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Vermelding onderdeel organisatie
1
A Multibody Dynamics Benchmark on the Equations of Motion of an Uncontrolled Bicycle
Fifth EUROMECH Nonlinear Dynamics ConferenceENOC-2005, Eindhoven, The Netherlands, 7-12 August 2005
Laboratory for Engineering MechanicsFaculty of Mechanical EngineeringDelft University of Technology The Netherlands
Arend L. SchwabGoogle: Arend Schwab [I’m Feeling Lucky]
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Aug 9, 2005 2
Acknowledgement
TUdelft:Jaap Meijaard 1
Jodi Kooiman
Cornell University:Andy RuinaJim Papadopoulos 2
Andrew Dressel
1) School of MMME, University of Nottingham, England, UK2) PCMC , Green Bay, Wisconsin, USA
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Aug 9, 2005 3
Motto
Everybody knows how a bicycle is constructed …
… yet nobody fully understands its operation!
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Aug 9, 2005 5
Experiment
Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park
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Aug 9, 2005 6
Some Advice
Don’t try this at home !
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Aug 9, 2005 7
Contents
• Bicycle Model• Equations of Motion• Steady Motion and Stability• Benchmark Results• Experimental Validation• Conclusions
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Aug 9, 2005 8
The Model
Modelling Assumptions:
• rigid bodies• fixed rigid rider• hands-free• symmetric about vertical
plane• point contact, no side slip• flat level road• no friction or propulsion
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Aug 9, 2005 9
The Model
4 Bodies → 4*6 coordinates(rear wheel, rear frame (+rider), front frame, front wheel)
Constraints:3 Hinges → 3*5 on coordinates2 Contact Pnts → 2*1 on coordinates
→ 2*2 on velocities
Leaves: 24-17 = 7 independent Coordinates, and24-21 = 3 independent Velocities (mobility)
The system has: 3 Degrees of Freedom, and4 (=7-3) Kinematic Coordinates
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Aug 9, 2005 10
The Model
3 Degrees of Freedom:
4 Kinematic Coordinates:
lean angle
steer angle
rear wheel rot.
d
r
q
r
r
front wheel rot.
yaw angle rear frame
rear contact pnt.
rear contact pnt.
f
k
x
y
q
Input File with model definition:
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Aug 9, 2005 11
Eqn’s of Motion
1
dd
d
d d
k dt
q M f
q q
q Aq b
State equations:
with TM T MT and T f T f Mh
For the degrees of freedom eqn’s of motion:
and for kinematic coordinates nonholonomic constraints:
dq
kq
T d T T MTq T f Mh
k d q Aq b
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Aug 9, 2005 12
Steady Motion
0d
constantd
constant
d
d
kt
q
q
q
Steady motion:
Stability of steady motion by linearized eqn’s of motion:
and linearized nonholonomic constraints:
d d d d k k M q C q K q K q 0
k d d d k k q A q B q B q
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Aug 9, 2005 13
Linearized State
d d k d
d d
k d k k
M 0 0 q C K K q 0
0 I 0 q I 0 0 q 0
0 0 I q A B B q 0
1
dd
d
d d
k dt
q M f
q q
q Aq b
Linearized State equations:
State equations:
with, d
T T q
C T CT T Mh
, , , ,d k T T T q q q qK K K T KF T Mx f T Mh Cvand
and ,d k qB B B b
Green: holonomic systems
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Aug 9, 2005 14
Straight Ahead Motion
d d k d
d d
k d k k
M 0 0 q C K K q 0
0 I 0 q I 0 0 q 0
0 0 I q A B B q 0
Turns out that the Linearized State eqn’s:
Upright, straight ahead motion :
lean angle 0
steer angle 0
rear wheel rot. speed / constantr v r
0
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Aug 9, 2005 15
Straight Ahead Motion
d d k d
d d
k d k k
M 0 0 q C K K q 0
0 I 0 q I 0 0 q 0
0 0 I q A B B q 0
Linearized State eqn’s:
Moreover, the lean angleand the steer angle are decoupled from the rear wheel rotation r (forward speed ), resulting in:
0
rv r
x x 0 x x 0 x x 0
x x 0 , x x 0 , x x 0
0 0 x 0 0 0 0 0 0
d
M C K
lean angle
steer angle
rear wheel rot.
d
r
qwith
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Aug 9, 2005 16
Stability of Straight Ahead Motion
with and a constant forward speed
Linearized eqn’s of motion for lean and steering:
1 0 2
130 3 0 40 1003 27 0 96, , ,
3 0.3 0.6 1.8 27 8.8 0 2.7
M C K K
21 0 2( ) ( ) 0d d dv v Mq C q K K q
lean
steer d
q rv r
For a standard bicycle (Schwinn Crown) :
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Aug 9, 2005 17
Root Loci Parameter: forward speed
rv r
v
vv
Stable forward speed range 4.1 < v < 5.7 m/s
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Aug 9, 2005 18
Check Stability by full non-linear forward dynamic analysis
Stable forward speed range 4.1 < v < 5.7 m/s
forward speedv [m/s]:
01.75
3.53.68
4.9
6.3
4.5
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Aug 9, 2005 19
Comparison
A Brief History of Bicycle Dynamics Equations
- 1899 Whipple- 1901 Carvallo- 1903 Sommerfeld & Klein- 1948 Timoshenko, Den Hartog- 1955 Döhring- 1967 Neimark & Fufaev- 1971 Robin Sharp- 1972 Weir- 1975 Kane- 1983 Koenen- 1987 Papadopoulos
- and many more …
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Aug 9, 2005 20
ComparisonFor a standard and distinct type of bicycle + rigid rider combination
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Aug 9, 2005 21
ComparePapadopoulos (1987) with Schwab (2003) and Meijaard (2003)
1: Pencil & Paper 2: SPACAR software 3: AUTOSIM software
Relative errors in the entries in M, C and K are
< 1e-12
Perfect Match!
21 0 2( ) ( ) 0d d dv v Mq C q K K q
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Aug 9, 2005 22
Experimental Validation
Instrumented Bicycle, uncontrolled
2 rate gyros:
-lean rate
-yaw rate
1 speedometer:
-forward speed
1 potentiometer
-steering angle
Laptop + Labview
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Aug 9, 2005 23
Experimental Validation
Linearized stability of the Uncontrolled Instrumented Bicycle
Stable forward speed range:
4.0 < v < 7.8 [m/s]
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Aug 9, 2005 24
An Experiment
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Aug 9, 2005 25
Measured Data
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Aug 9, 2005 26
Extract EigenvaluesStable Weave motion is dominant
Nonlinear fit function on the lean rate:
11 2 2 3 2e [ cos( ) sin( )]tc c t c t
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Aug 9, 2005 27
Extract Eigenvalues & Compare
Nonlinear fit function on the lean rate:
11 2 2 3 2e [ cos( ) sin( )]tc c t c t
2 = 5.52 [rad/s]
1 = -1.22 [rad/s]
forward speed:
4.9 < v <5.4 [m/s]
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Aug 9, 2005 28
Compare around critical weave speed
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Aug 9, 2005 29
Just below critical weave speed
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Aug 9, 2005 30
Compare at high and low speed
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Aug 9, 2005 31
Conclusions
- The Linearized Equations of Motion are Correct.
Future Investigation:
- Add a controller to the instrumented bicycle -> robot bike.
- Investigate stability of steady cornering.
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Aug 9, 2005 32
MATLAB GUI for Linearized Stability
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Aug 9, 2005 33
Myth & Folklore
A Bicycle is self-stable because:
- of the gyroscopic effect of the wheels !?
- of the effect of the positive trail !?
Not necessarily !
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Aug 9, 2005 34
Myth & Folklore
Forward speedv = 3 [m/s]:
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Aug 9, 2005 35
Steering a Bike
To turn right you have to steer …
briefly to the LEFT
and then let go of the handle bars.
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Aug 9, 2005 36
Steering a BikeStandard bike with rider at a stable forward speed of 5 m/s, after 1 second we apply a steer torque of 1 Nm for ½ a secondand then we let go of the handle bars.
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Aug 9, 2005 37
Conclusions
- The Linearized Equations of Motion are Correct.
- A Bicycle can be Self-Stable even without Rotating Wheels and with Zero Trail.
Future Investigation:
- Validate the modelling assumptions by means of experiments.
- Add a human controller to the model.
- Investigate stability of steady cornering.