State University of New York at Buffalo
Department of Mechanical and Aerospace Engineering
MAE571: System Analysis
Fall Semester – 2006
Final Project Report
STATE SPACE ANALYSIS OF BICYCLE MODEL
Name:
Person #:
Instructor:
Date:
Qiushi Fu
34326816
Dr. John Crassidis
December 18 2002
State Space Analysis of Bicycle Model
Abstract
This project will present an investigation into the motion of the daily used two wheelers —
Bicycles. The emphasis is on stability and controllability of the system for small lateral deviations
from straight running with a rider. A discussion of the kinematics, assumptions and geometry of a
rider–bicycle system is presented, followed by a statement of the equations of motion. The state
space model is then formalized and the concept of stability and controllability will be introduced
and investigated in various situations.
Index
Abstract............................................................................................................................................2
1
Introduction...............................................................................................................................3
1.1
Bicycle Analysis............................................................................................................3
1.2
Scope of Report.............................................................................................................4
2
Bicycle Model ...........................................................................................................................4
2.1
Parameter Description...................................................................................................5
2.2
Benchmark parameters..................................................................................................5
2.3
Equation of motion........................................................................................................6
2.4
State space model..........................................................................................................8
3
Stability observation..................................................................................................................9
3.1
MATLAB code verification ..........................................................................................9
3.2
Asymptotical stability ...................................................................................................9
4
Controllability observation .....................................................................................................11
4.1
Control of bicycle........................................................................................................11
4.2
Controllability .............................................................................................................11
5
Conclusion ..............................................................................................................................14
6
Acknowledgement ..................................................................................................................14
References.......................................................................................................................................15
2
State Space Analysis of Bicycle Model
1 Introduction
1.1 Bicycle Analysis
The ability of bicycles with seated riders to remain close to vertical during running has fascinated
many people for many years, and authored articles span a century to date. How does a bicycle
rider avoid falling over? There are at least two aspects to the problem: first, we may ask what rider
does to counteract an incipient fall, and second, it is possible that a bicycle may be self-stabilized
at a sufficient high speed (so that the rider need do nothing to maintain his bike upright).
Today these same two basic features of bicycle balance are clear:
A. Some uncontrolled bicycles can balance themselves. If an appropriate typical bicycle is given a
push to about 6 m/s, it steadies itself and then progresses stably until its speed gets too low. The
torques for the self-correcting steer motions can come from various geometric, inertial and
gyroscopic features of the bicycle.
B. A controlling rider can balance a forward-moving bicycle by turning the front wheel in the
direction of an undesired lean. This moves the ground-contact points back under the rider, just like
an inverted broom or stick can be balanced on an open hand by accelerating the support point in
the direction of lean.
Beyond these two generalities, there is little that has been solidly accepted in the literature,
perhaps because of the lack of need. [1]
Feature A is known as the self-stability of bicycles on ‘open-loop’ or free motion characteristics,
while feature B is a continuous ‘close-loop’ process, which includes the rider’s perception of
bicycle response to a set of body torques at the handlebars, the saddle and the foot pedals applied
by the rider. It is complicated to analysis this behavior because of the significance of rider mass in
the system. A simple model is an inverted pendulum. People maintain it upright by controlling the
movement of its base. The base velocity is expressed in terms of the forward speed and angle of
steer. A control law relates the lean angle to lateral movement assuming that the rider steers in
proportion to lean. [2] More accurate studies use the equations of motion for a rider-bicycle
system between ‘output’ and ‘input’. The input includes the rider body torques and the output is
the rider sensory result which is available for feedback to the input. The equations can be specified
either in Laplacian domain or as a state space model.
Overall, the behavior of bicycles has been well understood. However, state space method,
especially the concept of controllability, is less used. The controllability of any system describes
3
State Space Analysis of Bicycle Model
the ability of the input altering the system ‘states’. The output and knowledge of a closed loop are
neither considered. Moreover, controllability could reflect the potential effectiveness of the
controller in maintaining dynamic equilibrium or restoring it from some perturbed state.
1.2 Scope of Report
The aim of this project is to apply the state space technique to the analysis of a well known bicycle
model. Section 2 begins by formulating equations of motion of the bicycle model. Then the state
space model will be constructed. In section 3, stability analysis will be done, and in section 4, the
controllability of the bicycle will be studied. In the last section, I will make a conclusion.
2 Bicycle Model
The bicycle model used in this report is obtained from [1]. Figure 1 illustrates the model. This
system has non-holonomic kinematic constraints. Start with the 24 DOF (degrees of freedom) of
the 4 rigid bodies, each with 3 translational and 3 rotational DOF. Then subtract out 5 DOF for
each of the three hinges and one more for each wheel touching the ground plane: 24−3×5−2=7.
(a)
(b)
Figure 1 bicycle model
This seven-dimensional configuration space can be parameterized as follows: the location of the
rear-wheel contact with the ground is ( xP , yP ) relative to a global fixed coordinate system with
origin O; a yaw rotationψ , about the z–axis; a lean rotation φ , about the x–axis. The steering
angle δ is the rotation of the front handlebar frame with respect to the rear frame about the
steering axis. A right turn of a forward moving bicycle has δ > 0. Finally, the rotation of the rear R
and front F wheels with respect to their respective frames B and H are θ R and θ F . In summary,
the configuration space is parameterized here with ( xP , yP ,ψ , φ , δ , θ R , θ F ) .
However, the 4 non-holonomic rolling constraints reduce the seven-dimensional accessible
configuration space to 7−4=3 velocity degrees of freedom. This 3-dimensional kinematically
4
State Space Analysis of Bicycle Model
accessible velocity space can be parameterized by the lean rate φ of the rear frame, the steering
rate δ and the rotation rate θ R of the rear wheel R relative to the rear frame B.
2.1 Parameter Description
This well studied bicycle model is fully described by 25 parameters (Table 1). The bicycle design
parameters are defined in an upright reference configuration with both wheels on the level flat
ground and with zero steer angle. The reference coordinate origin is at the rear wheel contact point
P. Conventions is used with positive x pointing towards the front contact point, positive z pointing
down and the y axis pointing to the rider’s right.
The radii of the circular wheels are rF and rR . The wheel masses are mF and mR with their
centers of mass at the wheel centers. The moments of inertia of the rear and front wheels about
their axles are I Fyy and I Ryy . The moments of inertia of the wheels about any diameter in the xz
plane are I Fxx and I Rxx . All front wheel parameters can be different from those of the rear
Tire contact is modeled as non-slipping rolling contact between the ground and the knife-edge
wheel perimeters. The frictionless wheel axles are orthogonal to the wheel symmetry planes and
are located at the wheel centers. In the reference configuration the front wheel ground contact Q is
located at a distance w in front of the rear wheel contact P. The front wheel ground contact point
trails a distance c behind the point where the steer axis intersects with the ground.
The rear wheel R is connected to the rear frame assembly B (which includes the rider body) at the
rear axle. The centre of mass of B, with mass mB , is located at ( xB , yB = 0, z B <0). The moment
of inertia of the rear frame about its centre of mass is represented by a 3 × 3 moment of inertia
matrix where all mass is symmetrically distributed relative to the xz plane, but not necessarily on
the plane. The centre of mass of the front frame assembly H is at ( xH , y H = 0, zH <0) relative to
the rear contact P. H has mass mH .
The steer axis tilt angle λ is measured back from the upwards vertical, positive when tipped back
as on a conventional bicycle with −π / 2 ≤ λ ≤ π / 2 (all angles in radians). The steer axis
location is implicitly defined by the wheel base w, trail c and steer axis tilt angle λ .
2.2 Benchmark parameters
A benchmark bicycle is also defined in [1] with all parameter values given in table 1. The
parameter values were chosen to minimize the possibility of cancellation that could occur if used
in an incorrect model. In the benchmark bicycle the two wheels are different in all properties and
5
State Space Analysis of Bicycle Model
no two angles, masses or distances match.
Table 1 benchmark parameters
2.3 Equation of motion
I used the subscript R for the rear wheel, B for the rear frame incorporating the rider Body, H for
the front frame including the Handlebar, F for the front wheel, T for the Total system, and A for
the front Assembly which is the front frame plus the front wheel.
The total mass and the corresponding centre of mass location (with respect to the rear contact
point P) are
mT = mR + mB + mH + mF
xT = ( xB mB + xH mH + ω mF ) / mT
zT = ( − rR mR + z B mB + z H mH − z F mF ) / mT
For the system as a whole, the relevant mass moments and products of inertia with respect to the
rear contact point P along the global axes are
ITxx = I Rxx + I Bxx + I Hxx + I Fxx + mR rR2 + mB z B2 + mH zH2 + mF rF2
ITxz = I Bxz + I Hxz − mB xB zB − mH xH z H + mF ω rF
The dependent moments of inertia for the axisymmetric rear wheel and front wheel are
6
State Space Analysis of Bicycle Model
I Rzz = I Rxx , I Fzz = I Fxx
Then the moment of inertia for the whole bicycle along the z-axis is
ITzz = I Rzz + I Bzz + I Hzz + I Fzz + mB xB2 + mH xH2 + mF ω 2
The same properties are similarly defined for the front assembly A:
mA = mH + mF
x A = ( xH mH + ω mF ) / mA
z A = ( z H mH − rF mF ) / mA
The relevant mass moments and products of inertia for the front assembly with respect to the
centre of mass of the front assembly along the global axes are
I Axx = I Hxx + I Fxx + mH ( z H − z A ) + mF (rF + z A ) 2
2
I Axz = I Hxz − mH ( xH − x A )( z H − z A ) + mF (ω − xA )(rF + z A )
I Azz = I Hzz + I Fzz + mH ( xH − x A ) + mF (ω − x A ) 2
2
G
Let λ = ( sin λ , 0, cos λ )
T
be a unit vector pointing down along the steer axis where λ is the
angle in the xz-plane between the downward steering axis and the +z direction. The centre of mass
of the front assembly is ahead of the steering axis by perpendicular distance
u A = ( xA − ω − c ) cos λ − z A sin λ
For the front assembly three special inertia quantities are needed:
I Aλλ = mAu A2 + I Axx sin 2 λ + 2 I Axz sin λ cos λ + I Azz cos 2 λ
I Aλ x = − mAu A z A + I Axx sin λ + I Axz cos λ
I Aλ z = mAu A z A + I Axz sin λ + I Azz cos λ
The ratio of the mechanical trail (i.e., the perpendicular distance that the front wheel contact point
is behind the steering axis) to the wheel base is
μ = ( c / ω ) cos λ
The rear and front wheel angular momentum along the y-axis, divided by the forward speed,
together with their sum form the gyrostatic coefficients:
S R = I Ryy / rR ,
S F = I Fyy / rF ,
ST = S R + S F
Define a frequently appearing static moment term as
S A = mAu A + μ mT xT
The linearized equations of motion for the lean angle φ and the steer angle δ , are two coupled
second-order constant-coefficient ordinary differential equations. Define the canonical form below
by insisting that the right-hand sides of the two equations consist only of Tφ and Tδ , respectively.
The first of these two equations is the lean equation and the second is the steer equation. That we
7
State Space Analysis of Bicycle Model
+ Cq + Kq = f .
have a mechanical system requires that the linear equations have the form Mq
For the bicycle these equations can be written as (Papadopoulos 1987)
Mq + vC1q + [ gK 0 + v 2 K 2 ]q = f
Where the time-varying variables are,
(1)
⎡φ ⎤
⎡T ⎤
q = ⎢ ⎥ and f = ⎢ φ ⎥ ,
⎣δ ⎦
⎣Tδ ⎦
I Aλ x + μ ITxz
⎤
,
2
I Aλλ + 2μ I Aλ z + μ ITzz ⎥⎦
ITxx
⎡
M =⎢
⎣ I Aλ x + μ ITxz
−S A ⎤
,
− S A sin λ ⎥⎦
⎡m z
K0 = ⎢ T T
⎣ −S A
⎡0
K2 = ⎢
⎣⎢0
(( S
− mT zT ) / ω ) cos λ ⎤
⎥,
( ( ST + SF sin λ ) / ω ) cos λ ⎦⎥
T
I cos λ
⎡
⎤
0
μ ST + S F cos λ + Txz
− μ mT zT ⎥
⎢
ω
⎥
C1 = ⎢
I Aλ z cos λ
ITzz cos λ ⎞ ⎥
⎛
⎢ − μ S + S cos λ
+ μ ⎜ SA +
)
F
⎟ ⎥
⎢ ( T
ω
ω
⎝
⎠ ⎦
⎣
2.4 State space model
The equations of motion are now recast as a set of coupled first-order equations using the method
of state space from linear systems theory. The elements of a state variable vector X are motion
coordinates and velocities, such that x = [φ
δ φ δ ]T , and the set of equations described in
equation (1) may now be rewritten with input vector u = ⎡⎣Tφ
T
Tδ ⎤⎦ ,
x = Ax + Bu
(2)
The linear time invariant matrices A and B are given by
⎡ −vM −1C1 − M −1[ gK 0 + v 2 K 2 ]⎤
⎡ M −1 ⎤
=
A=⎢
B
,
⎢
⎥
⎥
02×2
⎣ 02×2 ⎦
⎣ I 2×2
⎦
The numbers in the subscripts in each of the zero- and identity matrices , identified by 0 and I,
respectively, correspond to the number of matrix rows and columns.
The output of the system is
y = Cx + Du
(3)
Where C and D matrix represents the way in which the rider interprets the dynamic environment
by sensing reactions from seat and handlebars or by integration of body accelerations. In any close
loop analysis with human control, identifying C and D is a difficult task.
8
State Space Analysis of Bicycle Model
3 Stability observation
3.1 MATLAB code verification
Substitution of the values of the design parameters for the benchmark bicycle from table 1 in the
expressions from Section 2 using MATLAB results in the following values for the entries in the
matrices in the equations of motion (1):
33.8664 ⎤
⎡80.8172 2.3194 ⎤
⎡ 0
, C1 = ⎢
M =⎢
⎥
⎥,
⎣ -0.8504 1.6854 ⎦
⎣ 2.3194 0.2978 ⎦
⎡-80.9500 -2.5995⎤
⎡0 76.5973⎤
K0 = ⎢
, K2 = ⎢
⎥
⎥
⎣ -2.5995 -0.8033⎦
⎣0 2.6543 ⎦
They are the same as the value given in [1], this result shows that my MATLAB mode code is
correct and can be used in the following analysis.
3.2 Asymptotical stability
Now I study the stability of the zero-input response or the asymptotical stability of the bicycle
model. The equation x = Ax is said to be asymptotically stable if and only if all eigenvalues of A
have negative real parts, that is, for no input and in the event of a perturbation all the state
variables decay harmlessly to steady values. The asymptotical stability can be considered as the
‘self-stability’ of a bicycle-rider system. Using the value obtained above to formulate the state
space model, I plot the eigenvalues of A matrix in Figure 2. In the forward speed range of 0<v<10
m/s, the speed range for the asymptotic stability of the benchmark bicycle is about
4.292m/s<v<6.024m/s, in which the rider can perform off-handle riding without falling down.
Table 2 shows several specific eigenvalues at different speed.
At near-zero speeds 0<v<0.5 m/s, there are two pairs of real eigenvalues corresponds to an
inverted-pendulum-like falling of the bicycle. When speed is increased to 0.684m/s two real
eigenvalues become identical and form a complex conjugate pair; where the oscillatory motion
emerges. At first this motion is unstable but at 4.292 m/s, these eigenvalues cross the imaginary
axis and this mode becomes stable. At a higher speed 6.024 m/s one eigenvalue becomes mildly
unstable.
9
State Space Analysis of Bicycle Model
Figure 2 eigenvalues of A matrix
v [m/s]
Eigenvalue1
Eigenvalue2
Eigenvalue3
Eigenvalue4
0.5
4.5482
3.3143
-3.1177
-6.3400
2
2.6823 + 1.6807i
2.6823 - 1.6807i
-3.0716
-8.6739
5
-0.7753 + 4.4649i
-0.7753 - 4.4649i
-0.3229
-14.0784
8
-2.6935 + 8.4604i
-2.6935 - 8.4604i
0.1433
Table 2 eigenvalues at different speed
-20.2794
This result is the same as the result obtained in Lapacian domain[1], which clearly shows that
when a bicycle with a rider is pushed to a speed about 5~6 m/s, the rider need to do nothing to
keep the bicycle upright until the speed gets too low. Another interesting thing is that higher speed
does not mean better stability. For a speed higher than 6, the bicycle would fall down under
perturbation.
Figure 3 Stability analysis
The steer axis tilt is chosen to analyze the influence of changing the configuration of the bicycle.
Figure 3 shows the contour plot of the maximum real part of the four eigenvalues with two
10
State Space Analysis of Bicycle Model
variables speed v and steer axis tilt λ . We can see that if the steer axis tilt is not well designed
( −5D < λ < 9D ), the bicycle-rider system will not be self-stable at any speed. However, a larger
tilt cannot improve the self-stability as the asymptotically stable speed range is not expanded but
only move to a slightly higher speed region.
4 Controllability observation
4.1 Control of bicycle
This section presents a final investigation using state space method. For the bicycle-rider system,
there are two sets of physical process which take place during control. First, there are the
dynamics of lateral deviation from vertical of the bicycle-rider during riding, which have been
studied in detail. The input is a set of torques applied to the frame and handlebar. Second, there are
neural-muscular activities of the rider. Through perception, the rider finds the coordinates and
velocities of motion which are then compared to a desired state of motion in mind. After a finite
length of time, the rider effects the proper amounts of torque and ‘closes’ the loop by feedback.
Figure 4 shows a schematic layout of this process.
Figure 4 bicycle control
Riding and controlling of bicycles in reality is unique from one individual to the next. The
pathways from output back through the feedback loop to input in Figure 4 may have different
layouts between riders. Only the dynamics of the bicycle-rider system remains the same.
Therefore one approach to investigate the control of a bicycle is to make use of the concepts of
controllability from state space method. [3]
4.2 Controllability
A system is said to be controllable if for all initial state x (0) = x0 and any final state x1 , there
exists an input that transfers x0 to x1 in a finite time, that is, all the state variables can be affected,
to some degree, by the input. Formalizing controllability matrix defines
Q = ⎡⎣ B
AB "
Ak −1 B ⎤⎦
(4)
where k is the order of the system and equal to the number of state variables. For the system
11
State Space Analysis of Bicycle Model
described by equation (1) to be controllable, the rank of matrix Q must be equal to k; otherwise,
the system is uncontrollable. A further embellishment is that singular values are indicative of
matrix rank. Thus, the system is also uncontrollable if the lowest singular value is zero. If this is
not the case, the ’degree’ of controllability may be given by the ratio of highest singular value
S max to lowest singular value Smin of Q. In essence, as the ratio becomes larger, it is more
‘difficult to influence the system states; for bicycles, this corresponds to decreasing
controllability.[3][8] Thus, an ‘index’ D associated with the difficulty in control (riding) may be
defined as
D = log10
Smax
Smin
(5)
Figure 5a shows the plot of controllability index D over forward moving speed v for a benchmark
model shown in the previous section, Figure 5b shows the contour plot of controllability index D
with vertical axis v and horizontal axis steer tilt λ . We can see, interestingly, that as the speed
becomes larger, the controllability gets worse. And as the tilt angle increases, the bicycle-rider
system becomes less controllable. If the tilt angle falls into the range 0D < λ < 3D , speed falls into
the range 0m / s < v < 1m / s , the bicycle-rider system has the best controllability..
(a)
(b)
Figure 5 controllability analysis 1
Figure 6 shows the controllability plot when one of the input torques is set to zero. Note that B
matrix is a 4x2 matrix, we can write as B = [ B1
B2 ] . When steer torque Tδ is set to zero, we
are studying on the controllability of ( A, B1 ) , and when lean torque Tφ is set to zero, we are
studying on the controllability of ( A, B2 ) . We can observe a peak in steer-only riding, and two
peaks in lean-only riding, which means that the system is hard to control at speed 1.5m/s with
steer torque only and at speed 0.5m/s and 2.5m/s with lean torque only.
Consider now when the rider takes a finite length of time to responds to the outputs, in ordre to
apply the input torque. An input delay factor with transfer function Gd = e −τ s was added in front
of the bicycle dynamics block (Figure 4). A first order Pade approximation [7], for a finite length
12
State Space Analysis of Bicycle Model
of reaction time equal to 0.2s [6], is used in equation (6)..
e −0.2 s =
− s + 10
s + 10
(6)
Both input torque Tφ and Tδ are applied a time delay, such that the state space model of the input
delay factor is
⎡ −10 0 ⎤
⎡4 0⎤
x = ⎢
x + ⎢
⎥
⎥u
⎣ 0 −10 ⎦
⎣0 4⎦
⎡5 0 ⎤
⎡1 0 ⎤
u = ⎢
x + ⎢
⎥
⎥u
⎣0 5 ⎦
⎣0 1 ⎦
(7)
, B , C , D ) is the desired torque, and the output is the delayed
The input of the delay model ( A
torque which will be the input of the bicycle model given in equation (2) and (3). The serial
combination of the two would be:
⎡ x ⎤ ⎡ A
⎢ ⎥=⎢ ⎣ x ⎦ ⎣ BC
0 ⎤ ⎡ x ⎤ ⎡ B ⎤
⎥⎢ ⎥+⎢ ⎥u
A⎦ ⎣ x ⎦ ⎣ BD ⎦
(8)
Figure 6 controllability analysis 2
Using Equation (8), we can then plot the controllability index D of the input delayed bicycle
model which is shown in Figure (7). Compare to Figure 5a, the overall controllability is weakened,
which means that the bicycle is less controllable considering time delay on input torque. Another
interesting things is the best controllability is obtained in the range 1m / s < v < 3m / s , instead of
0m/s which is obtained without time delay.
Taking a look at the stability analysis in Section 3 and comparing with the result in this section, it
is very interesting that the more stable the bicycle-rider system is, the less controllable the system
is, and vice versa.
13
State Space Analysis of Bicycle Model
Figure 7 controllability analysis 3
5 Conclusion
In this project, I used state space method to analyze the benchmark bicycle model. The concept of
asymptotic stability and controllability has been introduced. In stability analysis, the results meet
the former result given by Laplacian domain method. There exists a speed range that the
bicycle-rider system can be self-stabilized. But in controllability analysis, the result is different
from neither what I expected nor Dr. Seffen’s result [3]. (The model used in [3] is not the one I
used in this project.). There are several possible reasons: first, I might make some mistakes in
programming or understanding the concept; second, the benchmark model (or model in [3]) has
limitations; third, there are differences between the controllability in human experience and the
mathematical ‘degree of controllability’.
There are many further work that can be done. In the future work, the result of the project will be
checked. The singular value decomposition would be introduced to analyze the controllability of
each state. A better bicycle model would be studied.
6 Acknowledgement
Thanks very much for the help from Dr. Crassidis, Dr. Seffen, Dr. Krovi, Chin Pei, Pat, Hao and
Baro.
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State Space Analysis of Bicycle Model
References
[1] J. P. Meijaard, J. M. Papadopoulos, A. Ruina and A. L. Schwab. Linearized dynamics
equations for the balance and steer of a bicycle: a benchmark and review. Draft v37, Oct 15, 2006
[2] R. E. Klein. Using bicycles to teach systems dynamics. IEEE Control Systems Mag., April
1989, 4–9.
[3] K. A. Seffen. Bicycle-rider dynamics: equations of motion and controllability. Technical
reports in Cambridge University. 1999
[4] Chi-Tsong Chen. Linear System Theory and Design. Third edition. 1999
[5] George M. Swisher. Introduction to linear systems analysis.
[6] Wu, J. C. Liu, T. S. Stabilization control for rider-motorcycle model in Hamiltonian form.
Vehicle Systems Dynamics. 26:431-448, 1996
[7] http://mathworld.wolfram.com/PadeApproximant.html
[8] J. L. Crassidis. Class notes. UB Fall 2006 MAE550 System Analysis
[9] Mathworks. MATLAB
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