1. No category

Control Systems 2
Lecture 5: RHP poles and zero limitations
& how to design and ride a bike
Roy Smith
2015-3-18
5.1
Non-minimum phase behaviour (stable systems)
Right-half plane zeros
Can arise from fast and slow responses of opposite sign:
G(s) =
1.0
1
2
3−s
−
=
.
s+1
s+5
(s + 1)(s + 5)
Amplitude
step
1
s+1
step (G(s))
0.5
0
1
− step
−0.5
2015-3-18
2
s+5
2
3
time
4 (sec)
5.2
Non-minimum phase behaviour
Can also be interpreted as a negative derivative response:
G(s) =
3
s
−
(s + 1)(s + 5)
(s + 1)(s + 5)
g(t) = 3
1.0
1 −t −1 −5t
e +
e
4
4
d
−
dt
1 −t −1 −5t
e +
e
4
4
Amplitude
3 step
0.5
1
(s+1)(s+5)
step (G(s))
1
0
d
− dt
step
2
1
(s+1)(s+5)
4 time
(sec)
3
−0.5
2015-3-18
5.3
Non-minimum phase systems in feedback
Non-minimum phase response in closed-loop
G(s) =
NG (s)
,
DG (s)
T (s) =
L(s)
1 + L(s)
=
=
2015-3-18
1
K(s) =
NK (s)
,
DK (s)
L(s) =
NG (s)NK (s)
DG (s)DK (s)
NG (s) NK (s)
DG (s) DK (s)
NG (s) NK (s)
+D
G (s) DK (s)
NG (s)NK (s)
DG (s)DK (s) + NG (s)NK (s)
5.4
Non-minimum phase systems: r.h.p. zeros
1
Magnitude
1
10
Gmp (s) =
0.1
log ω
(rad/sec)
100
(s+10)
(s+1)(s+50)
Gnmp1 (s) =
0.01
(10−s)
(s+1)(s+50)
0.001
0
1
10
log ω
(rad/sec)
100
Gmp (s) =
−90
(s+10)
(s+1)(s+50)
−180
−270
Gnmp1 (s) =
Phase (deg.)
(10−s)
(s+1)(s+50)
2015-3-18
5.5
Non-minimum phase systems: delays
1
Magnitude
1
10
Gmp (s) =
0.1
log ω
(rad/sec)
100
(s+10)
(s+1)(s+50)
Gnmp1 (s) =
0.01
(10−s)
(s+1)(s+50)
e−0.05s (s+10)
(s+1)(s+50)
Gnmp2 (s) =
0.001
0
1
10
100
Gmp (s) =
−90
Gnmp2 (s) =
−180
−270
2015-3-18
log ω
(rad/sec)
Phase (deg.)
(s+10)
(s+1)(s+50)
e−0.05s (s+10)
(s+1)(s+50)
Gnmp1 (s) =
(10−s)
(s+1)(s+50)
5.6
Non-minimum phase systems in feedback
Delays in feedback
G(s) = e−θs
T (s) =
=
NG (s)
,
DG (s)
K(s) =
NK (s)
DK (s)
L(s) = e−θs
NG (s)NK (s)
DG (s)DK (s)
L(s)
1 + L(s)
NG (s) NK (s)
e−θs D
G (s) DK (s)
NG (s) NK (s)
1 + e−θs D
G (s) DK (s)
−θs
=e
NG (s)NK (s)
DG (s)DK (s) + e−θs NG (s)NK (s)
2015-3-18
5.7
Performance limitations from delays
If G(s) contains a delay, e−θs , then T (s) also contains e−θs .
Under these circumstances the ideal T (s) ≈ e−θs ,
5
Magnitude
1
1/θ
log ω
(rad/sec)
0.1
0.01
|S(jω)| = 1 − e−jθω Which implies that we must have ωc < 1/θ.
2015-3-18
5.8
Controllability (summary)
Actuation constraints: from disturbances
|G(jω)| > |Gd (jω)|
for frequencies where |Gd (jω)| > 1.
Actuation constraints: from reference
|G(jω)| > R
up to frequency: ωr .
Disturbance rejection
ωc > ω d
or more specifically
|S(jω)| ≤ |1/Gd (jω)|
for all ω.
Reference tracking
|S(jω)| ≤ 1/R
up to frequency: ωr .
2015-3-18
5.9
Controllability (summary)
Right-half plane zeros
For a single, real, RHP-zero:
ωB < z/2.
Time delays
Approximately require:
ωc < 1/θ.
Phase lag
Most practical controllers (PID/lead-lag):
G(jω180 ) = −180 deg.
ωc < ω180
Unstable real pole
Require ωc > 2p.
Also require |G(jω)| > |Gd (jω)| up to ω = p.
2015-3-18
5.10
Example: controllability analysis
d
Gd (s)
y
+
G(s)
u
K(s)
+
r
−
n
G(s) = k
ym
+
e−θs
1 + τs
Gd (s) = kd
e−θd s
,
1 + τd s
|kd | > 1.
What are the requirements on k, kd , τ , τd , θ and θd in order to obtain good
performance. And how good is it?
2015-3-18
5.11
Example: controllability analysis
Objective:
|e| ≤ 1 for all |u| < 1, |d| < 1.
Disturbance rejection (satisfying actuation bound)
|G(jω)| > |Gd (jω)| for all ω < ωd .
=⇒ k > kd and k/τ > kd /τd .
Disturbance rejection
ωc > ωd ≈ kd /τd .
Delay constraints
ωc < 1/θ
2015-3-18
(assuming θ is the total delay in the loop).
5.12
Example: controllability analysis
Delay and disturbance rejection requirements.
θ < τd /kd .
Plant requirements:
k > kd and
θ < τd /kd .
k/τ > kd /τd
Required/achievable bandwidth
kd /τd < ωc < 1/θ.
2015-3-18
5.13
F E A T U R E
Bicycle dynamics
U R E
By Karl J. Åström,
Richard E. Klein, and
Anders Lennartsson
T
his article analyzes the dynamics of b
cles from the perspective of cont
Models of different complexity are
sented, starting with simple ones
ending with more realistic models ge
ated from multibody software. We
sider models that capture essential behavior suc
self-stabilization as well as models that dem
strate difficulties with rear wheel steering.
relate our experiences using bicycles in con
education along with suggestions for fun
thought-provoking experiments with pro
student attraction. Finally, we describe bicy
and clinical programs designed for children
disabilities.
Adapted bicycles for
education and research
Karl J. Åström,
hard E. Klein, and
ders Lennartsson
T
his article analyzes the dynamics of bicycles from the perspective of control.
Models of different complexity are presented, starting with simple ones and
ending with more realistic models generated from multibody software. We consider models that capture essential behavior such as
self-stabilization as well as models that demonstrate difficulties with rear wheel steering. We
relate our experiences using bicycles in control
education along with suggestions for fun and
thought-provoking experiments with proven
student attraction. Finally, we describe bicycles
and clinical programs designed for children with
disabilities.
IEEE Control Systems Magazine
vol. 25, no. 4, pp. 26–47,
2005.
The Bicycle
The Bicycle
2015-3-18
CLELLAN/THOMPSON-MCCLELLAN PHOTOGRAPHY
Bicycles are used everywhere—for transportation, exercise, and recreation. The bicycle’s evolution over time
has been a product of necessity, ingenuity, materials, and
industrialization. While efficient and highly maneuverable,
the bicycle represents a tantalizing enigma. Learning to
ride a bicycle is an acquired skill, often obtained with some
difficulty; once mastered, the skill becomes subconscious
and second nature, literally just “as easy as riding a bike.”
Bicycles display interesting dynamic behavior. For
example, bicycles are statically unstable like the inverted pendulum, but can, under certain conditions, be stable in forward motion. Bicycles also exhibit nonminimum
phase steering behavior.
Bicycles have intrigued scientists ever since they appeared in
the middle of the 19th century. A thorough presentation of the
history of the bicycle is given in the recent book [1]. The papers
[2]–[6] and the classic book by Sharp from 1896, which has
recently been reprinted [7], are good sources for early work.
Notable contributions include Whipple [4] and Carvallo
[5], [6], who derived equations of motion, linearized around the
1066-033X/05/$20.00©2005IEEE
IEEE Control Systems Magazine
August 2005
Adapted bicycles fo
education and researc
LOUIS MCCLELLAN/THOMPSON-MCCLELLAN PHOTOGRAPHY
26
Bicycles are used everywhere—for transportation, e
cise, and recreation. The bicycle’s evolution over
has been a product of necessity, ingenuity, materials,
industrialization. While efficient and highly maneuvera
the bicycle represents a tantalizing enigma. Learnin
ride a bicycle is an acquired skill, often obtained with s
difficulty; once mastered, the skill becomes subconsc
and second nature, literally just “as easy as riding a bik
Bicycles display interesting dynamic behavior.
example, bicycles are statically unstable like the inv
ed pendulum, but can, under certain conditions, be
ble in forward motion. Bicycles also exhibit nonminim
phase steering behavior.
Bicycles have intrigued scientists ever since they appeare
the middle of the 19th century. A thorough presentation of
history of the bicycle is given in the recent book [1]. The pa
[2]–[6] and the classic book by Sharp from 1896, which
recently been reprinted [7], are good sources for early w
Notable contributions include Whipple [4] and Carv
[5], [6], who derived equations of motion, linearized around
1066-033X/05/$20.00©2005IEEE
IEEE Control Systems Magazine
August
5.14
Bike parameter
considered explicitly, but we often assume that the forward
the
forces
between
ground and
wethat
do
velocity
is acting
constant.
To summarize,
we wheel.
simply Since
assume
not
consider
extreme
conditions
and
tight
turns,
we
the bicycle moves on a horizontal plane and that the
assume
that the
bicyclecontact
tire rolls
without
longitudinal or
wheels always
maintain
with
the ground.
definitions
lateral slippage. Control of acceleration and braking is not
considered explicitly, but we often assume that the forward
velocity is constant. To summarize, we simply assume that
λ plane and that the
the bicycle moves on a horizontal
wheels always maintain contact with the ground.
h
h
λ
C1
C2
CP11
P2
P3
C2
a
b
coordinate system xyz has
Coordinates
P1 of the rear wheel and the
Figure 1. Parameters defining the bicycle geometry. The
The
coordinates
used
is aligned
with the
line to
of ac
a
points of the wheels with the
points P1 and P2 are the contact
low horizontal
the ISO 8855
standard,
the
plane.
The x
of the steer axis with
ground, the point P3 is the intersection
b
c the
is an inertial
system
P3 , which
point
is thewith
inte
horizontal plane, a is the distance from a vertical line through
xyz has
coordinate
system
axis
and
the
horizontal
pl
the center of mass to P1 , b is the wheel base, c is the trail, h is
P1 ofwheel
the rear
wheel
and th
rear
plane
is defined
λ is thegeometry.
the height
the centerdefining
of mass,the
andbicycle
head angle.
Figure
1. of
Parameters
The
is
aligned
with
the
line
of ca
angle between the ξ -axis
5.15
points P1 and P2 are the contact points of the wheels with the
the horizontal
plane.
The
vertical,
and y is
perpendic
ground, the point P3 is the intersection of the steer axis with the
P
point
,
which
is
the
left side3 of the bicycle int
so
horizontal plane, a is the distance from a vertical line through
ζ
axis
and
the
horizontal
ϕp
obtained.
The
roll
angle
ϕ
the center of mass to P1 , b is the wheel base, c is thef trail, h is
definitions
rear wheel
plane
is defined
when
leaning
to the
right. T
the height of the center of mass, and λ is the
η head angle.
ξ
angle
between
the
-axis
ϕ
plane
is
.
The
steer
anglea
z
f
ϕ
y is perpendic
vertical, and
between
the rear
and front
left left.
sideThe
of the
bicycle
so
ing
effective
steer
ζ
the
lines ofThe
intersection
obtained.
roll angleofϕt
ϕf
the
horizontal
plane.
when leaning to the right. T
η
δf
C2
plane is ϕf . The steer angle
z
x
C1
ϕ
Simple
Second-Ord
between the
rear and front
Second-order
models will
no
P
ing
left.
The
effective
steer
P2 3
P1
tional
simplifying
assumpt
ψ
the lines
of intersection
of t
bicycle
rolls onplane.
the horizon
the horizontal
δf
ξ
fixed position and orientati
C2
x
C1
that
the forward
velocity at
Simple
Second-Ord
For
simplicity,models
we assume
Second-order
will nt
Figure 2. Coordinate systems. The orthogonal
P2 P3 system ξ ηζ is
which
implies
that
the
hea
P
tional simplifying assumpt
ψ is vertical. The orthogo1 ζ -axis
fixed to inertial space, and the
c is rolls
trail
zero. on
Wethe
alsohorizo
assum
bicycle
nal system xyz has its origin at the contact point of the rear
control
variable.
The
rotatio
wheel with the ξ η plane. The x axis passes through the points
ξ
fixed position and orientati
P1 and P3 , while the z axis is vertical and passes through P1 .
ated
withforward
the frontvelocity
fork then
that the
at
For simplicity, we assume
Figure 2. Coordinate systems. The orthogonal system ξ ηζ is
which implies that the hea
fixed
to inertial space, and the ζ -axis is vertical. The
orthogoIEEE Control Systems Magazine
28
c is zero. We also assum
trail
nal system xyz has its origin at the contact point of the rear
control variable. The rotatio
wheel with the ξ η plane. The x axis passes through the points
P1 and P3 , while the z axis is vertical and passes through P1 .
ated with the front fork then
P1
2015-3-18
Reference frame
2015-3-18
c
shaped so that the contact
Geometry
the
road is behind the exten
The parameters
that descr
defined
as the horizontal
di
are
defined
in
Figure
1.
The
k
point and the steer axis wh
λ,
c.
head
angle
and
trail
zero steer angle. The riding
shaped so
that the
strongly
affected
bycontact
the tra
the
road
is
behind
the
improves stability but exten
make
definedfor
asc the
horizontal
values
range
0.03–0.08dm
point
and the steer
wh
Geometrically,
it axis
is conv
zero
steer
angle.
The
riding
composed of two hinged pla
strongly
by the
tr
front
forkaffected
plane. The
frame
improves
stability
but
mak
frame plane, while the fron
values The
for c planes
range 0.03–0.08
m
plane.
are joined
is convp
P1 Geometrically,
and P2 are the itcontact
composed
of
two
hinged
pl
horizontal plane, and the po
front axis
fork with
plane.
frame
steer
theThe
horizonta
frame plane, while the fron
plane. The planes are joined
Coordinates
P1 and
P2 are theused
contact
p
The
coordinates
to an
horizontal
plane,
and
the
po
low the ISO 8855 standard,
steer
with system
the horizonta
is
an axis
inertial
with
28
P2
P3
IEEE Control Systems Magazine
5.16
Na¨ıve analysis
only degree of freedom. All
ll so that the equations can
cycle are shown in Figure 3.
ates around the vertical axis
Vδ/b, where b is the wheel
coordinate system xyz expeeleration of the coordinate
e.
f the system. Consider the
wheels, the rider, and the
o the rear frame with δ = 0,
inertia of this body with
D = − Jxz denote the inertia
axes. Furthermore, let the x
r of mass be a and h, respecof the system with respect
z
y
O
ϕ
δ
P1
x
P2
a
b
(a)
= J
dϕ
VD
−
δ.
dt
b
tem are due to gravity and
ngular momentum balance
y
(b)
Figure 3. Schematic (a) top and (b) rear views of a naive
(λ = 0) bicycle. The steer angle is δ, and the roll angle is ϕ.
2015-3-18
typographical error: λ = 90.
DV dδ
mV 2h
+ Na¨
δ.
b dt
b ıve
It follows from (1) that the transfer function from steer
angle δ to tilt angle ϕ is
5.17
(1)
analysis:
simple second order models
V(Ds + mVh)
b( Js2 − mgh)
mVh
V
aV s + a
VD s + D
≈
.
=
b J 2 mgh
bh s2 − g
angle, sφ,−transfer
function
h
J
G ϕδ (s) =
generated by gravity. The
of (1) are the torques genfirst term due to inertial
(4)
due to centrifugal
forces.angle, δ, to tilt
Steering
verted pendulum model
the linearized equation for
dφ Notice that both
dφ theVgain
D and the zero of this transfer func2 and
=
J
−
Dω
=
J
−
δ V. Angular momentum about x
t of inertia as J ≈Lmh
tion
depend
on
the
velocity
x
dt
dt
b
, the model becomes
The model (4) is unstable and thus cannot explain why
it
is
possible to ride with2no hands. The system (4), howevd2 φ
DV dδ
mV h
er, can
control using
the proporJ 2 − mghφ
= be stabilized
+ by active
δ Torque
balance
V dδ
V2
dt
b
dt
b
+
δ.
tional feedback law
h dt
bh
J ≈ mh2 and D ≈ mah
d [21], is a linear dynamical
wo real poles
d2 φ
g
dt2
!
gh
g
≈±
J
h
h
V
≈− .
a
−
Inertia approximations
δ = −k2 ϕ,
(5)
aV dδ
V2
φ=
+
δ
Simplified model
h which
bh yields
dt the
bhclosed-loop system
(2)
d2 ϕ
DVk2 dδ
J 2 +
+
b dt
dt
2015-3-18
(3)
"
#
mV 2hk2
− mgh ϕ = 0.
b
(6)
This closed-loop system is asymptotically stable if and only
if k2 > bg/V 2 , which is the case when V is sufficiently large.
5.18
Na¨ıve analysis: simple second order models
Steering angle, δ, to tilt angle, φ, transfer function
Transfer function:
Gφδ (s) =
V (Ds + mV h)
φ(s)
aV (s + V /a)
=
≈
2
δ(s)
b(Js − mgh)
bh (s2 − g/h)
poles: p1,2 = ±
zero: z1 = −
2015-3-18
Bike parameter
r
r
mgh
g
≈±
J
h
V
mV h
≈−
D
a
the forces acting between ground and wheel. Since we do
not consider extreme conditions and tight turns, we
assume that the bicycle tire rolls without longitudinal or
lateral slippage. Control of acceleration and braking is not
considered explicitly, but we often assume that the forward
velocity is constant. To summarize, we simply assume that
the bicycle moves on a horizontal plane and that the
wheels always maintain contact with the ground.
definitions
λ
h
C1
C2
P1
P2
P3
c
Figure 1. Parameters defining the bicycle geometry. The
points P1 and P2 are the contact points of the wheels with the
ground, the point P3 is the intersection of the steer axis with the
horizontal plane, a is the distance from a vertical line through
the center of mass to P1 , b is the wheel base, c is the trail, h is
the height of the center of mass, and λ is the head angle.
2015-3-18
The parameters that descri
are defined in Figure 1. The ke
head angle λ, and trail c. T
shaped so that 5.19
the contact
the road is behind the exten
defined as the horizontal di
point and the steer axis wh
zero steer angle. The riding
strongly affected by the tra
improves stability but make
values for c range 0.03–0.08 m
Geometrically, it is conv
composed of two hinged pla
front fork plane. The frame
frame plane, while the fron
plane. The planes are joined
P1 and P2 are the contact p
horizontal plane, and the po
steer axis with the horizonta
Coordinates
a
b
Geometry
The coordinates used to an
low the ISO 8855 standard,
is an inertial system with
coordinate system xyz has
P1 of the rear wheel and the
is aligned with the line of c
the horizontal plane. The x
point P3 , which is the inte
axis and the horizontal pl
rear wheel plane is defined
angle between the ξ -axis a
5.20
vertical, and y is perpendic
left side of the bicycle so
Front fork model
Handlebar torque, T , to tilt angle, φ, transfer function
Model the actuation as a torque to the handlebars, T .
mg 2 (bh cos λ − ac sin λ)
d2 φ
DV g
dφ
J 2 + 2
+
φ
dt
V sin λ − bg cos λ dt
V 2 sin λ − bg cos λ
b(V 2 h − acg)
DV b
dT
=
+
T
acm(V 2 sin λ − bg cos λ) dt
ac(V 2 sin λ − bg cos λ)
√
The system is stable if V > Vc = bg cot λ and bh > ac tan λ
Gyroscopic effects could be included (giving additional damping).
2015-3-18
5.21
Front fork model
Torque to steering angle transfer function
With a stabilizable bicycle going at sufficiently high speed, V ,
k1 (V )
δ
= GδT (s) =
,
T
1 + k2 (V )Gφδ (s)
where, as before,
So,
Gφδ (s) =
GδT (s) =
s2
2015-3-18
V (Ds + mV h)
aV (s + V /a)
≈
2
b(Js − mgh)
bh (s2 − g/h)
mgh
2
k1 (V ) s − J
k2 (V )DV
k2 (V )V 2 mh
mgh
+
s +
− J
bJ
bJ
5.22
Front fork model
Torque to path deviation transfer function
If η is the deviation in path,
GηT (s) =
k1 (V ) V 2
b
mgh
s − J
2
k
(V
)DV
mgh
2
V
s2 s2 +
s + J
− 1
bJ
Vc2
2
2015-3-18
5.23
Non-minimum phase behaviour
Counter-steering
“I have asked dozens of bicycle riders how they turn to the
left. I have never found a single person who stated all the
facts correctly when first asked. They almost invariably
said that to turn to the left, they turned the handlebar to
the left and as a result made a turn to the left. But on
further questioning them, some would agree that they first
turned the handlebar a little to the right, and then as the
machine inclined to the left they turned the handlebar to
the left, and as a result made the circle inclining inwardly.”
Wilbur Wright.
2015-3-18
5.24
Non-minimum phase behaviour
Counter-steering
2015-3-18
5.25
Non-minimum phase behaviour
Aircraft control
“Men know how to construct airplanes. Men also know
how to build engines. Inability to balance and steer still
confronts students 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.”
Wilbur Wright, 1901.
2015-3-18
5.26
Rear-wheel steered bicycles
Klein’s Ridable Bike
c K. J. Åström, Delft, June, 2004
!
32
2015-3-18
5.27
Rear-wheel steered bicycles
Stabilization: simple model
The sign of V is reversed in all of the equations.
mV h
2
−s
+
−V Ds + mV h
VD
D
Gφδ (s) =
=
2
mgh
b(Js − mgh)
bJ
s2 − J
≈
aV (−s + V /a)
bh (s2 − g/h)
This now has a RHP pole and a RHP zero.
The zero/pole ratio is:
2015-3-18
z
mV h
=
p
D
r
J
V
≈
mgh
a
s
h
g
5.28
Rear-wheel steered motorbikes
NHSA Rear-steered Motorcycle
I
I
1970’s research program sponsored by the US National Highway Safety
Administration.
Rear steering benefits:
Low center of mass.
Long wheel base.
Braking/steering on different wheels
I
Design, analysis and building by South Coast Technologies, Santa
Barbara, CA.
I
Theoretical study: real(p) in range 4 – 12 rad/sec. for V of 3– 50 m/sec.
I
Impossible for a human to stabilize.
2015-3-18
5.29
Rear-wheel steered motorbikes
The NHSA
Rear
NHSA Rear-steered
Motorcycle
Steered Motorcycle
c K. J. Åström, Delft, June, 2004
!
2015-3-18
37
5.30
Rear-wheel steered motorbikes
about what knowledge is required to avoid this trap,
NHSA Rear-steered
Motorcycle
emphasizing
the role of dynamics
and control. You can
spice up the presentation with the true story about the
“The outriggers were essential; in fact, the only way to
NHSA rear-steered motorcycle. You can also briefly menkeep
theinmachine
upright
forcrucial
any measurable period of time
tion that poles and
zeros
the right-half
plane are
was to start
out down
on one
outrigger,
apply a steer input
concepts for understanding
dynamics
limitations.
Return
to
yawinvelocity
a discussion of to
the generate
rear-steeredenough
bicycle later
the courseto pick up the outrigger,
and has
then
attempt
toTell
catch
it as
the machine approached
been
presented.
students
how
a recumbent when more material
recognize Analysis
systems thatofarefilm
difficult
to conon that has a important it is tovertical.
data
indicated that the longest
inherentlyonbad
dynamics.
that2.5 seconds.”
ave students trol because of stretch
two
wheelsMake
was sure
about
that the presence
of poles
and zeros
the
y, “I have a everyone knowsRobert
Schwartz,
South
Coastin Technology,
1977.
e and try it.” right-half plane indicates that there are severe difficulties in
with the rear- controlling a system and also that the poles and zeros are
The riding influenced by sensors and actuators.
This approach, which has been used by one of the authors
verly courain
introductory
classes on control, shows that a basic knowlte of repeated attempts, edge of control is essential for all engineers. The approach also
a discussion. illustrates the advantage of formulating a simple dynamic
a static point model at an early stage in a design project to uncover potential
problems caused by unsuitable system dynamics.
a discussion2015-3-18
gned based
ount for staof studying
on design
at an early
ccessfully in
Rear-wheel
steered
bicycles
aligns with the
frame when
the speed is sufficiently large.
Another experiment is to ride a bicycle in a straight path
on a flat surface, lean gently to one side, and apply the
steer torque to maintain a straight-line path. The torque
required can be sensed by holding the handlebars with a
UCSB
bike
light
fingered
grip. Torque and lean can also be measured
with simple devices as discussed below. The functions
The UCSB Rideable Bike
KARL ÅSTRÖM
the bicycle,
mple experile and lean
he front fork
experiment
e front fork
This bicycle
ravity and
ass of the
heel with
5.31
Figure 20. The UCSB rear-steered bicycle. This bicycle is ridable as demonstrated by Dave Bothman, who supervised the
construction of the bicycle. Riding this bicycle requires skill
and dare because the rider has to reach high speed quickly.
2015-3-18
IEEE Control Systems Magazine
41
c K. J. Åström, Delft, June, 2004
!
5.32
Rear-wheel steered bicycles
An unridable bike
Klein’s Unridable Bike
2015-3-18
5.33
c K. J. Åström, Delft, June, 2004
!
31
Rear-wheel steered bicycles
Yet to be determined ...
2015-3-18
5.34
Notes and references
Skogestad & Postlethwaite (2nd Ed.)
Control limitations: sections 5.7 – 5.11
Practical controllability examples: sections 5.13 – 5.15.
More on bicycles
TU Delft: http://bicycle.tudelft.nl/schwab/Bicycle/index.htm
Article: Karl J. ˚
Astr¨
om, Richard E. Klein & Anders Lennartsson,
“Bicycle dynamics and control,” IEEE Control Systems
Magazine, vol. 25, no. 4, pp. 26–47, 2005.
2015-3-18
5.35