# Real-time simulation of multibody systems with applications for working mobile vehicles

##### Baharudin, Mohamad Ezral (2016-04-01)

Väitöskirja

Baharudin, Mohamad Ezral

01.04.2016

Lappeenranta University of Technology

Acta Universitatis Lappeenrantaensis

**Julkaisun pysyvä osoite on**

https://urn.fi/URN:ISBN:978-952-265-933-0

#### Tiivistelmä

This dissertation describes an approach for developing a real-time simulation for working

mobile vehicles based on multibody modeling. The use of multibody modeling allows

comprehensive description of the constrained motion of the mechanical systems involved

and permits real-time solving of the equations of motion. By carefully selecting the

multibody formulation method to be used, it is possible to increase the accuracy of the

multibody model while at the same time solving equations of motion in real-time.

In this study, a multibody procedure based on semi-recursive and augmented Lagrangian

methods for real-time dynamic simulation application is studied in detail. In the semirecursive

approach, a velocity transformation matrix is introduced to describe the

dependent coordinates into relative (joint) coordinates, which reduces the size of the

generalized coordinates. The augmented Lagrangian method is based on usage of global

coordinates and, in that method, constraints are accounted using an iterative process.

A multibody system can be modelled as either rigid or flexible bodies. When using

flexible bodies, the system can be described using a floating frame of reference

formulation. In this method, the deformation mode needed can be obtained from the finite

element model. As the finite element model typically involves large number of degrees

of freedom, reduced number of deformation modes can be obtained by employing model

order reduction method such as Guyan reduction, Craig-Bampton method and Krylov

subspace as shown in this study

The constrained motion of the working mobile vehicles is actuated by the force from the

hydraulic actuator. In this study, the hydraulic system is modeled using lumped fluid

theory, in which the hydraulic circuit is divided into volumes. In this approach, the

pressure wave propagation in the hoses and pipes is neglected. The contact modeling is

divided into two stages: contact detection and contact response. Contact detection

determines when and where the contact occurs, and contact response provides the force

acting at the collision point. The friction between tire and ground is modelled using the

LuGre friction model, which describes the frictional force between two surfaces.

Typically, the equations of motion are solved in the full matrices format, where the

sparsity of the matrices is not considered. Increasing the number of bodies and constraint

equations leads to the system matrices becoming large and sparse in structure. To increase

the computational efficiency, a technique for solution of sparse matrices is proposed in

this dissertation and its implementation demonstrated. To assess the computing

efficiency, augmented Lagrangian and semi-recursive methods are implemented

employing a sparse matrix technique. From the numerical example, the results show that

the proposed approach is applicable and produced appropriate results within the real-time

period.

mobile vehicles based on multibody modeling. The use of multibody modeling allows

comprehensive description of the constrained motion of the mechanical systems involved

and permits real-time solving of the equations of motion. By carefully selecting the

multibody formulation method to be used, it is possible to increase the accuracy of the

multibody model while at the same time solving equations of motion in real-time.

In this study, a multibody procedure based on semi-recursive and augmented Lagrangian

methods for real-time dynamic simulation application is studied in detail. In the semirecursive

approach, a velocity transformation matrix is introduced to describe the

dependent coordinates into relative (joint) coordinates, which reduces the size of the

generalized coordinates. The augmented Lagrangian method is based on usage of global

coordinates and, in that method, constraints are accounted using an iterative process.

A multibody system can be modelled as either rigid or flexible bodies. When using

flexible bodies, the system can be described using a floating frame of reference

formulation. In this method, the deformation mode needed can be obtained from the finite

element model. As the finite element model typically involves large number of degrees

of freedom, reduced number of deformation modes can be obtained by employing model

order reduction method such as Guyan reduction, Craig-Bampton method and Krylov

subspace as shown in this study

The constrained motion of the working mobile vehicles is actuated by the force from the

hydraulic actuator. In this study, the hydraulic system is modeled using lumped fluid

theory, in which the hydraulic circuit is divided into volumes. In this approach, the

pressure wave propagation in the hoses and pipes is neglected. The contact modeling is

divided into two stages: contact detection and contact response. Contact detection

determines when and where the contact occurs, and contact response provides the force

acting at the collision point. The friction between tire and ground is modelled using the

LuGre friction model, which describes the frictional force between two surfaces.

Typically, the equations of motion are solved in the full matrices format, where the

sparsity of the matrices is not considered. Increasing the number of bodies and constraint

equations leads to the system matrices becoming large and sparse in structure. To increase

the computational efficiency, a technique for solution of sparse matrices is proposed in

this dissertation and its implementation demonstrated. To assess the computing

efficiency, augmented Lagrangian and semi-recursive methods are implemented

employing a sparse matrix technique. From the numerical example, the results show that

the proposed approach is applicable and produced appropriate results within the real-time

period.

##### Kokoelmat

- Väitöskirjat [975]