1. Simulation Software: Performances and Examples Dr. Mario Acevedo Multibody Systems and Mechatronics Laboratory Engineering School, UNIVERSIDAD PANAMERICANA (Mexico City)

2. Agenda

Objective and scope

Simulation software: overview

Kinematics simulation

Dynamics simulation

Simulation using web technology

Final remarks

3. Objective and Scope 4. About this Presentation

Objectives:

• Introduce the topic of simulation software for robotic multibody systems

• • Explain the problems that can be solved

• • Show an idea of the implementation

• • Motivate collaboration in the study of problems, prototypes:

• • • The development ofa common language to describe systems (XML)

• • • Use of WEB technologies for publications and collaboration

Scope

• All theory and examples will be treated in 2D.

• • 3D systems are treated in similar way

5. Simulation Software: Overview 6. Simulation of MBS

Many computer codes have been developed but they differ in:

• Model description

• Choice of basic principles of mechanics

• Topological structure

• Numeric vs. Symbolic

• Formulations

7. MBS Simulation Options Modeling Cartesian coordinates Relative coordinates Fully Cartesian coordinates Graph theory Spatial algebra Principles of Mechanics Virtual Power Newton-Euler Hamiltons Principle Lagranges Equations Gibbs-Apell Equations Formulations Spatial Algebra Velocity Transformations Recursive Methods Baumgarte Stabilization Penalty Methods Augmented Lagrangian Numerical Integration ODE Methods Implicit Integrators Explicit Integrators Single step vs Multistep DAE Methods Backward Difference Implicit Runge-Kutta Intelligent Simulator 8. Software for Multibody Systems Simulation 1

ADAMSby Mechanical Dynamics Inc., United States

alaska, by Technical University of Chemnitz, Germany

AUTOLEV, by OnLine Dynamics Inc., United States

AutoSimby Mechanical Simulation Corp., United States

COMPAMMby CEIT, Spain

DADSby CADSI, United States

Dynawizby Concurrent Dynamics International

DynaFlexby University of Waterloo, Canada

HyperviewandMotionviewby Altair Engineering, United States

MECANOby Samtech, Belgium

9. Software for Multibody Systems Simulation 1

MBDynby Politecnico di Milano, Italy

MBSoftby Universite Catholique de Louvain, Belgium

NEWEULby University of Stuttgart, Germany

RecurDynby Function Bay Inc., Korea

Robotranby Universite Catholique de Louvain, Belgium

SAMby Artas Engineering Software, The Netherlands

SD/FASTby PTC, United States

SIMPACKby INTEC GmbH, Germany

Universal Mechanismby Bryansk State Technical University, Russia

Working Modelby Knowledge Revolution, United States

10. Actual State of MBS Simulators Model Description User Interface SOLVER Post-processor 1 Model Description User Interface SOLVER Post-processor 2 Model Description User Interface SOLVER Post-processor n 11. Desired Goal of MBS Model Description ( Neutral Data Format ) User Interface SOLVER Signal Analysis 1 User Interface SOLVER Animation 2 User Interface SOLVER Strength Analysis n Standardized Result Description Visualization 12. Kinematics Simulation

• Modeling:

• • Coordinates, Constraints and Joints library

• Analysis:

• • Positions, Velocities and Accelerations

13. Coordinates for Modeling

Relative coordinates

• Minimum set of coordinates

Cartesian coordinates

• Also known as Reference Point coordinates

Fully Cartesian coordinates

• Also known as Natural Coordinates

Mixed coordinates

14. Constraints Equations

If the selected set of coordinates is dependent, a set of constraint equations can be found

• Constraint equations relate the dependent coordinates and define the movement geometry

• NoC = NoDC NoDOF

• • NoC: Number of Constraints

• • NoDC: Number of Dependent Coordinates

• • NoDOF: Number of Degrees of Freedom

• Constraint equations generally are not linear

15. Relative Coordinates

Open kinematic chain

• Model

Close loop

• Constraints

• • No constraint equations since it is an open kinematic chain

1 2 X Y 16. Relative Coordinates

Close kinematic chain

• Model

Constraints

• NoDC = 3

• NoDOF = 1

• NoC = 3 - 1 = 2

• Non linear

• Transcendental functions

1 X Y 2 O D 3 17. Cartesian Coordinates

Open kinematic chain

• Model

Constraints

• NoDC = 6

• NoDOF = 2

• NoC = 6 - 2 = 4

• Non linear

1 2 X Y y ( x y ( x y 18. Cartesian Coordinates

Close kinematic chain

• Model

Constraints

• NoDC = 9

• NoDOF = 1

• NoC = 9 - 1 = 8

• Non linear

2 O D 1 X Y ( x y ( x y ( x y 3 19. Fully Cartesian Coordinates

Open kinematic chain

• Model

Constraints

• NoDC = 4

• NoDOF = 2

• NoC = 4 - 2 = 2

• Non linear

1 X Y y ( x y ( x y 2 ( x y 20. Fully Cartesian Coordinates

Close kinematic chain

• Model

Constraints

• NoDC = 4

• NoDOF = 1

• NoC = 4 - 1 = 3

• Non linear

2 O D 1 X Y ( x y ( x y 3 21. Constraints Origin

Constraint equations generally are obtained from:

• Close loop equations

• • Relative coordinates

• The rigid body condition of the elements

• • Fully Cartesian coordinates

• Joints definition

• • Cartesian and Fully Cartesian coordinates

Joints definition can be part of a joints library

• Treat the multibody system as a LEGO

• Use computational tools in multibody systems

22. Kinematics of MBS

Set of dependent coordinates:q

Positions analysis.

• Set of constraint equations:

• • Solution using iterative procedures (Newton Raphson)

Velocity analysis:

Acceleration analysis:

( 3 ) ( 2 ) ( 1 ) 23. Joints Definition

Limited to systems in plane (2D)

• Cartesian coordinates

Lower-pairs :revoluteandprismatic

• Show the general modeling for the joint

• Identify the corresponding elements inand

Higher-pairs : gears, cams, etc.

• Require some information on the shape of the connected bodies

• Require to know the shape or curvature of a slot in one of the bodies

24. Modeling of the Revolute Joint X Y i j r i r j P 25. Modeling of the Prismatic Joint 1 X Y i j r i r j P i Q i P j 26. Modeling of the Prismatic Joint 2 27. Kinematics Simulation Computer Implementation Examples 28. Dynamics Simulation

• Constraint Dynamics

• • Lagrange multipliers

• • Velocity transformations

• • Numerical integration

29. Lagrange Multipliers

The general form of equations of motion using Lagrange multipliers is

• This equation representsmequations innunknowns, it is necessary to givenmore equations, a possibility are acceleration equations

Equations to solve

( 4 ) ( 5 ) 30. Velocity Transformations

Based on the fact that it is possible to express equations (5) in terms of a different set coordinates by a linear transformation (velocity transformation)

• Open loop systems

• Close loop systems

• • Lagrange multipliers

• • Second velocity transformation

( 7 ) ( 6 ) ( 8 ) ( 9 ) ( 10 ) 31. Numerical vs Symbolical Model description Data input Formalism Numerical equations Simulation Local output Global result Next time step Model variation Parameter variation Model description Data input Formalism Symbolical equations Simulation Local output Global result Next time step Model variation Parameter variation 32. Dynamics Simulation

Lagrange Multipliers

• Computer Implementation

• Examples

• • Inverse dynamics

• • Direct dynamics

33. Dynamics Simulation

Velocity transformations

• Computer Implementation

• Examples

• • Inverse dynamics

• • Direct dynamics

34. Simulation using WEB WEB Server Active Pages Java/JavaScript 35. References 1

Cuadrado, J. et.al. Modeling and Solution Methods for Efficient Real-Time Simulation of Multibody Dynamics , Multibody Systems Dynamics, Vol. 1, No. 3, 1997.

Garca de Jaln, J. and Bayo E.,Kinematic and Dynamic Simulation of Multibody Systems, The Real-Time Challenge , Springer-Verlag, 1993.

Schiehlen, S., Multibody Systems Dynamics: Roots and Perspectives , Multibody Systems Dynamics, Vol. 1, No. 2, 1997.

Shabana, A.,Computational Dynamics , Wiley, 1994.