Control of Robot Motion in Radial Mass Density Field
BRANKO NOVAKOVICa, DUBRAVKO MAJETIC JOSIP KASAC, DANKO BREZAK
Faculty of Mechanical Engineering and Naval Architecture,
University of Zagreb,
Ivana Lučića 5. Zagreb,
CROATIA
aORCiD: https://orcid.org/0000-0003-0735-770X
Abstract: - T In this article, a new approach to control of robot motion in the radial mass density field is
presented. This field is between the maximal and the minimal radial mass density values. Between these two
limited values, one can use n points (n = 1, 2, . . . nmax) that can be included in the related algorithm for control
of the robot motion. The number of the points nstep can be calculated by using the relation nstep = nmax / nvar ,
where nvar is the control parameter. The radial mass density is maximal at the minimal gravitational radius and
minimal at the maximal gravitational radius. This is valid for Planck scale and for the scales that are less or
higher of that one. Using the ratio of Planck mass and Planck radius it is generated the energy conservation
constant κ = 0.99993392118.
Key-Words: - robot motion control; radial mass density field; maximal (minimal) radial mass density; energy
conservation constant; macro (micro, nano) robot control; electrical robots; magnetic robots;
chemical actuated robots; bio/soft robots.
Received: March 12, 2023. Revised: November 29, 2023. Accepted: December 15, 2023. Published: December 31, 2023.
1 Introduction
Generally, the very large structures of the robots
have a lot of application areas as in the precise
production processes, in the medicine for cell
manipulation, drug delivery, medical image
acquisition, and non-invasive intervention. For those
applications, one can use electrical, magnetic,
chemical actuated robots and the bio/soft robots, [1],
[2]. Genetic algorithms and unsupervised machine
learning for predicting robotic manipulation failures
for force-sensitive tasks discussed in [3]. An
integrated design and fabrication strategy for
entirely soft, autonomous robots is presented in [4].
Versatile soft-grippers with intrinsic electro-
adhesion based on multifunctional polymer
actuators is point out in [5]. Magnetic actuation
methods in bio/soft robotics are discussed in [6].
Efficient constant-time addressing scheme for
parallel-controlled assembly of stress-engineered
MEMS micro-robots is present in [7].
In this article, the control of the robot's motion
is described in the radial mass density field. This
field is in the region from the minimal radius (with
the maximal radial mass density (ρr max) and
maximal radius (with the minimal radial mass
density (ρr min). Between these two limited values,
one can choose n points (n=1,2,..nmax ). In the case
of the precise robot motion the number nmax should
be bigger. Contrary, for the less precise robot
motion, the number nmax may be smaller. In that
sense, one can introduce the related steps number
(nstep) between maximal and minimal radiuses in a
gravitational field. This value can be calculated by
using the relation (nstep = nmax / nvar ). If one uses the
smaller parameter (nvar) than the number of the steps
(nstep) is bigger and vice versa. In that way, one can
obtain the most precise control of the robot's
motion.
The very important consequence of the solution
of the field equations by including gravitational
energy-momentum tensor (EMT) on the right side of
the field equation is that the gravitational field
exhibits repulsive (positive) and attractive
(negative) gravitational forces. The minimum time
transition between quantum states in the
gravitational field is present in [8]. To precisely
follow the desired trajectory of the robot motion one
can include the new Relativistic Radial Density
Theory (RRDT), [9]. The particle transition and
correlation in quantum mechanics are discussed in
[10]. Independent position control of two identical
magnetic micro-robots in a plane using permanent
magnets and magnetically powerful microrobots is
presented in [11]. This application represents the new
approach to the medical revolution epoch.
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Magnetically powered micro-robots are discussed in
[12], [13].
Further, the robust control of micro-robot motion
is presented in [14]. A conjugate gradient-based
BPTT – like optimal control algorithm with vehicle
dynamics control application is discussed in [15].
Robust motion control with anti-windup scheme for
electromagnetic actuated micro-robot using time-
delay estimation is presented in [16]. The two
independent position controls of two equally micro-
robots motion in a plane are realized by using
rotating permanent magnets, [17]. Magnetically
powered micro-robots and the robust motion
control, with an anti-windup scheme for
electromagnetic actuated micro-robots, are
presented in [18] and [19], respectively. Robotic-
assisted minimally invasive surgery is illustrated in
[20]. The design of a novel haptic joystick for the
teleoperation of continuum-mechanism-based
medical robots is presented in [21]. In this reference,
a novel mechanism with a series of coupled gears,
that aims for the control of continuum robots for
medical applications is pointed out. Positioning
control of robotic manipulators subject to excitation
from non-ideal sources is discussed in [22]. Further,
tractor-robot cooperation is illustrated in [23].
Indoor positioning systems of mobile robots are
present in [24]. A new single–leg lower-limb
rehabilitation robot motion is presented in [25].
Multi-robot task scheduling for consensus-based
fault resilient intelligent behavior in smart factories
is discussed in [26]. A new single-leg lower limb
rehabilitation robot with design, analysis, and
experimental evolution is presented in [27]. It is also
important to know how the portable surveillance
robots can be used in IoT applications, [28]. The
recent trends in robot learning and evolution for
swarm robotics are presented in [29]. Finally, the
proactivity of fish and leadership of self-propelled
robotic fish during interaction and bio-inspiration
with biomimietics is discussed in [30].
2 Dynamics of Autonomous Robot
Motion in the Electromagnetic and
Gravitational Radial Mass Density
Field
The problem of the nonlinear control of robot
motion is discussed as the function of the maximal
radial mass density value. To simplify the related
calculation, here it started with the concept of the
external linearization of the nonlinear control of the
robot motion in the radial mass density field. In that
case, in the closed regulation loop, one obtains the
linear behaviour of the hole-system. Thus, the
problem of the robot position control in the radial
mass density field can be started by the calculation
of the control of the error vector, e(t). This vector is
a function of the radial mass density,
r
, and can be
presented by the relations:
2
2
2
2
1
11
ww
w w p t I
r max min
w
w p t Iw
r max min
d e n
e X X , r (t ) F F NF ,
dt r c
d X / n
r (t ) F F NF .
dt r c
(1)
Here n=1,2,..,nmax and nmax =
r max
/
r min
, while
the subscript w denotes the desired robot motion.
The variables without this subscript present the real
autonomous robot motion. Further, Fp is a potential
force, Ft is a time - variation force, Fi is the
interaction force and N is the related connection
parameter. At the same time, the relations (1) also
describe the canonical differential equations of the
robot motion in the combination of the
electromagnetic and gravitational fields. Vector rw(t)
presents the desired (nominal) acceleration of the
robot motion in the radial mass density field.
Now following the idea of external
linearization, one can introduce the following
substitution:
2
2
1
w p t I
r max min
T
x y z
d e n
u(t ) r (t ) F F NF ,
dt r c
u(t ) (u (t )u (t )u (t )) .
(2)
Here u(t) is the internal control vector of the
robot motion in the radial mass density field.
Further, one can apply the state-space phase
variables, (z1 z2 z3)T , that from (1) gives the related
state-space model of the robot motion in the radial
mass density field:
1 2 3
456
TT
x y z I
yTT
xzII
e (e e e ) Z ( z z z ) ,
de
de de
de ( ) Z ( z z z ) ,
dt dt dt dt
(3)
and
00 111
00
dZ / dt AZ(t ) Bu(t ),
I
A ,B , I diag , , .
I
(4)
In (4), parameters A and B are constant matrices
with dimension (6x6) and (6x3), respectively. Here,
it is supposed that the disturbances in a state-space
model of the robot motion in the radial mass density
field (3) and (4) are of the initial condition types. To
eliminate the control error of the robot motion in the
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radial mass density field, which is caused by the
disturbances, one can introduce the following
internal control:
1
p r max min w I I II II t I
N
F r r (t ) K Z K Z F F ,
nc
u(t ) K Z .
(5)
Here, K is the state space controller, Z is the
control error, Fp is the potential force, Ft is the time
variable force, FI is an interaction force, N is a
constant and c is the speed of the light in vacuum.
Including the internal control relations (3) and (4)
into (5), one obtains the related equation of the
potential force as a function of the radial mass
density value in the linear form:
1
p r max min w I I II II t I
N
F r r (t ) K Z K Z F F .
nc
(6)
Now starting from the previous relations one
can generate the new equations of the potential force
Fp as the functions of the potential energies Uj an Uc
:
xy
z
jj
cc
pp
jc
p
UU
UU
F , F ,
x x y y
UU
F .
zz
(7)
Here j = g for the gravitational field, j = e for
the electromagnetic field and Uc is the related
control potential field. It is followed by the
inclusion of the control potential force, Fcp , that is
derived by the artificial control field with potential
control energy Uc . After inclusion of the relation (7)
into the relation (6), one obtains the nonlinear
control of the robot motion in the multi-potential
field as the function of the maximal radial mass
density
r max
:
1F
cp r max min I I II II dp t I
N
F r r(t ) K Z K Z F F .
nc
(8)
Now, using (8), the control of the nonlinear
system is solved by employing the concept of
external linearization in the radial mass density
field. Here the obtained equations are functions of
the radial mass density values.
The general approach to control the dynamics of
the robot motion in radial mass density field for
more potential fields, given in (8), can also be
applied to the two potential electromagnetic and
gravitational fields. In this sense, let a robot be an
electrically charged particle with charge q and rest
mass m0 that is moving with a non-relativistic
velocity (v << c) in combined electromagnetic and
gravitational potential fields. Further, it is also
assumed that the gravitational field is produced by
the spherically symmetric (non-charged) body with
mass M. In that case, the total potential energy U of
the robot motion in the two potential radial mass
density fields is described by the relation:
00
01
e g e
r max e r max min
min
GM
U qV m V qV m ,
r
mGM
, U qV r .
r n r
(9)
Here Ve and Vg are the related scalar potentials
of the electromagnetic and gravitational radial mass
density fields, respectively. Parameter G is the
gravitational constant and r is the radius as the
distance between the autonomous robot and the
center of the mass M and n=1,2,..,nmax , nmax =
r max
/
r min
. Now applying (1) and using the notations,
(Ee,He) for an electromagnetic field and (Eg,Hg) for
the gravitational field, one can generate the vector
equation as the explicit functions of the Lorentz
forces:
2
2
11
11
r max min e e
r max min g g
dX
r q E v H
n dt c
r E v H .
nc
(10)
The parameters Ee, Eg, He and Hg are vectors
described by the relations:
x x x x
y y y y
z z z z
e g e g
e e g g e e g g
e g e g
E E H H
E E , E E , H H , H H .
E E H H
(11)
In this example, a robot is a particle with charge
q and rest mass m0 and, therefore, this robot
interacts with both electromagnetic and gravitational
radial mass density fields. In that sense, the relations
(10) and (11) describe the dynamic of the robot
motion in two – potential electromagnetic and
gravitational field. The components of the vector Ee
and Eg can be calculated by using the following
equations:
11
1
xx
xx
y
y
eg
g
e
eg
e
e
e
AA
V
V
E , E ,
x c t x c t
A
V
E ,
y c t
(12)
and
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11
1
yz
yz
z
z
ge
ge
ge
g
g
g
AA
VV
E , E ,
y c t z c t
A
V
E .
z c t
(13)
The components of vectors Ae, Ag, He, and Hg in
(12) and (13) are given by the relations:
y
z
i i x
y
z
x
e
e
ig
ie
e g e
g
g
g
A
A
vV
vV
A , A , H ,
c c y z
A
A
H , i x,y,z,
yz
(14)
and
xx
zz
yy
yy
xx
zz
eg
eg
eg
eg
eg
eg
AA
AA
H , H ,
z x z x
AA
AA
H , H .
x y x y
(15)
Applying (6) and (7) to the canonical
differential equations of the autonomous robot
motion in the two-potential radial mass density
field, one obtains the control error model of the
robot motion as a function of the maximal radial
mass density value:
1v
1v
w e e
r max min
gg
nq
e(t ) r (t ) E H
rc
E H ,
c
(16)
and
1v
1v
ww
ww
w e w e
r max min
g w g
nq
r (t ) E H
rc
E H .
c
(17)
In (17) rw is the vector of desired acceleration of
the robot motion. The subscript w denotes the
desired values of the related variables. The next step
is the application of the concept of external
linearization to transform the equation (16) into the
new relation:
1
1
w e e
r max min
gg
nq
u(t ) r (t ) E v H
rc
E v H .
c
(18)
Here u(t) is the internal control vector and
n=1,2,..nmax is the number of the robot steps from
the minimal to the maximal radiuses in radial mass
density field. From (17) and (18), one obtains the
related equivalent of the linear control error model
of the robot motion in the combined electromagnetic
and gravitational radial mass density field, given by
(6) and (7). Applying (18), one obtains the new
relation as the function of the maximal radial mass
density in the form:
1v
1v
r max min
e w I I II II e
r max min gg
r
E r (t ) K Z K Z H
nq c
r
E H .
nq c
(19)
Now, let the electric field Ee consist of the two
electric components Ee = Ede + Ece. Here Ede is a
disturbance electric field that is caused by the
influence of a two-potential field on the motion of
the robot in the radial mass density field. The
component Ece is an artificial electric control field
that should control robot motion in the two potential
fields. Including Ee = Ede + Ece into (19), one obtains
the nonlinear electric control of the robot motion in
the two-potential radial mass density field as the
function of the maximal radial mass density:
11
vv
r max min
ce w I I II II
r max min
de e g g
r
E r (t ) K Z K Z
nq
r
E H E H .
c nq c
(20)
Taking into account the relation (10), the
canonical differential equations of the robot motion
in the two-potential radial mass density field can be
rewritten as a function of the maximal radial mass
density:
2
2
1v
1v
de ce e
r max min
gg
d X nq E E H
dt r c
E H .
c
(21)
Applying the nonlinear control Ece from (20) to
the nonlinear dynamical model of the robot motion
(21), one obtains the closed-loop system in the
linear form:
2
2w I I II II
dX r (t ) K Z K Z .
dt
(22)
Thus, equation (20) is the nonlinear control,
which in the closed loop with the nonlinear
canonical differential equations of the robot motion
(21), results in the linear behaviour of the hole
system (22). On that way the problem of controlling
the robot motion in the combination of an
electromagnetic and gravitational radial mass
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density field has been solved by employing the
concept of external linearization. This is very
important for application of the micro and
nanorobots in the drag delivery across the human
body.
3 The Other Methods of Application
of the Radial Mass Density to
Robot Control
The global positioning of robot manipulators with
mixed revolute and prismatic joints is presented in
[19]. In this section, it is illustrated how one can
apply the maximal radial mass density theory to the
mentioned class of the robots. In that sense, the
dynamic model of the robot with the n-link rigid
body can be described as the function of the
maximal radial mass density:
2
02
2
2
0
r max min
r max min
d q dq dq
m ( q ) C( q, ) q(q ) U ,
dt dt dt
d q dq dq
r ( q ) C( q, ) q(q ) U ,
dt dt dt
m ( q ) r .
(23)
Here q is (nx1) vector of robot joints
coordinates, dq/dt is the related vector of joints
velocities, U is a vector of applied joint torques and
forces,
0
m
(q) is (nxn) inertia matrix, and
C(q,dq/dt)dq/dt is (nx1) vector of centrifugal and
Coriolis torques. Further q(q) is the vector of
gravitational torques and forces and
r max
is the
maximal radial mass density at the minimal radius.
If the robot, described by (23), is in the closed loop
with the nonlinear PID controller, described by the
relation:
2
2
p d I
d q dq dq
U(t ) ( K K K ),
dt dt
dt
(24)
Then the closed loop system of the relations
(23) and (24) resulted in the form that is the function
of the maximal radial mass density:
2
2
2
2
r max min
p d I
d q dq dq
r ( q ) C( q, ) q(q )
dt dt
dt
d q dq dq
( K K K ).
dt dt
dt
(25)
The relation (25) can be applied for the
parameter n =1,2,…,
r max r min
/
. Now one can use
the relation (25) in the new form:
2
2
2
2
r max min
step
step max varp d I
d q dq dq
(C( q, ) q(q )
dt r ( q ) dt dt
d q dq dq
K K K ), .
dt dt d
n
n n / n
t
(26)
Thus, using the relation (26) it is possible to
control the robot’s acceleration by changing the
numerical parameter nstep. In that way by changing
the parameter nvar it is possible the realization of the
most precise robot motion control. This means that
the radial distance between two points should be
minimal if the nvar is maximal.
The dynamics of the robot motion can also be
described as the function of the alpha field
parameters derived in the Relativistic Alpha Field
Theory (RAFT), [7]. In this theory, one can start
with the potential energy of the robot (particle) in
the combination of the electromagnetic and
gravitational fields, Ue and Ug, respectively. Now let
q, m, Ve, and Vg are the robot’s (particle’s) charge,
mass, electrical potential, and gravitational
potential, respectively. Further, G is the gravitational
constant, M is the mass of the gravitational field and
c is the speed of light in a vacuum. The potential
energy of the robot in combination with the
electromagnetic and gravitational fields is given by
the relations:
2 2 2
e g e
e
mGM
U U U qV ,
r
qV
U GM
,
mc mc rc
(27)
and
2 2 2
e
r max min r max min
r max min
qV
U GM ,
r c r c rc
m r .
(28)
The relation (28) can also be described as the
function of the parameter n:
2 2 2
1step m
e
r max min r max
ax v
min
ma arx
nqV
nU GM ,
r c r c rc
n ...,n , .n n / n
(29)
If one wants to use RAF theory in robotics then
it requires the introduction of the related alpha field
parameters. The solution of the field parameters for
an electron in the two-potential electromagnetic and
gravitational fields are given as follows. In that
sense parameters
1
and
1
'
are given by the
relations:
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122
122
1
1
e
r max min
e
r max min
nq V GM
i - ,
r c rc
nq V GM
'
i - .
r c rc
(30)
For parameters
2
and
2
'
:
222
222
1
1
e
r max min
e
r max min
nqV GM
i - ,
r c rc
nqV GM
'
i - .
r c rc
(31)
For parameters
3
and
3
'
:
322
322
1
1
e
r max min
e
r max min
nqV GM
i - ,
r c rc
nqV GM
' i ,
-
r c rc
(32)
and for parameters
4
and
4
'
:
422
422
1
1
e
r max min
e
r max min
nqV GM
i - ,
r c rc
nqV GM
' i .
-
r c rc
(33)
Now one can introduce the generalized Lorentz-
Einstein parameters, for an electron in a two-
potential electromagnetic and gravitational field.
These parameters are described by the following
equations:
12
2
1
2
12 222
2
/
v
nqV GM
e
crr
r max min
nqV
H.
GM
e
,i c v
r c rc
r max min
nqV GM
e
crr
r max min
(34)
The solutions of the H3,4 are symmetric to the
solutions of the parameters H12.
The previously presented two potential fields
can be generalized by the application of the multi-
potential field as the function of the field parameters
α and α’. Now, for derivation of a four-potential
vector A of the related potential field, one can recall
the general Hamilton function, H, for the weak
potential fields:
12
22
2
32
1
p x p y
xy
r max min
pz
step max
zp
max var
H
n n /
U v U v
c p c p
cc
r
Uv
c p c U ,
n
c
n ...,n , .n
(35)
Here Up is a potential energy, (px, py, pz) is a
three-momentum vector, (vx, vy, vz) is a three-
velocity vector and ζ1, ζ2, ζ3 and β are the well-
known Dirac’s matrices. If an electron is moving
with a constant velocity v << c in an
electromagnetic field with a scalar potential, V, then
one should use the following relations:
22
2
p x p y y
xxy
pz zz
U v U v V v
Vv
q q q q
A , A ,
c c c c c c
cc
Uv q V v q A.
c c c
c
(36)
Here q is an electric charge of an electron and
(Ax , Ay , Az ) is a three-potential vector of the
electromagnetic field. Including (36) into (35), one
obtains the well-known Hamilton function for
Dirac’s electron in an electromagnetic field:
12
2
3
x x y y
r max min
zz
qq
H c P A c P A
cc
r
q
c P A c qV .
cn
(37)
On the other hand, if a robot (particle) is
moving with constant velocity v << c in a
gravitational field, then, according to the previous
procedure, one should use the following relations:
2x
r max min r max min
pg
r max min r max min
p x g x g
r GM r
U V ,
nr n
rr
U v V v A.
nc c nc
c
(38)
and
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2
2
y
z
r max min r max min
p y g y g
r max min r max min
p z g z g
rr
U v V v A,
nc c nc
crr
U v V v A.
nc c nc
c
(39)
In the relations (38) and (39) G is a gravitational
constant, M is a gravitational mass, Vg is a
gravitational scalar potential and (Agx , Agy , Agz ) is a
three-potential vector of the gravitational field.
Including (39) into the equation (37), one obtains
the Hamilton function Hg for the particle in a
gravitational field:
1
23
2
x
yz
r max min
g x g
r max min r max min
y g z g
r max min r max min g
r
H c P A
nc
rr
c P A c P A
nc nc
rr
c V .
nn
(40)
Generally, if the robot velocity v in a potential
field is constant, then the four-potential vector A can
be derived as a function of the field parameters α
and α′:
2
0 1 2 3 0
10
1
x
x
c
A A ,A ,A ,A , A ,
v
A A A ,
c
'
(41)
and
1 0 2 0
30
y
x
xy
z
z
v
v
A A A , A A A ,
cc
v
A A A .
c
(42)
Now, the components of the field tensor Fij of
the potential field can be calculated by using
relations (41) and (42) and the well-known
procedure:
0 1 2 3
0 1 2 3
ji
ij ij
AA
F , i, j , , , ,
xx
X x ,x ,x ,x ct,x,y,z .
(43)
As the result of this calculation, one obtains the
well-known anti-symmetric tensor Fij of the
potential field in the following form:
01 02 03 01 02 03
10 12 13 01 12 13
20 21 23 02 12 23
30 31 32 03 13 23
00
00
00
00
ij
F F F F F F
F F F F F F
F.
F F F F F F
F F F F F F
(44)
This tensor can be employed for the derivation
of the related Maxwell’s like equations in a vacuum.
Following the previous consideration, one can
introduce the normalized scalar potential
0
m
A
of a
multi-potential field in the dimension of specific
potential energy:
0 0 2
1 1 2
m j j j A A ' c , j , ,..,n.
(45)
Here term αα′ has to be calculated by employing
the relations (30) to (33):
2
1 1 2
j
p
r max min
nU
, j , ,..,n.
rc
'=
(46)
The relations (45) and (46) tell us what the
normalized scalar potential
0
m
A
really is:
01 2 1
j
p
m ma
step max va
x
r max min
r
nU
A , j , ,..,n, n ...,n ,
r
n n / n .
(47)
In recent decades, it has been created a wide
range of robotic systems mostly inspired by animals.
In that sense, engineers have created a wide range of
robotic systems like four - legged robots, snake
robots, insect robots, and fish robots, [37].
Following the previous consideration, it is possible
to control that class of robots by using the radial
mass density field theory.
4 Calculation of the Robot Motion in
Radial Mass Density Field
Gravitational field with the mass Mg has the
maximal and minimal gravitational radial mass
densities given in [12]:
2
27
max
min
(1 ) 2.693182 10 / ,
g
rm
Mckg m
rG
(48)
and
2
23
min
max
(1 ) 0.888779 10 / .
g
rm
Mckg m
rG
(49)
The numerical values in (48) and (49) are
constant and are valued for all amounts of the
gravitational mas Mg. In relations (48) and (49) the
parameter κ is the energy conservation constant that
has been calculated in the 12, by using Planck’s
mass and Planck’s length:
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.56
Branko Novakovic,
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E-ISSN: 2224-2856
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Volume 18, 2023
22
22
1 0 99993392118
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pp
GM GM
L , . .
( )c L c
(50)
Thus, the value of κ is close to one but less than
it. Using the combination of equations (21) and (48)
one obtains the canonical differential equations of
the robot motion at the minimal gravitational radius
with the maximal radial mass density:
2
2 27
1v
2.693182 10
1v / , / .
de ce e
step max vargg
d X nq E E H
c
dt
E H m k n n ng
c
(51)
By changing parameter n = 1, 2, .., nmax and
using the parameter nvar , one can obtain the desired
acceleration and precise control of the robot motion
in the related control region. Further, using the
combination of (21) and (48) one obtains the
canonical differential equations of the robot motion
at the maximal gravitational radius with the minimal
radial mass density:
2
2 23
1v
0.888779 10
1v / , / .
de ce e
step max vargg
d X nq E E H
dt c
E H m k n n ng
c
(52)
Now, by changing parameter n = 1, 2, .., nmax
and variable parameter nvar , one can obtain the
desired acceleration and precise motion of the robot
control in the related region. The ratio between the
maximal and minimal radial mass densities can be
calculated by using the relation:
27
max 4
max 23
min
2.693182 10 3.030204 10 .
0.888779 10
rm
rm
n
(53)
This ratio is the constant and is valued for the
all amounts of the gravitational masses.
Following the previous equations, one can
calculate of the maximal steps, nstep , between
maximal and minimal radiuses in a gravitational
field. For the calculation of the precise motion of the
robots in the gravitational radial direction, one can
introduce the variable step of the robot motion, nvar.
In that case, it is possible to select (change) the scale
of the desirable step of the robot motion in the radial
mass density field.
For an example, let the variable step of the robot
motion in the radial direction be given by the
amount nvar =100. In that case the number of the
robot steps nstep from the minimal to the maximal
radiuses has the value:
2
max
var
303.0204 10 303.0204 .
100
step n
n steps
n
(54)
In this calculation a robot needs 303 steps of the
motion from the minimal to the maximal radiuses in
the radial direction. In the case where the robot
motion is not in the radial direction one should use
the related projection of the radial trajectory to the
desired robot trajectory.
The next example is related to the possibility
that one wants to introduce the potential energies at
the minimal and maximal gravitational radiuses, Ug
max and Ug min, respectively. In that case it is possible
to calculate the minimal and the maximal radial
lengths, Lgmin and Lgmax , respectively, by using the
relations:
max min
0 max min 0 max min
min 2
min
,
2,
(1 )
g rm
rm rm p
g
g
Mr
m G r m G r
LUc
(55)
and
0 min max 0 min min
max 2
max
2.
(1 )
rm rm p
g
g
m G r m G r
LUc
(56)
From relations (55) and (56) one can see how
the potential energies in the gravitational field can
influence the robot's motion in that field.
5 Conclusion
This article is based on the new Relativistic Radial
Density Theory (RRDT) that has been applied to the
control of the robot motion in potential fields. The
robot motion is calculated from the minimal to the
maximal gravitational radiuses and vice-verse. In
the case where the robot motion is not in the radial
direction, it is necessary to transform the radial
coordinates into the rectangular ones by using
related projection. It is shown that the maximal
radial mass density occurs at the minimal
gravitational radius. On the other hand, the minimal
radial mass density happens at the maximal
gravitational radius. Furthermore, the both maximal
and minimal radial mass densities can also be
described as the functions of the energy
conservation constant κ. In that sense, the related
gravitational length, time, energy, and temperature
can be represented as functions of the Planck length,
time energy, and temperature, respectively.
Finally, it is concluded that the precise control
of the robot motion in combination with the
electromagnetic and gravitation fields can be
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.56
Branko Novakovic,
Dubravko Majetic Josip Kasac, Danko Brezak
E-ISSN: 2224-2856
547
Volume 18, 2023
controlled by the introduction of the variable step of
the robot motion. In that sense, one can introduce
the steps number, nstep, that is function of the
variable term, nstep = nmax / nvar, between maximal
and minimal radiuses in gravitational field. On that
way, the smaller value of the parameter nvar gives
the bigger number of the steps, nstep, and vice versa.
Thus, the bigger nstep gives more precise control of
the robot motion in the radial mass density field.
Acknowledgments:
The authors of this article are thanked for the all
administrative and technical support during the
writing of this article.
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Volume 18, 2023
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Branko Novakovic was responsible for the concept
of the research and for the organization of all
contribution to this scientific article. Dubravko
Majetic carried out for the computer simulation.
Josip Kasac and Danko Brezak have implemented
the computer algorithm for the simulation of the
presented article.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research was supported by the European
Regional Development Fund (ERDF) through the
grant KK.01.1.1.07.0016 (ARCOPS).
Conflict of Interest
The authors have no conflicts of interest to declare.
Ethical Compliance: The all procedures performed
in studies involving human participants were by the
ethical standards of the institutional and/or national
research committee and with the 1964 Helsinki
Declaration and its later amendments or comparable
ethical standards.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.56
Branko Novakovic,
Dubravko Majetic Josip Kasac, Danko Brezak
E-ISSN: 2224-2856
549
Volume 18, 2023
