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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]

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

Aug 9, 2005 3

Motto

Everybody knows how a bicycle is constructed …

… yet nobody fully understands its operation!

Aug 9, 2005 5

Experiment

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

Aug 9, 2005 6

Some Advice

Don’t try this at home !

Aug 9, 2005 7

Contents

• Bicycle Model• Equations of Motion• Steady Motion and Stability• Benchmark Results• Experimental Validation• Conclusions

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

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

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:

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

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

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

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

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

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) :

Aug 9, 2005 17

Root Loci Parameter: forward speed

rv r

v

vv

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

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

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 …

Aug 9, 2005 20

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

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

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

Aug 9, 2005 23

Experimental Validation

Linearized stability of the Uncontrolled Instrumented Bicycle

Stable forward speed range:

4.0 < v < 7.8 [m/s]

Aug 9, 2005 24

An Experiment

Aug 9, 2005 25

Measured Data

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

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]

Aug 9, 2005 28

Compare around critical weave speed

Aug 9, 2005 29

Just below critical weave speed

Aug 9, 2005 30

Compare at high and low speed

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.

Aug 9, 2005 32

MATLAB GUI for Linearized Stability

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 !

Aug 9, 2005 34

Myth & Folklore

Forward speedv = 3 [m/s]:

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.

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.

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.