modeling and adaptive control of an omni-mecanum-wheeled robot · drives wheels to move the robot...

Intelligent Control and Automation, 2013, 4, 166-179 http://dx.doi.org/10.4236/ica.2013.42021 Published Online May 2013 (http://www.scirp.org/journal/ica)

Modeling and Adaptive Control of an Omni-Mecanum-Wheeled Robot

Lih-Chang Lin, Hao-Yin Shih Department of Mechanical Engineering, National Chung Hsing University, Taichung, Chinese Taipei

Email: [email protected]

Received December 14, 2012; revised February 6, 2013; accepted February 13, 2013

Copyright © 2013 Lih-Chang Lin, Hao-Yin Shih. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

The complete dynamics model of a four-Mecanum-wheeled robot considering mass eccentricity and friction uncertainty is derived using the Lagrange’s equation. Then based on the dynamics model, a nonlinear stable adaptive control law is derived using the backstepping method via Lyapunov stability theory. In order to compensate for the model uncertainty, a nonlinear damping term is included in the control law, and the parameter update law with σ-modification is considered for the uncertainty estimation. Computer simulations are conducted to illustrate the suggested control approach. Keywords: Mecanum-Wheeled Mobile Robot; Dynamics Model; Backstepping Adaptive Control; Lyapunov Stability

1. Introduction

In 1973, BengtIlon invented the Mecanum wheel (also called Ilon wheel) when he was an engineer with the Swedish company Mecanum A.B. [1,2]. The Mecanum wheel is designed with passive rollers mounted around the wheel circumference at an angle of 45 degrees to the wheel plane, thus it allows for in place rotation with small ground friction and low driving torque. Usually the mobile robots using Mecanum wheels, such as an intelligent wheelchair, a forklift, or the URANUS omni-directional robot, are designed with four wheels to provide agile mobility in any direction without changing its orientation. This omni-directional capability provides greater flexibility in congested environments. Ould-Khessal [3] applied it in a robot soccer team design.

Although the benefit of omni-directionality of a standard Mecanum wheel, it has an unfortunate side effect of reducing the motor effective driving force through the rollers by projecting a portion of the motor force into a force perpendicular or at an angle to that produced by the motor. Thus, it may be inefficient when the platform travels in a straight line, especially when travels diago- nally. Diegel et al. [4] proposed an improved Mecanum wheel design with a “twist” mechanism for adjusting and locking the angle of the passive rollers to best suit the direction the platform is traveling in. Since a planar mobile robot consisting of four Mecanum wheels has only three degrees of freedom (DOF): two translational mo- tions along X- and Y-axes, and one rotation about Z-axis,

it has one redundant degree of freedom. Asama et al. [5] proposed a transmission mechanism such that four wheels can be driven by only three actuators, each of which drives wheels to move the robot for a certain DOF, respectively.

The most popular approach to the control of an omni- directional robot considers the kinematic control relying only on the kinematics model of the platform [e.g. 6,7]. The kinematic control neglects the dynamics effect and thus lowers the effective moving speed that could be obtained. Based on the Newton’s second law, Tlale and de Villiers [8] developed and verified a dynamic model for the omni-directional robot with four Mecanum wheels using the resolved force method. In their developed mobile platform, each wheel was fitted with encoder for measuring the wheel rotation/velocity, and 3D gyrometer and accelerator are also installed in the platform for measuring its orientation and translation motion. Vi- boonchaicheep et al. [9] presented a position rectification method including symptomatic and preventive rectifica- tions during the position and orientation control. Their control system is based on kinematics and joint-space linear dynamics model. Recently, Han et al. [10], Park et al. [11], and Tsai and Wu [12] proposed fuzzy control systems for Mecanum wheeled robots based on kinematic model or joint-space dynamic model.

In this work, we will propose a more complete kinematics and dynamics modeling of an omni-directional mobile robot with four Mecanum wheels considering

Copyright © 2013 SciRes. ICA

L.-C. LIN, H.-Y. SHIH 167

both the friction and load eccentricity effects. After es- tablishing the kinematics model, the 3-DOF dynamics model in the Cartesian space is derived using the La- grange’s equation, which can be used for arbitrary translational and rotational dynamic control. Then an adaptive control is constructed using the backstepping method via Lyapunov stability theory. The derived adaptive controller with uncertainty compensator has excellent three-axis arbitrary trajectory tracking performance, even the platform is encountered a not small eccentricity uncertainty. Finally, simulation results are presented to illustrate the suggested control system performance. The suggested nonlinear controller is more complex than traditional PID control, however it could have more satisfactory and faster arbitrary-trajectory tracking capability.

YR

π

2y R

α π

2

2. Kinematics and Dynamics Modeling of a Mecanum-Wheeled Mobile Robot

As shown in Figure 1, a Swedish wheel consists of a fixed standard wheel with passive rollers attached to the wheel circumference. The Mecanum wheel is a type of Swedish wheels with , where 45 is the angle between the passive-roller rotation axis and the wheel plane. Complete kinematics and dynamics modeling of an omni-directional robot with four Mecanum wheels will be considered in this section.

2.1. Kinematics of a Four-Wheeled Mecanum Robot

Consider a Mecanum wheel mounted on a mobile robot with local coordinate frame {R}: R R RX Y Z , as shown in Figure 1, where point A is its center and the other geometric parameters are defined as follows. is the angle of the vector , from the robot frame origin G to the wheel center A, with respect to the

GA

RX axis, and

G

l A

XR

α

β

γroller’s axis

v r

Rx

GA

l

Figure 1. Parameters of a mecanum wheel.

is the angle between the vector and the main wheel axis. The distance from the geometric center G to the wheel center A is l, and the main wheel’s radius is r. And and SW

r

are respectively the rotation speeds of the main wheel and the passive roller contacted with the flat floor.

Assume that the contact point between the Mecanum wheel and the floor is an instantaneous rotation center, that is, the contact is in a pure rolling condition without slipping, then the corresponding velocity of the wheel center A is

cosr

along the tangential direction as shown in Figure 1. So the wheel center A’s velocity component along the contact roller’s axis is

T.

Let the robot’s instantaneous translation velocity in terms of local frame {R} be R Rx y , and the rotation velocity about RZ axis be

, ,

. Then the wheel center A’s velocity can also be computed by summing the translational velocity vectors R Rx y

l and the relative

velocity due to the rotation velocity shown in Fig- ure 1. Thus, the wheel center A’s velocity component along the contact roller’s axis can be expressed as follows (computed via the platform’s velocity):

π π π πcos cos

2 2 2 2

cos π

l

l

π π πcos

2 2 2

πcos cos π

2

sin cos

R R

R R

R R

x y

x y

x y

sin cos

T

cos

cos R R

l

l x y

.

(1)

If no slipping occurs along the contact roller’s axis, the

same velocity can also be computed from the wheel’s rotation speed

T

cos cos cosR Rl x y r

Hence, we have the following constraint equation for a Swedish wheel:

sin (2)

Since the rotation matrix representing the orientation of the inertia frame {I} with respect to the robot frame



{R} can be expressed as

cos

sinRI R

sin 0

cos 0

0 0 1

,

where is the angle between axes and R IX X

T

R R Rx y

, and the robot’s velocity vector in terms of the robot frame {R},

ξ

can be computed as:

RR I Iξ R ξ

T

I I Ix y

,

where ξ

is the robot velocity vector

in terms of the inertia frame {I}, Equation (2) can be transformed to as follows [6]:

sin cos cos cosRI Il r

R ξ (3)

In the direction orthogonal to the contact roller’s axis,

the motion is not constrained because of the free rotation of the passive contact roller, thus we have the following velocity relation:

Tcos sin sin sin

sw sw

R R swl x y r

π π π π π π πsin sin sin

2 2 2 2 2 2 2

sin

0

R R

sw

x y l

r r

r

Thus, the above rolling condition can be transformed as:

cos sin sin sin 0RI I sw swl R r r

(4)

Consider the omni-directional robot with four Mecanum

wheels shown in Figure 2. The angles , , and i i i 1, 2,3, 4,i

of the mounted four Mecanum wheels, are

shown in Table 1. From Equation (3), we have the following four constraint equations for the centers of the four Mecanum wheels:

1 1 1 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3 3 3 3

sin cos cos cos

sin cos cos cos

sin cos cos cos

sin cos cos cos

RI I

l r

l

l r

l r

4 4 4 4 4 4 4 4 4 4 4 4

r

R ξ

(5)

YR

l

Wheel 4

XR

α b

G

l l

a

l

Wheel 1Wheel 2

Wheel 3 YI

XI O

-β

, , 1, 2,3, 4,i ir r l l i

Figure 2. A four-Mecanum-wheeled robot.

Assuming that each Mecanum wheel has equal radius and mounting distances, and by

substituting the parameters in Table 1 into Equation (5), we can obtain the inverse kinematics equation as follows:

1

2

3

4

2 2 2 2 sin π 4

2 2 2 2 sin π 42

2 2 2 2 sin π 4

2 2 2 2 sin π 4

cos sin 0

sin cos 0

0 0 1

I

I

l

lr

l

l

x

y

(6)

where 1tan .b a Define the Jacobian matrix as:



Table 1. Parameters of the Mecanum wheels.

Wheels i i i

1 1tan b a 1tan b a π 2 π 4

2 1tan b aπ 1tan b a π 2 π 4

3 1tan b aπ 1tan b a π 2 π 4

4 1tan b a2π 1tan b a π 2 π 4

sin π 4

sin π 4

sin π 4

sin π 4

2 2 2 2

2 2 2 2

2 2 2 2

2 2 2 2

cos sin 0

sin cos 0

0 0 1

l

l

l

l

The forward kinematics equation of the four-wheeled Mecanum mobile robot can be obtained as follows:

J

(7)

1

2

3

4

2 2I

I

x

y r

J

1T T

(8)

J J J

TG I I

is the pseudoinverse of J. Jwhere

2.2. Dynamics of the Mecanum Robot

Consider the four-wheeled Mecanum mobile robot shown in Figure 3, where G is the geometric center with

position vector I x yr

G

in terms of the inertia

is the mass center of the moving plat- frame {I}, and

form with relative position vector T1 2R

G G d d rR

Gv

Tcos sincos sin sin cos

sin cosIR

G I I I II

xx y x y

y

in terms of robot frame {R}. The velocity of point

G, in terms of robot frame {R} can be expressed as

v (9)

where and I Ix y - and -

are the velocity components of G along the I IX Y axes, respectively, and is the orientation angle of the platform relative to the reference

frame {I}. Hence the velocity of the mass center RGv

G in terms of robot frame {R} can be obtained as follows:

2 1cos sin sin cos

R R RG' G R G' G

I I R I I Rx y d x y d

v v k r

i j (10)

The total kinetic energy T of the mobile robot includ-

ing those of the platform and four Mecanum wheels can be computed as below:

2T 2b wi i

i i

I m r4 4

2

1 1

1

2 b G' G' i iT m I

v v

bm wim1, 2,3, 4i b

(11)

where is the mass of the platform, and is the mass of the ith wheel, ; I is the moment of

inertia of the platform about RZ axis ( parallel to RZ ) through point , and G iI is the moment of inertia of the ith wheel about its main axis; is the rotational speed of the platform, and i is the rotational speed of the ith wheel about its main axis; and is the radius of each Mecanum wheel. Since the mobile robot is assumed moving in a plane, the total potential energy

r

0V . As- sume that the four Mecanum wheels are identical and thus let wi w and i,m m ,I 1, 2,3, 4.

L T V T can be obtained as below:

I i After substituting Equation (2) and some computations, the La-

grangian

1

4

2

3

GV

'RY

'RX

G

1f

4f

3f

2f

3Wheel

4Wheel

1Wheel

2Wheel

'G

2d1d

RY

RX

IY

O X I

Figure 3. Schematic of the Mecanum robot.



2 2

2 1

2

2

os

4

I d

21 1cos sin sin c

2 21

cos sin sin cos 2 sin π2

cos sin sin cos 2 sin π 4


b b I I I

w I I

I I

I I

L I m x y d x y

m x y l

x y l

x y l

2

2

2

2

2

4

π 4

π 4

12

22

32


1 1cos sin sin cos 2 sin π

21 1

cos sin sin cos 2 sin21 1

cos sin sin cos 2 sin2

I I

I I

I I

I I

x y l

I x y lr

I x y lr

I x y lr

2

π 4 42

1 1cos sin sin cos 2 sin

2 I II x y lr

(12)

The dynamics model can then be derived using the

Lagrange’s equations:

d

d i i

L L

t q q

, 1, 2,3iF i

iq i

(13)

where is the ith generalized coordinate, and

the ith generalized force/torque. The generalized coordi-nate vector of the mobile robot can be defined as:

1 2 3 I I T Tq q q x y q . Refer to Figure 3,

where if is the contact friction force of the ith Me-canum wheel with the floor, the generalized force/torque

, 1,2,3iF i , can be derived as follows [13]:F is

4 4

11 1

sgn sgni ii i i i i i

i iI I

F r f r fx x

(14)

By Equation (6), we can obtain

31 2 41 1 1 1cos sin ; cos sin ; cos sin ; cos sin .

I I I Ix r x r x r

x r

Thus,

3 3 3 4 4 4sgn cos sin sgn cos sinr f r fr r

1 1 1 1 2 2 2

1 1sgn cos sin sgn cos sin

1 1

F r f r fr r

(15)

Similarly,

2 1 1 11

2 2 2 3 3 3

4 4 4

sgn sgn sin cos

1sgn sin cos sgn

1sgn sin cos

ii i i

i I

F r f r fy r

r f r fr r

r f

4 1

1sin cos

r

(16)



4 4

in π 4l

4

3 1 2 31

1 1 2 2 3 3 4 4

2sgn sin π

sgn sgn sgn sgn 2 s

ii i i

i

F r f lr

f f f f

(17)

After some straightforward computations, the equa-

tions of motion of the mobile robot can be expressed in matrix/vector form as:

T T1

rB Sf B τ, M q q C q q q (18)

where

1 2 3 4 1 2

1 2diag sgn sgn sgn

f f

τ f

S

T T

3 4

3 4

, ,

sgn ,

f f

113 3ij b 2, 4 ;w

Im m m

M m

r

22 12 212

13 31 1 2

23 32 1 2

2 2 2 233 1 2 2

4 ; 0;

sin cos ;

cos sin ;

8 sin 4 ;

b w

b

b

b b w

Im m m m m

r

m m m d d

m m m d d

Im m d d I m l

r

1 2

1 2

0 0 cos sin

0 0 sin cos

0 0 0

b

b

m d d

m d d

C

;

sin sin cos 2 sin π 4

sin sin cos 2 sin π 4.



l

l

l

l

ˆ

cos

cos

cos

cos

B

3. Stable Adaptive Control of a

Mecanum-Wheeled Robot

3.1. Modeling Uncertainty

In practice, the mass of the platform carrying payload, and the contact friction forces may be varied, thus we can model their uncertainty by letting , and ˆb b bm m m f f f , where and ˆ bm

T

,1 ,2 ,3 ,4

1ˆ ˆ 44

ˆ ˆ ˆ ˆ

b w

rr rr rr rr

m + m

g u u u u

f (19)

are the nominal platform mass and friction vector, respectively. Here g is the gravitational constant, and

,rr i , are the nominal rolling friction coeffi- cients of the four wheels. Substituting into Equation (18), the dynamics of the mobile robot considering uncertainty can be summarized as follows:

ˆ ,u 1, 2,3, 4i

T T1 2

1 ˆˆr

Mq B τ B Sf D D (20)

where

2 2

2

ˆ ˆdiag 4 , 4 ,

8 sin π 4 ,

b w b w

b w

I Im m m m

r r

II l m

r

M

2 2

2 2

T

1 2

T1 2

sin cos cos sin

cos sin sin cos ,

sin cos cos sin

,

, .

I

m I

I I I I

m b b b

m m

x

y

0 x y x y

m m d m d

2 2

ˆ

H

w

D H w D B S f

0

3.2. Stable Adaptive Control of a Mecanum-Wheeled Robot

3.2.1. Nominal Control Law Derivation In order to synthesize the adaptive control law, we can first neglect the uncertainty effect, i.e., let 1 2 D D in Equation (20), and consider the following nominal dynamics model:

T T1 ˆˆr

Mq B τ B Sf

T TT T T T1 2

T

I I I Ix y θ x y θ

(21)

Defining the state vector as:

z z z q q

the state equation of the nominal system can be written as

1 2

1 T T2

1 ˆˆr

z z

z M B B S

(22)

f



Based on the backstepping method, a stable nonlinear nominal control law can be obtained as follows.

First consider the 1z subsystem, 1 2.z z

1 1

Let

z v

v

1 1 d e z q

T I d dt t

1d d q v q

(23)

where 1 is a virtual input. Define the tracking error vector as

(24)

where is the desired , ,, ,d I dt x t yq

trajectory vector for the platform. Differentiating Equa-tion (24), we have

1 1 e z (25)

Considering the Lyapunov function candidate

T1 1 1 1

1

2V e K e

3 3K R

(26)

where 1 is symmetric and positive definite, and differentiating Equation (26), we have

T T1 1 1 1 1

T1 1 1

1 1

2 2

d

V

e K e e

e K v q

T1 1 1 1 1 K e e K e

1 1d v q e

T1 1 1 1 0 e K e

(27)

Thus, we can choose

(28)

and obtain

V (29)

By Equation (29), we know that 1lim 0,t e

2 2 1 e z v

t that is,

the subsystem is asymptotically stable. Further, the whole nonlinear system (22) is considered.

After introducing new error vector

(30)

we can obtain

1

2 1d

v

q e e

1 1 2 1

2 1 2 1

2 2 1 2 1

1 T T1 ˆˆ

d d

d

d

r

e z q z q v

e v q e e

e z v z q + e

M B B Sf

(31)

Then by considering the Lyapunov function candidate as

T T2 1 2 2 2 1

T

1 1

2 21

2 a

V V

e K e e K

e K e

T1 1 2 2 2

1

2e e K e

3 3K RTT T

1 2 , e e e

2V

(32)

where is symmetric and positive definite, 2

1 2diag ,a K K K and taking the time

derivative of , we have

T T T2 1 1 1 2 2 2 1 1 2 1

T 1 T 1 T2 2 2 1

1 ˆˆ ˆd

V

r

e K e e K e e K e e

e K M B M B Sf q e e

(33)

Thus, we can consider the nominal control law as

1T

1 T 12 2 1 2 1 1

ˆ

ˆˆ

n

d

r

B B B M

q M B S f e e e K K e

T T T2 1 1 1 2 2 2 22 0aV V

(34)

and obtain

e K e e K e e K e (35)

Since V t0

2 is negative definite, we know that the equilibrium point e

1 2,

is exponentially stable.

3.2.2. Adaptive Control of a Mecanum-Wheeled Robot with Uncertainty

Consider the Mecanum-wheeled robot dynamics model with uncertainty D D :

T T1 2

1 ˆˆr

Mq B B Sf D D

1 1

2 2 1 1 1

d d

d

(20)

Using the error vectors defined before in the nominal control design,

e z q q q

e z v q v q q e

(36)

the system’s error dynamics can be obtained by direct differentiating Equation (36) as below:

T1 2

1, ,t

r

e q q q B τ D D

2 1

1 T2 1

, , ,ˆˆd

t

(37)

where

e eq q

M B Sf q e e

1

0.

ˆ

qM

Consider the following Lyapunov function for adaptive control design,

T 12

1

2a m mV V w w

ˆ

(38)

where m m m w w w

aV

and is a symmetric and posi-tive definite matrix. And taking the time derivative of

, we have

T21 2

T 1

1, ,a

m m

VV t

r

q q q B D De

w w

(39)


L.-C. LIN, H.-Y. SHIH


173

By the definition of 2D in Equation (20), we know that it is bounded, i.e., 2 2

law as D 0, 2

. Since

1 m mD H w

dτ

n d is in linear parametrized form, we can in- troduce a compensating term and choose the control

(40)

with

T

2

1 1T T2

21

ˆd m m

Vq

r rV

q c

eB B B B B B H w

e

1 0c

(41)

where . Substituting Equations (40) and (41) into Equation (39), we have

T T T 12 21 2

T

2

2

1 1, ,a n d m m

m m

V VV t

r r

VV

T 1 22 2

21

T

2

22 2 2

21

ˆ2

2

m m m mVV

c

VV

VV

c

q q q B τ q B τ

D D w we e

qw D

ew w q H w H

eq

e

qe

q De

qe

2m m

V

q H we T 1

m m w w

(42)

Choose the parameter adaptation law as

T

02ˆ ˆm m m m m

Vσ

w w q H w w

e

0,

(43)

0w mwand is the best guess for the unknown parameter vector . Since where m

T T

2 2 21

2 22 2 2 2

2 2 21 1 1

2

( )

V V Vc

V VD q D

V V Vc c c

q q qe e e

qe e

q q qe e e

1c

2 2

2 22 2 1 2 1

2 21 1

V VV V

c cV V

c c

q qe e

q qe e

q qe e

Since 2V e satisfies the following inequality equa-

tion [14], we have

2

222m

aV V d w (44)

where

20

2

m mw w1d c (45)

1 22e eV e e e (46)

where we can choose 1 2

T1

2e e a e e e K e

1e

, thus

K and 2e are class- function, and

2 1eV e e , we can obtain


2

2m d

w12a eV e (47)

Let 1 e2 0d e , then 1 2e d e , and thus

11 2e ed b e . Similarly, while

2

02m d

w ,

we have 2

m

db

w w . Hence, if ebe or

mw wb

0aV

, then

(48)

Since

T T 1

21ax

m m

e m

w w

e w

2 m

1 1

2 21

2

a aV

e K e (49)

by substituting ebe and m into the right side of Equation (49), we can define

wbw

1maxd

12 1 2r e e

dV

(50)

and thus,

0 ,a rV V V1 maxe a e

Hence, we have

11 max e 0 ,e a rV V (51)

Define

ax 0 ,a rV V

6 1

1: me eR B e e (52)

and we thus have ,t t e B, , ,c

e

By properly choosing the parameters: 2 1

.

, , and

1 2 ,K K constants eb and w can be made suf- ficiently small, and the norm of the tracking error

b e is

bounded. And thus the adaptive control system is stable.

4. Results and Discussion

In this section, two computer simulation examples are given to illustrate the performance of the proposed adaptive control for the Mecanum-wheeled mobile robot.

The first considers a pure translation along a rectangu- lar desired trajectory in the I IX Y

plane with fixed

orientation 0.t d The platform’s geometric center is planned to move forward from the origin of the inertia frame along

IX axis 1 m, then leftward along IY axis 1 m, and then backward along IX axis 1m, and finally move rightward along IY

and axis 1 m and return to the

origin. The desired , ,I d I d x t y t

ˆ 0.25,u

are obtained using the cubic spline method [15] and shown as the dashed lines in Figure 4(a).

The parameters of the mobile robot are selected as follows: mb = 12 kg, I = 0.5 kg·m2, Iw = 4.0378 × 10−4 kg·m2, a = 0.2 m, b = 0.3 m, l = 0.25 m, mw = 0.313 kg, r = 0.0508 m, rr and And the adaptive controller parameters are chosen as:

29.8 m/s .g

1 diag 9000 9000 100 ,K

2 diag 2000 1800 3.25 ,K

0.002,

1 0.0001,c

0.002 ,I T0 0 0 0 ,w

and

diag 8 2 4.75 .

1 2 0.02 m,d d

In the simulation, we consider the platform having ec-centricity with

3 kg.mand the mass has varia-

tion b

T0.05 0.05 0.05 0.05 Nf

The wheels’ contact frictions are as-sumed with uncertainty

.

The simulation results are shown in Figure 4. Figure 4(a) depicts the tracking performance of the IX - and

IY -axes translation, and the orientation variation, and Figure 4(b) shows their tracking errors. We know that the tracking errors along the IX - and IY -axes are within ‒0.0063 - 0.0056 m and ‒0.0065 - 0.0062 m, re-spectively, and the orientation error is within 0.57 - 0.64

. The corresponding control torques of the four

Mecanum wheels are shown in Figure 4(c). The adaptation processes of the uncertainty compensation term’s parameters vector ˆ twm are shown in Figure 4(d). And the geometric center’s moving trajectory in the

I I plane is shown in Figure 4(e). X YConsidering the same uncertainties as the first case, a

second simulation is considered to show the pure rotational control performance. The adaptive control parameters are selected as below:

1 0.0001,c 0.002,

1 diag 9000 9000 2000 ,K

2 diag 2000 900 1000 ,K

diag 0.0001 0.03 0.03 , 0 0,w

and

diag 5 0.001 0.03 .

Simulation results of the pure rotation case are shown in Figure 5. Figure 5(a) depicts the tracking perform- ances of the IX - and IY -axes translation, and the orientation , and Figure 5(b) shows their tracking errors. We know that the undesired displacement along the -IX


L.-C. LIN, H.-Y. SHIH


175


Figure 4. Simulation results of the pure translational case. (a) and I Ix y, , responses; (b) Tracking errors; (c) Control

torques, 1 4~ N m ; (d) Compensator parameters adaptation ˆm t

y t, m

w ; (e) Tracking results of geometric center,

. I Ix t

and IY -axes can be kept within ‒0.0257 - 0.0235 m and ‒0.0143 - 0.0270 m, respectively, and the orientation error is within ‒0.0574 - 0.0362 rad. The correspond-ing control torques of the four Mecanum wheels are shown in Figure 5(c). The adaptation processes of the uncertainty compensation term’s parameters vector ˆ twm are shown in Figure 5(d). And the geometric center’s displacement in I IY

-

plane is shown in Figure 5(e). X

5. Conclusions

In this paper, Cartesian-space dynamics modeling and

stable adaptive control for a four-Mecanum-wheeled robot are considered. Based on the derived 3-DOF dynamics model considering the platform mass variation, eccentricity, and friction uncertainty, a nonlinear stable adaptive control law is derived using the backstepping method via Lyapunov stability theory. A nonlinear damping term is included in the control law to compensate for the estimation error, and the parameters adaptation law with modification is considered for the un-certainty estimation. Computer simulations are presented to illustrate the control system performance. Real



Figure 5. Simulation results of the pure rotational case. (a) and I Ix y, , responses; (b) Tracking errors, (c) Control

torques, 1 4~ N m ; (d) Compensator parameters adaptation ˆmw t

, m

; (e) Displacement of geometric center

. I Ix t y t

implementation study using the suggested control law with microcontroller deserves future consideration.

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http://dx.doi.org/10.1109/MMVIP.2008.4749608

http://dx.doi.org/10.1002/0471221139

modeling and adaptive control of an omni-mecanum-wheeled robot · drives wheels to move the robot...

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