[IEEE 2006 International Conference on Computational Inteligence for Modelling Control and Automation and International Conference on Intelligent Agents Web Technologies and International

A Robotic Model (ROBI) of Autonomous Bicycle System

Himanshu Dutt Sharma Umashankar N

Scientist, Embedded Systems Lab, Central Electronics Engineering Research Institute (CEERI) Pilani, Rajasthan, INDIA

Department of Mechanical Engineering, Birla Institute of Technology and Science

(BITS) Pilani, Rajasthan, INDIA [email protected],

[email protected] [email protected], [email protected]

Abstract

The authors have applied robotics based approach

in formulating the generalized dynamic equations of motion of a bicycle system. The developed model is considerably more comprehensive than any of the existing models. The model formulation considers a bicycle system to be composed of three rigid bodies connected by revolute joints. The presented model facilitates study of various parameters of motion namely- lean, steer, angular speed, lean-rate, steer-rate, lean-torque, steer-torque, speed-torque, inertial forces, coriolis and centrifugal forces, gravity forces, kinetic energy, potential energy and total energy of various rigid body parts forming the system. The model equations are verified by simulation in MATLAB for the bicycle’s behavior when running autonomously. 1. Introduction

Bicycle stability problem has remained difficult to analyze for decades. The main reason for this is the non-availability of a mathematical model that closely represents a practical bicycle’s dynamics. Existing models have lots of approximations and assumptions, which drive them away from the practical one.

The works of Whipple [1], Klein and Sommerfeld [2] had small-angle approximations. Jones [3] and Le Hénaff [4] concentrated on geometrical considerations that neglected dynamical forces on the steering system. Frank et. al [5] derived non-linear equations of motion of a rigid-rider bicycle system using Newtonian mechanics. Papadopoulos [6] and Hand [7] developed the linearized equations of motion for general bicycle geometry and obtained ranges of stable motion depending on the velocity and parameters of the bicycle. In their recent work on the same problem, the authors studied various existing bicycle models in

literature and simulated these in MATLAB; several experiments were conducted to observe their performance and limitations [8]. Further, using these models intelligent controllers were also designed to balance autonomous bicycle system [9, 10].

Significant information is lost for stability analysis as these models assume constant velocity of bicycle. Thus, to eliminate most of the limitations and more closely represent bicycle’s dynamics, a generalized bicycle model is developed. It uses standard techniques from robotics wherein the bicycle is modeled as a robotic arm and later Lagrangian-Euler method is used to find motion equations representing the dynamics [11].

The generalized model equations are used to simulated bicycle’s motion in MATLAB and time variation is observed for several important parameters i.e. lean, steer, angular velocity, lean rate, steer rate, lean torque, steer torque, angular velocity torque, KE & PE of all links, and forces on these links namely- inertial, gravity, coriolis and centrifugal force .

Finally, a set of model equations is found out by curve fitting to represent a simplified but sufficiently accurate model for the autonomous bicycle system.

The paper is organized as follows- section 2 deals with physical model of bicycle, section 3 develops Lagrangian Euler dynamics, section 4 describes results of simulation experiments and, discussions and conclusions are presented in sections 5 & 6 respectively. 2. Bicycle Model

The bicycle has been designed to closely resemble a practical bicycle. The entire bicycle is constructed using rigid rods and blocks. The three-dimensional model, designed in Pro-Engineer is shown in Figure 1.

International Conference on Computational Intelligence for ModellingControl and Automation,and International Conference onIntelligent Agents,Web Technologies and Internet Commerce (CIMCA-IAWTIC'06)0-7695-2731-0/06 $20.00 © 2006

The design of the bicycle model considers the following-

• All rods have been designed as hollow cylinders with constant wall thickness.

• The wheels are also designed as hollow cylinders. • The mass of the spokes are accounted for in that of

the wheel centre, which is a solid cylinder. • The mass of all parts have been calculated based

on their density and volume values. • The material used for all parts, except wheels and

seat, is mild steel. • Wheels and seat are considered to be made of

synthetic rubber. • Both the front and rear wheels are identical. • The pedaling set-up has been reduced to a circular

disk, with the chains neglected. • The bicycle is considered to be autonomous and

rider-free and hence the mass of the rider is neglected in this model.

Further, the bicycle model is considered under lab

conditions. It is supposed to be moving on a running belt, as shown in figure 1, whose speed is equal to bicycle’s linear speed and friction is sufficient to produce motion without slipping. This avoids need of considering its translatory motion with reference to the World/ Base coordinate frame

Figure 1&2: 3-D Bicycle Model.& Frames attached For analysis the bicycle is assumed to consist of

three rigid bodies i.e. links, namely, rear part, front part and front wheel. The components that constitute the three links are as follows. Link 1 (Rear Part) Wheel, Wheel centre, Carrier, Carrier supports, Connectors, Seat, Seat support, Middle bar, Slant bars and Pedal disk. Link 2 (Front Part) Handle bar, Handle holds, Steer rod, Wheel hold centre and Wheel holds. Link 3 (Front Wheel) Wheel and Wheel centre 2.1. Frame and Joint Considerations

The bicycle has been modeled as a robotic arm (Figure 2) consisting of a sequence of three rigid bodies i.e. Link1, Link2 and Link3 connected by revolute joints. To describe the relationships between adjacent links, the Denavit – Hartenberg (D-H)

algorithm [8] have been used for establishing a coordinate system to each link. The base frame has been fixed at the contact point of the rear wheel to the ground and it is virtually fixed to that point.

The joint angles, namely, lean, steer and angle of rotation of the front wheel correspond to the three degrees of freedom of the bicycle model.

2.2. Links: Mass, Inertia and Centre of Mass

The parameters calculated for different links, with

the corresponding frames, have been tabulated in Table 2. The values correspond to the default bicycle parameters used in the simulation experiments.

Table 2: Mass, Inertia and Centre of Mass calculations

Link Mass (in kg )

Inertia Terms

(in kg m2 )

Coordinates of Centre of

Mass (in m)

Ixx 29.7644

Iyy 17.7778

x -0.005

Izz 12.0837 Ixy -0.0042

y 0.6902

Iyz 13.9992

Rear Part

22.9029

Izx -0.0056 z 0.8605

Ixx 0.6936 Iyy 0.0348

x 0.0201

Izz 0.6722 Ixy 0.0379

y 0.4292

Iyz 0

Front Part

2.5480

Izx 0 z 0

Ixx 0.1268 Iyy 0.1268

x 0

Izz 0.2527 Ixy 0

y 0

Iyz 0

Front Wheel

2.5411

Izx 0 z 0

2.3. Graphical User Interface

A graphic user interface (Figure 3) has been created using MATLAB – GUIDE; it can be used to change the dimensions of various components of the bicycle and the initial conditions and input functions, which are used for solving the dynamic equations (See Section 3) of the bicycle model. The model has been named as ROBI (RObot BIcycle), as we are modeling the bicycle as a robot arm.

Figure 3: Graphical User Interface.


3. Bicycle Dynamics Model using Lagrange – Euler (L-E) Formulation

The derivation of the dynamic model based on the L-E formulation is simple and systematic. It utilizes the D-H matrix representation to describe the spatial displacement between the neighboring link coordinate frames to obtain the link kinematic information, and then employs the Lagrangian dynamics technique to derive the dynamic equations. The direct application of the Lagrangian dynamics formulation, together with the D-H link coordinate representation, results in a convenient and compact algorithmic description of the bicycle equations of motion.

The algorithm is expressed by matrix operations and facilitates both analysis and computer implementation. The derivation of the dynamic equations of bicycle [11] is based on following:

1. The 4 X 4 homogeneous coordinate transformation

matrix, i-1Ai, which describes the spatial relationship between the ith and the (i - 1)th link coordinate frames.

2. the Lagrange – Euler equation

.3,2,1==∂∂−

∂∂ i

qL

qL

dtd

iii

τ ………….(1)

where L = lagrangian function = K – P K = total kinetic energy of the bicycle P = total potential energy of the bicycle qi = generalized coordinates of the bicycle τi = generalized force (torque) at joint i

In the case of ROBI, there are three generalized

coordinates, namely, χ (lean), ψ (steer) and Ө (angle of rotation of the front wheel). The Homogeneous Coordinate Transformation Matrices [11] are given below:

−−−−

=

1000sincos0

0sincossinsincos0coscoscossinsin

11

0

aA

ααχαχαχχαχαχ

−

=

1000010

0sin0cos0cos0sin

22

1

aA

ψψψψ

−

=

1000010000cossin00sincos

32 θθ

θθ

A

21

10

20 AAA = ;

32

20

30 AAA = ;

32

21

31 AAA =

where α = slant angle of the steering axis with the vertical, a1 = distance between two wheel-ground contact points + (Rwheelout * tan α); a2 = Rwheelout / cos α

The Kinetic Energy of the bicycle system K is:

( )[ ]rpTiriip

i

r

i

pi

qqUJUTrK ∑∑∑===

=11

3

12

1

Where, χ=1q ; ψ=2q ; ω=3q

Terms, ijU , iQ and iJ are given in Appendix A. The Potential Energy of the bicycle system is:

( )ii

iii

rAgmP 03

1−∑=

=

Where, [ ] 2/81.9;,000 smggg =−=

[ ]Tiiii

i zyxr 1= mi = mass of link i

3.1. Motion Equations of the Bicycle

The Lagrangian function: L = K – P is given by

( )[ ] ( )ii

iii

rpTiriip

i

r

i

pi

rAgmqqUJUTrL 03

111

3

12

1====

∑+= ∑∑∑ Applying the Lagrange – Euler formulation, the expression for generalized torque can be written a (procedure shown in Appendix - A) in as:

3,2,13

1

3

1

3

1

=++= ∑∑∑===

icqqhqD imkikmmk

kikk

iτ

or in matrix form as ))(())(),(()())(()( tqctqtqhtqtqDt ++=τ

where, )(tτ is generalized torque vector applied at joints i=1, 2, 3. i.e.

[ ]Ttttt )(),(),()( θψχ ττττ =

=)(tq [ ]Tttt )(),(),( θψχ Joint variable vector

=)(tq [ ]Tttt )(),(),( θψχ Joint velocity vector

=)(tq [ ]Tttt )(),(),( θψχ Joint acceleration vector

D(q) = 3 X 3 inertial acceleration-related symmetric matrix whose elements are

== ∑=

kiUJUTrDn

kij

Tjijjkik ,)(

),max(

1, 2, 3

h(q, q )= 3 X 1 nonlinear Coriolis and centrifugal force vector i.e. [ ]Thhhqqh 321 ,,),( =

where == ∑∑==

iqqhh mkikmmk

i

3

1

3

1

1, 2, 3


and == ∑=

mkiUJUTrhn

mkij

Tjijjkmikm ,,)(

),,max(

1, 2, 3

c(q) = [ ]Tccc 321 ,, gravity loading force vector . where )(

3

jj

jijij

i rUgmc −∑==

i=1, 2, 3

4. Simulation Experiments for Model Validation

All existing bicycle models assume the velocity of bicycle to be a constant in their formulation; in contrast the ROBI model allows time variation in bicycle’s velocity. In the simulation experiments the autonomous bicycle is left with some initial velocity and during the self balancing efforts the time variations in velocity are also observed along with several other parameters. However, no external control torque is applied to cause a velocity change. The results are compared with the already studied models’ [9, 10, 11] behavior and characteristic of bicycle for validation.

Experiments have been performed with different initial angular velocities, from 10 rad/sec, to 100 radians/second in the steps of 10 rad/sec,. Two cases of initial angular velocity values - 10 radians/second and 40 rad/sec, (Table 3) are only illustrated here for the reason of space constraint. For the simulation experiments conducted Lean, steer and angular velocity plots are given below. Other parameters of interest i.e. lean-rate, steer-rate, lean-torque, steer-torque, speed-torque, inertial forces, coriolis and centrifugal forces, gravity forces, kinetic energy, potential energy and total energy are provided in Table 4. (Appendix B)

5. Discussions ● At lower velocities, e.g. 10 rad/sec, the bicycle

system is unstable and falls asymptotically i.e. lean and steer both sharply fall to one side. Thus at lower velocities there is no tendency to stabilize itself. The same is observed in practice that bicycle would fall without oscillation of steer and lean to one side at lower velocities. Slight reduction in angular velocity is noticed as it falls and at the same time, the lean and steer rates increase [Table 4 - Appendix B].

Table 3: Simulation Results of Autonomous Bicycle

● At higher velocities, e.g. 40 rad/sec,, the bicycle

system is unstable but it shows tendency to stabilize itself through oscillations in steer, lean and velocity, eventually the oscillating system falls exponentially to one side. The inherent stabilization efforts are thus noticeable in the system in the form of oscillations which delay the falling phenomenon. This represents the inherent stability characteristic present in bicycle system. ● The frequency of aforementioned steer, lean and

velocity oscillations in the system is high at higher velocities and low at relatively lower velocities. Such a behavior is consistent with practical observations. ● The forces generated in the system, show that at

lower velocities the centrifugal and coriolis effects are smaller in magnitude as compared to gravity generated force and thus the bicycle tends to fall quickly. In contrast, the forces of coriolis and centrifugal origin have larger magnitude than gravity force at higher velocities thus the motion is controlled by Centrifugal and Coriolis forces; therefore, falling due to gravity is delayed, showing the balancing effort of the system. ● The total energy of the system remains constant

but KE and PE oscillate as the parameters, lean rate, steer rate and velocity change with time. ● The system shows two distinct regions in its

operation. First region is at low velocities i.e. upto 20 radians per second, where it is non-oscillatory and sharply unstable and the second region starts after 20 radians per second, which is oscillatory and there is a delayed falling out of equilibrium of the bicycle, thus,

Plot

s Initial Velocity 10 rad/s (= 3.55 m/s)

Initial Velocity 40 rad/s

(=14.2 m/s)

Lean

Stee

r

Ang

ular

V

eloc

ity


it can be said to be relatively stable. But in no case the system shows exact stable behavior in any velocity range as long as it is an autonomous system. So it can be said that bicycle is by nature unstable at all velocities but at higher velocity the system tries to balance itself through inherent lean-steer feedback relationship, which at max can delay the fall.

● The data generated is at several velocities to

confirm the system behavior viz. 10, 20,30,40,50, 60, 70, 80, 90 and 100 rad/sec, though the presented data is for selected velocity for the reason of space constraint. The interesting nature of the curves motivated us to write down simplified model equations by curve fitting method to represent motion of a bicycle system, Simplified Lean Equation

)1(*)()*)((*)(),( *)(0000

0 −+= tvbevatvSinvAtv ωχwhere,

00 0365.01719.00 *001828.0*00445.0)( vv eevA −− +=

8403.0*5270.0)( 00 −= vvω 00 2865.00331.05

0 *002616.0*10*928.4)( vv eeva −− +−= 00 0348.02206.0

0 *8408.0*594.4)( vv eevb −− += Simplified Steer Equation

)1(*)()*)((*)(),( *)(0000

0 −+= tvbevatvSinvAtv ωψwhere,

202020 )7097.3

6402.14()4723.13

8951.23()8804.0

5779.9(

0 *01106.0*007863.0*074.0)(−−−−−−

++=vvv

eeevA

52.2*5769.0)( 00 −= vvω 00 3730.00321.0

0 *3815.0*002464.0)( vv eeva −+−= 00 0214.01612.0

0 *1096.0*218.5)( vv eevb += − One could use these equations for simulating

bicycle’s kinematics accurately. RMSE plots for the simplified lean and steer curve fit equations are given below:

Figure 16&17: RMSE plot for Simplified Lean

Equation and Steer Equation respectively. .

6. Conclusions Robotics approach in developing dynamic

equations of motion of a bicycle results into a generalized set of equations. The motion equations for the bicycle are highly nonlinear and consist of inertia forces, coupling reaction forces between joints (coriolis and centrifugal), and gravity effects. Furthermore,

these torques and forces depend on the bicycle’s physical parameters, instantaneous joint configuration, joint velocity and acceleration.

The plots for different velocities verify that it is not stable in autonomous mode at any velocity.

The lean and steer response graphs follow a particular trend, which upon curve fitting results into simplified model equations to represent bicycle behavior. The resulting model equations are very good fit at higher velocities i.e. 10 m/s and above. More experiments have been conducted on the same model with intelligent controller to balance it; the results are communicated to IEEE International Conference on Robotics and Biomimetics (ROBIO 2006), China.

7. Acknowledgements

We are thankful to Dr. Chandrashekhar, Director CEERI Pilani for useful discussion and providing support to the work. Authors also thank CEERI library and publication committee and administration to assist in timely processing of the presentation of the research work.

8. References [1] Whipple F J W, “The Stability of the Motion of a

Bicycle,” in Quarterly Journal of Pure and Applied Mathematics 30, 1899, pp. 312-348.

[2] F. Klein and A. Sommerfeld, “Über die Theorie des Kreisels,” Teubner Verlag Stuttgart, 1965.

[3] D. H. Jones, “The stability of a bicycle,” Physics Today, 23(4), 1970, pp.34-40.

[4] Y Le Hénaff, “Dynamic Stability of the bicycle,” European Journal of Physics 8, 1987, pp. 207-210.

[5] G Franke, W Suhr and F Rieß, “An advanced model of bicycle,” European Journal of Physics 11, 1990, pp. 116-121.

[6] Papadopoulos J, preprint CBRP-JP-88b, Cornell University, 1988.

[7] R. S. Hand, “Comparisons and Stability Analysis of Linearized Equations of Motion for a Basic Bicycle Model,” PhD thesis, Cornell University, 1988.

[8] Himanshu Dutt Sharma, Sameer M. Kale and Umashankar N, “Simulation Model for Studying Inherent Stability Characteristics of Autonomous Bicycle,” Proceedings of IEEE International Conference on Mechatronics and Automation (ICMA 2005) , July 2005, pp. 193-198.

[9] Himanshu Dutt Sharma and Umashankar N, “Stabilization of Autonomous Bicycle using Fuzzy Controller with Maximum Allowable Lean Constraint,” Proceedings of 3rd International Conference on Computational Intelligence, Robotics and Autonomous Systems (CIRAS 2005), December 2005.

[10] Himanshu Dutt Sharma and Umashankar N, “A Fuzzy Controller Design for an Autonomous Bicycle System,” Proceedings of IEEE International Conference on


Engineering of Intelligent Systems (ICEIS 2006), pp. 498 - 503, April 2006.

[11] K.S. Fu, R.C. Gonzalez, and C.S.G. Lee, Robotics: Control, Sensing, Vision, and Intelligence, McGraw-Hill Book Co., 1987.

8. Appendix A. Terms used in Equations of Motion

>≤

=−

−

ijijAQA

U ij

jjij ;0

;11

0

−

=

0000000000010010

iQ

<<≥≥≥≥

= −−

−−

−−

−−

kiorjiforkjiforAQAQAjkiforAQAQA

Ui

jjj

kkk

ik

kkj

jj

ijk

0

11

11

0

11

11

0

−++−

++−

=

iiiiiii

iizzyyxxyzxz

iiyzzzyyxxxy

iixzxyzzyyxx

i

mzmymxmzmIIIIIymIIIIIxmIIIII

J)(5.0

)(5.0)(5.0

B. Simulation Results Note: i) C&C impleis Coriolis & centrifugal force/torque, ii) More graphs of link parameters are available at different velocity too but unable to show due to space constraints. Interested researchers may kindly write an email to authors for more graphs.

Table 4: Simulation Results

Plot

s Angular Velocity: 10 rad/s (= 3.55 m/s)

Angular Velocity:40 rad/s (=14.2 m/s)

Lean

Rat

e St

eer R

ate

Gra

vity

L1

Gra

vity

L2

C&

C L

1 C

& C

L2

C&

C L

3 K

E T

otal

PE

Tot

al

Tota

l Ene

rgy


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[IEEE 2006 International Conference on Computational Inteligence for Modelling Control and Automation and International Conference on Intelligent Agents Web Technologies and International