Page 1

Experimental results on

LPV stabilization of a riderless bicycle

D. Andreo, V. Cerone, D. Dzung, D. Regruto

Abstract—In this paper the problem of designing a control

system aiming at automatically balancing a riderless bicycle in

the upright position is considered. Such a problem is formu-

lated as the design of a linear-parameter-varying (LPV) state-

feedback controller which guarantees stability of the bicycle

when the velocity ranges in a given interval and its derivative

is bounded. The designed control system has been implemented

on a real riderless bicycle equipped with suitable sensors and

actuators, exploiting the processing platform ABB PEC80. The

obtained experimental results showed the effectiveness of the

proposed approach.

Index Terms—Bicycle Dynamics, LPV control.

I. INTRODUCTION

Modeling, analysis and control of bicycle dynamics has

been an attractive area of research since the end of the

19th century. Pioneering works on bicycle mathematical

modeling were presented in papers [1], [2] and [3]. Rankine

in [1] presents a qualitative analysis of both roll and steer

dynamics of a bicycle. A quantitative stability analysis was

independently developed by Whipple [2] and Carvallo [3]

which derived equations of motion linearized around the

upright vertical equilibrium. The obtained model was used

to formally show that bicycles can balance themselves when

running in a proper speed range (self-stability property).

Although papers on bicycle modeling regularly appeared

also during the first half of the 20th century (see, e.g., the

extensive literature reviews presented in [4] and [5]), bicycle

dynamics has received a significantly renewed attention since

1970 mainly due to the fast improvement of computers

performance and the development of effective software pack-

ages for the simulations of complex mathematical models.

Computer simulations of a nonlinear bicycle model were

presented in the paper by Roland [6]. A good deal of

remarkable works about modeling and analysis of bicycle

dynamics have been recently conducted by Schwab and co-

workers at Delft Bicycle Dynamics Lab (see [7], [8], [9]

and the references therein). More specifically they provide a

benchmark linearized model of the bicycle dynamics on the

basis of the results derived from the comparison between

the original model proposed by Whipple and numerical lin-

earization of nonlinear models obtained exploiting different

softwares for multibody systems modeling. Recently bicycle

dynamics attracted the attention of the automatic control re-

search community. Since bicycle dynamics strongly depends

V.Cerone and

Informatica,

D. Regruto

Politecnico

are with

Torino,

Dipartimento

Italy

di

Automatica

vito.cerone@polito.it, diego.regruto@polito.it

D. Dzung and D. Andreo are with ABB Switzerland Ltd. email:

dacfey.dzung@ch.abb.com, davide.andreo@ch.abb.com

e di

email:

on the forward velocity and, under certain conditions, it

can show both right half plane poles and zeros (see, e.g.,

[4]), the design of feedback controllers for either balancing

the bicycle in the upright position (stabilization) or moving

the bicycle along a predefined path (trajectory tracking)

is indeed a challenging problem. A deep analysis of the

bicycles dynamics from the perspective of control can be

found in the paper [4] by Astrom, Klein and Lennartsson,

where, through models of different complexity, they show

a number of interesting dynamics properties and highlight

the main difficulties in controlling bicycles. An input-output

feedback linearization approach is proposed by Getz and

Marsden in [10] in order to design a controller which allows

the bicycle to track planar trajectories. Lee and Ham in [11]

presented a control strategy for bicycle balancing based on

the sliding patch and stuck phenomena of 2nd order nonlinear

control system, while Yamakita and Utano in [12] discuss

the design of an input-output feedback linearization control

for bicycle equipped with a balancer used to guarantee

stability also when the speed is zero. A number of different

fuzzy control algorithms have been recently proposed in the

literature. Guo, Liao and Wei [13] proposed a fuzzy sliding

mode controller for bicycles described by a proper nonlinear

model, while an adaptive neuro-fuzzy controller for bicycle

stability has been presented by Umashankar and Himanshu

Dutt Sharma in [14].

In this paper the problem of designing a control sys-

tem aiming at automatically balancing a riderless bicycle

in the upright position is considered. Such a problem is

formulated as the design of linear-parameter-varying (LPV)

state-feedback controller which guarantees stability of the

bicycle when the velocity varies within a given range and its

derivative is bounded. The paper is organized as follows.

Section II provides a detailed description of the physical

plant, i.e. an instrumented bicycle, considered in this work.

Then, the mathematical model of the plant is discussed

and analyzed in Section III. Further, the control problem is

formulated in Section IV where a detailed description of the

proposed solution is also presented. Finally, the experimental

results obtained testing the design controller on the real bike

are reported and discussed in Section V. Concluding remarks

can be found in Section VI.

II. PLANT DESCRIPTION

The plant to be controlled is an instrumented riderless

bicycle built during the Master of Science Thesis [15] devel-

oped at ABB Corporate Research, Baden, Switzerland in col-

laboration with the Dipartimento di Automatica e Informatica

2009 American Control Conference

Hyatt Regency Riverfront, St. Louis, MO, USA

June 10-12, 2009

ThB15.3

978-1-4244-4524-0/09/$25.00 ©2009 AACC 3124

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(DAUIN) of Politecnico di Torino, Italy. The prototype of

the instrumented bicycle was obtained from a standard city

bicycle removing all the unnecessary parts (brakes, chain,

crankset and pedals). The bicycle (Figure 1) is equipped

with three sensors, one actuator and an industrial processing

platform for the implementation of the controller and the

signals processing algorithms. The rear wheel is equipped

with an encoder in order to measure the bicycle forward

speed, while measurements of roll velocity and steering angle

are respectively provided by a gyroscope and a potentiome-

ter. Indirect measurements of the roll angle are obtained

through numerical integration from the gyroscope output.

Since the gyroscope provides a relative measure, the initial

condition of the integration is given by a pendulum connected

to the bicycle frame through a linear potentiometer. The

initial condition of the roll angle is provided by the mean

value of the pendulum angle oscillations. The steering an-

gular velocity measurements are indirectly obtained through

filtered numerical differentiation of the steering angle. The

actuator is a torque driven servo-motor which provides a

torque on the steering axis. The maximum value of the

torque is ±1.5Nm. The bicycle is not equipped with active

traction on the wheels. The processing platform is a general

purpose ABB PEC80 which includes both a computational

unit and a data acquisition and communication unit. The

control implementation software level of the ABB PEC80

is programmed through MatlabTM/SimulinkTMenvironment

and Matlab Real Time WorkshopTM. Thanks to a proprietary

toolbox (ABB AC PEC800 Toolbox ver.500) which contains

a number of Simulink blocks and functions compatible with

the available hardware, a huge number of control structures

can be easily implemented. The sample time used for the

implementation of the controller presented in this paper is

0.01 s. A detailed description of the ABB PEC80 can be

found in [15].

III. PLANT MODELING

In this paper the bicycle mathematical model proposed by

Schwab and co-workers in [7], [9] has been considered for

both the analysis of the plant dynamics and the design of the

controller. The mechanical model of the bicycle consists of

four rigid bodies: the rear frame, the front fork and handlebar

assembly, the rear and the front knife-edge wheels. The four

bodies are interconnected by revolute hinges and, in the

reference configuration, they are all symmetric relatively to

the bicycle longitudinal axis. The contact between the stiff

non-slipping wheels and the flat level surface is modelled

by holonomic constraints in the normal direction and by

nonholonomic constraints in the longitudinal and lateral

direction. In spite of its relative simplicity, the model ade-

quately describes the main dynamics of the bicycle as proved

by the experimental validation performed in [16]. This model

considers three degrees of freedom (the roll, the steer, and the

forward speed) and has been obtained through linearization

of the equations of motion for small perturbation around

the so-called constant-speed straight-ahead upright trajectory.

The linearized equations of motion are two coupled second-

order ordinary differential equations which depends on the

forward speed v. The equations, written in matrix form, are

the following:

M¨ q + [vC1] ˙ q +?K0+ v2K2

?q = f

(1)

with:

qT= [φ δ],

fT= [Tφ Tδ]

where φ is the roll angle, δ is the steering angle, v is

the bicycle velocity, Tφis a possible exogenous roll torque

disturbance and Tδ is the steering torque provided by the

actuator. The remaining quantities involved in the equations

are the symmetric mass matrix M, the damping matrix vC1

which is linear in the forward speed v, and the stiffness

matrix which is the sum of a constant (symmetric) part, K0,

and a part, v2K2, which is quadratic in the forward speed

v. These matrices depends on the geometrical parameters

of the bicycle (see [7] for details). The obtained linearized

equations have been rewritten in state-space form choosing

the roll angle φ, the steering angle δ and their derivatives,

˙φ and˙δ respectively, as state variables. The control input

u(t) is the torque applied on the handlebar axis, Tδ(t). The

measured output y(t) are all the four state variables. Due

to the complexity and non-linear characteristic (with respect

to the velocity v) of the equations of motion, the following

state-space equations can be straightforwardly derived from

(1):

?

y(t) = Cx(t) + Du(t)

˙ x(t) = A(v)x(t) + Bu(t)

(2)

where:

xT(t) =

?

φ δ

˙φ ˙δ

?

,u(t) = Tδ,y(t) = φ

(3)

and:

A =

0

0

0

0

1

0

0

1

13.67

4.857

0.225 − 1.319v2

10.81 − 1.125v2

−0.164v

3.621v

−0.552v

−2.388v

B =

0

0

−0.339

7.457

,C =

1

0

0

0

0

1

0

0

0

0

1

0

0

0

0

1

,D = 0 (4)

The numerical values of matrices in equation (4) have been

derived in [15]. As can be seen from equation (4), matrix A

depends on the bicycle forward speed v. Such a dependence

qualifies system (2) as a Linear-Parameter-Varying (LPV)

model. For each constant value of time-varying parameter v,

equation (2) describes an LTI system. The location of the

eigenvalues of system (2) for a number of different values

of v in the interval [0,10] m/s is depicted in Figure 2. When

the bicycle forward speed v is zero, the system has four

real poles at p1 = 3.23, p2 = −3.23, p3 = 3.74 and

p4= −3.74, marked with circles in Figure 2. The first pair

of poles corresponds to the so-called pendulum-like poles;

the notation is due to the fact that in this condition the

bicycle would fall over just like an inverted pendulum. The

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remaining pole pair are related to the front fork dynamics.

As the velocity increases, the poles p1 and p3 meet (at

v ≈ 0.5 m/s) and become complex conjugate. The real part

of the complex pole-pair decreases as the velocity increases.

Following Sharp ([17]) this mode is called the weave mode,

which becomes stable at the critical velocity vw ≈ 3.4

m/s. The pole p2 remains real and moves toward the right

half plane with increasing velocity. It becomes unstable at

the velocity vc ≈ 4.1 m/s. The pole p4 moves to the

left as velocity increases. For large velocity v, the pole p2

approaches zero. The values of vw and vcstrongly depend

on the value of the bicycle trail (see [4] for details). From

the above analysis it is clear that the task of controlling a

riderless bicycle is a challenging problem, since the plant

dynamics strongly depends on the bicycle forward speed.

IV. CONTROL PROBLEM FORMULATION AND PROPOSED

SOLUTION

The problem we are dealing with in this paper is the

design of a control system in order to automatically balance

a riderless bicycle in the upright position. Such a problem

can be formulated as the design of a feedback controller

able to guarantee stability of the LPV system described by

the equations (2) when the velocity v is allowed to vary in

the interval [0,γ] and its derivative ˙ v satisfies the constraint

|˙ v| ≤ ρ where γ and ρ are known constants. As is well

known stability of all the frozen LTI systems obtained for

each fixed values of v is not a sufficient condition for the

stability of the LPV system. As stated in Section II, the

bicycle under consideration is not equipped with a wheels

active traction system, thus, in the experimental tests, a

human operator makes the bicycle run by pushing it for a

few seconds. According to such experimental conditions,

reasonable values for γ and ρ are respectively γ = 5 m/s and

ρ = 0.5 m/s2. Since all the state variables of systems (2) are

available for measurements, a static state-feedback control

structure has been chosen. The dynamics of the system to

be controlled depends on the time-varying parameter v and

on-line measurements of such a parameter are available

(see Section II), thus we look for an LPV controller of the

form u = K(v)x = [kφ(v) kδ(v) k˙φ(v) k˙δ(v)]xTwhich

on the basis of the measurements of v and x(t) provides

the control input u(t) that guarantees stability of the closed

loop LPV systems when ∈ [0,γ] and |˙ v| ≤ ρ. The following

proposition, derived from the application of Theorem 14 of

paper [18] to the problem considered in this work, provides

the basis for the design of such a controller.

Proposition 1 [18] The closed loop system described

by equation

˙ x = Acl(v)x,Acl(v) = A(v) + BK(v)

(5)

is exponentially stable for all the trajectory of the velocity v

satisfying v ∈ [0,γ] and |˙ v(t)| ≤ ρ if there exists a matrix

X(v) = XT(v) > 0 such that

X(v)Acl(v) + AT

cl(v)X(v) + ˙ vdX(v)

dv

< 0

(6)

for all v ∈ [0,γ] and |˙ v| ≤ ρ.

Proposition 1 clearly states that the problem of designing a

feedback controller able to guarantee exponential stability

of the LPV system described by the equations (2) can

be solved by performing a numerical search for a pair of

parameter dependent matrices X(v) and K(v) which satisfy

the matrix inequality (6). Unfortunately, since inequality (6)

is a nonconvex constraint in the variables X(v) and K(v),

a direct numerical search leads to a nonconvex optimization

problem. However, by means of some simple manipulations

(see [18] for details), it can be shown that the existence of

a pair of matrices X(v) and K(v) satisfying the nonconvex

constraint (6) is equivalent to the existence of the matrices

Y (v) = X−1(v) and K(v) = K(v)X−1(v)

(7)

which satisfy the following linear matrix inequality (LMI):

A(v)Y (v)+BK(v)+Y (v)AT(v)+K

T(v)BT− ˙ vdY (v)

dv

< 0

(8)

As is well known, computing solutions to LMIs leads to

a special kind of convex optimization problem for which

efficient numerical solutions are available (see, e.g., [19]).

Due to the dependence on the parameter v, equation (8)

actually represents an infinite family of LMIs. As properly

discussed in [18], a possible approach to reduce the problem

to the solution of a finite set of LMIs is the parameter

discretization. More precisely, by dividing the interval [0,γ]

into N subintervals of width h and exploiting a forward

difference approximation for the derivative, the constraint (8)

is approximately converted to the following finite collection

of LMIs:

A(jh)Y (jh) + BK(jh) + Y (jh)AT(jh) + K

± ρY (jh + h) − Y (jh)

h

T(jh)BT+

< 0, j = 0,...,N − 1

(9)

The discretization has been performed in this work choosing

N = 100 and h = 0.05 m/s.

In order to bound the damping ζ and the natural frequency

ωn of the eigenvalues of the matrix Acl(v) for each fixed

value of v = jh obtained from the discretization of the

interval [0,γ], the following two sets of constraints are added

to the design problem:

?

for j = 0,...,N − 1

−rY (jh)A(jh)Y (jh) + B(jh)K(jh)

−rY (jh)Y (jh)A(jh)T+ K

T(jh)BT

?

< 0,

(10)

and

?

sinθ(A(jh)Y (jh) + BK(jh) + Y (jh)A(jh)T+ K

cosθ(Y (jh)A(jh)T− K

T(jh)BT)

T(jh)BT− A(jh)Y (jh) − BK(jh))

cosθ(A(jh)Y (jh) + BK(jh) − Y (jh)A(jh)T− K

sinθ(A(jh)Y (jh) + BK(jh) + Y (jh)A(jh)T+ K

for j = 0,...,N − 1

T(jh)BT)

T(jh)BT)

?

< 0,

(11)

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Equations (10) and (11) are LMIs in the variables Y (jh) and

K(jh) which guarantee a minimum damping ratio ζ = cosθ

and a maximum natural frequency ωn = rsinθ for the

eigenvalues of the closed loop systems computed at each

fixed value of v (see paper [20] and the references therein

for a detailed description on how to describe constraints on

eigenvalues location in terms of LMIs). In this work the

values ζ = 0.6 and ωn = 20 rad/s have been considered

which correspond to the values r ≈ 25 rad/s and θ ≈ 0.93

rad. The value ζ = 0.6 has been chosen in order to avoid

significantly under-damped oscillation in the response; the

value of ωnhas been chosen on the basis of the simulation

performed exploiting the mathematical model of the plant

which revealed that a value greater than 20 rad/s leads to

saturation of the actuator. The set of LMIs described by

equations (9), (10) and (11) have been solved using the LMI

toolbox of MatLabTM. The obtained solution is a collection

of discretized variables Y (jh) and K(jh). A continuous

solution Y (v), K(v) has been formed by interpolation.

Finally, the controller gains K(v) have been obtained from

(7) as K(v) = K(v)Y (v)−1. The location of the closed loop

eigenvalues when v ranges in [0,5] m/s is depicted in Fig.

3.

Remark – An alternative approach to reduce problem (8)

to the solution of a finite set of LMIs is to impose a fixed

structure on the search variables Y (v) = X−1(v) and K(v)

avoiding parameter discretization. However, in that case,

some conservatism is introduced in the problem (see, e.g.,

[18] for details). For such a reason, in this work we have

chosen to exploit the gridding approach. Besides, we would

like to remark that the bicycle dynamics depends on a single

parameter and, thus, a fine discretization can be performed.

V. EXPERIMENTAL RESULTS AND DISCUSSION

In this section we report the experimental results obtained

testing the controlled systems. As stated in Section II, the

bicycle under consideration is not equipped with a wheels

active traction system; thus, in all the experimental tests, a

human operator pushes the bicycle for a few seconds, then

the bicycle is released. All the experimental data reported in

the figures of this section have been collected from the time

the human operator releases the bicycle. Since the tests were

performed in a room of limited dimension, the maximum

bicycle velocity considered was about 2.1 m/s. However, we

would like to remark that the control of a riderless bicycle

is more challenging at such low velocities, since, as stated

in Section III, the system is open loop unstable for values

of velocity v less that 3.4 m/s. Three different experimental

tests, described below, have been performed.

A. Uncontrolled bicycle

In the first test the bicycle ran with the control system

switched off. Such a test was performed to show that the

open-loop riderless bicycle is unstable when the velocity

v is less than 3.4 m/s. The bicycle has been pushed to a

velocity of about 3m/s and then released. The evolution of

the roll angle φ(t) is shown in Figure 4. As expected, after

few seconds φ(t) rapidly increases reaching a value of about

20 degrees just before falling down.

B. Controlled bicycle, 1m/s ≤ v ≤ 1.7m/s

In this test the bicycle was pushed by the human operator

until the velocity reached the value of 1.7 m/s; then it was

released. The test stopped when the bicycle reached the end

of the room. The final velocity was 1 m/s. The evolution

of the roll angle φ(t) is shown in Figure 5, from which it

can be seen that the designed LPV control system effectively

balances the riderless bicycle in the upright position. As a

matter of fact, after about 10 s, the amplitude of the bicycle

roll angle φ(t) is driven to a value of about 0.1 degrees

starting from the initial condition φ(0) ≈ −2.25 degrees.

The steering torque Tδ applied on the handlebar axis to

control the bicycle is reported in Figure 6. Figure 7 reports

the velocity of the bike, while Figure 8 shows the values of

the state-feedback gains K(v).

C. Controlled bicycle, 1.7m/s≤ v ≤ 2.1m/s and external

impulsive roll torque disturbance

In this test the bicycle was pushed by the human operator

until the velocity reached the value of 2.1 m/s, then it was

released. The test stopped when the bicycle reached the

end of the room. The final velocity was 1.7 m/s. In order

to evaluate the robustness of the designed control system

against exogenous disturbances (like for example a side wind

gust), the human operator slapped the rear frame of the

bicycle. Such a “slap” can be modeled as an impulsive roll

torque disturbance of about 26 Nm (see [15] for details on

the approximate characterization of such a disturbance). The

lateral disturbance was applied just after the time the bicycle

was released. The experimental data reported in the figures of

this section have been collected from the time the roll torque

disturbance has been applied to the bicycle. The evolution

of the roll angle φ(t) is shown in Figure 9, from which it

can be seen that the designed LPV control system effectively

balances the riderless bicycle in the upright position, also in

the presence of the roll torque disturbance. As a consequence

of the “slap”, the bike roll angle reaches a maximum value of

about −2.75 degrees (t ≈ 0.6s), then it is rapidly attenuated

by the control system to a value less than 0.75 degrees

(t ≈ 6.5s). The steering torque Tδapplied on the handlebar

axis to control the bicycle is reported in Figure 10. Figure

11 reports the velocity of the bike, while Figure 12 shows

the values of the state-feedback gains K(v).

VI. CONCLUSIONS

In this paper the problem of designing a control system

aiming at automatically balancing a riderless bicycle in the

upright position is considered. Such a problem is formu-

lated as the design of linear-parameter-varying (LPV) state-

feedback controller able to guarantee stability of the bicycle

when the velocity is allowed to vary in a known interval and

its derivative is bounded. The controller has been designed

solving an LMI optimization problem. The designed control

system has been implemented on a real riderless bicycle

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equipped with suitable sensors and actuators, and exploiting

the processing platform ABB PEC80. The obtained exper-

imental results showed the effectiveness of the proposed

approach.

ACKNOWLEDGEMENTS

We thank Mats Larsson of ABB Corporate Research for

his crucial contributions to the bicycle modeling, and Paul

Rudolf for the support in setting up the hardware.

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Fig. 1.The instrumented bicycle used in this work

−20 −15−10−50

−10

0

10

eigenvalue real part

eigenvalue imaginary part

Fig. 2.

(0,10)m/s). The poles marked with circles correspond to v = 0m/s.

Poles trajectory for the uncontrolled bike (speed range

−15 −10−50

−4

−2

0

2

4

eigenvalue real part

eigenvalue imaginary part

Fig. 3.

[0,5] m/s) and damping constraints (straight thin lines)

Root locus for the controlled bike eigenvalues (speed range

0 0.51 1.52 2.53

−10

0

10

20

time [s]

φ [degrees]

Fig. 4.Uncontrolled bicycle: roll angle

3128