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Научная статья
ПРИМЕНЕНИЕ МЕТОДА «ЧИСТОЕ ПРЕСЛЕДОВАНИЕ»
(PURE PURSUIT) ДЛЯ УПРАВЛЕНИЯ БЕСПИЛОТНЫМ АВТОГРЕЙДЕРОМ
Р. Ю. Сухарев
Сибирский государственный автомобильно-дорожный университет,
г. Омск, Россия
suharev_ry@mail.ru, http://orcid.org/ 0000-0002-2627-8110
АННОТАЦИЯ
Введение. Актуальная задача создания перспективных систем беспилотного управления дорожно-строительных машин может быть решена путем проведения теоретических исследований на математических моделях.
Материалы и методы. Одна из важных проблем при создании системы управления движением беспилотной машины - это составление алгоритма следования заданной траектории. Наиболее известным методом следования траектории является метод «чистое преследование», успешно применяемый для управления движением мобильных роботов.
В связи с этим была сформулирована цель исследования - адаптировать метод «чистое преследование» для управления беспилотным автогрейдером. Для достижения поставленной в работе цели были решены следующие задачи: разработана математическая модель движения автогрейдера с передними управляемыми колесами, разработана математическая модель системы управления движением автогрейдера, предложен интегральный критерий для оценки эффективности работы системы управления движением беспилотного автогрейдера, проведены теоретические исследования математической модели и получены зависимости интегрального критерия от конструктивных и эксплуатационных параметров автогрейдера и от параметра метода управления (дальность видимости), найдены оптимальные значения дальности видимости при различных значениях длины базы, коэффициента базы и скорости машины по предложенному критерию эффективности.
Результаты. В результате аппроксимации полученных оптимальных значений метод «чистое преследование» был модифицирован для управления беспилотным автогрейдером с учетом его конструктивных особенностей и скорости передвижения.
Полученные результаты могут быть использованы при создании опытных образцов систем беспилотного управления дорожно-строительных машин.
КЛЮЧЕВЫЕ СЛОВА: автогрейдер, беспилотный, траектория, машина, управление, алгоритм, метод управления, курс, чистое преследование.
Статья поступила в редакцию 08.03.2022; одобрена после рецензирования 15.03.2022; принята к публикации 12.04.2022.
Автор прочитал и одобрил окончательный вариант рукописи.
Прозрачность финансовой деятельности: автор не имеет финансовой заинтересованности в представленных материалах и методах. Конфликт интересов отсутствует.
Для цитирования: Сухарев Р Ю. Применение метода «чистое преследование» (Pure Pursuit) для управления беспилотным автогрейдером / РЮ. Сухарев // Вестник СибАДИ. 2022. Т19, № 2(84). С. 156-169. https://doi.org/10.26518/2071-7296- 2022-19-2-156-169
© Сухарев Р Ю., 2022
Контент доступен под лицензией Creative Commons Attribution 4.0 License.
https://doi.org/10.26518/2071-7296-2022-19-2-156-169
https://elibrary.ru/FUWXGR
Original article
PURE PURSUIT METHOD USE TO CONTROL UNMANNED MOTOR GRADER
Roman Yu. Sukharev
Siberian State Automobile and Highway University (SibADI),
Omsk, Russia
suharev_ry@mail.ru, http://orcid.org/0000-0002-2627-8110
ABSTRACT
Introduction. A relevant objective of implementing the advanced systems of self-driving road construction vehicles can be accomplished by mathematical modelling. One of the important issues when creating a motion control system for a self-driving vehicle is to develop a trajectory following algorithm. The most well-known method of following the trajectory is a pure pursuit method, which is successfully used to control the movement of mobile robots. Materials and methods. Hence, the research objective has been defined and is to adapt the pure pursuit method to control an autonomous grader. To achieve the research objective, the task of a mathematical model of the motor grader movement with front steering wheels has been developed, and a mathematical model of the motor grader motion control system has been compiled. Besides, we propose an integral criterion to evaluate the efficiency of the motion control system of a unmanned grader. Some theoretical studies of the mathematical model have been carried out and the dependencies of the integral criterion on the design and operational parameters of the grader, as well as on the parameter of the control method (visibility range) have been obtained. Moreover, the optimal values of the visibility range for various values of the base length, base coefficient and machine speed have been defined according to the proposed efficiency criterion.
Results. As a result of approximating the obtained optimal values, the pure pursuit method has been modified to control a self-driving motor grader, taking into account its design features and travel speed.
The results obtained can be used to create the prototypes of unmanned control systems for road construction vehicles.
KEYWORDS: motor grader, unmanned vehicle, trajectory, vehicle, control, algorithm, control method, course, pure pursuit.
The article was submitted 08.03.2022; approved after reviewing 15.03.2022; accepted for publication 12.04.2022.
The authors have read and approved the final manuscript.
Financial transparency: the authors have no financial interest in the presented materials or methods. There is no conflict of interest.
For citation: Roman Yu. Sukharev. Pure pursuit method use to control unmanned motor grader. The Russian Automobile and Highway Industry Journal. 2022; 19 (2): 156-169. https://doi.org/10.26518/2071-7296-2022-19-2-156-169
© Sukharev R. Y., 2022
Content is available under the license Creative Commons Attribution 4.0 License.
INTRODUCTION
Nowadays, unmanned technologies are widely used in various branches of industry and economy. The adoption of self-driving technologies in the construction industry, namely in heavy equipment, is a promising direction that will develop quite rapidly in the next few years [1]. A motor grader is a construction machine similar in control algorithm to a self-driving vehicle. One of the well-known methods of controlling the self-driving cars is the «pure pursuit» method1· 2. This method is used to control mobile robots3 [2,3,4], unmanned vehicles4· 5 [5], agricultural machines [6], logging harvesters [7], underwater uninhabited vehicles [8], etc. However, there are no studies of this method when driving a grader.
Driving an unmanned grader differs from driving an unmanned vehicle mainly by purpose [9,10,11]. The main purpose of the grader movement is the movement of the working body in accordance with the project of the earthen structure. The trajectories of the basic machine and its parts are secondary in this case. In this case, the trajectory is an alternation of sections of rectilinear motion during the working stroke and reversals6 [10,12].
The setting parameter of the «pure pursuit» method is the look-ahead distance. With an increase in this parameter, the machine tends to follow the trajectory more precisely and, thereby, the yaw along the trajectory increases. With a decrease in this parameter, the movements of the machine become smoother, but with sharp turns of the trajectory, the machine begins to “cut the corners”.
Several works were devoted to finding the optimal value of the look-ahead distance for the
car7 [5]. Some authors even suggested using a dynamic look-ahead distance, i.e. a change depending on the speed and accuracy of the trajectory7.
PROBLEM STATEMENT
The purpose of this work is to adapt the «pure pursuit» method to control an unmanned grader. To achieve this goal, it is necessary to solve a number of tasks: to make mathematical models of the movement of a grader with front steerable wheels and a motion control system, to justify a criterion for evaluating the effectiveness of the motion control system of an unmanned grader, to conduct theoretical studies of the mathematical model and to obtain the dependences of the efficiency criterion on the design and operational parameters of the grader and on the parameter of the control method (visibility range), to find optimal values of the visibility range at different values of the base length, the coefficient of the base and the speed of the machine according to the proposed efficiency criterion and on the basis of the data obtained to propose a modified method of «pure pursuit».
MATHEMATICAL MODEL
The mathematical model of movement was developed on the basis of a two-dimensional grid of turning a motor grader with front steering wheels in the Earth coordinate system
OEARTHXEARTH YEARTH (figure 1).
When turning a motor grader with front steering wheels, the elementary displacement of the midpoint of the rear axle OR can be calculated by the following formula8 [13,14]:
1 Omead Amidi. Integrated Mobile Robot Control. Technical Report CMU-RI-TR-90-17, Robotics Institute, Carnegie Mellon University, Pittsburgh, PA, May 1990.
2 R. Craig Coulter. Implementation of the «pure pursuit» Path Tracking Algorithm. Technical Report CMU-RI-TR-92-01, Robotics Institute, Carnegie Mellon University, Pittsburgh, PA, January 1992.
3 Alessandro De Luca, Giuseppe Oriolo. Feedback Control of a Nonholonomic Car-like Robot. 2004.
4 Jarrod M. Snider Automatic Steering Methods for Autonomous Automobile Path Tracking Technical Report CMU-RI-TR-09-08, Robotics Institute, Carnegie Mellon University, Pittsburgh, PA, February 2009.
5 Matthew J. Barton. Controller Development and Implementation for Path Planning and Following in an Autonomous Urban Vehicle. Undergraduate thesis, University of Sydney, November 2001.
6 Gorbov I. A., Leonard A.V. Planning the trajectory of a vehicle when bypassing an obstacle // XXVIII International Innovation-oriented Conference of Young Scientists and Students (MICMUS - 2016) : proceedings of the conference, Moscow, 07-09 December 2016. - Moscow: Federal State Budgetary Institution of Science A.A. Blagonravov Institute of Machine Science of the Russian Academy of Sciences, 2017. pp. 236-239.
7 Wu, Yiyang & Xie, Zhijiang & Lu, Ye. (2021). Steering Wheel AGV Path Tracking Control Based on Improved «pure pursuit» Model. Journal of Physics: Conference Series. 2093. 012005. 10.1088/1742-6596/2093/1/012005.
8 Portnova A. A. the Problem of minimizing the turning radius grader with articulated / Innovation, quality and service in engineering and technology. Kursk: Closed Joint Stock Company "University Book", 2014. pp. 97-99.
Theturningradii Df th^Lî 'ra^r Lam ba defined bT tlNefollowing foirriDteSIl'IS, M]
whence it follows that:
Vdt = RRdy;
άγ _ V dt s Rr’
(2) Rr = - ■ (4)
K tg aw ’
ЙҒ =---’ (5)
(3) sin aw
where S isthemidpoint reovLAiDnt 0LN1e PDotaxIe, Rr is the turning radius of the rear axle midpoint, γ isthe heading angleof the motorgrader, V isthe velocity of the motor grader.
where RF is theturning radius ofthe frontaxle midpoint, L is the length of the grader base, aW is the steering angle of the front wheels.
Figure 1 - Two-dimensional grid of turning the motor grader with front steering wheels [14]
Substituting formula (4) into formula (3) the following ec|uation is obtaineH:
The velouity vectorof ttin roar axle midpoint con be decomposeC iato tne veiacityprojactions Tloag hie axisXE/1/Tc:
Vx c Vt( ny, (7)
or:
I c Vsinr, (8)
and along the axis YEARTH-
VY c ^so^]/·, (9)
or
İÇ citsoT]/. (10)
The set of formulas (4)c .8) anC (10) c^n be represented in the formf a mathematical model (figure 2) compiled in :he MATLAB Simulink software.
ana (he HreOictiod veefor.Uning the data of )|) ure :3, we con record Ithe followmg foemelas1o:ioeg [У-9].
Г-0)’ (11)
_iL
2 ΔΥ1 (12)
where L0 isthe look-aheaddistance, ΔΥ1 isthe deviationof the midpointof the rearaxlefrom the trajectory, φ is the anglebetweenthe longitudinal axisofthe motor graderandthedirection to the target point.
The required turning angle of the motor grader can be calculated using the appropriate equations for a specific type of a machine. For example, for а gaader wide CronC sCeering wheelSr a taming angle is defined by tSe following formula1·2’3’4’5’7 [2-9]:
— = — ■. (13)
h^C tgaw 4 '
Accordingly, the wheel turning angle is defined bythe formula:
/ 2ΔΥ-1 L\
aw = arctan ^ ^-J. (14)
PURE PURSUIT METHODDESCRIPTION
The «pure pursuit» method cionisistis of a geometric calculation of t»e ea^dsoHhocioculao occ connecting the location of °)e reat axle wOh tfio target point on a trajeefor)) in frontofthe ,i,^(ii(oln. The target point is detxrmined baoed or аІ:е fod· ahead distance L0 fromtho midftokit ot ttio od^r axis to the trajectory1’2'1·^·7 [d-8],
The turning angle of te motor grader can be defined using only th location of the target point and angle φ between the machine course vector
Thus’ the «pure pursuit» method is a proportional regulator of the grader transverse displacement error.
EFFICIENCY CRITERION
The main purpose of the grader movement is to movetheblade in accordance with theearth structure project. Consequently’ the deviation of the midpoint of the blade (B) from the specified trajectory should be used as a performance criterion of the selectedcontrolmethod [9J5].
Figure 2 - Mathematical model of turning a motor grader with front steering wheels [14]
OeaRTH
Xe,
Yearih
Figure3-Design diagramofthe«purepursuit»algorithm1·2·3·4·5·7[2-9]
The given para meter can be quantified using the infeernl indiaatorf i.e. lie arnu Odtwgas too eiancifige trajactos/ tund tie tr-tecto- or tde blade midqoiat end is nalculated d. too forncuin
[a.-eqe]
ET = 5<,°°HyOO|dX( (15)
etoareit >г(ПсЄ^) — 3/(tnn^) is the reajactory deviatisie nt hue Plaae rni-tarin^ Гоопр toe eade toat
cor/aspoads to tra ppeaif;ap tccijt^(p1;ary [9,111,16].
Тігє сгіргіоп rsfp paomatrically refosente tde shaded area in figure 4a. The transient process shown in figure 4a is caused by a disturbance, forexample,astepwise change ofthegiven trajectory. The smaller the shaded area, the more preferablethetransitionprocess[9,15,16].
This integral criterion can be used not only to assess the control quality, IdU also to oiutimine toe variable para meters forthe controlsystem synthesis. Moreover, :he absolute value of the criterion ET is not significant.Using the equations for ET and system transfer fuoctі oys,the dependencea of the criterionSh-OfShe voriabie corameters of the control systom and 1^ses<^f^tyr^cl aal ues can be obtained [9,15,16].
When using the module, the integralcriterion ET can be applied to the systems which transients have oscillation and change the sign (figure 4b) [9,15,16].
Figure 4 - Integral quality criteria [9,15]
More complex integral criteria based on the second and following derivatives of ΔΥ can be used. Their application will bring the transients closer to the second and higher order curves [9,15,16].
RESEARCH RESULTS
During theoretical investigation of the mathematical model, a step change of the trajectory by 1 m was used as an input signal.
Figure 5 - The influence of the look-ahead distanceL0 ontheintegralcriterionET
at different velocities ofthemotor graderV Source: compiled by the author
The studies have shown that the following pa -rameters of them athematical model,namelythe look-ahead distance LO the length of the motor grader base, the coefficient of the mohoe tgte^c^^r base, the motor grader velocity have the strongest influence on the integral criterion ET (figures 5, 6, 7).
The coeffioinnh othhh mohoreeadeobaee determines hhc pasiboft oSthe blade in the baec and is dofinhd bg bhe followinh foomule:
^=7 06)
where L1 isthe distance feom the front axleto the blade.
The main objecOive ot terrrbtoal sSud ies of the mathematicalmbdel obt^fie tnofor grader it a dynamicmodewasto determine tooemal tu^ mcrieal volues nf me roatrol rnemod narcr^^-^r^^, meic dependc oces oo ths dosign tod bperation al pnrnm cteto oftfie? erefer.
Thsrmohol parombtets weooc ibided iott three groups: fixed parameters, stochastic parameters, vari able parameters [17,18].
The variable parameters have been divided, in urn, into three subgroups:
d
Et 6 5 4 V
0 2 4 6 en ОП
Figure6-Theinfluence ofthelook-aheaddistance L0 on the integral criterion ET
atdifferentvaluesofthe basecoefficientKb Source:compiledbythe author.
/////
Figure 7 - The influence of the look-aheaddistanceL0 on theintegral criterionET
at different valuesofthebase lengthL Source: compiled by the author.
Design pbramotpbs oUtlee motbb graeeb(base lebb)Uh,baog poefficient).
Oyera(ion al |oabameUees ob t-e erotoe bue<e^r (vehiclp velucS).
Paeametors ebfleeeenCr^ei mothoOeeok-ahead Sistanco te.
T0u ebtainel dupendences have been presented in the becm of a graphical complex of the -(faces for different base lengths and different va 1 ues of the motor grader velocity (figures 8, 9, 10, 11, Of).
Et
0 6 70
0
2.5
Figure 8 - Dependences oftheintegral criterion ET onthe look-aheaddistance L0 and velocity V at different values of thebase coefficient(L=5m)
Source: compiled by the author.
Et
20
1а
16
14
12
10
а
6
4
2
0.5
2.5 9
0,6
0,5
0,4
0,3
0,2
Figure 9 - Dependences ofthe integralcriterionET onthelook-aheaddistance L0 and velocity V atdifferent values ofthe base coefficient (L=6 m)
Source: com°iled by the author.
0,(5
0,5
0,4
0,3
0,2
Figure 10 - Dependences oftheintegralcriterionET onthelook-ahead distance L0 and velocity V at different valuesofthebase coefficient (L=7 m)
Source: compiled by the author.
Et
0,6
0,5
0,4
0,3
0,2!
Figure 11 - Dependencesofthe integralcriterion ET on the look-aheaddistance L0 and velocity V at differentvaluesof the base coefficient (L=8 m)
Source: cyrufil еП by the author.
Et
2.5 10
0,6
0,5
0,4
0,3
0,2
Figure 12 -Dependencesoftheintegral quality criterion on thelook-ahead distance L0 andvelocityVatdifferentvaluesofthebasecoeffident(L=9 m)
Source: compiledby the author.
Based on the obtained dependences, we can conclude that the grader speed significantly affects the efficiency criterion. Since speed is no takeninto account intheoriginalmethod of «pure pursuit»,we propose tomodifythe method and inkroducespeek ink) it.
The ol:^ta^ined dependences have been approximated with the 4th degree poiynomiel swith determination coefficients R2 of at least 0,99
ET = α4· Lq + a3 · L0 + α2· L20 + % · L0 + a0. (17)
Lo
2.5 м/с 2 м/с
1.5 м/с 1 м/с 0,5 м/с
Figure13 -Dependencesofthe optimal values ofthe look-ahead distance L0 at different values of speed,base length andbase coefficient Source: compiled by the author.
Due to the optimization carried out by the Newton method, for the minimality rondition of the iutogralcritnrion, tie obtimri walnut oU tOe look-ahemd distance On have been obfoinw0 fou varioobvolues otta^^ louse lun^W iD^^tu t^ca^ff'ibi^nl; and steroli Fig ure 13 shom/e 1^^^ha^ın^f5^^nn)^^1;htэ obtained optimal values оН°еюок-аЬеаи .istence.
The obtained depieu^noes of the optimal values of the look-ahad distance onthespeed, base length and base coefficient havm been ap-
prenimateh with -egression tcid^tior^^ ondaro hrewented in fable t i
waereoreheiuo oonotigno hecw tnafotm df ^(g = a0 1 u +- a1.|nrom tee rewrehoiow e°uoa -іиьи, ft u°al lee oOvinysttrat the multipfiotWo den °»er^ddor^ly un°he base length L aed cun beafa-aeow^oted withted following equafigy
a0 = 1,6 + 0,04 · E (18)
Table 1
The regression equations of the look-ahead di stance L0
Serce:comead di tt^e^L^thor.
o Kb TSe regression equation R2
0,2 L0 = 1,4 -V + 4,338 0,967
0,3 L0 = 1,4 -V + 4,07 0,999
5 0,4 Ln = 1,4 -V + 3,666 0,974
0,5 L0 = 1,4 -V + 3,282 0,985
0,6 L0 = 1,4 -V + 2,976 0,984
0,2 L0 = 1,36 -V + 5,102 0,968
0,3 L0 = 1,36- V + 4,616 0,961
6 0,4 L0 = 1,36 - V + 4,146 0,976
0,5 L0 = 1,36 - V + 3,742 0,972
0,6 L0 = 1,36 - V + 3,332 0,971
0,2 L0 = 1,32 - V + 5,774 0,956
0,3 L0 = 1,32 - V + 5,282 0,991
7 0,4 L0 = 1,32 - V + 4,688 0,986
0,5 L0 = 1,32 -V + 4,176 0,983
0,6 L0 = 1,32 -V + 3,676 0,976
0,2 L0 = 1,28 - V + 6,36 0,872
0,3 L0 = 1,28 - V + 5,762 0,958
8 0,4 L0 = 1,28 - V + 5,24 0,975
0,5 L0 = 1,28 - V + 4,624 0,975
0,6 L0 = 1,28 - V + 4,054 0,981
0,2 L0 = 1,24 - V + 6,876 0,844
0,3 L0 = 1,24 - V + 6,362 0,881
9 0,4 L0 = 1,24 - V + 5,708 0,958
0,5 L0 = 1,24 - V + 5,084 0,982
0,6 L0 = 1,24 - V + 4,41 0,973
Table 2
The coefficients a1 in the regression equations of the look-ahead distance L0
Source: compiled by the author.
K 0,2 0,3 0,4 0,5 0,6
L
0,5 4,388 4,07 3,666 3,282 2,976
1 5,102 4,616 4,146 3,742 3,332
1,5 5,774 5,282 4,688 4,176 3,676
2 6,36 5,762 5,24 4,624 4,054
2,5 6,876 6,362 5,708 5,084 4,41
"Iffıefree coeffidentH a1 ictOere g гениіноер ue-tioner depend сп fpH base length and Астр coeffici ent. Pot fu rther i3|Dp»rHxic^i^tion, tti eywere -і nm-menidep in txfle 2 andt he oes eltieg depp xCecu(x was a^s^n^fonCd^c^fncI bnthe folluwing nqun^Sigf
% = 3,2 - u · f(t + 0 ,x · L. ( 1 c·
T"her<hUocc, u moditiHh meSltoul оГ «pu rx ρνρ suit» adapted to со nSrol thx motor grad en han been proposedi =ho dopendesco of the optimal value oi ffielook-atiead elistopce on tiew ufe^ed, Irane loie<:)tlh md bawu co<offici<^n1; is reurecestgcI .є;3 foh ■fc5l|uifii^n formufo
L0 = α0·ν + аг. (20)
hfıere ae = 1,6 S- 0,04 · L,
αχ = 0.2 - O · Kt + 0,0 · L, are applied for the
m eter o rede- with fe-ft thee tinpwnocls.
After substituting formula (20) into formula (14), we cbtein a form ulafcrcalculatine the franc wnee ls turning angle adaptedto the grader:
O = atctan Г 2аУі\2\ (21)
( ',(a0Vv-a1)2a n C
DISCUSSION AND CONCLUSION
The developed mathematical models of the movementofamotorgraderwithfrontsteerable wheels and amotorgrader motion control system made it possible to conduct theoretical studies andidentifythedependenciesoftheintegral cri-teriononthe designandoperational parameters ofthegraderandontheparameterofthe control method (look-ahead distance). After optimization, optimal valuesofthelook-ahead distance were found for different values of the base length, base coefficient and machine speed according to the proposed efficiency criterion. As a result of the approximation of the obtained optimal values, the «pure pursuit» method was modified to control an unmanned grader, taking into account its design features and speed of movement. The results obtained can be used to create prototypes of unmanned control systems for road construction vehicles.
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9. Sukharev R. Yu. Methods of controlling the coursefor anself-drivinggrader. The Russian Automobile and Highway Industry Journal. 2022;19(1): 48-60. (In Russ.) https://doi.org/10.26518/2071-7296-2022-19-1-48-60.
10. Sukharev R. Y Trajectory plotting algorithm for a self-driving road grader. Journal of Physics: Conference Series, Vladivostok, 2021. P 012181. DOI 10.1088/1742-6596/2096/1/012181.
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14. Sukharev R. Yu. Mathematical models of the processes of turning wheeled road-building machines. Nauchno-Tekhnicheskiy Vestnik Bryanskogo Gosu-darstvennogo Universiteta. 2021; 3: 259-269. DOI 10.22281/2413-9920-2021-07-03-259-269.
15. Sukhareva S. V., Sukharev R. Yu. Substantiation of integral criteria for the quality of earthworks performed by chain trench excavators. Vestnil SibADI. 2015; 5(45): 52-55. (in Russ.)
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17. Shcherbakov V. S., Sukharev R. Yu., Korytov M. S. Development of the theory of optimal control of road and construction vehicles based on satellite navigation systems: Electronic resource: monograph. Omsk: SibADI, 2017. 155 p. ISBN 978-5-93204-929-7.
18. Shcherbakov V. S., Portnova A. A., Sukharev R. Yu. Improving the steering of a grader with a articulated frame: monograph. Omsk: SibADI, 2016. ISBN 978-593204-971-6.
INFORMATION ABOUT THE AUTHORS
Roman Yu. Sukharev - Cand. of Sci., Associate Professor.
ИНФОРМАЦИЯ ОБ АВТОРЕ
Сухарев Роман Юрьевич - канд. техн. наук, доц, кафедры «Автоматизация производственных процессов и электротехника».
