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

21

tfLL

sLLt

θβθ

βθ

−−=⇒−=

0010sincos1cossin

21

21

βθβθ

LLLL

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

FBn

RBn

T

n

T WWW δδδδδ rFθT

( )( ) dVdVfW

mW

V

T

Vb

TFB

TTRB

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

+−−=

3222

22222

211

2/cos02/cos3/0

003/

mmLmLmLm

Lm

ββM

−−=

0sincos1cossin

21

21

βθβθ

LLLL

qΦ


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)

A

B

D

C

a

b

c

d e

f

g


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:

T

T

yxyxyxmmmrrTreesshhrrTree

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

],,[},,,{

33322211132174

111081110874

θθθ

βθ

=⇒−=

=⇒−=

qq


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

dtdKv

T

v

θ

θ

−=

++=


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,

22

22

222

22

)(21

)()(

vds

sdCF

vdtds

dssdC

dtdvsCi

=

+=


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

– where:

0)(21

0)()(

22

25665

2222

2

=−+++

=−+

vds

sdCgmsksdsm

ivdtds

dssdC

dtdvsC

)(42

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:

( )nf

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...

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