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Benchmark Results on the Stability of an Uncontrolled BicycleMechanics Seminar

May 16, 2005 DAMTP, Cambridge University, UK

Laboratory for Engineering MechanicsFaculty of Mechanical Engineering

Arend L. SchwabGoogle: Arend Schwab [I’m Feeling Lucky]

May 16, 2005 2

Acknowledgement

TUdelft:Jaap Meijaard 1

Cornell University:Andy RuinaJim Papadopoulos 2

Andrew Dressel

1) School of MMME, University of Nottingham, England, UK2) PCMC , Green Bay, Wisconsin, USA

May 16, 2005 3

Motto

Everbody knows how a bicycle is constructed …

… yet nobody fully understands its operation!

May 16, 2005 4

Experiment

Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park

May 16, 2005 5

Experiment

Cornell University, Ithaca, NY, 1987: Yellow Bike in the Car Park

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Experiment

Don’t try this at home !

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Contents

• Bicycle Model• Equations of Motion• Steady Motion and Stability• Benchmark Results• Myth and Folklore• Steering• Conclusions

May 16, 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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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

May 16, 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:

May 16, 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

May 16, 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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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

May 16, 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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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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Stability of Straight Ahead Motion

with and the 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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Root Loci Parameter: forward speed

rv r

v

vv

Stable forward speed range 4.1 < v < 5.7 m/s

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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

May 16, 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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ComparisonFor a standard and distinct type of bicycle + rigid rider combination

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ComparePapadopoulos (1987) with Schwab (2003) and Meijaard (2003)

pencil & paper SPACAR software 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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MATLAB GUI for Linearized Stability

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Myth & Folklore

A Bicycle is self-stable because:

- of the gyroscopic effect of the wheels !?

- of the effect of the positive trail !?

Not necessarily !

May 16, 2005 24

Myth & Folklore

Forward speedv = 3 [m/s]:

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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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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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Conclusions

- The Linearized Equations of Motion are Correct.

- A Bicycle can be Self-Stable even without Rotating Wheels and with Zero Trail.

Future Investigation:

- Add a human controler to the model.

- Investigate stability of steady cornering.