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<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" article-type="research-article" dtd-version="1.2" xml:lang="en"><front><journal-meta><journal-id journal-id-type="publisher-id">Tractors and Agricultural Machinery</journal-id><journal-title-group><journal-title xml:lang="en">Tractors and Agricultural Machinery</journal-title><trans-title-group xml:lang="ru"><trans-title>Тракторы и сельхозмашины</trans-title></trans-title-group></journal-title-group><issn publication-format="print">0321-4443</issn><issn publication-format="electronic">2782-425X</issn><publisher><publisher-name xml:lang="en">Eco-Vector</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="publisher-id">631801</article-id><article-id pub-id-type="doi">10.17816/0321-4443-631801</article-id><article-categories><subj-group subj-group-type="toc-heading" xml:lang="en"><subject>Theory, designing, testing</subject></subj-group><subj-group subj-group-type="toc-heading" xml:lang="ru"><subject>Теория, конструирование, испытания</subject></subj-group><subj-group subj-group-type="article-type"><subject>Research Article</subject></subj-group></article-categories><title-group><article-title xml:lang="en">The optimal control law of the power supplied to the wheeled vehicle in case of curvilinear movement</article-title><trans-title-group xml:lang="ru"><trans-title>Закон оптимального управления подводимой мощностью к движителю колёсной машины при криволинейном движении</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0001-7884-157X</contrib-id><contrib-id contrib-id-type="spin">8963-6431</contrib-id><name-alternatives><name xml:lang="en"><surname>Kotiev</surname><given-names>Georgy O.</given-names></name><name xml:lang="ru"><surname>Котиев</surname><given-names>Георгий Олегович</given-names></name></name-alternatives><address><country country="RU">Russian Federation</country></address><bio xml:lang="en"><p>Dr. Sci. (Engineering), Professor, Director of the Naberezhnye Chelny Institute</p></bio><bio xml:lang="ru"><p>д-р техн. наук, профессор, директор Набережночелнинского института КФУ</p></bio><email>kotievgo@yandex.ru</email><xref ref-type="aff" rid="aff1"/></contrib><contrib contrib-type="author"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-2171-6302</contrib-id><contrib-id contrib-id-type="spin">1455-9984</contrib-id><name-alternatives><name xml:lang="en"><surname>Gorelov</surname><given-names>Vasiliy A.</given-names></name><name xml:lang="ru"><surname>Горелов</surname><given-names>Василий Александрович</given-names></name></name-alternatives><address><country country="RU">Russian Federation</country></address><bio xml:lang="en"><p>Dr. Sci. (Engineering), Professor, Head of the Multipurpose Tracked Vehicles and Mobile Robots Department</p></bio><bio xml:lang="ru"><p>д-р техн. наук, профессор, заведующий кафедрой «Многоцелевые гусеничные машины и мобильные роботы»</p></bio><email>gorelov_va@bmstu.ru</email><xref ref-type="aff" rid="aff2"/></contrib><contrib contrib-type="author"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-2131-2738</contrib-id><contrib-id contrib-id-type="spin">2005-7528</contrib-id><name-alternatives><name xml:lang="en"><surname>Kositsyn</surname><given-names>Boris B.</given-names></name><name xml:lang="ru"><surname>Косицын</surname><given-names>Борис Борисович</given-names></name></name-alternatives><address><country country="RU">Russian Federation</country></address><bio xml:lang="en"><p>Dr. Sci. (Engineering), Associate Professor, Professor of the Wheeled Vehicles Department</p></bio><bio xml:lang="ru"><p>д-р техн. наук, доцент, профессор кафедры «Колёсные машины»</p></bio><email>kositsyn_b@bmstu.ru</email><xref ref-type="aff" rid="aff2"/></contrib><contrib contrib-type="author"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-4022-9286</contrib-id><contrib-id contrib-id-type="spin">3145-4190</contrib-id><name-alternatives><name xml:lang="en"><surname>Gazizullin</surname><given-names>Ruslan L.</given-names></name><name xml:lang="ru"><surname>Газизуллин</surname><given-names>Руслан Ленарович</given-names></name></name-alternatives><address><country country="RU">Russian Federation</country></address><bio xml:lang="en"><p>Cand. Sci. (Engineering), Senior Lecturer of the Wheeled Vehicles Department</p></bio><bio xml:lang="ru"><p>канд. техн. наук, старший преподаватель кафедры «Колёсные машины»</p></bio><email>rlgazizullin@bmstu.ru</email><xref ref-type="aff" rid="aff2"/></contrib><contrib contrib-type="author"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-9922-5810</contrib-id><contrib-id contrib-id-type="spin">7678-7778</contrib-id><name-alternatives><name xml:lang="en"><surname>Byakov</surname><given-names>Konstantin   E.</given-names></name><name xml:lang="ru"><surname>Бяков</surname><given-names>Константин Евгеньевич</given-names></name></name-alternatives><address><country country="RU">Russian Federation</country></address><bio xml:lang="en"><p>Cand. Sci. (Engineering), Associate Professor of the Multipurpose Tracked Vehicles and Mobile Robots Department</p></bio><bio xml:lang="ru"><p>канд. техн. наук, доцент кафедры «Многоцелевые гусеничные машины и мобильные роботы»</p></bio><email>byakov@bmstu.ru</email><xref ref-type="aff" rid="aff2"/></contrib></contrib-group><aff-alternatives id="aff1"><aff><institution xml:lang="en">Kazan Federal University</institution></aff><aff><institution xml:lang="ru">Казанский федеральный университет</institution></aff></aff-alternatives><aff-alternatives id="aff2"><aff><institution xml:lang="en">Bauman Moscow State Technical University</institution></aff><aff><institution xml:lang="ru">Московский государственный технический университет им. Н.Э. Баумана</institution></aff></aff-alternatives><pub-date date-type="preprint" iso-8601-date="2025-01-15" publication-format="electronic"><day>15</day><month>01</month><year>2025</year></pub-date><pub-date date-type="pub" iso-8601-date="2024-12-21" publication-format="electronic"><day>21</day><month>12</month><year>2024</year></pub-date><volume>91</volume><issue>6</issue><issue-title xml:lang="en"/><issue-title xml:lang="ru"/><fpage>723</fpage><lpage>739</lpage><history><date date-type="received" iso-8601-date="2024-05-08"><day>08</day><month>05</month><year>2024</year></date><date date-type="accepted" iso-8601-date="2024-08-01"><day>01</day><month>08</month><year>2024</year></date></history><permissions><copyright-statement xml:lang="en">Copyright ©; 2024, Eco-Vector</copyright-statement><copyright-statement xml:lang="ru">Copyright ©; 2024, Эко-Вектор</copyright-statement><copyright-year>2024</copyright-year><copyright-holder xml:lang="en">Eco-Vector</copyright-holder><copyright-holder xml:lang="ru">Эко-Вектор</copyright-holder><ali:free_to_read xmlns:ali="http://www.niso.org/schemas/ali/1.0/" start_date="2028-02-21"/></permissions><self-uri xlink:href="https://journals.eco-vector.com/0321-4443/article/view/631801">https://journals.eco-vector.com/0321-4443/article/view/631801</self-uri><abstract xml:lang="en"><p><bold>BACKGROUND:</bold> Highly-mobile wheeled vehicles are designed to move on roads and terrain in various road and soil conditions, which is accompanied by frequent and significant changes in traction forces and rolling resistance forces. In this regard, in order to maintain vehicle mobility and to ensure low energy costs when performing transport tasks, it is necessary to continuously change the operating mode during motion from the completely locked mode to the differential mode in the case of a mechanical transmission. At the same time, the transmission operating mode selected by the driver is not always reasonable. Thus, the development of a law for controlling the power supplied to the propulsion system, ensuring minimal energy losses while maintaining vehicle mobility in widely varying road conditions, is a relevant task.</p> <p><bold>OBJECTIVE:</bold> Increasing the energy efficiency of highly-mobile wheeled vehicles by applying a control law for the power supplied to the propulsion system, adaptive to driving conditions.</p> <p><bold>METHODS:</bold> Increasing the energy efficiency of motion can be achieved by means of reducing losses due to wheel slipping by controlling the power supplied to the propulsion system. It is advisable to obtain the control law as a result of solving an optimization problem, where the loss power is chosen as the objective function, and the traction forces developed on each of the wheels are chosen as the varied values. At the same time, in order to maintain the ability of vehicle to move, it is necessary to take into account that the total traction force on all wheels must be determined by external conditions and be provided by the powertrain. To solve the optimization problem, the Lagrange multiplier method was used.</p> <p><bold>RESULTS:</bold> The conducted studies made it possible to obtain a unified law of adaptive control of the power supplied to the propulsion system in analytical form, applicable in a wide range of road conditions in both straight and curvilinear motion, ensuring the distribution of moments across the driving wheels of the machine close to optimal.</p> <p><bold>CONCLUSIONS:</bold><italic> </italic>The application of the developed law for controlling the power supplied to the propulsion system, based on the use of information about the longitudinal and vertical forces, rotational velocities and steer angles of the wheels during the motion, will improve the efficiency of performing transport tasks when moving a vehicle in continuously changing road conditions. This is achieved due to reducing the workload on the driver in terms of controlling differential locks in comparison with a manual transmission both in straight and curvilinear motion.</p></abstract><trans-abstract xml:lang="ru"><p><bold>Введение.</bold> Высокоподвижные колёсные машины предназначены для движения по дорогам и местности в различных дорожно-грунтовых условиях, что сопровождается частым и значительным по величине изменением тяговых сил и сил сопротивлений качению. В этой связи, для сохранения показателей подвижности машины и обеспечения низких затрат энергии при выполнении транспортных задач необходимо непрерывно в процессе движения изменять режим работы от полностью блокированного до дифференциального в случае механической трансмиссии. При этом выбираемый водителем режим работы трансмиссии не всегда является рациональным. Таким образом, разработка закона управления подводимой мощностью к движителю, обеспечивающего минимальные потери энергии при сохранении подвижности машины в широко изменяющихся дорожных условиях является актуальной задачей.</p> <p><bold>Цель работы</bold> — повышение энергоэффективности высокоподвижных колёсных машин путём применения адаптивного к условиям движения закона управления подводимой мощностью к движителю.</p> <p><bold>Методология и методы.</bold> Повышение энергоэффективности движения может быть достигнуто за счёт снижения потерь на буксование колёс путём управления подводимой мощностью к движителю. Закон управления целесообразно получить в результате решения задачи оптимизации, где в качестве целевой функции выбрана мощность потерь, а в качестве варьируемых величин — силы тяги, развиваемые на каждом из колёс. При этом для сохранения возможности движения машины необходимо учесть, что суммарная тяговая сила на всех колёсах должна определяться внешними условиями и обеспечиваться силовой установкой. Для решения задачи оптимизации применён метод множителей Лагранжа.</p> <p><bold>Результаты и научная новизна.</bold> Проведённые исследования позволили в аналитическом виде получить единый закон адаптивного управления подводимой мощностью к движителям, применимый в широком диапазоне дорожных условий, как при прямолинейном, так и при криволинейном движении, обеспечивающий близкое к оптимальному распределение моментов по ведущим колёсам машины.</p> <p><bold>Практическая значимость.</bold> Применение разработанного закона управления подводимой мощностью к движителям, основанного на использовании в процессе движения информации о продольной и вертикальной силах на осях колёс, их частотах вращения и углах поворота, а также оценке коэффициента сцепления с опорной поверхностью, позволит повысить эффективность выполнения транспортных задач при движении машины в непрерывно изменяющихся дорожных условиях. Это достигается за счёт снижения нагрузки на водителя в части управления блокировками дифференциалов в сравнении с механической трансмиссией, как при прямолинейном, так и при криволинейном движении.</p></trans-abstract><kwd-group xml:lang="en"><kwd>wheeled vehicle</kwd><kwd>law of control of power supplied to the propulsion system</kwd><kwd>optimization</kwd><kwd>Lagrange function</kwd><kwd>specific power loss coefficient</kwd><kwd>mathematical model</kwd><kwd>transmission</kwd></kwd-group><kwd-group xml:lang="ru"><kwd>колёсная машина</kwd><kwd>закон управления подводимой мощностью к движителю</kwd><kwd>оптимизация</kwd><kwd>функция Лагранжа</kwd><kwd>коэффициент удельных потерь мощности</kwd><kwd>математическая модель</kwd><kwd>трансмиссия</kwd></kwd-group><funding-group/></article-meta></front><body></body><back><ref-list><ref id="B1"><label>1.</label><mixed-citation>Petrushov VA, Moskovkin VA, Shuklin SA. Rolling resistance of cars and road trains. Mechanical engineering, 1975. 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