Разработка адаптивных манипуляционных и энергоэффективных локомоционных роботов тема диссертации и автореферата по ВАК РФ 05.02.05, кандидат наук Борисов Иван Игоревич

• Борисов Иван Игоревич
• кандидат наук
• 2019

Оглавление диссертации кандидат наук Борисов Иван Игоревич

Реферат

Synopsis

Введение

Глава 1. Обзор манипуляционных и локомоционных

робототехнических систем в задачах контактного

взаимодействия

1.1 Принципы построения роботов для взаимодействия с окружающей средой

1.2 Обзор захватных устройств

1.3 Обзор галопирующих роботов

1.4 Обобщённая постановка задачи

Глава 2. Разработка адаптивных захватных устройств со

звеньями переменной длины

2.1 Структурный синтез механизма пальца

2.2 Геометрический синтез механизма пальца

2.2.1 Синтез механизма со звеньями фиксированной длины

2.2.2 Синтез механизма со звеньями переменной длины

2.2.3 Верификация кинематической схемы

2.3 Силовой анализ механизмов пальцев

2.3.1 Силовой анализ в режиме щипкового захвата

2.3.2 Силовой анализ в режиме обхватывающего захвата

2.4 Синтез системы управления движением пальца

2.4.1 Динамическая модель объекта управления

2.4.2 Управление входным звеном

2.4.3 Результаты моделирования

2.5 Выводы по главе

Стр.

Глава 3. Разработка энергоэффективных галопирующих

роботов со звеньями переменной длины

3.1 Структурный синтез механизма ноги

3.2 Синтез редуцированного механизма ноги

3.2.1 Синтез механизма со звеньями фиксированной длины

3.2.2 Синтез механизма со звеньями переменной длины

3.3 Синтез целого механизма ноги

3.3.1 Синтез механизма со звеньями фиксированной длины

3.3.2 Синтез механизма со звеньями переменной длины

3.4 Энергетический анализ механизмов

3.4.1 Анализ по упрощённой модели перевёрнутого пружинного маятника

3.4.2 Анализ по динамической модели механизма ноги

3.5 Имитационное моделирование

3.5.1 Моделирование привода ноги

3.5.2 Моделирование редуцированного механизма ноги

3.5.3 Моделирование полного механизма ноги

3.6 Синтез системы управления движением ноги

3.6.1 Управление скоростью движения кривошипов

3.6.2 Управление смещением фаз между кривошипами

3.7 Выводы по главе

Глава 4. Проектирование опытных образцов роботов со

звеньями переменной длины

4.1 Проектирование захватных устройств

4.1.1 Проектирование индустриального захватного устройства

4.1.2 Проектирование кистевого протеза

4.2 Проектирование ноги галопирующего робота

4.3 Выводы по главе

Заключение

Словарь терминов

Графические обозначения

Стр.

Список литературы

Список рисунков

Список таблиц

Приложение А. Публикации по теме диссертации

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Общая характеристика работы

Введение диссертации (часть автореферата) на тему «Разработка адаптивных манипуляционных и энергоэффективных локомоционных роботов»

Актуальность темы.

Роботы, получили большое распространение при решении производственных, транспортных, сервисных и иных задач, частично или полностью заменяя труд человека. При этом, ставшая традиционной парадигма построения и использования робототехнических систем подразумевающая, что состояние робота и объектов окружающей среды должно быть заранее определено и постоянно для эффективного и гарантированного выполнения поставленных задач, ведёт к необходимости использовать человеческий труд даже в опасных и физически тяжёлых операциях, требующих работы в неструктурированном окружении, в особенности в задачах контактного взаимодействия.

В частности, для решения задач манипулирования в условиях неструктурированной среды необходимо использовать новые подходы, способные обеспечить адаптацию поведения системы к силам взаимодействия. При этом необходимо отметить, что функциональность конечного решения во многом определяется не самим манипулятором, а используемой оснасткой, в частности захватными устройствами, которые также должны обеспечивать надёжное удержание объектов произвольной формы.

При проектировании мобильных роботов окружающая среда, в частности геометрия и физические свойства подстилающей поверхности, также априори не известна. Помимо этого актуальной является задача создания энергоэффективных локомоционных роботов, способных обеспечить больший запас хода за счёт рекуперации энергии.

Следовательно, крайне востребованы методы и алгоритмы, позволяющие создавать робототехнические системы, способные автоматически обеспечивать желаемый характер контактного взаимодействия с объектами окружающей среды. Одним из способов решения этой задачи является использование сложных алгоритмов планирования и управления движением по силе или импедансу для относительно простых, полноприводных конструкций. Однако, этот путь сопряжён с рядом сложностей, в частности, необходимостью настройки нелинейных регуляторов, которые к тому же оказываются чувствительны к точности и шумам в измерениях, задержкам и квантованию сигналов по времени, а также производительности вычислительных систем, особенно в режиме реального времени.

Как следствие, востребованы альтернативные подходы, лишённые перечисленных недостатков, когда необходимые принципы управления движением реализуются не на сигнальном, а на физическом уровне, при этом алгоритмы управления просты и понятны для настройки и применимы в реальной инженерной практике.

Степень разработанности темы. Для роботов, механизмы которых представляют собой совокупность абсолютно твёрдых звеньев и сочленений, в большинстве случаев достаточно сложно добиться стабильного характера силового взаимодействия с обеспечением требуемой силы и характера контакта, используя только регуляторы по положению и скорости. В задачах, где необходимо силовое физическое взаимодействие между роботом и объектом производства, применяются методы управления по силе, в том числе с использованием дорогостоящих силомоментных датчиков. Для динамического силового воздействия применяют методы управления по импедансу, впервые представленные в трёх статьях Нэвила Хогана1 как подход управления роботами-манипуляторами. При этом многие ро-бототехнические системы оснащены относительно простой механической конструкцией в виде открытой кинематической цепи, что требует сложной системы управления движением каждого сочленения.2 Помимо создания алгоритмов управления движением для роботов, испытывающих динамический силовой контакт, проектируют специальные электродвигатели с высокой удельной плотностью момента на единицу объёма.3 В этом контексте отдельную ценность представляют методы и алгоритмы анализа

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и синтеза неполноприводных механизмов, а также механизмов включающих элементы переменной геометрии, в частности эластичные звенья и сочленения.6

Принимая это во внимание, в настоящем диссертационном исследовании развиваются методы и алгоритмы структурно-параметрической оптимизации, силового и энергетического анализа механизмов, а также синтеза регуляторов для роботов, где в качестве ключевого элемента предлагается использовать звенья переменной длины (ЗПД), способные обеспечить механическую адаптацию к объектам манипулирования среды и энергоэффективность перемещения. Под ЗПД понимается гибкое тело или система твёрдых тел, входящее в две вращательные кинематические

1 Hogan, N. Impedance control: An approach to manipulation / N. Hogan // 1984 American control conference. IEEE. 1984. С. 304—313.

2Platform portable anthropomorphic grasping with the bielefeld 20-dof shadow and 9-dof tum hand / F. Rothling [et al.] // 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems. IEEE. 2007. P. 2951—2956.

3Proprioceptive Actuator Design in the MIT Cheetah: Impact Mitigation and High-Bandwidth Physical Interaction for Dynamic Legged Robots / P. M. Wensing [et al.] // IEEE Transactions on Robotics. 2017. June. Vol. 33, no. 3. P. 509—522.

4Stable precision grasps by underactuated grippers / G. A. Kragten [et al.] // IEEE Transactions on Robotics. 2011. Vol. 27, no. 6. P. 1056—1066.

5Борисов, А. В. Динамика механических стержневых систем со звеньями переменной длины применительно к эндо- и экзоскелетам [Текст] : дис. .. . д-ра физ.-мат. наук : 01.02.01 / Борисов А. В. Смоленск, 2018. 324 с.

6Bicchi, A. Hands for dexterous manipulation and robust grasping: A difficult road toward simplicity / A. Bicchi // IEEE Transactions on robotics and automation. 2000. Vol. 16, no. 6. P. 652—662.

пары, расстояние между которыми может быть либо зафиксированным либо переменным, при этом длина звена напрямую не контролируется. При наличии возможности полностью зафиксировать длину с помощью дополнительного актуатора этот характер регулировки называется активным, в противном случае пассивным. Характер изменения длины звена переменной длины обуславливается только голономными связями механизма и внешней нагрузкой, приложенной к выходному звену робота.

Целью диссертационной работы является разработка адаптивных манипуляционных и энергоэффективных локомоционных робототехниче-ских систем, способных решать задачи надёжного удержания объектов произвольной формы и собственного перемещения при внешних статических и динамических контактных воздействиях соответственно, система управления движением которых может быть реализована на основе относительно простых регуляторов по положению и скорости благодаря использованию в конструкции решений, позволяющих механически декомпозировать каналы управления и автоматически изменять характер движения, в частности, за счёт введения звеньев переменной длины.

Для достижения поставленной цели диссертационного исследования необходимо было решить следующие задачи:

1. Разработать алгоритм структурно-параметрического синтеза механизмов пальцев электромеханических захватных устройств, способных осуществлять фундаментальные типы захватов, такие как точный щипковый и силовой обхватывающий, изменяя количество степеней свободы с помощью звена переменной длины с активным регулированием.

2. Провести силовой анализ синтезированного механизма пальца манипуляционного захватного устройства со звеном переменной длины в обоих режимах работы для определения значений и направлений реакционных сил в кинематических парах и определения уравновешивающего момента на входном звене.

3. Разработать алгоритм структурно-параметрического синтеза механизмов ног галопирующих роботов, способных осуществить стабильную энергоэффективную динамическую локомоцию с помощью звена переменной длины с пассивным регулированием для обеспечения автоколебательных процессов.

4. Провести энергетический анализ конструкции галопирующего робота, позволяющий исследовать процессы перехода энергии между его составными элементами при движении, рассчитать энергию, расходуемую на перемещение и запасаемую в эластичных элемента, а также определить потери энергии при соударении робота с поверхностью пола.

5. Оптимизировать расположение и параметры эластичных элементов звеньев переменной длины для механизма ноги галопирующего

робота для достижения наилучшей энергоэффективности и наиболее высокой достижимой скорости горизонтального передвижения с помощью динамической модели по критериям регулярности траектории движения робота и минимизации коэффициента затрат энергии на перемещение.

6. Разработать и экспериментально протестировать опытные макеты промышленного захватного устройства и персонального протеза кисти человека с предлагаемым адаптивным механизмом пальца, а также ноги галопирующего робота со звеном переменной длины, оснащённым пассивным гибким элементом.

Научная новизна: Основу концепции диссертационного исследования составляет разработка адаптивных манипуляционных и энергоэффективных локомоционных робототехнических систем, механизмы которых содержат звенья переменной длины, что позволило, благодаря совместному использованию геометрических, аналитических, графоаналитических и численных методов синтеза, моделирования, анализа и оптимизации получить следующие новые научные результаты:

1. Метод проектирования, позволяющий решать задачи управления движением манипуляционных и локомоционных роботов в условиях контактного силового взаимодействия на базе простых регуляторов по положению и по скорости.

2. Алгоритм структрурно-параметрического синтеза механизмов пальцев адаптивных захватных устройств на основе оптимизации с учётом кинематических и динамических ограничений и результатов силового анализа.

3. Алгоритм структрурно-параметрического синтеза механизмов ног галопирующих роботов на основе оптимизации с учётом кинематических и динамических ограничений и результатам энергетического анализа.

Теоретическая и практическая значимость. Результаты, полученные в диссертационной работе, имеют теоретическую и практическую ценность для разработки манипуляционных и локомоционных ро-бототехнических систем. Метод проектирования широкого класса роботизированных захватных устройств и галопирующих локомоционных роботов, испытывающих контактное взаимодействие, заключающийся в декомпозиции каналов управления за счет механической конструкции и использованию звеньев переменной длины, позволяет решать задачи управления движением на базе простых регуляторов по положению и скорости.

Предложенные алгоритмы структурно-параметрического синтеза механизмов со звеньями переменной длины позволяют создавать мехатрон-ные и робототехнические системы, способные захватывать и надежно удерживать объекты произвольной формы без потери своей грузоподъемности или энергоэффективно и устойчиво перемещаться в пространстве.

Разработанные прототипы захватных устройств, конструкция протеза кисти, основанные на синтезированном механизме со звеном переменной

длины, могут быть доведены по коммерческого использования, что под-

7

тверждает подданная заявка на международный патент .

Галопирующий робот служит платформой для разработки и отладки новых методов синтеза, анализа и управления робототехническими системами для промышленного, сервисного и медицинского применения, в частности, локомоционных роботов для мониторинга местности и перемещения грузов, а также промышленных и медицинских экзоскелетов.

Методология и методы исследования. При решении поставленных задач был использован широкий спектр методов современной робототехники, теории механизмов и механики машин, теории автоматического управления, оптимизации и моделирования систем. Оптимизация геометрических параметров была проведена методами кинематического синтеза по дискретным положениям и математического программирования. Силовой анализ механизмов был проведён графоаналитическим методом с построением планов сил. Для описания динамических систем использовался порт-Гамильтонов подход. Для апробации характеристик и поведения разработанных робототехнических систем было проведено имитационное моделирование роботов в среде MATLAB с использованием модуля Simscape и библиотеки Contact Modelling, которая использует для моделирования контакта метод штрафов (penalty-based approach), а также проведено тестирование опытных образцов.

Положения, выносимые на защиту:

1. Метод проектирования адаптивных захватных устройств и энергоэффективных галопирующих роботов, позволяющий, благодаря механической декомпозиции каналов управления и использованию звеньев переменной длины, решать задачи управления движением в условиях контактного взаимодействия на базе простых регуляторов по положению и по скорости.

2. Алгоритм структрурно-параметрического синтеза механизмов пальцев адаптивных захватных устройств со звеньями переменной длины по кинематическим ограничениям, а также критерию минимизации реакционных сил в кинематических парах и уравновешивающего момента на входном звене.

3. Алгоритм структрурно-параметрического синтеза механизмов ног энергоэффективных галопирующих роботов со звеньями переменной длины по кинематическим ограничениям, а также критерию повторяемости траектории и минимизации энергии на перемещение.

7An advanced mechanism for finger adaptive gripper operation mode switching : application PCT/RU2019/000167 / I. I. Borisov [et al.]. 03/2013. OOO "TRA Robotics".

Достоверность полученных результатов, предоставленных в диссертационной работе, подтверждается:

1. Корректным использованием математического аппарата и классических методов синтеза и анализа;

2. Представленными в диссертационной работе результатами численного и имитационного моделирования в программной среде MATLAB;

3. Представленными в диссертационной работе результатами экспериментальных исследований разработанных опытных образцов захватных устройств и ноги галопирующего робота;

4. Печатными работами, а также статьями в сборниках трудов международных конференций.

Апробация работы. Основные результаты работы докладыва-

лись на 7 международных и 3 всероссийских конференциях:

— 20th IFAC World Congress, IFAC WC 2017 (20-ый всемирный конгресс международной федерации автоматического управления, 9-14 июля 2017, Тулуза, Франция) [1];

— VII Конгресс молодых учёных Университета ИТМО (VII КМУ, 2018, Санкт-Петербург, Россия);

— Юбилейная XX конференция молодых учёных с международным участием «Навигация и управление движением» (XX КМУ 2018, 20-23 марта 2019, Санкт-Петербург, Россия).

— 1st IEEE International Conference on Industrial Cyber-Physical Systems, ICPS 2018 (1-ая международная конференция по индустриальным кибер-физическим системам, 15-18 мая 2018, Санкт-Петербург, Россия) [2];

— 16th IFAC Symposium on Information Control Problems in Manufacturing, INCOM 2018 (16-ый симпозиум по проблемам информационного управления в промышленности, 11-13 июня 2018, Бергамо, Италия) [3];

— The seventh IEEE International Conference on Biomedical Robotics and Biomechatronics, BioRob 2018 (7-я международная конференция по биомедицинской робототехники и биомехатроники, 26-29 августа, Энсхеде, Нидерланды) [4];

— 12th IFAC Symoisium on Robot Control, SYROCO 2018 (12-ый симпозиум по управлению роботами, 26-30 августа 2018, Будапешт, Венгрия) [5];

— VIII Конгресс молодых учёных Университета ИТМО (VIII КМУ, 15-19 апреля 2019, Санкт-Петербург, Россия);

— International Conference Cyber-Physical Systems and Control, CPS&C 2019 (международная конференция «Киберфизические системы и управление», 10-11 июня 2019, Санкт-Петербург, Россия) [6];

— 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems, IROS 2019 (международная конференция по интеллектуальным роботам и системам, 4-8 ноября 2019, Макао, Китай) [7].

Результаты диссертационного исследования были использованы в следующих НИР:

— «Разработка адаптивных методов очувствления, планирования и управления движением биомехатронных систем» (грант Российского научного фонда, соглашение № 17-79-20341 от 25.07.2017);

— «Управление киберфизическими системами» (проект в рамках программы повышения конкурентоспособности ведущих российских университетов среди ведущих мировых научно-образовательных центров, субсидия 08-08);

— «Нелинейное и адаптивное управление сложными системами» (проект в рамках программы повышения конкурентоспособности ведущих российских университетов среди ведущих мировых научно-образовательных центров, субсидия 074-U01);

— «Идентификационные методы синтеза наблюдателей в задачах адаптивного управления нелинейными системами» (проект по Заданию № 8.8885.2017/БЧ от 26.01.2017, заключенному с Министерством образования и науки Российской Федерации);

— «Адаптивные и оптимальные алгоритмы планирования и управления движением манипуляционных и мобильных робототехни-ческих систем» (НИР, финансируемая Университетом ИТМО, соглашение от 22.08.2017);

— «Разработка методов синтеза и параметрической оптимизации механизмов локомоционных биомехатронных систем» (НИР, финансируемая Университетом ИТМО, соглашение от 31.08.2018);

— «Разработка методов создания и внедрения киберфизических систем» (Магистерско-аспирантский грант Университета ИТМО);

— «Исследование и разработка конструкции и системы управления опытного макета адаптивного промышленного захвата» (договор с ООО «ТРА Роботикс»);

— «Разработка и изготовление опытного макета универсального захватного устройства» (договор с ООО «ТРА Роботикс»);

— «Разработка устройства адаптивной кинематики для захвата объектов различных форм и размеров» (грант Правительства Санкт-Петербурга, диплом ПСП № 18563, распоряжение от 25.09.2018 № 124);

Личный вклад. Содержание диссертационной работы и основные

положения, выносимые на защиту, отражают персональный вклад автора в опубликованных работах [1—10]. Соискатель принимал непосредственное участие на всех этапах диссертационного исследования, включая разработку алгоритмов синтеза механизмов со звеньями переменной длины,

алгоритмов силового и динамического анализов механизмов, а также проведения имитационного моделирования и проектирования опытных образцов захватных устройств и галопирующих роботов.

Публикации. Основные результаты по теме диссертации изложены в 10 печатных изданиях, 3 из которых изданы в журналах, рекомендованных ВАК, 7 — в периодических научных журналах, индексируемых Web of Science и Scopus.

Содержание работы

Во введении диссертационной работы обоснована актуальность проводимых исследований. Сформулирована цель, задачи и положения, выносимые на защиту. Изложена научная новизна и практическая значимость полученных в диссертационной работе результатов.

Первая глава посвящена анализу современных тенденций в области проектирования манипуляционных и локомоционных роботов. Отмечено, что для стабильной работы роботов в условиях априорно неизвестной и неструктурированной среды необходимо обеспечить их способностью осуществлять контактное статическое/динамическое силовое взаимодействие

Рассмотрены общие принципы разработки робототехнических систем и отмечено, что свойство адаптивности может быть достигнуто как с помощью алгоритмов управления движением роботов, так и быть заложено в механической конструкции. Проанализированы достоинства и недостатки данных подходов. Проведено обоснование целесообразности использования методов бионики при проектировании роботов для контактного взаимодействия с объектами окружающей среды.

Приведены обзор электромеханических адаптивных захватных устройств (ЗУ). Представлена классификация механизмов пальцев ЗУ по принципу приведения в движение (рисунок 2), а также обзор различных индустриальных захватов и протезов кисти, реализованных на основе данных механизмов.

с объектами окружения.

б)

а) Точный и б) силовой

Рисунок 1 -Фундаментальные захваты, осуществляемые кистью

O

A

а) б) в) г)

Рисунок 2 - Примеры механизмов пальцев электромеханических ЗУ

Приведено сравнение достоинств и недостатков полноприводных (рисунок 2, а) и неполноприводных (рисунок 2, б) пальцев ЗУ в виде открытых кинематических цепей, а также неполноприводных (рисунок 2, в) и полноприводных (рисунок 2, г) пальцев ЗУ в виде закрытых кинематических цепей. Сделан вывод, что для комбинирования в одном устройстве способности осуществлять фундаментальные виды захватов, на которые способна человеческая рука: точный щипковый и силовой обхватывающий (рисунок 1), обеспечения высокой грузоподъёмности и минимизации массы самого ЗУ за счет использования минимального количества управляющих двигателей, приводимых в движение простыми алгоритмами управления, необходимо синтезировать механизм, способный реконфигурироваться из неполноприводного (рисунок 2, в) в полноприводный (рисунок 2, г) и обратно, повторяя человеческую руку на функциональном уровне.

Представлен обзор педипуляторов, т.е. робототехнических систем имитирующих опорно-двигательную систему человека либо животных, с фокусировкой на способных осуществлять динамическую энергоэффективную локомоцию галопирующих роботов. Приведена базовая модель, используемая для описания динамики галопирующих роботов,8 а также охарактеризованы основные фазы данного вида перемещения (рисунок 3).

Полет

Приземеление

Стойка

Полет

наивысшее положение

момент до удара

возможное нога в контакте проскальзывание с полом

наивысшее положение

\ \

M

_*

' v.

I, m

Рисунок 3 - Фазы галопирующего движения на примере перевёрнутого

пружинного маятника

g

i

k

8The effect of swing leg retraction on running energy efficiency / M. Haberland [et al.] // 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems. IEEE. 2011. P. 3957—3962.

в

г)

а) б) в)

Рисунок 4 - Примеры механизмов ног

Отмечено, что энергоэффективность может быть достигнута за счёт введения в конструкцию гибких звеньев, способных рекуперировать энергию. Описаны методы расчёта энергоэффективности, а также способы её повышения.

Приведена классификация механизмов ног галопирующих роботов по принципу приведения в движение (рисунок 4), а также обзор наиболее релевантных роботов. Представлены механизмы в виде открытых полноприводных (рисунок 4, а) и неполноприводных (рисунок 4, б) кинематических цепей, а также замкнутых неполноприводных (рисунок 4, в) и полноприводных (рисунок 4, г) кинематических цепей. Сделан вывод, что для обеспечения одновременно стабильного характера динамического контактного взаимодействия, энергоэффективности, минимизации массы и габаритов робота при использовании минимального количества управляющих двигателей, приводимых в движение простыми алгоритмами управления, необходимо синтезировать неполноприводные механизмы, оснащённые гибкими элементами и способные входить в режим автоколебаний.

В заключение главы приводится описание предлагаемого метода проектирования робототехнических систем, подразумевающего следующие этапы:

1. Задание желаемого характера работы робототехнического устройства: опорной траектории и рабочей области выходных звеньев, кинематических и динамических ограничений, а также дополнительных функциональных критериев (например, адаптация к форме объектов для захватных устройств и минимизация затрат энергии на перемещение для галопирующих роботов). Выбор типа регулирования ЗПД: активное или пассивное.

2. Выбор базисного плоского механизма исполнительного элемента робототехнической системы с замкнутой кинематической цепью и способа управления движением входных звеньев.

3. Структурно-параметрический синтез механизма исполнительного элемента робототехнической системы включающий:

а) Расчёт конфигурации присоединяемой группы звеньев к базисному механизму, с определением звеньев-кандидатов на замену звеньями переменной длины.

б) Оптимизация геометрических параметров всего механизма и получение его кинематической схемы в полноприводном режиме со звеньями фиксированной длины.

в) Замена одного из имеющихся двухвершинных звеньев присоединённой группы на ЗПД, при этом минимальное и максимальное значения параметров ЗПД выбираются так, что при их изменении в данном диапазоне все возможные траектории выходных звеньев принадлежат заданной рабочей области.

г) Задание динамических параметров всех звеньев механизма (массы, моменты инерции, координаты центров масс) с учётом применяемых конструкционных материалов и технологии изготовления устройства.

д) Оптимизация параметров ЗПД механизма-кандидата с учётом введённых динамических ограничений и на основе показателей заданных функциональных критериев: силовой анализ для захватных устройств и энергетический анализ для галопирующих роботов.

е) Выбор наилучшей модификации механизма с ЗПД.

4. Синтез алгоритмов управления движением, при которых конечный функционал устройства с учётом типа регулирования ЗПД достигается за счёт независимого решения задач стабилизации или слежения по скорости или положению входных звеньев.

5. Верификация полученной системы на соответствие желаемому характеру работы на основе имитационного моделирования.

Вторая глава посвящена задаче проектирования адаптивных ЗУ, способных благодаря возможности изменения количества степеней свободы пальцев обеспечить оба фундаментальных вида захватов: точный щипковый и силовой обхватывающий. В главе рассмотрен предлагаемый метод проектирования робо-тотехнических ЗУ в задачах статического контактного взаимодействия с объектами манипулирования, при этом управление движением сводится к стабилизации скорости и положения входных звеньев механизма.

В качестве примера рассмотрен случай использо-Неполноприводный вания ЗПД с активным регулированием, что изменяет механизм пальца количество степеней свободы пальца, тем самым

обеспечивая реконфигурацию механизма пальца из неполноприводного в полноприводный и обратно.

Рисунок 5 -

Список литературы диссертационного исследования кандидат наук Борисов Иван Игоревич, 2019 год

Литература

Seok S., Wang A., Chuah M.Y., Hyun D.J., Lee J., Otten D.M., Lang J.H., Kim S. Design principles for energy-efficient legged locomotion and implementation on the MIT cheetah robot // IEEE/ASME Transactions on Mechatronics. 2015. V. 20. N 3. P. 1117-1129. doi: 10.1109/TMECH.2014.2339013 Seok S., Wang A., Chuah M.Y., Otten D., Lang J., Kim S. Design principles for highly efficient quadrupeds and implementation on the MIT Cheetah robot // Proc. 2013 IEEE International Conference on Robotics and Automation, ICRA. 2013. P. 33073312. doi: 10.1109/ICRA.2013.6631038 Bhounsule P.A., Cortell J., Ruina A. Design and control of ranger: an energy-efficient, dynamic walking robot // Adaptive Mobile Robotics — Proc. 15th International Conference on

References

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5. Raibert M., Blankespoor K., Nelson G., Playter R Bigdog, the rough-terrain quadruped robot // IFAC Proceedings Volumes. 2008. V. 41. N 2. P. 10822-10825. doi: 10.3182/20080706-5-KR-1001.01833

6. Duindam V., Stramigioli S. Modeling and control for efficient bipedal walking robots: A port-based approach. Springer, 2009. 214 p. (Springer Tracts in Advanced Robotics book series; V. 53). doi: 10.1007/978-3-540-89918-1

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10. Hutter M., Gehring C., Bloesch M., Hoepflinger M.A., Remy C.D., Siegwart R. StarlETH: A compliant quadrupedal robot for fast, efficient, and versatile locomotion // Adaptive Mobile Robotics - Proc. 15th International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, CLAWAR 2012. 2012. P. 483-490. doi: https://doi.org/10.1142/9789814415958_0062

11. Folkertsma G.A. Energy-based and biomimetic robotics. University of Twente, 2017. doi: 10.3990/1.9789036543163

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13. Snippe M. Cheetah robot leg mechanism: analysis, design and cost of transport. University of Twente, 2017.

14. Kenneally G., De A., Koditschek D.E. Design principles for a family of direct-drive legged robots // IEEE Robotics and Automation Letters. 2016. V. 1. N 2. P. 900-907. doi: 10.1109/LRA.2016.2528294

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Climbing and Walking Robots and the Support Technologies for Mobile Machines, CLAWAR 2012, 2012, pp. 441-448. doi: 10.1142/9789814415958_0057

4. Sakagami Y., Watanabe R., Aoyama C., Matsunaga S., Higaki N., Fujimura K. The intelligent ASIMO: System overview and integration. Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems, 2002, vol. 3, pp. 2478-2483. doi: 10.1109/ IRDS.2002.1041641

5. RaibertM., Blankespoor K., NelsonG., Playter R Bigdog, therough-ter-rain quadruped robot. IFAC Proceedings Volumes, 2008, vol. 41, no. 2, pp. 10822-10825. doi: 10.3182/20080706-5-KR-1001.01833

6. Duindam V., Stramigioli S. Modeling and control for efficient bipedal walking robots: A port-based approach. Springer, 2009. 214 p. (Springer Tracts in Advanced Robotics book series, vol. 53). doi: 10.1007/978-3-540-89918-1

7. Tedrake R., Zhang T.W., Fong M.-F., Seung H.S. Actuating a simple 3D passive dynamic walker. Proc. IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04, 2004, vol. 5. pp. 4656-4661. doi: 10.1109/R0B0T.2004.1302452

8. McGeer T. Passive bipedal running. Proceedings of the Royal Society of London. B. Biological Sciences, 1990, vol. 240, no. 1297, pp. 107-134. doi: 10.1098/rspb.1990.0030

9. Hurst J.W., Chestnutt J.E., Rizzi A.A. The actuator with mechanically adjustable series compliance. IEEE Transactions on Robotics, 2010, vol. 26, no. 4, pp. 597-606. doi: 10.1109/TR0.2010.2052398

10. Hutter M., Gehring C., Bloesch M., Hoepflinger M.A., Remy C.D., Siegwart R. StarlETH: A compliant quadrupedal robot for fast, efficient, and versatile locomotion. Adaptive Mobile Robotics - Proc. 15th International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, CLAWAR 2012, 2012, pp. 483-490. doi: https://doi. org/10.1142/9789814415958_0062

11. Folkertsma G.A. Energy-based and biomimetic robotics. University of Twente, 2017. doi: 10.3990/1.9789036543163

12. Folkertsma G.A., Kim S., Stramigioli S. Parallel stiffness in a bounding quadruped with flexible spine. Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems, 2012, pp. 2210-2215. doi: 10.1109/IR0S.2012.6385870

13. Snippe M. Cheetah robot leg mechanism: analysis, design and cost of transport. University of Twente, 2017.

14. Kenneally G., De A., Koditschek D.E. Design principles for a family of direct-drive legged robots. IEEE Robotics and Automation Letters, 2016, vol. 1, no. 2, pp. 900-907. doi: 10.1109/LRA.2016.2528294

15. Timofeev G.A., Musatov A.K., Popov S.A. Theory of mechanisms and mechanics of machines. Moscow, Publishing house of MSTU. N.E. Bauman, 2017, 556 p. (in Russian)

Авторы

Борисов Иван Игоревич — ассистент, Университет ИТМО, Санкт-Петербург, 197101, Российская Федерация, Scopus ID: 57200417949, ORCID ID: 0000-0003-0168-6609, [e-mail] Монич Даниил Сергеевич — ассистент, Университет ИТМО, Санкт-Петербург, 197101, Российская Федерация, Scopus ID: 57204657372, ORCID ID: 0000-0001-9098-8873, [e-mail] Колюбин Сергей Алексеевич — доктор технических наук, доцент, Университет ИТМО, Санкт-Петербург, 197101, Российская Федерация, Scopus ID: 35303066700, ORCID ID: 0000-0002-8057-1959, [e-mail]

Authors

Ivan I. Borisov — Assistant, ITMO University, Saint Petersburg, 197101, Russian Federation, Scopus ID: 57200417949, ORCID ID: 0000-0003-0168-6609, [e-mail]

Daniil S. Monich — Assistant, ITMO University, Saint Petersburg, 197101, Russian Federation, Scopus ID: 57204657372, ORCID ID: 0000-0001-9098-8873, [e-mail]

Sergey A. Kolyubin — D. Sc, Associate Professor, ITMO University, Saint Petersburg, 197101, Russian Federation, Scopus ID: 35303066700, ORCID ID: 0000-0002-8057-1959, [e-mail]

2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Macau, China, November 4-8, 2019

Study on Elastic Elements Allocation for Energy-Efficient Robotic Cheetah Leg

Ivan I. Borisov 12, Ivan A. Kulagin1, Anastasiya E. Larkina1, Artem A. Egorov1, Sergey A. Kolyubin1 and Stefano Stramigioli1'3

Abstract— The biomimetic approach in robotics is promising: nature has found many good solutions through millions of years of evolution. However, creating a design that enables fast and energy-efficient locomotion remains a major challenge. This paper focuses on the development of a full leg mechanism for a fast and energy-efficient 4-legged robot inspired by a cheetah morphology. In particular, we analyze how the allocation of flexible elements and their stiffness affects the cost of transport and peak power characteristics for vertical jumps and a galloping motion. The study includes the femur and full leg mechanism's locomotory behavior simulation, capturing its interaction with the ground.

I. Introduction

Wheel-based mobile robots are used in many applications providing advantages, such as low energy consumption, high forward speed, and precision. Additionally, they are much easier to construct and control. They have nevertheless lack of versatility, especially for a priori undefined generic terrain. A quadruped robot is an alternative solution and our current research centers on high speed and energy-efficient running of legged robots. The focus of this paper is on the biomimetic legged locomotion and development of an energy-efficient cheetah robot.

The main direction of our research is to find ways to reduce cost of transport of legged locomotion. This is defined as the ratio of the energy spent E to the product of weight W and covered distance d (CoT):= E/Wd [1].

The cheetah is the fastest animal on Earth, which makes it an interesting source of inspiration. There are several advanced robotic platforms trying to mimic its features. We mention here just a few, closely related to our study. The design principles for the energy-efficient legged locomotion and implementation on the MIT Cheetah Robot is presented in [1]. using energy storage elements to reversibly store the negative work performed during a running cycle and achieve better energy efficiency is described in [2].

This work is supported by the Russian Science Foundation grant (project №17-79-20341). Work of students Ivan A. Kulagin, Anastasiya E. Larkina, and Artem A. Egorov and is also supported within ITMO University project No.619296

1Ivan I. Borisov, Ivan A. Kulagin, Anastasiya E. Larkina, Artem A. Egorov, Sergey A. Kolyubin, and Stefano Stramigioli are with the Biomechatronics and Energy-Efficient Robotics Lab, ITMO University, Saint Petersburg, Russia e-mail: {borisovii, s.kolyubin}@itmo.ru

2Ivan I. Borisov is also with the Center for Technologies in Robotics and Mechatronics Components, Innopolis University, Innopolis, Russia

3 Stefano Stramigioli is also with the Department of Electrical Engineering, Mathematics and Computer Science, University of Twente, The Netherlands

Fig. 1. A render of the proposed design of 4-legged cheetah-robot from the Biomechatronics and Energy-Efficient Robotics Lab, ITMO University

In [3] authors have proven that the cheetah locomotion can be described as a SLIP (spring-loaded inverted pendulum) model. Many walking and running legged robots have fully actuated joints to induce locomotion and this result in complicated control algorithms. An alternative way is to create a system with embedded mechanical intelligence: its desired behavior is programmed by a mechanical design, and simple controllers are sufficient and efficient.

Implementation of motor control for jumping and landing, which exploits the synergy between the control and mechanical structure for a pneumatically actuated bipedal robot called "Mowgli" with an artificial musculoskeletal system, is presented in [4]. The design principles of a highly dynamic biped robot with mechanically adjustable series compliance are described in [5]. An asymmetric antagonistic actuation scheme characterized by large energy storage capacity that enables efficient execution of motions for single degree-of-freedom knee-actuated hopping robot is presented in [6].

This paper is mostly inspired by the robotic cheetah project at the University of Twente[7], [8]. The original task was to create a leg mechanism for a cheetah robot, which is able to run with the least amount of control. These works mostly cover the hip subsystem (femur) development and its analysis. We advance this study to a cheetah-like full leg design and its analysis towards designing a 4-legged robot, which is shown in Fig. 1. As the following analysis shows, the full leg can achieve higher acceleration and speed as well as provide more variability of different walking and running gaits keeping the CoT characteristic close the one of the Minitaur mechanism.

978-1-7281-4003-2/19/$31.00 ©2019 IEEE

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Fig. 2. The femur mechanism of the cheetah robot leg along with their change in trajectory. (1) cranks, (2) connecting rods, (3) crank arm, (4) brick, (5) frame, B is a point to be attached with elastic element

We suggest adding elasticity for light-weight parallel kinematics legs as the key factor for better energy-efficiency of the entire system. The challenge is finding the best position and parameters for such elements. This task becomes nontrivial because of the multiple-link structure.

The rest of the paper is organized in as follows. Section II gives a description of the cheetah robot and leg mechanisms evolved from the femur to full assembly. Section III explains how the simulation model was created for the mechanism itself and for actuator dynamics and discusses results of analysis of the femur and full leg structures for different simulation scenarios like vertical jumps and a galloping free run. The jumping height, distance covered within 20 seconds, and CoT for the femur mechanism, which based on the structure called "Minitaur", and for the full leg are presented for various cases of flexibility allocation.

II. Mechanism Design Description A. Desired behavior

Let us consider a real cheetah galloping gait. It has a rotary gallop, in which the feet touche the ground in a circular pattern [9]. When a rear foot touches the ground, the knee joint bends to absorb the impact force. Then the legs push the body to the flight phase taking the body forward over the rear leg.

The rear and front cheetah legs look different and have different functions. The rear legs are more muscular than the front and they are mostly responsible for the push off motion. As for the front legs, their main function is to keep the body at a certain distance from the ground. In any case, most of leg muscles are located close to the body to reduce the inertia of the legs as much as possible, while a lot of propulsion comes from elasticity of tendons, muscles and even bones bending.

Trying to mimic these features and reproduce animals running abilities, we should design robot legs as light-weight, but stiff structures with the center of mass strongly shifted towards the hip rotary joint and small feet that can follow variable-shaped trajectories. Embedded elasticity is also a strict requirement.

Fig. 3. The mechanism of the leg along with their change in trajectory. (1) cranks, (2) connecting rods, (3) crank arm, (4) brick, (5) sartorius, (6) tibia, (7) fibula, (8) metatarsal, (9) frame, F is a foot

B. Prior art in femur mechanism design

The goal is to design a simple planar leg mechanism with a minimal number of actuators and links, able to change the trajectory of the foot by switching between jumping in place and fast running.

A good candidate for this goal is the Minitaur mechanism firstly used in Ghost Robotics Minitaur Quadruped Robot [10]. [8] also suggests using the "Minitaur" structure for the femur mechanism (Fig. 2). The main difference is that the mechanism in [8] has two constantly rotating cranks, compared to pulsing aligned inputs of the Ghost Robotics design, and the leg's actuation principle is based on the resonance of an elastic element attached to the point B (Fig. 2).

In this case, the femur structure represents a crank-slider mechanism attached to another mirrored crank-slider mechanism in a slider revolute joint B. The cranks AO1 and AO3 (1) rotate against each other. The mechanism has two degrees of freedom (DOF). 1 DOF is used to actuate the mechanism, the second DOF is needed to change the phase between the cranks. When the cranks move identically, the slider B joint travels vertically in a straight line, but if there is a phase difference between the cranks, the step size increases. A mechanism with a crank arm BE and brick E is needed as a guideline for an elastic element.

In general the Minitaur mechanism has several advantages. It can change the step size via adjusting an angle between cranks, it is able to provide a smooth push motion of a foot during a stance phase (when the foot touches the ground) and to retract itself as close to the body as possible during the flight phase to decrease the rotational inertia of the leg to save the energy. However, the main advantage of the Minitaur is that the vertical displacement is almost uncoupled from the step size. The vertical size is almost the same for all output trajectories (see Fig. 2).

C. Full leg mechanism design

In [8] a foot mechanism was not studied. Instead, an elastic element was used to absorb the impact forces, also employing a passive energy storage. In contrast, our paper elaborates the idea to create a more cheetah-like full leg mechanism, which is able to provide the faster acceleration and speed. Here, we describe a full leg design based on the "Minitaur" structure and real cheetah anatomy.

Following a cheetah anatomy, a tibia, a fibula, and a metatarsal should be added to the Minitaur femur mechanism to create a full leg structure. The full leg mechanism has been designed using three position synthesis method as an assembly of the Minitaur, two rockers 4 bar mechanism HNFNB, and a rocker-slider mechanism BCDE. (see Fig. 3). A 4 bar rocker-slider mechanism BCDE was added as a knee. The slider brick E is fixed with DE, the link DC is a connecting rod, CB is a rocker in the description of a rockerslider mechanism or the tibia in the description of a cheetah leg structure. The two rockers 4 bar mechanism was added as an ankle. In the description of a 4 bar mechanism, BM and HN are rockers, HB is a frame, NMF is a connecting rod. In the description of the cheetah leg BM is the second part of the tibia, HN is a fibula, and NMF is a metatarsal.

The changes in the trajectory between points B and F can be seen in Fig. 3. It can be observed that the foot mechanism acts as a motion converter, while the ratio between both trajectories equals 2. As a result, the gait can be twice wider and twice higher resulting in higher speed of locomotion.

In terms of actuation, the idea is to use one powerful DC motor to rotate both cranks of the mechanism and an additional servo motor to control the phase between the cranks. The movement translation from the main DC motor to both cranks can be implemented via a planetary gearbox. Thus, the leg mechanism can be divided into a linkage mechanism and a planetary gearbox.

III. Simulation-Based Mechanism Analysis

As mentioned above, a leg mechanism has to store and release mechanical energy as well as soften impact shock during the galloping motion, therefore, it is to be equipped with elastic elements. For a Minitaur the allocation of a spring seems quite obvious, but it is really a question in terms of the full leg mechanism.

Let us consider how to make the best design choices for the allocation of elastic elements and their parameters based on a dynamic simulation analysis.

A. Model Description and Simulation Scenarios

A detailed simulation is a good tool to evaluate and support fundamental design choices, study interaction between different parts of a complicated mechanical structure and environment, and further plan reference trajectories and tune controllers [11]. Since a mechatronic system is an interaction between mechanics, electronics, and information, simulation should be performed with respect to all domains. In order to do that, we have implemented a simulation in MATLAB simscape Multibody using Contact Forces Library.

Fig. 4. The actuators block consisting of 2 revolute joints, gearbox, DC motors, electrical, and controllers blocks with the sources of input signals

The mechanisms' models were implemented according to the kinematic schemes shown in Fig. 2 and Fig. 3. The robot is considered as a planar mechanism connected with a world frame with a rectangular joint. This means it is able to jump in place or run along a horizontal line. Links were modeled as rigid bodies using solid blocks combining a geometry, inertia and mass, graphics component, and rigidly attached frames into a single unit. The inertia tensors were obtained from the link geometry assuming uniform material density distribution. The planetary gearbox is implemented via Matlab simscape Driveline. The model designed also allows to capture leg-ground dynamic interaction using the penalty-based approach for contact modeling [12]. For better models of spatial visco-elastic contacts [13] can be used.

The phase control between the cranks is essential for the reconfiguration of the mechanism for the trajectory changing. The simulation was conducted for the Minitaur and the full leg with various elastic elements allocation with the wide range of phases in order to understand what design is better in terms of jumping height, speed, and energy efficiency.

B. Actuator model

The model of an actuator block is shown in Fig. 4. There are two rotational joints Oi and O3, which actuate the cranks. The joint O1 is connected via simscape Interface to one of the two the gears of the Simple Gear 0. The Main DC Motor transmits the motion to the second gear of Simple Gear 0 via Motor's Gear Box. The shaft of the Motor's Gear Box is attached to a sun gear S of the Planetary Gear. The second revolute joint O3 is connected to a small gear of the Simple Gear 1 block. The bigger gear of the Simple Gear 1 (ratio equals to 1/3) is fixed with the planetary gearbox's ring R (ratio equals to 3). The additional servo motor transmits the motion through a Servo's Gear Box and a Simple Gear 2 to a Carrier C of the Planetary Gearbox. Thus, we obtain the same motion on both cranks, with the phase being controlled via a servo motor. To control the positions of the cranks two PID controllers are implemented for the servo and the DC motor respectively. Control torques are calculated

14 14.5 15 15.5 16 14 14.5 15 15.5 16 14 14.5 15 15.5

t [s] t [s] t [s]

Fig. 5. Simulation results for jumping in a place of the femur mechanism "Minitaur" (K1 = 1.8 -—m)

based on the difference between the desired and computed motor's and servo's angles and then sent into electrical blocks modeling the PWM transformer and the H-Bridge. Then calculated voltages are fed to both drives. This additional model, which takes into account electromagnetic effects, is important because of two reasons. At first, this is the way to estimate the electrical energy consumed, which influences batteries capacity. Moreover, it enables direct calculation of voltages and currents that are efforts and flows within the port-Hamiltonian approach framework, which is planned to be used later for energy-aware motion controllers design.

C. Femur mechanism case study

As the first step of the simulation let us consider the Minitaur femur mechanism, which represent a typical SLIP model. In order to achieve the hopping behavior, a spring has to be attached to the joint B, since it is a symmetrical structure. If the hopping in a place is a desired behavior, the phase between cranks must be zero. Otherwise, the robot starts to run.

In order to model impacts between the mechanism and the ground surface we have implemented a penalty force approach, which allows a small overlap of the bodies, using the MATLAB's Contact Library. The simulation results for jumping in place for the Minitaur is presented in Fig. 5. The vertical position sensor is attached to the center of a contact sphere, which is connected to a contact point. Therefore, it shows values above zero at the lowest impact position for the jumping height plot. Within the context of this work, the most interesting characteristic is the power consumption. For the example selected its peak value is almost 12 W per impact. As seen, the servo power is quite small, since the servo is only responsible for phase changing; it means impact forces do not affect a moving low-inertia part.

The simulation was conducted for a wide range of phases from -2.9 rad to 2.9 rad, with various spring stiffness and damping coefficients. Step size is 0.26 rad. Free-run experiments are worth making a closer look at. The running sequence is shown in Fig. 6, (a). They revealed that a stable behavior is possible within a narrow range of coefficients. The results of the free-run scenario simulation with the spring stiffness coefficients K1 = 1.8 --, K2 = 1.6 —, K3 = 1.6 N

}■ mm' ^ mm' J mm

and damping fi = 1 —js are shown in Fig. 7. The spring natural length is 5 cm. The main mass is concentrated in the robot's body, the total robot mass equals 0.875 g.

The first plot shows the relation between the phase and the jumping height. It can be seen that the height is stable and almost uncoupled to the phase. The stable jumping height is almost 14 cm. The second plot shows the distance covered within 20 seconds of the simulation; the maximum horizontal velocity is approximately 1 m/s. The third plot is the relation between CoT and the phase. The CoT is the ratio of the spent energy to the product of weight and covered distance. CoT decreases with the increasing in phase, because of the greater distance. If the distance is almost zero (jumping in a place) than the CoT tends to go to infinity.

D. Leg mechanism case study

When the whole leg mechanism has to be designed, one of the tasks is to understand where an elastic element should be located to obtain the best characteristics in terms of the jumping height, horizontal velocity and energy efficiency.

Evolving from the femur mechanism to the full leg mechanism, only four links were added. Since link 6 CM acts as a tibia it has to be rigid to hold external forces. Therefore, only three links remain candidates to be replaced by flexible elements: either a spring-damper with a guide or a flexible link. Namely, rising parallels with cheetah anatomy, we consider the following cases (see Fig. 3):

1) Link MF can be considered as a metatarsal, it can be studied as a flexible body, while others links are rigid bodies.

2) Link DC can be considered as sartorius and built as a spring on a prismatic joint, while others links are rigid bodies.

3) Link HN can be presented as a fibula and built as a spring on a prismatic joint, while others links are rigid bodies.

1) Metatarsal-allocated flexibility: As the second step of the simulation let us analyze the full leg simulation results with a metatarsal-allocated flexible link (Fig. 6, b). The simulation was performed for different materials as nylon (Young's modulus E = 2 GPa, Poisson's ratio v = 0.39), rubber (E = 0.5 GPa, v = 0.48), and glass fiber (E = 72 GPa, v = 0.21). Corresponding simulation results are shown in Fig. 7. These results are similar to the Minitaur: the narrow region of jumping in the middle, it is able to run forward and backward (but these areas are reversed), jumping height is almost uncoupled with the phase. However, since there is not

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(a) Minitaur (c) Sartorius

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(b) Metatarsal (d) Fibula

Fig. 6. Animation of the running sequence for free-run of the femur mechanism "Minitaur" (a), free-run of the full leg mechanism with metatarsal-allocated flexibility (b), sartorius allocated flexibility (c), and fibula-allocated flexibility (d). The red element indicates an elastic element

Average jumping height Metatarsal Sartorius

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Stable running forward Stable run Jumping in a place

—0—1.8 N/mm --S--1.6 N/mm —©— 1.4 N/mm

101 Phase [rad]

—©— Fiber Glass - -©- - Rubber —O—Nylon

0 -Phase [rad]

—G— 3.2 N/mm —0— 3.0 N/mm —©— 2.8 N/mm

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—Q— 4.1 N/mm --©-• 3.9 N/mm —©— 3.7 N/mm

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Fig. 7. The simulation results show relation between the jumping height, distance covered within 20 second, cost of transport criterion and the phase between the cranks. The yellow area indicates stable jumping in a place with a relativity small horizontal velocity, the green area means running forward, the violet area displays running backward, and the red zone means unstable behavior

enough elasticity in the system the jumping height is very small and CoT for flexible metatarsal is too high. It leads to the conclusion that the use of one flexible link without additional springs is insufficient.

2) Sartorius-allocated flexibility: Now let us consider the running of the whole leg mechanism with a spring on a prismatic joint, which is located inside the link DC, which we call 'sartorius' (Fig. 6, c). The simulation was also performed within a range of spring stiffness coefficients from

K1 = 3.2 mm through K2 = 3.° mm until K3 = 2.8 m wto same damping coefficient ft = 1 mjs. The natural length of the spring is 3.5 cm. Corresponding simulation results are shown in Fig. 7. It was concluded that the behavior is significantly different comparing with Minitaur and Metatarsal-allocated flexibility. It has a very wide area for jumping in a place and a very short transition region to the running mode. It is hopping in a place if the phase equals 1.04 rad, but if will change it a little bit and make it 1.3 rad than the robot will

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Fig. 8. The trajectories of a contact point with the ground for the Sartorius-allocated flexibility and the Minitaur femur mechanism in bodies frames

run at 0.7 m/s. The maximum horizontal velocity is 1.5 m/s if the phase is 2.09 rad, which is much faster comparing to the Minitaur.

The main disadvantage is that we got lacks in controllability. If we want slower velocity we have to change the sign of the phase, since instead of backwards movement there is a different gait for the forward moving.

Fig. 8 shows the trajectories of a contact point with the ground for the Sartorius-allocated flexibility and the Mini-taur femur mechanism in bodies frames. The blue dashed lines show the trajectories without contact modeling as in Figures 2 and 3. The red lines are the trajectories of the contact points with contact modeling. Point A indicates the initial moment of contact, point B indicates the takeoff. The dashed-dotted curve AB describes the spring compression, the dotted curve BC means disclosure of the spring. In order to get the stable running behavior the trajectory must be similar to the depicted trajectories.

3) Fibula-allocated flexibility: Finally, we have simulated the leg mechanism with a spring located inside the link HN, which we call 'fibula' (Fig. 6, d). The simulation was also conducted with a range of stiffness coefficients

from Ki = 4.1 mm toough K2 =3.9 mm untn K3 = 3л mm

and damping coefficient в = 1 mN/s. The natural length of the spring is 3.5 cm. Corresponding simulation results are shown in Fig. 7. The behavior is much better that metatarsal-allocated flexibility, but much worse than the Minitaur. The controllability is better in previous example. However, it does not have any advantages comparing with the Minitaur.

IV. Conclusion and Future Work

The paper presented the analysis of the Minitaur femur mechanism and the cheetah-inspired full leg structure for a energy-efficient galloping motion. The femur mechanism from [8] was considered in the paper as a benchmark that should be outperformed by the proposed full leg design.

We have discussed the galloping robot leg structures inspired by cheetah morphology and studied the best flexibility allocation for it. To do so we performed extensive simulations considering three major scenarios (metatarsal-, sartorius-, and fibula-allocated flexibility) for a wide range of phase differences, stiffness and damping coefficients as well as natural lengths for spring elements or Young's modulus and Poisson's ratio for flexible beams. Simulation models have been developed using the MATLAB Simscape Multibody package with the Contact Forces Library for leg-ground

contact modelling. It was concluded, that the best design in terms of horizontal velocity and energy efficiency is the sartorius-allocated flexibility with stiffness coefficient K3 = 2.8 N/mm, actual damping coefficient N=/s, and phase difference 2.09 rad. Modelling taking into account real mass distribution, body inertia, and drives' dynamics shows that such a leg is able to run with the highest achievable velocity given assumed design constraints of up to 1.5 m/s with the best achievable among all considered cases CoT= 0.3, which is comparable with the best-in-class results for actuated legged robots reported in [1].

As the next step to make the quadruped design following the cheetah morphology we are planning to experiment with front and rear legs' mechanisms and focus on motion synchronization and control.

ACKNOWLEDGMENT

The authors would like to express their deepest appreciation to Geert Folkertsma and Martijn Snippe for their early contributions on cheetah robots that have inspired authors for this article.

References

[1] S. Seok, A. Wang, M. Y. (Michael) Chuah, D. J. Hyun, J. Lee, D. M. Otten, J. H. Lang, and S. Kim, "Design principles for energy-efficient legged locomotion and implementation on the mit cheetah robot," IEEE/ASME Transactions on Mechatronics, vol. 20, no. 3, pp. 11171129, June 2015.

[2] G. A. Folkertsma, S. Kim, and S. Stramigioli, "Parallel stiffness in a bounding quadruped with flexible spine," in 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, Oct 2012, pp. 2210-2215.

[3] H. Yu, M. Li, W. Guo, and H. Cai, "Stance control of the slip hopper with adjustable stiffness of leg spring," in 2012 IEEE International Conference on Mechatronics and Automation, Aug 2012, pp. 20072012.

[4] R. Niiyama, A. Nagakubo, and Y. Kuniyoshi, "Mowgli: A bipedal jumping and landing robot with an artificial musculoskeletal system," in Proceedings 2007 IEEE International Conference on Robotics and Automation, April 2007, pp. 2546-2551.

[5] J. W. Hurst, J. E. Chestnutt, and A. A. Rizzi, "Design and philosophy of the bimasc, a highly dynamic biped," in Proceedings 2007 IEEE International Conference on Robotics and Automation, April 2007, pp. 1863-1868.

[6] W. Roozing, Z. Li, G. A. Medrano-Cerda, D. G. Caldwell, and N. G. Tsagarakis, "Development and control of a compliant asymmetric antagonistic actuator for energy efficient mobility," IEEE/ASME Transactions on Mechatronics, vol. 21, no. 2, pp. 1080-1091, April 2016.

[7] G. Folkertsma, "Energy-based and biomimetic robotics," Ph.D. dissertation, University of Twente, Netherlands, 4 2017.

[8] M. Snippe, "Cheetah robot leg mechanism: analysis, design and cost of transportt," Netherlands, 6 2017.

[9] J. E. Bertram and A. Gutmann, "Motions of the running horse and cheetah revisited: fundamental mechanics of the transverse and rotary gallop," Journal of the Royal Society Interface, vol. 6, no. 35, pp. 549-559, 2008.

[10] G. Kenneally, A. De, and D. E. Koditschek, "Design principles for a family of direct-drive legged robots," IEEE Robotics and Automation Letters, vol. 1, no. 2, pp. 900-907, 2016.

[11] R. Hyde and J. Wendlandt, "Tool-supported mechatronic system design," in 2008 34th Annual Conference of IEEE Industrial Electronics. IEEE, 2008, pp. 1674-1679.

[12] C. Rengifo, Y. Aoustin, C. Chevallereau, and F. Plestan, "A penalty-based approach for contact forces computation in bipedal robots," in 2009 9th IEEE-RAS International Conference on Humanoid Robots. IEEE, 2009, pp. 121-127.

[13] S. Stramigioli and V. Duindam, "Port based modeling of spatial visco-elastic contacts," European Journal of Control - EUR J CONTROL, vol. 10, 12 2004.

Static Force Analysis of a Finger Mechanism for a Versatile Gripper

Ivan I. Borisov,1' 2 Sergey A. Kolyubin1 and Alexey A. Bobtsov1

'Faculty of Control Systems and Robotics, ITMO University, 49 Kronverksky Pr., St. Petersburg, 197101, Russia; [e-mail]

2Center for Technologies in Robotics and Mechatronics Components, Innopolis University, Innopolis, Russia

Abstract. In the development of a mechanism, it is important to know the magnitudes, directions, and locations of the constraint forces between the connected links of the kinematic chain in order to design a mechanism with desired characteristics. This paper presents an approach of a static force analysis of the novel complex mechanism consisting of 8 links, which belongs to the VI class of the Assur group. It means that it can be only separated into an input link and a system of 6 links which cannot be divided into smaller groups; thus, traditional methods for graphical analyses cannot be used. The mechanism is used to implement a finger of a versatile bio-inspired industrial gripper, which can change the degree of freedom (DOF) in order to change the mode of grasping. It is possible to change DOF via breaking/reconnecting the kinematic chain of the finger. When the mechanism is intact, it has only 1 DOF and it represents a fully kinematically defined structure that allows performing a precision grasp. When the kinematic chain is broken, the finger gets underactuated, thus it has 2 DOF, and an underactuated power grasp can be performed. The finger represents different types of a mechanism in precision and power grasps. Force analyses of the finger in both modes were carried out in order to get information about the relationship between the torque applied to a driving link and forces applied to surfaces of phalanges. The paper is concerned with the force analysis and a design of a prototype of the gripper.

Keywords: grasping; grippers; mechanisms; underactuation; robotics

1 Introduction

Articulated robots became a universal automation tool for manufacturing processes. They are intended to increase the quality of products, decrease production costs by reducing scrap, and increase quantities of products since

they can work non-stop for long periods of time. Robots are broadly used in order to reduce routine low-skilled manual labor and in cases of frequent changes in objects to produce.

However, a robot alone is not able to perform any task without an end effector. Therefore, a design of the performance device at the end of a robotic arm, designed to interact with the environment, is very important. There are two basic categories of end effectors: a gripper to grasp and hold objects for pick-n-place operations or a tool to perform various manufacturing processes. This paper is concerned with the first category of end effectors. An analysis of the finger mechanism and a design of the versatile gripper are presented.

A lot of grippers are available on the market. There are electric, pneumatic, magnetic grippers and suction cups which can be used to grasp and hold objects [1]. But when it comes to universal devices, electric grippers seem to be the best option, since they are capable to

• Control the position of the gripper finger using encoded motors. Thus, position control can be performed

• Detect the finger position during a contact with an object using encoders

• Control the grip force and speed through detecting current supply

• Use the electric gripper without additional pneumatic hardware.

The challenge for researches is to create a universal gripper which can perform all or almost all manufacturing operations. The idea is to equip all robots with the same device or a narrow range of similar devices.

Since human beings have been a kind of textbook for the development of robots because they perform the tasks that humans routinely do [2], a real human hand is proposed as an example of a "true universal gripper". The level of dexterous manipulation by robots is currently far from that of human beings. It is possible to improve the abilities of robots by transferring human functions to robotic manipulation. The structure of a human finger and hand plays an important role for dexterous manipulation. Let us suppose that a gripper which acts like a human hand will be the best option. There are several grasping styles, which are presented here [3], but the most popular and fundamental ones are the precision and power grasps [2]. For the precision grasp, only fingertips are used for grasping, in-hand movements are available. For the power grasp, internal areas of all parts of the hand are utilized to envelope an object, the hand and the object can be considered as one rigid body. Since a real human hand can perform a great number of grasps [4], the robotic hand also must be able to perform both the precise pinch and power encompassing grasps.

This paper is devoted to an approach of a static force analysis of a novel complex mechanism consisting of 8 links, which can be separated only into an input link and a system of 6 links which cannot be divided into smaller parts. The mechanism is applied to implement a finger for an industrial anthropomorphic gripper, which can perform both precision grasps and power grasps. The concept and the first prototype of the industrial gripper is described in [5]. The paper is organized as follows. Section 2 describes commonly used strategies to develop a finger mechanism for a bio-inspired gripper or artificial hand. Section 3 proposes the finger mechanism and closing sequences in both modes. Section 4 is devoted to the force analysis of the finger mechanism. Section 5 proposes a mechanical design of the gripper and describes the operation process. The conclusion is the final part of the paper.

2 Current Status and Challenges in the Development of Versatile Grippers

A lot of the electrical grippers have been created over the past two decades. But despite everything, these devices are still far away from human-like movements, functionality, and dexterity of a real human hand. Since a human hand is composed of a fixed palm and actuated digits, it is enough to create a mechanism for fingers. There are several commonly employed strategies to implement a mechanism for a digit of an anthropomorphic gripper or artificial hand, which are described underneath.

2.1 Coupling Linkage Mechanism

Coupling linkage mechanism is the one that is fully kinematically defined and has only one degree of freedom. A finger represents a closed kinematic chain with one link fixed. This type of mechanism is used to create an artificial hand to achieve human-like movements [6]. Precision grasp can be performed since the motions of all points on the linkage can be measured with respect to the fixed link. However, no adaptive grasps can be performed.

2.2 Multi DOF Mechanism

Multi DOF mechanism is the one that has more than one degree of freedom. The mechanism represents an open kinematic chain. Each link is actuated by its own motor. Thus, both grasp and adaptive grasp and precision can be performed. However, the payload capacity is limited by motors. There are commercial robotic hands, such as Barrett [7] or Schunck SDH [8]. They both are equipped with three 2-phalange fingers and each of their joints is actuated by a servo DC motor; as a result, these grippers are very precise, but payload capacity is low.

2.3 Underactuated Mechanism

One of the most famous and suitable variants of finger mechanisms is the underactuated one. An underactuated mechanism is the one that has fewer actuators than degrees of freedom [9]. A mechanical finger based on an un-deractuated mechanism makes the gripper self-adaptive. Such fingers envelop the objects to grasp them and automatically adapt to the shape of the objects having only one motor. According to [10], underactuation is able to help to overcome mechanical complexity that is often caused by the need to independently actuate and control each DOF individually. The authors state that underactuated mechanisms are an interesting approach because they offer increased functionality ensuring good adaptability. According to [11], the concept of underactuation allows to reduce a number of actuators without affecting DOFs by using passive elements, such as springs and mechanical limits. Such approach is able to ensure the capability of self-adaptation without any help from control algorithms when the object is grasped. But this type of the mechanism has its own disadvantages too. It is reported that a precision grasp is difficult to achieve by underactuated fingers [9]. It is stated, that grasping of small objects by underactuated hands is impossible, unless specific design modifications are applied [12].

2.4 Hybrid Mechanism

There are devices which can produce both modes of operation. The MultiModal (M2) Gripper [13] can produce both underactuated and fully actuated behaviors. The hand can adaptively grasp objects of varying geometries, pinch-grasp smaller items, and perform some degree of in-hand manipulation. An underactuated soft gripper utilizing fluid fingertip based on the hybrid soft and hard structure of human fingertips was developed to implement versatile grasping [14]. The gripper can perform pinching grasp and enveloping grasp.

2.5 Proposed Design

The main feature of the proposed mechanism for a gripper finger is the ability to lock or run free a special link of the finger mechanism that allows the finger to either be fully defined kinematically, or underactuated (the gripper design is the subject for PTC patent application PCT/RU2019/000167 19.03.2019). The closing/opening of the kinematic chain is implemented with a BK special link (see Fig. 2-4), which can fix its length or make it variable. The fixing/unfixing can be implemented in plenty of ways, such as via electromagnet, pneumatic pin-to-hole setup or with an elastic element with variable stiffness. This approach can be used to create grippers for different tasks both, for industrial grippers and prostheses. This feature allows to choose between a power-adaptive encompassing grasp or a pinch precision grasp. The ability to fix the number of degrees of freedom can help to prevent un-

predictable behavior of underactuated modes, resulting in a more stable and accurate pinch grasp.

3 Finger Implementation

Let us consider the mechanism of the finger which is composed of 8 links (see Fig. 2-4). The finger represents a six-class Assur mechanism which consists of a driving link AB and an Assur group of class VI. The Assur group is a closed kinematic chain with zero degrees of mobility; it cannot be divided into smaller groups.

W = 3n - 2P5 = 0,

where W is the DOF, n is the number of movable links, P5 is the number of kinematic pairs with 5 constraints. If W is zero, then n is even. In this case, there are only a few combinations of n and P5 (see Table 1).

The input link with W = 1 is an exception (see Fig. 1a). All other Assur groups are even. The typical Assur group class VI is shown in Figure 1d. The class of the whole mechanism equals the class of the highest group of the mechanism.

Table 1. The Assur groups

n 1 2 4 6

P5 1 3 6 9

Class of Assur I II IV VI

group

The finger mechanism is shown in Figure 4. Points O, A and Q are frames and represent the eighth fixed link. The link AB is a binary driving link, which is connected with the frame in the point A and with ternary link BC in the point B. The link BC is ternary. In Figure 2, there are two joints in the point B since it connects three links. The ternary link BC is connected to the binary link BK also in the joint B. The finger consists of two phalanges. The distal phalanx CDH is a binary link which connects with the link BC in the point C and with the proximal phalanx DEO in the point D. The proximal phalanx DEO is connected with the frame in the joint O and with the binary link EF in

the joint E. The ternary link KFQ connects with the frame in the joint Q, with the binary link EF in the joint F, and with the binary link BK in the joint K.

Figure 1. Examples of Assur groups: (a) the I class (an input link), (b) the II class, (c) the IV, (d) the VI class of Assur group

A desirable closing sequence is as follows: the distal phalanx CDH has to keep perpendicular orientation with respect to the palm surface during the whole closing sequence. A closing sequence of the finger in the pinch mode is shown in Figure 2. Figure 2a presents the finger in the initial position, Figure 2b presents the finger is the transitional position, and Figure 2c presents the finger in the final position. As one can see, the orientation of the distal phalanx to the palm surface is constant during the closing sequence of the finger, which allows performing a pinch grasp.

A grip force is applied only to the distal phalanx during a pinch grasp. The links are supposed to be rigid. It means that there can be no relative motion between two randomly chosen points in the same link.

/

a) ' !b)

A

c)

Figure 2. Initial, transitional and final positions of the finger in the pinch mode

When a kinematic chain of a digit is closed (see Fig. 2), it has only one DOF:

W = 3n - 2P5 = 3 • 7 - 2-10 = 1.

Thus, a pinch grasp can be performed. But if the kinematic chain of a digit is opened by breaking the kinematic chain, the digit will have two DOFs:

W = 3n - 2P5 = 3 ■ 6 - 2 • 8 = 2.

That allows accomplishing power and adaptive underactuated grasps of unknown random objects. Closing/opening of the kinematic chain is implemented with a special link with an electromagnet that can fix its length or make it variable. Thus, a digit can choose between different operation modes of the closing sequence. Figure 3 presents the closing sequence in an under-actuated mode. Figure 3a presents the finger in the initial position, Figure 3b shows the first contact of the proximal phalanx with the object to grasp, and Figure 3c shows the finger in the final position. As it can be seen, the magnitude of the special link BK has changed.

4 Force Analysis

It is necessary to understand the relationships between the geometry and motions of the parts of a mechanism and forces that produce these motions [15]. The finger represents a complex mechanism. It has to be established, what is the relationship between the motion of the input link and the motion of output links or phalanges, what are the loads on the surfaces. The synthesis of the proposed mechanism was presented in [16]. The topic of the paper is to present the approach that allows examining the proposed design.

4.1 Force Analysis of the Finger in Pinch Mode

In the pinch mode, the mechanism represents a coupling linkage mechanism with only one degree of freedom. The links are supposed to be rigid bodies. The finger mechanism is shown in Figure 4a. Several forces are applied to the mechanism: a force of the contact with the object F at the point H and gravitational forces of each link. Since the motion of the finger is slow and the mechanism has got a fixed structure during the grasp, it is possible to use only the static equations of equilibrium, without taking into account D'Alembert's principle. It means, there is no need to calculate inertial forces and inertial torques, which makes the calculations much easier.

Figure 4. a) The finger mechanism with applied forces, b) The finger mechanism with applied forces and reaction forces in fixed points A, O and Q, and the position of point S24

For the initial position of the finger, we found all the constraint forces and their reactions necessary for this to be a position of equilibrium. The force analysis was carried out using Assur group theory [17-20]. However, since the finger is a class VI Assur mechanism, it can only be divided into three parts: the frame, the input link, and the VI class group. Since it is impossible to separate the class VI group into smaller parts, the whole group should be considered.

Let us consider the equilibrium of the class VI group. It is impossible to calculate full reaction forces before the preparation. The task of the first step is to calculate tangential constraint forces of each link of the group. In order to do this, each link must be considered in the following order 3, 5, 7, 4, 6, 2.

Then the equilibrium of whole group should be considered. The moment at point S24, which is an interaction of lines BS2 and OS4 (see Fig. 4b), should be found in order to find the full reaction force in joint Q. The final step is to consider each link once again and get full reactions in each joint using the method of vector diagrams.

The graphic solution for the mechanism is presented: 1. Let's begin drawings of the free-body diagram of the link 3 or the distal phalanx, which is shown in Figure 5a. There are 4 forces: the applied force F,

the gravitation force G3 and two tangential reaction forces R34 and R32. Therefore, taking counterclockwise moments as positive, we obtain the static equations of equilibrium for link 3:

EMC = -G3C + R\4CD - FIDH = 0, EMd = G3lD}+ R\2ICD - FIDH = 0,

where Mc is the moment about joint C, MD is the moment about joint D, G3 is the magnitude of the gravitation force G3, F is the magnitude of the applied

force F, R3r4 is the magnitude of tangential reaction force R34, R|"2 is the magnitude of tangential reaction force R32, lC is the moment arm of G3

about joint C, lD is the moment arm of G3 about joint D, lDH is the length

of DH, and Icd is the length of CD. The counterclockwise moments taken as positive, the Equations (1) give:

R34 - (FlDH + G3lC3 ) / lCD, R32 = (FlDH - G3lD3 )/ lCD ■

(2)

Figure 5. Free-body diagrams of the links 3, 5, 7, 4, 6 and 2 with applied forces

2. Next we proceed to a free-body diagram of the link 5, which is shown in Figure 5b. There are 3 forces: the gravitation force G5 and two tangential reaction forces R^6 and R^4 . The counterclockwise moments taken as positive, the static equations of equilibrium give:

E ME =~G5lE5 + R16IEE = 0, (3)

ZMe = G5E - R^EE = 0,

which yields to

R56 = G5lE5 !lEE , (4)

R54 = G5lE5 !lEE ,

where Me is the moment about joint E, Mf is the moment about joint F, l„ is the moment arm of G5 about joint E, l„ is the moment arm of G5

E5 e5

about joint F, lEE is the length of EF.

3. Also, 3 forces are applied to link 7: the gravitation force G7 and two tangential reaction forces R76 andR72 (see Fig. 5c). The counterclockwise moments taken as positive, the static equations of equilibrium give:

which yields to

E ML = -GjIL7 + R^KL = 0, Z Mk = GA7- E^KL = 0,

R76 - G7lL7 ! lKL ,

(5)

(6)

where ML is the moment about joint L, Mk is the moment about joint K, L is the moment arm of G7 about joint L, lK is the moment arm of G7 about

joint K, lLK is the length of LK.

4. Let us consider the equilibrium of the ternary link 4, which is shown in Figure 4d. The gravitation force G4 and three tangential reaction forces R48,

R43 and R45 are applied to link 4. The method for continuation is to measure the moment arms about point 54, which can be found as an interaction of lines CD and FE (see Fig. 4a).

ZMS4 =~R45lES4 + R43lOS4 + G4lS4 " R48lOS4 = 0, (7)

which yields to

R!s = (-R45lES4 + R43lOS4 + G4lS4 )!lOS4 = 0- (8)

R72 _ /1

T K

7

Where lES is the moment arm of R45 about point S4, lOS is the moment arm of R43 about point S4, lS^ is the moment arm of G4 about point S4, and lOS is the moment arm of R48 about point S4.

5. We proceed next to link 6. Applied forces Ga, R67, R£s and R65 are shown in Fig. 5e. The magnitude of the reaction force R68 can be calculated like this:

EMs6 = RVks, + RVFS, -RIzIqs, + G6lS6 = 0, (9)

which yields to

R(,8 = (RVKS, + RVFS, + G6ls6 ) / IQS6 = 0, (10)

here lKS^ is the moment arm of R67 about point S6, lFS is the moment arm of R65 about point S6, Iqs is the moment arm of R68 about point S6 and l^ is the moment arm of G6 about point S6. Point S6 can be found as an interaction of lines LK and FE (see Fig. 4a).

6. Figure 5/shows the equilibrium of the last link 2. Forces G2, R23, R27 and R2j are applied to the ternary link 2. The magnitude of the reaction force R£j is as follows:

EMS2 = ~Rt21IBS2 -RT27ILS2 + R2VCS6 -G2ls2 = 0, (11)

R2l = (~R27lLS2 + R23lCS2 " G2lS2 ) / lBS2 = 0, (12)

where lBS2 is the moment arm of R^1 about point S2, lLS2 is the moment arm of R27 about point S2, lCS is the moment arm of R23 about point S2 and l^ is the moment arm of G2 about point S2. Point S2 is an interaction of

lines LK and CD (see Fig. 4a).

That is the way how the magnitudes of tangential reaction forces can be calculated.

7. At this stage, the equilibrium of the chain of links BCDOEFQKL should be considered. Since the action and reaction forces have equal magnitudes and opposite directions, the sum of the action and reaction forces equals zero. When the moment arms of gravitation forces, the grip force and reaction

forces R21, R4s, R£s and R"s are found, equations (13) should be measured. Therefore, taking counterclockwise moments as positive, we obtain:

EMS24 = G2lS24G2 + G3lS24G, + G4lS24G4 -

c l c /""■ ^ & /"■■ ^ CG-71 o Z"1 Fl & c ^

5 S24G5 0 S24G6 7 S24G7 S24F

R21^B + R48lS24O _ R08lS24Q _ R68lS24Q =

(13)

which yields

R68 _ (G2lS24G2 + G3lS24G3 + G4lS24G4

G5lS24G5 + G6lS24Go + G7lS24G7 - FlS24F + (14)

R21lS24B + R48lS24O _ R68lS24Q Vl S24Q ,

where R6S is the normal reaction force of frame g, lS G is the moment arm of G; about point S24, lS F is the moment arm of F, lS B is the moment arm of R21, is the moment arm of R48, 1Sq is the moment arm

of R68 about point S24 .

All others normal reaction forces of each link can be calculated by vector diagrams. At this stage, we proceed to vector diagrams showing the graphic solution for each link. Let's begin drawings of the vector diagram for each link.

The equation of the static equilibrium for links 6, 7, 5, 4, 3, and 2, respectively gives:

°f ff ff ff ff ff °f R(57 + R67 + R6S + R68 + G6 + R(55 + R65 = ° (15)

where the first symbol before V above each vector indicates whether the magnitude is known or unknown respectively; the second symbol before V or above each vector indicates whether the direction is known or unknown,

respectively. R 6n7 means that the direction of the vector is known, but the

magnitude is unknown. A two-dimensional vector equation can be easily solved for 2 unknown variables (2 magnitudes, 2 directions, or 1 magnitude and 1 direction) [12].

The similar is true for all other links:

ff ff ff °f

r76 + G7 + r72 + r62 = (16)

ff ff ff °f

R56 + G5 + R54 + R54 = (17)

°f ff ff ff ff °f R 43 + R43 + R45 + G4 + R48 + R43 = 0, (18)

ff ff ff ff

R34 + G3 + F + R32 + Rn2 = 0, (19)

ff ff ff ff °f

R23 + R27 + °2 + R21 + Rni = 0. (2°)

The vector diagrams for each link is shown in Fig. 6. Thus, it is possible to calculate full constraint forces in all joints and to figure out the torque, which has to be applied to the input link to get the equilibrium position of the finger mechanism during a pinch mode.

4.2 An Example of Calculations

To grasp an object with the weight G0, the magnitude of the grip force F is:

F = G02^ = 1kg • 9.8m/s2 • 2 • 0.6 = 8.2N, (21)

where n is the coefficient of friction between surfaces of the distal phalanx and an object, the mass of the object to grasp is 1 kg. We proceed next to a torque applied to the input link. The moment of R21 about point A:

M1 = R21h21 = 15.854N ■ 0.014m = 0.22Nm, (22)

where h21 is the moment arm of the R21. Thus, the information about the relationships between the torque applied to a driving link and forces applied to surfaces of phalanges are obtained.

4.3 Note for the Static Force Analysis of the Finger Mechanism in an Underactuated Mode

The finger mechanism in an underactuated mode has two degrees of freedom. The first one is responsible for the input link AB, which drives the proximal phalanx, the second is responsible for the distal phalanx. After the first contact with the object (see Fig. 3b), the degree of freedom is decreasing to 1. Thus, the finger mechanism after the first contact represents a typical four-bar mechanism. A static force analysis for such kind of mechanisms is presented in [12].

Figure 6. Vector diagrams showing the graphical solution for each link: a) Link 6, b) Link 7, c) Link 5, d) Link 4, e) Link 3 and f) Link 2

5 Design of the Gripper

The grasping tool consists of two fingers and a palm. Both fingers are actuated by one worm gear DC motor with encoder and both fingers are equipped with clutches to break/reconnect the kinematic chains of the fingers. The torque of the motor is 0.2 Am. The main feature of the gripper is the ability to change the mode of the finger mechanism in order to use power or precision grasps. The gripper was designed to perform variable types of grasps to grip

objects of variable shapes and sizes using the suggested hybrid structure. Figure 7 shows the gripper performing both kinds of grasps. The sizes of the gripper and the working area are shown in Figure 8. The gripper opening varies from 0 to 120 mm. The gripper weight is 2 kg. Object diameter for encompassing is 10-100 mm. Max payload in the adaptive grip is 4 kg and 1 kg in the pinch mode. The prototype of the gripper was manufactured via 3D printer from ABS plastic. Bars, which are located inside joints, are made from steel. The gripper has a flange to be connected as an end effector to an industrial robot.

Figure 7. The design of the gripper. The pinch mode is on the left, the underactuated mode is on the right

ia)

Figure 8. The sizes of the gripper and the working area of the fingers in pinch mode (a) and underactuated mode (b)

b)

Conclusion

In this paper, a graphical approach of static force analysis of a finger mechanism of versatile bio-inspired industrial gripper is presented. The proposed mechanism represents a complex structure consisting of 8 links. In performing the static analysis, the proposed mechanism can only be separated into an input link and a system of 6 links which cannot be divided into smaller parts. The proposed approach allows examining a static force analysis for such complex mechanisms.

The force analysis is needed to calculate the magnitudes and directions of the constraint forces between the connected links of the mechanism and to understand the relationship between the motion of the input link and the motion of phalanges. According to the obtained magnitudes, adjustments can be made to the geometry of the mechanism to optimize the magnitudes of constraint forces. In order to design a finger mechanism with desired characteristics, it is necessary to know the relationship between the geometry and motions of the parts of a mechanism and the forces that produce these motions. The force analysis was carried out using Assur group theory. The finger, which is class VI Assur mechanism, can be only divided into three parts: the frame, the input link and the class VI group. Since it is impossible to separate the class VI group into smaller parts, the equilibrium of the whole class VI group has been discussed. Such kind of mechanisms is needed to enable the finger to change a DOF in order to switch the mode of grasping through breaking/reconnecting the kinematic chain of the finger. The prototype of the gripper was created in order to apply the proposed mechanism to a real device.

Acknowledgment

This work is supported by the Russian Science Foundation grant (project №17-79-20341). The authors would like to express their deepest appreciation to TRA Robotics Ltd. Company for the technical assistance and support of this study.

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ELSEVIER

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CONFERENCE PAPER ARCHIVE

Design of versatile gripper with robust control*

Ivan I. Eîorisov * Oleg I. Eîorisov* "V^adMav S. C^romov" Sergey M. niasov * Dmiïriï Dolt)riborscC * Sergey A.. KmlyuMn*

* Faculty of Control Systems and Robotics, ITMO Universiti/, St. Petersburg, Russia (e-mail: borisovii@corp.i/mo.rM).

Abstract: This paper presents a novel approach to the development of finger mechanisms for anthropomorphic grippers. The proposal approach allows to design a gripper finger that is able to break/reconnect a kinematic chain of the finger in order to change a degree of freedom, thot will enable to choose between (different operation modes of the grappmg deoiee. This approrch wan be used tn (create gaippers for different; tanks suoh industrial grippers os well as prosthesis. In this paper, the peopoead approed was used to our gripper named "UHVAT" (Usable

HoMing Versatile _A.clj tushalbl«t Too^ for industrial 1ltrtsl<ltз. The gripper consists of three fingers and a palm and eqmpped with four motors. The robueS regulator was designed foA positioning control of the motors. The control law includes cancellation of steady ^t;at^ error and anfi-windnp partions.

© 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

rOeifworAs: Rot>otic Sjestoms, Factory and Industry Automation, Robust Control, Industrial rolbots, Adaptioe Graspiing

1. INTRODUCTION

Dextecous artificial hand enables amputees to execute their daild lève activitiee and to perform thefr worOing duties and tasks. Universal grasping devices are aimed to mimic a human hand in order to provide the same dexterity an o reae human hand. Researchers all oves the world are interested in y robotic device to act likn a real humao hand or even beyond the abilities of humkn handl A robotic nand or an anthropomorphic gripper with nearly m he same deateritc as a human uand is usetul fot to mmimizing human labor in industry or even compleeely replace humus in dangerons itnIS hazardous clreironment. DeueSodmen1 of such devicn, took Einc^ software solutions Dould allow to decrease human involvement in industrial dperations, increasa s^^^^yy and peeУormnnce ot thefr exe-2ution. hh lot of ismping devices have been develoded iy recent yhars. Gripping is a vevy important issue in industry mid requires receeech and exyeriments in order to create more efficient and more productive grippers.

A lot offi robotic pevicei developeg have anthropomorphic rtructure. ]^lr[u.lt:i]:>l^ ttate-ot-the-art j-oteo^i^ hii^nds wil1lls morc degrees of freedom cttOF- and sensors have been Aeveloped since 1990s, such as (Gosseiin et al. 1199S-, !L^iIlil:>«з:^1liз and GoQselm (( 1 £li9S(2)^ Gooselm et al. (200S)-, CyberHand ^etozzu et al. (2006))i[ GCUA HiintI gChe and Zhang (OrOL), a fi'^^fing^i'^d, multi-sensory a:nd biomimehic hand (Wang vt al. 12010-U HI"]? hand (Lin eti a. (2014-- and tt0li3 EMG-controll2d prosth2tic hand wfth sensosy system (Borisov et ah ^ Howevei, a

device does not have to loov exactly like a r^al humai

* This work is supported by the Russian Science Foundation grant; (project №lS-S9-2030r).

Ii^uiC. It has to act like a human hand only on thc frmctional level. Unéverskl grippers with anthropomorphic fingers have been developed since the turn oi the centuries, such as underactuaeed eobotic hands (LalibertO et al. 12002--, a reconfignrabIe three-finger robotic gripper(Li ett j i20i5-n Multi-Modal (M2)] Gripper Yale O^Hal Project 1Ma et al. (l2016)))) which can produce efther underactuated or tully-actuated behaviors. Researcherr nre trying too create a robotic hand wlnch is abce to miintc human hand movements. There are leveral typn of m^cli^n^sm which for implementing a fi^vser design. hh]n underoctuated mechanism is one oO the most common And suitable variants. TOu principle of undeeactuation is a commonly employeb strategy for tUe ilnaer d^^^gn. hh mtîcli^n^^m is said to be underactuated when ri lias fewer actuators than degrees ou freedom (DOFs-t hi m^(sllan^ca^ finger Ija^ed on an underastuated m^lt^s the

nripper fnuc]LL fingeils envelopl ^lrn olitiecîts tto

b)^ grasped and automatically adapt tto the shape of the objrcts with only one motor. However, this type os the m^cls^ni^m has its own advantagas ^nd disadvantages.

Fig. 1. The UHVhhT Gripper

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Peer review© under responsibility of International Federation of Automatic Control.

10.1016/j.ifacol.2018.11.518

This paper suggests an approach to develop a different design of the finger mechanism that is able to change the modes of grasping via breaking/reconnecting a kinematic chain of the finger in order to change the degrees of freedom. The finger has one DOF when the kinematic chain is closed, thus the trajectory of the finger is predictable, which allows to execute the precise pinch grasp. The finger has 2 DOF, when the kinematic chain is open, which allows to accomplish power adaptive encompassing underactu-ated grasp of unknown random objects irrespective of of their orientation, which helps with pick-and-place tasks. Closing/opening of the kinematic chain is implemented with the special link, which is able to fix its length or make it variable. In this paper the approach proposed has been used to design the Usable Holding Versatile Adjustable Tool (Fig. 1) to grasp objects with variety size, shape, mass, kind of material and design firmness. The first prototype of UHVAT industrial gripper is described Borisov et al. (2018).

This paper is organized as follows. Section 2 proposes the finger mechanism. Section 3 analyses the gripper mechanism. Section 4 proposes a mechanical design of the gripper and describes the operation process. The control system is proposed in Section 5. Section 6 presents and discusses experimental results. Conclusion is a final part of the paper.

2. FINGER IMPLEMENTATION

The mechanism of the gripper finger is inspired by the biological design of human fingers. The gripper developed (Fig. 1) is able to use two different modes of the fingers mechanisms in order to choose between power adaptive or pinch precision grasp for a certain object. This is possible through changing the mode of finger mechanism from underactuation mode to pinch mode and back via special link, which will be discussed in section 4. When kinematic chain is closed the pinch mode is on. Pinch mode makes it possible to perform precision grasps, thus assembly tasks can be performed.

A closing sequence of the finger with the pinch mode is shown in Fig. 2. The finger is actuated through the link AC. In a, the finger is in the initial position, there are no external forces. An input link is located on the finger base and drives two bars to transmit the motion to the first phalanx, while the first phalanx drives the second phalanx (b). Using the coupled mechanism a determined system is obtained at any time during finger bending process.

The force applied by the motor is distributed among all phalanges when the object is fully grasped (c). When kinematic chain is opened via breaking link CCi an under-actuation mode is on, which allows to perform an adaptive power grasp of random objects irrespective of their orientation, which helps with pick-and-place tasks. Each finger becomes selfadaptive, while relative configuration of the finger at any time is determined by external constraints associated with the object. Since the kinematic chain is open, the finger has 2 DOFs, thus a torsion spring has to be embedded in the mechanism to keep the phalanges aligned under the action of this spring when no external forces are applied to the finger. A closing sequence of the finger in underactuation mode is shown in Fig. 3. In a, the finger is in the initial position, no external forces are present. All phalanges of the finger are rotating about point A (Fig. 3) as one rigid body until (b) the proximal phalanx ABD is blocked by an object or reaches in the final position. When the phalanx ABD gets in a contact with the object, only the distal phalanx EDH rotates around the proximal joint D (c) until it is blocked by the object or reaches the final position. Thus, the finger can grasp objects self-adaptively.

3. ANALYSIS OF THE FINGER MECHANISM

The kinematic behavior of the finger mechanism has been analyzed. The both finger mechanisms are shown in Fig. 4. Points A and Ai are fixed. Point A represents encoder rotation joint, link AC is able to rotate around point A. Link AC is the driving link and it drives the whole finger mechanism. The first phalanx ABD, which is connected to point A, connects with link BBi at point B, link BBi connects with link B1C1 at point B1, link B1C1 connects with the finger base in point Ai and connects with link CC1 at point C1 , link AC connects with link CC1 and link EC at point C, the second phalanx EDH is connected with the first phalanx at point D and with link CE at point E. The both modes are considered below.

3.1 Analysis of Coupled Mechanism

The finger mechanism represents a coupled mechanism and it depends on only one input variable to control the whole finger. Thus, the position and the orientation of the fingertip is always determined during task execution, and it is possible to predict the trajectory of the fingertip. Consider the relationship between the angles of the mechanism shown in Fig. 4. a is the angle between horizontal line (palm) and the link AC. a represents the input variable.

Fig. 2. Closing sequence in pinch mode