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
Mechatronic multibody system dynamics J. McPhee
1. Multibody Mechanical Systems a collection of rigid and flexible bodies connected by
joints, e.g.
Serial Robots Parallel Robots (PKMs)
Mechatronic multibody system dynamics J. McPhee
– Walking robots:
http://real.uwaterloo.ca/~robot
Mechatronic multibody system dynamics J. McPhee
– Mechanisms and machinery:
Spatial slider-crank mechanism
Mechatronic multibody system dynamics J. McPhee
– Vehicles (road, rail, aerospace):
Lola World Sports Car
Mechatronic multibody system dynamics J. McPhee
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
Mechatronic multibody system dynamics J. McPhee
Example multibody system:
Planar slider-crank mechanism (f=1DOF)
Mechatronic multibody system dynamics J. McPhee
– 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
Mechatronic multibody system dynamics J. McPhee
– 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
Mechatronic multibody system dynamics J. McPhee
– 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
Mechatronic multibody system dynamics J. McPhee
– 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
Mechatronic multibody system dynamics J. McPhee
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
Mechatronic multibody system dynamics J. McPhee
4th-year (senior) design projects:
Hexplorer 6-legged walking robot
Mechatronic multibody system dynamics J. McPhee
Hexplorer 3-DOF leg in extended position
Mechatronic multibody system dynamics J. McPhee
ADAMS simulation of walking maneuver
Mechatronic multibody system dynamics J. McPhee
SAE mini-Baja vehicle:
Winner of ADAMS modelling award
Mechatronic multibody system dynamics J. McPhee
Research into vehicle stability:
Grapple Skidder (Timberjack Inc)
Mechatronic multibody system dynamics J. McPhee
Research into vehicle stability:
ADAMS simulation of roll-over
Mechatronic multibody system dynamics J. McPhee
Research into mechanisms and machinery:
6-bar mechanism designs
Mechatronic multibody system dynamics J. McPhee
Research into vehicle suspension design:
Lola World Sports Car (Multimatic Inc)
Mechatronic multibody system dynamics J. McPhee
– ADAMS model of four-post test:
Mechatronic multibody system dynamics J. McPhee
– no chassis flexibility or joint compliance – rear left-hand suspension:
Mechatronic multibody system dynamics J. McPhee
Research into biomechanics: – Investigation of metabolic energy consumption
for normal and prosthetic gaits.
Mechatronic multibody system dynamics J. McPhee
Research into biomechanics: – Forward dynamic simulation, realistic friction
Mechatronic multibody system dynamics J. McPhee
Research into biomechanics: – Forward dynamic simulation, low friction
Mechatronic multibody system dynamics J. McPhee
3. Modelling using Linear Graph Theory Origins in Koningsburg, Prussia, 1732:
Mechatronic multibody system dynamics J. McPhee
Topology = 4 land masses connected by 7 bridges:
Leonhard Euler’s sketch of Koningsburg topology
Mechatronic multibody system dynamics J. McPhee
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
Mechatronic multibody system dynamics J. McPhee
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)
Mechatronic multibody system dynamics J. McPhee
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
Mechatronic multibody system dynamics J. McPhee
– 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 ==
Mechatronic multibody system dynamics J. McPhee
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
Mechatronic multibody system dynamics J. McPhee
– n branch coordinates q are defined by tree selection:
T
T
yxyxyxmmmrrTreesshhrrTree
],,,,,,,,[},,,{
],,[},,,{
33322211132174
111081110874
θθθ
βθ
=⇒−=
=⇒−=
Mechatronic multibody system dynamics J. McPhee
Mechatronic Multibody Systems
Mechatronic multibody system dynamics J. McPhee
Robot control system to capture moving payload tracked by overhead vision system:
http://real.uwaterloo.ca/~watflex
Mechatronic multibody system dynamics J. McPhee
Electrical network + multibody system, coupled by transducer elements:
Mechatronic multibody system dynamics J. McPhee
Linear graph representation: – The DC-motors (transducers) have an edge in both
the mechanical and electrical sub-graphs.
Mechatronic multibody system dynamics J. McPhee
– 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
−=
++=
Mechatronic multibody system dynamics J. McPhee
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
Human and Robotic Slapshots
Downswing Puck contact
Synthesis of 4-bar hockey robot “Thor”
Human and Robotic Slapshots
MapleSim model
Human and Robotic Slapshots
MapleSim animation
Mechatronic multibody system dynamics J. McPhee
Modelling of mechatronic systems
Modelling of contact dynamics
Vehicle dynamics and tires
4. Advanced Topics:
Mechatronic multibody system dynamics J. McPhee
Mechatronic system models:
Condensator microphone (Hadwich and Pfeiffer, 1995)
Mechatronic multibody system dynamics J. McPhee
– Linear graph representation:
– where, for the moving-plate capacitor,
22
22
222
22
)(21
)()(
vds
sdCF
vdtds
dssdC
dtdvsCi
=
+=
Mechatronic multibody system dynamics J. McPhee
– 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 multibody system dynamics J. McPhee
Mechatronic system models:
Mechatronic multibody system dynamics J. McPhee
– Linear graph of multibody system + induction motor :
Mechatronic multibody system dynamics J. McPhee
– Rotation of input link:
Mechatronic multibody system dynamics J. McPhee
– Current through one rotor inductor:
Mechatronic multibody system dynamics J. McPhee
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]:
Mechatronic multibody system dynamics J. McPhee
Contact dynamic applications:
Mechatronic multibody system dynamics J. McPhee
Vehicle dynamics and tires:
– Application to planetary rovers [Petersen, 2011]:
Juno rover
MapleSim model
– Application to planetary rovers [Petersen, 2011]:
MapleSim model, with tire/soil interactions
– Application to planetary rovers [Petersen, 2011]:
– Application to planetary rovers [Petersen, 2011]:
Dynamics of Multibody Systems: Conventional and
Graph-Theoretic Approaches
SD 652 John McPhee Systems Design Engineering University of Waterloo, Canada
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