SINGLE-TRACK VEHICLE MODELING AND CONTROL

© PHOTODISC & HARLEY-DAVIDSON 2001 DIGITAL PRESS KIT

DAVID J.N. LIMEBEER and ROBIN S. SHARP

®
Bicycles, Motorcycles,
and Models
he potential of human-powered transportation was so cumbersome that the next generation of machines was

T recognized over 300 years ago. Human-propelled

vehicles, in contrast with those that utilized wind
power, horse power, or steam power, could run on
that most readily available of all resources:
willpower. The first step beyond four-wheeled horse-drawn
vehicles was to make one axle cranked and to allow the
rider to drive the axle either directly or through a system of
fundamentally different and based on only two wheels.
The first such development came in 1817 when the
German inventor Baron Karl von Drais, inspired by the
idea of skating without ice, invented the running machine,
or draisine [3]. On 12 January 1818, von Drais received his
first patent from the state of Baden; a French patent was
awarded a month later. The draisine shown in Figure 1
cranks and levers. These vehicular contraptions [1], [2] were features a small stuffed rest, on which the rider’s arms are

34 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006 1066-033X/06/$20.00©2006IEEE

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 laid, to maintain his or her balance. The front wheel was caught their legs when cornering. As a result, machines
steerable. To popularize his machine, von Drais traveled with centrally hinged frames and rear-steering were tested
to France in October 1818, where a local newspaper but with little success [4].
praised his skillful handling of the Speed soon became an obses-
draisine as well as the grace and sion, and the velocipede suf-
speed with which it descended a fered from its bulk, its harsh
hill. The reporter also noted that ride, and a poor gear ratio to the
the baron’s legs had “plenty to driven wheel. In 1870, the first
do” when he tried to mount his light all-metal machine
vehicle on muddy ground. Despite appeared. The “ordinary” or
a mixed reception, the draisine penny farthing had its pedals
enjoyed a short period of Euro- attached directly to a large front
pean popularity. In late 1818, the wheel, which provided
draisine moved to England, where improved gearing (see Figure 3).
Denis Johnson improved its Indeed, custom front wheels
design and began manufacturing were available that were as
the hobby horse. Despite the pub- large as one’s leg length would
lic’s enduring desire for rider-pro- allow. Solid rubber tires and the
pelled transportation, the draisine long spokes of the large front
was too flawed to survive as a wheel provided a smoother ride
viable contender; basic impedi- than its predecessors. This
ments were the absence of drive machine, which was the first to
and braking capabilities. be called a “bicycle,” was the
Although the history of the world’s first single-track vehicle
invention of the pedal-drive bicycle to employ the center-steering
is riven with controversy [2], tradi- FIGURE 1 The draisine, or running, machine. This head that is still in use today.
vehicle, which was first built in Germany in 1816, is
tional credit for introducing the early in a long line of inventions leading to the con- These bicycles enjoyed great
first pedal-driven two wheeler, in temporary bicycle. (Reproduced with permission of popularity among young men of
approximately 1840, goes to the the Bicycle Museum of America, New Bremen, Ohio.) means during their hey-day in
Scotsman Kirkpatrick Macmillan the 1880s. Thanks to its
[1]. Another account has it that adjustable crank and several
pedals were introduced in 1861 by other new mechanisms, the
the French coach builder Pierre penny farthing racked up
Michaux when a customer brought record speeds of about 7 m/s.
a draisine into his shop for repairs As is often said, pride comes
and Michaux instructed his son before a fall. The high center of
Ernest to affix pedals to the broken gravity and forward position of
draisine. In September 1894, a the rider made the penny far-
memorial was dedicated in honor thing difficult to mount and dis-
of the Michaux machine. Shown in mount as well as dynamically
Figure 2, this vehicle weighed an challenging to ride. In the event
unwieldy 60 pounds and was FIGURE 2 Velocipede by Pierre Michaux et Cie of that the front wheel hit a stone
Paris, France circa 1869. In the wake of the draisine,
known as the velocipede, or bone the next major development in bicycle design was the or rut in the road, the entire
shaker. This nickname derives velocipede, which was developed in France and machine rotated forward about
from the fact that the velocipede’s achieved its greatest popularity in the late 1860s. The its front axle, and the rider, with
construction, in combination with velocipede marks the beginning of a continuous line of his legs trapped under the han-
the cobblestone roads of the day, developments leading to the modern bicycle. Its most dlebars, was dropped uncere-
significant improvement over the draisine was the
made for an extremely uncomfort- addition of cranks and pedals to the front wheel. Dif- moniously on his head. Thus
able ride. Although velocipedoma- ferent types of (not very effective) braking mecha- the term “taking a header”
nia only lasted about three years nisms were used, depending on the manufacturer. In came into being.
(1868–1870), the popularity of the the case of the velocipede shown, the small spoon Another important invention
machine is evidenced by the large brake on the rear wheel is connected to the handlebar was the pneumatic tire intro-
and is engaged by a simple twisting motion. The
number of surviving examples. A wheels are wooden wagon wheels with steel tires. duced by John Boyd Dunlop in
common complaint among veloci- (Reproduced with permission of the Canada Science 1899. The new tires substantially
pedists was that the front wheel and Technology Museum, Ottawa, Canada.) improved the cushioning of the

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 35

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 ride and the achievable top speed. Dunlop sold the market- [1] gives a detailed account of the early history of the bicy-
ing rights to his pneumatic tire to the Irish financier Harvey cle and a thorough account of bicycle design as it was
Du Cros, and together they launched the Pneumatic Tyre understood in the 19th century. Archibald Sharp was an
Company, which supplied inflatable tires to the British instructor in engineering design at the Central Technical
bicycle industry. To make their tires less puncture prone, College of South Kensington (now Imperial College).
they introduced a stout canvas lining to the inner surface of Although Sharp’s dynamical analysis of the bicycle is only
the tire carcass while thickening the inner tube [2]. at a high school physics course level, it is sure footed and
A myriad of other inventions and developments have of real interest to the professional engineer who aspires to
made the bicycle what it is today. For bicycles using wheels a proper appreciation of bicycle dynamics and design.
of equal size, key innovations include chain and sprocket
drive systems, lightweight stiff steel frames, caliper brakes, EARLY POWERED MACHINES
sprung seats, front and rear suspension systems, free-running If one considers a wooden frame with two wheels and a
drive hubs, and multispeed Derailleur gear trains [1], [5]. steam engine a “motorcycle,” then the first one was probably
A comprehensive and scholarly account of the history American. In 1867, Sylvester Howard Roper demonstrated a
of the bicycle can be found in [2]. Archibald Sharp’s book motorcycle (Figure 4) at fairs and circuses in the eastern
United States. His machine was powered by a charcoal-fired,
two-cylinder engine, whose connecting rods drove a crank
on the rear wheel. The chassis of the Roper steam velocipede
was based on the bone-shaker bicycle.
Gottlieb Daimler is considered by many to be the inven-
tor of the first true motorcycle, or motor bicycle, since his
machine was the first to employ an internal combustion
engine. After training as a gunsmith, Daimler became an
engineer and worked in Britain, France, and Belgium before
being appointed technical director of the gasoline engine
company founded by Nikolaus Otto. After a dispute with
Otto in 1882, Daimler and Wilhelm Maybach set up their
own company. Daimler and Maybach concentrated on pro-
ducing the first lightweight, high-speed gasoline-fueled
engine. They eventually developed an engine with a surface-
mounted carburetor that vaporized the petrol and mixed it
with air; this Otto-cycle engine produced a fraction of a kilo-
watt. In 1885 Daimler and Maybach combined a Daimler
engine with a bicycle, creating a machine with iron-banded,
wooden-spoked front and rear wheels as well as a pair of
FIGURE 3 Penny farthing, or ordinary. This bicycle is believed to
smaller spring-loaded outrigger wheels (see Figure 5).
have been manufactured by Thos Humber of Beeston, Notting-
hamshire, England, circa 1882. The braking limitations of this vehi- The first successful production motorcycle was the
cle’s layout are obvious! (Reproduced with permission of the Glynn Hildebrand and Wolfmueller, which was patented in
Stockdale Collection, Knutsford, England.) Munich in 1894 (see Figure 6). The engine of this vehicle
was a 1,428-cc water-cooled, four-stroke parallel twin,
which was mounted low on the frame with cylinders in a
fore-and-aft configuration; this machine produced less than
2 kW and had a top speed of approximately 10 m/s. As
with the Roper steamer, the engine’s connecting rods were
coupled directly to a crank on the rear axle. The Hildebrand
and Wolfmueller, which was manufactured in France under
the name Petrolette, remained in production until 1897.
Albert Marquis de Dion and his engineering partner
Georges Bouton began producing self-propelled steam
vehicles in 1882. A patent for a single-cylinder gasoline
engine was filed in 1890, and production began five years
later. The De Dion Bouton engine, which was a small,
FIGURE 4 Sylvester Roper steam motorcycle. This vehicle is pow-
ered by a two-cylinder steam engine that uses connecting rods fixed lightweight, high-rpm four-stroke “single,” used battery-
directly to the rear wheel. (Reproduced with permission of the and-coil ignition, thereby doing away with the trouble-
Smithsonian Museum, Washington, D.C.) some hot-tube ignition system. The engine had a bore of 50

36 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 mm and a stroke of 70 mm, giving rise to a swept volume
of 138 cc. De Dion Bouton also used this fractional kilowatt
engine, which was widely copied by others including the
Indian and Harley-Davidson companies in the United
States, in road-going tricycles. The De Dion Bouton engine
is arguably the forerunner of all motorcycle engines.
Testosterone being what it is, the first motorcycle race
probably occurred when two motorcyclists came across
each other while out for a spin. From that moment on, the
eternal question in motorcycling circles became: “How do I
make my machine faster?” As one would imagine, the
quest for speed has many dimensions, and it would take
us too far afield to try to analyze these issues in detail. In
the context of modeling and control, it is apparent that the
desire for increased speed as well as the quest to more FIGURE 5 Daimler petrol-powered motorcycle. Gottlieb Daimler, who
fully utilize machine capability, requires high-fidelity later teamed up with Karl Benz to form the Daimler-Benz Corpora-
models, control theory, and formal dynamic analysis. One tion, is credited with building the first motorcycle in 1885. (Repro-
also needs to replace the fractional kilowatt Otto-cycle duced with permission of DaimlerChrysler AG, Stuttgart, Germany.)
engine used by Daimler with a much more powerful one.
Indeed, modern high-performance two- and four-stroke
motorcycle engines can rotate at almost 20,000 rpm and
produce over 150 kW. In combination with advanced
materials, modern tires, sophisticated suspension systems,
stiff and light frames, and the latest in brakes, fuels, and
lubricants, these powerful engines have led to Grand Prix
machines with straight-line speeds of approximately 100
m/s. Figure 7 shows Ducati’s Desmosedici GP5 racing
motorcycle currently raced by Loris Capirossi.
The parameters and geometric layout that characterize
the dynamic behavior of modern motorcycles can vary
widely. Ducati’s Desmosedici racing machine has a steep
steering axis and a short wheelbase. These features pro- FIGURE 6 Hildebrand and Wolfmueller motorcycle. This machine,
duce the fast steering and the agile maneuvering required patented in 1894, was the first successful production motorcycle.
for racing. The chopper motorcycle, such as the one (Reproduced with permission of the Deutsches Zweirad- und NSU-
shown in Figure 8, is at the other extreme, having a heav- Museum, Neckarsulm, Germany.)
ily raked steering axis and a long wheelbase. “Chopped”
machines are not just aesthetically different; they also
have distinctive handling properties that are typified by a
very stable feel at high straight-line speeds as compared
with more conventional machine geometries. However,
as with many other modifications, this stable feel is
accompanied by less attractive dynamic features such as
a heavy feel to the front end and poor responsiveness at
slow speeds and in corners.
Web sites and virtual museums dedicated to bicycles
and motorcycles are ubiquitous. See, for example, [6]–[10]
for bicycles and [11]–[15] for motorcycles.

BICYCLE MODELING AND CONTROL

Background FIGURE 7 Loris Capirossi riding the Ducati Desmosedici GP5.

Ducati Corse’s MotoGP racing motorcycle is powered by a V-4 four-
From a mathematical modeling perspective, single-track
stroke 989-cc engine. The vehicle has a maximum output power of
vehicles are multibody systems; these vehicles include bicy- approximately 161 kW at 16,000 rpm. The corresponding top speed
cles, motorcycles, and motor scooters, all of which have is in excess of 90 m/s. (Reproduced with permission of Ducati
broadly similar dynamic properties. One of the earliest Corse, Bologna, Italy.)

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 37

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 bly. The front and rear wheels are attached to the front and
rear frames, respectively, and are free to rotate relative to
them. The rider is described as an inert mass that is rigidly
attached to the rear frame. The rear frame is free to roll and
translate in the ground plane. Each wheel is assumed to be
thin and thus touches the ground at a single ground-con-
tact point. The wheels, which are also assumed to be non-
slipping, are modeled by holonomic constraints in the
normal (vertical) direction and by nonholonomic con-
straints [18] in the longitudinal and lateral directions.
FIGURE 8 “Manhattan” designed and built by Vic Jefford of Des- There is no aerodynamic drag representation, no frame
tiny Cycles. Manhattan received the Best in Show award at the flexibility, and no suspension system; the rear frame is
2005 Bulldog Bash held at the Shakespeare County Raceway, assumed to move at a constant speed. Since Whipple’s lin-
Warwickshire, England. Choppers, such as the one featured, are ear straight-running model is fourth order, the corre-
motorcycles that have been radically customized to meet a par-
sponding characteristic polynomial is a quartic. The
ticular taste. The name chopper came into being after the Sec-
ond World War when returning GIs bought up war surplus stability implications associated with this equation are
motorcycles and literally chopped off the components they did deduced using the Routh criteria.
not want. According to the taste and purse of the owner, high Concurrent with Whipple’s work, and apparently inde-
handle bars, stretched and heavily raked front forks, aftermarket pendently of it, Carvallo [19] derived the equations of
exhaust pipes, and chrome components are added. Custom-built
motion for a free-steering bicycle linearized around a
choppers have extreme steering-geometric features that have a
significant impact on the machine’s handling properties. These straight-running equilibrium condition. Klein and
features include a low head angle, long forks, a long trail, and a Sommerfeld [20] also derived equations of motion for a
long wheelbase. The extreme steering geometry of Manhattan straight-running bicycle. Their slightly simplified model (as
includes a steering head angle of 56°! (Reproduced with the per- compared with that of Whipple) lumps all of the front-
mission of Destiny Cycles, Kirkbymoorside, Yorkshire, England.)
wheel assembly mass into the front wheel. The main pur-
pose of their study was to determine the effect of the
attempts to analyze the dynamics of bicycles appeared in gyroscopic moment due to the front wheel on the
1869 as a sequence of five short articles [16]. These papers machine’s free-steering stability. While this moment does
use arguments based on an heuristic inverted-pendulum- indeed stabilize the free-steering bicycle over a range of
type model to study balancing, steering, and propulsion. speeds, this effect is of only minor importance because the
Although rear-wheel steering was also contemplated, it rider can easily replace the stabilizing influence of the front
was concluded that “A bicycle, then, with the steering wheel’s gyroscopic precession with low-bandwidth rider
wheel behind, may possibly be balanced by a very skillful control action [21].
rider as a feat of dexterity; but it is not suitable for ordinary An early attempt to introduce side-slipping and force-
use in practice.” These papers are interesting from a histori- generating tires into the bicycle literature appears in [22].
cal perspective but are of little technical value today. Other classical contributions to the theory of bicycle
The first substantial contribution to the theoretical bicy- dynamics include [23] and [24]. The last of these refer-
cle literature was Whipple’s seminal 1899 paper [17], ences, in its original 1967 version, appears to contain the
which is arguably as contributory as anything that fol- first analysis of the stability of the straight-running bicycle
lowed it; see “Francis John Welsh Whipple.” This remark- fitted with pneumatic tires; several different tire models
able paper contains, for the first time, a set of nonlinear are considered. Reviews of the bicycle literature from a
differential equations that describe the general motion of a dynamic modeling perspective can be found in [25] and
bicycle and rider. The possibility of the rider applying a [26]. The bicycle literature is comprehensively reviewed
steering torque input by using a torsional steering spring is from a control theory perspective in [27], which also
also considered. Since appropriate computing facilities describes interesting bicycle-related experiments.
were not available at the time, Whipple’s general nonlinear Some important and complementary applied work
equations could not be solved and consequently were not has been conducted in the context of bicycle dynamics.
pursued beyond simply deriving and reporting them. An attempt to build an unridable bicycle (URB) is
Instead, Whipple studied a set of linear differential equa- described in [21]. One of the URBs described had the
tions that correspond to small motions about a straight- gyroscopic moment of the front wheel canceled by
running trim condition at a given constant speed. another that was counterrotating. The cancellation of
Whipple’s model, which is essentially the model con- the front wheel’s gyroscopic moment made little differ-
sidered in the “Basic Bicycle Model” section, consists of ence to the machine’s apparent stability and handling
two frames—the rear frame and the front frame—which qualities. It was also found that this riderless bicycle
are hinged together along an inclined steering-head assem- was unstable, an outcome that had been predicted

38 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 theoretically in [20]. Three other URBs described in [21] radius and the magnitude and sign of the fork offset.
include various modifications to their steering geometry. Experimental investigations of bicycle dynamics have
These modifications include changes in the front-wheel also been conducted in the context of teaching [28].

Francis John Welsh Whipple

F
rancis John Welsh Whipple (see Figure A) was born on 17 Figure B, which was generated directly from a quartic equa-
March 1876. He was educated at the Merchant Taylors’ tion given in Whipple’s paper, shows the dynamic properties of a
School and was subsequently admitted to Trinity College, Cam- forward- and reverse-running bicycle as a function of speed.
bridge, in 1894. His university career was brilliant, and he Whipple found the parameters by experiment on a particular
received his B.A. degree in mathematics in 1897 as second machine. It is surely the case that Whipple would have loved to
wrangler. (Wrangler is a term that refers to Cambridge honors have seen this figure—derived from the remarkable work of a
graduates receiving a first-class degree in the mathematics tri- young man of 23, working almost 100 years before the wide-
pos; the senior wrangler is the first on the list of such gradu- spread availability of MATLAB!
ates.) In 1898, he graduated in the first class in Part II of the
mathematics tripos. Whipple received his
M.A. degree in 1901 and an Sc.D. in 1929.
FIGURE A Francis John Welsh Whipple by
In 1899, he returned to the Merchant Tay- Elliot and Fry. Francis Whipple was assistant
lors’ School as mathematics master, a director of the Meteorological Office and
post he held until 1914. He then moved to Superintendent of the Kew Observatory from
the Meteorological Office as superinten- 1925–1939. He served as president of the
Royal Meteorological Society from 1936–
dent of instruments.
1938. Apart from his seminal work on bicycle
Upon his death in 1768, Robert Smith, dynamics, he made many other contribu-
master of Trinity College, Cambridge and tions to knowledge, including identities for
previously Plumian professor of astrono- generalized hypergeometric functions, sev-
my, left a bequest establishing two annual eral of which have subsequently become
known as Whipple’s identities and transfor-
prizes for proficiency in mathematics and
mations. He devised his meteorological slide
natural philosophy to be awarded to junior rule in 1927. He introduced a theory of the
bachelors of arts. The prizes have been hair hygrometer and analyzed phenomena
awarded every year since, except for 1917 related to the great Siberian meteor. (Picture
when there were no candidates. Through- reproduced with the permission of the
National Portrait Gallery, London.)
out its existence, the competition has
played a significant role by enabling grad-
uates considering an academic career, and the majority of prize
winners have gone on to become professional mathematicians 10
or physicists. In 1883, the Smith Prizes ceased to be awarded 8
through examination and were given instead for the best two 6
essays on a subject in mathematics or natural philosophy.
4
... (1/s) xxx (rad/s)

On 13 June 1899, the results of the Smith Prize competition Auto-Stable Region
2
were announced in the Cambridge University Reporter [84].
Whipple did not win the prize, but it was written: “The adjudica- 0
tors are of the opinion that the essay by F.J.W. Whipple, B.A., −2
of Trinity College, ‘On the stability of motion of a bicycle,’ is −4
worthy of honorable mention.” −6
The main results of this essay depend on the work of anoth- −8
er Cambridge mathematician, Edward John Routh, who
−10
received his B.A. degree in mathematics from Cambridge in −15 −10 −5 0 5 10 15
1854. He was senior wrangler in the mathematical tripos exami- Speed (m/s)
nations, while James Clerk Maxwell placed second. In 1854,
FIGURE B Stability properties of the Whipple bicycle. Real and
Maxwell and Routh shared the Smith Prize; George Gabriel imaginary parts of the eigenvalues of the straight-running Whip-
Stokes set the examination paper for the prize, which included ple bicycle model as functions of speed. Plot generated using
the first statement of Stokes’ theorem. equation (XXVIII) in [17].

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 39

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 Point-Mass Models The vehicle’s entire mass m is concentrated at its mass
Bicycles and motorcycles are now established as nonlinear center, which is located at a distance h above the ground
systems that are worthy of study by control theorists and and distance b in front of the rear-wheel ground-contact
vehicle dynamicists alike. In most cases, control-theoretic point. The acceleration due to gravity is denoted g, and w
work is conducted using simple models, which are special is the wheelbase. The motion of the bicycle is assumed to
cases of the model introduced by Whipple [17]. An early be constrained so that there is no side slipping of the vehi-
example of such a model can be found in [29] (see equa- cle’s tires and thus the rolling is nonholonomic. The kine-
tions (e) and (j) on pages 240 and 241, respectively, of [29]). matics of the planar motion are described by
These equations describe the dynamics of a point-mass
bicycle model of the type shown in Figure 9; [29] presents ẋ = v cos ψ, (1)
both linear and nonlinear models. Another early example ẏ = v sin ψ, (2)
of a simple nonholonomic bicycle study in a control sys- v tan δ
tems context can be found in [30], which gives a servo- ψ̇ = , (3)
w cos ϕ
related interpretation of the self-steer phenomenon. A
more contemporary nonholonomic bicycle, which is essen- where v is the forward speed.
tially the same as that presented in [29], was introduced in The roll dynamics of the bicycle correspond to those of
[31] and [32]. This model is studied in [32] and [33] in the an inverted pendulum with an acceleration influence
context of trajectory tracking. A model of this type is also applied at the vehicle’s base and are given by
examined in [27] in the context of the performance limita-

tions associated with nonminimum phase zeros. hϕ̈ = g sin ϕ− (1 − hσ sin ϕ)σ v2
  
The coordinates of the rear-wheel ground-contact point ϕ̇
+ b ψ̈ + v̇ σ − cos ϕ, (4)
of the inverted pendulum bicycle model illustrated in Fig- v
ure 9 are given in an inertial reference frame O-xyz. The
Society of Automotive Engineers (SAE) sign convention is where the vehicle’s velocity and yaw rate are linked by the
used: x-forward, y-right, and z-down for axis systems and curvature σ satisfying vσ = ψ̇ . Using (3) to replace ψ̈ in (4)
a right-hand-rule for angular displacements. The roll angle by the steer angle yields
ϕ is around the x-axis, while the yaw angle ψ is around the

z-axis. The steer angle δ is measured between the front v2 bv̇
frame and the rear frame. hϕ̈ =g sin ϕ − tan δ +
w w
 
vb hv2 bvδ̇
+ tan ϕ ϕ̇ − 2 tan δ − . (5)
w w w cos2 δ

Equation (5) represents a simple nonholonomic bicycle

with the control inputs δ and v. The equation can be lin-
earized about a constant-speed, straight-running condition
to obtain the simple small-perturbation linear model

h δ g v2 bv
ϕ̈ = ϕ− δ− δ̇ . (6)
h hw wh

In the constant-speed case, the only input is the steer angle.

b Taking Laplace transforms yields the single-input, sin-
w
(x, y) gle-output transfer function
ψ

R bv s + v/b
Hϕδ (s) = − , (7)
wh s2 − g/ h

FIGURE 9 Inverted pendulum bicycle model. Schematic diagram of which has the speed-dependent gain (−bv)/(wh), a speed-

an elementary nonholonomic bicycle with steer δ, roll ϕ, and yaw ψ dependent zero at −v/b, and fixed poles at ± g/ h; the

degrees of freedom. The machine’s mass is located at a single unstable pole g/ h corresponds to an inverted-pendulum-
point h above the ground and b in front of the rear-wheel ground-
type capsize mode. The zero −v/b, which is in the left-half
contact point. The wheelbase is denoted w. Both wheels are
assumed to be massless and to make point contact with the ground. plane under forward-running conditions, moves through
Both ground-contact points remain stationary during maneuvering the origin into the right-half plane as the speed is reduced
as seen from the rear frame. The path curvature is σ (t) = 1/R(t). and then reversed in sign. Under backward-running

40 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 conditions, the right-half plane zero, which for some model has three degrees of freedom—the roll angle ϕ of
speeds comes into close proximity to the right-half plane the rear frame, the steering angle δ, and the angle of
pole, is associated with the control difficulties found in
rear-steering bicycles [34].

Basic Bicycle Model δ

We use AUTOSIM [35] models, which are derivatives of
that given in [26], to illustrate the important dynamic prop-
erties of the bicycle. As with Whipple’s model, the models
we consider here consist of two frames and two wheels.
Figure 10 shows the axis systems and geometric layout
of the bicycle model studied here. The bicycle’s rear frame θf
assembly has a rigidly attached rider and a rear wheel that θr
is free to rotate relative to the rear frame. The front frame,
Head
which comprises the front fork and handlebar assembly, Angle
has a front wheel that is free to rotate relative to the front
O
frame. The front and rear frames are attached using a
hinge that defines the steering axis. In the reference config- X Trail
uration, all four bodies are symmetric relative to the bicy- Z Wheelbase
cle midplane. As with Whipple’s model, the nonslipping
ψ
road wheels are modeled by holonomic constraints in the
normal (vertical) direction and by nonholonomic con-
FIGURE 10 Basic bicycle model with its degrees of freedom. The
straints in the longitudinal and lateral directions. There is model comprises two frames pinned together along an inclined
no aerodynamic drag, no frame flexibility, no propulsion, steering head. The rider is included as part of the rear frame. Each
and no rider control. Under these assumptions, the bicycle wheel is assumed to contact the road at a single point.

TABLE 1 Parameters of the benchmark bicycle. These parameters are used to populate the AUTOSIM model described in [26]
and its derivatives. The inertia matrices are referred to body-fixed axis systems that have their origins at the body’s mass
center. These body-fixed axes are aligned with the inertial reference frame 0 − xyz when the machine is in its nominal state.

Parameters Symbol Value

Wheel base w 1.02 m
Trail t 0.08 m
Head angle α arctan(3)
Gravity g 9.81 N/kg
Forward speed v variable m/s
Rear wheel (rw)
Radius Rrw 0.3 m
Mass mrw 2 kg
Mass moments of inertia (Axx ,Ayy ,Azz) (0.06,0.12,0.06) kg-m2
Rear frame (rf)
Position center of mass (xrf ,yrf ,zrf ) (0.3,0.0,-0.9) m
Mass mrf 85 kg



Bx x 0 Bx z 9.2 0 2.4
Mass moments of inertia Byy 0 11 0 kg-m2
sym Bzz sym 2.8

Front frame (ff)

Position center of mass (xff ,yff ,zff ) (0.9,0.0,-0.7) m
Mass mff 4 kg



Cx x 0 Cx z 0.0546 0 −0.0162
Mass moments of inertia Cyy 0 0.06 0 kg-m2
sym Czz sym 0.0114
Front wheel (fw)
Radius Rf w 0.35 m
Mass mf w 3 kg
Mass moments of inertia (Dxx ,D yy ,Dzz) (0.14,0.28,0.14) kg-m2

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 41

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 rotation θr of the rear wheel relative to the rear frame. The To study (8) in the frequency domain, we introduce the
steering angle δ represents the rotation of the front frame matrix-valued polynomial
with respect to the rear frame about the steering axis.
The dimensions and mechanical properties of the P(s, v) = s2 M + svC + (v2 K2 + K0 ) , (9)
benchmark model are taken from [26] and presented in
Table 1. All inertia parameters use the relevant body-mass which is quadratic in both the forward speed v and in the
centers as the origins for body-fixed axes. The axis direc- Laplace variable s. The associated dynamic equation is
tions are then chosen to align with the inertial O-xyz axes     
when the bicycle is in its nominal state, as shown in Figure P11 (s) P12 (s, v) ϕ(s) mϕ (s)
= , (10)
10. Products of inertia Axz , Bxz and so on are defined as P21 (s, v) P22 (s, v) δ(s) mδ (s)



− m(x, z)xzdxdz.
As derived in [17] and explained in [26], the linearized where P11 is independent of v. When studying stability,
equations of motion of the constant-speed, straight-running the roots of the speed-dependent quartic equation
nonholonomic bicycle, expressed in terms of the general-
ized coordinates q = (ϕ, δ)T , have the form det(P(s, v)) = 0 (11)

Mq̈ + vCq̇ + (v2 K2 + K0 )q = mext , (8) need to be analyzed using the Routh criteria or found by
numerical methods. Figure 11 shows the loci of the roots of
where M is the mass matrix, the damping matrix C is mul- (11) as functions of the forward speed. The basic bicycle
tiplied by the forward speed v, and the stiffness matrix has model has two important modes—the weave and capsize
a constant part K0 and a part K2 that is multiplied by the modes. The weave mode begins at zero speed with the two
square of the forward speed. The right-hand side mext con- real, positive eigenvalues marked A and B in Figure 11. The
tains the externally applied moments. The first component eigenvector components corresponding to the A-mode
of mext is the roll moment mϕ that is applied to the rear eigenvalue have a steer-to-roll ratio of −37; the negative
frame. The second component is the action-reaction steer- sign means that as the bicycle rolls to the left, for instance,
ing moment mδ that is applied between the front frame the steering rotates to the right. This behavior shows that
and the rear frame. This torque could be applied by the the motion associated with the A mode is dominated by the
rider or by a steering damper. In the uncontrolled bicycle, front frame diverging toward full lock as the machine rolls
both external moments are zero. This model, together with over under gravity. Because real tires make distributed con-
nonslipping thin tires and the parameter values of Table 1, tact with the ground, a real bicycle cannot be expected to
constitute the basic bicycle model. behave in exact accordance with this prediction. The eigen-
vector components corresponding to the B-mode eigenval-
ue have a steer-to-roll ratio of −0.57. The associated motion
involves the rear frame toppling over, or capsizing, like an
10
unconstrained inverted pendulum to the left, for instance,
8 while the steering assembly rotates relative to the rear
6 A frame to the right with 0.57 of the roll angle.
... (1/s) xxx (rad/s)

4 Note that the term “capsize” is used in two different

B Capsize contexts. The static and very-low-speed capsizing of the
2
0
bicycle is associated with the point B in Figure 11 and the
associated nearby locus. The locus marked capsize in Fig-
−2
ure 11 is associated with the higher-speed unstable top-
−4
pling over of the machine. This mode crosses the stability
−6 boundary and becomes unstable when the matrix
Weave
−8 v2 K2 + K0 in (8) is singular.
−10 As the machine speed builds up from zero, the two
0 2 4 6 8 10 12 14 16 18 20 unstable real modes combine at approximately 0.6 m/s to
Speed (m/s)
produce the oscillatory fish-tailing weave mode. The
FIGURE 11 Basic bicycle straight-running stability properties. The basic bicycle model predicts that the weave mode fre-
real and imaginary parts of the eigenvalues of the straight-running quency is approximately proportional to speed above 0.6
basic bicycle model are plotted as functions of speed. The (blue) m/s. In contrast, the capsize mode is a nonoscillatory
dotted lines correspond to the real part of the eigenvalues, while the motion, which when unstable corresponds to the rider-
(red) crosses show the imaginary parts for the weave mode. The
weave mode eigenvalue stabilizes at vw = 4.3 m/s, while the cap-
less bicycle slowly toppling over at speeds above 6.057
size mode becomes unstable at vc = 6.1 m/s giving the interval of m/s. From the perspective of bicycle riders and design-
auto-stability vc ≥ v ≥ vw . ers, this mode is unimportant because it is easy for the

42 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 rider to stabilize it using a low-bandwidth steering con- Point-Mass Model with Trail and Inclined Steering
trol torque. In practice, the capsize mode can also be sta- Interesting connections can now be made between the
bilized using appropriately phased rider body motions, Timoshenko-Young-type point-mass model and the more
as is evident from hands-free riding. complex basic bicycle model. To forge these links, we set to
In the recent measurement program [36], an instrument- zero the masses of the wheels and the front frame, as well
ed bicycle was used to validate the basic bicycle model as all the inertia terms in (10). The trail and steering incli-
described in [17] and [26]. The measurement data show nation angle are left unaltered.
close agreement with the model in the 3–6 m/s speed We first reconcile (7) and the first row of equation (10),
range; the weave mode frequency and damping agreement which is
is noteworthy. The transition of the weave mode from sta-
ble to unstable speed ranges is also accurately predicted by −P12 (s, v)
ϕ(s) = δ(s), (14)
the basic bicycle model. These measurements lend credibili- P11 (s, v)
ty to the idea that tire and frame compliance effects can be
neglected for benign maneuvering in the 0–6 m/s range. when the roll torque is mϕ (s) = 0. As in [26], we denote the
trail by t and the steering inclination angle as measured
Special Cases from the vertical by λ. Direct calculation gives
Several special cases of the basic bicycle model are now used
to illustrate some of the key features of bicycle behavior. −P12
Hϕδ (s, v) = (s, v) (15)
These cases include the machine’s basic inverted-pendulum- P11
like characteristics, as well as its complex steering and self- cos(λ)(tbs2 + sv(b + t) + v2 − gtb/ h)
=− . (16)
stabilizing features. Some of these features are the result of wh(s2 − g/ h)
carefully considered design compromises.
Equation (16) reduces to (7) when λ and t are set to zero. It
Locked Steering Model follows from (10) and mϕ = 0 that
The dynamically simple locked steering case is considered
first. If the steering degree of freedom is removed, the −P12
Hϕmδ (s, v) = (s, v) , (17)
steering angle δ(s) must be set to zero in (10), and conse- det(P)
quently the roll freedom is described by
which reduces to

mϕ (s) = P11 (s)ϕ(s) w(tbs2 + sv(t + b) + v2 − gtb/ h)

Hϕmδ (s, v) = (18)
= (s2 Txx + gmtzt)ϕ(s) . (12) mtbg(s2 − g/ h)(hw sin(λ) − tb cos(λ))

The roots of P11 (s) are given by under the present assumptions. In contrast to the analysis

given in [27], (18) shows that the poles of Hϕmδ (s, v) are

gmtzt fixed at ± g/ h and that the steering inclination and trail
p± = ± , (13)
Txx do not alone account for the self-stabilization phenomenon
in bicycles.
where mt is the total mass of the bicycle and rider, zt is the We now compute Hδmδ (s, v) as
height of the combined mass center above the ground,
and Txx is the roll moment of inertia of the entire machine P11
Hδmδ (s, v) = (s, v)
around the wheelbase ground line. In the case of the basic det P
bicycle model, p± = ±3.1348 . For the point-mass, hw2
= , (19)
Timoshenko-Young model, zt = h and Txx = mh2 and so mtbg cos(λ)(hw sin(λ) − tb cos(λ))

p± = ± g/ h.
Since the steering freedom is removed, the A mode (see which is a constant. Equation (19) shows that in a point-
Figure 11) does not appear. The vehicle’s inability to steer mass specialization of the Whipple model, the steer angle δ
also means that the weave mode disappears. Instead, the and the steering torque mδ are related by a virtual spring
machine’s dynamics are fully determined by the speed- whose stiffness depends on the trail and steering axis incli-
independent, whole-vehicle capsize (inverted pendulum) nation. Physically, this static dependence means that the
mode seen at point B in Figure 11 and given by (13). Not steer angle of the point-mass bicycle responds instanta-
surprisingly, motorcycles have a tendency to capsize at neously to steering torque inputs. It also follows from (19)
low speeds if the once-common friction pad steering that this response is unbounded in the case of a zero-trail
damper is tightened down far enough to lock the steering (t = 0) machine [29] because in this case the connecting
system; see [37]. spring has a stiffness of zero.

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 43

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 No Trail or Steering Inclination
10
We now modify the previous special case by including
8 front-frame mass offset effects (xff
= w). As before, the first
6 Weave row of (10), which relates the roll angle to the steering
angle when mϕ = 0, represents steer angle forcing of the
... (1/s) xxx (rad/s)

4
2
inverted pendulum dynamics. The second row of (10) in
this case is shown in (21), found at the bottom of the page.
0
The quadratic self-steering term in (21) contains a new
−2 Capsize term involving xff − w that comes from the fact that the
−4 front-frame mass is no longer on the steering axis, imply-
−6 ing an increase in the effective xz-plane product of inertia
−8 of the front frame. The constant self-steering term in (21)
represents a mass-offset-related gravitational moment,
−10
0 5 10 15 20 which is proportional to the roll angle. The steering mass
Speed (m/s) offset also increases the moment of inertia of the steering
system, enhances the steering damping, and introduces a
FIGURE 12 Bicycle straight-running stability properties. This plot
new speed-dependent stiffness term.
shows the real and imaginary parts of the eigenvalues of the
straight-running basic bicycle model with the gyroscopic moment By comparing (20) and (21), it is suggested that the
associated with the front road wheel removed by setting D yy = 0. bicycle equations become too complicated to express in
The (blue) dotted lines correspond to the real parts of the eigen- terms of the original data set when trail and steering incli-
values, while the (red) pluses show the imaginary parts for the nation influences are included. Indeed, when these elabo-
weave mode.
rations are introduced, it is necessary to resort to the use of
intermediate variables and numerical analysis procedures
No Trail, Steering Inclination, or Front-Frame Mass Offset [26]. In the case of state-of-the-art motorcycle models, the
We now remove the basic bicycle’s trail (by setting equations of motion are so complex that they can only be
t = 0), the inclination of the steering system (by setting realistically derived and checked using computer-assisted
λ = 0), and the front-frame mass offset by setting multibody modeling tools.
xff = w. This case is helpful in identifying some of the
key dynamical features of the steering process. The first Gyroscopic Effects
row in (10) relates the roll angle to the steer angle when Gyroscopic precession is a favorite topic of conversation in
mϕ = 0, and shows how the inverted pendulum system bar-room discussions among motorcyclists. While it is not
is forced by the steer angle together with δ̇ and δ̈. The surprising that lay people have difficulty understanding
second row of (10) is these effects, inconsistencies also appear in the technical lit-
erature on single-track vehicle behavior. The experimental
(s2 Cxz − sfw D yy)ϕ(s) + {s2 (Czz + Dzz ) evidence is a good place to begin the process of under-
+s(Czz + Dzz )v/w)}δ(s) = mδ (s), (20) standing gyroscopic influences. Experimental bicycles
whose gyroscopic influences are canceled through the
where fw (s) is the angular velocity of the front wheel. inclusion of counterrotating wheels have been designed
The ϕ(s) term in (20), which is the self-steering term, and built [21]. Other machines have had their gyroscopic
shows how the roll angle influences the steer angle. influences exaggerated through the use of a high-moment-
The first component of the self-steering expression is a of-inertia front wheel [27]. In these cases, the bicycles were
product of inertia, which generates a steering moment found to be easily ridable. As with the stabilization of the
from the roll acceleration. The second self-steering capsize mode by the rider, the precession-canceled bicycle
term represents a gyroscopic steering moment generat- appears to represent little more than a simple low-band-
ed by the roll rate. The expression for P22 in (20) relates width challenge to the rider. As noted in [21], in connection
the steering torque to the steering angle through the with his precession-canceled bicycle, “. . . Its ‘feel’ was a bit
steered system inertia and a physically obscure speed- strange, a fact I attributed to the increased moment of iner-
proportionate damper, apparently coming from the tia about the front forks, but it did not tax my (average) rid-
rear-wheel ground-contact model. ing skill even at low speed . . . ”. It is also noted in [21] that


s2 (Cxz + mff zff (w − xff )) − sfw D yy + gmff (w − xff ) ϕ(s)+


s2 (mff (w − xff )2 + Czz + Dzz ) + sv(Czz + Dzz − mff xff (w − xff ))/w + v2 mff (w − xff )/w δ(s) = mδ (s). (21)

44 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 the precession-canceled bicycle has no autostable speed range, w2
thereby verifying by experiment the findings reported in [20]. k1 (v) = , (25)
tmb cos λ(v2 cos λ − gw sin λ)
When trying to ride this particular bicycle without hands, wg
however, the rider could only just keep it upright because k2 (v) = 2 . (26)
v cos λ − wg sin λ
the vehicle seemed to lack balance and responsiveness.
In their theoretical work, Klein and Sommerfeld [20] Although this stiffness-only model represents the low-
studied a Whipple-like quartic characteristic equation frequency behavior of the steering system, the approxima-
using the Routh criteria. While the basic bicycle model has tion obscures some of the basic bicycle model’s structure.
a stable range of speeds, which Klein and Sommerfeld The poles and zeros of Hδmδ (s, v), as a function of speed,
called the interval of autostability, this model with the are shown in Figure 14. Except for the pair of speed-inde-
spin inertia of the front wheel set to zero is unstable up to pendent zeros, this diagram contains the same information
a speed of 16.4 m/s. This degraded stability can be seen in
Figure 12, where the capsize mode remains stable with the
mδ(s)
damping increasing with speed; due to its stability, the
capsize nomenclature may seem inappropriate in this
case. In contrast, the weave mode is unstable for speeds
below 16.4 m/s, and the imaginary part is never greater
than 1.8 rad/s. Klein and Sommerfeld attribute the stabi- 1
P22
lizing effect of front-wheel precession to a self steering
effect; as soon as a bicycle with spinning wheels begins to
roll, the resulting gyroscopic moment due to the s
fw D yy
term in (20) causes the bicycle to steer in the direction of P21
the fall. The front contact point, consequently, rolls − Σ
P22 δ(s)
towards a position below the mass center.
The Klein and Sommerfeld finding might leave the
impression that gyroscopic effects are essential to auto-sta-
bilization. However, it is shown in [38] that bicycles without
trail or gyroscopic effects can autostabilize at modest speeds P12
−
by adopting extreme mass distributions, but the design P11
choices necessary do not make for a practical machine.

A Feedback System Perspective FIGURE 13 Block diagram of the basic bicycle model described in
[26]. The steer torque applied to the handlebars is mδ (s), ϕ(s) is the
roll angle, and δ(s) is the steer angle.
Basic Bicycle as a Feedback System
To study the control issues associated with bicycles, we
use the second row of (10) to solve for δ(s), which yields 10
8
−P21 (s, v) 1
δ(s) = ϕ(s) + mδ (s). (22)
P22 (s, v) P22 (s, v) 6
Imaginary Part (rad/s)

4
Equations (22) and (14) are shown diagrammatically in the 2
feedback configuration given in Figure 13. Eliminating ϕ(s)
0
yields the closed-loop transfer function
−2
P11 −4
Hδmδ (s, v) = (s, v) . (23)
det(P) −6
−8
In [27], (22) is simplified to
−10
−8 −6 −4 −2 0 2 4 6 8
δ(s) = k1 (v)mδ (s) + k2 (v)ϕ(s) , (24) Real Part (1/s)

FIGURE 14 Poles and zeros of Hδmδ (s, v) as functions of speed.

in which the mass and damping terms are neglected. If the
The speed v is varied between ±10 m/s. The poles are shown as
wheel and front frame masses, as well as all of the inertia blue dots for forward speeds and red crosses for reverse speeds.
terms, are set to zero, these velocity-dependent gains are There are two speed-independent zeros shown as black squares
given by at ± 3.135 1/s.

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 45

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 as that given in Figure 11. As the speed of the bicycle perturbation yaw rate response for the model described in
increases, the unstable poles associated with the static cap- [26] can be calculated using
size modes coalesce to form the complex pole pair associat-
ed with the weave mode. The weave mode is stable for v cos λδ
ψ̇ = ,
speeds above 4.3 m/s [26]. As the machine’s speed increas- w + t/ cos λ
es further, it becomes unstable due to the dynamic capsize
mode at 6.06 m/s. which corresponds to (3) for the Timoshenko-Young bicy-
The zeros of Hδmδ (s, v), which derive from the roots of cle with small perturbation restrictions. In the case of small
P11 (s) as shown in (13) [see (23)], are associated with the perturbations from straight running, (2) becomes
speed-independent whole-vehicle capsize mode. The back-
ward-running vehicle is seen to be unstable throughout ẏ = vψ.
the speed range, but this vehicle is designed for forward
motion and, when running backwards, it has negative trail It now follows that the transfer function linking the lateral
and a divergent caster action. See “Caster Shimmy” and displacement to the steer angle is
note that the cubic terms of (38) and (39) are negative for
negative speeds, indicating instability in this case. v2 cos λ
Hyδ (s, v) = (27)
A control theoretic explanation for the stabilization difficul- s2 (w + t/ cos λ)
ties associated with backward-running bicycles centers on the


positive zero fixed at + gmtzt/Txx , which is in close proximi- and that the transfer function linking the lateral displace-
ty to a right-half plane pole in certain speed ranges [34]. ment to the steering torque is given by Hyδ (s, v)Hδmδ (s, v),
with Hδmδ (s, v) given in (23).
This transfer function is used
F (s)
in the computation of
responses to step steering
r (s) mδ(s) δ(s) torque inputs.
Σ ν
To study the basic bicycle
model’s steering response at
different speeds, including
those outside the autostable
ν
speed range, it is necessary to
introduce stabilizing rider
control. The rider can be
FIGURE 15 Steering torque prefilter F(s) described in (29). This filter is an open-loop realization of the
emulated using the roll-angle
roll-angle-plus-roll-rate feedback law described in (28). As readers familiar with control systems are
aware, open- and closed-loop systems can be represented in equivalent ways if there are no distur- plus roll-rate feedback law
bances and no modeling uncertainties.

Steering mδ (s) = r(s) + (kϕ + skϕ̇ )ϕ(s) , (28)

An appreciation of the subtle nature of bicycle steering
goes back over 100 years. Archibald Sharp records [1, p. in which r(s) is a reference torque input and kϕ and kϕ̇ are
222] “. . . to avoid an object it is often necessary to steer for the roll and roll-rate feedback gains, respectively. This
a small fraction of a second towards it, then steer away feedback law can be combined with (17) to obtain the
from it; this is probably the most difficult operation the open-loop stabilizing steer-torque prefilter
beginner has to master. . . ” While perceptive, such histori-
cal accounts make no distinction between steering torque det(P(s, v))
F(s) = , (29)
control and steering angle control. They do not highlight det(P(s, v)) + (kϕ + skϕ̇ )P12 (s, v)k(s)
the role played by the machine speed, and timing esti-
mates are based on subjective impressions rather than which maps the reference input r(s) into the steering
experimental measurement. torque mδ (s) as shown in Figure 15. In the autostable
As Whipple [17] surmised, the rider’s main control speed range, the stabilizing prefilter is not needed and
input is the steering torque. While in principle one can F(s) is set to unity in this case. The bicycle’s steering
steer through leaning (by applying a roll moment to the behavior can now be studied at speeds below, within, and
rear frame), the resulting response is too sluggish to be above the autostable speed range. Prior to maneuvering,
practical in an emergency situation. The steer-torque-to- the machine is in a constant-speed straight-running trim
steer-angle response of the bicycle can be deduced from condition. For an example of each of the three cases, the
(23). Once the steer angle response is known, the small filtered steering torque and the corresponding roll-angle

46 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 responses are shown in Figure 16, while the steer angle in the steer angle and lateral displacement responses is
and lateral displacement responses are shown in Figure attributable to the right-half plane zero in Hδmδ (s, v) given
17. In each case, the filter gains are chosen to be stabilizing by the roots of P11 (s) = 0 and corresponding to the locked-
and to achieve approximately the same steady-state roll steering whole-machine capsize mode as illustrated in
angle; numerical gain values appear in the figure captions. (13). Toward the end of the simulation shown, the steer
The autostable case is considered first, because no stabiliz- angle settles into an equilibrium condition, in which the
ing torque demand filtering is required. In this case, the bicycle turns left in a circle with a fixed negative roll
clockwise (when viewed from above) unit-step steer angle. In relation to the nonminimum phase response in
torque demand is applied directly to the bicycle’s steering the lateral displacement behavior, the reader is reminded
system (see Figure 16). The machine initially steers to the of the control difficulty that arises if one rides near to a
right and the rear wheel ground-contact point starts mov- curb or a drop [39]; to escape, one has to go initially closer
ing to the right also (see Figure 17). Following the steer to the edge. Body lean control is unusually useful in such
torque input, the bicycle immediately rolls to the left (see circumstances.
Figure 16) in preparation for a left-hand turn. After At speeds below the autostable range, a stabilizing steer-
approximately 0.6 s, the steer angle sign reverses, while ing-torque prefilter must be utilized to prevent the machine
the rear-wheel ground-contact point begins moving to the from toppling over. In the low-speed (3.7 m/s) case, the steer
left after approximately 1.2 s. The oscillations in the roll torque illustrated in Figure 16 is the unit-step response of the
angle and steer angle responses have a frequency of about prefilter, which is the steer torque required to establish a
0.64 Hz and are associated with the weave mode of the steady turn. The output of the prefilter is unidirectional apart
bicycle (see Figure 11). Therefore, to turn to the left, one from the superimposed weave-frequency oscillation required
must steer to the right so as the make the machine roll to to stabilize the bicycle’s unstable weave mode. In the case
the left. This property of the machine to apparently roll in considered here, the steady-state steer torque is more than
the wrong direction is sometimes referred to as counter- twice the autostable unit-valued reference torque required to
steering [39], [27], but an alternative interpretation is also bring the machine to a steady-state roll angle of approximate-
possible, as seen below. The nonminimum phase behavior ly −0.65 rad. To damp the weave oscillations in the roll and

2.5 0

2 −0.1 0.1 0.5

1.5 −0.2 0.05 0

Steer Torque (N-m)

Roll Angle (rad)

Lateral Displacement (m)

0
1 −0.3 −0.5
Steer Angle (rad)

−0.05
0.5 −0.4 −1
−0.1
0 −0.5 −1.5
−0.15
−2
−0.5 −0.6 −0.2

−0.25 −2.5
−1 −0.7
0 5 10 15 20 0 5 10 15 20
Time (s) Time (s) −0.3 −3
0 0.5 1 1.5 2 0 0.5 1 1.5 2
Time (s) Time (s)
FIGURE 16 Step responses of the prefilter and the roll angle of the
basic bicycle model. The steering torque and roll angle response at
the autostable speed of 4.6 m/s are shown in blue; the prefilter gains
are kϕ = 0 and kϕ̇ = 0. The low-speed 3.7 m/s case, which is below FIGURE 17 Response of the simple bicycle model to a steering
the autostable speed range, is shown in red; the stabilizing preflter moment command. The steer angle and the rear-wheel ground-con-
gains are kϕ = −2 and kϕ̇ = 3. The high-speed 8.0-m/s case, which tact point displacement responses at the autostable speed of 4.6
is above the autostable speed range, is shown in green; the stabiliz- m/s are shown in blue; the prefilter gains are kϕ = 0 and kϕ̇ = 0.
ing prefilter gains are kϕ = 2.4 and kϕ̇ = 0.02. In each case, a clock- The responses at a speed of 3.7 m/s, which is below the autostable
wise steering moment (viewed from above) causes the machine to speed range, are shown in red; the stabilizing prefilter gains are
roll to the left. This tendency of the machine to apparently roll “in the kϕ = −2 and kϕ̇ = 3. The responses at a speed of 8.0 m/s, which is
wrong direction” is sometimes referred to as countersteering. In the above the auto-stable speed range, are shown in green; the stabiliz-
high-speed case (green curves), the steering torque is positive initially ing prefilter gains are kϕ = 2.4 and kϕ̇ = 0.02. The steer angle and
and then negative. This need to steer in one direction to initiate the lateral displacement responses show the influence of the right-half-
turning roll response, and then to later apply an opposite steering plane zero of P11 (s). This zero is associated with the unstable
torque that stabilizes the roll angle, is a high-speed phenomenon. whole-vehicle capsize mode. See point A in Figure 11 and (13).

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 47

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 steer angle responses, the torque demand filter, which mim- which is clearly minimum phase and represents the
ics the rider, introduces weave-frequency fluctuations into response one would expect when applying a torque to a
the steering torque. The steer angle and lateral displacement pure inertia with a damper to ground.
responses are similar to those obtained in the autostable case.
If the trim speed is increased to the upper limit of the Pneumatic Tires, Flexible Frames, and Wobble
autostable range (in this case 6.1 m/s; see Figure 11), then A modified version of the basic bicycle model is now con-
the steady-state steering torque required to maintain an sidered in which a flexible frame and side-slipping tires
equilibrium steady-state turn falls to zero; this response is are included. The flexibility of the frame is modeled by
due to the singularity of the stiffness matrix v2 K2 + K0 at including a single rotational degree of freedom located
this speed. At speeds above the autostable range, stabiliz- between the steering head and the rear frame. In the model
ing rider intervention is again required. As before, in studied here, the twist axis associated with the frame flexi-
response to a positive steer torque input, the steer angle bility freedom is in the plane of symmetry and perpendic-
and lateral displacement initially follow the steer torque ular to the steering axis, and the associated motion is
(see Figure 17). At the same time the machine rolls to the restrained by a parallel spring-damper combination. In
left (see Figure 16). Moments later, one observes the non- this modified model, the nonholonomic lateral ground
minimum phase response in the steer angle and the lateral contact constraints are replaced by (31) and (32); see “Tire
displacement responses. The interesting variation in this Modeling.” These equations represent tires that produce
case is in the steering torque behavior. This torque is ini- lateral forces in response to sideslip and camber, with time
tially positive and results in the machine rolling to the left. lags dictated by the speed and the tires’ relaxation lengths.
However, if this roll behavior were left unchecked, the The tire and frame flexibility data used in this study are
bicycle would topple over, and so to avoid the problem the given in Table 2; two representative values for the frame
steer torque immediately reduces and then changes sign stiffness KP and frame damping CP are included. The high-
after approximately 4 s. The steer torque then approaches a er values of KP and CP are associated with a stiff frame,
steady-state value of −0.6 N-m to stabilize the roll angle while the lower values correspond to a flexible frame.
and maintain the counterclockwise turn. This need to steer First, we examine the influence of frame compliance
in one direction to initiate the turning roll angle response, alone on the system eigenvalues, which can be seen in Fig-
and then to later apply an opposite steering torque that ure 18. The dotted curve corresponds to the rigid frame that
stabilizes the roll angle is a high-speed phenomenon, pro- was studied in Figure 11 and is included here for reference
viding the alternative interpretation of countersteering purposes. The cross- and circle-symbol loci correspond to
mentioned earlier. Countersteering in the first sense is the stiff and flexible frames, respectively. The first important
always present, while in the second sense it is a high-speed observation is that the model predicts wobble when frame
phenomenon only. It is interesting to observe that the pre- compliance is included; see “Wobble.” In the case of the
filter enforces this type of countersteering for all stabilizing flexible frame, the damping of the wobble mode reaches a
values of kϕ and kϕ̇ . First note that the direct feedthrough minimum at about 10 m/s and the mode has a resonant fre-
(infinite frequency) gain of F(s) is unity. Since kϕ and kϕ̇ quency of approximately 6 Hz. In the case of the stiff frame,
are stabilizing, all of the denominator coefficients of F(s) the wobble mode’s resonant frequency increases, while its
have the same sign as do all of the numerator terms in the damping factor decreases, with increasing speed. Figure 18
autostable speed range. As the speed passes from the also illustrates the impact of frame flexibility on the damp-
autostable range, det(v2 K2 + K0 ) changes sign, as does the ing of the weave mode. At low speeds, frame flexibility has
constant coefficient in the numerator of F(s). Therefore, at no impact on the characteristics of the weave mode. At
speeds above the autostable range, F(s) has a negative intermediate and high speeds, the weave-mode damping is
steady-state gain, thereby enforcing the sign reversal in the
steering torque as observed in Figure 16.
We conclude this section by associating the basic bicy- TABLE 2 Bicycle tire and frame flexibility parameters. Tire
cle model’s nonminimum phase response (in the steer parameters include relaxation lengths and cornering- and
camber-stiffness coefficients. The frame flexibility is
angle) with its self-steering characteristics. To do this, con-
described in terms of stiffness and damping coefficients. All
sider removing the basic bicycle’s ability to self-steer by of the parameter values are given in SI units.
setting α = π/2, t = 0, Cxz = 0, D yy = 0, and xff = w. With
these changes in place, it is easy to see from (20) that Parameters Value
P21 (s, v) = 0. This identity means that σ f , σr 0.1, 0.1
C f s , Cr s 14.325, 14.325
1
Hδmδ = C f c , Cr c 1.0, 1.0
P22 (s, v) KPlow ,
high
KP 2,000, 10,000
1 high
= , (30) CPlow , CP 20, 50
s(Czz + Dzz )(s + v/w)

48 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 Tire Modeling
vf
C
lassical bicycle models, such as those developed by Whipple ssf = ,
[17], and Timoshenko and Young [29], describe the wheel- < vfus , i >
road contact as a constraint. The wheel descriptions involve rota-
where < ·, · > denotes the inner product. The slip is in the j direction in
tional coordinates to specify the wheels’ orientation and
the case of no longitudinal slipping, as is assumed here. If the bristle
translational coordinates that describe the location of the road
bending stiffness is constant and the frictional coupling between the
contact points. The rolling constraints connect these coordinates
bristle tips and the ground is sufficient to prevent sliding, the lateral
so that translational changes are linked to rotational ones. In the
force developed is proportional to ssf and acts to oppose the slip.
case of general motions, the rotational and translational coordi-
When the rolling wheel is leaned over, then even with no slip,
nates cannot be linked algebraically, since this linkage is path
the tread base material becomes distorted from its unstressed state.
dependent; thus the nomenclature “nonholonomic,” or incomplete
This distortion leads to the development of a lateral force that is
constraint [18, p. 14]. Instead, it is the rotational and translational
approximately equal to the normal tire load multiplied by the camber
velocities that are linked, and the rolling constraint renders the
angle [51], [52]. If the tire is not working hard, the force due to cam-
wheels’ ground-contact points, or lines, absolutely stationary [24],
ber simply superimposes on the force due to slip. The elemental lat-
[51], [52]. During motion, the wheel-ground contact points change
eral forces due to camber are distributed elliptically over the contact
with time, with each point on the wheel periphery coming into con-
length, while those due to sustained slip increase with the longitudi-
tact with the ground once per wheel revolution. In the case of the
nal distance of the tire element from the point of first contact. As the
bicycle, it is illustrated that (nonholonomic) tire constraint modeling
sideslip increases, the no-sliding condition is increasingly chal-
limits the fidelity of the vehicle model to low speeds only.
lenged as the rear of the contact patch is approached. Thus, as the
By 1950, the understanding of tire behavior had improved sub-
tire works harder in slip, sliding at the rear of the contact patch
stantially, and it had become commonplace, although not universal,
becomes more pronounced. Force saturation is reached once all
to regard the rolling wheel as a force producer rather than as a con-
the tire elements (bristles) in contact with the road begin to slide.
straint on the vehicle’s motion. With real tire behavior, the tread
When the tire operates under transient conditions, following for
material at the ground contact “slips” relative to the road and so has
example a step change in steering angle, the distortion of the tire
a nonzero absolute velocity and the linkage between the wheels’
tread material described above does not develop instantly. Instead,
rotational and translational velocities is lost. To model this behavior,
the distortion builds up in a manner that is linked to the distance cov-
it is necessary to introduce a slip-dependent tire force-generation
ered from the time of application of the transient. For vehicle model-
mechanism.
ing purposes, a simple approximation of this behavior is to treat the
To understand the underlying physical mechanisms underpin-
dynamic force development process as a speed-dependent first-
ning tire behavior, it is necessary to analyze the interface between
order lag. The characterizing parameter, called the relaxation length
the elastic tire tread base and the ground. This distributed contact σ f , is similar to a time constant except that it has units of length
involves the tire carcass and the rubber tread material, which can
rather than time. The relaxation length is a tire characteristic that can
be thought of as a set of bristles that join the carcass to the ground.
be determined experimentally. The lateral force response of the tire
Under dynamic conditions, these bristles move, as a continuous
due to steering, and therefore side slipping, is a dynamic response to
stream, into and out of the ground-contact region. Under free-
the slip and camber angles of the tire, which is modeled as
rolling conditions, in common with the nonholonomic rolling model,
σf
the tread-base material is stationary; consequently, the bristles Ẏf + Yf = Zf (Cfs ssf + Cf c ϕf ) , (31)
|vfus |
remain undeformed in bending as they pass through the contact
region. When rolling resistance is neglected, no shear forces are where Zf is the normal load on the front tire and ϕf is the front wheel’s
developed. Free-rolling corresponds to zero slip, and, if a slip is camber angle relative to the road (pavement). The force Yf acts in the j
developed, it has in general both longitudinal and lateral compo- direction and opposes the slip. The product Zf Cf s is the tire’s cornering
nents [50], [51]. In contrast to the physical situation, tire models stiffness, while Zf Cf c is its camber stiffness. The sideforce associated
usually rely on the notion of a ground-contact point. with the rear tire is given analogously by
To assemble these ideas in a mathematical framework, let vf
σr
denote the velocity of the tread base material at the ground contact Ẏr + Yr = Zr (Cr s ssr + Cr c ϕr ) , (32)
|vus
r |
point. In the case of no longitudinal slipping, vf is perpendicular to
the line of intersection between the wheel plane and the ground where each term has an interpretation that parallels that of the front
plane; the unit vector i lies along this line of intersection and the wheel. Equations (31) and (32) are suitable only for small perturbation
unit vector j is perpendicular to it. The velocity of a tread base point modeling.
with respect to the wheel axle is given by vf r = ωf Rf i, where ωf is Contemporary large perturbation tire models are based on
the wheel’s spin velocity and Rf is the wheel radius. If we now magic formulas [51] and [53]–[55], which can mimic accurately mea-
associate with this ground contact point an “unspun” point, its sured tire force and moment data over a wide range of operating
velocity is vfus = vf + vf r . The slip (for the front wheel) is defined as conditions.

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 49

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 Wobble

A
phenomenon known variously as speedman’s wobble, wobble, raising both the frequency and the worst-case speed,
speed wobble, or death wobble is well known among but is not necessarily advantageous overall.
cyclists [85] and [86]. As the name suggests, wobble is a steer- In an important paper from a practical and experiential view-
ing oscillation belonging to a more general class—wheel shim- point, [37] implies that wobble was a common motorcycling phe-
my. The oscillations are similar to those that occur with nomenon in the 1950s. Machines of the period were usually fitted
supermarket trolley wheels, aircraft nose wheels, and automo- with a rider-adjustable friction-pad steering damper. The idea was
bile steering systems. Documentation of this phenomenon in that the rider should make the damper effective for high-speed
bicycles is sparse, but a survey [86] suggests that wobble at running and ineffective for lower speeds; see also [88]. Refer-
speeds between 4.5–9 m/s is unpleasant, while wobble at ence [37] offers the view that steering dampers should not be
speeds between 9–14 m/s is dangerous. The survey [86] also necessary for speeds under 45 m/s, indicating that, historically,
suggests a wide spread of frequencies for the oscillations with wobble of motorcycles has been a high-speed problem. Refer-
the most common being between 3–6 Hz, somewhat less than ence [37] also points to the dangers of returning from high speed
for motorcycles. The rotation frequency of the front wheel is to low speed while forgetting to lower the preload on the steering
often close to the wobble frequency, so that forcing from wheel damper. A friction lock on the steering system obliges a rider to
or tire nonuniformity may be an added influence. Although use fixed (steering position) control, which we have earlier
rough surfaces are reported as being likely to break the regular- demonstrated to be difficult. The current status of motorcycle
ity of the wobble and thereby eliminate it, an initial event is nor- wobble analysis is covered in the “Motorcycle Modeling” section.
mally needed to trigger the problem. Attempting to damp the Wheel shimmy in general is discussed in detail in [51], where
vibrations by holding on tightly to the handlebars is ineffective, a a whole chapter is devoted to the topic. Ensuring the stability of
result reproduced theoretically for a motorcycle [87]. The sur- wheel shimmy modes in aircraft landing gear, automotive steer-
vey [86] recommends “pressing one or both legs against the ing systems, and single-track vehicles is vital due to the potential
frame, while applying the rear brake” as a helpful practical pro- violence of the oscillations in these contexts. An idea of how
cedure, if a wobble should commence. The possibility of accel- instability arises can be obtained by examining simple cases
erating out of a wobble is mentioned, suggesting a worst-speed (see “Caster Shimmy”), but systems of practical importance are
condition. The influences of loading are discussed with special sufficiently complex to demand analysis by automated multibody
emphasis on the loading of steering-frame-mounted panniers. modeling tools and numerical methods.
Evidently, these influences are closely connected with the first A simple system quite commonly employed to demonstrate
term in each of (20) and (21), representing roll-acceleration-to- wheel shimmy, both experimentally and theoretically [51], [89],
steer-torque feedback. Sloppy wheel or steering-head bearings is shown in Figure C. If the tire-to-ground contact is assumed
and flexible wheels are described as contributory. Increasing to involve nonholonomic rolling, the characteristic equation of
the mechanical trail is considered stabilizing with respect to the system of Figure C is third order, and symbolic results for

compromised by the flexible frame, although this mode surement. Comparison with Figure 18 shows that the intro-
remains well damped. As is now demonstrated, the more duction of the side-slipping tire model causes a marked
realistic tire model has a strong impact on the predicted reduction in the wobble-mode frequency for the stiff-framed
properties of both wobble and weave. machine, while the impact of the side-slipping tires on the
Figure 19 shows the influence of frame flexibility in com- wobble mode of the flexible-framed machine is less marked.
bination with relaxed side-slipping tires. Again, the cross- As with the flexible frame, side-slipping tires have little
and circle-symbol loci correspond to the stiff and flexible impact on the weave mode at very low speeds. However, as
frames, respectively, while the dotted loci belong to the the speed increases, the relaxed side-slipping tires cause a
rigid-framed machine. As can be seen from this dotted locus, significant reduction in the intermediate and high-speed
the introduction of side-slipping tires also produces a wob- weave-mode damping. By extension from measured motor-
ble mode, which is not a property of the basic bicycle. The cycle behavior, there is every reason to suspect that the accu-
predicted resonant frequency of the wobble mode varies rate reproduction of bicycle weave- and wobble-mode
from approximately 12.7–4.8 Hz, depending on the frame behavior requires a model that includes both relaxed side-
stiffness. Lower stiffnesses correspond to lower natural fre- slipping tires and flexible frame representations.
quencies. With a rigid frame, the wobble damping is least at
low and high speeds. With a compliant frame, the damping MOTORCYCLE MODELING
is least at an intermediate speed. The frame flexibility can be
such that the resonant frequency aligns with the practical Background
evidence. Frame flexibility modeling can also be used to Several factors differentiate bicycles from motorcycles. A
align the wobble-mode damping with experimental mea- large motorcycle can weigh at least ten times as much as a

50 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 the conditions for stability are
obtained in “Caster Shimmy.” For
higher levels of complexity, the Forks υ
system order is increased and
m, Jz
analytical stability conditions k
become significantly more com-
δ
plex. In [51], a base set of para-
meter values is chosen, and
y
stability boundaries are found
numerically for systematic varia- f
v
tions in speed and mechanical e King-Pin
e
trail . The resulting stability Y Bearing
boundaries are plotted in the (v ,
FIGURE C Plan view of a simple system capable of shimmy. This example is adapted from [51]
e) parameter space for several and [89, pp. 333, 334, ex. 215 p. 414]. The wheel is axisymmetric and free to spin relative to
values of the lateral stiffness k of the forks that support it; the wheel is deemed to have no spin inertia. The wheel has mechani-
the king-pin mounting. The least cal trail e and mass offset f with respect to the vertical king-pin bearing. The king-pin is free to
oscillatory system is that having translate laterally with displacement y from static equilibrium, while the whole assembly
moves forward with constant speed v. The king-pin mounting has stiffness k, while the moving
the highest stiffness, with the
assembly has mass m. The steer angle is δ. The king-pin is assumed massless so that analy-
king-pin compliance contributing sis deals with only one body; see “Caster Shimmy.” The tire-ground contact can be treated on
to the system behavior in much one of three different levels. First, pure (nonholonomic) rolling, implying no sideslip, can be
the same way tire lateral compli- assumed. Second, the tire may be allowed to sideslip thereby producing a proportionate and
ance contributes. instantaneous side force. Third, the side force may be lagged relative to the sideslip by a first-
order lag determined by the tire relaxation length; see “Tire Modeling.”
Significant from the point of
view of single-track vehicles, and
aircraft nose-wheels, is the lateral compliance at the king-pin. If [51] to create a second area of instability in the (v , e) space at
this compliance allows the assembly to rotate in roll about an higher speeds, which have a substantially different mode
axis well above the ground, as with a typical bicycle or motorcy- shape. The gyroscopic mode involves a higher ratio of lateral
cle frame or aircraft fuselage, lateral motions of the wheel contact point velocity to steer velocity than occurs in situations
assembly are accompanied by camber changes. If, in addition, in which a roll freedom is absent. This new phenomenon is
the wheel has spin inertia, gyroscopic effects have an important called gyroscopic shimmy, and it is this shimmy variant that is
influence on the shimmy behavior. These effects are shown in particularly relevant to the single-track vehicle [40], [41], [47].

bicycle, and, consequently, in the case of a motorcycle, the also dimensionless. These changes of variable allowed
rider’s mass is a much smaller fraction of the overall rider- Whipple to establish that the roots of the quartic character-
machine mass. A modern sports motorcycle can achieve istic equation, which represents the small perturbation
top speeds of the order 100 m/s, while a modern sports behavior around a straight-running trim state, are inde-
bicycle might achieve a top speed of approximately 20 pendent of the mass units used. Therefore, for the nonho-
m/s. As a result of these large differences in speed, our lonomic bicycle model, increasing the masses and inertias
understanding of the primary modes of bicycles must be of every body by the same factor makes no difference to
extended to speeds that are usually irrelevant to bicycle the roots of the characteristic equation. In this restricted
behavioral studies. At high speeds, aerodynamic forces are sense, a grown man riding a motorcycle is dynamically
important and need to be accounted for. equivalent to a child riding a bicycle.
In his study of bicycles, Whipple [17] introduced a Whipple then showed that the characteristic equation
nondimensional approach to bicycle dynamic analysis, p(λ, v) = 0 can be replaced with p̃(ξ, ) = 0 using a change
which is helpful when seeking to deduce the behavior of of variables. In the first case, the speed v has units such as
motorcycles from that of bicycles. The dimensionless m/s, the characteristic equation has roots λi having the
model was obtained by representing each mass by units of circular frequency (rad/s for example). The new
m = αw, where α is dimensionless and w has the units of variables: ξ = λb/v and  = gb/v 2 , where g is the gravita-
mass (kilograms, for example) and each length quantity by tional constant, are dimensionless as are the polynomial’s
l = β b, where β is dimensionless and b has the units of coefficients. Therefore, all of the nondimensional single-
length (meters, for example). As a result, the moments and track vehicles corresponding to p̃(ξ, ) = 0, where  is a
products of inertia are expressed as J = γ wb2 , where γ is constant, have the same dynamical properties in terms of

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 51

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 Caster Shimmy

C
aster wheel shimmy can occur in everyday equipment such where det[·] comes from (37). It follows from (38) and the Routh
as grocery trolleys, gurneys, and wheelchairs. These self- criterion that shimmy occurs if m f (e − f ) ≤ Jz , and in the case
excited oscillations, which are energetically supported by the that m f (e − f ) = Jz the frequency of oscillation is

vehicle prime mover, are an important consideration in the ω = (ke)/(m(e − f )). These results show the role played by the
design of aircraft landing gear and road vehicle suspension steering system geometry, and the mass and inertia properties of
and steering systems. In the context of bicycles and motorcy- the moving assembly in determining the stability, or otherwise, of
cles, this quantitative analysis is conducted by including the the system. The king-pin stiffness influences the frequency of oscil-
appropriate frame flexibility freedom and dynamic tire descrip- lation. The case of m f (e − f ) = Jz corresponds to a mass distri-
tions in the vehicle model. The details are covered in the bution in which the rolling contact is at the center of percussion
“Pneumatic Tires, Flexible Frames, and Wobble” section. relative to the kingpin. In this situation the rolling constraint has no
By its nature, a caster involves a spinning wheel, a king-pin influence on the sping force.
bearing, and a mechanical trail sufficient to provide a self-center- In the case of a rigid assembly
ing steering action. Our purpose here is to demonstrate how
oscillatory instability can be predicted for the simple system of
det[·] σ (f 2 m + Jz )s3
Figure C. In the case of small perturbations, the tire sideslip is lim =
k →∞ kC Cv
eδ̇ − ẏ (f 2 m + Jz )s2 e2 s
s=δ+ . (33) + + + e. (39)
v C v

It follows from (31) that the resulting tire side force F is given by It follows from (39) that shimmy occurs if e ≤ σ , and in the case

that e = σ the frequency of oscillation is ω = Ce/(f 2 m + Jz ).
σ
Ẏ + Y = Cs, (34) The tire properties dictate both conditions for the onset of shimmy
v
and its frequency when it occurs. Interestingly, the tire relaxation
in which C is the tire’s cornering stiffness and σ is the relaxation length alone determines the onset, or otherwise, of shimmy, while
length. The equations of motion for the swivel wheel assembly in the frequency of oscillation is dictated by the tire’s cornering stiff-
Figure D are ness alone. The Pirelli company reports [90] on a tire tester that
m(ÿ − f δ̈) + ky − Y = 0 (35) relies on this precise result. The test tire is mounted in a fork trailing
and a rigidly mounted king-pin bearing and runs against a spinning
Jz δ̈ + (e − f )Y + kyf = 0, (36) drum to represent movement along a road. Following an initial
steer displacement of the wheel assembly, the exponentially
where Jz is the yaw-axis moment of inertia of the swiveled wheel decaying steering vibrations are recorded, and the decrement
assembly around the mass center. The characteristic polynomial yields the tire relaxation length, while the frequency yields the cor-
associated with small motions in the system in Figure D is derived nering stiffness. Unlike the bicycle case, the shimmy frequency is
directly from (33)–(36). The resulting quintic polynomial is independent of speed.
As discussed in the “Basic Bicycle Model” section, in connec-

 tion with the zero-speed behavior, the simple caster does not in
ms2 + k −f ms2 −1
det kf s2 Jz e−f . (37) reality oscillate at vanishingly small speeds due to the distributed
Cs/v −C(1 + (es)/v ) 1 + σ s/v contact between the tire and the ground. The energy needed to
increase the amplitude of unstable shimmy motions comes from
Two interesting special cases can be deduced from the gen- the longitudinal force that sustains the forward speed of the king-
eral problem by making further simplifying assumptions. In the pin. This longitudinal force is given by
case of the nonholonomic wheel, the cornering stiffness becomes
arbitrarily large for all values of σ, thereby preventing tire sideslip F = m(δ ÿ + f δ̇ 2 ) + δky

for the small perturbation problem described in (33)–(36). In the

det[·] (m(e − f )2 + Jz )s3
lim = case of a pure-rolling (nonholonomic) tire, δ̇ should be eliminated
C→∞ kC kv
m(e − f )s2 e2 s from the above equation using the zero-sideslip constraint
+ + + e, (38) δ̇ = (ẏ − v δ)/e.
k v

ξ . The modal frequencies and decay/growth rates scale gle nondimensional vehicle. For example, if b is halved so


according to eλi t translating to e(ξi v/bt ) , where t
is dimen- as to represent a child’s bicycle in this alternative length-
sionless. This analysis provides a method for predicting scaling sense,√then a simultaneous reduction of the speed
the properties of a family of machines from those of a sin- by a factor of 2 leaves the roots of p̃(ξ, ) unchanged. The

52 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 associated variation in √ the time domain response comes slowly divergent instability of the whole vehicle, which
from λi translating to 2λi . corresponds to the machine toppling over onto its side.
Whipple’s scaling rules, in combination with observa- This mode is relatively unimportant because it is easily
tions, lead one to conclude that a viable motorcycle model (and subconsciously) controlled by the rider. As with the
1) must be consistent with bicycle-like behavior at low bicycle, weave is a low-frequency (2–3 Hz) oscillation of
speed, 2) must reproduce the autostability properties pre- the whole vehicle involving roll, yaw, and steer motions,
dicted by Whipple [17], 3) must reproduce the motorcycle’s and is well damped at moderate speeds but becomes
inclination to wobble at intermediate and high speeds, and
4) must reproduce the observed high-speed weave charac-
teristics of modern high-performance motorcycles. 100
High-powered machines with stiff frames have a ten- 90
dency to wobble at high speeds [40]–[42]; see “Tommy 80
Smith’s Wobble.” A primary motivation for studying wob-

Imaginary Part (rad/s)

70
ble and weave derives from the central role they play in
60
performance and handling qualities. These modes are also
50 Wobble
associated with a technically challenging class of stability-
related road accidents. Several high-profile accidents of this 40
type are reviewed and explained in the recent literature 30
[43]. Central to understanding the relevant phenomena is 20 Weave
the ability to analyze the dynamics of motorcycles under Capsize
10
cornering, where the in-plane and out-of-plane motions,
which are decoupled in the straight-running situation, 0
become interactive. Consequently, cornering models tend −40 −30 −20 −10 0 10
to be substantially more complex than their straight-run- Real Part (1/s)
ning counterparts. This added complexity brings computer-
FIGURE 18 Root loci of the basic bicycle model with a flexible frame.
assisted multibody modeling to the fore [42], [44], [45]. The speed is varied from 0–20 m/s; the zero-speed end is repre-
In the remainder of this article, we study several contri- sented by a square and the high-speed end by a diamond. The
butions, both theoretical and experimental, that have (blue) dotted loci correspond to the rigid frame, the (black) crosses
played key roles in bringing the motorcycle modeling art to the high frame stiffness and damping values, and the (red) circles
to the low stiffness and damping values.
to its current state of maturity. Readers who are interested
in the early literature are referred to the survey paper [46],
which reviews theoretical and experimental progress up to 100
the mid 1980s. That material focuses almost entirely on the
90
straight-running case, which is now considered.
80
Imaginary Part (rad/s)

Straight-Running Motorcycle Models 70

An influential contribution to the theoretical analysis of the 60
straight-running motorcycle is given in [47]. The model
50
developed in [47] is intended to provide the minimum level
40
of complexity required for predicting the capsize, weave,
and wobble modes. This research is reminiscent of Whip- 30 Wobble
ple’s analysis in terms of the assumptions concerning the 20
Weave
rider and frame degrees of freedom. In contrast to Whipple, Capsize
10
[47] treats the tires as force generators, which respond to
0
both sideslip and camber; tire relaxation is included (see
“Tire Modeling”), while aerodynamic effects are not. −40 −30 −20 −10 0 10
Real Part (1/s)
A linearized model is used for the stability analysis
through the eigenvalues of the dynamics matrix, which is FIGURE 19 Root loci of the basic bicycle model with flexible frame
a function of the vehicle’s (constant) forward speed. Two and relaxed sideslipping tires. The speed is varied from 0–20 m/s;
cases are considered: one with the steering degree of free- the zero-speed end is represented by a square and the high-speed
dom present, giving rise to the free-control analysis, and end by a diamond. The (blue) dotted loci correspond to the rigid
the other with the steering degree of freedom removed, frame, the (black) crosses to the high frame stiffness and damping
values, and the (red) circles to the low stiffness and damping val-
giving rise to the fixed-control analysis. The free-control ues. The properties illustrated here for the limited speed range of
model predicts the existence of capsize, weave, and wob- the bicycle are remarkably similar to those of the motorcycle, with
ble modes. As with the bicycle, the capsize mode is a its extended speed capabilities.

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 53

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 Tommy Smith’s Wobble

T
ommy Smith was born in 1933. He started riding motorized bicy- On 25 August 1952, Tommy made his third high-speed run.
cles at the age of 13 and was racing motorcycles professionally Everything started normally—the floating sensation was the same
by the age of 17. In 1952, Tommy had the opportunity to ride a as it had been on previous tests. Suddenly, the motorcycle went
modified 650 cc Triumph Thunderbird at the Bonneville Salt Flats in into a high-speed wobble and Tommy held tightly onto the handle-
Utah, United States. At that time, fuel (as opposed to gasoline) bars to prevent himself from falling off. After a period of 3–5 s, the
motorcycles used about 70% methanol and 30% nitro methane; wobble was so violent that Tommy “hit the salt” and slid through
crankcase explosions occurred when higher nitro percentages were the first 1/10 mi speed trap at an official speed of 139 mi/h. The
tried. To further increase the motorcycle’s engine power, the cylin- speed of the motorcycle was not recorded! Although the motorcy-
der head was reversed, so that the intake ports were pointing for- cle was only slightly damaged, Tommy’s abrasion injuries were
ward to achieve a ram air effect. This engine configuration made it severe enough to keep him out of the Korean War. At the time, it
impossible to sit on the machine in a conventional manner. For this was suggested that Tommy’s light weight contributed to the
reason, the motorcycle was fitted with a plywood board for the rider motorcycle’s instability, because heavier riders did not experience
to lie prone on. Leathers were heavy and uncomfortable and so wobble at similar speeds. This suggestion that light riders might
Tommy rode the bike wearing a fiberglass helmet, goggles, tennis be prone to instability has been investigated by computer simula-
shoes (with socks), and a Speedo bathing suit (see Figure D). tion studies [43]. The mobility of the rider relative to the motorcy-
On the first high-speed run, the machine produced an eerie cle, as well as his rearward positioning, which led to a reduction in
“floating” sensation that was probably the result of a veneer of the front wheel load, are likely to have been important influences
loose salt on the running track combined with a lightly loaded on the machine problem treated above.
front wheel, resulting from the high speed and unusual riding
position. Engine revolution and speed measurements taken at
the time suggested that there was approximately 4.5 m/s of lon-
gitudinal tire-slip velocity. An accompanying lateral drifting phe-
nomenon had to be corrected with small handlebar inputs that
were required every 5–10 s. The need for continuous steering
corrections may have also been associated with an unstable
capsize mode with an unusually large growth rate and the lack
of constraint between the rider and machine, both related to the
riding position. Detailed calculations relevant to the situation
described have not been carried out, so far as the authors are
aware. The official one-way speed achieved was 147.78 mi/h, FIGURE D 24 Modified Triumph Thunderbird. Tommy Smith riding
a modified 650-cc Triumph Thunderbird at the Bonneville Salt
which was not to be exceeded by a 650-cc-motorcycle rider for
Flats in Utah. Note the forward-facing air intake ports.
another ten years.

increasingly less damped and possibly unstable at higher offset from the steering axis are also demonstrated. A recur-
speeds. Wobble is a higher frequency (typically 7–9 Hz) ring theme is the need to find compromises under varia-
motion that involves primarily the steering system. In tions in these critical parameters.
contrast to the bicycle study presented in this article, [47] Leaving briefly the constant-forward-speed case, [48]
predicts that the wobble mode is well damped at low represents the first attempt to study the effects of accelera-
speeds, becoming lightly damped at high speeds. tion and deceleration on the stability of motorcycles. A
In particular, the study shows that tire relaxation is an rather simple approach, in which the longitudinal equation
important contributor to the prediction of wobble and the of motion is decoupled from the lateral equations, gives the
quantitative characteristics of high-speed weave. The influ- longitudinal acceleration as a parameter of the lateral
ences of parameter variations on the vehicle’s dynamic motion. The acceleration parameter contributes to longitudi-
behavior are also studied, and the results obtained are for nal inertia forces, which are included in standard stability
the most part aligned with the behavior of vehicles of the computations. Such computations lead to some tentative
time. Of particular importance is the predicted influence of conclusions, which depend on knowledge of the influence
the steering damper on the wobble-mode damping and the of loading on tire force and moment properties. More recent
destabilizing effect that the damper has on the weave mode. results [49], which are based on a higher fidelity model, are
The positive effect of moving the rear frame mass center for- not supportive of the conclusions given in [48]. In [49], it is
ward, the critical impact on stability of the steering-head found that braking and acceleration have little influence on
angle, the mechanical trail, and the front frame mass center the frequency and damping of the weave mode. It is also

54 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 concluded in [49] that descending a hill or braking have a ity of production motorcycles are minor. However, the drag,
substantial destabilizing effect on the wobble mode. Con- lift, and pitching moments contribute significantly to
versely, the wobble-mode damping increases substantially changes in the posture of the machine on its suspension and
under acceleration or ascending an incline, for small pertur- also to the tire loads. Aiming to explain the high-speed
bations from straight running. An open issue is the influ- weave stability problem, [61] introduces these aerodynamic
ence of acceleration or braking on a cornering machine. effects into the model of [47] using aerodynamic parameters
corresponding to a streamlined machine. These results yield
Tire Modeling the conclusions that aerodynamic effects lead to only minor
Modeling the generation of shear forces and moments by changes in the wobble mode and that high-speed weave dif-
pneumatic tires has been approached in various ways, which ficulties cannot be attributed entirely to steady-state aerody-
recognize the physics of the situation in more or less detail. At namic loading. As a result, it is postulated in [61] that the
one extreme, physical models [50]–[52] contain detailed problem may involve nonsteady aerodynamic influences. To
descriptions of the tire structure and the tread-ground interac- fully appreciate aerodynamic effects, it is necessary to
tions, while, at the other, empirical formulas [50]–[53] come employ a state-of-the-art model that includes the suspension
from fitting curves to measured data. In the middle ground, system as well as tire models that recognize the influences
simple physical models provide good representations of the of load changes. In such models [44], the aerodynamic drag
basic geometry and the distributed tire-ground rolling contact. and lift forces and the pitching moments are represented as
The detailed models are effective in terms of accuracy and being proportional to the square of the speed.
range of behavior covered but are computationally demand-
ing to use. Contemporary high-fidelity models, which can be Structural Flexibility
used over a wide variety of operating conditions, are almost Motivated by the known deterioration in the steering behavior
exclusively of the empirical variety. An overview of many of resulting from torsional compliance between the wheels, [62]
these ideas in the context of car tires is given in [51]. extends the model of [47] by allowing the rear wheel to cam-
The basis for contemporary tire models are magic formu- ber relative to the rear frame. This freedom is constrained by a
las [51], [53]–[55], which are empirical models favored for parallel spring-damper arrangement. It was found that
their ability to accurately match tire force and moment data swingarm flexibility had very little influence on the capsize
covering a full range of operating conditions. The original and wobble modes, but it reduced the weave mode damping
development was for car tires [56], in which context magic at medium and high speeds. The removal of the damping
formula models are now dominant. These models describe associated with the swingarm flexibility made no material dif-
the steady-state longitudinal forces, side forces, aligning ference to these findings. The results indicate that a swingarm
moments, and overturning moments as functions of the lon- stiffness of 12,000 N-m/rad for a high-performance machine
gitudinal slip, sideslip, camber angle, and normal load. The would bring behavior approaching closely that for a rigid
extension of magic formula ideas to motorcycle tires is rela- frame. Product development over the last 30 years has clearly
tively recent, with substantial changes needed to accommo- involved substantial stiffening of the swingarm structure,
date the changed roles of sideslip and cambering in the such that most contemporary designs are probably deep into
force and moment generation process. When finding the diminishing returns for additional stiffness.
parameters that populate the magic formulas, constraints Experimental work [63]–[66] shows that the theory
must be placed on the parameter set to ensure that the tire existing at the time overpredicted the wobble-mode damp-
behavior is reasonable under all operating conditions, some ing at moderate speeds, at which the damping is often
of which may be beyond those used in the parameter identi- quite small. In particular, [65] associates the low medium-
fication process. Although limited tire-parameter informa- speed-wobble damping with front fork compliance and
tion can be found in the literature, models can be shows improved behavior from stiffer forks. It is also
augmented with available experimental force and moment shown in [65] that stiffening the rear frame with additional
data. A full set of parameters for modern front and rear structures increased the damping of the weave mode.
high-performance motorcycle tires can be found in [42]. The discrepancy between theory and experiment, mainly
Additional data are available in [51], [53]–[55], and [57]–[60]. with respect to the damping of the wobble mode and its
variation with speed, is substantially removed by the results
Aerodynamic Forces of [40] and [41], where mathematical models were extended
The importance of aerodynamic forces on the performance to include front frame compliances. In particular, [40]
and stability of high-powered motorcycles at high speeds employs three model variants A, B, and C. The A model
was demonstrated in [61]. Wind tunnel data were obtained allows the front wheel to move laterally along the wheel
for the steady-state aerodynamic forces acting on a wide spindle. The B model allows torsional compliance in the
range of motorcycle-rider configurations. It appears from front frame about an axis parallel to the steer axis, while the
the results in [61] that the effects of aerodynamic side forces, C model allows twisting of the front frame relative to the
yawing moments, and rolling moments on the lateral stabil- rear frame about an axis perpendicular to the steering axis.

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 55

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 In each case, the new compliance involves a parallel spring- Rider Modeling
damper arrangement. The parameters from four different In early motorcycle and bicycle models, the rider is consid-
large production motorcycles are used. The following con- ered to be no more than an inert mass rigidly attached to the
clusions are drawn. 1) The torsional freedom parallel to the rear frame [17], [47]. In [57] and [58], the rider’s lower body is
steering axis makes very little difference to the results represented as an inert mass attached to the rear frame, while
obtained from the stiff-framed model. 2) The front-wheel the upper body is represented as an inverted pendulum that
lateral compliance results in a decrease in the wobble-mode has a single roll freedom constrained by a parallel spring-
damping, but the associated speed dependence is not sup- damper arrangement. The parameter values come from sim-
ported by experiment. This flexibility also results in ple laboratory experiments, which show that values can vary
improved weave-mode damping at moderate speeds but significantly from rider to rider [73]. This single-degree-of-
worsens it for high speeds, which is where it matters. It is freedom inverted pendulum rider model is also used in [70].
suggested that the lateral stiffness should be made large but The straight-running stability of a combined motorcy-
that such stiffening brings diminishing returns beyond an cle rider model, which focuses on the frame flexibilities
intermediate stiffness level. 3) The C-model freedom leads and the rider’s dynamic characteristics, is studied in [73].
to the prediction of the observed intermediate-speed low This 12-degree-of-freedom model includes two rider free-
damping of the wobble mode, with higher damping at high doms. The first is associated with the rolling motion of the
speeds deriving from the frame compliance. Thus, the com- rider’s upper body, while the second allows the rider’s
pliance may to some extent contribute to good behavior. In lower body to translate laterally relative to the motorcy-
an independent study, [41] confirms the findings described cle’s main frame. Both bodies associated with the rider are
above. Apart from varying the torsional stiffness, the effect restored to their nominal positions by linear springs and
of changing the height of the lateral fork bending joint was dampers. The system parameters are found experimental-
also examined. The analysis concluded that the lateral bend- ly, and the rider data, in particular, is measured by means
ing of the front fork should be reduced by stiffening and of forced vibration experiments, whereby the frequency
that the bending axis should be located as close to the pave- responses from vehicle roll to rider body variables are
ment as possible. It also concluded that the “best” front- obtained. The frequency and damping ratios of the wob-
wheel suspension system should be designed to have high ble and weave modes are calculated at various speeds and
lateral stiffness without being excessively heavy. compared with results obtained from experiments con-
Measured static torsional stiffness data for motorcycle ducted with four motorcycles covering a range of sizes. A
frames are given in [58] and [67]–[69], while [68] and [69] model without rider freedoms (a reduction of two degrees
also include the results of dynamic testing. Stiffnesses for of freedom) is used for comparison. In general terms,
large motorcycles from the past apparently lie in the range there is very good agreement between the experimental
of 25,000–150,000 N-m/rad, where the influence on stabili- results for each of the four machines and the detailed
ty properties is marked. Predicting the wobble mode prop- model, with a tendency for the measured damping factors
erly and understanding the need for a steering damper to be a little greater than those predicted.
depend on accounting for frame torsional compliance in The effect of individual rider parameters on the rider-
the steering-head region and lateral fork bending. motorcycle system stability is also investigated analytical-
This frame flexibility work is consolidated in [70], where ly. It is found that the rider’s vibration characteristics
a motorcycle model is developed for straight-running stud- influence both wobble and weave. The parameters of the
ies with design parameters and tire properties obtained rider’s upper body motion are most influential on weave,
from laboratory experiments. The model constituents are, in while those concerned with the rider’s lower body primar-
addition to those given in the earlier model [47], lateral and ily influence the wobble mode.
frame twist flexibilities at the steering head, a flexibility of The role of the rider as an active controller is studied in
the rear wheel assembly about an inclined hinge, a roll free- some detail in [74], where it is recognized that inadvertent
dom associated with the rider’s upper body, in-plane aero- rider motions can have a significant influence on the vehi-
dynamic effects, and more elaborate tire modeling. cle’s behavior. The focus of [74] is to treat the rider as a feed-
Hands-on and hands-off cases are presented, and the results back compensator that maps the vehicle’s roll angle errors
are in agreement with empirical observations and experi- into a steering torque, where the controller’s characteristics
mental findings of [71]. The results show the advantage that are chosen to mimic those of the rider’s neuromuscular sys-
can be derived in respect of the weave mode damping from tem. The rider is modeled (roughly) as being able to control
a long wheelbase and a large steering-head angle. The his upper body roll angle as well as the steering torque; the
model of [70] was subsequently rebuilt using a modern steering torque influence is found to be dominant.
multibody simulation package [72], confirming the original. A motorcycle rider model similar to that studied in [73] is
In the context of contemporary high-performance machines, investigated in [75] to find those aspects of the rider’s control
the only frame flexibility deemed to be important is that action that are most important in the description of single-
associated with the steering head and front forks [42], [44]. lane-change maneuvering behavior. In this case, the rider

56 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 model comprises upper and lower body masses that are both are constant, while the suspension posture of the machine,
free to roll relative to the motorcycle’s main frame. The rider is the tire force system, and the aerodynamic forces are all
assumed to generate three control torques that are applied to functions of the lean angle. Essential components of high-
the steering system from the rider upper body, the upper fidelity cornering models include [44]: 1) a rigid rear frame,
body from the lower body, and the lower body from the rear which has six degrees of freedom; 2) a front frame joined to
frame. The rider representation, which plays the role of a feed- the rear frame using an inclined steering system with a com-
back controller tasked with tracking a heading, is as a propor- pliance included between the steering head and the rear
tional controller. Simulations for a single-lane-change frame; 3) spinning road wheels, which include thick profiled
maneuver are compared with measurements generated by 12 tire descriptions, where the dynamic migration of the road-
different riders. The results show that, for a running speed of tire ground contact point under cornering is modeled; 4) an
17 m/s, a good match can be obtained between the simulation elaborate tire force and moment representation informed by
model with suitably chosen controller parameters and the extensive measurements; 5) lag mechanisms by which tire
measured responses of the different riders. The results also forces are delayed with respect to the slip phenomena that
suggest that the most important control input is the steering produce them; 6) aerodynamic effects, which allow the tire
torque. While it is possible to control the motorcycle with loads and machine posture to be properly represented under
lower body lean movement, much larger torques are required speed variations; 7) a realistic suspension model; and 8) the
in this case. Normally, lower body control is utilized to assist freedom for the rider’s upper body to roll relative to the rear
steering torque control, while the upper body is controlled frame of the vehicle. The accuracy of predicted behavior
only to keep the rider in the comfortable upright position. depends not only on effective conceptual modeling and
A complex rider model that comprises 12 rigid bodies multibody analysis but also on good parameter values.
representing the upper and lower body, the upper and The in-plane modes present under straight-running con-
lower arms, and the upper and lower legs, with appropri- ditions are shown in Figure 20; similar plots can be found
ate mass and inertia properties is introduced in [76]. The in [44]. The in-plane modes that are associated with the sus-
various rider model masses are restrained by linear pension and tire flexibilities are referred to as the front
springs and dampers so that rider motions such as steer- suspension pitch mode, the rear suspension pitch mode,
ing, rolling, pitching, weight shifting and knee gripping the front wheel-hop mode, and the rear wheel-hop mode.
are possible. Rider control actions associated with these These modes are insensitive to speed variations and are
degrees of freedom are also modeled using proportional decoupled from the out-of-plane modes described above.
control elements. Steady-state cornering and lane-chang- The cornering situation is considerably more complex
ing maneuvers are studied. than the straight-running case, since the in-plane and out-of-
plane motions are coupled and these interactions tend to
Suspension and Cornering Models increase with roll angle. As a consequence, several straight-
Under steady-state cornering it is clear that a motorcycle’s running modes merge together to form combined cornering
forward speed, yaw rate, lateral acceleration, and lean angle modes with the particular characteristics shown in the right-

Front
70 70 Front Wheel Hop
Wheel
Hop
60 60
Wobble
Wobble
Imaginary (rad/s)

50 50

40 40

30 30
Weave
20 Rear 20 Weave
Suspension Front Front Suspension Pitch
10 Pitch Suspension 10
Pitch
0 0
−18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2
Real (1/s) Real (1/s)
(a) (b)

FIGURE 20 Motorcycle root locus plots: (a) straight-running and (b) 30° of roll angle with speed the varied parameter. The speed is increased
from (a) 5 m/s (
), (b) 6 m/s (
) to 60 m/s (
).

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 57

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 hand root locus plot in Figure 20. Cornering weave is simi- sion system, indicate a frequency of 2.2 Hz, while at 36
lar in frequency to straight-running weave at high speeds, m/s the frequency is 2.6 Hz. It is also found that the weave
but for the machine studied here, the weave-mode damping oscillations die out once the rider reduces the roll angle.
decreases as the lean angle increases. The suspension sys- Further, [66] also demonstrates that reduced rear suspen-
tem contributes significantly to the machine characteristics, sion damping, increased rear loading, and increased speed
as observed experimentally. The influence of suspension increases the tendency for the motorcycle to weave. As
damping on the weave mode is demonstrated both analyti- predicted by theory, the frequency of wobble varies little
cally and experimentally in [66] and [77]. Under cornering, with speed, while that of weave increases with speed.
the wobble mode involves suspension movement, and the Significant steps in the theoretical analysis of motorcycle
previously speed-independent suspension-pitch and wheel- behavior are documented in [57] and [58]. The model devel-
hop modes now vary markedly with speed. An interaction oped considers small perturbations about straight-running
between the front wheel-hop and wobble modes occurs conditions but also for the first time about steady-cornering
when the two modes are close enough in terms of natural conditions. The model in [57] is used to calculate the eigen-
frequency. This interaction is possibly linked to wheel patter, values of the small-perturbation linearized motorcycle,
which is known anecdotally [78]. The coupling of the in- where the results for straight running are consistent with
plane and out-of-plane motions also suggests the possibility the conventional wisdom. The way the weave- and wobble-
of road excitation signals being transmitted into the lateral mode characteristics are predicted as varying with speed is
motions of the vehicle, causing steering oscillations [43]. conventional, with new front- and rear-suspension pitch
The early literature [77] discusses the existence of a and wheel-hop modes almost independent of speed appear-
modified weave mode that occurs under cornering condi- ing. Under cornering conditions, the interaction of these
tions, where the suspension system plays an important role otherwise uncoupled modes produces more complicated
in its initiation and maintenance. To investigate the effect of modal motions. The cornering weave and combined wheel-
suspension damping on cornering weave, [77] benchmarks hop/wobble modes are illustrated, and root loci are plotted
several front and rear suspension dampers in laboratory to observe the sensitivity of the results to parameter varia-
experiments and riding tests. Motorcycle stability is found tions. Surprisingly, it is predicted that removing the suspen-
to be sensitive to suspension damping characteristics, while sion dampers hardly affects the stability of the cornering
cornering weave instability is to some extent controllable weave mode, contrary to the experiences of [66] and [77].
through rear suspension damper settings. As stated in [77], One of the original aims of [44] is to investigate the
“. . . slight road surface undulations exacerbate the problem, apparent conflict between the results of [80] on the negligi-
which is generally confined to speeds above 60 mph and ble influence of suspension damping on the stability of
roll angles in excess of 25 deg from pavement-perpendicu- cornering weave and the anecdotal and experimental evi-
lar . . . ”. It is also found that, as the speed is increased, cor- dence of [66] and [77]. Cornering root loci with the rear
nering weave is produced at smaller roll angles. A separate suspension damping varied are reproduced and the damp-
study [79] demonstrates, using a simple analysis, the possi- ing is found to have a significant influence, indicating a
bility of interaction between pitch and weave modes at probable error in the calculations in [80]. The model pre-
high speeds, where the lightly damped weave-mode natur- sented in [44] is enhanced in [42] to include magic formula
al frequency approaches that of the pitch mode. Although tire models with the additional features included in [81].
for straight running the coupling of in-plane and out-of The influence of the front suspension system on the ride
plane motions is weak, for steady-state cornering the cou- qualities of a motorcycle is studied in [82]. A typical suspen-
pling between the two modes increases with increased roll sion unit is modeled on the basis of its inner structure and
angle, indicating that the inclusion of pitch and bounce functionality, which give rise to the spring forces, viscous
freedoms in motorcycle models is essential for handling damping forces, friction forces, and oil lock forces. Sine-wave
studies involving cornering. excitation experiments show that the model represents the unit
Cornering experiments described in [66] quantify the accurately. Further experiments are conducted, this time to
influences of various motorcycle design parameters and check the validity of the fork unit model combined with a sim-
operating conditions on wobble and weave. Tests with a plified motorcycle model that comprises the vertical and longi-
range of motorcycles and riders are carried out for both tudinal dynamics. The results obtained for riding over bumps
straight running and steady-state cornering. The wobble and under braking agree with measurements. The influence of
mode, which is excited by a steering torque pulse input the suspension characteristics on riding qualities of the vehicle
from the rider, is seen to be self-sustained during hands- are found by simulation; experiments verify the findings.
off straight running at a moderate speed of 18 m/s; the Experimental cornering results obtained from an instru-
measured wobble frequency is 5.4 Hz, which is lower than mented motorcycle are presented in [83]. The motorcycle is
the theoretical prediction. More importantly, under steady- fitted with steering torque and angle transducers. Fiber-
state cornering, measurements of cornering weave optic gyros are used to measure the roll rate and yaw rate,
responses at 27 m/s, involving oscillations in the suspen- and strain gauges provide tire force and moment data.

58 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 This paper provides experimental data that are used for operation of a real single-track vehicle, are completely
model qualification. absent. Nevertheless, steer-displacement control inputs that
A study of the effects of road profiling on motorcycle allow a prescribed path to be followed while the rolling
steering responses is presented in [43]. The results show that motion is properly stabilized have been optimized on sim-
under cornering conditions, regular low-amplitude road ple Timoshenko-Young-type models and applied to sophis-
undulations that would not trouble four-wheeled vehicles ticated machine models with some success.
can be a source of considerable difficulty to motorcycle rid- Whipple’s model, when linearized for constant-speed
ers. At low machine speeds, the wobble and front suspen- straight running, yields two second-order equations of
sion pitch modes are likely to respond vigorously to motion in rolling and steering. The Whipple model, while
resonant vertical-displacement road forcing, while, at higher simple enough for control system optimization studies,
speeds, the weave and front wheel-hop modes may be simi- contains a sufficient level of physical realism to make it
larly affected. Connections between resonant responses and credible. Physical influences deriving from the vehicle’s
a class of single-vehicle loss-of-rider-control accidents are design can be seen to combine in complex ways to give an
postulated. This work has several practical consequences. effective steering inertia, steering damping, and steering
First, these results appear to explain the key features of many stiffness. The Whipple model also provides an appreciation
stability-related road traffic accidents reported in the popu- of the complex interactions between the roll angle, roll
lar literature and help to show why motorcycles that behave velocity, and roll acceleration, and the steering torque. Lin-
perfectly well for long periods can suddenly suffer serious ear versions of Whipple-type models are useful for explain-
and dangerous oscillation problems. Such oscillations are ing nonminimum phase responses, the benefits of feedback,
likely to be difficult to reproduce and study experimentally. and achievable robustness margins in single-track vehicles.
Second, road builders and maintainers, as well as motorcycle In models of the Timoshenko-Young type, the roll dynam-
manufacturers, should be aware of the possibility of strong ics are driven kinematically by the steer angle, the steer veloc-
resonant responses to small but regular undulations under ity, the steer acceleration, and the vehicle acceleration inputs.
certain critical running conditions. These conditions are char- The advantage of these models is that considerable insight
acterized by the machine speed, the lean angle, the rider’s into the stability and steering control of single-track vehicles,
mass and posture, and the road profile wavelength. within the model applicability boundaries, can be gained
from their separable design parameter influences. Although
CONCLUSIONS such insights cannot easily be developed by referring to the
Research and scholarship relating to single-track vehicles numerical results derived from more complex models, com-
involves, to a large extent, two separate communities that prehensive models surely have their place in enabling effec-
can benefit from a higher level of interaction. One group tive virtual product design and testing across the full
favors the use of simple bicycle models, while the other is operating envelope. The essential features of modern motor-
concerned with high-performance motorcycles and the cycle design models include: 1) multiple rigid bodies and a
development of models with a high level of quantitative complex set of allowed motion freedoms; 2) detailed tire force
predictive capability over a wide operating envelope. and moment models, incorporating static behavior up to and
The simple models can be regarded as derivatives or possibly beyond the tire saturation limits, as well as transient
simplifications of Whipple’s model. In these models, the behavior; 3) case-dependent frame and rider compliances; 4)
lateral motion constraints at the road contact are nonholo- suspension systems; 5) aerodynamic forces and moments;
nomic and thus special techniques may be needed to form and 6) detailed geometric models for accurately describing all
correct equations of motion. When the tire is regarded as of the external forces. When motorcycles are ridden “on the
constraining the motion of the vehicle, the model validity limit,” the stability and performance of the machine are
is restricted to low speeds (< 10 m/s), low frequencies restricted by the properties of the tires, the suspension setup,
(< 1.0 Hz), and low tire-force utilization associated with the weight distribution, the frame stiffness properties, and the
benign maneuvering (< 20% of capacity). steering damping. A practical virtual design and testing facil-
The model of Timoshenko and Young [29] represents the ity must be able to accurately predict every feature of this
lowest level of complexity of any potential usefulness; their limit behavior.
model has no rake, no trail, no inertias, no front frame mass, In relation to trajectory tracking on the boundaries of
and a point-mass representation of the rear frame. The Tim- the vehicle’s capability—essentially the racing problem—it
oshenko-Young model leads one to conclude that the steer is clear that performance is limited by tire force saturation
angle and speed completely determine the lateral motion of and transient dynamics, among other things. It appears to
the base point of an inverted pendulum that represents the be a considerable act of faith to regard the ultimate perfor-
vehicle’s roll dynamics. In terms of understanding single- mance as calculable on the basis of nonholonomic rolling
track vehicle steering, this level of modeling complexity is constraints! In the future, we hope to see more elaborate
too low, since unrealistic steer angle control must be accom- models, of the motorcycle-fraternity type, applied to mini-
modated. The self-steering influences, which are vital to the mum lap time and optimal-trajectory tracking problems.

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 59

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 Complexity-related difficulties, implicit in comprehensive ber of The Proceedings of the Institution of Mechanical Engineers,
motorcycle modeling activities, are an exciting opportunity, Part D, Journal of Automobile Engineering. From 1990–2002, he
rather than a threat to be feared and avoided. Indeed, the was professor of automotive product engineering at Cranfield
thoughtful use of powerful multibody modeling tools makes University. He was a visiting associate research scientist at the
routine the study of problems that would have been deemed University of Michigan Transportation Research Institute, Ann
intractable only a decade ago. The challenges facing modelers Arbor. His research covers topics in automotive dynamics and
include a systematic approach to removing redundancy in control and in control and stability of single-track vehicles,
nonlinear models and the retention of key insights, which tend unmanned air vehicles, and the application of optimal preview
to be obscured or even destroyed, in model reduction exercis- and learning control to road vehicle driving/riding.
es. The challenges facing control theorists include: 1) the devel-
opment of general theories for reducing complex nonlinear
REFERENCES
models that guarantee the reduced-order model’s dynamic [1] A. Sharp, Bicycles and Tricycles: An Elementary Treatise on Their Design and
fidelity; 2) removing assumptions that currently make general Construction. White Plains, NY: Longman, 1896. (Reprinted as: Bicycles and
control theories inapplicable to nonlinear mechanics problems, Tricycles: A Classic Treatise on Their Design and Construction. Mineola, NY:
Dover, 1977.)
and 3) the parallel development of computational platforms [2] D.V. Herlihy, Bicycle: The History. New Haven, CT: Yale Univ. Press, 2004.
that support complex controller synthesis applications. [3] M. Hamer, “Brimstone and bicycles,” New Scientist, issue 2428, pp. 48–49,
Jan. 2005.
[4] N. Clayton, Early Bicycles. Princes Risborough, Bucks, UK: Shire Publica-
ACKNOWLEDGMENTS tions, 1986.
The authors would like to thank Prof. Karl Åström, Prof. Mal- [5] F.J. Berto, The Dancing Chain: History and Development of the Derailleur
colm Smith, and the reviewers for their helpful suggestions. Bicycle. San Francisco, CA: Van der Plas Publications, 2005.
[6] “The Bicycle Museum of America” [Online]. Available: http://www.
bicyclemuseum.com/
AUTHOR INFORMATION [7] “Pedaling history bicycle museum” [Online]. Available: http://www.
David J.N. Limebeer (d.limebeer@imperial.ac) received the pedalinghistory.com/
[8] “Metz bicycle museum” [Online]. Available: http://www.
B.Sc. degree in electrical engineering from the University of metzbicyclemuseum.com/
Witwatersrand, Johannesburg, in 1974, the M.Sc. and Ph.D. [9] “Canada science and technology museum” [Online]. Available:
degrees in electrical engineering from the University of Natal, http://www.sciencetech.technomuses.ca/
[10] “Canberra bicycle museum” [Online]. Available: http://
Durban, South Africa, in 1977 and 1980, respectively, and the canberrabicyclemuseum.com.au/
D.Sc. degree from the University of London in 1992. He has [11] “National motorcycle museum” [Online]. Available: http://www.
been with Imperial College London since 1984, where he is nationalmcmuseum.org/
[12] “London motorcycle museum” [Online]. Available: http://www.motor-
currently the head of the Department of Electrical and Elec- cycle-uk.com/lmm/beginning.html
tronic Engineering. He has published over 100 papers and a [13] “Allen vintage motorcycle museum” [Online]. Available: http://www.
textbook on robust control theory. Three of his papers have allenmuseum.com/
[14] “Sammy Miller museum” [Online]. Available: http://motorcycle.com/
been awarded prizes, including the 1983 O. Hugo Schuck mo/mcfrank/sammymuseum.html
Award. He is a past editor of Automatica and a past associate [15] “Motorcycle hall of fame museum” [Online]. Available: http://
editor of Systems and Control Letters and the International Jour- home.ama-cycle.org/
[16] W.J.M. Rankine, “On the dynamical principles of the motion of veloci-
nal of Robust and Nonlinear Control. He is a Fellow of the IEEE, pedes,” The Engineer, pp. 2, 79, 129, 153, 175, 1869/1870.
the IEE, the Royal Academy of Engineering, and the City and [17] F.J.W. Whipple, “The stability of the motion of a bicycle,” Q. J. Pure
Guilds Institute. His research interests include control system Appl. Math., vol. 30, pp. 312–348, 1899.
[18] H. Goldstein, Classical Mechanics, 2nd ed. Reading, MA: Addison-Wes-
design, frequency response methods, H-infinity optimization ley, 1980.
and mechanical systems. He is qualified as an IAM senior [19] M.E. Carvallo, “Théorie du movement du monocycle, part 2: Théorie de
motorcycle instructor and received a RoSPA certificate for la bicyclette,” J. L’Ecole Polytechnique, vol. 6, pp. 1–118, 1901.
[20] F. Klein and A. Sommerfeld, Über die Theorie des Kreisels. Teubner,
advanced motorcycling. He can be contacted at Imperial Col- Leipzig, 1910, Chap. IX, Sec. 8, “Stabilität des Fahrrads,” Leipzig, Germany:
lege London, Department of Electrical and Electronic Engi- B.G. Teubner, pp. 863–884.
neering, Exhibition Road, London SW7 2AZ U.K. [21] D.E.H. Jones, “The stability of the bicycle,” Phys. Today, vol. 23, no. 4, pp.
34–40, 1970.
Robin S. Sharp is a professorial research fellow in the [22] R.D. Roland, “Computer simulation of bicycle dynamics,” in Proc.
Department of Electrical and Electronic Engineering at Imperial ASME Symp. Mechanics Sport, 1973, pp. 35–83.
College London. He is a member of the Dynamical Systems and [23] E. Döhring, “Stability of single-track vehicles,” Institut für Fahrzeugtech-
nik, Technische Hochschule Braunschweig, Forschung Ing.-Wes., vol. 21, no. 2,
Mechatronics Working Group of the International Union of pp. 50–62, 1955.
Theoretical and Applied Mechanics, the editorial board of Vehi- [24] J.I. Neĭmark and N.A. Fufaev, “Dynamics of nonholonomic systems,”
cle System Dynamics, and the editorial advisory board of Multi- (Amer. Math. Soc. Translations Math. Monographs, vol. 33), 1972.
[25] R.S. Hand, “Comparisons and stability analysis of linearized equations
body System Dynamics. He was first vice-president and secretary of motion for a basic bicycle model,” M.Sc. Thesis, Cornell Univ., 1988.
general of the International Association for Vehicle System [26] A.L. Schwab, J.P. Meijaard, and J.M. Papadopoulos, “Benchmark results
Dynamics, editorial panel member and book review editor for on the linearized equations of motion of an uncontrolled bicycle,” in Proc.
2nd Asian Conf. Multibody Dynamics, Aug. 2004, pp. 1–9.
The Proceedings of the Institution of Mechanical Engineers, Journal of [27] K.J. Åström, R.E. Klein, and A. Lennartsson, “Bicycle dynamics and con-
Mechanical Engineering Science, Part C, and editorial panel mem- trol,” IEEE Control Syst. Mag., vol. 25, no. 4, pp. 26–47, 2005.

60 IEEE CONTROL SYSTEMS MAGAZINE » OCTOBER 2006

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.
 [28] R.E. Klein, “Using bicycles to teach system dynamics,” IEEE Control curves,” in Proc. 7th IAVSD Symp. Dynamics Vehicles Roads Railway Tracks,
Syst. Mag., vol. 6, no. 4, pp. 4–9, 1989. Cambridge, 1981, pp. 53–65.
[29] S. Timoshenko and D.H. Young, Advanced Dynamics. New York: [59] V. Cossalter, A. Doria, R. Lot, N. Ruffo, and M. Salvador, “Dynamic
McGraw-Hill, 1948. properties of motorcycle and scooter tires: Measurement and comparison,”
[30] A.M. Letov, Stability in Nonlinear Control Systems. Princeton, NJ: Prince- Vehicle Syst. Dyn., vol. 39, no. 5, pp. 329–352, 2003.
ton Univ. Press, 1961. [60] H. Sakai, O. Kanaya, and H. Iijima, “Effect of main factors on dynamic
[31] N.H. Getz, “Control of balance for a nonlinear nonholonomic nonminimum properties of motorcycle tires,” Soc. Auto. Eng., paper 790259, 1979.
phase model of a bicycle,” in Proc. American Control Conf., 1994, pp. 148–151. [61] K.R. Cooper, “The effects of aerodynamics on the performance and sta-
[32] N.H. Getz and J.E. Marsden, “Control of an autonomous bicycle,” in bility of high speed motorcycles,” in Proc. 2nd AIAA Symp. Aerodynamics
Proc. IEEE Conf. Robotics Automation, 1995, pp. 1397–1402. Sport Competition Automobiles, Los Angeles, 1974.
[33] J. Hauser, A. Saccon, and R. Frezza, “Achievable motorcycle trajectories,” [62] R.S. Sharp, “The influence of frame flexibility on the lateral stability of
in Proc. 43rd CDC, Paradise Island, Bahamas, 14–17 Dec. 2004, pp. 3944–3949. motorcycles,” J. Mech. Eng. Sci., vol. 16, no. 2, pp. 117–120, 1974.
[34] K.J. Åström, “Limitations on control system performance,” Euro. J. Con- [63] D.J. Eaton, “Lateral dynamics of the uncontrolled motorcycle,” in Proc.
trol, vol. 6, no. 1, pp. 2–20, 1980. 2nd Int. Congr. Automotive Safety, San Fransisco, 1973.
[35] Anon., Autosim 2.5+ Reference Manual. Mech. Simulation Corp., Ann [64] M.K. Verma, R.A. Scott, and L. Segel, “Effect of frame compliance on the lat-
Arbor MI, 1998 [Online]. Available: http://www.carsim.com eral dynamics of motorcycles,” Vehicle Syst. Dyn., vol. 9, no. 3, pp. 181–206, 1980.
[36] A.L. Schwab, J.P. Meijaard, and J.D.G. Kooijman, “Experimental valida- [65] G.E. Roe and T.E. Thorpe, “A solution of the low-speed wheel flutter
tion of a model of an uncontrolled bicycle,” in Proc. III European Conf. Compu- instability in motorcycles,” J. Mech. Eng. Sci., vol. 18, no. 2, pp. 57–65, 1976.
tational Mechanics: Solids, Structures Coupled Problems Engineering, Lisbon, [66] D.H. Weir and J.W. Zellner, “Experimental investigation of the transient
Portugal, 5-9 June 2006, paper 98. behaviour of motorcycles,” Soc. Auto. Eng., paper 790266, 1979.
[37] R.A. Wilson-Jones, “Steering and stability of single-track vehicles,” in [67] A. Clerx, “Stijfheid en sterkte van motorfietsframes,” Dept. Mech. Eng.,
Proc. Auto. Div. Institution Mechanical Engineers, 1951, pp. 191–199. Tech. Univ. Eindhoven, Tech. Rep., 1977.
[38] A.L. Schwab, J.P. Meijaard, and J. M. Papadopoulos, “A multibody [68] C.G. Giles and R.S. Sharp, “Static and dynamic stiffness and deflection
dynamics benchmark on the equations of motion of an uncontrolled bicy- mode measurements on a motorcycle, with particular reference to steering
cle,” in Proc. ENOC-2005, Eindhoven, The Netherlands Aug. 2005, pp. 1–11. behaviour,” in Proc. Inst. Mech. Eng./MIRA Conf. Road Vehicle Handling, Lon-
[39] J. Fajans, “Steering in bicycles and motorcycles,” Amer. J. Phys., vol. 68, don, 1983, pp. 185–192.
no. 7, pp. 654–659, 2000. [69] M.P.M. Bocciolone, F. Cheli, and R. Vigano, “Static and dynamic proper-
[40] R.S. Sharp and C.J. Alstead, “The influence of structural flexibilities on ties of a motorcycle frame: Experimental and numerical approach,” Dept.
the straight running stability of motorcycles,” Vehicle Syst. Dyn., vol. 9, no. 6, Mech. Eng., Politecnico di Milano, Tech. Rep., 2005.
pp. 327–357, 1980. [70] R.S. Sharp, “Vibrational modes of motorcycles and their design parame-
[41] P.T.J. Spierings, “The effects of lateral front fork flexibility on the vibra- ter sensitivities,” in Proc. Int Conf. Vehicle NVH Refinement, Birmingham, 3–5
tional modes of straight-running single-track vehicles,” Vehicle Syst. Dyn., May 1994, pp. 107–121.
vol. 10, no. 1, pp. 21–35, 1981. [71] B. Bayer, “Flattern und pendeln bei krafträdern,” Automobil Industrie,
[42] R.S. Sharp, S. Evangelou, and D.J.N. Limebeer, “Advances in the modelling vol. 2, pp. 193–197, 1988.
of motorcycle dynamics,” Multibody Syst. Dyn., vol. 12, no. 3, pp. 251–283, 2004. [72] S. Evangelou and D.J.N. Limebeer, “Lisp programming of the ‘sharp
[43] D.J.N. Limebeer, R.S. Sharp, and S. Evangelou, “Motorcycle steering oscil- 1994’ motorcycle model,” 2000 [Online]. Available: http://www.ee.ic.ac.
lations due to road profiling,” J. Appl. Mech., vol. 69, no. 6, pp. 724–739, 2002. uk/control/motorcycles
[44] R. S. Sharp and D.J.N. Limebeer, “A motorcycle model for stability and [73] T. Nishimi, A. Aoki, and T. Katayama, “Analysis of straight running sta-
control analysis,” Multibody Syst. Dyn., vol. 6, no. 2, pp. 123–142, 2001. bility of motorcycles,” in Proc. 10th Int. Technical Conf. Experimental Safety
[45] V. Cossalter and R. Lot, “A motorcycle multi-body model for real time Vehicles, Oxford, 1–5 July 1985, pp. 1080–1094.
simulations based on the natural coordinates approach,” Vehicle Syst. Dyn., [74] D.H. Weir and J.W. Zellner, “Lateral-directional motorcycle dynamics
vol. 37, no. 6, pp. 423–447, 2002. and rider control,” Soc. Auto. Eng. paper 780304, pp. 7-31, 1978.
[46] R.S. Sharp, “The lateral dynamics of motorcycles and bicycles,” Vehicle [75] T. Katayama, A. Aoki, and T. Nishimi, “Control behaviour of motorcy-
Syst. Dyn., vol. 14, no. 6, pp. 265–283, 1985. cle riders,” Vehicle Syst. Dyn., vol. 17, no. 4, pp. 211–229, 1988.
[47] R.S. Sharp, “The stability and control of motorcycles,” J. Mech. Eng. Sci., [76] H. Imaizumi, T. Fujioka, and M. Omae, “Rider model by use of multibody
vol. 13, no. 5, pp. 316–329, 1971. dynamics analysis,” Japanese Soc. Auto. Eng., vol. 17, no. 1, pp. 75–77, 1996.
[48] R.S. Sharp, “The stability of motorcycles in acceleration and [77] G. Jennings, “A study of motorcycle suspension damping
deceleration,” in Proc. Inst. Mech. Eng. Conf. Braking Road Vehicles, London: characteristics,” Soc. Auto. Eng., paper 740628, 1974.
MEP, 1976, pp. 45–50. [78] R.S. Sharp and C.G. Giles, “Motorcycle front wheel patter in heavy brak-
[49] D.J.N. Limebeer, R.S. Sharp, and S. Evangelou, “The stability of motorcycles ing,” in Proc. 8th IAVSD Symp. Dynamics Vehicles Roads Railway Tracks,
under acceleration and braking,” J. Mech. Eng. Sci., vol. 215, no. 9, pp. 1095–1109, 2001. Boston, 1983, pp. 578–590.
[50] H.B. Pacejka and R.S. Sharp, “Shear force development by pneumatic [79] R.S. Sharp, “The influence of the suspension system on motorcycle
tyres in steady state conditions: A review of modelling aspects,” Vehicle Syst. weave-mode oscillations,” Vehicle Syst. Dyn., vol. 5, no. 3, pp. 147–154, 1976.
Dyn., vol. 20, no. 3–4, pp. 121–176, 1991. [80] C. Koenen, “The dynamic behaviour of motorcycles when running straight
[51] H.B. Pacejka, Tyre and Vehicle Dynamics. Oxford, U.K.: Butterworth ahead and when cornering,” Ph.D. dissertation, Delft Univ. Technol., 1983.
Heinemann, 2002. [81] R.S. Sharp, D.J.N. Limebeer, and M. Gani, “A motorcycle model for sta-
[52] S.K. Clark, Ed., Mechanics of Pneumatic Tires, 2nd ed. Washington DC: bility and control analysis,” in Proc. Euromech Colloquium 404, Advances Com-
NTIS, 1981. putational Multibody Dynamics, 1999, pp. 287–312.
[53] E.J.H. de Vries and H.B. Pacejka, “Motorcycle tyre measurements and [82] T. Kamioka, N. Yoshimura, and S. Sato, “Influence of the front fork on the
models,” Vehicle Syst. Dyn., vol. 29, pp. 280–298, 1998. movement of a motorcycle,” in Proc. SETC’97, Yokohama, 1997, pp. 397–403.
[54] E.J.H. de Vries and H.B. Pacejka, “The effect of tyre modeling on the sta- [83] H. Ishii and Y. Tezuka, “Considerations of turning performance for
bility analysis of a motorcycle,” in Proc. AVEC’98, Nagoya, Soc. Automotive motorcycles,” in Proc. SETC’97, Yokohama, 1997, pp. 383–389.
Engineers paper Japan, 1998, pp. 355–360. [84] A. Hill, “Smith’s prizes: award,” Cambridge Univ. Reporter, p. 1027, 1899.
[55] Y. Tezuka, H. Ishii, and S. Kiyota, “Application of the magic formula tire [85] D.G. Wilson, Bicycling Science. Cambridge, MA: MIT Press, 2004.
model to motorcycle maneuverability analysis,” J. Soc. Auto. Eng. paper Rev., [86] C. Juden, “Shimmy,” Cycletouring, pp. 208–209, June/July 1988.
vol. 22, no. 3, pp. 305–310, 2001. [87] R.S. Sharp and D.J.N. Limebeer, “On steering wobble oscillations of
[56] E. Bakker, L. Nyborg, and H.B. Pacejka, “Tyre modelling for use in vehi- motorcycles,” J. Mech. Eng. Sci., vol. 218, no. 12, pp. 1449–1456, 2004.
cle dynamics studies,” Soc. Auto. Eng., paper 870421, 1987. [88] T. Wakabayashi and K. Sakai, “Development of electronically controlled
[57] C. Koenen and H.B. Pacejka, “Vibrational modes of motorcycles in hydraulic rotary steering damper for motorcycles,” in Proc. Int. Motorcycle
curves,” in Proc. Int. Motorcycle Safety Conf., Washington, Motorcycle Safety Safety Conf., Munich, 2004, pp. 1–22.
Foundation, 1980, vol. II, pp. 501–543. [89] J.P. Den Hartog, Mechanical Vibration. New York: Dover, 1985.
[58] C. Koenen and H.B. Pacejka, “The influence of frame elasticity, simple [90] P. Bandel and C. Di Bernardo, “A test for measuring transient character-
rider body dynamics, and tyre moments on free vibrations of motorcycles in istics of tires,” Tire Sci. Technol., vol. 17, no. 2, pp. 126–137, 1989.

OCTOBER 2006 « IEEE CONTROL SYSTEMS MAGAZINE 61

Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY DELHI. Downloaded on January 05,2022 at 07:14:40 UTC from IEEE Xplore. Restrictions apply.