An extended divide-and-conquer algorithm for a generalized class of multibody constraints

An extension to the divide-and-conquer algorithm (DCA) is presented in this paper to model constrained multibody systems. The constraints of interest are those applied to the system due to the inverse dynamics or control laws rather than the kinematically closed loops which have been studied in the literature. These imposed constraints are often expressed in terms of the generalized coordinates and speeds. A set of unknown generalized constraint forces must be considered in the equations of motion to enforce these algebraic constraints. In this paper dynamics of this class of multibody constrained systems is formulated using a Generalized-DCA. In this scheme, introducing dynamically equivalent forcing systems, each generalized constraint force is replaced by its dynamically equivalent spatial constraint force applied from the appropriate parent body to the associated child body at the connecting joint without violating the dynamics of the original system. The handle equations of motion are then formulated considering these dynamically equivalent spatial constraint forces. These equations in the GDCA scheme are used in the assembly and disassembly processes to solve for the states of the system, as well as the generalized constraint forces and/or Lagrange multipliers.     

A divide-and-conquer direct differentiation approach for multibody system sensitivity analysis

In the design and analysis of multibody dynamics systems, sensitivity analysis is a critical tool for good design decisions. Unless efficient algorithms are used, sensitivity analysis can be computationally expensive, and hence, limited in its efficacy. Traditional direct differentiation methods can be computationally expensive with complexity as large as O(n ⁴+n ² m ²+nm ³), where n is the number of generalized coordinates in the system and m is the number of algebraic constraints. In this paper, a direct differentiation divide-and-conquer approach is presented for efficient sensitivity analysis of multibody systems with general topologies. This approach uses a binary tree structure to traverse the topology of the system and recursively generate the sensitivity data in linear and logarithmic complexities for serial and parallel implementations, respectively. This method works concurrently with the forward dynamics problem, and hence, requires minimal data storage. The differentiation required in this algorithm is minimum as compared to traditional methods, and is generated locally on each body as a preprocessing step. The method provides sensitivity values accurately up to integration tolerance and is insensitive to perturbations in design parameter values. This approach is a good alternative to existing methodologies, as it is fairly simple to implement for general topologies and is computationally efficient.     

An efficient direct differentiation approach for sensitivity analysis of flexible multibody systems

This paper presents a recursive direct differentiation method for sensitivity analysis of flexible multibody systems. Large rotations and translations in the system are modeled as rigid body degrees of freedom while the deformation field within each body is approximated by superposition of modal shape functions. The equations of motion for the flexible members are differentiated at body level and the sensitivity information is generated via a recursive divide and conquer scheme. The number of differentiations required in this method is minimal. The method works concurrently with the forward dynamics simulation of the system and requires minimum data storage. The use of divide and conquer framework makes the method linear and logarithmic in complexity for serial and parallel implementation, respectively, and ideally suited for general topologies. The method is applied to a flexible two arm robotic manipulator to calculate sensitivity information and the results are compared with the finite difference approach.     

An efficient computational method for dynamic stress analysis of flexible multibody systems

This paper presents an efficient computational method of dynamic stress history calculation for a general three-dimensional flexible body by combining flexible multibody dynamic simulation and quasi-static finite element analysis (FEA). In the dynamic simulation of flexible multibody systems, flexible components can undergo nonsteady gross motion and small elastic deformation that is described with respect to the body reference frame by using the assumed mode method. D'Alembert inertia loads from the gross body motion and the elastic deformation are expressed as a combination of space-dependent and time-dependent terms that are obtained from the dynamic simulation. D'Alembert inertia loads that are associated with each unit value of the time-dependent terms are then distributed to all finite element nodes in order to compute a corresponding stress influence coefficient through quasi-static structural analyses. Total dynamic stresses due to D'Alembert inertia loads are obtained by multiplying actual magnitude of time-dependent terms with the associated stress influence coefficients. By the proposed method, it is shown that, for a general three-dimensional component, the required number of FEAs can be significantly reduced.     

Analytical approach for the evaluation of the torques using inverse multibody dynamics

A mathematical model for the analysis of human motion is presented in this paper. This model is based on linkage dynamics in order to understand trajectory and internal moment of force coordination. Mobility at the base of the supporting limb is a critical factor in the freedom to fall forward. The approach used to state a coupled system of differential equations of motion consists in introducing the displacement of the center of mass together with the displacement of each segment of the body and to evaluate the final system as a whole. The resultant methodology is task independent. The main goal of this study is to assist the work of health care professionals in the determination of the torques at the joints generated to maintain the movement. The evaluation of a weighted average of all the forces has served as a basis for other authors to obtain the center of pressure. In this work, the resultant ground reaction force passes through the center of mass of the body system enabling the calculation of the location of this force.     

Non-linear dynamics of flexible multibody systems

In this work we set to examine several important issues pertinent to currently very active research area of the finite element modeling of flexible multibody system dynamics. To that end, we first briefly introduce three different model problems in non-linear dynamics of flexible 3D solid, a rigid body and 3D geometrically exact beam, which covers the vast majority of representative models for the particular components of a multibody system. The finite element semi-discretization for these models is presented along with the time-discretization performed by the mid-point scheme. In extending the proposed methodology to modeling of flexible multibody systems, we also present how to build a systematic representation of any kind of joint connecting two multibody components, a typical case of holonomic contraint, as a linear superposition of elementary constraints. We also indicate by a chosen model of rolling contact, an example of non-holonomic constraint, that the latter can also be included within the proposed framework. An important aspect regarding the reduction of computational cost while retaining the consistency of the model is also addressed in terms of systematic use of the rigid component hypothesis, mass lumping and the appropriate application of the explicit-implicit time-integration scheme to the problem on hand. Several numerical simulations dealing with non-linear dynamics of flexible multibody systems undergoing large overall motion are presented to further illustrate the potential of presented methodology. Closing remarks are given to summarize the recent achievements and point out several directions for future research.     

Linear dynamics of flexible multibody systems

We consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path.Enforcing this condition directly in form of a servo constraint leads to differential-algebraic equations (DAEs) of arbitrarily high index. Typically, the model equations are of index 5, which already poses high regularity conditions. If we relax the servo constraints and consider the system from an optimal control point of view, the strong regularity conditions vanish, and the solution can be obtained by standard techniques.By means of the well-known \(n\)-car example and an overhead crane, the theoretical and expected numerical difficulties of the direct DAE and the alternative modeling approach are illustrated. We show how the formulation of the problem in an optimal control context works and address the solvability of the optimal control system. We discuss that the problematic DAE behavior is still inherent in the optimal control system and show how its evidences depend on the regularization parameters of the optimization.     

A logarithmic complexity divide-and-conquer algorithm for multi-flexible-body dynamics including large deformations

A new algorithm is presented for the modeling and simulation of multi-flexible-body systems. This algorithm is built upon a divide-and-conquer-based multibody dynamics framework, and it is capable of handling arbitrary large rotations and deformations in articulated flexible bodies. As such, this work extends the current capabilities of the flexible divide-and-conquer algorithm (Mukherjee and Anderson in Comput. Nonlinear Dyn. 2(1):10–21, 2007), which is limited to the use of assumed modes in a floating frame of reference configuration. The present algorithm utilizes the existing finite element modeling techniques to construct the equations of motion at the element level, as well as at the body level. It is demonstrated that these equations can be assembled and solved using a divide-and-conquer type methodology. In this respect, the new algorithm is applied using the absolute nodal coordinate formulation (ANCF) (Shabana, 1996). The ANCF is selected because of its straightforward implementation and effectiveness in modeling large deformations. It is demonstrated that the present algorithm provides an efficient and robust method for modeling multi-flexible-body systems that employ highly deformable bodies. The new algorithm is tested using three example systems employing deformable bodies in two and three spatial dimensions. The current examples are limited to the ANCF line or cable elements, but the approach may be extended to higher order elements. In its basic form, the divide-and-conquer algorithm is time and processor optimal, yielding logarithmic complexity O(log(N _b )) when implemented using O(N _b ) processors, where N _b is the number of bodies in the system.     

An enhanced inverse dynamic and joint force analysis of multibody systems using constraint matrices

In this paper, a new way of computing the constraint transfer matrix for the inverse dynamic and joint force analysis of multibody systems is developed. The method is based on the Newton–Euler method and the screw theory notations. This method is first developed in (Taghvaeipour et al. in Multibody Syst. Dyn. 29(2):139–168, 2013), however, in this study, it is efficiently modified by incorporating a unified constraint transfer matrix for all types of joints. This change makes both the derivation of the equations and the computations less time consuming. Moreover, in the foregoing procedure, the constraint wrenches of a system are obtained in one reference frame, namely, the global reference frame. As a case study, the proposed method is carried out on the agile wrist which is a three-legged spherical parallel robot with three degrees of freedom. At the end, the results obtained by the modified method are verified with the ones calculated by the original procedure and a software package.

     

On the cosimulation of multibody systems and hydraulic dynamics

Multibody System Dynamics - The simulation of mechanical devices using multibody system dynamics (MBS) algorithms frequently requires the consideration of their interaction with components of a...     

Natural coordinates for the computer analysis of multibody systems

In this paper we will describe a new method for the computer kinematic and dynamic analysis of a wide range of three-dimensional mechanisms or multibody systems. This method is based on a new system of non-independent coordinates that use Cartesian coordinates of points and Cartesian components of unitary vectors in order to describe the position and the motion of the system. Angular coordinates are not used. The kinematic constraint equation comes in two ways, from the rigid-body condition for each element and from the joints or kinematic pairs. The consideration of unitary vectors facilitates considerably the formulation of pair constraints when the pair is associated with a particular direction, as is the case with revolute (R), cylindrical (C), or prismatic (P) pairs. The constraint equations are quadratic in the problem coordinates and they never involve transcendental functions. The dynamic differential equations are obtained in a very simple and effective way from the theorem of virtual power. Finally, two examples will be presented.     

Real-time inverse dynamics control of parallel manipulators using general-purpose multibody software

This work deals with the problem of computing the inverse dynamics of complex constrained mechanical systems for real-time control applications. The main goal is the control of robotic systems using model-based schemes in which the inverse model itself is obtained using a general purpose multibody software, exploiting the redundant coordinate formalism. The resulting control scheme is essentially equivalent to a classical computed torque control, commonly used in robotics applications. This work proposes to use modern general-purpose multibody software to compute the inverse dynamics of complex rigid mechanisms in an efficient way, so that it suits the requirements of realistic real-time applications as well. This task can be very difficult, since it involves a higher number of equations than the relative coordinates approach. The latter is believed to be less general, and may suffer from topology limitations. The use of specialized linear algebra solvers makes this kind of control algorithms usable in real-time for mechanism models of realistic complexity. Numerical results from the simulation of practical applications are presented, consisting in a “delta” robot and a bio-mimetic 11 degrees of freedom manipulator controlled using the same software and the same algorithm.     

Nonsmooth spatial frictional contact dynamics of multibody systems

Multibody System Dynamics - Computational prediction of 3D crutch-assisted walking patterns is a challenging problem that could be applied to study different biomechanical aspects of crutch walking...     

Absolute nodal coordinate plane beam formulation for multibody systems dynamics

A new plane beam dynamic formulation for constrained multibody system dynamics is developed. Flexible multibody system dynamics includes rigid body dynamics and superimposed vibratory motions. The complexity of mechanical system dynamics originates from rotational kinematics, but the natural coordinate formulation does not use rotational coordinates, so that simple dynamic formulation is possible. These methods use only translational coordinates and simple algebraic constraints. A new formulation for plane flexible multibody systems are developed utilizing the curvature of a beam and point masses. Using absolute nodal coordinates, a constant mass matrix is obtained and the elastic force becomes a nonlinear function of the nodal coordinates. In this formulation, no infinitesimal or finite rotation assumptions are used and no assumption on the magnitude of the element rotations is made. The distributed body mass and applied forces are lumped to the point masses. Closed loop mechanical systems consisting of elastic beams can be modeled without constraints since the loop closure constraints can be substituted as beam longitudinal elasticity. A curved beam is modeled automatically. Several numerical examples are presented to show the effectiveness of this method.     

Improved bushing models for general multibody systems and vehicle dynamics

The development and computational implementation, on a multibody dynamics environment, of a constitutive relation to model bushing elements associated with mechanical joints used in the models of road and rail vehicles is presented here. These elements are used to eliminate vibrations in vehicles, due to road irregularities, to allow small misalignment of axes, to reduce noise from the transmission, or to decrease wear of the mechanical joints. Bushings are made of a special rubber, used generally in energy dissipation, which presents a nonlinear viscoelastic relationship between the forces and moments and their corresponding displacements and rotations. In the methodology proposed here a finite element model of the bushing is developed in the framework of the finite element code ABAQUS to obtain the constitutive relations of displacement/rotation versus force/moment for different loading cases. The bushing is modeled in a multibody code as a nonlinear restrain that relates the relative displacements between the bodies connected with the joint reaction forces, and it is represented by a matrix constitutive relation. The basic ingredients of the multibody model are the same vectors and points relations used to define kinematic constraints in any multibody formulation. One particular, and relevant, characteristic of the formulation now presented is its ability to represent standard kinematic joints, clearance, and bushing joints just by defining appropriate constitutive relations. Spherical, revolution, cylindrical, and translational bushing joints are modeled, implemented, and demonstrated through the simulation of two multibody models of a road vehicle, one with perfect kinematic joints for the suspension sub-systems, and other with bushing joints. The tests conducted include an obstacle avoidance maneuver and a vehicle riding over bumps. It is shown that the bushing models for vehicle multibody models proposed here are accurate and computationally efficient so that they can be included in the vehicle models leading reliable simulations.     

Efficient numerical solution of constrained multibody dynamics systems

A numerical algorithm for conducting coupled system dynamical simulation is presented. The interconnected system, comprising numerous modules, is treated as a constrained multibody dynamics system. Of particular focus is the efficient solution of coupled system simulation without sacrificing the independence of the separate dynamical modules. The proposed algorithm, Maggi’s equations with perturbed iteration (MEPI) emanates from numerical methods for differential-algebraic equations. Separate treatment of the constraint equations from the resolution of subsystem dynamical responses marks MEPI’s main characteristic.     

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 parallel Hamiltonian formulation for forward dynamics of closed-loop multibody systems

This paper presents a novel recursive divide-and-conquer formulation for the simulation of complex constrained multibody system dynamics based on Hamilton’s canonical equations (HDCA). The systems under consideration are subjected to holonomic, independent constraints and may include serial chains, tree chains, or closed-loop topologies. Although Hamilton’s canonical equations exhibit many advantageous features compared to their acceleration based counterparts, it appears that there is a lack of dedicated parallel algorithms for multi-rigid-body system dynamics based on the Hamiltonian formulation. The developed HDCA formulation leads to a two-stage procedure. In the first phase, the approach utilizes the divide and conquer scheme, i.e., a hierarchic assembly–disassembly process to traverse the multibody system topology in a binary tree manner. The purpose of this step is to evaluate the joint velocities and constraint force impulses. The process exhibits linear \(O(n)\) (\(n\) – number of bodies) and logarithmic \(O(\log_{2}{n})\) numerical cost, in serial and parallel implementations, respectively. The time derivatives of the total momenta are directly evaluated in the second parallelizable step of the algorithm. Sample closed-loop test cases indicate very small constraint violation errors at the position and velocity level as well as marginal energy drift without any additional form of constraint stabilization techniques involved in the solution process. The results are comparatively set against more standard acceleration based Featherstone’s DCA approach to indicate the performance of the HDCA algorithm.     

Geometrically nonlinear analysis of multibody systems

A method for the dynamic analysis of geometrically nonlinear inertia-variant flexible systems is presented. Systems investigated consist of interconnected rigid and flexible components that undergo large rigid body rotations as well as nonlinear elastic deformations. The differential equations of motion are formulated using Lagrange's equation and nonlinear constraint equations describing mechanical joints in the system are adjoined to the system differential equations of motion using Lagrange's multipliers. A computer program that systematically constructs and numerically solves the system equations of motion is used to predict the effect of the geometric elastic nonlinearities on the dynamic response of flexible multibody systems. The automated formulation presented imposes no limitations on the size of the mechanical systems to be treated. Two examples, namely a slider crank and six-bar mechanisms, are presented to illustrate the effect of introducing geometric nonlinearities to the dynamics of flexible multibody systems.     

An efficient formulation for flexible multibody dynamics using a condensation of deformation coordinates

Multibody System Dynamics - In this work a new approach to deal with non-ideal operative aspects of spatial revolute joints by means of a three-dimensional finite element analysis (3D-FEA) is...