a study of bicycle dynamics via system identification
DESCRIPTION
A Study of Bicycle Dynamics via System IdentificationTRANSCRIPT
A Study of Bicycle Dynamics via System Identification
Chih-Keng Chen, Trung-Kien Dao Dept. of Mechanical and Automation Engineering
Dayeh University Changhua, Taiwan
Abstract — This study investigates bicycle dynamic properties by using system identification approaches. The nonlinear bicycle model with configuration parameters from a previously developed benchmark model is studied. The roll angle of the bicycle is controlled at different speeds to generate input-output data including steering torque, roll and steering angles. The collected data are then used to identify the one-input two-output linear model by a prediction-error identification method using parameterization in canonical state-space form. Numerous properties for various speed ranges are discussed from the pole and zero locations of the identified linear model. The system stability, limit-cycle phase portraits of the roll and steering angles, and the non-minimum phase property of the nonlinear system are further investigated and compared.
Keywords — bicycle dynamics, bicycle control, system identification
I. INTRODUCTION
The dynamics of bicycles is a classical topic in mechanics. For the first time, differential equations of motion of a bicycle were established by Whipple in 1899 [1]. However, since massive computational facilities were not available at the time, Whipple’s equations could not be solved. Jones [2] pointed out that, in order to balance a ridden bicycle, an sufficient centrifugal force could be generated to correct its fall by steering the fork into the direction of the fall. Schwab and Meijaard et al. [3][4] developed linearized equations of motion for a bicycle as a benchmark. In their study, the results obtained by pencil-and-paper, the numerical multibody dynamics program SPACAR and the symbolic software AutoSim®, were compared for validation. Limitations due to the simplification of the benchmark model were later discussed by Sharp [5], including acceleration effects, finite cross-section tires, tire forces and dynamics, frame compliance and rider compliance. Among studies related to two-wheel vehicles, Sharp et al. [6] presented a study analyzing straight-running motorcycles. Unlike the Whipple’s model, the tires were modeled as “force and moment producers” rather than as rolling constraints. In others studies, Sharp applied optimal linear preview control theory in the steering control of bicycles [5] via the benchmark model developed in [3] and [4] with extensions to the limitations pointed out in the same paper. The benchmark bicycle dynamic model presented by Schwab and Meijaard et al. [3][4] was reproduced from Whipple’s linear model [1] with certain assumptions which, in principle, cause loss in dynamic properties.
In this study, a different approach using system
identification techniques has been applied to determine the dynamic behaviors of bicycles from the input-output data of a nonlinear bicycle model. This study points out and discusses the pole-location portraits vs. forward speed and the corresponding dynamic properties. Starting from the eleven-generalized-coordinates dynamic model described in a previous study [7] with configuration parameters adopted from the benchmark bicycle [4], input steering torque signal and output data including roll and steering angles were generated from simulations. System identification has been used for different speeds to obtain speed-specific linear models. From identification results and simulations with the nonlinear bicycle model, important nonlinear dynamic behaviors of bicycles are discussed and compared with the linearized benchmark model.
II. SYSTEM-IDENTIFICATION APPROACH
The speed-dependent benchmark model introduced by Schwab et al. [3] was derived from Whipple’s linear model [1] with two parameters in configuration space and two in velocity space defined by
� �21 0 2v g v� � � �Mq K q K K q f�� � , (1)
where [ , ]T� ��q is the vector composed of roll and steering angles, respectively, [0, ]T��f is the force vector consisting of the steering torque � , v is the bicycle forward speed, g is the gravity, and M, C1, K0 and K2 are bicycle-dependant constant coefficient matrices. In this study, a prediction-error method [8] is used to identify the bicycle state-space model in canonical from. To start off, equation (1) can be rewritten as
� �1 2 11 0 2v g v �� � � �� q M K q K K q M f�� � . (2)
By choosing the state vector [ , , , ]T� � � ��x � � , the state-space model can be expressed in the canonical form as
,,
�� ��
x Ax By Cx�
(3)
where 1 2 3 4
5 6 7 8
0 1 0 0
0 0 0 1a a a a
a a a a
�� �� ��� �� ��
A , 1
2
0
0b
b
�� �� ��� �� ��
B and
1 0 0 00 0 1 0 �
� � ��
C . It can be noticed from (2) and (3) that
2 , 1,3,5,7,, 2, 4,6,8,
i ii
i
v ia
v i� ��� � ��� �
��� (4)
2010 International Symposium on Computer, Communication, Control and Automation
978-1-4244-5567-6/10/$26.00 ©2010 IEEE 3CA 2010
where i� and i� are constants dependent on M , 0,1,2i�Kand 1,2ib � are constants dependent on M .
Figure 1. System-identification schematic
Canonical parameterization represents a state-space system in its minimal form, i.e., the system dynamics are expressed by using a minimal number of free parameters. In system (3), free parameters 1..8 1,2
Ti ia b� � �� � � appear
in only the second and forth rows in the system matrices A and B . By this parameterization, numerical Gauss-Newton method can be used to search for the optimal parameters which minimize the error in least-squares sense
defined by 2
1( ) ( ) ( )
n
iE i i
�
� � �� y y , where ( )iy is the ith
original output data sample, ( )i�y is the estimated ith
output sample from the simulated model using parameters � with the original input data, n is the number of data samples. The initial parameter values 0� required in the Gauss-Newton method can be estimated using subspace methods [8]. This estimation procedure has been implemented in the PEM function within MATLAB.
The identification process is shown in Figure 1. To identify the model given by (3), or more specifically, the matrices A and B , at a given speed, the time history of the input steering torque � and the corresponding outputs composed of � and � needs to be generated. However, as the bicycle can be unstable at certain speeds, a roll-angle controller is necessary to produce sufficiently long simulations. The controller can be of any type, such as PID or fuzzy controllers [7][9], and the control accuracy is not an important issue since the only requirement is to keep the bicycle from not falling down during these simulations to ensure enough input-output data are obtained. Nevertheless, as the input-output identification data are from closed-loop simulations, the input signal (steering torque) may not be persistently exciting enough. To reinforce the excitation of the identification data, random signals are generated and added to the input torque.
III. SYSTEM-IDENTIFICATION RESULTS
A. Nonlinear-Bicycle Identification For the simulations in this study, parameters are
derived from the benchmark bicycle provided in [10]. The bicycle is controlled to follow a sinusoidal roll-angle with increasing frequency using a fuzzy-logic controller [9]. The uniformly distributed pseudo-random number signals with amplitude smaller than 1Nm are generated and added to the input steering torque to make the input signal
persistently exciting. Figure 2 shows the identification data from a simulation with the initial forward a speed of 4.2m/s and time step of 0.01s. This data is then plugged into the PEM function in MATLAB to identify the continuous canonical state-space model. This resulted
0 1 0 0 05.392 0.432 12.570 3.177 19.711
, .0 0 0 1 0
13.645 20.944 20.913 17.000 233.588
� �� � � � � � � �� �� � � �� � � �
� �
A B (5)
0 5 10 15 20 25 30 35 40 45 50-2
0
2
(Nm
)
0 5 10 15 20 25 30 35 40 45 50-6
-4
-2
0
2
4
6
(deg)
0 5 10 15 20 25 30 35 40 45 50
-0.2
0
0.2
Time (s)
(deg)
Steering torque
Roll angle data
Steer angle data
Id. model roll angle
Id. model steer angle
Roll angle error
Steer angle error
Figure 2. Identification at speed of 4.2m/s
The four poles calculated from A are 15.101 ,1.668 , and 0.331 3.343 j � . The obtained linear model
is then used in an open-loop steering simulation by giving the original steering torque for verification. The result presented in Figure 2 shows that the output roll and steering angles follow the original data with small error. The largest errors in the total 50s simulation time are 0.231° for roll angle and 0.172° for steering angle. This comparison verifies the fitting accuracy of the identified model.
From the identified state-space model, transfer functions G� and G� corresponding to the roll and steering angles, respectively, with respect to steering torque are obtained as follows,
2
4 3 2
2
4 3 2
19.711 1077.263 2524.109 ,17.431 47.586 205.934 284.279
233.588 311.882 1528.374 .17.431 47.586 205.934 284.279
s sGs s s s
s sGs s s s
�
�
� � �� � � � ��
� �� � � � �
(6)
The corresponding zeros of G� are 52.199 and 2.453 , and those of G� are 3.311 and 1.976 . G� is
identified to be a non-minimum phase system which contains one right-half-plane zero. This effect in two-wheeled vehicles is conventionally referred to as countersteering behavior [11].
B. Pole Locations and Dynamics The procedure in Subsection 3.1 is repeated over the
speed range from 1 to 15m/s. For the speeds lower than 1m/s, collected data is insufficient for the identification process since simulations could not be implemented in sufficient time. The pole locations of the identified model are shown in Figure 3(a). Compared with the poles from the linearized benchmark bicycle [3][4], the speeds can also similarly be divided into four ranges A, B, C and D, and the pole locations have similar variations. At very low speeds, all of the four poles are real, including two stable and two unstable poles. As the speed increases, the two unstable real poles coalesce before splitting to form a
complex conjugate pair, leading to the oscillatory weave mode. The real part of these poles decreases then becomes stable, whereas the imaginary part keeps increasing. At first, the capsize pole moves from the stable to unstable regions but stays close to zero as the speed increases. The castering pole is always stable and its absolute value keeps increasing at high speeds.
0 2 4 6 8 10 12 14 15-30
-25
-20
-15
-10
-5
0
5
10
15
20
v (m/s)
�
vcvwvl
B1 B2 C D
vd
A
(a)
3 3.2 3.4 3.6 3.8 4 4.2-16
-14
-12
-10
-8
-6
-4
-2
0
2
4
v (m/s)
�
vl vw
B1 B2 C
(b) Figure 3. Identified eigenvalues from nonlinear model for speeds (a)
from 1 to 15m/s (b) from 3 to 4.2m/s
There are, however, important differences which should be pointed out. The boundaries of ranges A, B, C and D are different from the benchmark model with
1.465m/sdv � , 3.994m/swv � and 10.525m/scv � ,which extend self-stabilizing region C. Unlike the castering eigenvalue of the benchmark model which always moves far away from zero, the variation of the pole identified from the nonlinear model is more complicated. The most remarkable difference is the variation of the capsize mode within range B2 from 3.152m/slv � to wv ,in which the capsize pole goes down to 9.123 at a speed of 3.280m/s then up to 0.797 at a speed of 3.453m/s very quickly, where the difference in speed is 0.173m/s. Within this range, the capsize pole passes to a stable region at 3.401m/s and back to an unstable region at 3.568m/s. This is made clear in Figure 3(b).
C. Limit Cycles The existence of periodic orbits is shown in Figure 4
obtained at different initial conditions and speeds within range B2. In these simulations, the forward speed is set as 3.5m/s, initial steering angles � as 10°, 5° and 0.1°, initial roll angles � as zero, and no steering torque is applied. The initial steering angle speeds �� are zero in all three
cases, and initial roll angle speeds �� of 55.236 ,27.291 and 0deg/s. It appears that for these three cases,
both steering and roll angles approach very close to periodic orbits. This can be explained by looking at the total energy which is constant during the simulations since there is no external non-conservative force applied.
-25 -20 -15 -10 -5 0 5 10 15 20 25-60
-40
-20
0
20
40
60
Roll angle (deg)
Roll
angle
speed (
deg/s
)
�0=10�, �0
=0�
�0=5�, �0
=0�
�0=0.1�, �0
=0�
Figure 4. Limit cycles of steering angle at speed of 3.5m/s for different initial steering angles
-30 -20 -10 0 10 20 30-80
-60
-40
-20
0
20
40
60
80
Roll angle (deg)
Roll
angle
speed (
deg/s
)�0
=0�, �0=0.1�
�0=0�, �0
=15�
�0=0�, �0
=25�
Figure 5. Limit cycles of roll angle at speed of 3.5m/s for different initial roll angle
Figure 5 shows the roll-angle phase portraits for simulations at the same speed of 3.5m/s for different initial roll angles of 0.1°, 15°, and 25° while keeping the initial steering angle at zero in all cases. It appears that in different cases, the roll angle approaches different orbits, which can also be realized by looking at the total energy. With increasing initial roll angle, the potential energy decreases remarkably, thereby, also making the total energy decrease. When the bicycle is at lower energy, its roll angle swings at a larger amplitude corresponding to the point at which its potential energy reaches its minimum.
IV. ROLL-ANGLE-TRACKING CONTROL
As shown in Figure 6, the controller is composed of two paths: a state-feedback control, and a feedforward path between the error comparator and the plant. The control law is
Ik� �� �Kx , (7)
where 1 4��K � and Ik are constant control gains, and �is the output of the integrator whose input is the tracking error, that is
*ref ref� � � �� � C x� , (8)
o real root x real part of complex root * imaginary part of complex root
o real root x real part of complex root * imaginary part of complex root
where ref� is the reference roll angle signal, and *C is the first row of C in Eq. (3). The dynamics of the system can be described by an equation that is a combination of Eqs. (3) and (8):
ref*
( ) ( )( ) ( )
( ) 0 ( ) 0 1t t
t tt t
� �� � � � � � �
� � �� � � � � � � � � �� � � � �
x A 0 x B 0C
�� . (9)
Note that matrices A , B and C can be determined for a certain speed by using system identification.
[ , , , ]T� � � ��x � �
Figure 6. Pole-placement controller structure
At steady state, Eq. (9) yields
ref*
( ) ( )( ) ( )
( ) 0 ( ) 0 1� �
� � � � � � �
� � � � � � � � � � � � � � � � � �
x A 0 x B 0C
�� . (10)
To design an asymptotically stable controller, ref� is supposed to be step input, i.e., ref ( )t const� � . Subtracting Eq. (10) from Eq. (9) yields
! "*
( ) ( ) ( ) ( )( ) ( )
( ) ( ) 0 ( ) ( ) 0t t
tt t
� �� � � �
� � � �� � � � � � � � � � � � � �
x x A 0 x x BC
� �� �
. (11)
Define ( ) ( )e t� � �� (12)
and a new fifth-order error vector ( )te by ( ) ( )
( )( ) ( )t
tt� � �
� � � �
x xe , (13)
then Eq. (11) becomes
e�� �e Ae B� �� , (14) where
* 0 �
� � ��
A 0A
C� and
0 �
� � ��
BB� . (15)
0 5 10 15 20 25 30 35 40 45 50
-15
-10
-5
0
5
10
15
Roll
angle
(deg)
Reference
Response
Error
0 5 10 15 20 25 30 35 40 45 50-5
0
5
Ste
ering a
ngle
(deg)
0 5 10 15 20 25 30 35 40 45 50
0
5
10
Ste
ering t
orq
ue (
Nm
)
Time (s)
Figure 7. Control results at 15km/h
In a similar fashion, from Eq. (7) ! " ! "( ) ( ) ( ) ( ) ( ) ( )It t k t� � � � � � K x x , (16)
or,
e� � Ke� , (17)
where ! "Ik�K K� are the controller gains, which can be
determined by solving a pole-placement problem from the fifth-order state equation (14).
As a property of pole-placement technique, the bicycle can asymptotically approach a constant reference input. However, for a time-varying reference input, as shown in Figure 7, the bicycle is controlled at 15km/h, there is always a delay time when comparing the response and the reference input. The delay time is affected by the choice of the closed-loop poles. More stable poles can shorten the delay time, but also require larger control steering torque. In consequence, the poles should be chosen so that the generated steering torque is feasible and the tracking error is acceptable.
V. CONCLUSIONS
In this study, system identification has been applied to identify the dynamics properties of a nonlinear bicycle model. Compared with linearization results of previous works [3][4], compatible modes were identified for different speed ranges. Furthermore, differences in variation of pole locations were also found and discussed, of which the most remarkable is the existence of the limit-cycle phase portraits of roll and steering angles in the nonlinear model within the speed range from 3.152 to 3.994m/s.
The approach in this paper has potential in several applications. First, system identification can be applied directly to experimental data from a physical bicycle to obtain its linear model. Furthermore, the identified linear model can be used in controller design of real bicycles with higher accuracy in areas such as pole placement, adaptive control, optimal control and so on.
REFERENCES
[1] F.J.W. Whipple, “The stability of the motion of a bicycle,” Q. J. Pure Appl. Math., vol. 30, pp. 312–348 (1899).
[2] D.E.H. Jones, “The stability of the bicycle,” Physics Today,vol. 23, pp. 34–40, American Institute of Physics (1970).
[3] A.L. Schwab, J.P. Meijaard, J.M. Papadopoulos, “Benchmark results on the linearized equations of motion of an uncontrolled bicycle,” KSME Int. J. of Mechanical Science and Technology, vol. 19, pp. 292–304 (2005).
[4] J.P. Meijaard, J.M. Papadopoulos, A. Ruina, A.L. Schwab, “Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review,” Proceedings of the Royal Society, Series A, vol. 463, pp. 1955–1982 (2007).
[5] R.S. Sharp, “On the stability and control of the bicycle,” Applied Mechanics Reviews (2008).
[6] R.S. Sharp, “The stability and control of motorcycles,” J. of Mech. Eng. Sci., vol. 13, pp.316–329 (1971).
[7] C.K. Chen, T.S. Dao, “Fuzzy control for equilibrium and roll-angle tracking of an unmanned bicycle,” MultibodySystem Dynamics, vol. 15, pp. 325–350 (2006).
[8] L. Ljung, System Identification: Theory for the User,Prentice-Hall, Inc., Upper Saddle River, New Jersey (1999).
[9] C.K. Chen, T.K. Dao, “Speed-adaptive roll-angle-tracking control of an unmanned bicycle using fuzzy logic,” Vehicle System Dynamics, (2010) (in print).
[10] J.M. Papadopoulos, “Bicycle steering dynamics and self-stability: a summary report on work in progress,” Technical report, Cornell Bicycle Research Project, Cornell University, Ithaca, New York (1987).
[11] J. Fajans, “Steering in bicycles and motorcycles,” American Journal of Physics, vol. 68, pp. 654–659 (2000).