Experimental results on LPV stabilization of a riderless bicycle

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 formulated 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.

Full text

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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
arewith
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
edi
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 AACC3124

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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
00.51 1.522.53
−10
0
10
20
time [s]
φ [degrees]
Fig. 4.Uncontrolled bicycle: roll angle
3128