DYNAMICAL MODELS FOR OMNI-DIRECTIONAL ROBOTS WITH

3 AND 4 WHEELS

elder P. Oliveira, Armando J. Sousa, A. Paulo Moreira and Paulo J. Costa

Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias s/n 4200-465, Porto, Portugal

Keywords:

Identiﬁcation, simulation, modeling and omni-directional mobile robots.

Abstract:

Omni-directional robots are becoming more and more common in recent robotic applications. They offer

improved ease of maneuverability and effectiveness at the expense of increased complexity. Frequent appli-

cations include but are not limited to robotic competitions and service robotics. The goal of this work is to

ﬁnd a precise dynamical model in order to predict the robot behavior. Models were found for two real world

omni-directional robot conﬁgurations and their parameters estimated using a prototype that can have 3 or 4

wheels. Simulations and experimental runs are presented in order to validate the presented work.

1 INTRODUCTION

Omni-directional robots are becoming a much sought

solution to mobile robotic applications. This kind of

holonomic robots are interesting because they allow

greater maneuverability and efﬁciency at the expense

of some extra complexity. One of the most frequent

solutions is to use some of Mecanum wheels (Diegel

et al., 2002) and (Salih et al., 2006). A robot with 3

or more motorized wheels of this kind can have al-

most independent tangential, normal and angular ve-

locities. Dynamical models for this kind of robots are

not very common due to the difﬁculty in modeling

the several internal frictions inside the wheels, mak-

ing the model somewhat speciﬁc to the type of wheel

being used (Williams et al., 2002).

Frequent mechanical conﬁgurations for omni-

directional robots are based on three and four wheels.

Three wheeled systems are mechanically simpler but

robots with four wheels have more acceleration with

the same kind of motors. Four wheeled robots are

expected to have better effective ﬂoor traction, that

is, less wheel slippage – assuming that all wheels

are pressed against the ﬂoor equally. Of course four

wheeled robots also have a higher costs in equipment,

increased energy consumption and may require some

kind of suspension to distribute forces equally among

the wheels.

In order to study and compare the models of the

3 and 4 wheeled robots, a single prototype was built

that can have both conﬁgurations, that is, the same

mechanical platform can be used with 3 wheels and

then it can be disassembled and reassembled with a 4

(a) Three wheeled robot. (b) Four wheeled robot.

Figure 1: Omni-directional robot.

wheel conﬁguration, see ﬁgure 1.

Data from experimental runs is taken from over-

head camera. The setup is taken from the heritage

of the system described in (Costa et al., 2000) that

currently features 25 fps, one centimeter accuracy in

position(XX and YY axis) and about 3 sexagesimal

degrees of accuracy in the heading of the robot.

In order to increase the performance of robots,

there were some efforts on the studying their dy-

namical models (Campion et al., 1996)(Conceic¸

et al., 2006)(Khosla, 1989)(Tahmasebi et al.,

2005)(Williams et al., 2002) and kinematic models

(Campion et al., 1996) (Leow et al., 2002)(Loh et al.,

2003)(Muir and Neuman, 1987)(Xu et al., 2005).

Models are based on linear and non linear dynami-

cal systems and the estimation of parameters has been

the subject of continuing research (Conceic¸

ao et al.,

2006)(Olsen and Petersen, 2001). Once the dynam-

ical model is found, its parameters have to be esti-

mated. The most common method for identiﬁcation

189

P. Oliveira H., J. Sousa A., Paulo Moreira A. and J. Costa P. (2008).

DYNAMICAL MODELS FOR OMNI-DIRECTIONAL ROBOTS WITH 3 AND 4 WHEELS.

In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - RA, pages 189-196

DOI: 10.5220/0001489201890196

 SciTePress

of robot parameters are based on the Least Squares

method and Instrumental Variables.

However, the systems are naturally non-linear

(Julier and Uhlmann, 1997), the estimation of pa-

rameters is more complex and the existing meth-

ods (Ghaharamani and Roweis, 1999)(Gordon et al.,

1993)(Tahmasebi et al., 2005) have to be adapted to

the model’s structure and noise.

1.1 Structure

This paper starts by presenting the mechanical proto-

type and studied mechanical conﬁgurations for the 3

and 4 wheeled robots. Finding the dynamical model is

discussed and then, an initial approach in estimating

model parameters for each robot is done. The need

for additional accuracy drives the comparative study

on relative importance of the estimated parameters.

An additional experiment is done for estimating ﬁ-

nal numerical values for the conﬁgurations of 3 and

4 wheels. Conclusions and future work are also pre-

sented.

2 MECHANICAL

CONFIGURATIONS

Figures 2 and 3 present the conﬁguration of the three

and four wheeled robots respectively, as well as all

axis and relevant forces and velocities of the robotic

system. The three wheeled system features wheels

separated by 120 degrees.

Figure 2: Three Wheeled Robot.

Figures 2 and 3 show the notation used through-

out this paper, detailed as follows:

• x,y,θ - Robot’s position (x,y) and θ angle to the deﬁned

front of robot;

• d [m] - Distance between wheels and center robot;

• v

[m/s] - Wheels linear velocity;

• ω

,ω

[rad/s] - Wheels angular velocity;

• f

, f

[N] - Wheels traction force;

Figure 3: Four Wheeled Robot.

• T

[N ·m] - Wheels traction torque;

• v, vn [m/s] - Robot linear velocity;

• ω [rad/s] - Robot angular velocity;

• F

[N] - Robot traction force along v and vn;

• T [N ·m] - Robot torque (respects to ω).

3 MODELS

3.1 Kinematic

The well known kinematic model of an omni-

directional robot located a (x,y, θ) can be written as

(t) = dx(t)/dt , v

(t) = dy(t)/dt and ω(t) = dθ(t)/dt

(please refer to ﬁgures 2 and 3 for notation issues).

Equation 1 allows the transformation from linear ve-

locities v

and v

on the static axis to linear velocities

v and vn on the robot’s axis.





v(t)

vn(t)

ω(t)





; X





(t)

ω(t)









cos(θ(t))

−sin(θ(t))

sin(θ(t))

cos(θ(t))





·X

(1)

3.1.1 Three Wheeled Robot

Wheel speeds v

, v

and v

are related with robot’s

speeds v, vn and ω as described by equation 2.





(t)









−sin(π/3)

sin(π/3)

cos(π/3)

−1

cos(π/3)









v(t)

vn(t)

ω(t)





(2)

Applying the inverse kinematics is possible to obtain

the equations that determine the robot speeds related

the wheels speed. Solving in order of v, vn and ω, the

following can be found:

v(t) = (

√

3/3) ·(v

(t)−v

(t)) (3)

vn(t) = (1/3)·(v

(t)+ v

(t))−(2/3)·v

(t) (4)

ω(t) = (1/(3 ·d))·(v

(t)+ v

(t)) (5)

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190

3.1.2 Four Wheeled Robot

The relationship between the wheels speed v

, v

and v

, with the robot speeds v, vn and ω is described

by equation 6.







(t)













−1











v(t)

vn(t)

ω(t)





(6)

It is possible to obtain the equations that determine the

robot speeds related with wheels speed but the matrix

associated with equation 6 is not square. This is be-

cause the system is redundant. It can be found that:

v(t) = (1/2)·(v

(t)−v

(t)) (7)

vn(t) = (1/2)·(v

(t)−v

(t)) (8)

ω(t) = (v

(t)+ v

(t))/(4 ·d) (9)

3.2 Dynamic

The dynamical equations relative to the accelerations

can be described in the following relations:

M ·

dv(t)

∑

(t)−F

(t) (10)

M ·

dvn(t)

∑

(t)−F

Bvn

(t)−F

Cvn

(t) (11)

J ·

dω(t)

∑

T (t) −T

Bω

(t)−T

Cω

(t) (12)

where the following parameters relate to the robot as

follows:

• M [kg] - mass;

• J [kg ·m

] - inertia moment;

• F

Bvn

[N] - viscous friction forces along v and vn;

• T

Bω

[N ·m] - viscous friction torque with respect to the

robot’s rotation axis;

• F

Cvn

[N] - Coulomb frictions forces along v and vn;

• T

Cω

[N ·m] - Coulomb friction torque with respect to

robot’s rotation axis.

Viscous friction forces are proportional to robot’s

speed and as such F

(t) = B

·v(t), F

Bvn

(t) = B

·vn(t)

and T

Bω

(t) = B

·ω(t), where Bv,Bvn [N/(m/s)] are the

viscous friction coefﬁcients for directions v and vn

and Bω [N ·m/(rad/s)] is the viscous friction coefﬁ-

cient to ω.

The Coulomb friction forces are constant in am-

plitude F

(t) = C

·sign(v(t)), F

Cvn

(t) = C

·sign(vn(t))

and T

Cω

(t) = C

· sign ω(t), where Cv,Cvn [N] are

Coulomb friction coefﬁcient for directions v e vn and

Cω [N ·m] is the Coulomb friction coefﬁcient for ω.

3.2.1 Three Wheeled Robot

The relationship between the traction forces and rota-

tion torque of the robot with the traction forces on the

wheels is described by the following equations:

∑

(t) = ( f

(t)− f

(t))·sin(π/3) (13)

∑

(t) = −f

(t)+ ( f

(t)+ f

(t))·cos(π/3)(14)

∑

T (t) = ( f

(t)+ f

(t))·d (15)

The traction force on each wheel is estimated by trac-

tion torque, which can be determined using the motor

current, as described in the following equations:

(t) = T

(t)/r (16)

(t) = l ·K

·i

(t) (17)

• l - Gearbox reduction;

• r [m] - Wheel radius;

• K

[N ·m/A] - Motor torque constant;

• i

[A] - Motor current (j=motor number).

3.2.2 Four Wheeled Robot

The relationship between the traction forces and rota-

tion torque of the robot with the traction forces on the

wheels, is described by the following equations:

∑

(t) = f

(t)− f

(t) (18)

∑

(t) = f

(t)− f

(t) (19)

∑

T (t) = ( f

(t)+ f

(t))·d (20)

As above, the traction force in each wheel is esti-

mated using the wheels traction torque, which is de-

termined by the motor current, using equations 16 and

17, where j=0,1,2,3.

3.3 Motor

The prototype uses brushless motors for the locomo-

tion of the robot. The model for brushless motors is

the similar to the common DC motors, based on (Pil-

lay and Krishnan, 1989).

(t) = L ·

(t)

+ R ·i

(t)+ K

·ω

m j

(t) (21)

m j

(t) = K

·i

(t) (22)

• L [H] - Motor inductance;

• R [Ω] - Motor resistor;

• K

[V /(rad/s)] - EMF motor constant;

• u

[V ] - Motor voltage (j=motor number);

• ω

m j

[rad/s] - Motor angular velocity (j=motor num-

ber);

• T

m j

[N ·m] - Motor torque (j=motor number).

DYNAMICAL MODELS FOR OMNI-DIRECTIONAL ROBOTS WITH 3 AND 4 WHEELS

191

4 PARAMETER ESTIMATION

The necessary variables to estimate the model param-

eters are motor current, robot position and velocity.

Currents are measured by the drive electronics, posi-

tion is measured by using external camera and veloc-

ities are estimated from positions.

The parameters that must be identiﬁed are the vis-

cous friction coefﬁcients (Bv,Bvn, Bω), the Coulomb

friction coefﬁcients (Cv,Cvn,Cω) and inertia moment

J. The robot mass was measured, and it was 1.944 k g

for the three wheeled robot and 2.34 kg for the four

wheeled robot.

4.1 Experience 1 - Steady State Velocity

This method permits to identify the viscous friction

coefﬁcients Bω and the Coulomb friction coefﬁcients

Cω. The estimation of the coefﬁcient ω was only im-

plemented because inertia moment is unknown, and it

is necessary to have an initial estimate of these coef-

ﬁcients. The experimental method relies on applying

different voltages to the motors in order to move the

robot according his rotation axis - the tests were made

for positive velocities. Once reached the steady state,

the robot’s speed ω and rotation torque T can be mea-

sured. The robot speed is constant, so, the accelera-

tion is null, and as such equation 12 can be re-written

as follows:

∑

T (t) = B

·ω(t) +C

(23)

This linear equation shows that it is possible to test

different values of rotation speed and rotation torques

in multiple experiences and estimate the parameters.

4.2 Experience 2 - Null Traction Forces

This method allows for the estimation of the viscous

friction coefﬁcients (Bv, Bvn), the Coulomb friction

coefﬁcients (Cv,Cvn) and the inertia moment J. The

experimental method consists in measuring the robot

acceleration and speed when the traction forces were

null. The motor connectors were disconnected and

with a manual movement starting from a stable posi-

tion, the robot was pushed through the directions v,

vn and rotated according to his rotation axis. During

the subsequent deceleration, velocity and acceleration

were measured. Because the traction forces were null

during the deceleration equations 10, 11, and 12 can

be re-written as follows:

dv(t)

= −

·v(t) −

(24)

dvn(t)

= −

·vn(t) −

(25)

dω(t)

= −

·ω(t) −

(26)

These equations are also a linear relation and estima-

tion of all parameters is possible.

The inertia moment J is estimated using the values

obtained previously in section 4.1. To do this, equa-

tion 26 must be solved in order of J:

J = −

ω(t)

(dω(t)/dt)

·B

−

(dω(t)/dt)

·C

(27)

4.3 DC Motor Parameters

The previous electrical motor model (equation 21) in-

cludes an electrical pole and a much slower, dominant

mechanical pole - thus making inductance L value

negligible. To determinate the relevant parameters K

and R, a constant voltage is applied to the motor. Un-

der steady state condition, the motor’s current and the

robot’s angular velocity are measured. The tests are

repeated several times for the same voltage, chang-

ing the operation point of the motor, by changing the

friction on the motor axis.

In steady state, the inductance L disappears of the

equation 21, being rewritten as follows:

(t) = R ·i

(t)+ K

·ω

m j

(t) (28)

As seen in equation 29, by dividing (28) by i

(t), a

linear relation is obtained and thus estimation is pos-

sible.

(t)

= K

m j

(t)

+ R (29)

5 RESULTS

5.1 Robot Model

By combining previously mentioned equations, it is

possible to show that model equations can be rear-

ranged into a variation of the state space that can be

described as:

(dx(t)/dt) = A ·x(t) +B ·u(t) + K ·sign(x) (30)

x(t) = [v(t) vn(t) w(t)]

(31)

This formulation is interesting because it shows ex-

actly which part of the system is non non-linear.

5.1.1 Three Wheeled

Using equations on section 3.2, 13 to 17 and 28, the

equations for the three wheeled robot model are:

A =









(32)

= −

3 ·K

·l

2 ·r

·R ·M

−

= −

3 ·K

·l

2 ·r

·R ·M

−

= −

3 ·d

·K

·l

·R ·J

−

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192

B =

l ·K

r ·R





−

√

3/(2 ·M)

1/(2 ·M)

d/J

1/M

d/J

√

3/(2 ·M)

1/(2 ·M)

d/J





(33)

K =





−C





(34)

5.1.2 Four Wheeled

Using equations on section 3.2 and equations 16 to

20 and 28 we get the following equations to the four

wheeled robot model.

A =









(35)

= −

2 ·K

·l

·R ·M

−

= −

2 ·K

·l

·R ·M

−

= −

4 ·d

·K

·l

·R ·J

−

B =

l ·K

r ·R





1/M

d/J

−1/M

d/J

−1/M

d/J

1/M

d/J





(36)

K =





−C





(37)

5.2 Experimental Data for Robot Model

Experience 1 was conducted using an input signal cor-

responding to a ramped up step. This way wheel

sleeping was avoided, that is, wheel - traction prob-

lems don’t exist.

Shown in Figure 4 are the experimental plots re-

garding the 4 wheeled system. Due to space con-

straints only T vs. ω and results of the experiment 2

along the v direction are shown.

The motor model was presented earlier in equa-

tion 21.

Experimental tests to the four motors were made

to estimate the value of resistor R and the constant

. The numerical value of the torque constant K

identical to the EMF motor constant K

Figure 5 plots experimental runs regarding motor

0. Other motors follow similar behavior.

5.3 Numerical Results

Table 1 presents the experimental results to the fric-

tion coefﬁcients and inertial moment. From the exper-

imental runs from all 4 motors, the parameters found

are K

= 0.0259 V /(rad/s) and R = 3.7007 Ω .

(a) Experience 1 - T and ω.

(b) Experience 2 - Direction v.

Figure 4: Experimental Results for the four wheeled robot.

Figure 5: Experimental tests for motor 0.

Table 1: Friction coefﬁcients and inertia moment.

Parameters 3 wheels 4 wheels

J(kg ·m

) 0.015 0.016

(N/(m/s)) 0.503 0.477

(N/(m/s)) 0.516 0.600

N ·m/(rad/s) 0.011 0.011

(N) 1.906 1.873

(N) 2.042 2.219

(N ·m) 0.113 0.135

5.4 Sensitivity Analysis

To understand which model parameters have more in-

ﬂuence on the robot’s dynamics, a comparison was

made between the matrices of the models.

The model equation 30 is a sum of fractions. Ana-

lyzing the contribution of each parcel and of the vari-

able portion within each fraction, a sensitivity analy-

sis is performed, one estimated parameter at a time.

DYNAMICAL MODELS FOR OMNI-DIRECTIONAL ROBOTS WITH 3 AND 4 WHEELS

193

1. Matrix A, robot moving along v direction;

• Three wheeled robot:



3 ·K

·l

2 ·r

·R ·M



= 3.3110

/M) = K

·B

= 0.3245

• Four wheeled robot:



2 ·K

·l

·R ·M



= 3.6676

/M) = K

·B

= 0.2041

2. Matrices B and K, robot moving along v direction with

constant voltage motor equal to 6V ;

• Three wheeled robot:

√

3 ·l ·K

2 ·r ·R ·M

·12 =

·12 = 5.7570

/M) = K

·C

= 0.8728

• Four wheeled robot:



l ·K

r ·R ·M



·12 =

·12 = 5.5227

/M) = K

·C

= 0.7879

The same kind of analysis could be taken further by

analyzing other velocities (vn and ω). Conclusions

reafﬁrm that motor parameters have more inﬂuence

in the dynamics than friction coefﬁcients. This means

that it is very important to have an accurate estima-

tion of the motor parameters. Some additional ex-

periences were designed to improve accuracy. The

method used previously does not offer sufﬁcient ac-

curacy to the estimation of R. This parameter R is

not a physical parameter and includes a portion of the

non-linearity of the H bridge powering the circuit that,

in turn, feeds 3 rapidly switching phases of the brush-

less motors used. In conclusion, additional accuracy

in estimating R is needed.

5.5 Experience 3 - Parameter

Estimation Improvement

The parameter improving experience was made using

a step voltage with an initial acceleration ramp.

As seen in 5.1 the model was deﬁned by the equa-

tion 30 and we can improve the quality of the estima-

tion by using the Least Squares method. The system

model equation can be rewritten as:

y = θ

·x

+ θ

·x

+ θ

·x

(38)

Where x

= x(t), x

= u(t), x

= 1 and y = dx(t)/dt. The

parameters θ are estimated using:

θ =



·x



−1

·x

·y (39)

x = [x

(1)...x

(n) x

(1)...x

(n) x

(1)...x

(n)]

(40)

Estimated parameters can be skewed and for this rea-

son instrumental variables are used to minimize the

error, with vector of states deﬁned as

z = [x

(1)...x

(n) x

(1)...x

(n) x

(1)...x

(n)]

(41)

The parameters θ are now calculated by:

θ =



·x



−1

·z

·y (42)

Three experiments were made for each conﬁguration

of 3 and 4 wheels, along v, vn and ω. For the v and

vn experiments values C

and C

are kept from pre-

vious analysis. For the ω experiment, the value of

the R parameter used is the already improved version

from previous v and vn experimental runs of the cur-

rent section.

The numerical value of R for each motor was

estimated for each motor and then averaged to ﬁnd

R=4.3111 Ω. The results are present on followings ta-

bles. Table 2 shows values estimated by the experi-

ment mentioned in this section.

Table 2: Parameters estimated using the method 3.

Parameters 3 wheels 4 wheels

J(kg ·m

) 0.0187 0.0288

(N/(m/s)) 0.5134 0.5181

(N/(m/s)) 0.4571 0.7518

N ·m/(rad/s) 0.0150 0.0165

(N ·m) 0.0812 0.1411

The ﬁnal values for friction and inertial coefﬁ-

cients are averaged with results from all 3 experimen-

tal methods and the numerical values found are pre-

sented in Table 3.

Table 3: Parameters of dynamical models.

Parameters 3 wheels 4 wheels

d(m) 0.089

r (m) 0.0325

l 5

(V /(rad/s)) 0.0259

R (Ω) 4.3111

M(kg) 1.944 2.34

J (kg ·m

) 0.0169 0.0228

(N/(m/s)) 0.5082 0.4978

(N/(m/s)) 0.4870 0.6763

(N ·m/(rad/s)) 0.0130 0.0141

(N) 1.9068 1.8738

(N) 2.0423 2.2198

(N ·m) 0.0971 0.1385

5.6 Model Validation Experiences

The models were validated with experimental tests on

using a step voltage with an initial acceleration ramp.

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194

Due to space constraints in this document, ﬁgures 6

to 9 show plots for some of the runs only. Other runs

conﬁrm the global validity of the model as simulation

follows reality closely.

Figure 6: 3 wheel model validation - velocity along vn.

Figure 7: 3 wheel model validation - angular velocity ω.

Figure 8: 4 wheel model validation - velocity along v.

Figure 9: 4 wheel model validation - velocity along vn.

6 CONCLUSIONS

This paper presents models for mobile omni-

directional robots with 3 and 4 wheels. The derived

model is non-linear but maintains some similarities

with linear state space equations. Friction coefﬁcients

are most likely dependent on robot and wheels con-

struction and also on the weight of the robot. The

model is derived assuming no wheel slip as in most

standard robotic applications.

A prototype that can have either 3 or 4 omni-

directional wheels was used to validate the presented

model. The test ground is smooth and carpeted. Expe-

rience data was gathered by overhead camera capable

of determining position and orientation of the robot

with good accuracy.

Experiences were made to estimate the parame-

ters of the model for the prototypes. The accuracy of

the presented model is discussed and the need for ad-

ditional experiences is proved. The initial estimation

method used two experiences to ﬁnd all parameters

but a third experience is needed to improve the accu-

racy of the most important model parameters. Sen-

sitivity analysis shows that the most important model

parameters concern motor constants.

Observing estimated model parameters, the four

wheel robot has higher friction coefﬁcients in the vn

direction when compared to the v direction. This

means of course higher maximum speed for move-

ment along v axis and higher power consumption for

movements along the vn direction. This difference in

performance points to the need of mechanical suspen-

sion to even wheel pressure on the ground.

The found model was shown to be adequate for the

prototypes in the several shown experimental runs.

7 FUTURE WORK

The work presented is part of a larger study. Fu-

ture work will include further tests with different pro-

totypes including prototypes with suspension. The

model can also be enlarged to include the limits for

slippage and movement with controlled slip for the

purpose of studying traction problems. Dynamical

models estimated in this work can be used to study the

limitations of the mechanical conﬁguration and allow

for future enhancements both at controller and me-

chanical conﬁguration level. This study will enable

effective full comparison of 3 and 4 wheeled systems.

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195

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