Time optimal and PID controller for armed manipulator

robots

FARIDEH GIV, ALAEDDIN MALEK

Department of Applied Mathematics, Faculty of Mathematical Sciences

Tarbiat Modares University

Tehran

IRAN

Abstract: This paper is fourfold. First, three dierent time-optimal control problems for simulating the

manipulator robots with one, two, and three arms are mathematically formulated. Corresponding to the

related dynamical systems, the nonlinear system of ordinary dierential equations is derived. It has been

found that these problems are time-free and have distinct initial and boundary conditions, making them

hard to solve. To nd the minimum time with dierent payloads, a successful numerical method based

on the nite dierence and the three-stage Lobatto formula is proposed. Secondly, the related torque-

controlling problems are simulated, and then for one, two, and three armed manipulator robots, they

are solved using the PID controller method. Thirdly, it is shown that, compared to the time-optimal

controlling problem, the optimal PID torque controller solution takes more time to do the job than was

expected. However, the solution in the PID controller method shows less oscillation than the time-optimal

control problem. Fourthly, mathematical theories are used, and the numerical results for both methods

and dierent payloads are compared.

Key-Words: armed robot dynamics, time optimal control, proportional-integral-derivative (PID)

controller

Received: May 15, 2023. Revised: April 17, 2024. Accepted: May 15, 2024. Published: June 26, 2024.

1 Introduction

The robot is an automatic device that performs

functions ordinarily ascribed to human beings, [1],

[2].A two-jointed arm light robot without pay-

load (RR-type robot) is introduced by [3].Robots

for manipulating with several rigid links that are

controlled by a computer are used to produce

things such as electronics, medical devices, optics,

and watches, [4].Most industries today can im-

prove time and facility energy eciency with light-

armed robots, reducing operating costs. Smaller,

lighter parts take less time and energy to acceler-

ate, enabling these machines to work faster than

their competitors. This speed is ideal for pick-

ing, placing, part assembly, sorting, and carrying

light payloads. This kind of robot is lighter, so it

has less inertia when moving. Their lower weight

also means that if a collision occurs, it won’t be

as damaging, [5].

Since manipulators are typically used to repeat

a prescribed task a large number of times, even

small improvements in their performances may re-

sult in large monetary savings. Here, an attempt

is made to reduce the movement time (time op-

timal control strategy) between two points. Any

trajectory that can be realized by applying the

available driving forces and connecting the start-

ing point with the target point can be used to

implement the maneuver, [6], [7]. Among the op-

timal control methods, the application of Pontrya-

gin’s maximum principle is certainly one of the

most widely used. It provides an optimal condi-

tion that must be met at each time during the

trajectory. The method generally involves restric-

tions in the form of ordinary dierential equations

(ODE), which are usually of the rst order, there-

fore limiting the computational cost of their solu-

tion up to a certain extent, [8], [9], [10], [11], [12],

[13]. One of the most common control algorithms

used in the industry is the proportional-integral-

derivative (PID) controller (smooth payload trans-

portation strategy), partly because of its robust

performance over a wide range of operating con-

ditions as well as its simplicity of operation, [14],

[15], [16].

The paper is organized as follows: In Section

Armed robot dynamic systems, dynamic systems

of armed robots is proposed. The statement of

the time optimal control is proposed for armed

robots in Section Time optimal control problem .

In Section PID controller problem , the problem

is described with the PID controller. Numerical

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

185

Volume 19, 2024

simulation and the corresponding results are given

in Section Work Examples. Comprising the results

of time optimal control and PID controller discuss

in Section Numerical results and discussions and

nal result is in Section Conclusions.

2 Armed robot dynamic systems

In the year 2020 the behavior of armed robot dy-

namical systems by using Euler-Lagrange equa-

tions are described, [2]:

D(θ(t))¨

θ(t) + C(θ(t),˙

θ(t)) + g(θ(t)) = τ(t),(1)

where τis the generalized force associated with

angle θ(rad). The variable θis stated as gen-

eralized coordinates θ1for a robot with one arm,

(θ1, θ2)for a robot with two arms and (θ1, θ2, θ3)

for a robot with three arms. D(θ)is the inertia

matrix that is symmetric and positive denite in

the form, [2].

D(θ) =

i=1

(miJνi(θ)TJνi(θ)

+Jωi(θ)TRi(θ)IiRi(θ)TJωi(θ)) ,

(2)

inwhich miis the mass of the i-th arm, Jνiand

Jωiare Jacobian matrices, Riis the orientation

transformation between the body attached frame

and the inertial frame, and Iiis the i-th inertial

torque. C(θ,˙

θ)is the Christopher matrix

ckj =

i=1

cijk(θ)˙

θi=

i=1

2(∂dkj

∂θi

+∂dki

∂θj

−∂dij

∂θk)˙

θi

(3)

inwhich dij are components of D(θ)and g(θ)is

the gravity vector

g(θ) = [g1(θ), ..., gn(θ)]T, gk=∂P

∂θk

.(4)

inwhich Pis a potential energy. For R-type armed

robots n= 1,n= 2 for RR-type armed robots

and n= 3 for RRR-type armed robots. The dy-

namical system (1) consists of a typical second-

order nonlinear dierential equation (the number

of equations corresponds to the number of arms).

The physical characteristics of an armed robot

without payload for a robot with two arms are

given by [3]. Here, we assume that for R, RR and

RRR type robots, Liis the length for the i-th arm

and lci is the i-th center of mass for i= 1,2,3(see

Table 1).

(a) Robot with 1 arm

(b) Robot with 2 arms

Figure 1: Manipulator robot with payload M, (a)

R-type armed (b) RR-type armed, (c) RRR-type

armed.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

186

Volume 19, 2024

Table 1: Physical characteristics of an armed

robot without payload.

Arms #i Li(m)lci (m)mi(kg)Ii(kg.m2)

10.250 0.198 0.193 1.15 ×10−3

20.234 0.143 0.115 4.99 ×10−4

30.218 0.109 0.100 6.58 ×10−4

According to (1) the dynamic equation for one

armed robot (see Figure 1(a)) is

τ1= (m1l2

c1+I1)¨

θ1+m1glc1cos (θ1)(5)

inwhich τ1is the corresponding torque for one

armed robot. The dynamical system for an RR-

type robot in matrix form is as follows (see Figure

1(b) and [2], [17], [18])

τ1

τ2=d11 d12

d21 d22 ¨

θ1

θ2!

+ h˙

θ2h˙

θ1+h˙

θ2

−h˙

θ10! ˙

θ1

θ2!+g1

g2

(6)

where

d11 =m1l2

c1+m2(L2

1+l2

c2+ 2 L1l2

+ 2 L1lc2cos (θ2)) + I1+I2,

d12 =d21 =m2(l2

c2+L1lc2cos (θ2)) + I2,

d22 =m2l2

c2+I2,

c121 =c211 =1

2δd11

δθ2=−m2L1lc2sin (θ2) = h,

c221 =δd12

δθ2−1

2δd22

δθ1=h,

c112 =δd21

δθ1−1

2δd11

δθ2=−h,

g1= (m1lc1+m2L1)gcos (θ1) + m2g lc2cos (θ1+θ2),

g2=m2g lc2cos (θ1+θ2).

The dynamical system for RRR-type robot in ma-

trix form is as follows (see Figure 1(c), [14])





τ1

τ2

τ3

=



d11 d12 d13

d21 d22 d23

d31 d32 d33







θ1

θ2

θ3







+



c11 c12 c13

c21 c22 c23

c31 c32 c33







θ1

θ2

θ3





+



g3

.

(7)

where

d11 =I1+I2+I3+1

4m1L2

1+m2(l2

c1+1

4L2

+L1L2cos(θ2)) + m3(L2

1+L2

2+1

4L2

3+ 2 L1

L3cos(θ2) + L1L3cos (θ2+θ3) + L1L3cos(θ3)),

d12 =I2+I3+1

2m2(1

2l2

c2+L1L2cos(θ2)) + m3

(2 L2

2+1

2L2

3+ 2 L1L2cos(θ2) + L1L3

cos (θ2+θ3)+2L2L3cos(θ3)),

d13 =I3+1

2(1

2L2

3+L1L3cos (θ2+θ3) + L2L3

cos(θ3)),

d21 =d31 =d32 = 0,

d22 =I2+I3+1

4m2L2

2+m3(L2

2+1

4L2

3+L2L3

cos(θ3)),

d23 =I3+1

2m3(L2

3+L2L3cos(θ3)),

d33 =I3+1

4m3L2

c11 =c21 =c31 = 0,

c12 =−1

2m2L1L2sin (θ2)[ ˙

θ1+˙

θ2+1

2˙

θ3]−m3L1

sin (θ2) [L3˙

θ1+L2(2 ˙

θ2+˙

θ3)] −m3L1L3

sin (θ2+θ3)[1

2˙

θ1+˙

θ2+1

2˙

θ3]−1

4L1L3

sin (θ2+θ3)˙

θ3,

c13 =L3sin (θ3)[−1

2L2(1

2˙

θ2+˙

θ3)−1

2m3L1˙

θ1

−3

4m3L2˙

θ2],

c22 =c32 =1

2˙

θ1[L1L2sin (θ2)(1

2m2+ 2 m3)

+m3L1L3sin (θ2+θ3)],

c23 =L2L3sin (θ3)[−1

2m3(1

2˙

θ1+˙

θ2+˙

θ3) + 1

4˙

θ1],

c33 =1

4L2L3sin (θ3)[ ˙

θ1−m3˙

θ2],

g1=1

2m1g L1cos(θ1) + m2g(L1cos(θ1) + 1

2L2

cos (θ1+θ2)) + m3g(L1cos(θ1) + L2cos (θ1

+θ2) + 1

2L2cos (θ1+θ2+θ3)),

g2=1

2m2g L2cos (θ1+θ2) + m3g(L2cos (θ1

+θ2) + 1

2L3cos (θ1+θ2+θ3)),

g3=1

2m3g L3cos (θ1+θ2+θ3).

Now, additional payloads M (see Table 2) will

be added to the last arm to make a sensitivity

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

187

Volume 19, 2024

analysis for the additional payloads with dierent

masses. Note that if one changes the load miinto

mi+M,Listays xed, Iiand lci will change cor-

respondingly.

Table 2: Masses M that added to the last arm in

grams (g).

M (g) 21.2 31.2 51.8 100.6 121.9 130.7 150.5 200 500 1000

3 Time optimal control problem

The aim of designing time optimal control prob-

lem here is to consider how one transfer the end-

eector of an armed manipulator robot from an

initial position to the xed destination in the min-

imum possible time and optimal eort. The time

optimal control problem is represented by the fol-

lowing performance index:

J(u) = Ztf

dt =tf−t0,(8)

where the nal time tfis unknown, [2], [19], sub-

ject to the following nonlinear rst order dynamic

constraint:











˙x(t) = f(x(t),τ(t))

x0= (θ0,˙

θ0)T

xtf= (θtf,˙

θtf)T

(9)

inwhich x(t) = (x1(t),x2(t))T= (θ(t),˙

θ(t))T,

that is derived from (5) and (6), for certain initial

x0= (θ0,˙

θ0)Tand nal desired xtf= (θtf,˙

θtf)T

conditions, [14]. The minimum time t∗

fdenotes

the rst time that xposits in the above nal de-

sired condition. Note that θis radian and ˙

θis

radian

second . Here, for a robot with one arm the vec-

tor points x0,xtfbelong to R2. The function f

depends upon the control parameter τis belong-

ing to a set A⊂R, so that f:R2×A→R2,

x: [−π, π]→R2and function τ: [−3,3] →A

are given. If the robot has two arms R2converts

to R4and if the robot has three arms R2converts

to R6.

By applying the Routh-Hurwitz criteria to an-

alyze dierent stability zones, the stability of the

dynamical system is investigated. As well as the

modulation equations and detuning parameters,

the obtained results oer insights into stable or

unstable xed points, which are illustrated by time

history graphs of solutions, phases, and ampli-

tudes.

In the following, we propose two dierent ap-

proaches to solve the optimal control problem (8)

and (9). We call this problem as time optimal

control problem with the corresponding dynami-

cal system for certain initial x0= (θ0,˙

θ0)Tand

nal desired xtf= (θtf,˙

θtf)Tconditions.

The rst approach: Although, one can con-

sider the simple dynamics of a vibrating spring

where the control is interpreted as an exterior force

acting on an oscillating weight of unit mass hang-

ing from a spring, here we use this idea for control-

ling the armed manipulator robots. For one armed

manipulator robot, according to (5) one can write

θ1=−m1glc1

m1l2

c1+I1cos(θ1) + ˆτ1,

inwhich ˆτ1=1

m1l2

c1+I1τ1. The dynamical system

˙x(t) = x2

−m1glc1

m1l2

c1+I1cos(x1) + ˆτ1!

= x2

−m1glc1

m1l2

c1+I1cos(x1)!+0

ˆτ1.(10)

We use the rst two terms of the Taylor expansion

around point −π

2for cos(x1)thus

˙x(t) = x2

rx1+π

2+0

k τ1,(11)

inwhich r=−m1glc1

m1l2

c1+I1and k=1

m1l2

c1+I1. According

to Pontryagin minimum principle, there exists a

nonzero vector hsuch that

(hTx−1(t)N)τ∗

1(t) = max

τ1(.)n(hTx−1(t)N)τ1(t)o,

(12)

where τ∗

1(t)is the optimal torque for one arm

robot, for each time 0≤t≤t∗

f. Hence

˙x(t) = 0 1

r0 x1

x2+0

kτ1+0 0

rπ

20.

(13)

Naming

M=0 1

r0,N=0

k,

yields

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

188

Volume 19, 2024

Mn=





rn−1

2Mn= 1,3,5, ...

2I n = 2,4,6, ... .

In order to derive x−1(t)in (12) we use the expan-

sion of etM as

etM =I+tM +t2

2!M2+t3

3!M3+...

=cos(rt) sin(rt)

rsin(rt) cos(rt)

consequently

x−1(t) = cos(rt) sin(rt)

rsin(rt) cos(rt)−1

det cos(rt)−sin(rt)

−rsin(rt) cos(rt)

inwhich det = cos2(rt)−rsin2(rt). Finally, ac-

cording to (12) we have

τ∗

1(t) = k

det sgn (−h1sin(rt) + h2cos(rt)) .(14)

To simplify further, we choose δsuch that

h1=−cos(δ)and h2= sin(δ)

when h12+h22= 1. By recalling the trigonometry

identity sin(a+b) = sin(a) cos(b) + sin(b) cos(a),

one can write

τ∗

1(t) = k

det sgn (sin(δ+rt)) ,(15)

ie., the optimal control torque is found and τ∗

switches from k

det to −k

det and vice versa, every

2π

runits of time. Thus, our goal, which is to de-

sign an optimal exterior torque τ∗

1(.)that brings

the motion to a stop in minimum time t∗

fis suc-

cessfully achieved. Numerical results for a robot

with one arm is given in Table 3.

The second approach, [20]: The authors

considered the Variation Evolving Method (VEM)

that was developed for the classic time-optimal

control problem by initializing the transformed

IVP with arbitrary initial values of variables.

Here, we use their idea to solve the problem of con-

trolling armed manipulator robots in the shortest

Table 3: First approach for a robot with one arm.

M (g) 2π

r±k

det

21.2 0.166 2.662

31.2 0.177 2.653

51.8 0.198 2.625

100.6 0.237 2.552

121.9 0.258 2.491

130.7 0.267 2.459

150.5 0.290 2.367

200 0.324 2.232

500 0.452 1.614

1000 0.627 0.354

amount of the shortest amount of time. Math-

ematically, the dierence is that we face initial-

boundary-value (IBVP) problems instead of IVP

problems. Moreover, we use the rst optimize,

then discretize technique for a nite-dimensional

IBVP as follows:

To solve the time optimal control problems (8)

and (9) the Hamiltonian equation

H= 1 + ρT˙x (16)

is used and for 0≤t≤tf

∗the response

(x∗,τ∗,ρ∗, tf

∗)according to

H(x∗(t),τ∗(t),ρ∗(t)) = min

τH(x∗(t),τ(t),ρ∗(t)),

(17)

can be computed by solving the following canoni-

cal system of equations (optimality conditions)











∂H

∂x(x∗(t),τ∗(t),ρ∗(t)) = −˙ρ∗(t)

∂H

∂ρ(x∗(t),τ∗(t),ρ∗(t)) = ˙x∗(t)

H(x∗(t),τ∗(t),ρ∗(t))|t=t∗

f= 0,

(18)

where τ∗and x∗are optimal torque and optimal

state, τand xare admissible torque and state, ρ

is Lagrange multiplier and ρ∗is optimal costate.

For a robot with one arm the Hamiltonian

equation is:

H= 1 + ρ1x2+ρ2rcos(x1) + ρ2kτ.

According to (17) one can write

τ∗

1=−ρ2.(19)

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

189

Volume 19, 2024

For a robot with two arms the Hamiltonian equa-

tion is:

H=ρ2(¯

d11(τ1−c1−g1) + ¯

d12(τ2−c2−g2))

+ρ4(¯

d21(τ1−c1−g1) + ¯

d22(τ2−c2−g2))

+ρ1x2+ρ3x4+ 1,

then from (17) we get







τ∗

1=−(ρ2¯

d11 +ρ4¯

d21)

τ∗

2=−(ρ2¯

d12 +ρ4¯

d22).(20)

For a robot with three arms the Hamiltonian equa-

tion is:

H=ρ2(¯

d11(τ1−c1−g1) + ¯

d12(τ2−c2−g2)

+¯

d13(τ3−c3−g3)) + ρ4(¯

d21(τ1−c1−g1)

+¯

d22(τ2−c2−g2) + ¯

d23(τ3−c3−g3))

+ρ6(¯

d31(τ1−c1−g1) + ¯

d32(τ2−c2−g2)

+¯

d33(τ3−c3−g3)) + ρ1x2+ρ3x4+ρ5x6+ 1,

according to (17) we have











τ∗

1=−(ρ2¯

d11 +ρ4¯

d21 +ρ6¯

d31)

τ∗

2=−(ρ2¯

d12 +ρ4¯

d22 +ρ6¯

d32)

τ∗

3=−(ρ2¯

d13 +ρ4¯

d23 +ρ6¯

d33)

(21)

where ¯

dij for i, j = 1,2,3are components of

D−1(θ). Thus, the response (x∗,τ∗,ρ∗, tf

∗)can

be computed by replacing (19), (20) and (21) into

(18) for a robot with one, two and three arms re-

spectively.

4 PID controller problem

In general, the main purpose of a PID controller

is to bring the nal result of the system process

closer to the desired value.

In the year 2015 robust adaptive PID control

schemes with known or unknown upper bound

of the external disturbances are applied to solve

the strong nonlinearity and coupling problems in

robot manipulator control problems, [21].

In the year 2016 a PID controller for the simula-

tion of a robotic force control application demon-

strating well-damped control with no requirement

for a force-rate signal is applied, [22].

In the year 2022 a PID controller for the trajec-

tory tracking of robotic manipulators with known

or unknown upper bounds of the uncertainties is

applied, [16].

Here, the goal is to carry some payloads from

the given starting position to the given nal po-

sition. Thus, for a dened torque ˆτ(θ(t)) as a

controller output, the nal form of parallel PID

algorithm for an armed manipulator robot is

Output(t)≡ˆτ(θ(t)) = D−1(θ)τ

=kPe(θ(t)) + kD

de(θ(t))

d(t)+kIZt

e(α)dα

(22)

where kP,kIand kDare the proportional, integral

and derivative gains as tuning parameters, respec-

tively, tis the present time or instantaneous time,

αis the variable of integration and e(θ(t)) stands

for error. Therefore, the conversion or Laplace

transformation function G(s)for the PID con-

troller is

G(s) = kP+kI

s+kDs, (23)

inwhich sis the complex frequency (Laplace trans-

formation variable), [23], [24]. The proportional

term involving the kPcoecient increases system

speed and reduces the permanent state dierences

(but does not take zero). Adding an integral term

involving the kIcoecient vanishes the perma-

nent state dierences, but adds a lot of overshoot

to the transient response. The derivative term in-

volving kDcoecient attenuates the transient re-

sponse uctuations and brings the step response

closer to the ideal desired angle in radians, [18].

Combination of robot dynamic system (1) and

PID controller (22) yields

θ=D−1(θ)[−C(θ,˙

θ)−g(θ)] + ˆτ(24)

where the n-vector ˆτis

ˆτ=D−1(θ)τ(25)

and the error or dierence is considered as

e(θi) = ˜

θi−θi, i = 1, ..., n (26)

inwhich ˜

θiis the given nal position and θiis the

instantaneous nal position of each joint for the

arm robot in radians.

From (22) and (23), the matrix form for the PID

controller of each conversion function vector input

ˆτ=kPe+kD˙e +kIZedt. (27)

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

190

Volume 19, 2024

According to robot dynamic (24) and PID con-

troller algorithm (27) for one armed robot (R-

type) we have

θ1=−m1glc1cos(θ1)

m1l2

c1+I1

+kP1(˜

θ1−θ1) + kD1˙

θ1

+kI1Ze(θ1)dt, (28)

for two armed robot (RR-type) we get

θ1

θ2!=d11 d12

d21 d22−1− h˙

θ2h˙

θ1+h˙

θ2

−h˙

θ10!

θ1

θ2!−g1

g2

+ kP1(˜

θ1−θ1) + kD1˙

θ1+kI1Re(θ1)dt

kP2(˜

θ2−θ2) + kD2˙

θ2+kI2Re(θ2)dt !

(29)

and for three armed robot (RRR-type) we can

write







θ1

θ2

θ3





=



d11 d12 d13

d21 d22 d23

d31 d32 d33



−1"−



c11 c12 c13

c21 c22 c23

c31 c32 c33









θ1

θ2

θ3





−



g3

#

+





kP1(˜

θ1−θ1) + kD1˙

θ1+kI1Re(θ1)dt

kP2(˜

θ2−θ2) + kD2˙

θ2+kI2Re(θ2)dt

kP3(˜

θ3−θ3) + kD3˙

θ3+kI3Re(θ3)dt 



.

(30)

The problem here is dened as the determina-

tion of the best possible control strategy (optimal

control torque vector ˆτ), which minimizes a per-

formance index as the dierence between the given

desired position and the instantaneous nal posi-

tion of each joint arm. The last arm has a payload

of mass M and the goal is to transport the mass

M from the initial position into the desired posi-

tion. The robot dynamic system under control is

described in (28), (29) and (30) for a robot with

one, two and three arms respectively. Thus, the

general corresponding optimal parallel PID con-

troller problem is as follows:











minτJ=Rtf

t0e(θ,˙

θ)dt,

S.t.

θ=D−1(θ)[−C(θ,˙

θ)−g(θ)]

+kPe+kD˙e +kIRedt

x0= (θ0,˙

θ0)T

xtf= (θtf,˙

θtf)T.

(31)

Note that, this problem is not considered the

minimum time control problem since the termi-

nal time is not free. This problem can be called

a xed endpoint for a xed time to some great

enough time extension. Moreover, components of

the n-vector ˆτincrease as the robot’s arms in-

crease. Thus, we face a nonlinear problem that is

far too dicult to solve, since the corresponding

dynamical problem is IBVP. To overcome these

diculties, the parallel PID controller technique

for manipulator robots with one, two and three

arms (R, RR and RRR types) is proposed, [14].

5 Work Examples

For one, two and three armed manipulator robots

with the assumptions in Table 1 and Table 2.

Firstly, we consider three optimal time control

problems in Work Examples 1, 2 and 3. The per-

formance index is stated by (8) while in the nonlin-

ear dynamical system, initial and nal states are

given by (9). Secondly, in Work Examples 4, 5 and

6 we consider three parallel PID controller prob-

lems as described by (31). In both cases, control is

the torque τ(t)while x(t)and ρ(t)are state and

costate, respectively. Then, we compare and dis-

cuss the numerical results for all the above cases.

5.1 Example 1 (time optimal for

R-type robot)

In this example, the one armed manipulator robot

(see Figure 1(a)) for carrying 10 dierent loads in

Table 2 from angle position −π

2to angle position

20 is considered. The initial and nal desired con-

ditions are:

θ1(0)

θ1(0) =−π

0and θ1(t∗

θ1(t∗

f)!=π

0

respectively. The optimal total time t∗

fis unknown

and will be found. The optimal angle θ∗(t), veloc-

ity ˙

θ∗(t)and torque τ∗(t)for t∈[0, t∗

f]must be

found.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

191

Volume 19, 2024

5.2 Example 2 (time optimal for

RR-type robot)

Here, the two armed manipulator robot (see Fig-

ure 1(b)) for carrying 10 dierent loads in Table

2 from angle position (−π

2,π

84 )to angle position

(π

20 ,−π

84 )is considered. The initial and nal de-

sired conditions are:







θ1(0)

θ2(0)

θ1(0)

θ2(0)







=





−π







and 





θ1(t∗

θ2(t∗

θ1(t∗

θ2(t∗







=





−π







respectively. The optimal total time t∗

fis unknown

and will be found. The optimal angle θ∗(t), veloc-

ity ˙

θ∗(t)and torque τ∗(t)for t∈[0, t∗

f]must be

found.

5.3 Example 3 (time optimal for

RRR-type robot)

In this Example the state is X(t) =

(θ1(t), θ2(t), θ3(t),˙

θ1(t),˙

θ2(t),˙

θ3(t))T.

The three armed manipulator robot (see Fig-

ure 1(c)) for carrying 10 dierent loads in Table 2

from angle position (−π

2,π

84 ,−π

84 )to angle position

(π

20 ,−π

84 ,π

84 )is considered. The initial and nal de-

sired conditions are:







θ1(0)

θ2(0)

θ3(0)

θ1(0)

θ2(0)

θ3(0)













−π







and







θ1(t∗

θ2(t∗

θ3(t∗

θ1(t∗

θ2(t∗

θ3(t∗













−π







respectively. The optimal total time t∗

fis unknown

and will be found. The optimal angle θ∗(t), veloc-

ity ˙

θ∗(t)and torque τ∗(t)for t∈[0, t∗

f]must be

found.

Examples 4, 5 and 6 are about PID control for

one-armed two-armed and three-armed robots, re-

spectively. In Section 4, we described the method

of calculating PID controller problem is proposed.

In Examples 4, 5 and 6, the initial and nal po-

sition values are xed while the optimal torque

is computed by specifying PID coecients. The

speed and position of the one and two armed

robots with the optimal torque are shown in Fig-

ure 3, Figure 5 and Figure 7 respectively.

5.4 Example 4 (PID controller for

R-type robot)

The one armed manipulator robot for carrying 10

dierent loads from angle position θ0=−π

2to an-

gle position θtf=π

20 , with the initial and nal

velocity equal to zero (˙

θ0=˙

θtf= 0,rad

sec )is consid-

ered. Suppose that t∈[0,10], i.e., tf= 10 is long

enough time to complete the robot work process.

The coecients of the PID controller are:

kP1= 30, kD1= 10, kI1= 50 (32)

5.5 Example 5 (PID controller for

RR-type robot)

Here, the two armed manipulator robot for car-

rying 10 dierent loads from angle position θ0=

(−π

84 ,π

84 )to angle position θtf= ( π

84 ,−π

84 )and the

initial and nal velocity equal to zero (˙

θ0=˙

θtf=

0,rad

sec )is considered. Suppose that t∈[0,10], i.e.,

tf= 10 is long enough time to complete the robot

work process. The PID coecients are:

kPi= 30, kDi= 10, kIi= 50, i = 1,2(33)

5.6 Example 6 (PID controller for

RRR-type robot)

The three armed manipulator robot for carry-

ing 10 dierent loads from angle position θ0=

(−π

2,π

84 ,−π

84 )to angle position θtf= ( π

20 ,−π

84 ,π

84 ),

with the initial and nal velocity equal to zero

(˙

θ0=˙

θtf= 0,rad

sec )is considered. Suppose that

t∈[0,10], i.e., tf= 10 is long enough time to com-

plete the robot work process. The PID coecients

are:

kPi= 30, kDi= 10, kIi= 50, i = 1,2,3(34)

6 Numerical results and

discussions

In this section, we discuss the numerical results

calculated for Examples 1, 2, 3, 4, 5 and 6. We

remind you that in each of the examples in the

Work Example section, 10 dierent masses in Ta-

ble 2 are added to the end of the last arm of each of

the one, two, and three armed robots. The results

related to the total time process are presented in

Table 4, Table 5 and Table 6. In these tables, the

rst column is related to the mass added to the

last arm, and the second column is the computed

minimum total time of the solution in problems

(8) and (9). The third column is related to the so-

lution for the problem (31). Total times in Table

4, Table 5 and Table 6 show the period time when

the robot is placed in the given nal position for

the rst time. Examples 1, 2 and 3 are solved.

Figure 2, Figure 4 and Figure 6 are optimal posi-

tion, velocity and torque diagrams obtained from

solving the time optimal control problems (8) and

(9). Examples 4, 5 and 6 are solved. Figure 3,

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

192

Volume 19, 2024

Figure 5 and Figure 7 are optimal position, veloc-

ity and torque diagrams obtained from solving the

PID controller problem (31). Note that all 10 pay-

loads are used. However, the graphs are presented

for the mass of 100.6 (g)and for the rest of the

masses, only the times are calculated and given in

tables.

In Section 6.1, results for Examples 1 and 4 are

compared when the one-armed robot is used. Ex-

ample 1 is the optimal time control problem while

Example 4 is a PID controller problem. Accord-

ing to Figure 2 and Figure 3, some resulting data

are compared in Table 4. Total time, Oscillations,

Range of speeds, Eect of payloads, and Stability

are discussed and compared for Examples 1 and 4.

6.1 Results comparison for Example 1

and 4

Numerical results for the solutions in Examples 1

and 4 for manipulator robots with one arm (see

Table 4, Figure 2 and Figure 3) show that:

(i) (Total time) Due to Figure 2, the minimum

time is t∗

f= 0.3319 seconds. In Example 4, for

the PID controller problem (see Figure 3), it takes

about 4.317 seconds for the payload to be in the

given nal position for the rst time (comparison

can be made by considering Table 4, Figure 2 and

Figure 3).

(ii) (Oscillations) The range of changes for posi-

tion, velocity and torque in Example 1 (see Figure

2) are greater than the corresponding results for

the PID controller problem (see Figure 3). In an

optimal time control problem, one needs that the

robot move very quickly to reach the nal given

position in the posible minimum time. In Exam-

ple 4, one needs to minimize the states during the

total time with the best possible torque. Thus

we expect a longer period of time with smoother

graphs compared to the resultes in Example 1.

(iii) (Range of speeds) Velocity comparison

(see Figure 2(b) and Figure 3(b)) shows that the

range of the velocity values in Example 1 is greater

than the velocity values in Example 4.

(iv) (Eect of payloads) Both problems for dif-

ferent payloads are solved. As the payloads are

increased, the time in both Examples 1 and 4 in-

creases (see Table 4).

(v) (Stability) As it is shown in Figure 3(a), Fig-

ure 3(b) and Figure 3(c) the solutions stay stable

even after the minimum torque is found. In Ex-

ample 1, after t=t∗

fthe process of computing is

stopped.

In the minimum time control problem, one

reaches the nal destination in the least time,

(with almost 3 oscillations in 0.312 to 0.405 sec-

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Time (sec)

-1.5

-1

-0.5

0.5

(rad)

Optimal Position

join1

(a) Optimal position value for robot with 1 arm

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Time (sec)

-1

Velocity (rad/s)

Optimal Velocity

join1

(b) Optimal velocity value for robot with 1 arm

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Time (sec)

-0.2

0.2

0.4

0.6

0.8

1.2

Torque (Nm)

Optimal Torque

join1

Figure 2: (Example 1) Time optimal control so-

lutions for an R-type armed manipulator robot

with a payload of 100.6 (g). As it is shown in

the three dierent graphs, the optimal minimum

time is t∗

f= 0.3319 (sec)and it is computed in the

process of solving the problem in Example 1: (a)

Optimal Position values against time are depicted

(b) Optimal Velocity values against time are de-

picted (c) Optimal Torque values against time are

depicted.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

193

Volume 19, 2024

0 2 4 6 8 10

Time (sec)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.1

(rad)

Position

join1

(a) Position value for robot with 1 arm

0 2 4 6 8 10

Time (sec)

-2

-1.5

-1

-0.5

0.5

velocity (rad/s)

Velocity

velocity1

(b) Velocity value for robot with 1 arm

0 2 4 6 8 10

Time (sec)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Torque (Nm)

Optimal Torque

torque1

Figure 3: (Example 4) PID controller numeri-

cal solutions for R-type armed manipulator robot

with payload 100.6 (g). As it is shown in the three

dierent graphs, the optimal minimum torque

is 0.3891 (Nm)at time 4.317 seconds (see Fig-

ure 3(c)) and it is computed in the process of

solving the problem in Example 4: (a) Posi-

tion values (corresponding to the optimal torque

value) against time are depicted (b) Velocity val-

ues (corresponding to the optimal torque value)

against time are depicted (c) Optimal Torque val-

ues against time are depicted.

Table 4: Total time (sec) for R-type manipulator

robot.

M (g) Total time (Example 1) Total time (Example 4)

21.2 0.312 4.260

31.2 0.321 4.271

51.8 0.330 4.301

100.6 0.331 4.317

121.9 0.347 4.328

130.7 0.358 4.336

150.5 0.363 4.345

200 0.382 4.369

500 0.395 4.415

1000 0.405 4.502

onds for mass between 21.2 to 1000 grams) how-

ever this itself causes wear and tear and sometimes

damage to the robot.

6.2 Results comparison for Example 2

and 5

Numerical results for the solutions in Examples 2

and 5 for a manipulator robot with two arms (see

Table 5, Figure 4 and Figure 5) show that:

(i) (Total time) Due to Figure 4, the minimum

time is t∗

f= 0.3837 seconds. In Example 5, for

the PID controller problem (see Figure 5), it takes

about 4.607 seconds for the payload to be in the

given nal position for the rst time (comparison

can be made by considering Table 5, Figure 4 and

Figure 5).

(ii) (Oscillations) The corresponding uctua-

tions of the position, velocity and torque graphs

for Example 2 (see Figure 4) are greater than the

results of the PID controller problem (see Figure

5). Since, in an optimal time control problem, one

needs the robot to move quickly enough to reach

the nal given position in the minimum time. As

it was expected in Example 5, (one needs to do

the work with the best torque), thus we expect a

longer period of time with smoother graphs com-

pared to the results of Example 2, while the opti-

mal torque value by solving problem (31) is greater

(see Figure 4(c) and Figure 5(c)).

(iii) (Range of speeds) Velocity comparison

(see Figure 4(b) and Figure 5(b)) shows that the

range of the velocity values in Example 2 is greater

than the velocity values in Example 5.

(iv) (Eect of payloads) Both problems for dif-

ferent payloads are solved. As the payloads are

increased, the time in both Examples 2 and 5 in-

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

194

Volume 19, 2024

creases (see Table 5).

(v) (Stability) As it is shown in Figure 5(a), Fig-

ure 5(b) and Figure 5(c) the solutions stay stable

even after the minimum torque is found. In Ex-

ample 2, after t=t∗

fthe process of computing is

stopped.

Table 5: Total time (sec) for RR-type manipulator

robot.

M (g) Total time (Example 2) Total time (Example 5)

21.2 0.342 4.213

31.2 0.360 4.320

51.8 0.379 4.568

100.6 0.383 4.607

121.9 0.399 4.779

130.7 0.410 4.880

150.5 0.432 4.995

200 0.449 5.005

500 0.560 5.012

1000 0.584 5.029

6.3 Results comparison for Example 3

and 6

Numerical results for the solutions in Examples 3

and 6 for a manipulator robot with three arms (see

Table 6, Figure 6 and Figure 7) show that:

(i) (Total time) Due to Figure 6, the minimum

time is t∗

f= 0.6257 seconds. In Example 6, for

the PID controller problem (see Figure 6), it takes

about 6.235 seconds for the payload to be in the

given nal position for the rst time (comparison

can be made by considering Table 6, Figure 6 and

Figure 7).

(ii) (Oscillations) The corresponding uctua-

tions of the position, velocity and torque graphs

for Example 3 (see Figure 6) are greater than the

results of the PID controller problem (see Figure

7). Since, in an optimal time control problem, one

needs that the robot move quickly enough to reach

the nal given position in the minimum time. As

it was expected in Example 6, (one needs to do

the work with the best torque), thus we expect

a longer period of time with more smooth graphs

compared to the results of Example 3, while the

optimal torque value by solving problem (31) is

greater (see Figure 6(c) and Figure 7(c)).

(iii) (Range of speeds) Velocity comparison

(see Figure 6(b) and Figure 7(b)) shows that the

range of the velocity values in Example 3 is greater

than the velocity values in Example 6.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Time (sec)

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0.2

(rad)

Optimal Position

join1

join2

(a) Optimal position value for robot with 2 arms

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Time (sec)

-20

-15

-10

-5

Velocity(rad/s)

Optimal Velocity

join1

join2

(b) Optimal velocity value for robot with 2 arms

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Time (sec)

-1

-0.5

0.5

1.5

Torque (Nm)

Optimal Torque

join1

join2

Figure 4: (Example 2) Time optimal control so-

lutions for an RR-type armed manipulator robot

with a payload 100.6 (g). As it is shown in the

three dierent graphs, the optimal minimum time

is t∗

f= 0.3837 (sec)and it is computed in the

process of solving the problem in Example 2: (a)

Optimal Position values against time are depicted

(b) Optimal Velocity values against time are de-

picted (c) Optimal Torque values against time are

depicted.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

195

Volume 19, 2024

0 2 4 6 8 10

Time (sec)

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.6

(rad)

Position

join1

join2

(a) Position value for robot with 2 arms

0 2 4 6 8 10

Time (sec)

-2.5

-2

-1.5

-1

-0.5

0.5

1.5

2.5

velocity (rad/s)

Velocity

velocity1

velocity2

(b) Velocity value for robot with 2 arms

0 2 4 6 8 10

Time (sec)

0.2

0.4

0.6

0.8

1.2

1.4

Torque (Nm)

Optimal Torque

torque1

torque2

Figure 5: (Example 5) PID controller numerical

solutions for RR-type armed manipulator robot

with payload 100.6 (g). As it is shown in the three

dierent graphs, the optimal minimum torque for

the rst joint 1.242 (N m)and the second joint

0.3396 (Nm)at time 4.607 seconds is calculated

(see Figure 6(c)) and it is computed in the pro-

cess of solving the problem in Example 5: (a) Po-

sition values (corresponding to the optimal torque

value) against time are depicted (b) Velocity val-

ues (corresponding to the optimal torque value)

against time are depicted (c) Optimal Torque val-

ues against time are depicted.

(iv) (Eect of payloads) Both problems for dif-

ferent payloads are solved. As the payloads are

increased, the time in both Examples 3 and 6 in-

creases (see Table 6).

(v) (Stability) As it is shown in Figure 7(a), Fig-

ure 7(b) and Figure 7(c), the solutions stay stable

even after the minimum torque is found. In Ex-

ample 1, after t=t∗

fthe process of computing is

stopped.

Table 6: Total time (sec) for RRR-type manipu-

lator robot.

M (g) Total time (Example 3) Total time (Example 6)

21.2 0.566 6.202

31.2 0.582 6.218

51.8 0.601 6.229

100.6 0.625 6.235

121.9 0.651 6.248

130.7 0.683 6.260

150.5 0.701 6.275

200 0.719 6.298

500 0.723 6.354

1000 0.758 6.407

6.4 Total torques computation for

Examples 1 to 6

The trapezoidal numerical integral method is used

to calculate the total torque for each arm.

Ztf

|τtotal(t)|dt =tf−t0

n=1

[|τtotal(tn)|+|τtotal(tn+1)|]

(35)

where the spacing between each point is equal to

the scalar value tf−t0

N= 1.

The numerical results for robots of R, RR and

RRR types with payload of 100.6 (g)are given in

Table 7. According to Table 7, the total torque

increases with the increase in the number of arms

in Examples 1, 2, ..., 6. In the PID controller

strategy for RR and RRR type robots, it is found

that the rst arm consumes more torque compared

to the other arms.

7 Conclusions

This paper presents the mathematical simulation

of an armed manipulator robot of R, RR, RRR

type for the application of lifting and moving light

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

196

Volume 19, 2024

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time (sec)

-2.5

-2

-1.5

-1

-0.5

0.5

1.5

(rad)

Optimal Position

join1

join2

join3

(a) Optimal position value for robot with 3 arms

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time (sec)

-25

-20

-15

-10

-5

Velocity(rad/s)

Optimal Velocity

join1

join2

join3

(b) Optimal velocity value for robot with 3 arms

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time (sec)

-10

-8

-6

-4

-2

Torque (Nm)

Optimal Torque

join1

join2

join3

Figure 6: (Example 3) Time optimal control so-

lutions for an RRR-type armed manipulator robot

with a payload of 100.6 (g). As it is shown in the

three dierent graphs, the optimal minimum time

is t∗

f= 0.6257 (sec)and it is computed in the

process of solving the problem in Example 3: (a)

Optimal Position values against time are depicted

(b) Optimal Velocity values against time are de-

picted (c) Optimal Torque values against time are

depicted.

0 2 4 6 8 10

Time (sec)

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.1

(rad)

Position

join1

join2

join3

(a) Position value for robot with 3 arms

0 2 4 6 8 10

Time (sec)

-4

-3

-2

-1

velocity (rad/s)

Velocity

velocity1

velocity2

velocity3

(b) Velocity torque value for robot with 3 arms

0 2 4 6 8 10

Time(sec)

-0.5

0.5

1.5

2.5

3.5

Torque(Nm)

Optimal Torque

torque1

torque2

torque3

Figure 7: (Example 6) PID controller numerical

solutions for RRR-type armed manipulator robot

with payload 100.6 (g). As it is shown in the three

dierent graphs, the optimal minimum torque

for the rst joint 1.356 (N m), the second joint

0.5411 (Nm)and the third joint 0.1338 (N m)at

time 6.235 seconds is calculated (see Figure 7(c))

and it is computed in the process of solving the

problem in Example 6: (a) Position values (cor-

responding to the optimal torque value) against

time are depicted (b) Velocity values (correspond-

ing to the optimal torque value) against time are

depicted (c) Optimal Torque values against time

are depicted.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

197

Volume 19, 2024

Table 7: Total torque (Nm) for Examples 1 to 6.

Robot type Example #Total torque

R-type 1|τ1|=0.0977

4|τ1|=1.9141

RR-type

2|τ1|=0.2171

|τ2|=0.1062

5|τ1|=4.7883

|τ2|=1.3450

RRR-type

|τ1|=0.1062

3|τ2|=0.5022

|τ3|=0.2673

|τ1|=8.8336

6|τ2|=2.5010

|τ3|=0.5668

material objects between 21 to 1000 grams. The

torque controller is used to control the servo mo-

tor in combined motion and the robot end-eector.

The robotic problem is mathematically formulated

as: (i) The Time Optimal Control Problem, (ii)

The PID Controller Problem. Numerical results

show that the rst strategy gives better solutions

considering time, speed and torque consumption.

However, the PID controller strategy will give a

smoother solution with relatively greater torque,

a lower speed and a longer time. In both cases,

manipulator robots can successfully move objects

with a maximum weight of 1 kg based on the given

robotic assumptions and problem hypotheses. The

nal conclusion in the application of method 1 and

2 is that although method 1 announces smaller nu-

merical results, in practice it may cause the failure

of the devices, while in method 2, although the

numerical results are larger, but from the point

of view of damage to the devices, it is reasonable

It looks better. Moreover, computational results

show that when the number of jointed arms is in-

creased both total time and total torque increase,

respectively. However, in the PID strategy, when

we add an arm to the robot, the operational torque

needed for the rst arm is greater than the others.

8 Future work

We intend to investigate the following topics in the

future:

(i) The use of articial intelligence as well as

interval calculations for the problem of the least

optimal time for the robot dynamic system.

(ii) Combination of the methods with Luen-

berger observers.

(iii) Combinations of the methods with compu-

tational Intelligence.

(iv) Stability and the stability margin of the

system.

References:

[1] Isabel M, Ferreira A, Sequeira J and Virk G

(2020) Robotics and Well-Being. Springer.

[2] Spong M, Hutchinson W and Vidyassagar S

(2020) Robot Modeling and Control. Wilely.

[3] Gallant A and Gosselin C (2018) Extending

the capabilities of robotic manipulators using

trajectory optimization. Mechanism and Ma-

chine Theory. 121. 502-514.

[4] Weber A (2020) Small robots play a big tole

in automation. Assembly Magazine.

[5] Partida D (2021) Benets of miniature indus-

trial robots. Robotics Tomorrow. 21 Septem-

ber 2021.

[6] Malek A, Jafarian-Khaled Abad L and

Khodayari-Samghabadi S (2015) Semi-

Innite programming to solve armed robot

trajectory problem using recurrent neural

network. International Journal of Robotics

and Automation. 30 (2).

[7] Jazar RN (2022) Time optimal control. In:

Theory of Applied Robotics (pp. 731-757),

Springer, Cham.

[8] Gattringer H, Mueller A, Oberherber M and

Kaserer D (2022) Time-optimal robotic ma-

nipulation on a predened path of loosely

placed objects. Modeling and experiment.

Mechatronics. 84, 102753.

[9] Rojas-Quintero JA, Dubois F and Ramirez-

de-Avila HC (2022) Riemannian formula-

tion of Pontryagin’s maximum principle for

the 0ptimal control of robotic manipulators.

Mathematics. 10(7) 1117.

[10] Troltzsch F (2024) Optimal Control of Partial

Dierential Equations Theory, Methods and

Applications.Amer Mathematical Society.

[11] Rojas-Quintero JA, Rojas-Estrada JA,

Villalobos-Chin J, Satibanez V and

Bugarin E (2022) Optimal controller

applied to robotic systems using covariant

control equations. International Journal of

Control. 95(6): 1576-89.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

198

Volume 19, 2024

[12] Ferrentino E, Cioppa A D, Marcelli A

and Chiacchio P (2020) An Evolution-

ary Approach to Time-Optimal Control of

Robotic Manipulators. Journal of Intelligent

&Robotic Systems. 99: 245-260.

[13] Jin W, Kulic D, Mou S and Hirche S (2021)

Inverse optimal control from incomplete tra-

jectory observations. The International Jour-

nal of Robotics Research. 40(6-7):848-865.

[14] Giv F and Malek A (2024) Proportional-

integral-derivative controller for armed ma-

nipulator robots. International journal of

robotics and automation. 39(3): 170-180.

[15] Sachan S, Goud H and Swarnkar P (2022)

Performance and stability analysis of in-

dustrial robot manipulator. In Intelligent

Computing Techniques for Smart Energy .

Springer. pp. 473-481.

[16] Qiao L, Zhao M, Wu C, Ge T, Fan R

and Zhang W (2022) Adaptive PID control

of robotic manipulators without equality/in-

equality constraints on control gains. Interna-

tional Journal of Robust and Nonlinear Con-

trol. 32(18): 9742-9760.

[17] Alonso E and Garcia M (2015) Numerical

Modelling in Robotics. OmniaScience.

[18] Verl A, Schaer A, Raatz A and Brock O

(2015) Soft Robotics Transferring Theory to

Application. Springer Verlag.

[19] Aschepkov LT, Dolgy DV, Kim T and Agar-

wal P (2016) Optimal control. Springer.

[20] Zhang S and Qian W (2017) Computation of

time-optimal control problem with variation

evolution principle. arXiv. 1711.02998.

[21] Xu J and Qiao L (2015) Robust adaptive PID

Control of robot manipulator with bounded

disturbances. Robust Control with Engineer-

ing Applications. 535437.

[22] Armstrong B and Wade BA (2016) Nonlin-

ear PID control with partial state knowledge:

Damping without derivatives. International

Journal of Robotics Research. 19(8).

[23] Daniel Abebe Beyene (2024) Modern Control

System Lecture Note. Addis Ababa Science

and Technology University.

[24] Wang T, Wang H, Wang C and Hu H

(2022) A novel PID controller for BLDCM

speed control using dual fuzzy logic systems

with HSA optimization. Scientic Reports.

12, 11316.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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

_US

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2024.19.20

Farideh Giv, Alaeddin Malek

E-ISSN: 2224-2856

199

Volume 19, 2024