Bicycles - A Mechatronics View · Bicycles - A Mechatronics View K. J. Åström Lund University 1. Introduction 2. Modeling 3. Stabilization 4. Rear wheel steering 5. Maneuvering

Bicycles - A Mechatronics View

K. J. ÅströmLund University

1. Introduction

2. Modeling

3. Stabilization

4. Rear wheel steering

5. Maneuvering and stabilization

6. Conclusions

c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 1

But don’t forget Control!c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 2

The US Army Motor Bike - A Design Disaster

F. R. Whitt and D. G. Wilson (1974) Bicycling Science - Er-gonomics and Mechanics. MIT Press Cambridge, MA.

“Many people have seen theoretical advantages in the fact thatfront-drive, rear-steered recumbent bicycles would have simplertransmissions than rear-driven recumbents and could have thecenter of mass nearer the front wheel than the rear. The U.S.Department of Transportation commissioned the constructionof a safe motorcycle with this configuration. It turned out to besafe in an unexpected way: No one could ride it.”

The Santa Barbara Connection


Why Bicycles are Interesting?

• Very common efficient means of transportation andrecreation

• Interesting non-trivial behavior

• Suitable to illustrate many issues in mechatronics

– Modeling– Dynamics and control– Integrated systems and control design– The role of sensors and actuators

• Useful for education

– Simple and illustrative experiments– High student attraction


Some Interesting Questions

• How do you stabilize a bicycle?

– By steering or by leaning?

• Do you normally stabilize a bicycle when you ride it?

• Why is it possible to ride without touching the handle bar?

• How is stabilization influenced by the design of the bike?

• Why does the front fork look the way it does?

• The main message:

– A bicycle is a feedback system, the action of the frontfork is the key!



K. J. Åström

1. Introduction

2. Modeling

3. Stabilization



6. Conclusions


Bicycle Modeling

Sommerfeld (1952) Mechanics. Lectures on TheoreticalPhysics, Vol 1. Academic Press 1952.

F. Klein and A. Sommerfeld Theorie d’s Kreisels, Vol IV, pp880.

“A bicycle is a doubly non-holonomic system; it has fivedegrees of freedom in finite motion, but only three suchdegrees i infinitesimal motion (rotation of the rear wheel in itsinstantaneous plane, to which the rotation of the front wheelis coupled by the condition of pure rolling; rotation aboutthe handle bar axis; and common rotation of front and rearwheel about the line connecting their points of contact with theground), as long as we do not consider the degrees of freedomof the cyclist himself.”


Arnold Sommerfeld on Gyroscopic Effects

That the gyroscopic effects of thewheels are very small comparedwith these (centripetal forces) canbe seen from the constructionof the wheel: if one wanted tostrengthen the gyroscopic effects,one should provide the wheelswith heavy rims and tires insteadof making them as light as possi-ble. It can nevertheless be shownthat these weak effects contributetheir share to the stability of thesystem.


Bicycle Modeling

Whitt and Wilson Bicycling Science MIT Press:

“The scientific literature (Timoshenko, Young, DenHartog et al)shows often complete disagreement even about fundamentals.One advocates that a high center of mass improves stability,another concludes that a low center of mass is desirable.”

F. J. W Whipple, The stability of motion of a bicycle.Quart. JMath vol 30, 1899

Jones, The Stability of the bicycle. Physics Today. April 1970

More references later

Control theory gives very nice insights!


Coordinates and Variables


Major Simplifications

• Neglect elasticities.

• Five subsystems: frame, wheels, rider and front fork

• Configuration space has dimension 8: position of rearwheel x and y orientation ψ , roll of frame and front wheelϕ an ϕ f , steering angle δ , wheel angles γ r and γ f .

• Four non-holonomic constraints reduces degrees offreedom to 3.

• Assume constant forward velocity (-1 dof)

• Assume that rider is fixed to the bicycle (-1dof)

• Let steering angle δ be the control variable (-1dof)

• Resulting model has one dof, roll angle ϕ• Assume small tilt angles ; linear model.


Tilt Dynamics

Linearized momentum balance around ξ -axis

Jd2ϕdt2 − D

dωdt= mnhϕ + hF, ω = V

bδ , F = mV 2

bδ


Linearized Tilt Dynamics

Jd2ϕdt2 − D

dωdt= mnhϕ + hF, ω = V

bδ , F = mV 2

bδ

Substituting and rearrangign gives

Jd2ϕdt2 −mnhϕ = mhV 2

bδ + D

bdδdt

Transfer function: P(s) = DVbJ

s+mhV/Ds2 −mnh/J

• Poles: p = ±√

mnh/J � ±√n/h

• Zero: z = −mhV/D � V/a

• Gain: k = amhVbJ

� aVbh

Approximations J � mh2, D � mahc& K. J. Åström, Mekatronikmöte, Göteborg, 030828 13

Summary

• Steering angle is the control variable

• Two states tilt and tilt rate

• Transfer function with one zero and two poles one in theRHP

• What is the model good for?

• The model cannot explain that it is possible to ride withouttouching the handle bar

• What assumptions should be changed?

• Teaching kids to bike (Don’t use stiff arms!)



K. J. Åström

1. Introduction

2. Modeling

3. Stabilization



6. Conclusions


A Slightly More Complicated Model

• It is not a good idea to assume that the steering angle isthe control variable!

• The driver influences the bicycle primarily by applyingtorques to the handle bar.

• The front fork creates a feedback because leaning createsa torque on the front fork.

• A simple experimental verification.

• Important to understand the basic mechanisms!

• Improved model

– Two subsystems: frame and front fork– One control variable: Torque on front fork– Dynamic model for frame static model for front fork


Block Diagram of a Bicycle

Control variable: Handlebar torque T

Process variables: Steering angle δ , tilt angle ϕ

PSfrag replacements

T

ϕδ

?

T

Front fork

Frame?

−1

Σ


Overview

• The model of the frame should describe how the thetilt of the frame ϕ is influenced by the steering angle δinfluences (already done)

– Dynamics is essential– What is the time scale? (ω =

√mnh/J)

– Storage of angular momentum (ϕ and dϕ/dt)

• The model of the front fork should tell how the steeringangle ϕ is influenced by the torque on the handle bar Tand the tilt angle ϕ– A static model will be used since the front fork dynam-

ics faster than tilt dynamics– Tire-road interaction and gyroscopic effects will be

neglected


The Front Fork

The front fork has many interesting features that were devel-oped over a long time. Its behavior is complicated by geometry,the trail, tire-road interaction and gyroscopic effects. We willdescribe it by a very simple static linear model.

With a positive trail the front wheellines up with the velocity, castercamber. The trail also creates atorque that turns the front fork intothe lean. This torque is opposedby the caster-camber action andby torques generated by tire roadinteraction. A simple model is

δ = k1T − k2ϕ

d

PSfrag replacements

t

Experimental verifications?c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 19

Block Diagram of a Bicycle

PSfrag replacements

T

ϕδ

k1

Handlebar torque T

Front fork

k(s+V/a)s2−mnh/Jk2

−1

Σ


The Closed Loop System

Combining the equations for the frame and the front fork gives

Jd2ϕdt2 −mnhϕ = DV

bdδdt+ mhV 2

bδ

δ = −k2ϕ + k1T

we find that the closed loop system is described by

Jd2ϕdt2 +

k2 DVb

dϕdt+mnh

(k2V 2

bn −1)

ϕ = k1 DVb

(dTdt+mhV

DT)

This equation is stable if

V > Vc =√

bn/k2

where Vc is the critical velocity. Physical interpretation. Thinkabout this next time you bike!


Gyroscopic Effects

Gyroscopic effects has a little influence on the front fork

Jd2ϕdt2 −mnhϕ + DV

bdδdt+ mhV 2

bδ

δ = −k2ϕ−kndϕdt+ k1T

we find that the closed loop system is described by(

J + knDVb

)d2ϕdt2 +

(k2 DVb

+ mhknV 2

bn)dϕ

dt+mnh

(k2V 2

bn − 1)

ϕ

= k1 DVb

(dTdt+ mhV

DT)

Damping can be improved drastically, but the stability conditionis the same as before

V > Vc =√

bn/k2


Summary

Modeling and feedback analysis gives good insight into sta-bilization of bicycles. A model of second order captures theessence.

• Simple models

– Stiff rider– Static front fork model

• Interesting conclusion

– Bicycle is self stabilizing if velocity is larger than thecritical velocity. Design of the front fork is the key!

• Model can be augmented in several ways

– Better front fork model centrifugal forces ...– Tire road interaction


Major Drawback of Simple Model

Parameters k1 and k2 in front fork model δ = k1T − k2ϕ arevelocity dependent. A static model givesXS

k1 =b

acmn cos λ(V 2 cos λ

bn − sin λ) , k2 =

1V 2 cos λ

bn − sin λ

Closed loop system(V 2 cos λ

bn − sin λ)

Jd2ϕdt2 +

DV cos λb

dϕdt+(

mnh sin λ − acmn cos λb

)ϕ

= DV hacmn

dTdt+ V 2h

acn T

Bifurcations when velocity changes are very different.


Root LociEmpirical front fork model Physical front fork model

Effects of additional dynamics


More Complicated Models

• Front fork dynamics gives a fourth order model

• Linearized models have been obtained by many tech-niques

– Complicated error proned calculations– Computational tools highly desirable

• Other factors

– Rider lean– Tire road interaction

• Nonlinear models and parameter dependencies

• Tools like Modelica are indispensable

• Expressions for critical velocity complicated



K. J. Åström

1. Introduction

2. Modeling

3. Stabilization



6. Conclusions


Rear Wheel Steering

F. R. Whitt and D. G. Wilson (1974) Bicycling Science - Er-gonomics and Mechanics. MIT Press Cambridge, MA.

“Many people have seen theoretical advantages in the fact thatfront-drive, rear-steered recumbent bicycles would have simplertransmissions than rear-driven recumbents and could have thecenter of mass nearer the front wheel than the rear. The U.S.Department of Transportation commissioned the constructionof a safe motorcycle with this configuration. It turned out to besafe in an unexpected way: No one could ride it.”

The Santa Barbara Connection


The Tilt Equation for Rear Wheel Steering

PS

fragreplacem

entsORCA


bδ−D

bdδdt


Compare with Front Wheel Steering

Front wheel steering:


bδ+D

bdδdt

Rear wheel steering:


bδ−D

bdδdt


The Linearized Tilt Equation

The linearized equation becomes


bδ−D

bdδdt

The transfer function of the system is

P(s) = DVbJ

−s+ mhVD

s2 − mnhJ

One pole and one zero in the right half plane.

Physical interpretation.


Rear Wheel Steering

The transfer function from steering angle to tilt is

P(s) = DVbJ

−s+ mhVD

s2 − mnhJ

One RHP pole at p =√

mnhJ

� 3 rad/s

One RHP zero at z = mhVD

Pole position independent of velocity but zero proportional tovelocity. When velocity increases from zero to high velocity youpass a region where z = p and the system is uncontrollable.The system is impossible to control when z = p.


Does Feedback from Rear Fork Help?

Combining the equations for the frame and the rear fork gives

Jd2ϕdt2 −mnhϕ − amhV

bdδdt+ mhV 2

bδ

δ = −k2ϕ + k1T

we find that the closed loop system is described by

Jd2ϕdt2 −

k2

bdϕdt+mnh

(k2V 2

bn − 1)

ϕ = k1Vb

(dTdt+ mhV

DT)

where Vc =√

bn/k2. This equation is unstable for all k2.There are several ways to turn the rear fork but it makes littledifference.

Can the system be stabilized robustly with a more complexcontroller?


Limitations due to Non-minimum PhaseDynamics

Factor process transfer function as P(s) = Pmp(s)Pnmp(s) suchthat hPnmp(iω )h = 1 and Pnmp has negative phase. Requiring aphase margin ϕ m we get

arg L(iω nc) = arg Pnmp(iωnc) + arg Pmp(iωnc) + arg C(iω nc)≥ −π +ϕm

With Pmp(s)C(s) = (s/ω nc)n (Bode’s ideal loop transferfunction) we get arg PmpC = nπ/2 and

arg Pnmp(iω nc) ≥ −π +ϕ m − nπ2

This equation holds approximatively for other designs if n is theslope at the crossover frequency.


Bode Plots Should Look Like This

10−1

100

101

10−2

10−1

100

101

102

10−1

100

101

−300

−200

−100

0PSfrag replacements

Gain curve

ω

hP(iω)h

Phase curve

ω

arg

P(iω)


Summary of Limitations - Part 1

• A RHP zero z gives an upper bound to bandwidth

ω nc

z≤{

0.5 for Ms, Mt < 20.2 for Ms, Mt < 1.4.

Slow RHP zeros difficult! Why e−sT � 1−sT/21+sT/2 , s = 1/2T

• A RHP pole p gives a lower bound to bandwidth

ω nc

p≥{

2 for Ms, Mt < 25 for Ms, Mt < 1.4.

Fast RHP poles difficult!

• RHP poles and zeros must be sufficiently separated

zp≥{

6.5 for Ms, Mt < 214.4 for Ms, Mt < 1.4.


System with RHP Pole and Zero Pair

Pnmp(s) =(z− s)(s+ p)(z+ s)(s− p)

For z > p the cross over frequency inequality becomes

ωnc

z+ p

ωnc≤ (1− p

z) tan

(π2− ϕm

2+ nnc

π4

)

ϕm < π + nncπ2− 2 arctan

√p/z

1− p/z

With nnc = −0.5 we get

z/p 2 2.24 3.86 5 5.83 8.68 10 20

ϕm -6.0 0 30 38.6 45 60 64.8 84.6

Systems with RHP poles and zeros can be controlled robustlyonly if poles and zeros are well separated


Return to Rear Wheel Steering

The zero-pole ratio is

zp= V

D

√mhJn � V

a

√hn

The system is difficult to control robustly if this ratio is greaterthan 6.

To make the ratio large you can

• Make a small by leaning forward

• Make V large by biking fast (takes guts)

• Make h large by standing upright

• Move back and sit down when the velocity is sufficientlylarge


Richard Klein and His Bikes


Bicycles in Education

• Not High Tech but all students are familiar with them

• Good illustration of many interesting issues in control

• Special projects in Control Lund 1968, 1972, Richard KleinIllinois ACC Atlanta 1988 and IFAC CE Boston 1991

• Summer course for European student in Lund 1996

• Regular use in introductory courses at Lund and UCSB,Hannover 2000 (Lunze), Michigan (Stefanopoulou)

• How bicycles are used?

– Motivation– Concrete illustration of ideas and concepts– Experiments

• Very high student attractionc& K. J. Åström, Mekatronikmöte, Göteborg, 030828 40

The Santa Barbara experience

• Why should a mechanical engineer learn control

• Argue the design of a recliner

• Demo

• Motivate limitations and why it is worth while to knowabout poles and zeros

• Student feedback

– Control theory really gives insight– I did not know that a bike could be self stabilizing– Now I know why RHP poles and zeros are important– Even if I will not specialize in control I have a feel for

the questions I should ask if I became involved with acontrol system


The UCSB Bike


The Hamburg-Bochum Bike



K. J. Åström

1. Introduction

2. Modeling

3. Stabilization



6. Conclusions


Maneuvering and Stabilization - A ClassicProblem

Lecture by Wilbur Wright 1901:

Men know how to construct airplanes.Men also know how to build engines.

Inability to balance and steer still confrontsstudents of the flying problem.

When this one feature has been worked out,the age of flying will have arrived, for

all other difficulties are of minor importance.

The Wright Brothers figured it out and flew the Kitty Hawk onDecember 17 1903!

Minorsky 1922:

It is an old adage that a stable ship is difficult to steer.c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 45

Maneuvering

Having understood stabilization of bicycles we will now investi-gate steering for the bicycle with a rigid rider.

• Key question: How is the path of the bicycle influenced bythe handle bar torque?

• Steps in analysis, find the relations

– How handle bar torque influences steering angle– How steering angle influences velocity– How velocity influences the path

We will find that the instability of the bicycle frame causessome difficulties in steering (dynamics with right half planezeros). This has caused severe accidents for motor bikes.


Block Diagram Analysis Useful

PSfrag replacements

T ϕδ

k1

Handlebar torque T

Front fork

k(s+V/a)s2−mnh/J

k2

−1

Σ

Transfer function from T to δ is

k1

1+ k2P(s) =k1

1+ k2k(s+V/a)

s2−mnh/J

= k1s2 −mnh/J

s2 + amhk2VbJ s+ mnh

J

(V2

V2c− 1)


Steering Angle to Path

dydt= Vψ

dψdt

= Vb

δ

Lδ = Gδ TLT

Gδ T =k2

1+ k1Gϕδ (s)

Transfer function from steering torque to path deviation

GyT =k1V 2

b(1+ k1Gϕδ (s))=

k1V 2(

s2 − mnhJ

)

bs2

(s2 + amhk1V0

bJ s+ mnhJ

(V2

0V2

c− 1))


Simulation

Assume that the bicycle is driven along a straight line and thata constant torque is applied to the handle bar. What path willthe bicycle take?

0 0.5 1 1.5 2 2.5 3−4

−3

−2

−1

0

1

2

PSfrag replacements

x

y

Physical interpretation!


Summary

• The simple second order model with a rigid rider canexplain stabilization. The model indicates that steeringis difficult due to the right half plane zero in the transferfunction from handle bar torque to steering angle.

• The right half plane zero has some unexpected conse-quences which gives the bicycle a counterintuitive behav-ior. This has caused many motorcycle accidents.

• How can we reconcile the difficulties with our practicalexperience that a bicycle is easy to steer?

• The phenomena depends strongly on the assumption thatthe rider does not lean.

• The difficulties can be avoided by introducing an extracontrol variable (leaning).


Coordinated Steering

An experienced rider uses both lean and the torque on thehandle bar for steering. Intuitively it is done as follows:

• The bicycle is driven so fast so that it is automaticallystabilized.

• The turn is initiated by a torque on the handle bar, therider then leans gently into the turn to counteract thecentripetal force which will tend to lean the bike in thewrong direction. This is particularly important for motorbikes which are much heavier than the rider.

A proper analysis of a bicycle where the rider leans requirea more complex model because we have to account for twobodies instead of one. There are also two inputs to deal with.Accurate modeling of a bicycle also has to consider tire roadinteraction and a more detailed account of the mechanics.


Sensors, Actuators Poles and Zeros

• Watch our for time delays, and RHP poles and zeros

• Zeros can be eliminated by adding sensors and actuators!The zeros of the standard linear system (A,B,C) are thevalues of s where the matrix

sI − A B

C 0

looses rank

• Big difference between using observers and sensors as faras robustness are concerned!

• A naive observation on the rear steered bike

Jd2ϕdt2 −

k2 DVb

dϕdt+mnh

(k2V2

bn − 1)

ϕ = k1 DVb

(dTdt+ V

aT)

Adding a rate gyro solves the problem!c& K. J. Åström, Mekatronikmöte, Göteborg, 030828 52


K. J. Åström

1. Introduction

2. Modeling

3. Stabilization



6. Conclusions


Conclusions

• Bicycles illustrate many issues in mechatronics

– Modeling– Integrated system and control design– The role of sensors and actuators

• Bicycles are cheap and very useful in education

– Stabilization and steering– Nice way to illustrate zeros

• Lesson 1: Dynamics is important! A system may look OKstatically but un-tractable because of dynamics.

• Lesson 2: A system that is difficult to control because ofzeros in the right half plane can be improved significantlyby introducing more sensors and actuators.


References

Bode, H. W. (1945): Network Analysis and Feedback AmplifierDesign. Van Nostrand, New York. F. R. Whitt and D. G. Wilson(1974) Bicycling Science - Ergonomics and Mechanics. MIT PressCambridge, MA.

R.E. Klein (1988) Novel Systems and Dynamics Teaching Tech-niques Using Bicycles. Proc ACC, Atlanta, GA.

R.E. Klein (1989) Using bicycles to teach system dynamics. IEEEControl Systems Magazine, CSM Vol 9 No 3 pp, 4–9.

Seron, M. M., J. H. Braslavsky, and G. C. Goodwin (1997): Funda-mental Limitations in Filtering and Control. Springer, London.

K. J. Åström (2000) Limitations on control system performance.European Journal of Control Vol 6 No 1 pp 2–20.


Documents

Bicycles - A Mechatronics View · Bicycles - A Mechatronics View K. J. Åström Lund University 1. Introduction 2. Modeling 3. Stabilization 4. Rear wheel steering 5. Maneuvering