Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

56, 4, pp. 1193-1204, Warsaw 2018
DOI: 10.15632/jtam-pl.56.4.1193

ROBUST NEURAL NETWORKS CONTROL OF OMNI-MECANUM

WHEELED ROBOT WITH HAMILTON-JACOBI INEQUALITY

Zenon Hendzel

Rzeszow University of Technology, Faculty of Mechanical Engineering and Aeronautics, Rzeszów, Poland

e-mail: zenhen@prz.edu.pl

This paper presents a novel approach to the problem of controlling mechanical objects of
unspecified description, considering variable operating conditions. The controlled object is a
mobile robotwithmecanumwheels (MRK M).To solve the control task, taking into account
compensation for nonlinearity and the object variable operating conditions, the Lyapunov
stability theory is applied, including theHamilton-Jacobi (HJ) inequality. A neural network
with basic sigmoid functions is used to compensate for the nonlinearity and variable ope-
rating conditions of the robot. A simulation example is provided in order to evaluate the
analytical considerations. The simulation results obtained confirmed high accuracy of the
predicted robot motion in variable operating conditions.

Keywords: mechatronics, mobile robot, mecanumwheels, Hamilton-Jacobi inequality

1. Introduction

The motivation for analytical considerations of the behaviour of a mobile robot with Swedish
wheels (Canudas deWit et al., 1996), referred to in the literature asmecanumwheels (Becker et
al., 2014; Abdelrahman et al., 2014; Hendzel and Rykała, 2015) comes from the fact that there
is a relatively small amount of literature in this area, especially with regards to the impact of
resistance to motion and variable operating conditions on the quality of motion and its control
in real time. Most often, control synthesis for these objects is carried out based on kinematics
equations (Siegwart et al., 2011; Tai et al., 2011). Kinematics control does not take into account
nonlinearity of the robot and variable operating conditions. The literature offers solutions which
take into consideration dynamic equations of motion (Han et al., 2009; Park et al., 2010; Tsai
et al., 2010) typically in a simplified form not including, for example, resistance to motion.
The work of (Lin and Shih, 2013) includes control synthesis for a mobile robot with mecanum
wheels based on the Lagrange equations using an adaptive control algorithm. The current paper
presents a new approach to the problem of controlling a mobile robot with mecanum wheels
as a mechanical object of unspecified description, considering variable operating conditions. To
solve the mobile robot control task, the Lyapunov stability theory is applied, including the HJ
inequality. The method of synthesis of robust neural networks control proposed in this paper
aims at extending the existing solutionswith a particular emphasis on its application in the field
of control methods for intelligent mobile wheeled robots. In Section 2, kinematic and dynamic
equations forMRK Mmotion are formulated. Section 3provides a theoretical basis of dissipation
of nonlinear systems andL2 stability. In Section 4, the assumed structure of the neural network
is discussed, and its description is given. The synthesis of robust neural networks control of
MRK M, including the HJ inequality, is provided in Section 5. Section 6 contains a simulation
example and simulation results of the adopted solution.



1194 Z. Hendzel

2. Mobile wheeled robot

When describingmotion of mobile wheeled robots, we are interested in the issues of kinematics
and dynamics of such systems (Żylski, 1996). From the perspective of controlling these objects
in the areas of both kinematics and dynamics, we solve reverse tasks. To describe kinematics
of a mobile robot with mecanum wheels, the model shown in Fig. 1 is adopted. In this figure,
x, y and z are the axes of the fixed system. The basic units of this model are: frame 5 and the
driving units. The driving unit consists of wheels 1,2,3,4, which aremounted on semi-axles and
set in motion by a drive module associated with a given wheel. The points A1,A2,A3,A4 lie
in their centres of symmetry. These wheels rotate around their own axes which do not change
their position relative to the frame. The rollers are placed on the periphery of the wheels, and
they are set at an angle of α= π/4 radians, relative to the axis of the driving wheel. Figure 1c
shows appropriate geometric dimensions and characteristic points of the system. Point S is the
centre of mass of the frame, pointH is the point lying on the axisA1A2 halfway between these
points. Point B, like point H, is the point belonging to the frame. The angle β is the angle of
temporary rotation of the frame. Figures 1a and 1b exemplify the angular velocity vectors of
wheel 2, ω2 and roller ωr2, respectively. To describe kinematics of any point in the system, it
is useful to give the kinematics equation. Knowing the geometry of the system and applying
classical methods used inmechanics, the description of the kinematics of the analysed system is
as follows

ẋscos(β−α)+ ẏs sin(β−α)− β̇(lcosα+ l1 sinα)=ω1(R+r)cosα

ẋscos(β+α)+ ẏs sin(β+α)+ β̇(lcosα+ l1 sinα)=ω2(R+r)cosα

ẋscos(β+α)+ ẏs sin(β+α)− β̇(lcosα+ l1 sinα)=ω3(R+r)cosα

ẋscos(β−α)+ ẏs sin(β−α)+ β̇(lcosα+ l1 sinα)=ω4(R+r)cosα

(2.1)

The adopted description will enable determination of the reverse kinematics task, which will be
the set motion trajectory for the S point of the robot in the control system.
When describing the dynamics of complex systems such asmobile wheeled robots, especially

robots with mecanum wheels, Lagrange’s equations with multipliers or Maggi’s equations can
be used (Giergiel et al., 2002). To determine the dynamic equations ofMRK Mmotion,Maggi’s
equations are used (Żylski, 1996) with an additional disturbance element, which are convenient
for synthesis of control. In the analysed case, take the general form

M(q)q̈+C(q, q̇)q̇+F(ω)+τd(t)=u (2.2)

where the vector of disturbances meets the restriction ‖τd(t)‖ < b, b = const > 0, matrices
M,C and vectors F, u take the following form

M(q) =



a1 sinβ+a2cos(β−α) a2 sin(β−α)−a1cosβ −a3

a2cos(β+α)−a8 sin(β−α) a2 sin(β+α)−a8cos(β−α) −a4
a1cosβ+a2cos(β−α) a2 sin(β−α)+a1 sin(β−α) a3




C(q, q̇)=



−a2 sin(β−α)β̇ a2cos(β−α)β̇ 0
−a2 sin(β+α)β̇ a2cos(β+α)β̇ 0
−a2 sin(β−α)β̇ a2 sin(β−α)β̇ 0




F(ω)=



a5 sgnω1
a6 sgnω3
a7 sgnω4


 q=



xs
ys
β


 u=



M1
M3
M4


 τd(t)=



τd1

τd3

τd4




(2.3)



Robust neural networks control of omni-mecanum wheeled robot... 1195

Fig. 1. MRK Mmodel

Since the analysed robot has 3 degrees of freedom and four drive modules, it is an object
of an “over actuated” type, the missing moment M2 is determined on the basis of the power
balance of the drive systems

M2 =M3+M4−M1 (2.4)

The vector a= [a1, . . . ,a8]T contains parameters resulting from the geometry, mass distribution
and the robot resistance to motion. We further assume that the trajectory of the robot motion
qd(t) ∈ R

3 and its derivatives are known. Equation (2.2) allows solving the direct and inverse
dynamics tasks. The direct task will be used in simulation of the controlled object, and the
inverse task in the feedback systemwill be used to determine the values of the drivingmoments
of the mobile robot wheels, i.e. the control vector.

3. Continuous dissipative systems and L2 gain

From the perspective of control theory, the dissipation andL2 gain properties of dynamic non-
linear systems are important in the control synthesis of these systems in the context of the



1196 Z. Hendzel

input-output relationship. Further considerations will focus on finite energy signals. This ap-
proach to control synthesis is important when considering the effects of disturbances in closed
control systems.

Definition2.1 (Abu-Khalaf andLewis, 2006; vanderSchaft, 1992, 2000).For agiven continuous
function f(t) : [0,∞)→ Rn, its normLp, ‖f(t)‖Lp is defined as

‖f(t)‖Lp =
( ∞∫

0

‖f(t)‖pp dt
)1/p

(3.1)

and for p=2we have

‖f(t)‖L2 =
( ∞∫

0

‖f(t)‖2 dt
)1/2

(3.2)

In order to definedissipation of the system, let us consider the description of the dynamic system
in the form

ẋo = fx(xo)+b(xo)d(t) z(t)=g(xo) (3.3)

where xo(t)∈ Rn is the state vector, fx(xo)∈ Rn, b(xo)∈ Rn×m, g(xo)∈ Rp and d(t)∈ Rm is
the disturbances signal, and z(t)∈ Rp is the system output signal, which can be interpreted as
a tracking control error, additionally fx(0)=0 and xo =0 is the system equilibrium point.

Definition 2.2 (Abu-Khalaf, Lewis, 2006; Fariwata et al., 2000; Nash, 1951; Slotine and Li,
1991; van der Schaft, 1992). The dynamic system described by equation (3.3) with the supplied
rate w(d,z) is dissipative if there exists a function V (xo)­ 0 which is interpreted as a storage
function, such that there is a dissipation inequality

V (xo(t1))−V (xo(t0))¬

t1∫

t0

w(d(t),z(t)) dt (3.4)

Generally, it can be stated that the dynamic system is dissipative if it loses cumulative energy
during operation until the state of the system reaches the equilibrium point.

Definition 2.3 (Abu-Khalaf and Lewis, 2006; Basar and Bernard, 1995; Lewis et al., 2012).
Dynamic system (3.3) hasL2 gain less than or equal to γ, for γ ­ 0, if the following inequality
is true

‖z(t)‖L2 ¬ γ‖d(t)‖L2 (3.5)

This means that the analysed system has L2 < γ gain if there exists 0 ¬ γ̂ < γ such that
equation (3.5) is true for γ̂. As has been demonstrated (van der Schaft, 2000), if the level of
energy supplied to the system is defined as w(d(t),z(t)) = γ2‖d(t)‖2 −‖z(t)‖2 and the system
is dissipative, then the dynamic system has gain L2, i.e. it is stable. The determination of
dissipativity of the system and, hence, the stability of the analysed nonlinearL2 system requires
demonstrating that the function V (xo) is bounded from below. Demonstrating this condition
requires solving the optimization problem, which comes down to solving the HJ equation or HJ
inequality (Basar and Bernard, 1995), written as

V̇ (xo)=
∂V (xo)
∂xo

[fx(xo)+b(xo)d(t)]¬
1
2
(γ2‖d(t)‖2 −‖z(t)‖2) (3.6)



Robust neural networks control of omni-mecanum wheeled robot... 1197

Satisfying this inequality leads to the solution V (xo)­ 0 for everyd(t)∈ Rm and z(t)=g(xo).
Determining the system gain L2 as

J =sup
d6=0

‖z(t)‖L2
‖d(t)‖L2

(3.7)

this equation can be interpreted as an indicator of the system resistance to interference. The
smaller it is, the more the designed control system is robust to disturbance, provided that
inequality (3.6) is satisfied, i.e. for

V̇ (xo)¬
1
2
(γ2‖d(t)‖2−‖z(t)‖2) (3.8)

J ¬ γ occurs. To solve the robust neural network control of MRK Mmotion, we will apply HJ
inequality (3.8) with the evaluation of the control by (3.7).

4. Neural network structure

Due to nonlinearity as well as complexity of the structure of dynamic equations of motion of
mobile wheeled robots and the need to include a mathematical model in motion control algo-
rithms for these objects, the application of artificial neural networks theory is an alternative and
attractive approach to solve these tasks. From the perspective of control theory, the possibilities
of approximation of nonlinear mappings are the most important properties of neural networks.
These properties will be used in synthesis of neural networks control for the implementation of
MRK Mtrackingmotion (Hendzel, 2007). In the real-time control, linear networks are used due
to weights, with the structure shown in Fig. 2.

Fig. 2. Neural network structure

The input layer weight matrix Vs is a constant matrix generated randomly. The network
input-outputmapping from Fig. 2 has the form (Lewis et al., 1999)

yi =
m∑

j=1

[
wijS

(
h∑

k=1

vjkxk+vj0

)
+wi0

]
i=1,2, . . . ,N (4.1)

Assuming the element of the input vector to be x0 ≡ 1 and the vector of threshold values
[v10,v20, . . . ,vm0]T as the first column of the matrix VTs , we get

y=WTS(VTsx)=W
T
ϕx(x) (4.2)



1198 Z. Hendzel

where x=VTsx, S= [1,S1(·),S2(·), . . . ,Sm(·)]
T is a vector describing neuron functions, whose

first element is equal to 1, and the vector [w10,w20, . . . ,wN0]T is the first column of the ma-
trix WT in order to account for the threshold value of linear neurons in the network output
layer.

5. Synthesis of robust neural networks control of MRK M motion

The purpose of the synthesis of the control algorithm is to determine the control rule and the
rule for adaptation of network weights, for which the trajectory of the selected point S of the
robot, Fig. 1, will coincide with the set trajectory (Hendzel and Szuster, 2012, 2015; Szuster
andHendzel, 2018), despite the occurrence of variable operating conditions of the robot. To this
end, the tracking error e∈ R3, the generalized error s and the auxiliary vector v are defined as

e=qd−q s= ė+Λe v= q̇d+Λe (5.1)

whereΛ is a diagonal matrix, positively defined. Then equation (2.2) can be transformed into

M(q)ṡ=−u−C(q, q̇)s+Mv̇+C(q, q̇)v+F(ω)+τd(t) (5.2)

If we define the nonlinear function

f(xr)=M(q)v̇+C(q, q̇)v+F(ω) (5.3)

where xr = [vT, v̇T, q̇T,qT,ωT]T, then equation (5.2) will be written as

M(q)ṡ=−u−C(q, q̇)s+ f(xr)+τd(t) (5.4)

In practice, an approximation of the control compensating for the robot nonlinearity f(xr) is
applied. For the approximation of the nonlinearity, the neural network described in Section 4 is
used, assuming x=xr, y= f(xr). The nonlinear function f(xr) is written as

f(xr)=W
T
ϕx(x)+ε (5.5)

where ε is the approximation error. The estimate the function f(xr) is written as

f̂(xr)=Ŵ
T
ϕx(x) (5.6)

where Ŵ is an estimate of weight of the ideal neural network. In addition, it is assumed that
ideal network weights are fixed,W= const. Using equation (5.6), we adopt the control rule in
the following form

u=ŴTϕx(x)+
1
2

(
1+
1
γ2

)
s (5.7)

It should be noted that the second component of equation (5.7) has an interpretation of the PD
type controller since the second component of (5.7) can be written as

KDs=KDė+KDΛe (5.8)

whereKD =0.5(1+1/γ2). Substituting (5.7) and (5.5) for (5.4), we obtain

M(q)ṡ+C(q, q̇)s+
1
2

(
1+
1
γ2

)
s= f̃(xr)+τd(t) (5.9)



Robust neural networks control of omni-mecanum wheeled robot... 1199

where f̃(xr) is a function of the approximation error f(xr), which is

f̃(xr)= f(xr)− f̂(xr)=W
T
ϕx(x)−Ŵ

T
ϕx(x)+ε=W̃

T
ϕx(x)+ε (5.10)

whereW̃=W−Ŵ is the error of estimation of neural network weights. Using equation (5.10),
equation (5.9) is written as follows

M(q)ṡ+C(q, q̇)s+
1
2

(
1+
1
γ2

)
s=W̃Tϕx(x)+ε+τd(t) (5.11)

In order to apply the considerations described in Section 3, the designations in equation (3.3)
are interpreted as

ẋo = ṡ xo = s b(xo)=M(q)
−1)

g(xo)= s d(t)= ε+τd(t)

fx(xo)=M(q)
−1
[
−
1
2

(
1+
1
γ2

)
s−C(q, q̇)s+W̃Tϕx(x)

]

In order to synthesize the neural network control of MRK M motion, the Lyapunov stability
theory isused.Asa candidate for theLyapunov function, the following functionhasbeen selected

V (s)=
1
2
s
T
M(q)s+

1
2
tr(W̃TP−1W̃) (5.12)

whereP=PT >0 is a designmatrix. The derivative of the functionV (s) relative to time along
the trajectory of system (5.11) is

V̇ (s)= sTM(q)ṡ+
1
2
s
T
Ṁ(q, q̇)s+ tr(W̃TP−1

˙̃
W) (5.13)

Substituting from equation (5.11) the expression M(q)ṡ and taking advantage of the fact that
0.5[Ṁ(q, q̇)−2C(q, q̇)] is a zero matrix (for β(t)= 0), the following is obtained

V̇ (s)=−sT
1
2

(
1+
1
γ2

)
s+ tr{W̃T[P−1

˙̃
W+ϕx(x)s

T]}+sT[ε+τd(t)] (5.14)

Selecting the following as the rule for adaptation of neural network weights
˙̃
W=−Pϕx(x)s

T (5.15)

equation (5.14) will be transformed into

V̇ (s)=−
1
2γ2
s
T
s−
1
2
s
T
s+sT[ε+τd(t)] (5.16)

LetD denote transformed equation (3.8)

D= V̇ (s)−
1
2
γ2‖ε+τd(t)‖

2+
1
2
‖s‖2 (5.17)

and then, on the basis of (5.16), equation (5.17) will be transformed into

D=−
1
2γ2
s
T
s−
1
2
s
T
s+sT[ε+τd(t)]−

1
2
γ2‖ε+τd(t)‖

2+
1
2
‖s‖2 (5.18)

Because −0.5sTs+0.5‖s‖2 =0, equation (5.18) will take the form

s
T[ε+τd(t)]−

1
2γ2
s
T
s−
1
2
γ2‖ε+τd(t)‖

2 =−
1
2

∥∥∥
1
γ
s+γ[ε+τd(t)]

∥∥∥
2
¬ 0 (5.19)

From inequality (5.19) it follows thatD¬ 0 and, based on equation (5.17), we get

V̇ (s)¬
1
2
γ2‖ε+τd(t)‖

2−
1
2
‖s‖2 (5.20)

Determining z(t) = s(t), d(t) = ε+τd(t), we have thus shown that condition (3.8) is satisfied,
i.e. inequality J ¬ γ is satisfied for the structure adopted as control rule (5.7) and, furthermore,
that the analysed system is stable according to the definition in (2.3).



1200 Z. Hendzel

5.1. Caution

In the proposed control synthesis, based on the Lyapunov stability theory, there is a compo-
nent of the form sT0.5[Ṁ(q, q̇)−2C(q, q̇)]swhich takes a value of zero if thematrix in brackets
is an obliquely symmetrical matrix. This situation takes place in the analysed case only for the
configuration of the robot frame β(t) = 0. In the case where β̇(t) 6= 0, the indicated equation
does not take place. Then, when determining the expression ε1 =0.5[Ṁ(q, q̇)−2C(q, q̇)]s, the
last component of equation (5.14) needs to bemodified to the form sT[ε+τd(t)+ε1] and, con-
sequently, equation (5.20) will be V̇ (s)¬ 0.5γ2‖ε+τd(t)+ε1‖

2−0.5‖s‖2, and the expression ε1
needs to be interpreted as a structural interference.
The next Section presents a simulation example, the purpose of which is to confirm the

solutions arrived at in the analytical considerations and to obtain quantitative solutions.

6. Simulation example

In the example, simulation of the suggested solution has been carried out for the movement of
the selected point S of the mobile robot on the trajectory set in the form of a loop, consisting
of five characteristic stages of motion: moving on a rectilinear track, starting, moving at a fixed
speed when, V ∗s = 0.4m/s, moving on a circular path with the radius R = 0.7m, exit from
the curve taking into account a transitional period, then moving on the rectilinear track at a
fixed speed and braking. For the assumed stages of motion, Figure 3 shows the trajectory set
for the point S with the assumed orientation angle of the robot, assuming the initial conditions
of motion: xs(0)= 1m, ys(0)= 5m, β(0)= 0.

Fig. 3. The trajectory set for motion of the point S and the robot orientation

Duringmotion, there is no change in the orientation angle of the robot frame β(t). Figure 4
presents a solution to the task of inverse kinematics. For the time t­ 7.5s, there is amovement
of the point S of the robot on a circular trajectory with the radiusR=0.7m.

Table 1.Values of ai parameters of a mobile robot

a1 a2 a3 a4 a5 a6 a7 a8

0.044 11.5708 2.4558 2.4545 3.001 3.001 3.001 0.0622

Then, the self-rotation angle of wheel 2 increases and the self-rotation angle of wheel 1
decreases resulting from the structure of the robot, as shown in Fig. 4a. The change of these
values is shifted in time. The change inmotion is confirmed by a change in the angular velocity
of the wheels, see Fig. 4b, and the course of angular accelerations of the wheels, see Fig. 4c.
When the point S turns an angle π/2rad then the angular velocities and angular accelerations
have the same values, which is followed by a change in the values and turns of vectors of these
kinematics parameters. The obtained parameters of motion of the driving wheels constitute the



Robust neural networks control of omni-mecanum wheeled robot... 1201

Fig. 4. The parameters set for the movement of the robot wheel and the path of the point S:
(a) values of self-rotation angles of the driving wheels, ϕ1, ϕ4 and ϕ2, ϕ3, (b) angular velocity of
self-rotation of the driving wheels ω1, ω4 and ω2, ω3, (c) angular accelerations ω̇1, ω̇4,ω̇2, ω̇3,

(d) set trajectory for the point S

set trajectory of motion for the control system. For simulation of the control system, values of
the robot parameters have been adopted as specified in Table 1 and the remaining data are:
Λ= diag[2,2,2],KD =0.5(1+1/γ),P= diag[100, . . . ,100], γ=0.2, τd =0. In order to check
the robustness of the proposed solution, it has been also assumed that for the time t­ 15s there
is aparametricdisturbancea+∆awhere∆a= [0,0,0,0,1,1,1,0]T, associatedwith thechangeof
the rolling friction of the robotwheels. Theneural network adopted in simulation to compensate
for the robot nonlinearity and the variable operating conditions consist of 6 sigmoidal bipolar
neurons whose first layer weights are generated by a random number generator from the range
[−0.1,0.1]. A separate neural network for each element is used for approximation of thenonlinear
function f(xr)∈ R3.Thecalculations aremadebyadopting theEulermethodof integrationwith
the time discretization step 0.01s. In the process of initializing the neural network, zero initial
values of the output layer weights are assumed. In order to evaluate quality of the generated
robust control and implementation of trackingmotion, the following quality indicators have been
adopted, restricted to the assessment of motion only for wheels 1 and 2:

• maximum value of the angle of rotation error e1 =(ϕ1d−ϕ1) [rad], e2 =(ϕ2d−ϕ2) [rad],
emax(·) =max |e(·)| [rad], ϕ1d, ϕ2d are the set angles of wheel rotation, Fig. 4a,
• maximum value of the angular velocity error ėmax(·) =max |ė(·)| [rad/s],

• root-mean-squared error for wheel rotation angles, ε1 =
√
(1/n)

∑n
k=1e

2
1k, ε2 =√

(1/n)
∑n
k=1e

2
2k [rad], where k is the number of subsequent discrete measurements,

n=3001 – total number of discrete measurements,

• root-mean-squared error for tracking the given angular velocity ε̇1 =
√
(1/n)

∑n
k=1 ė

2
1k,

ε̇2 =
√
(1/n)

∑n
k=1 ė

2
2k [rad/s],

• generalised root-mean-squared error: s1 = ė1+λe1, s2 = ė2+λe2, ν1 =
√
(1/n)

∑n
k=1s

2
1k,

ν2 =
√
(1/n)

∑n
k=1s

2
2k [rad/s],mean-squared error of distancedk,ρ=

√
(1/n)

∑n
k=1d

2
k [m],

where dk =
√
(xSk−xdSk)2+(ySk−ydSk)2 [m], k = n is the distance between the set



1202 Z. Hendzel

(xdS,ydS), Fig. 4d and implemented (xS,yS) locations of the selected pointS of themobile
robot in the xy plane duringmotion,
• maximum distance dmax =max(dk) [m].

Implementing adopted control algorithm (5.7), patterns of control errors have been obtained
as shown in Fig. 5.

Fig. 5. Results for the obtained errors: (a) patterns of errors of tracking the rotation angles of
wheels 1 and 2, (b) patterns of errors of tracking the angular velocity of wheels 1 and 2,

(c) pattern of error in the phase plane for the generalized error s1, (d) implementation of the set
trajectory for the point S

The quantitative assessment of the solution is given in Tables 2 and 3.

Table 2.Values of the quality indicators for the robust neural networks control

Indicator eimax [rad] εi [rad] ėimax [rad/s] ε̇i [rad/s] νi [rad/s]

wheel 1, i=1 0.05139 0.007379 0.1087 0.01529 0.02125
wheel 2, i=2 0.05184 0.007703 0.1097 0.01578 0.02205

Table 3.Values of the indicators dmax, ρ for the robust neural networks control

Indicator dmax [m] ρ [m]

Value 0.002839 0.0004151

Having analysed the changes in the individual indicators of the quality ofmotion implemen-
tation in the presence of variable operating conditions, it can be concluded that a high accuracy
in the implementation of the adopted method of solution has been obtained. However, Fig. 6
presents patterns of control signals in the presence of variable operating conditions of the object,
resulting from the assumed resistance to motion in the form of a change in the rolling friction.
The neural networks compensation of the robot nonlinearity plays a predominant role in the
control structure, as shown in Fig. 6b, in comparisonwith thePD control, Fig. 6c. The observed
parametric disturbance is particularly visible for driving wheel 2.



Robust neural networks control of omni-mecanum wheeled robot... 1203

Fig. 6. Results of simulation of the neural networks control system: (a) total control, (b) NN(·) neural
networks control, (c) PD control(γ), (d) evaluation of the system robustness γ̂¬ γ

The limitation of the signals shown in Fig. 6 confirms stability of the adopted solution in
accordance with Lyapunov’s interpretation of stability. Figure 6d presents the results of the
solution for the robust neural networks control of MRK M motion which changes over time,
with equation (3.7) constituting the evaluation of this control method. As indicated by the
patters shown in Fig. 6d, the analysed system has gain L2 <γ, for γ =0.2, and the condition
0¬ γ̂ < γ is met, thus equation (3.5) is satisfied.

7. Summary

This paper presents a novel approach to the problem of controllingmechanical objects of unspe-
cified description considering variable operating conditions. In order to solve the task of control-
ling amobile robotwithmecanumwheels taking into account the compensation for nonlinearity
and variable operating conditions of the object, Lyapunov’s stability theory has been applied
including the HJ inequality. A neural network with basic sigmoid functions has been used to
compensate for the nonlinearity and variable operating conditions of the robot. Additionally,
there is a PD(γ) controller in the control structure, i.e. the structure of moments driving the
wheels. The gain values of thePD(γ) controller have been determined based on theHJ inequali-
ty, thus formulating an algorithmwhich is stable and interference-resistant from the perspective
of input-output signals.

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Manuscript received June 27, 2017; accepted for print June 14, 2018