the geometric background of the optimized coordinates partitioning, a structure of the optimally partitioned vectors is discussed on the basis of a gradient analysis of the separate constraint sub-manifolds at the configuration and the velocity levels when holonomic and non-holonomic constraints are present in the system. It is shown that, for holonomic systems, the vectors of optimally partitioned coordinates have the same structure for the generalized positions and velocities. On the contrary, in the case of non-holonomic systems, the optimally partitioned coordinates generally differ at the configuration and the velocity levels. The conclusions of the paper are illustrated within the framework of the presented numerical example.     

Impulsive synchronization motion in networked open-loop multibody systems

This paper presents a procedure for studying impulsive synchronization motion in networked open-loop multibody systems formulated by Lagrange dynamics. Impulsive motion occurs when the networked systems are physically subject to either direct or indirect impulsive effects, or when subjected to both simultaneously. The impulsive effects are usually caused by impulsive forces or impulsive constraints. The governing equations of networked open-loop multibody systems are developed from Lagrange formulation. The procedure automatically incorporates a preliminary feedback control and the effects of impulsive constraints through its analysis. Some generic criteria on exponential synchronization of the system output with respect to generalized coordinates and its velocities over, respectively, undirected fixed and switching network topologies, are derived analytically. The procedure shows that impulsive synchronization motion in networked open-loop multibody systems can achieve by impulsive constraints strategies. Two examples and simulations are used to demonstrate and validate the analysis procedure.     

Geometric properties of projective constraint violation stabilization method for generally constrained multibody systems on manifolds

During numerical forward dynamics of constrained multibody systems, a numerical violation of system kinematical constraints is the important issue that has to be properly treated. In this paper, the stabilized time-integration procedure, whose constraint stabilization step is based on the projection of integration results to underlying constraint manifold via post-integration correction of the selected coordinates is discussed. A selection of the coordinates is based on the optimization algorithm for coordinates partitioning. After discussing geometric background of the optimization algorithm, new formulae for optimized partitioning of the generalized coordinates are derived. Beside in the framework of the proposed stabilization algorithm, the new formulae can be used for other integration applications where coordinates partitioning is needed. Holonomic and non-holonomic systems are analyzed and optimal partitioning at the position and velocity level are considered further. By comparing the proposed stabilization method to other projective algorithms reported in the literature, the geometric and stabilization issues of the method are addressed. A numerical example that illustrates application of the method to constraint violation stabilization of non-holonomic multibody system is reported. An erratum to this article can be found at     

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.     

Software for global dynamics evaluation of mechanisms

The paper deals with the multibody system software which implements the solution of the global dynamic problem for multibody systems described by redundant coordinates (DAE equations) and with a possibility of redundant number of actuators. The aim of the global dynamics evaluation is mainly the machine synthesis. The necessity of such formulation arises especially in robotics where the accessible velocities and accelerations on the given trajectory are important for the motion planning and for the design optimization of robots and manipulators. The analogical problem can be very important also for the design of any other machine type. The developed software is therefore one of the important tools for the multiobjective machine synthesis.     

Initial condition correction in multibody dynamics

A simple procedure is presented to correct initial conditions for the coordinates and velocities prior to performing a kinematic or forward dynamic analysis of multibody systems. Such corrections are crucial since slight amount of constraint violations at the start of any numerical integration of equations of motion can lead to erroneous results. The correction process is based on the well-known method of minimizing the sum-of-squares of adjustments in the coordinates or velocities. The process provides a solution that is closest to the estimated values. It should be a simple task to implement this methodology as a preprocessing step for any kinematic or forward dynamic analysis program regardless of the formulation. Commemorative Contribution.     

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.     

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.     

Subsystem synthesis method with approximate function approach for a real-time multibody vehicle model

This paper presents the subsystem synthesis method with approximate function approach for a real-time multibody vehicle dynamics model. In the subsystem synthesis method, equations of motion for the car body of a vehicle and the equations of motion for suspension subsystems are formed separately for efficient computation. Joint coordinates are used to construct suspension subsystem equations of motion. Since these joint coordinates must satisfy the loop closure constraint equations that represent suspension linkage kinematics, they are not all independent. Using the generalized coordinate partitioning method, suspension subsystem equations of motion can be represented only in terms of independent generalized coordinates. To represent dependent coordinates as a function of independent coordinates in the generalized coordinate partitioning method, expensive numerical approaches, such as the Newton–Raphson method, must be applied. For real-time computation of the multibody vehicle model, an approximate function approach is proposed to express the dependent coordinates as polynomial functions of the independent coordinates within the framework of the subsystem synthesis method. Different orders of candidate polynomial functions are investigated for solution accuracy. Efficiency of the proposed method has been studied theoretically by counting arithmetic operators. By measuring actual CPU times of the simulations with a quarter car and a full car model, efficiency of the proposed method has also been investigated.     

Evaluation of Dynamic Capabilities of Machines and Robots

The paper deals with the formulation and thesolution of the global dynamic problem for multibody systemsgenerally described by redundant coordinates (DAE equations) andwith a redundant number of actuators.Within the dynamics of multibody systems two basic well-known problems have been investigated: the solution of direct dynamicsand the solution of inverse dynamics. Both of these problems enableus to solve in each time instant the relation between the motiondescribed by positions, velocities, accelerations and the actingforces. However, such solutions obtained in isolated timeinstants do not provide us with the global overview about thedynamic capabilities of a mechanical system, i.e. about theaccessible motion described by accessible positions, accessiblevelocities, accessible accelerations and the required (given)forces. Thus, besides the two basic dynamical problems there isa third one: the global dynamic problem.The formulation of the global dynamic problem, the generalmethods for its solution and the specific methods for itssolution for robots and manipulators are included.     

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.     

Pulse control of flexible multibody systems

An active pulse control method is developed to reduce the vibrations of multibody systems resulting from impact loadings. The pulse, which is a function of system generalized coordinates and velocities, is determined analytically using energy and momentum balance equations of the impacting bodies. Elastic components in the multibody system are discretized using the finite element method. The system equations of motions and nonlinear algebraic constraint equations describing mechanical joints between different components are written in the Lagrangian formulation using a finite set of coupled reference position and local elastic generalized coordinates. A set of independent differential equations are identified by the generalized coordinate partitioning of the constraint Jacobian matrix. These equations are written in the state space formulation and integrated forward in time using a direct numerical integration method. Dependent coordinates are then determined using the constraint kinematic relations. Points in time at which impact occurs are monitored by an impact predictor function, which controls the integration algorithms and forces for the solution of the momentum relation, to define the jump discontinuities in the composite velocity vector as well as the system reaction forces. The effectiveness of the active pulse control in reducing the vibration of flexible multibody aircraft during the touchdown impact is investigated and numerical results are presented.     

A bicycle model for education in multibody dynamics and real-time interactive simulation

This paper describes the use of a bicycle model to teach multibody dynamics. The bicycle motion equations are first obtained as a DAE system written in terms of dependent coordinates that are subject to holonomic and non-holonomic constraints. The equations are obtained using symbolic computation. The DAE system is transformed to an ODE system written in terms of a minimum set of independent coordinates using the generalised coordinates partitioning method. This step is taken using numerical computation. The ODE system is then numerically linearised around the upright position and eigenvalue analysis of the resulting system is performed. The frequencies and modes of the bicycle are obtained as a function of the forward velocity which is used as continuation parameter. The resulting frequencies and modes are compared with experimental results. Finally, the non-linear equations of the bicycle are used to create an interactive real-time simulator using Matlab-Simulink. A series of issues on controlling the bicycle are discussed. The entire paper is focussed on teaching engineering students the practical application of analytical and computational mechanics using a model that being simple is familiar and attractive to them.     

Application of Runge–Kutta–Rosenbrock Methods to the Analysis of Flexible Multibody Systems

Numerical integration methods are discussed for general equations of motion for multibody systems with flexible parts, which are fairly stiff, time-dependent and non-linear. A family of semi-implicit methods, which belong to the class of Runge–Kutta–Rosenbrock methods, with rather weak non-linear stability properties, are developed. These comprise methods of first, second and third order of accuracy that are A-stable and L-stable and hence introduce numerical damping and the filtering of high frequency components. It is shown, both from theory and examples, that it is generally preferable to use deformation mode coordinates to global nodal coordinates as independent variables in the formulation of the equations of motion. The methods are applied to a series of examples consisting of an elastic pendulum, a beam supported by springs, a four-bar mechanism, and a robotic manipulator with collocated control.     

A computer-oriented method for reducing linearized multibody system equations by incorporating constraints

Consider a spatial multibody system with rigid and elastic bodies. The bodies are linked by rigid interconnections (e.g. revolute joints) causing constraints, as well as by flexible interconnections (e.g. springs) causing applied forces. Small motions of the system with respect to a given nominal configuration can be described by linearized dynamic equations and kinematic constraint equations. We present a computer-oriented procedure which allows to develop a minimum number of these equations. There are three problems. First: algorithmic selection of position coordinates; second: condensation of the dynamic equations; third: evaluation of the constraint forces. To demonstrate the procedure, a closed loop multibody system is used as an example.     

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.     

Performance of automatic differentiation tools in the dynamic simulation of multibody systems

Within the multibody systems literature, few attempts have been made to use automatic differentiation for solving forward multibody dynamics and evaluating its computational efficiency. The most relevant implementations are found in the sensitivity analysis field, but they rarely address automatic differentiation issues in depth. This paper presents a thorough analysis of automatic differentiation tools in the time integration of multibody systems. To that end, a penalty formulation is implemented. First, open-chain generalized positions and velocities are computed recursively, while using Cartesian coordinates to define local geometry. Second, the equations of motion are implicitly integrated by using the trapezoidal rule and a Newton–Raphson iteration. Third, velocity and acceleration projections are carried out to enforce kinematic constraints. For the computation of Newton–Raphson’s tangent matrix, instead of using numerical or analytical differentiation, automatic differentiation is implemented here. Specifically, the source-to-source transformation tool ADIC2 and the operator overloading tool ADOL-C are employed, in both dense and sparse modes. The theoretical approach is backed with the numerical analysis of a 1-DOF spatial four-bar mechanism, three different configurations of a 15-DOF multiple four-bar linkage, and a 16-DOF coach maneuver. Numerical and automatic differentiation are compared in terms of their computational efficiency and accuracy. Overall, we provide a global perspective of the efficiency of automatic differentiation in the field of multibody system dynamics.