Dynamics of Multibody Systems: Conventional and Graph...

Dynamics of Multibody Systems: Conventional and

Graph-Theoretic Approaches

SD 652 John McPhee Systems Design Engineering University of Waterloo, Canada

Mechatronic multibody system dynamics J. McPhee

Summary of Course:

1. Review of kinematics and dynamics

2. Conventional multibody dynamics 1. Planar systems 2. Spatial systems and MSC.Adams

3. Graph-theoretic modelling: 1. One-dimensional systems 2. Multibody systems and MapleSim

4. Advanced topics

1. Multibody Mechanical Systems a collection of rigid and flexible bodies connected by

joints, e.g.

Serial Robots Parallel Robots (PKMs)

– Walking robots:

http://real.uwaterloo.ca/~robot

– Mechanisms and machinery:

Spatial slider-crank mechanism

– Vehicles (road, rail, aerospace):

Lola World Sports Car

Multibody system dynamics: given only a description of the system as input, formulate the kinematic and dynamic equations needed to determine the system response.

System Equations

Dynamic simulation

Sensitivity + optimization

Model-based control

Example multibody system:

Planar slider-crank mechanism (f=1DOF)

– Coordinates: • Must be defined a priori • Selection affects the number and nature of equations • Absolute coordinates:

– Position and orientation of every body in system, e.g.

– Easy to formulate equations automatically – Very large systems of equations

• Joint (relative) coordinates: – Correspond to joints in system, e.g.

– Fewer in number (minimum for open-loop systems) – Requires more topological accounting

Tyxyxyx ],,,,,,,,[ 333222111 θθθ=q

Ts],,[ βθ=q

– Kinematic Analysis: • One prescribed motion per dof, e.g. if θ = f(t) for

slider-crank, solve the kinematic constraint equations for q(t)=[θ,β,s]T:

• For velocities, solve:

0qΦ =

−−

)(cossin

sincos),( 21

θβθ

−−=⇒−=

0010sincos1cossin

βθβθ

t qq ΦΦqΦ

– Dynamic equations from Newton-Euler, or from the Principle of Virtual Work:

– Expressing all variables in terms of q:

– Set the n generalized forces Q=0:

0=+++= ∑∑∑∑FBRBFT n

T WWW δδδδδ rFθT

( )( ) dVdVfW

σεar

θIωωωIra

∫∫ −−=

×+−−=

δρδδ

δδδ

FλΦqM q =+ T

0== qQ δδ TW

– Where, for the planar slider-crank:

A dynamic simulation is obtained by solving the n+m

differential-algebraic equations for q(t) and λ(t).

[ ]TLmF 2/sin,0,0 222 ββ−=F

+−−=

2/cos02/cos3/0

mmLmLmLm

−−=

0sincos1cossin

βθβθ

2. Conventional Methods for Multibody Systems

Automated modelling and simulation

Based on absolute coordinates Large systems of nonlinear DAEs Numerical data, not symbolic equations Commercial software:

– Working Model – MSC.Adams

4th-year (senior) design projects:

Hexplorer 6-legged walking robot

Hexplorer 3-DOF leg in extended position

ADAMS simulation of walking maneuver

SAE mini-Baja vehicle:

Winner of ADAMS modelling award

Research into vehicle stability:

Grapple Skidder (Timberjack Inc)

Research into vehicle stability:

ADAMS simulation of roll-over

Research into mechanisms and machinery:

6-bar mechanism designs

Research into vehicle suspension design:

Lola World Sports Car (Multimatic Inc)

– ADAMS model of four-post test:

– no chassis flexibility or joint compliance – rear left-hand suspension:

Research into biomechanics: – Investigation of metabolic energy consumption

for normal and prosthetic gaits.

Research into biomechanics: – Forward dynamic simulation, realistic friction

Research into biomechanics: – Forward dynamic simulation, low friction

3. Modelling using Linear Graph Theory Origins in Koningsburg, Prussia, 1732:

Topology = 4 land masses connected by 7 bridges:

Leonhard Euler’s sketch of Koningsburg topology

Euler’s “linear graph” representation of topology:

If there are more than 2 nodes with odd valence, then an “Eulerian path” cannot exist, Euler (1732)

Component models from measurements

– Edge = element, Nodes = connection points

– Through variable

– Across variable

– Constitutive equations, e.g.

)(current i=τ

τα ,

)(voltage v=α

dtdiLv =

“Graph-Theoretic Modelling” (GTM)

System model from e assembled components:

Electrical Circuit and Linear Graph

– e constitutive equations (linear or nonlinear) – e linear topological equations from system graph – primary variables determined by tree selection

– Linear graph (nodes=frames, edges=elements):

– Vector variables: – Cutsets = dynamic equilibrium for a subsystem – Circuits = summation of vector displacements

around closed kinematic chains

),(,),( θατ rTF ==

Advantages of G-T modelling approach: – Very systematic – Amenable to computer implementation – Leads to efficient systems of equations – Suitable for real-time simulation, e.g. for virtual

reality or hardware-in-loop experiments – Applicable to multiple physical domains – Unifying: coordinates may be absolute or joint or

other possibilities

– n branch coordinates q are defined by tree selection:

yxyxyxmmmrrTreesshhrrTree

],,,,,,,,[},,,{

],,[},,,{

33322211132174

111081110874

θθθ

=⇒−=

Mechatronic Multibody Systems

Robot control system to capture moving payload tracked by overhead vision system:

http://real.uwaterloo.ca/~watflex

Electrical network + multibody system, coupled by transducer elements:

Linear graph representation: – The DC-motors (transducers) have an edge in both

the mechanical and electrical sub-graphs.

– the equations for the individual domains are obtained using the electrical and multibody formulations.

– these equations are coupled by the constitutive equations for the transducers, e.g. for the DC-motor:

– from a single linear graph representation, the governing equations are automatically derived in symbolic form by a Maple program “DynaFlexPro”, now part of the MapleSim package.

dtdBiKT

dtdiLRi

Maple Algorithms (MapleSim)

Flowchart:

Model Description (ASCII File - *.dfp)

Mathematical Model (Maple Module)

1) Build Model

2) Build Equations

3) Build Sim Code

Optimized Simulation Code

Human and Robotic Slapshots

Motion capture analysis of slapshot

Downswing Puck contact

Synthesis of 4-bar hockey robot “Thor”

MapleSim model

MapleSim animation

Modelling of mechatronic systems

Modelling of contact dynamics

Vehicle dynamics and tires

4. Advanced Topics:

Mechatronic system models:

Condensator microphone (Hadwich and Pfeiffer, 1995)

– Linear graph representation:

– where, for the moving-plate capacitor,

dtdvsCi

– Selecting the trees shown, and using the current formulation, one obtains 2 equations in terms of and

– where:

=−+++

sdCgmsksdsm

ivdtds

dtdvsC

3212 tEdtdiLiRv +−−=

)(2 ti )(ts

Mechatronic system models:

– Linear graph of multibody system + induction motor :

– Rotation of input link:

– Current through one rotor inductor:

Contact dynamic models:

– Discrete versus continuous models [Gilardi and Sharf] – Hunt-Crossley model with modified damping:

– Sphere dropped inside cylinder:

( )nnn 1 xaxkf p +=

– Volumetric contact model [Gonthier et al, 2006]:

– Rolling resistance, etc, also a function of geometry:

fn 1 xaV

hkf +=

– Volumetric contact model [Gonthier et al, 2006]:

Contact dynamic applications:

Vehicle dynamics and tires:

– Application to planetary rovers [Petersen, 2011]:

Juno rover

MapleSim model

MapleSim model, with tire/soil interactions

Dynamics of Multibody Systems: Conventional and

Graph-Theoretic Approaches

SD 652 John McPhee Systems Design Engineering University of Waterloo, Canada

Dynamics of Multibody Systems: Conventional and Graph...

Documents

Transcript of Dynamics of Multibody Systems: Conventional and Graph...

Multibody Dynamics Modeling and System Identification for ...

Multidisciplinary Applications Using Multibody Dynamics as the … · 2015-03-17 · The flexible multibody dynamics represents another area where traditional multibody formalisms

Multibody dynamics notation (version 2)

A Quick Tutorial on Multibody DynamicsA good understanding of multibody dynamics is paramount for designing eﬀective controllers. There are many ways to learn multibody dynamics.

Dynamics of multibody systems

Flexible multibody dynamics Superelements using absolute ...€¦ · FLEXIBLE MULTIBODY DYNAMICS SUPERELEMENTS USING ABSOLUTE INTERFACE COORDINATES IN THE FLOATING FRAME FORMULATION

Multibody Dynamics in Natural Coordinates through ...

Multibody Dynamics Equations of Motion for Multibody ...

Multibody Dynamics Module - COMSOL Multiphysics

Dynamics and Control of Multibody Systems

Multibody Dynamics 2013eccomas-multibody.fsb.hr/docs/mbd2013-CD... · Multibody Dynamics 2013 Edited by Zdravko Terze ... and numerical methods for dynamic analysis, ... Multibody

Getting Started with Multibody Dynamics

Multibody dynamics - Georgia Institute of Technology · coordinates, multibody dynamics is a straightforward extension of a single rigid body

Multibody Dynamics Explain

Multibody Dynamics

Multibody Dynamics With Abaqus.pdf

Multibody Dynamics Module

MBDyn - Multibody System Dynamics · This presentation is split in two parts: • Multibody System Dynamics, with specific reference to MBDyn, the free general-purpose multibody solver

Dynamics Algorithms for Multibody Systems - Weebly

Multibody System Dynamics: MBDyn Overview · Multibody System Dynamics: MBDyn Overview ... Multibody dynamics ... • Static analysis as degeneration of IVP dynamic analysis: