MOTION CONTROL OF AN OMNIDIRECTIONAL MOBILE ROBOT

Xiang Li and Andreas Zell

Wilhelm-Schickard-Institute, Departmant of Computer Architecture, University of T

ubingen

Sand 1, 72076 T

ubingen, Germany

Keywords:

Mobile robots and autonomous systems, system identiﬁcation, actuator saturation, path following control.

Abstract:

This paper focuses on the motion control problem of an omnidirectional mobile robot. A new control method

based on the inverse input-output linearized kinematic model is proposed. As the actuator saturation and

actuator dynamics have important impacts on the robot performance, this control law takes into account these

two aspects and guarantees the stability of the closed-loop control system. Real-world experiments with an

omnidirectional middle-size RoboCup robot veriﬁes the performance of this proposed control algorithm.

1 INTRODUCTION

Recently, omnidirectional wheeled robots have re-

ceived more attention in the mobile robots applica-

tions, because the omnidirecitional robots “have full

mobility in the plane, which means that they can move

at each instant in any direction without any reorien-

tation ”(Campion et al., 1996). Unlike the nonholo-

nomic robots, such as car-like robots, having to rotate

before implementing any desired translation velocity,

omnidirecitonal robots have higher maneuverability

and are widely used in dynamic environment appli-

cations, for example, in the middle-size league of the

annual RoboCup competition.

Most motion control methods of mobile robots are

based on robots dynamic models (Watanabe, 1998;

Liu et al., 2003; Purwin and Andrea, 2006; Tsai et al.,

2006) or robots kinematic models (Muir and Neu-

man, 1990; Terashima et al., 2004; Rojas and F

orster,

2006). A dynamic model directly describes the rela-

tionship between the forces exerted by the wheels and

the robot movement, with the applied voltage of each

wheel the input and the robot movement in terms of

linear and angular accelerations as output. But the dy-

namic variations caused by the changes in the robot’s

inertia moment and perturbations from the mechanic

components (Scolari Conceic¸

ao et al., 2005) make the

controller design more complex. With the assump-

tion that no slippage of wheels occurs, sensors have

high accuracy and ground is planar enough, kinematic

models are widely used in designing robots behav-

iors because of the simpler structures. As the inputs

of kinematic models are robot wheels velocities, and

outputs are the robot linear and angular velocities,

the actuator dynamics of the robot are assumed fast

enough to be ignored, which means the desired wheel

velocities can be achieved immediately. However, the

actuator dynamics limit and even degrade the robot’s

performance in real situations.

Another important aspect of robot control in prac-

tice is actuator saturation. Because the commanding

motor speeds of the robot’s wheels are bounded by

the saturation limits, the actuator saturation can af-

fect the robot’s performance, even make robots mo-

tion become unstable (Indiveri et al., 2006; Scolari

Conceic¸

ao et al., 2006).

This paper presents a motion control method for

an omnidirectional robot, based on the inverse input-

output linearization of the kinematic model. It takes

into account not only the identiﬁed actuator dynamics

but also the actuator saturation in designing a con-

troller, and guarantees the stability of the closed-loop

control system.

The remainder of this paper introduces the

kinematic model of an omnidirectional middle-size

Robocup robot in section 2; Path following and ori-

entation tracking problems are solved based on the

inverse input-output linearized kinematic model in

125

Li X. and Zell A. (2007).

MOTION CONTROL OF AN OMNIDIRECTIONAL MOBILE ROBOT.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 125-132

DOI: 10.5220/0001644801250132

 SciTePress

section 3, where the actuator saturation is also ana-

lyzed; section 4 presents the identiﬁcation of actua-

tor dynamics and their inﬂuence on the control per-

formance; Finally the experiment results and conclu-

sions are discussed in sections 5 and 6, respectively.

2 ROBOT KINEMATIC MODEL

The mobile robot used in our case is an omnidirec-

tional robot, whose base is shown in ﬁgure 1. It has

three Swedish wheels mounted symmetrically with

120 degrees from each other. Each wheel is driven

by a DC motor and has a same distance L from its

center to the robot’s center of mass R.

Figure 1: Kinematics diagram of the base of an omnidirec-

tional robot.

Besides the ﬁxed world coordinate system

], a mobile robot ﬁxed frame [X

] is de-

ﬁned, which is parallel to the ﬂoor and whose origin

locates at R. θ denotes the robot orientation, which is

the direction angle of the axis X

in the world coor-

dinate system. α and ϕ denote the direction of the

robot translation velocity v

observed in the world

and robot coordinate system, respectively. The kine-

matic model with respect to the robot coordinate sys-

tem is given by :

v =





√

3/3 −

√

3/3 0

1/3 1/3 −2/3

1/(3L) 1/(3L) 1/(3L)





q, (1)

where v = [ ˙x

˙y

ω]

is the vector of robot veloc-

ities observed in the robot coordinate system; ˙x

and

˙y

are the robot translation velocities; ω is the robot

rotation velocity.

q is the vector of wheel velocities

[ ˙q

˙q

]

, and ˙q

(i = 1,2,3) is the i-th wheel’s ve-

locity, which is equal to the wheel’s radius multiplied

by the wheel’s angular velocity.

Introducing the transformation matrix from the

robot coordinate system to the world coordinate sys-

tem as



cosθ −sinθ

sinθ cosθ



, (2)

the kinematic model with respect to the world co-

ordinate system is deduced as:

x =





cos(θ+ δ) −

cos(θ−δ)

sinθ

sin(θ+ δ) −

sin(θ−δ) −

cosθ





(3)

where

x = [ ˙x

˙y

θ]

is the robot’s velocity vector

with respect to the world coordinate system; ˙x

and ˙y

are the robot translation velocities;

θ is the robot ro-

tation velocity; δ refers to the wheel orientation in the

robot coordinate system and is equal to 30 degrees.

It is important to notice that the transformation

matrix in model 1 is full rank, which denotes that

the translation and rotation of the robot are decou-

pled, and guarantees the separate control of these two

movements.

For the high level control laws without consider-

ing the wheel velocities, the kinematic model

x = Gv (4)

is used in our control method, where the transforma-

tion matrix G is equal to [

0 ; 0 1]. Because G is

full rank, the characteristics of decoupled movement

is also kept.

3 INVERSE INPUT-OUTPUT

LINEARIZATION BASED

CONTROL

The trigonometric functions of angle θ in the transfor-

mation matrix G determine the nonlinearities of the

kinematic model 4. Since the matrix G is full rank,

this nonlinear model can be exactly linearized by in-

troducing a simple compensator C = G

−1

. The lin-

earized system becomes

x = u with a new input vector

u = [u

]

Figure 2: Linearized system by the component C.

This linear system shown in ﬁgure 2 is completely

decoupled and allows the controlling of the robot’s

translation and rotation in a separate way. When a

controller K is designed based on this simple linear

system, the controller of the original system is gener-

ated as CK. The overall control loop, which consists

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126

Figure 3: Closed-loop control system.

of the nonlinear system, the compensator and the con-

troller, is shown in ﬁgure 3,

where x denotes the robot state vector [x

θ]

and x

is the desired state vector; x

and y

are robot

position observed in the world coordinate system.

Based on this input-output linearized system, path

following and orientation tracking problems are ana-

lyzed with respect to the robot translation and rotation

control in the following subsections. The inﬂuence of

actuator saturation is also accounted to keep the de-

coupling between the translation and rotation move-

ments.

3.1 Path Following Control

As one high-level control problem, path following is

chosen in our case to deal with the robot translation

control. The path following problem is illustrated in

ﬁgure 4. P denotes the given path. Point Q is the

orthogonal project of R on the path P. The path coor-

dinate system x

moves along the path P and the

coordinate axes x

and x

direct the tangent and nor-

mal directions at point Q, respectively. θ

is the path

tangent direction at point Q.

Figure 4: Illustration of the path following problem.

Based on the above deﬁnitions, the path follow-

ing problem is to ﬁnd proper control values of the

robot translation velocity v

and angular velocity

such that the deviation distance x

and angular error

= α−θ

tend to zero.

To solve this problem, a Lyapunov candidate func-

tion

V =

(5)

can be considered, where K

and K

are positive con-

stants. The time derivation of V results in

V = K

˙x

+ K

. (6)

Mojaev (Mojaev and Zell, 2004) presents a sim-

ple control law based on the deviation x

, where R

is controlled to move along an exponential curve and

to converge to the axis X

. The exponential curve is

expressed as

= x

exp(−kx

), (7)

where x

is the initial deviation and the positive con-

stant k determines the convergence speed of the devi-

ation. Differentiating (7) with respect to x

, we get the

tangent direction of the exponential curve as

= arctan(

) = arctan(−kx

). (8)

Therefore, for a non-zero constant desired velocity v

the translation velocity of robot in the coordinate sys-

tem x

results in

˙x

= v

sin

, (9)

˙x

= v

cos

. (10)

Substituting the time derivative of

into (6), we

get

V = K

˙x

+ kK

arctan(−kx

)

−˙x

1+ (kx

)

< 0,

(11)

because x

˙x

= x

sin(arctan(−kx

)) < 0 and

˙x

arctan(kx

) < 0. This solution of

V guarantees the

global stability of the equilibrium at x

= 0,

which means this control law solves the path follow-

ing problem.

Transforming the robot velocity into the world co-

ordinate system, we get the control values of the lin-

earized system as

= v

cosα, (12)

= v

sinα, (13)

where α =

+ θ

The input of controller 12 and 13 is the devia-

tion distance between point R and the given path,

which normally can be directly obtained by the sen-

sors on the robot. Moreover, the deviation converges

smoothly to zero with the speed controlled by param-

eter k, which can be chosen according to the perfor-

mance requirement.

3.2 Orientation Tracking

Unlike a car-like wheeled robot, the orientation of an

omnidirectional robot can be different from the di-

rection of the robot translation velocity by any angle

MOTION CONTROL OF AN OMNIDIRECTIONAL MOBILE ROBOT

127

ϕ. This relationship is denoted as α = θ + ϕ. That

means the robot orientation can track any angle when

the robot is following a given path. Based on the lin-

earized model, the orientation tracking task is to ﬁnd

a suitable u

, which is equal to the robot rotation ve-

locity ω, such that

lim

t→∞

(θ

(t) −θ(t)) = 0, (14)

where θ

(t) is the desired orientation.

As the system between input variable u

and out-

put variable θ is an integrator, a commonly used PD

controller can be designed to fulﬁll the orientation

tracking task.

3.3 Actuator Saturation

Based on the inverse input-output linearization, the

translation and rotation of an omnidirectional robot

can be easily achieved in a separate way. This lin-

earization is with respect to the input-output relation-

ship, which requires the internal parts having sufﬁ-

cient capability to achieve the desired inputs. How-

ever, the power of the robot’s motors is bounded and

the actuators will saturate when the commanding ve-

locities are too large. The presence of actuator satura-

tion can inﬂuence the decoupling between robot trans-

lation velocity and rotation velocity, such that the sys-

tem performance and stability is severely impacted.

Therefore, it is necessary to deal with the actuator sat-

uration in the controller design.

For our omnidirectional robot, the maximal veloc-

ity of each wheel is limited by ˙q

, namely

˙q

≤ ˙q

Substituting the above control values from equations

(12) (13) and u

into the inverse kinematic models (2)

and (1), the wheel velocities are computed as:





˙q









cos(α−θ−δ)+ Lu

−v

cos(α−θ+ δ)+ Lu

sin(θ−α) + Lu





, (15)

To achieve orientation tracking based on the above

path following control, the desired translation veloc-

ity’ magnitude V

is assumed to be less than ˙q

. Sub-

stituting ˙q

into (15), the lower and upper boundary

of each wheel’s velocity (L

and U

) can be calcu-

lated from the following three inequalities,

cos(α−θ−δ)+ Lu

< ˙q

−v

cos(α−θ+ δ)+ Lu

< ˙q

sin(θ−α) + Lu

< ˙q

(16)

Then the dynamic boundary values of u

are com-

puted as

= max(l

)

= min(u

(17)

where l

and u

are the low and up boundary.

Considering the saturation function







, if x

> u

, if l

≤ x

≤ u

, if x

< l

(18)

and its gain characteristics illustrated in ﬁgure 5, we

can take the saturation function as a dynamic gain

block k

, which has maximum value one and con-

verges to zero when the input saturates. Then the

closed-loop system of controlling the robot orienta-

tion is as shown in ﬁgure 6, in which a PD controller

is used to control the robot orientation converging to

the ideal θ

ω = k

+ k

˙e

), (19)

where k

and k

are the proportional and deriva-

tive gains, respectively. It can be obtained that the

closed-loop has only one pole

−k

1+k

and one zero

−1/k

. Therefore, when k

is negative and k

is cho-

sen such that the pole is negative too, the stability of

the closed-loop system can be guaranteed whenever

decreases.

Figure 5: Saturation function and its gain characteristics.

Figure 6: Closed-loop of robot orientation control.

4 ACTUATOR DYNAMICS

The results in the last section are only practical when

we assume that the low level actuator dynamics is

faster than the kinematics, or the delay of actuator dy-

namics can be ignored. It is necessary to analyze the

actuator dynamics and take it into account when de-

signing a controller. In the following subsections, the

actuator dynamics is identiﬁed based on the observed

input-output data, and its inﬂuence on the robot mo-

tion control is presented.

4.1 Actuator Dynamics Identiﬁcation

The system identiﬁcation problem is to estimate a

model based on the observed input-output data such

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128

that a performance criterion is minimized. Because

the full rank transformation matrix in the low level

dynamics model (1) denotes the outputs ˙x

, ˙y

and ω

are not relevant, we identify the actuator models for

these three values. The inputs of the actuator models

are required velocity values(˙x

, ˙y

and ω

), and the

outputs are corresponding measured values. As one

commonly used parametric model, ARMAX is cho-

sen as the identiﬁed model, which has the following

structure

A(z)y(t) = B(z)u(t −n

) +C(z)e(t), (20)

A(z) = 1+ a

−1

+ ... + a

−n

, (21)

B(z) = 1+ b

−1

+ ... + b

−n

, (22)

C(z) = 1+ c

−1

+ ... + c

−n

. (23)

denotes the delay from input u(t) to output y(t).

e(t) is white noise. z is the shift operator resulting

in q

−1

u(t) = u(t −1). n

, n

and n

are the orders

of polynomials A(z), B(z) and C(z), respectively. To

choose the optimal parameters of this model, we use

the prediction error method, which is to ﬁnd the op-

timal n

and parameters of A(z), B(z) and C(z) such

that the prediction error E is minimized, namely

[A(z),B(z),C(z),nk]

opt

= argmin

∑

t=1

(24)

E = y

(t) −A

−1

(z)(B(z)u(t −nk) +C(z)e(t)), (25)

where y

(t) denotes the measured output data.

The system identiﬁcation toolbox of Matlab has

been used to identify the actuator dynamics model.

Figures 7 8 and 9 show the optimal parameters and

comparison between models outputs and measured

outputs with respect to the actual inputs.

0 50 100 150 200 250 300

0.5

1.5

2.5

Data number

X−component velocity (m/s)

Identified ARMAX model (na =2, nb =1, nc = 1, nk = 1)

model outputs

measured outputs

Figure 7: Identiﬁed model for ˙x

To coincide with the robot’s continuous model, the

identiﬁed models are transformed from discrete ones

0 50 100 150 200 250 300

−2

−1.5

−1

−0.5

0.5

Data number

Y−component velocity (m/s)

Identified ARMAX model (na =3, nb =3, nc = 2, nk = 1)

model outputs

measured outputs

Figure 8: Identiﬁed model for ˙y

0 100 200 300 400 500 600 700

−1.5

−1

−0.5

0.5

1.5

2.5

Data number

Rotation velocity (rad/s)

Identified ARMAX model (na =2, nb =2, nc = 2, nk = 1)

model outputs

measured outputs

Figure 9: Identiﬁed model for ω.

into continuous ones using ’zoh’(zero-order hold)

method,

˙x

8.7948(s+ 58.47)

(s+ 73.66)(s+ 6.897)

˙y

, (26)

˙y

2.4525(s+ 48.83)(s+ 6.185)

(s+ 28.45)(s

+ 6.837s+ 25.97)

˙y

(27)

ω =

1.667(s+ 45.37)

+ 6.759s+ 76.11)

. (28)

4.2 Actuator Inﬂuence

With consideration of the actuator, thewhole structure

of the control system is shown in ﬁgure 10,

Figure 10: Closed-loop control system including actuator

dynamics.

where V

= [

˙y

] is the commanding robot

velocity vector with respect to the robot coordinate

system. Because the poles of the actuators dynam-

ics (26) and (27) have negative real parts, these two

MOTION CONTROL OF AN OMNIDIRECTIONAL MOBILE ROBOT

129

systems are stable. That means there exits a ﬁnite

short time t

∗

, after which the real velocities

and

can converge to the desired ones

and

, and

the inputs u

and u

begin to take effect. Therefore,

the above path following law can also guarantee the

robot approach to the reference path, although during

∗

the deviation distance x

and angular error

may

increase.

In the orientation tracking control, as the dynamic

system (28) adds another two poles to the closed-loop

system, shown in ﬁgure 11, the controller parameters

decided in the above section may result the system

losing the stability.

Figure 11: Closed-loop of robot orientation control includ-

ing actuator dynamics.

By setting the positions of poles and zeros of the

closed-loop system with the locus technique, we ob-

tain that the conditions k

> 0 and k

> 0.0515 can

guarantee the system stability, even when the actua-

tors saturate. Figure 12 shows the root locus of an

open-loop system in the critical situation with k

0.0515, where all the poles of closed-loop system

locate in the left-half plane whatever positive value

is. Otherwise, when k

is less than 0.0515,

the root locus may cross the imaginary axis, and the

poles of closes-loop system may move to the right-

half plane when k

goes to zero.

Figure 12: Root locus of open-loop model.

5 EXPERIMENT

The control algorithm discussed above has been

tested in our robot laboratory having a half-ﬁeld of the

RoboCup middle size league. The omnidirectional

robot used is shown in ﬁgure 13.

Figure 13: The real omnidirectional robot.

a AVT Marlin F-046C color camera with a reso-

lution of 780×580 is assembled pointing up towards

a hyperbolic mirror, which is mounted on the top of

the omnidirectional robot, such that a complete sur-

rounding map of the robot can be captured. A self-

localization algorithm described in (Heinemann et al.,

2004) based on the 50 Hz output signal of the cam-

era gets the robot’s position in the play ﬁeld in real

time. The wheels are driven by three 60W Maxon DC

motors and the maximum wheel velocity is 1.9m/s.

Three wheel encoders measure the real wheel veloci-

ties, which are steered by three PID controllers.

An eight-shaped path is adopted as the reference

path, whose geometrical symmetry and sharp changes

in curvature make the test challenging. With a scale

variable s, the chosen eight-shaped path is as

= 1.8sin(2s)

= 1.2sin(s),

(29)

The robot was controlled to follow the eight-

shaped path with a constant translation velocity v

1m/s, and the parameters of our control algorithm

were chosen as k = 2.5, k

= 4.15, k

= 3. The ﬁrst

experiment selected the path tangent direction θ

the desired robot orientation. Figures 14, 15, 16 and

17 show us that the proposed control method steers

the robot center R converging to the given path and

the robot orientation tracking the desired ones with

acceptable errors, where the actuator saturation did

not appear. In order to check the inﬂuence of the ac-

tuator saturation, the second experiment selected the

desired robot orientation as

= θ

+ 0.9c

, (30)

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130

where c

is the path curvature at point P. The results

illustrated in ﬁgures 18, 19, 20 and 21 show us that

the robot center R converges to the given path, even

though the wheels velocities come in the saturation

when the path turns sharply.

Figure 14: Reference path and robot path.

Figure 15: Distance error.

Figure 16: Orientation error.

6 CONCLUSION

In this paper a new motion control method for an

omnidirectional robot is presented. This approach

Figure 17: Real wheel velocities.

Figure 18: Reference path and robot path.

Figure 19: Distance error.

is based on the inverse input-output linearized robot

kinematic model, which completely decoupled the

robot translation and rotation. The robot translation

is steered to follow a reference path, and the robot

rotation is controlled to track the desired orientation.

Because the actuator dynamics and saturation can

greatly affect the robot performance, they are taken

into account when designing the controller. With the

Lyapunov stability theory, the global stability of the

path following control law has been proven. The lo-

cus technique is used to analyze and choose the suit-

able parameters of the PD controller, such that the

robot orientation can converge to the desired one even

MOTION CONTROL OF AN OMNIDIRECTIONAL MOBILE ROBOT

131

Figure 20: Orientation error.

Figure 21: Real wheel velocities.

when the wheels velocities saturate.

In real-world experiments, the robot was con-

trolled to follow an eight-shaped curve with a con-

stant translation velocity of 1m/s, and to track sharp

changing orientations. The result shows the effective-

ness of the proposed control method in the case of

both actuator saturation and non-saturation.

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