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The Transactions of the Korean Institute of Electrical Engineers

ISO Journal TitleTrans. Korean. Inst. Elect. Eng.

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The Transactions of the Korean Institute of Electrical Engineers

KIEE Vol. 73, No. 03, p.576-584

ISSN (print) :

1975-8359

ISSN (online) :

2287-4364

Received : 17 November 2023Revised : 01 February 2024Accepted : 11 February 2024

DOI :

https://doi.org/10.5370/KIEE.2024.73.3.576

궤적 제어 시스템을 이용한 움직이는 물체의 적응형 위치 결정 및 동적 모델 특성 합성 (멀티링크 매니퓰레이터의 경우)

Adaptive Posision-Determination and Dynamic Model Properties Synthesis of Moving Objects With Trajectory Control System (In the Case of Multi-Link Manipulators)

(Zaripov Oripjon Olimovich) ^†iD (Sevinova Dildora Usmonovna) ¹iD (Bobojanov Sukhrob Gayratovich) ²iD

(Doctoral student, Tashkent State Technical University, Tashkent city, Uzbekistan E-mail: sevinovadildora@gmail.com)
(Kumoh National Institute of Technology, Gumi-si, Gyeongsangbuk-do, South Korea E-mail: w.suxrob.w@gmail.com)

^†Corresponding Author: Professor, Tashkent State Technical University, Tashkent city, Uzbekistan E-mail: o.zaripov@edu.uz

License :

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0)which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Translated Abstract

Characteristics of the dynamic model of moving objects with an adaptive position-trajectory control system. In the example of industrial robots, many technological operations are determined by the combination of forces, accelerations, speeds, positions and directions possible in the processes of interaction of the working body with objects. In the case of multi-link manipulators of moving objects, in determining and synthesizing the characteristics of the dynamic model, in the process of determining the position of the manipulator, the accuracy of the absolute positions of the links and executive elements relative to each other, the speed of movement in different coordinate systems, angular accelerations and the high accuracy of its mobility are one of the important indicators. In this article, the dynamic characteristics of robots in motion of the type "Efort collaboration robot 5", PUMA and SCARA industrial robots are considered in the determination and synthesis of dynamic model characteristics on the example of multi-link manipulators of moving objects. When determining the characteristics of the dynamic model of multi-link manipulators, mathematical models for calculating the total kinetic energy, potential energy and acceleration of each link in terms of generalized coordinates were considered. Also presented are kinematic schemes that allow determining the inertial characteristics of the level of mobility, the component of Coriolis forces, the forces of gravity and the interaction between all links of the mobility of the manipulator during movement and calculating the levels of mobile movement for the industrial robots "Efort collaboration robot 5" PUMA and SCARA An algorithm for determining the dynamic characteristics of multi-link manipulators has been developed based on the analysis and calculated research results and the presented mathematical models.

Key words

adaptive position control, multi-link manipulator, dynamic model, mathematical model, industrial robot, movement dynamics, kinematic scheme.

1. Introduction

Intelligent robots and robotic technologies are widely used in the management of facility processes in automated technological complexes. The efficiency of the manufacturing industry is directly related to robots and robotic technologies or multi-link manipulators with adaptive position-trajectory control systems ^[1]. In recent years, not enough attention has been paid to the process of service and control of multiple machine tools of structural and kinematic synthesis of flexible industrial robots for adaptive position control systems. Including in this article, one of the urgent tasks is to determine and synthesize the dynamic characteristics of a multi-link industrial robot manipulator with an adaptive positional-trajectory control system of objects, and to determine the criteria characterizing the quality operation of the robot with the selected structures of the object. Also, one of the important tasks is to determine and synthesize the dynamic characteristics of moving objects with an adaptive position-trajectory control system, and to properly organize management processes in modern production enterprises.

Currently, Industry 4.0 technologies are widely used in production enterprises ^[1,^2]. It includes identifying and synthesizing the dynamic characteristics of a multi-link robot manipulator with an adaptive position-trajectory control system, modeling the dynamics of the manipulator as an executive body, solving a number of problems in the dynamics of the control system and the optimal control process ^[1,^3]. Analysis and synthesis of dynamic characteristics of multi-link robot manipulators in adaptive position-trajectory control systems requires the development of mathematical models using engineering computing methods. Also, on the basis of developed mathematical models, high accuracy of forces, accelerations, positions and directions of movement is achieved in the processes of interaction of the manipulator with the object ^[1,^2,^4]. The dynamic model of the manipulator allows to determine the driving forces and moments acting on the links of the object during movement.

2. Research methodology

Usually, the criteria for evaluating the dynamics of multi-link manipulators with an adaptive position-trajectory control system and achieving dynamic accuracy are the following variable parameters

$\dot{x}_{i}^{t}=\{ω_{1 i},\: ω_{2 i},\: ω_{3 i},\: υ_{1 i},\: υ_{2 i},\: υ_{3 i}\}$

(𝑖=1,2,…,𝑛) we define, where 𝑖− is the link number; 𝑛− the number of links in the kinematic chain of the manipulator; 𝜈𝑖𝑡 − is the speed of their center of mass; 𝑖− displacement angular speed of the 𝑖− th link; 𝑇− transposition sign; It is necessary to take into account that the turning angle of 𝑞𝑖− manipulator link 𝑗− th link (𝑗=1,2,…,𝑝) is interconnected. In most adaptive position-trajectory controlled industrial robots, the possible velocities of the elements of the dynamic system, when its configuration is set, generate the coupling equations ^[2,^3,^5].

$A_{\dot{x}}+ B_{\dot{q}}= C ,\: $

where $\dot{x}^{T}=\left\{\dot{x}_{1}^{T},\: …,\: \dot{x}_{n}^{T}\right\};$ 𝐴,𝐵,𝐶− matrices depending on position and time coordinates, respectively, 𝑟×6𝑛,𝑟×1 sizes; 𝑟− the number of connection equations.

In multi-link manipulators with an adaptive position-trajectory control system, active forces acting on the mechanism with real connections are: basic forces characterized by vectors

$F_{i}^{T}=\{F_{1i},\: F_{2i},\: F_{3i}\}$, 𝑖=1,2,…,𝑛

external forces acting on links characterized by initial moments $F_{i}^{T}=\{F_{1i},\: F_{2i},\: F_{3i}\}$, 𝑄𝑗(𝑗=1,2,…,𝑝) − control torques of drive motors. It is convenient to set 𝑣𝑖 and 𝐹𝑖 vectors in a fixed coordinate system, 𝑤𝑖, 𝑀𝑖− moving axes 𝑖− coincide with the main central inertia axes of the link.

3. Analysis and results

Kinematics is the first step in the study of the robot manipulator mechanism, considering only space-time relations and negligible forces and their effects. The study of the dynamic properties of the robot manipulator, as well as the organization of its control, requires the creation of a dynamic model of the robot of universal appearance, which is common to mechanisms with different kinematic structures. Including, we will consider the example of "Effort collaboration robot 5" (ECR5), PUMA and SCARA research object, built on the basis of the Lagrange-Euler equation in studying the dynamics of multi-link manipulators (MLM) ^[4,^5,^6]. Because in these robots, only mobile movement levels are considered. In addition, control parameters corresponding to the dynamic characteristics of movement levels in the specified coordinate systems are close to each other. The Lagrange-Euler equation is used to calculate the criterion for evaluating the dynamic characteristics of multi-link manipulators and the factors affecting their high accuracy in service, and its general form is as follows:

(1)

$P_{i}=\dfrac{d}{dt}\dfrac{d(T-П)}{d\dot{q}_{i}}-\dfrac{d(T-П)}{dq_{i}},\: (i=\overline{1,\: n}),\:$

where 𝑛− is the number of movement levels of MLM; 𝑖− mobility level number (𝑖=1,𝑛); 𝑇− total kinetic energy of the executive body and KLM; П−the total potential energy of the manipulator with the executive body; 𝑞𝑖− corresponding generalized coordinates of MLM; 𝑞̇𝑖 –derivatives of MLM generalized coordinates; The moments of the generalized forces created in the 𝑖− th position for the implementation of the given movement of the 𝑃𝑖− link are described.

With the help of this proposed ^[7,^8,^9] mathematical expression, it is convenient to express the effect of full moment in the 𝑖− th link of 𝑃𝑖 (1) MLM in the following form:

(2)

$P_{i}=H_{i}(q)\ddot{q}_{i}+h_{i}(q,\: \dot{q})\dot{q}_{i}+M_{E}i(q,\: \dot{q},\: \ddot{q)},\: (i=\overline{1,\: n}),\: $

where 𝐻𝑖(𝑞)− is the component describing the inert properties of the corresponding MLM mobility level; ℎ𝑖(𝑞,𝑞̇)− component of Coriolis forces; 𝑀𝑖(𝑞, 𝑞̇, 𝑞̈) − is a one-level movement that takes into account the interaction between gravity forces and all levels and positions of the mobility of the manipulator during movement; 𝑞̈𝑖− describes the generalized acceleration of MLM.

With the help of this mathematical expression, the kinematic schemes of ECR5, PUMA and SCARA robots of the MLM type, which allow to calculate the degrees of positional movement, are studied (Figures 1,2,3). The following symbols are given in these pictures: 𝑙𝑖− length, the 𝑖− th link of the MLM, the distance 𝑙𝑖∗− from the center of the 𝑖− th link to the axis of rotation, the 𝑚𝑖 mass of the 𝑖− th link, 𝑚𝑔− describes the weight of the manipulator's executive body load ^[10,^11,^12]. The total potential energy of MLM includes the potential energy of all its links and loads (Figures 1-2). Calculation of potential energy for robots of type ECR5, PUMA based on the mentioned kinematic schemes is carried out by the following relationship:

(3)

$П=П_{2}+П_{3}+П_{g}=g\left[m_{2}l_{2}^{*}\sin q_{2}+m_{3}\left(l_{2}\sin q_{2}+l_{2}^{*}\sin\left(q_{2}+q_{3}\right)\right)+\right .$

$\left . +m_{g}\left(l_{2}\sin q_{2}+l_{2}\sin\left(q_{2}+q_{3}\right)\right)\right],\:$

where П2,П3,П𝑔− are the potential energies of the second and third links of MLMs and the executive load, respectively; 𝑔− acceleration of free fall.

According to the specified coordinate systems, the kinetic energy of each MLM link is equal to the sum of the forward movements of their centers of mass and the kinetic energies of rotational movements relative to these centers of mass:

(4)

$T=\dfrac{1}{2}m_{i}v_{i}^{2}+\dfrac{1}{2}J_{ωi}ω_{i}^{2},\: (i=\overline{1,\: n}),\: $

where 𝑣𝑖2− square of the speed of the forward movement of the center of mass of the 𝑖− th link; 𝐽𝜔𝑖− the moment of inertia of the 𝑖− th link in relation to the axis passing through the mass center of the link; describes the square of the rotation speed 𝜔𝑖2− around the axis of rotation of the 𝑖− th link.

Taking into account (4), the following expression can be used to calculate the total kinetic energy of ECR5, PUMA type MLMs:

(5)

$T= T_{VR1}+ T_{VR2}+ T_{VR3}+ T_{П2}+ T_{П3}+ T_{Пg},\: $

where 𝑇_П2, 𝑇_П3, 𝑇_П𝑔− the kinetic energy of the forward movement of the corresponding link centers and load centers; 𝑇𝑉𝑅1, 𝑇𝑉𝑅2, 𝑇𝑉𝑅3 − describe the kinetic energies of the rotational movements of the corresponding links relative to their centers of mass.

그림 1. 다중 링크 산업용 로봇 매니퓰레이터 유형 ECR5의운동학 다이어그램.

Fig. 1. Kinematic diagram of multi-link industrial robot manipulator type ECR5.

그림 2. PUMA형 다중링크 산업용 로봇 매니퓰레이터의 운동학적 다이어그램.

Fig. 2. Kinematic diagram of PUMA-type multi-link industrial robot manipulator.

그림 3. SCARA형 다중링크 산업용 로봇 매니퓰레이터의 운동학적 다이어그램.

Fig. 3. Kinematic diagram of SCARA-type multi-link industrial robot manipulator.

In this case, the kinetic energies of rotational movements of the MLM links relative to the centers of mass 𝑇𝑉𝑅1 have the following form:

(6)

$T_{VR1}=\dfrac{1}{2}J_{Sl}\dot{q}_{l}^{2},\: $

𝐽𝑆𝑙 − is the moment of inertia of the first link of the manipulator MLM with respect to its longitudinal axis, and this link is considered symmetrical with respect to the axis.

The center of mass 𝑂𝑖 of each link is related to the correct coordinate system, whose axes 𝑂𝑖 , 𝑥𝑖 , 𝑦𝑖 , 𝑧𝑖 coincide with its main central inertial axes ^[13,^14].

The calculation of the moment of inertia with respect to an arbitrary axis of rotation forming angles 𝛼, 𝛽, 𝛾 − with the axes of this coordinate system can be written as follows ^[15,^16]:

(7)

$J_{ωi}= J_{\xi}\cos^{2}α_{i}+ J_{yi}\cos^{2}β_{i}+ J_{zi}\cos^{2}γ_{i},\: $

where 𝐽𝑥𝑖 , 𝐽𝑦𝑖 , 𝐽𝑧𝑖 − are the main central moments of inertia of the link; 𝛼𝑖 , 𝛽𝑖 , 𝛾𝑖 − describe the angles formed by the axis of rotation with the main central axes of inertia.

For ECR5 and PUMA type MLMs with an adaptive position- trajectory control system, taking into account (7), we can write the moment of inertia of their second link relative to the axis of rotation passing through the center of mass 𝑂2 in the following form:

(8)

$J_{ω2}= J_{S2}\dfrac{\dot{q}_{1}^{2}}{ω_{2}^{2}}\sin^{2}q_{2}+ J_{N2}\dfrac{\dot{q}_{1}^{2}}{ω_{2}^{2}}\cos^{2}q_{2}+ J_{N2}\dfrac{\dot{q}_{2}^{2}}{ω_{2}^{2}},\: $

Taking into account the mentioned moment of inertia interlink interaction (8), the rotational kinetic energy of the second link of ECR5, PUMA type MLMs is determined by the following expression:

(9)

$T_{VR2}=\dfrac{1}{2}\left[J_{S2}\sin^{2}q_{2}+ J_{N2}\cos^{2}q_{2}\right]\dot{q}_{1}^{2}+\dfrac{1}{2}J_{N2}\dot{q}_{2}^{2}.$

For ECR5 and PUMA-type KZMs with an adaptive position-trajectory control system, the moments of inertia of the third link relative to the axes of rotation 𝜔3, its center of mass passes through𝑂2, taking into account the expression (7), the calculation of the kinetic energy of the third link is as follows:

(10)

$J_{\omega 3}=J_{S3}\dfrac{\dot{q}_{1}^{2}}{\omega_{3}^{2}}\sin^{2}\left(q_{2}+q_{3}\right)+J_{N3}\dfrac{\dot{q}_{1}^{2}}{\omega_{3}^{2}}\cos^{2}\left(q_{2}+q_{3}\right)+J_{N3}\dfrac{\left(\dot{q}_{2}+\dot{q}_{2}\right)^{2}}{\omega_{3}^{2}},\:$

and taking into account the kinetic energy expression (10), and for ECR5 and PUMA type MLMs, the rotational movement of the third link is as follows:

(11)

$T_{VR3}=\dfrac{1}{2}\left[J_{S3}\sin^{2}\left(\left(q_{2}+q_{3}\right)+J_{N3}\cos^{2}\left(q_{2}+q_{3}\right)\right]\dot{q}_{1}^{2}+\dfrac{1}{2}J_{N3}\left(\dot{q}_{2}+\dot{q}_{2}\right)^{2}.\right .$

The coordinates of the centers of mass of the third links of the ECR5 and PUMA type MLMs with an adaptive position-trajectory control system are as follows:

$x_{2}= l_{2}^{*}\cos q_{1}\cos q_{2},\: $

$y_{2}= l_{2}^{*}\sin q_{2},\: $

$z_{2}=- l_{2}^{*}\sin q_{1}\cos q_{2}$

and their time derivatives are as follows:

(12)

$\dot{x}_{2}=-\dot{q}_{1}l_{2}^{*}\sin q_{1}\cos q_{2}-\dot{q}_{2}\cos q_{1}si n q_{2},\: $

$\dot{y}_{2}=\dot{q}_{2}l_{2}^{*}\cos q_{2},\: $

$\dot{z}_{2}=-\dot{q}_{1}l_{2}^{*}\cos q_{1}+\dot{q}_{2}l_{2}^{*}\sin q_{1}\cos q_{2}$.

For ECR5 and PUMA-type MLMs with an adaptive position-trajectory control system, taking into account (12), the second link of the manipulator has the square representation of the linear velocity of the center of mass,

$v_{2}^{2}=\dot{q}_{2}^{2}l_{2}^{*2}+\dot{q}_{1}^{2}l_{2}^{*2}\cos^{2}q_{2}$,

the kinetic energy of the forward movement of the center of mass of this link is equal to:

(13)

$T_{П2}=\dfrac{1}{2}m_{2}[\dot{q}_{2}^{2}l_{2}^{*2}+\dot{q}_{1}^{2}l_{2}^{*2}\cos^{2}q_{2}$.

The calculation of the coordinates of the centers of mass of the third links of the ECR5 and PUMA type MLMs with an adaptive position-trajectory control system looks like this:

$x_{3}=\cos q_{1}\left[l_{2}\cos q_{2}+ l_{3}^{*}\cos\left(q_{2}+ q_{3}\right)\right],\: $

$y_{3}= l_{2}\sin q_{2}+ l_{3}^{*}\sin\left(q_{2}+ q_{3}\right),\: $

$z_{3}=-\sin q_{1}\left[l_{2}\cos q_{2}+\cos\left(q_{2}+ q_{3}\right)\right],\: $

and their time derivatives are as follows:

(14)

$\dot{x}_{3}=-\cos q_{1}\left[\dot{q}_{2}l_{2}\sin q_{2}+\left(\dot{q}_{2}+\dot{q}_{2}\right)l_{3}^{*}\sin\left(q_{2}+ q_{3}\right)\right]-$

$-\dot{q}_{1}\sin q_{1}[l_{2}\cos q_{2}+ l_{3}^{*}\cos\left(q_{2}+ q_{3}\right)]$,

$\dot{y}_{3}=\dot{q}_{2}l_{2}\cos q_{2}+\left(\dot{q}_{2}+\dot{q}_{2}\right)l_{3}^{*}\cos\left(q_{2}+ q_{3}\right)$,

$\dot{z}_{3}=\sin q_{1}\left[\dot{q}_{2}l_{2}\sin q_{2}+\left(\dot{q}_{2}+\dot{q}_{2}\right)l_{3}^{*}\sin\left(q_{2}+ q_{3}\right)\right]-$

$-\dot{q}_{1}\cos q_{1}[l_{2}\cos q_{2}+ l_{3}^{*}\cos\left(q_{2}+ q_{3}\right)]$.

As a result, taking into account (14), the expression for determining the linear speed of the center of mass of the third link of ECR5 and PUMA type KZMs with an adaptive position-trajectory control system is as follows:

$\begin{align*} v_{3}^{2}=\dot{q}_{1}^{2}\left[l_{2}\cos q_{2}+l_{3}^{*}\cos ⁡\left(q_{2}+q_{3}\right)\right]^{2}+\dot{q}_{1}^{2}l_{2}^{2}+\left(\dot{q}_{2}+\dot{q}_{2}\right)^{2}l_{3}^{*2}\\ ++2\dot{q}_{2}\left(\dot{q}_{2}++\dot{q}_{3}\right)l_{2}l_{3}^{*}\cos q_{3} \end{align*},$

for ECR5 and PUMA type KZMs, the expression for determining the kinetic energy of the forward movement of the center of mass of the third link is as follows:

(15)

$T_{П3}=\dfrac{1}{2}m_{3}\left\{\dot{q}_{1}^{2}\left[l_{2}\cos q_{2}+l_{3}^{*}\cos ⁡\left(q_{2}+q_{3}\right)\right]^{2}+\dot{q}_{1}^{2}l_{2}^{2}+\left(\dot{q}_{2}+\dot{q}_{2}\right)^{2}l_{3}^{*2}+\right .$

$+2\dot{q}_{2}(\dot{q}_{2}+\dot{q}_{3})l_{2}l_{3}^{*}\cos q_{3}\}$.

Determining the load coordinates of ECR5 and PUMA type MLMs with an adaptive position-trajectory control system can be carried out with the following expressions ^[2,^17]:

$x_{g}=\cos q_{1}[l_{2}\cos q_{2}+ l_{3}\left(q_{2}+ q_{3}\right)]$

$y_{g}= l_{2}\sin q_{2}+ l_{3}\sin\left(q_{2}+ q_{3}\right),\: $

$z_{g}=-\sin q_{1}\left[l_{2}\cos q_{2}+ l_{3}\cos\left(q_{2}+ q_{3}\right)\right],\: $

and the time derivatives of the load coordinates are determined using the following expression:

(16)

$\dot{x}_{g}=-\cos q_{1}\left[\dot{q}_{2}l_{2}q_{2}+\left(\dot{q}_{2}+\dot{q}_{3}\right)l_{3}\sin\left(q_{2}+ q_{3}\right)\right]-$

$-\dot{q}_{1}\sin q_{1}[l_{2}\cos q_{2}+ l_{3}\cos\left(q_{2}+ q_{3}\right)]$,

$\dot{y}_{g}=\dot{q}_{2}l_{2}\cos q_{2}+\left(\dot{q}_{2}+\dot{q}_{3}\right)l_{3}\cos\left(q_{2}+ q_{3}\right),\: $

$\dot{z}_{g}=\sin q_{1}\dot{q}_{2}l_{2}\sin q_{2}+\left(\dot{q}_{2}+\dot{q}_{3}\right)l_{3}\sin\left(q_{2}+ q_{3}\right)]-$

$-\dot{q}_{2}\cos q_{1}\left[l_{2}\cos q_{2}+ l_{3}\cos\left(q_{2}+ q_{3}\right)\right].$

As a result, taking into account (16), the expression of the linear speed of the movement of ECR5 and PUMA type MLM with load will have the following form:

$\begin{align*} v_{g}^{2}=\dot{q}_{1}^{2}\left[l_{2}\cos q_{2}+l_{3}\cos ⁡\left(q_{2}+q_{3}\right)\right]^{2}+\dot{q}_{1}^{2}l_{2}^{2}\\ +\left(\dot{q}_{2}+\dot{q}_{2}\right)^{2}l_{3}^{2}++2\dot{q}_{2}\left(\dot{q}_{2}+\dot{q}_{3}\right)l_{2}l_{3}\cos q_{3} \end{align*},$

and the expression of the kinetic energy of its forward movement is written as follows:

(17)

$T_{Пg}=\dfrac{1}{2}m_{g}\left\{\dot{q}_{1}^{2}\left[l_{2}\cos q_{2}+l_{3}\cos\left(q_{2}+q_{3}\right)\right]^{2}+\dot{q}_{2}^{2}l_{2}^{2}+\left(\dot{q}_{2}+\dot{q}_{2}\right)^{2}l_{3}^{2}+\right .$

$\left . +2\dot{q}_{2}\left(\dot{q}_{2}+\dot{q}_{3}\right)l_{2}l_{3}\cos q_{3}\right\}$.

Taking into account the above expressions (3), (5), (6), (9), (11), (13), (15) and (17), the Lagrangian function for ECR5 and PUMA typed MLMs will have the following form:

$L=(T-П)=\dfrac{\dot{q}_{1}^{2}}{2}\left[J_{S1}+J_{S2}\sin^{2}q_{2}+J_{N2}\cos^{2}q_{2}+J_{S3}\sin^{2}\left(q_{2}+q_{3}\right)+\right .$

$+ J_{N2}\cos^{2}\left(q_{2}+ q_{3}\right)+\left(m_{2}l_{2}^{*2}+ m_{3}l_{2}^{2}+ m_{g}l_{2}^{2}\right)co s^{2}q_{2}+$

$+\left(m_{3}l_{2}^{*2}+ m_{g}l_{2}^{2}\right)\cos^{2}\left(q_{2}+ q_{3}\right)+2 l_{2}\left(m_{3}l_{3}^{*}+ m_{g}l_{3}\cos q_{2}\cos\left(q_{2}+ q_{3}\right)\right]{+}$

$+\dfrac{\dot{q}_{2}^{2}}{2}[J_{N2}+ J_{N3}+ m_{2}l_{2}^{*2}+ m_{3}\left(l_{2}^{2}+ l_{3}^{*2}\right)+ m_{g}\left(l_{2}^{2}+ l_{3}^{*2}\right)+$

$+2\left(m_{3}l_{2}l_{3}^{*}+ m_{g}l_{2}l_{3}\right)\cos q_{3}]+\dfrac{\dot{q}_{3}^{2}}{2}\left[J_{N3}+ m_{3}l_{2}^{*2}+ m_{g}l_{3}^{2}\right]+\dot{q}_{2}\dot{q}_{3}[J_{N3}+ m_{3}l_{3}^{*2}+$

$ \left.\left.+m_g l_3^2+m_3 l_2 l_2^{* 2}+m_g l_2 l_3\right) \cos q_3\right]-g\left[m_2 l_2^* \sin q_2+m_3\left(l_2 \sin q_2+\right.\right. $

$+ l_{3}^{*}\sin\left(q_{2}+ q_{3}\right))+ m_{g}(l_{2}\sin q_{2}+ l_{3}\sin\left(q_{2}+ q_{3}\right))]$.

Taking this function and expression (1) into account, the calculation of all parameters of equation (2) for all mobility levels of ECR5, PUMA-type MLMs will have the same form, and their expressions can be written as follows:

– for a manipulator with one degree of freedom:

(18)

$H_{1}= J_{S1}+ J_{S2}\sin^{2}q_{2}+ J_{N2}\cos^{2}q_{2}+ J_{S3}\sin^{2}\left(q_{2}+ q_{3}\right)+ J_{N3}\cos^{2}\left(q_{2}+ q_{3}\right)+$

$+ m_{2}l_{2}^{*2}+ m_{3}l_{2}^{2}+ m_{g}l_{2}^{2})\cos^{2}q_{2}+\left(m_{3}l_{3}^{*2}+ m_{g}l_{3}^{2}\right)\cos^{2}\left(q_{2}+ q_{3}\right)+$

$+2\left(m_{3}l_{2}l_{3}^{*}+ m_{g}l_{2}l_{3}\right)\cos q_{2}\cos\left(q_{2}+ q_{3}\right)$,

$h_{1}=-\dot{q}_{2}\{\left[J_{N2}- J_{S2}+ m_{2}l_{2}^{*2}+ l_{2}^{2}\left(m_{3}+ m_{g}\right)\right]\sin^{2}q_{2}+ J_{N3}- J_{S3}+$

$+ m_{3}l_{3}^{*2}+ m_{g}l_{3}^{2})\sin 2\left(q_{2}+ q_{3}\right)+2 l_{2}\left(m_{3}l_{3}^{*}+ m_{g}l_{3}\right)\sin ⁡(2 q_{2}+ q_{3})\}-$

(19)

$-\dot{q}_{3}\left[\begin{aligned}\left(J_{N3}- J_{S3}+ m_{3}l_{3}^{*2}+ m_{g}l_{3}^{2}\right)\sin 2\left(q_{2}+ q_{3}\right)+ \\ +2 l_{2}\left(m_{3}l_{3}^{*}+ m_{g}l_{3}\right)\cos q_{2}\sin\left(q_{2}+ q_{3}\right)\end{aligned}\right],\: $

(20)

$M_{E1}=0$;

– for a manipulator with two degrees of freedom:

(21)

$H_{2}= J_{N2}+ J_{N3}+ m_{2}l_{2}^{*2}+ m_{3}\left(l_{2}^{2}+2 l_{2}l_{3}^{*}c os q_{3}+ l_{3}^{*2}\right)+$

$+ m_{g}(l_{2}^{2}+2 l_{2}l_{3}\cos q_{3}+ l_{3}^{2})$,

(22)

$h_{2}=-\dot{q}_{3}2\sin q_{3}\left[m_{3}l_{2}l_{3}^{*}+ m_{g}l_{2}l_{3}\right],\: $

(23)

$M_{E2}=\ddot{q}_{3}\left[J_{N3}+ m_{3}l_{3}^{*}\left(l_{3}^{*}+ l_{2}\cos q_{3}\right)+ m_{g}l_{3}\left(l_{3}+ l_{2}\cos q_{3}\right)\right]-$

$-\dot{q}_{3}^{2}l_{2}(m_{3}l_{3}^{*}+ m_{g}l_{3}\sin q_{3}+\dfrac{\dot{q}_{1}^{2}}{2}\{\left[J_{N2}- J_{S2}+ m_{2}l_{3}^{*2}+ l_{2}^{2}\left(m_{3}+ m_{g}\right)\right]\sin 2 q_{2}+$

$+\left(J_{N3}- J_{S3}+ m_{3}l_{3}^{*2}+ m_{g}l_{3}^{2}\right)\sin 2\left(q_{2}+ q_{3}\right)+$

$ \left.+2 l_2\left(m_3 l_3^*+m_g l_3\right) \sin \left(2 q_2+q_3\right)\right\}+g\left[m_2 l_2^*+m_3 l_2+m_g l_2 \cos q_2+\right. $

$+(m_{3}l_{3}^{*}+ m_{g}l_{3})\cos\left(q_{2}+ q_{3}\right)]$;

– for a manipulator with three degrees of freedom:

(24)

$H_{3}= J_{N3}+ m_{3}l_{3}^{*2}+ m_{g}l_{3}^{2},\: $

(25)

$h_{3}=0,\: $

(26)

$M_{E3}=\ddot{q}_{2}\left(J_{N3}+ m_{3}l_{3}^{*2}+ m_{g}l_{3}^{2}+ l_{2}\left(m_{3}l_{3}^{*}+ m_{g}l_{3}\right)\cos q_{3}\right]{+}$

$+\dfrac{\dot{q}_{1}^{2}}{2}\left[\begin{aligned}\sin 2\left(q_{2}+ q_{3}\right)\left(J_{N3}- J_{S3}+ m_{3}l_{3}^{*2}+ m_{g}l_{3}^{2}\right)+ \\ +2 l_{2}\left(m_{3}l_{3}^{*}+ m_{g}l_{3}\right)\cos q_{2}\sin\left(q_{2}+ q_{3}\right)\end{aligned}\right]+\dot{q}_{2}^{2}l_{2}\sin q_{3}\left[m_{3}l_{3}^{*}+ m_{g}l_{3}\right]$.

The SCARA robot with an adaptive position-trajectory control system is usually a type of industrial robot working in the Cartesian coordinate system, and since ^[2,^14,^17,^119] it has two levels of positional mobility, the expressions for calculating the relevant elements of equation (2) for the kinematic scheme of the KZM (Fig. 3) are as follows only for the first and second links is written:

– for a manipulator with one degree of freedom:

(27)

$H_{1}= J_{N1}+ J_{N2}+ m_{1}l_{1}^{*2}+ m_{2}l_{2}^{*2}+\left(m_{2}+ m_{g}\right)l_{1}^{2}+$

$+ m_{g}l_{2}^{2}+2 l_{1}(m_{2}l_{2}^{2}+ m_{g}l_{2})\cos q_{2}$;

(28)

$h_{1}=2 l_{1}(m_{2}l_{2}^{*}+ m_{g}l_{2})\dot{q}_{2}\sin ⁡(\dot{q}_{2})$,

(29)

$M_{E1}= J_{N2}+ m_{2}l_{2}^{*2}+ m_{g}l_{2}^{2}+(m_{2}l_{2}^{*}+ m_{g}l_{2})l_{1}\cos q_{2}]{\ddot{q}}_{2}-$

$- l_{1}\left(m_{2}l_{2}^{*}+ m_{g}l_{2}\right)\dot{q}_{2}^{2}\sin q_{2}$;

– for the state of two degrees of freedom of the SCARA robot type MLM with an adaptive positional-trajectory control system:

(30)

$H_{2}= J_{N2}+ m_{2}l_{2}^{*2}+ m_{g}l_{2}^{2}$,

(31)

$h_{2}=0$,

(32)

$M_{E2}=\left[J_{N2}+ m_{2}l_{2}^{*2}+ m_{g}l_{2}^{2}+\left(m_{2}l_{2}^{*2}+ m_{g}l_{2}^{2}\right)l_{1}\cos q_{2}\right]\ddot{q}_{1}+$

$+ l_{1}\left(m_{2}l_{2}^{*}+ m_{g}l_{2}\right)\dot{q}_{2}^{2}\sin q_{2};$

Taking into account expression (2), it can be said that SCARA robots with adaptive position-trajectory control system perform movements mainly in coordinates. Therefore, the determination and synthesis of dynamic properties for one or two degrees of freedom is carried out.

Advantage of the chosen calculation method.

The use of this selected Lagrange-Euler equation in the calculation of dynamic characteristics of multi-link manipulators includes the following advantages:

- the main calculations are related to the kinetic and potential energy of the system;

– the dynamics of the multi-link manipulator is considered as a whole system;

- internal connection reaction forces are excluded from the equations;

- equations are written in symbolic form;

– the resulting expressions are more suitable for modeling and analyzing system properties.

Including the modeling of the dynamics of this multi-link manipulator can be seen in the example of the following structural scheme (Fig. 4). On the basis of this structural scheme, it is possible to determine and synthesize the dynamic characteristics of a multi-link manipulator, and to realize the dynamics ^[2,^17,^18,^19] of the manipulator as an integrated system on the basis of computer technologies.

그림 4. 다중 링크 매니퓰레이터 동역학 모델링의 구조 다이어그램.

Fig. 4. Structural diagram of multi-link manipulator dynamics modeling.

As a result of the computer modeling of the multi-link manipulator, it is possible to determine and synthesize the dynamic characteristics of the links according to the coordinates and the position of the manipulator (Fig. 5).

그림 5. 다중링크 매니퓰레이터의 동적 특성: a) 링크의 좌표에 따라 b- 위치별.

Fig. 5. Dynamic characteristics of multi-link manipulator: a) according to the coordinates of the links, b- by position.

These obtained dynamic characteristics allow defining and synthesizing the dynamic model characteristics of the multi-link manipulator in relation to its kinetic and potential energy as a whole system.

4. Conclusion

The considered mathematical expressions allow solving the problem of identifying and synthesizing adaptive devices and their dynamic characteristics for all automatic control systems. Because it serves to isolate the components of inertial and velocity forces of the moving links at each generalized moment, as well as the stability or other components of mobility between the levels of positional movement according to the specified coordinates to be performed. The article examines the issues of determining and synthesizing dynamic characteristics of MLMs with an adaptive position-trajectory control system. Also, when determining the characteristics of the dynamic model of multi-link manipulators, mathematical models were developed that allow the calculation of the total kinetic energy, potential energy and acceleration of each link in terms of generalized coordinates. allows to solve the problem. It is possible to develop a calculation algorithm based on the considered and proposed mathematical models. This allows computer calculation of the problem of determining and synthesizing the dynamic characteristics of ECR5, PUMA and SCARA type industrial robots with an adaptive position-trajectory control system.

Acknowledgements

Hello dear editorial members of The Transactions of the Korean Institute of Electrical Engineers (KIEE)! Thank you for taking the time to review our article on Adaptive Position-Determination and Dynamic Model Properties Synthesis of Moving Objects With Trajectory Control System (In the Case of Multi-Link Manipulators)! We would also like to thank the reviewers who have reviewed our article. We are pleased that the reviewers of the article, in turn, gave a fair and accurate assessment. We hope to co-publish more scientific articles with your journal in the future. We also wish the members of the magazine's editorial board and the magazine's activities good luck! In the future, we wish to increase the ranking of the magazine in the Scopus database, and in this regard, we, as the author, will promote your magazine and the published scientific articles more widely among the scientists of Uzbekistan.

Sincerely, the author, Dildora Sevinova!

References

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저자소개

Zaripov Oripjon Olimovich

Graduated from Tashkent State Technical University in 2000 with the specialty "Informatics and computing technology", in 2002 with the specialty "Automated systems of information and management processing", 05.13.2006 - Automation of technological processes and productions and He received a PhD degree in management and in 2012 he received a DSc degree in Automation and Control of Technological Processes and Manufactures on 05.01.08. In addition, in 2023, Knowledge Co-Creation Program on "Development of Advanced Industrial Human Resource through Japanese Style Engineering Education" from November 30, 2023 to December 15, 2023 at the Institute of China, to study the experiences of establishing engineering schools. Organized by the Japan International Cooperation Agency in collaboration with the Graduate School of Engineering, Mie University under the International Cooperation Program of the Government of Japan and Germany's Technische Universität Bergakademie Freiberg. He is currently a professor at the Tashkent State Technical University named after Islam Karimov. Under his leadership, 12 PhDs were trained. His research interests include the intellectualization of control processes for non-linear continuous- discrete dynamic objects, and the developed methods, and models used in the field of automation of electric power facilities, oil and gas, chemical-technological industries, and the light industry. In addition, he is a reviewer of leading scientific journals such as Vestnik TSTU, and Chemical Technology. Control and Management", "Technical science and innovation". He is the author or co-author of more than 130 refereed journals and conference articles, 5 monographs and 4 textbooks, and 8 scientific articles indexed in the Scopus database (Elsevier). He can be contacted at o.zaripov @edu.uz.

Sevinova Dildora Usmonovna

received a bachelor's degree in 2010-2014 in the field of "technological processes and production automation and management" at the Faculty of Electronics and Automation of the Tashkent State Technical University, and a master's degree in the field of "Mechatronics and Robotics" in 2017-2019. In 2019-2021, Tashkent State University, she worked as an assistant at the "Electronics and Automation" Department of the Technical University, "Mechatronics and Robotics" department. Currently a PhD student. He can be contacted at sevinovadildora @gmail.com.

Bobojanov Sukhrob Gayratovich

received his BS degree in Telecommunication Engineering from the Urgench branch of Tashkent University of Information Technologies, Uzbekistan, in 2015. He received his MS degree in Telecommunication Technologies from Tashkent University of Information Technologies named after Muhammad al-Khwarizmi, Uzbekistan, in 2017. From 2017 to 2018, he worked as an engineer in the Department of Technical Coordination and Support of State Events at Urgench Branch of "Uzbektelecom" JSC. From 2018 to 2020, he worked as an assistant teacher in the Department of Telecommunication Engineering at Urgench Branch of Tashkent University of Information Technologies named after Muhammad al-Khwarizmi in Urgench, Uzbekistan. He received his Ph.D. degree in Software Engineering from Kumoh National Institute of Technology, Gumi, South Korea in 2024. He can be contacted at w.suxrob.w@ gmail.com.

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Adaptive Posision-Determination and Dynamic Model Properties Synthesis of Moving Objects With Trajectory Control System (In the Case of Multi-Link Manipulators)

Translated Abstract

Key words

1. Introduction

2. Research methodology

3. Analysis and results

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4. Conclusion

Acknowledgements

References

저자소개

Zaripov Oripjon Olimovich

Sevinova Dildora Usmonovna

Bobojanov Sukhrob Gayratovich

Article Information (continued)

Key words

KIEEThe Transactions of
the Korean Institute of Electrical Engineers