______PROCEEDING OF THE 26TH CONFERENCE OF FRUCT ASSOCIATION

Calculation and Optimization of Industrial Robots Motion

Sergei Ivanov, Lubov Ivanova Zoia Meleshkova ITMO National Research University (ITMO University) ITMO National Research University (ITMO University) Saint Petersburg, Russian Federation Saint Petersburg, Russian Federation [email protected], [email protected] [email protected]

Abstract—This paper considers a problem of improving ULC Robotics, Universal Robotics, Inc., Vecna Technologies, automated industrial manufacturing systems with a help of low- Verb Surgical, VEX Robotics, Yamaha Robotics, Yaskawa, costs robotics manipulators with a programmed control. For Arkodim, Bit Robotics, Exoathlete, Ronavi Robotics, Eidos widespread use at various industrial enterprises of robots- Robotics. The international company KUKA produces dozens manipulators with programmed control without expensive of industrial robotic manipulators used in various industries sensors and elements of artificial intelligence, we use methods of with a wide range of carrying power parameters and handling determining the spatial and kinematic characteristics of the radius. For example, for the KR QUANTEC multipurpose working body of the manipulator. The method of determining the industrial robotic arm, the payload is: 120 - 300 kg, and the kinematic characteristics is based on the matrix method in the maximum handling radius is 270 - 310 cm. kinematics of robots and the second-order Lagrange matrix equations in dynamics. The method allows to calculate the Let us consider the urgent task of improving automated optimal modes of movement of the manipulator, to increase the production systems based on low-cost robotic manipulators speed of operations on the production line. The presented with programmed control. approach makes it possible to optimize the number of multipurpose manipulators with programmed control for various A lot of modern scientific works are devoted to the technological operations, and also allows to increase the problems of robotics and automation of production. productivity of a robotic production line. In [1], robot reconfiguration technologies for automating production lines are developed and the concept of self- I. INTRODUCTION adjusting robot skills for production is proposed. Skills for Industrial robots are used in automated production systems. performing various tasks obtained from the current instructions Automation of production with a help of robots allows to of the working wizard are transferred to industrial improve the quality of products, increase the accuracy of manipulators. technological operations, increase labor productivity and eliminate exposure of harmful factors for workers. Known methods of elastogeometric calibration to improve the accuracy of positioning of industrial robots 6 DOF [2]. In In the production process, industrial robots perform the the process of elastogeometric calibration, the optimal set of following main technological operations: machine robot configurations and external loads is determined. maintenance, loading, unloading, transportation, mechanical Elastogeometric calibration reduces the maximum position restoration, assembly, installation and loading of parts, transfer error to 0.96 mm, which, in contrast to the kinematic of parts and blanks, laser and spot welding, laser cutting, metal calibration, reduces the error to 2.557 mm. casting, stamping, forging and bending, spray paint and enamel coating, packaging and picking, product quality control. A methodology has been developed for planning and optimizing the movement of the robot, which provides a set of Industrial robots consist of a manipulator and a software robot trajectories for each task while minimizing the cycle time control system for the manipulator's working body - a gripping [3]. device (gripper). In [4], the problem of controlling the accuracy of following In industrial production, a huge number of specialized a given trajectory for an industrial robot is researched. The robotic manipulators of various companies are used: 3D considered approach is applied to the predictor controller for Robotics, ABB Robotics, Aethon, Alphabet, Inc. (Google), following the trajectory on the KUKA LWR IV robot. Amazon, Autonomous Solutions (ASI), CANVAS Technology, Carbon Robotics, Clearpath Robotics, Cyberdyne, Delphi The study [5] analyzes the safety of controllers and Automotive, DJI, Ekso Bionics, EPSON Robots, FANUC software vulnerabilities for industrial robots. Safety standards Robotics, Fetch Robotics, Foxconn Technologies Group, and safety problems in industrial robotics are considered. GreyOrange, IAM Robotics, Intuitive Surgical, iRobot, Jibo, For industrial robots, an algorithm for correcting the Kawasaki Robotics, Knightscope, KUKA, Lockheed Martin, welding path in real time has been developed [6]. Locus Robotics, Omron Adept Technologies Inc., Open Bionics, Rethink Robotics, ReWalk Robotics, Robotiq, An intelligent algorithm for determining the trajectory of an Samsung, Savioke, Schunk, Seegrid, Siasun Robot & industrial manipulator based on the analysis of alternative Automation Co.Ltd., SoftBank Robotics Corporation, Soil configurations while minimizing the operation time was Machine Dynamics Ltd., Swisslog, Titan Medical, Toyota, proposed in [7].

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In the study [8], an adaptive control method was proposed quaternions are a spatial transformation and are used for for industrial robots with six degrees of freedom, which kinematic analysis of the robot. The error model is obtained reduces errors with external noise and with parametric from direct kinematic equations using the algebra of double uncertainties. quaternions. The error model contains all the basic geometric parameters of the robot for kinematic calibration. The Denavit - In [9], a wireless three-dimensional manipulation system is Hartenberg error model in kinematic analysis gives an intuitive developed on a tablet PC based on augmented reality. The geometric meaning of kinematic parameters. system allows you to virtually control the movement of the robotic arm in three-dimensional space through augmented The study [18] presented a method for planning the reality. An intelligent system is being developed to build a trajectory to minimize synthesis errors and obtain stable motion robot trajectory without collisions. of industrial robots. The minimal synthesis error is determined under kinematic and dynamic constraints. A kinematic study of The study [10] proposed an improved A * algorithm for the dynamic model of the robot ER3A-C60 is carried out taking solving the robot path planning problem. Optimization of the into account the flexibility of all connections. The trajectory local path between the current and the end node is applied, planning technique can improve the tracking accuracy of the taking into account the safety of the path, the absence of final effector, while controlling the movement time and the collisions on the path and reduction of the path length. requirements of continuous trajectory control. In [11], to reduce the motion uncertainty of industrial robots Software-controlled robotic manipulators are widely used at with six degrees of freedom, a parameter identification method various industrial enterprises without expensive sensors and based on the Denavit-Hartenberg model, where excess artificial intelligence elements. parameters are taken into account during identification, is presented. For the kinematic model of an industrial robot with The paper presents a method for determining the spatial and six degrees of freedom, the linearization method and an kinematic characteristics of the working body of a manipulator alternative identification algorithm with a modified least- based on the matrix method in the kinematics of robots and squares scheme are used. second-order Lagrange matrix equations in dynamics. The method allows you to determine in space the correct position of In [12], the problems of dynamic identification for an the working body of the manipulator. The method allows you industrial robotic arm with six degrees of freedom are to calculate the working optimal operating modes of the investigated and a procedure for identifying parameters based manipulator, increase the speed of operations on the production on a modified cuckoo search algorithm is presented. The line. The presented matrix method was applied in [23] for the nonlinear model of the robotic arm takes into account friction study of manipulators with five degrees of freedom. in the joints. This approach optimizes the number of universal In [13], an adaptive robust controller is developed for manipulators with program control for various technological dynamically tracking a robot trajectory with 6 degrees of operations, and also allows to increase the productivity of a freedom with parametric uncertainties and external noise. The robotic production line while maintaining the quality of the controller is developed on the basis of a combination of sliding process. mode control, adaptive control and has the ability to adapt and resistance to parametric changes and uncertainties. The method allows you to calculate and optimize the sequence of movements of the links of the robot-manipulator to In the study [14], a smooth path planning method for move the grip of the manipulator to a given point in space. In industrial robots was proposed. The trajectory is planned for this case, it is possible to calculate and control the speed and joint movement of three parts: accelerated, slowed down part acceleration of the grip of the robot during movement, reducing and part with constant speed. In the accelerated part and the the time of the duration of the tact of the robot. slowed part, the acceleration is represented by a fourth-order polynomial with the property of multiplicity of roots. The time- optimal trajectory is obtained by maximizing the part at a II. MANIPULATOR MOTION MODE CALCULATION METHOD constant speed under kinematic constraints. To build a mathematical model of the dynamics of the In [15], 6DOF smart industrial manipulators were studied robot-manipulator, we use the matrix method and matrix and two compensation methods based on reinforced learning equations of the Lagrange of the second kind. In the method, were presented. The study compares the effectiveness of we determine the transition matrices from one local coordinate learning-based methods with typical tracking controllers, system of one manipulator link to another local coordinate proportional derivatives, predictive models, and iterative system of the next manipulator link. In accordance with [19], learning. by a special choice of local coordinate systems, one can In the study [16], a new method is proposed for a dynamic describe the transition from one local coordinate system of a model of a robotic arm with 6 degrees of freedom to increase link to another by means of four parameters, and not six as in the accuracy of movement. A centrosymmetric model of static general form. Four parameters correspond to rotation around friction is used to describe the hysteretic effect of friction in the axis, two movements along the axes and subsequent joints. The hybrid whale optimization algorithm and the genetic rotation around the axis. algorithm are used to determine the dynamic parameters of the 6 joints of the robot. Imagine a method for calculating the optimal movement of the manipulator. In [17], an error model for the kinematic calibration of a serial robot based on double quaternions was presented. Double Let the manipulator under study have n movable links.

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Let us introduce the notation of the beginning of the local ªºwwPP coordinate system associated with the i - link of the DP «»,..., - is a line-vector of derivatives of ¬¼wwqq1 i manipulator Oi . potential energy. For a global coordinate system associated with a fixed We determine the kinetic energy for each link of the coordinate system, we denote the origin O . 0 manipulator using transition matrices according to the Denote the generalized coordinates for the manipulator by formulas:

1 T qii ,( 1,..., n ) . TtrAHA , iiii2 00 We introduce the extended radius vector for the point in the i-th local coordinate system: where Hi is the link inertia matrix. T J JJmx Rxyziiii >@1 . ªºixx ixy ixz i Ci «»J JJmy «»iyx iyy iyz i Ci To move from the initial coordinate system O0 to the local Hi , «»JizxJJmz izy izz i Ci coordinate system of the first link O1 rotation around the axis «» ¬¼mxiCiiCiiCii my mz m by an angle D , two movements along the axes by a, b and a subsequent rotation around the axis by an angle E . where mi is the mass of the link,

The transition matrix from one coordinate system to the J ,,JJ are components of the link inertia tensor next has the form:: ixx iyy izz relative to the axes. ªºcos D  cos ED sin sin DE sin b cos D «»The coordinates of the center of gravity of the link in the sin D cos DE cos  cos DE sin b sin Dlocal coordinate system are determind as x ,,yz. A «»Ci Ci Ci ii,1 «»0sincos EE a «»The moments of inertia of the link relative to the axes are ¬¼«»00 0 1 denoted by J xiyizi,,JJ. The transition matrix Aii,1 connects the radii of the vectors of The equalities are valid for the axial moments of inertia: the coordinate systems i and i + 1 with the following formula:

J xiiyyizzyiixxizzziixxiyy JJJJJJJJ  ,  , . RARiiii ,1 1 The transition matrix from the global coordinate system Determine the total kinetic energy of all the links of the n O to the local coordinate system O is determined by the 0 i manipulator: TT ¦ i . i 1 product of the transition matrices: AAAA001121iii } We define the potential energy for the gravity of each link Thus, the position of the grip of the manipulator in the T global coordinate system is determined. in the form: PmGACiiii  , The links of the manipulator are connected by cylindrical T where CxyziCiCiCi > 1@ – is the column of hinges and sliding joints, in which low friction can be neglected. coordinates of the center of gravity of the link T To compose the differential equations of motion of the Ggi >00 0@ – column of acceleration of gravity. manipulator, we apply the matrix Lagrange equations in the form: We determine the total potential energy of all the links of n LTDPQ , the manipulator: PP ¦ i . i 1 where Substituting the total kinetic, potential energy and ªºdT§ ww· T dT§ ww· T generalized forces for the link drives in the Lagrange matrix L «»¨ ¸ ,..., ¨ ¸ - is the line vector equation, we obtain a system of differential equations of dtww qcc q dt ww q q ¬¼«»© 11¹¹© ii motion for an industrial robot. of the Lagrange operators, T is the total kinetic energy of all To formulate the Cauchy problem, we set the initial parts of the manipulator, Q is the vector line of the generalized conditions for the generalized coordinates and speeds of the forces created by the manipulator drives.. links of the manipulator for the system of nonlinear differential equations.

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To construct solutions to a system of second-order nonlinear differential equations, the method of transformations was applied [20,22]. Having obtained the solution, we determine the generalized coordinates, velocities, and accelerations for the manipulator. Having built an analytical solution for the system, we obtain the dependences for the generalized coordinates, velocities, and accelerations on the generalized forces of the manipulator electric drives. Using these dependencies, we determine the values of the generalized forces of the electric drives necessary to move the gripper to the end point.

Using the transition matrices Aii1, and the extended radius of the coordinate vector R001121 AA} Akkk R we obtain the grip coordinates of the manipulator in the global coordinate system, depending on the generalized coordinates. Substituting the generalized coordinates, we construct the trajectory of the gripper in time for given generalized forces of Fig. 1. The kinematic diagram of the manipulator the manipulator electric drives. A three-dimensional model of an industrial manipulator To optimize the grip movement of the manipulator, we was built and modeling was carried out in a specialized consider the law of uniformly variable grip movement at the computer program System-Modeler (Fig. 2). required travel time tk , the initial and final position of the grip of the form:

222 xkxkxkkykykkzkzk xvtwtyyvtwtzzvtwt00 ,,, 00  00 

vvx xxkyyykzzzk000 wtvv,,,  wtvv  wt where (,,)x000yz , (,,)xkkkyz is the initial and final

222 coordinates of the tong, v vvvx yzis the speed of

222 movement of the tong, w wwwx yz- is an acceleration of the tong. We take into account that with equal alternating motion, tangential acceleration is constant. Fig. 2. Three-dimensional model of the manipulator Having determined the solutions of the Lagrange differential equations, we find the dependences of the Define the coordinate system of the robot links at points generalized coordinates, velocities, and accelerations, and OOOOOO,,,,,. The absolute coordinate system is applying the law of uniformly variable grip motion, we 123456 determine the optimal manipulator grip motion. connected with the fixed base of the manipulator at a point O0 . III. MODELING OF A MANIPULATOR WITH SIX DEGREES OF FREEDOM Take as a generalized coordinates of the manipulator with six degrees of freedom - the angles of rotation of the links and Consider a multipurpose robotic manipulator with six the length of the extension of the arm. Here we measure angles degrees of freedom, which includes six links, a base, link in radians, lengths in centimeters. drives, and a gripper. We introduce the notation of periodic functions: The kinematic diagram of the robot is shown in Fig. 1 and consists of five rotational kinematic pairs and one translational CqSq cos> ()tt@ , sin> ()@ , pair. The robot model is applicable to industrial manipulators, iiii ªº the kinematic diagram of which is shown in Fig. 1. sin>@ 2qS27()ttt ,sin¬¼ 2 qqS 238 () () ,

When developing dynamic equations of the robot, we use sin>@>@ 2qq23()tt () S 9 ,cos 2 qq 23 () tt  () C 9. the matrix method and matrix dynamic Lagrange equations. ªº The links of the industrial manipulator are modeled by rods, cos>@ 2qC27()t ,cos¬¼ 2 qq 238 ()tt () C , the joints are modeled by cylindrical joints and sliding joints. sin>@>@qq(tt)() SqqC ,cos ( t )  (t ) We assume that the friction in the joints is small and is not 23 1023 11 taken into account when deriving the robot model. The transition matrices from the absolute coordinate system to the coordinate system are defined:

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CS 00 10 0 0 ªº11 ªº aSS312 a 4 CSS 312 CSS 213 q 5 CSS 312 CSS 213 º «»«» » SC00 00CS aCS a CCS CCS  q CCS CCS A «»11 , A «»22 , 312 4 132 123 5 132 123» 01 «»12 «» » 001a1 0 SCa222 aaaCaCCSSqCCSS     «»«» 12324232352323 » 0001 00 0 1 ¬¼¬¼ 1 ¼»

ªº10 0 0 ªºCS44 00 ªCC14 CCS 231  SS 123 S S 4 «»«» « 00CS33 SC4400 «»«» CS41 CCC 1 2 3 CSS 123 S 4 A23 , A34 , A « «»«» 06 0 SCa333 001a4 « CS CS S «»«» « 32 23 4 ¬¼00 0 1 ¬¼0001 ¬« 0 ªº100 0 ªº10 0 0 CC64  CCSSSS 231123  CS 14 CSSCSSS 3122136 «»010 0 «»00CS «»«»66 CCCCCCSS  SS CCSCCSS A45 , A56 . 6412312314 1321236 «»001q5 «»0 SCa665 «»«» CC46 CS 32 CS 23 CC 23 SS 23 S 6 000 1 00 0 1 ¬¼¬¼ 0

CCSSCSS C CCSSSS CSS The transition matrices from the absolute coordinate 6 312 213 4 2 31 123 14 6 system O to the coordinate system O are defined: 0 i CCCSCCSCCCCCSSSSS61321234123123146     CCCSS CCSCSS  AAAA } 62323 432236 001121iii 0

ªºCCSSS12112 0 «»aSS312 a 4 CSS 312 CSS 213 a 5 CSS 312 CSS 213 q 5 CSS 312 CSS 213 º SCCCS 0 » A «»11212 , aCS a CCS CC S  a CCS CCS  q CCS CC S 02 «»312 4 132 123 5 132 123 5 132 123» 0 SCaa2212 » «»aaaCaCCSSaCCSSqCCSS1232423235232352323     ¬¼00 0 1 » 1 »¼ The coordinates of the grip of the manipulator in the absolute ªC1 CCS 2 31 SSS 123 CSS 312  CSS 213 aSS 312 º « » coordinate system O are determined as a function of the S CCC CSS CCS CCS  aCS 0 A « 1123123132123 312» 03 « generalized coordinates of the manipulator 0 CS32 CS 23 CC 2 3 SS 23 a 1 a 2 aC 3 2» « » x aSS a CSS  CSS  ¬« 00 0 1¼ 06 3 1 2 4 3 1 2 2 1 3

a53122135312213 CSS CSS q CSS  CSS ª CCCCSSSSSC14 231  123 4 4  CCSSSSCS 231  123  14 yaCSaCCSCCS      « 06 3 1 2 4 1 3 2 1 2 3 C S CCC CS S S C CCC  CS S SS A « 41 1 2 3 123 4 4 1 2 3 123 14 a  CCS CC S  q CCS CC S 04 « CS CS S C CS CS 51321235132123 « 32 23 4 4 32 23 ¬« 00 zaaaCaCCSS06 1 2 3 2  4 2 3  2 3 

aCCSS5232352323  qCCSS  CSS312 CSS 213 aSS312 a 4 CSS 312 CSS 213 º » CCS132 CC 123 S aCS a CCS CCS The total kinetic energy of all the links of the manipulator 312 4 132 123» , » 6 CC23 SS 23 aaaCaCCSS123242323   » is determined: TT ¦ i 0 1 ¼» i 1 The total potential energy of all the links of the ª CC14 CCS 231  SS 123 S S 4 manipulator is determined: « CS CCC CSS S A « 41 1 2 3 123 4 6666 05 « CS32 CS 23 S 4 PPgamamCam ¦¦¦¦iiii ((21   23  « iiii 1213 ¬« 0 §·§·66 66 Cama3564¨¸¨¸¦¦ mii mq 5))  ama 564 ¦¦ m ii mqSS 523 CCCSSSSCS423112314  CSS312 CSS 213 ©¹©¹ii 45 ii 45

CCCCCSS4123123  SS 14CCS132 CC 123 S Substituting the kinetic, potential energy and generalized forces into the Lagrange equations, we obtain a system of CCSCS  CC23 SS 23 43223 equations of motion for a manipulator with six degrees of 0 0 freedom.

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ccccc2 0.5qimimimQcc 222 q1 k 5 kq 65 kq 565 S 8 q 2  q 3 q 1 1  C 8 k 5  kq 5655 q  4445566 4 (0.5kCkqmaCiaCqq 0.5 122 0.5 1  0.5 2 1 ) cc 15 8 56 5 6 5 8 6 5 8 5 1 mgC611458((0.51 k C 2 ak2 Sqcc q q c0.5 1  C q cc  c 44681 2 3 8 1 aCC3390.5   0.5  1 Cqq 851 )  22 akSqq2(cc 0.5 1 Cm 0.5 k 0.5 Ckq cc )  cccc 336712 7 3 46 7461 kaCqq45 3 3  5 2  kqq 45 5 3  q 5 )  c cc c 222 aq41(22 k 5 kqSq 5658 2qCkq38565 1  §·ccc mgCa5113(0.50.5¨¸ C 91 q C 3 q 1 q 2 ©¹ 1)Ckkqqcc  855651 222 ccc cc c aCqqq0.5 1    aq((2 k 2 kqSq k kq S S q 48123 3 1 5 565 92 5 565 3 9 3 222 ccc ccccc Ck9 56 Ck 3 56 q 5)  a 5 C 3  C 9 m 6  Ck 9 56 Ck 3 56 q 5 q 1  qCqqqqQ0.5 1    ) 58 1235 5 cc c cc 2 cc aq4(2 1 kSq 46 9 2 k 46 S 3 Sq 9 3  Ck 9 46 Ckq 3 46 1 )) Q 1 0.5imq666 Q 6

2 22c The fourth and sixth equations of the system are easily gkkqS 556510  0.5 amkq 5655   0.5 kqSq 56581  integrated: 22kkqqqkkqmaiaqqcc (0.5222 0.52 ) cc 5 56 5 2 5 25 56 5 6 5 6 5 5 2 2 2 tQ4 tQ6 22 qtQ ,, qtQ , . akSqkqakSqkq220.5ccc 0.5 ccc 4 im222 im im 6 im2 3367136244681462 44 55 66 66 222 a( gkS akSqccc  amSq  kqSq  To solve the remaining four differential equations of the 3 362 45691 5 69 1 5659 1 system, we apply the polynomial transformation method [20- cc cc cc 2a4k56323 Sqq22k 46323 Sqq k 565323 qSqq  23] with the following parameters: 2 §·ccc mmmmmm123456 200 , 100 , 30 , 20 , 10 , 30 , mgSaSq42491323¨¸ 2 Sqq  ©¹

cc cc QQQ123 100, 10, 10000, QQQ456 0.1, 0.1, 0.01, 22Ck3562 q q 5 C 3 am 5 6 ak 446 k 565 q q 2 ) 2 ccc aaaaaa123456 30 , 50 , 30 , 20 , 20 , 10 . agmSmgSaqSq4410610558125(( 2) qq  2 Initial Values: mgSqSqccc 222) qq  amkqqQ  cc 510581255656522 qq12 00 0, 3.14, q 3 0 4.71, q 4 0 0, 2 c mmqgS56510581(0.5 qSq  qq56 00 0, 3.14, qic 0 0

222 § ccccc· The transformation method [20] allows you to find a aSqSqq391312¨¸0.5 0.5  2 qq 35 ) © ¹ solution with all non-linear components of the original system. 2222 2cc The transformation method allows you to build a solution of a 0.5im33 0.5 im 44 0.5 im 55 0.5 im 66 m 5 m 6 q 5 q 3 nonlinear system of differential equations in an analytical form 2 am2 0.5 S qccc q with numerical coefficients. 56 8 1 3 2 The solution of the system of three differential equations ammmSqmmmq2 0.5 0.5 0.5 ccc  4456814563 was obtained by the method of transformations. 2 222 § ccc· qt 0.005  0.003 t  0.007 t , am(0.5 gSa§ Sq Sqq0.5 · 1 44¨¸ 103¨¸ 9 13 1 2 © © ¹¹ 2 2 qt2 3.1 0.128 t  0.011 t , c mgSaqSq6105581(  2 222 qt3 4.507 0.588 t  0.049 t , § ccccc· aSqSqq391312¨¸0.5 0.5  2 qq 35 ) © ¹ 2 qt4 0.167 t , 2222 mgSqSq(0.50.5cccc  a§ Sq  S q  q ·  5105813 ¨¸ 913 1 2 2 © ¹ qt5 2.298 10.948 t  8.342 t , cc cc 2)2qq35 am 56 m 5 m 6 q 5 q 3 )  2 qt6 0.239 t. 2 22 § c cc· am56(0.5 gS 10 a 3¨¸ S 9 q 1 Sqq310.5  2 © ¹ For industrial manipulators with a kinematic diagram 2 shown in Fig. 2, the kinematic characteristics are calculated 22)qqcc q S q c  q cc Q 35 5 8 1 3 3 for the values of the mass parameters and link lengths of the industrial manipulator.

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Model verification of an industrial manipulator is carried In figures 3,4,5 the following legend is used: out by parallel modeling in a specialized computer program. q q q Fig. 3 shows the rotation angles (in radians) for the links of 1 2 3 the manipulator. q4 q5 q6 q Substituting the coordinates of the manipulator gripper into 6 the equations of equal alternating movement of the manipulator grip, we obtain algebraic equations for the 5 optimal manipulator grip motion in time. tk .

4 2 SaS132  a 4 a 5 q 5 CS 32 Cvw 2St3  xk0  xk t 0, 3 2 CaS132 a 4 a 5 q 5 CS30 2CS 23  vyk t w yk t 0, 2 aaaCaaqCC   SS  vtwt 2 0, 1 1232455 2323 zk zk t Simplify the system of three equations, taking into account the 1 2 3 4 5 ªº ªº . notation Cii Cos¬¼ q tQ , , S ii Sin ¬¼ q tQ , :

Fig. 3. The rotation angles (in radians) of the manipulator links 2 Sin>qqaqqaaqvtwt123234550@ Sin> @  Sin> @ xk xk 0 Fig. 4 shows the extension of the arm of the manipulator (in 2 Cosqqaqqaaqvtwt Sin Sin 0 centimeters). >@123234550 >@ > @ yk yk 2 aa12Cos>@ qa 23  Cos > qqaaq 23455  @   vtwtzk 0  zk 0 q5 150 An analytical solution of the system of equations is obtained 22 ªºvt wt22 vt wt 1 «» yk00 yk xk xk qt1 ArcCos 22 100 2 «»vt wt22 vt wt ¬¼ yk00 yk xk xk

qt ArcTanª aPaP3222 a P P / P 50 2¬ 3 za 3 za 6 xy za xa 2 º aP224 a a 222 P P2/ a 2222 a P P P 3xy3 6xyza36xyzaxa» t ¼ 1 2 3 4 5 1 52 2 qt33xyza ArcTan[2 ( aP  P  222 Fig. 4. Extension of the arm of the manipulator (in centimeters) aa36 P xy P za 222222322 222 Fig. 5 shows the generalized speeds for the manipulator. aaaPPPP366xyzaxyza3xyza6xyza   aPP 2 aPP 

2 2(aaPaPa22 ) 224  aPP 2 2 2 2 aaPP 222 2 )] 36za3xy3 6xyza36xyza q 2 tQ5 2 aaa645 22, Pvtwtaa za0 zk  zk  12, im55 im 66 1.5 22 22222 Pxy vtwtyk 0 yk  vtwtP xk 0 xk , xa aPP 3 xy  za

1.0 The analytical solution for qtqtqt456 ,, is obtained from the matrix dynamic equations of Lagrange.

2222 0.5 q4() t ( tQ 4 )/( im 44 im 55 im 66 ), 222 qt5() ( tQ 5 )/( im 55 im 66 ), qt() ( tQ22 )/( im ). t 6666 1 2 3 4 5 Thus, the functions of the generalized coordinates of the Fig. 5. The speed of the links of the manipulator gripper of the manipulator are determined for uniform movement of the gripper.

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Fig. 6 shows the trajectories of the tong along the x, y, z dynamics. The method allows you to calculate the working axes relative to the fixed base of the rack.. optimal operating modes of the manipulator, increase the speed of operations on the production line. A kinematic x,y,z diagram of a manipulator with six degrees of freedom has been developed, and a mathematical model of the manipulator 40 has been investigated using the matrix method. As a result of 20 the analysis of the mathematical model, the coordinates, speeds and accelerations of the manipulator links are t 1 2 3 4 determined. 20 The presented approach optimizes the number of universal 40 manipulators with program control for various technological operations, and also allows to increase the productivity of a 60 robotic production line. 80

x y z REFERENCES [1] Pedersen, M. R., Nalpantidis, L., Andersen, R. S., Schou, C., Bøgh, Fig. 6. Manipulator grip coordinates S., Krüger, V., & Madsen, O. (2016). Robot skills for manufacturing: From concept to industrial deployment. Robotics and Computer- Fig. 7 shows the optimal spatial trajectory of the gripper Integrated Manufacturing, 37, 282-291. relative to the fixed base of the manipulator strut.. [2] Kamali, K., Joubair, A., Bonev, I. A., & Bigras, P. (2016, May). Elasto-geometrical calibration of an industrial robot under multidirectional external loads using a laser tracker. In 2016 IEEE International Conference on Robotics and Automation (ICRA) (pp. 4320-4327). IEEE. [3] Pellegrinelli, S., Orlandini, A., Pedrocchi, N., Umbrico, A., & Tolio, T. (2017). Motion planning and scheduling for human and industrial- 50 robot collaboration. CIRP Annals, 66(1), 1-4. [4] Faulwasser, T., Weber, T., Zometa, P., & Findeisen, R. (2016). Implementation of nonlinear model predictive path-following control for an industrial robot. IEEE Transactions on Control Systems Technology, 25(4), 1505-1511 0 [5] Quarta, D., Pogliani, M., Polino, M., Maggi, F., Zanchettin, A. M., & Zanero, S. (2017, May). An experimental security analysis of an industrial robot controller. In 2017 IEEE Symposium on Security and z Privacy (SP) (pp. 268-286). IEEE. [6] Guillo, M., & Dubourg, L. (2016). Impact & improvement of tool 50 deviation in friction stir welding: Weld quality & real-time compensation on an industrial robot. Robotics and Computer- Integrated Manufacturing, 39, 22-31. 40 [7] Kaltsoukalas, K., Makris, S., & Chryssolouris, G. (2015). On 100 20 generating the motion of industrial robot manipulators. Robotics and 0 Computer-Integrated Manufacturing, 32, 65-71 y [8] Yin, X., & Pan, L. (2018). Enhancing trajectory tracking accuracy for 20 industrial robot with robust adaptive control. Robotics and Computer- 40 Integrated Manufacturing, 51, 97-102. [9] Su, Y. H., Chen, C. Y., Cheng, S. L., Ko, C. H., & Young, K. Y. Fig. 7. The spatial trajectory of the grip of the manipulator (2018, October). Development of a 3D AR-Based Interface for Industrial Robot Manipulators. In 2018 IEEE International The presented approach allows us to construct an optimal Conference on Systems, Man, and Cybernetics (SMC) (pp. 1809- spatial trajectory of the uniform motion of the manipulator's 1814). IEEE. grip. All mathematical calculations were checked in [10] Fu, B., Chen, L., Zhou, Y., Zheng, D., Wei, Z., Dai, J., & Pan, H. (2018). An improved A* algorithm for the industrial robot path specialized computer mathematical packages. planning with high success rate and short length. Robotics and Autonomous Systems, 106, 26-37. IV. CONCLUSION [11] Gao, G., Sun, G., Na, J., Guo, Y., & Wu, X. (2018). Structural The work considers the problem of improving automated parameter identification for 6 DOF industrial robots. Mechanical Systems and Signal Processing, 113, 145-155. production systems based on low-cost robotic manipulators [12] Ding, L., Li, X., Li, Q., & Chao, Y. (2018). Nonlinear friction and with programmed control. For widespread use at various dynamical identification for a robot manipulator with improved industrial enterprises of robots-manipulators with programmed cuckoo search algorithm. Journal of Robotics, 2018. control without expensive sensors and elements of artificial [13] Pan, L., Bao, G., Xu, F., & Zhang, L. (2018). Adaptive robust sliding mode trajectory tracking control for 6 degree-of-freedom industrial intelligence, we use methods for determining the spatial and assembly robot with disturbances. Assembly Automation, 38(3), 259- kinematic characteristics of the working body of the 267. manipulator. The method for determining the kinematic [14] Wang, H., Wang, H., Huang, J., Zhao, B., & Quan, L. (2019). characteristics is based on the matrix method in the kinematics Smooth point-to-point trajectory planning for industrial robots with kinematical constraints based on high-order polynomial curve. of robots and the second-order Lagrange matrix equations in Mechanism and Machine Theory, 139, 284-293.

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[15] Pane, Y. P., Nageshrao, S. P., Kober, J., & Babuška, R. (2019). robots: in 3 books, Part 1: Kinematics and dynamics, M.: Higher. Reinforcement learning based compensation methods for robot school, 1988, p. 304. manipulators. Engineering Applications of Artificial Intelligence, 78, [20] G. I. Melnikov, S. E. Ivanov, V. G. Melnikov, “The modified 236-247. Poincare-Dulac method in analysis of autooscillations of nonlinear [16] Zhang, L., Wang, J., Chen, J., Chen, K., Lin, B., & Xu, F. (2019). mechanical systems”, Journal of Physics: Conference Series, 570(2), Dynamic modeling for a 6-DOF robot manipulator based on a IOP Publishing, 2014, p.022002. centrosymmetric static friction model and whale genetic optimization [21] T. V. Zudilova, S. E. Ivanov, L. N. Ivanova, “The automation of algorithm. Advances in Engineering Software, 135, 102684. electromechanical lift for disabled people with control from a [17] Li, G., Zhang, F., Fu, Y., & Wang, S. (2019). Kinematic calibration mobile device”, Computing Conference, IEEE, 2017, of serial robot using dual quaternions. Industrial Robot: the pp. 668-674. international journal of robotics research and application, 46(2), 247- [22] G.I. Melnikov, N.A. Dudarenko, K.S. Malykh, L.N. Ivanova, V.G. 258. Melnikov, “Mathematical models of nonlinear oscillations of [18] Liu, Z., Xu, J., Cheng, Q., Zhao, Y., Pei, Y., & Yang, C. (2018). mechanical systems with several degrees of freedom”, Nonlinear Trajectory planning with minimum synthesis error for industrial Dynamics and Systems Theory, 17(4), 2017, pp. 369-375. robots using screw theory. International Journal of Precision [23] S. E. Ivanov et al. “Automated Robotic System with Five Degrees of Engineering and Manufacturing, 19(2), 183-193. Freedom”, 25th Conference of Open Innovations Association [19] E. I. Vorobyov, S. A. Popov, G. I. Sheveleva, Mechanics of industrial (FRUCT), IEEE, 2019, pp. 139-145. 

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