The Jacobian conjecture Reduction of degree and formal expansion of the inverse

小学上册英语第一单元综合卷英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.I like to watch ________ in the summer.2.My favorite holiday is ________ (圣诞节). I like to decorate the ________ (圣诞树).3.My favorite book is ________.4.What do we call the place where you can buy groceries?A. StoreB. MarketC. MallD. Supermarket5.The _______ of a balloon can be affected by altitude.6.The _______ (兔子) hops around quickly when it is excited.7.What is the name of the game where you shoot hoops?A. SoccerB. BasketballC. BaseballD. TennisB8. A thermochemical reaction involves heat and chemical ______.9. (85) is a famous park in New York City. The ____10.The _______ (Apollo 11) mission successfully landed humans on the Moon.11.What is 100 - 25?A. 65B. 70C. 75D. 8012.What is the main ingredient in sushi?A. RiceB. NoodlesC. BreadD. PotatoesA13.The bear roams in the _____ woods.14.__________ are important for environmental sustainability.15.The chemical formula for table salt is ______.16.What is the capital of Honduras?A. TegucigalpaB. San Pedro SulaC. La CeibaD. CholutecaA17. A ______ (狗) has a keen sense of smell.18.The ancient Greeks created _______ to explain natural phenomena. (神话)19.The teacher, ______ (老师), guides us in our studies.20.The cake is _______ (刚出炉).21.The _____ (first) man-made satellite was Sputnik, launched by the USSR.22.The capital of Faroe Islands is __________.23.The __________ can provide critical insights into environmental health and stability.24.What do you call the place where we see many books?A. SchoolB. LibraryC. StoreD. Park25.What do you call the study of the Earth's atmosphere?A. MeteorologyB. GeologyC. AstronomyD. Ecology26.What is the term for the distance around a circle?A. AreaB. DiameterC. CircumferenceD. RadiusC27. A ___ (小蝴蝶) flutters gently in the air.28.My ________ (玩具) is made of plush material.29.What do we call the act of cleaning a room?A. TidyingB. OrganizingC. DeclutteringD. CleaningA30.What do we call the tool we use to write on paper?A. MarkerB. PenC. PencilD. All of the above31.The teacher gives _____ (作业) every week.32.The _______ of matter refers to whether it is a solid, liquid, or gas.33.What is the opposite of short?A. TallB. WideC. NarrowD. ThickA34.I like to play ___ (video games).35.I like to play ________ with my friends after school.36.My _____ (表妹) is visiting this weekend.37.The ________ was a famous treaty that settled disputes in Europe.38.What do you call the action of planting flowers in a garden?A. GardeningB. LandscapingC. CultivatingD. SowingA39.ts can live for ______ (数十年). Some pla40.My family lives near a __________ (水库).41.What is the opposite of right?A. WrongB. CorrectC. TrueD. AccurateA42.The _____ (羊) eats grass in the field.43.What is the term for a person who collects stamps?A. PhilatelistB. NumismatistC. CollectorD. HobbyistA44.Every year, we celebrate ______ (感恩节) with a big feast and share what we are thankful for.45.The ancient Egyptians created vast ________ (陵墓) for their pharaohs.46.I have a _____ (遥控车) that can go super fast. 我有一辆可以跑得非常快的遥控车。

中国科学技术大学硕士学位论文关于Jacobian猜测和坐标多项式姓名：***申请学位级别：硕士专业：基础数学指导教师：***20050501第１章Ｊａｃｏｂｉａｎ猜测§１．１多项式映射和Ｊａｃｏｂｉａｎ猜测在这篇文章理，除非特别声明外，我们采用阻下记号：Ｎ７－－－－：１，２，３，…自然数，Ｑ＝有理数，Ｒ一实数以及Ｃ＝复数．更进一步的，ｋ表示一个任意的域，Ｒ足任意的交换环．Ｆ＝（Ｒ，…，Ｒ）：ｋ”一ｋ”多项式映射，即，如下形式的映射（茹ｌ，···，。

１。

）¨（Ｆｌ（ｚｌ，…，。

：。

），…，Ｒ（ｚｌ，…，ｏ。

）），这里每个鼠属于多项式环ｋ［ｘ】：＝七陋ｌ，…，ａ：。

】线性映射是特须的多项式映射，也就足Ｒ（ｚｌ＇．一，ｚ。

）＝ａｉｌ３：１十…＋ａｉ。

ｚ。

，ａｉｊ∈ｋ对所有的ｉ，Ｊ．下面的结论足线性代数里面的结果．命题１．１．１．假设Ｆ：ｋ”＿七“是线性映射，那么ｒ砂Ｆ是双射，当且仅当Ｆ是单的．进一步的，其逆也是一个线性映射例Ｆ是可逆的，当且仅当ｄｅｔ（ａｉｊ）∈ｋ＋．ｒ圳如果Ｆ可逆，那么Ｆ是一些初等线性映射的乘积．似，如果Ｆ可逆，那么有公式可以描述Ｆ的逆例如，Ｃｒａｍｅｒ法则夕．研究多项式映射的上要只的就足考察上面的结果在仆么程度上可以推广到多项式映射的情形．罔此，让我们先行肴命题（１．１．１）的可能的推广，即，我们考虑：问题１．１．２．假设Ｆ：ｋ”＿ｋ“是单的多项式映射，那么Ｆ是双射吗穸１２００５年４月中国科学技术大学硕士学位论文第２页摹１搴３ａｃｏｂｉａｎ艟潮§ｌ１多项式映射扣ＪａｃｏｂｉａⅡ精洲下面的例子说明一般的，问题的答案足否定的：例１．１．３．假设Ｆ：Ｑ—Ｑ定义为Ｆ０）＝＝护，则Ｆ是单射，但不是双射显然，这里的问题在于域Ｑ不足代数ｆ｜拍々（域ｋ足代数闭的．如果每个非常数的多项式ｆ（ｘ）∈ｋｌｘｌ在ｋ都有根．）因此，我“】有下面的：问题１．１．４．假设ｋ是代数闭的ｒ枷ｌ｛ｏ，ｋ＝（＿，问一个单的多项式映射Ｆｋ”一ｋ”足双射吗Ｆ令人惊奇的结果足：答案足正确的，实际上，有下面更强的定理１．１．５．假设☆是代数闭的，Ｆ：ｋ“一驴是单的多项式映射，那么Ｆ也是双射，并且其逆也是一个多项式映射．因此在≈屉代数闭域的情形，我们得到丁命题（１１．１）巾（１）刘多项式映射的完全推广．特别的，我们得到：推论１．１．６．每个单的多项式映射ｊ１：护一ｋ”假设ｋ是代数闭的，Ｆ：舻一妒是单的多项式映射，那么Ｆ也是双射，并且其逆也是一个多项式映射．现在我们考虑命题（１．１１）的第二部分定义１．１．７．我们称一个多项式映射Ｆ：ｋ８一妒是可逆的，如果它有逆，并且逆也是多项式映射（因此定理¨．５断言：在代数闭域的情形，Ｆ是双射，’且仅斗Ｆ足可逆的，１注记１．１．８．我们可以证明？如果一个多项式映射Ｆ有一个左逆Ｇ，其中Ｇ也是一个多项式映射．那么Ｇ也是Ｆ的一个右逆饭．过来也对’．因此Ｆ可逆．２００５年４月中国科学技术大学硕士学位论文第３页摹１章．１ａｃｏｂｉａｎ待潮§１．１多项式映射和．１ａｃｏｂｉａｎ精测利用这个注记，我们可以得到如下一个很简单，但足又很有用的判别可逆的法则．引理１．１．９．假设Ｆ：ｋ“一ｋ８是多项式映射，邵么Ｆ可逆，当且仅当ｋ［ｘｉ：…，ｚ。

史上七大数学难题简述
1.黎曼猜想（Riemann Hypothesis）：提出于1859年，涉及到复变函
数的解析延拓与素数分布的关系。

2.Birch和斯沃德贝格猜想（Birch and Swinnerton-Dyer Conjecture）：
提出于1965年，猜想椭圆曲线的解的数量与该曲线上有理点的数量之间存在一种联系。

3.Navier-Stokes存在与光滑性问题（Navier-Stokes Existence and
Smoothness）：提出于1822年，关于流体动力学中描述流体运动的Navier-Stokes方程组的解的存在性与光滑性的问题。

4.Hodge猜想（Hodge Conjecture）：提出于1950年，涉及到拓扑学
和代数几何中的一些概念，尚未有一般性的解决方法。

5.P对NP问题（P versus NP Problem）：提出于1971年，是计算机
科学领域中一个著名的问题，涉及到算法复杂性理论，即在多项式时间内是否可以验证一个解。

6.黄俊舒猜想（The Huanf Junn Shu Conjecture）：提出于2001年，
是关于理论计算机科学中的一个问题，尚未得到解决。

7.雅克-米尔猜想（The Jacobian Conjecture）：提出于1939年，涉及
到多项式环上的一个代数问题。

这些问题都是当前数学领域的前沿难题，解决它们将对数学和相关领域产生深远的影响。

截至目前，其中一些问题已经被部分解决，但尚未有完整的解决方案。

要注意的是，随着时间的推移，这些问题
的解决状态可能发生了变化。

化学化工英语试题及答案一、选择题（每题2分，共20分）1. Which of the following is a chemical element?A. WaterB. OxygenC. HydrogenD. Carbon答案：B, C, D2. The chemical formula for table salt is:A. NaOHB. NaClC. HClD. NaHCO3答案：B3. What is the process called when a substance changes from a solid to a liquid?A. SublimationB. VaporizationC. MeltingD. Condensation答案：C4. In the periodic table, which group contains alkali metals?A. Group 1B. Group 2C. Group 17D. Group 18答案：A5. What is the name of the process where a substance decomposes into two or more substances due to heat?A. CombustionB. OxidationC. ReductionD. Decomposition答案：D6. Which of the following is a physical property of a substance?A. ColorB. TasteC. SolubilityD. Reactivity答案：A7. What is the term for a compound that releases hydrogen ions (H+) when dissolved in water?A. BaseB. AcidC. SaltD. Neutral答案：B8. The law of conservation of mass states that in a chemical reaction:A. Mass is lostB. Mass is gainedC. Mass remains constantD. Mass can be converted into energy答案：C9. Which of the following is a type of chemical bond?A. Ionic bondB. Covalent bondC. Hydrogen bondD. All of the above答案：D10. What is the name of the process where a substance absorbs energy and changes from a liquid to a gas?A. MeltingB. VaporizationC. SublimationD. Condensation答案：B二、填空题（每题2分，共20分）1. The symbol for the element iron is ________.答案：Fe2. The pH scale ranges from ________ to ________.答案：0 to 143. A compound that produces a basic solution when dissolvedin water is called a ________.答案：base4. The smallest particle of an element that retains its chemical properties is called a ________.答案：atom5. The process of separating a mixture into its individual components is known as ________.答案：separation6. The study of the composition, structure, and properties of matter is called ________.答案：chemistry7. The process of a substance changing from a gas to a liquid is called ________.答案：condensation8. A(n) ________ reaction is a type of chemical reactionwhere two or more substances combine to form a single product. 答案：synthesis9. The volume of a gas at constant temperature and pressureis directly proportional to the number of ________.答案：moles10. The process of converting a solid directly into a gas without passing through the liquid phase is known as ________. 答案：sublimation三、简答题（每题10分，共30分）1. Explain what is meant by the term "stoichiometry" in chemistry.答案：Stoichiometry is the calculation of the relative quantities of reactants and products in a chemical reaction.It is based on the law of conservation of mass and involvesthe use of balanced chemical equations and the molar massesof substances to determine the amounts of reactants needed to produce a certain amount of product or the amounts ofproducts formed from a given amount of reactant.2. Describe the difference between a physical change and a chemical change.答案：A physical change is a change in the state or form of a substance without altering its chemical composition. Examples include melting, freezing, and boiling. A chemical change, on the other hand, involves a change in the chemical composition of a substance, resulting in the formation of new substances. Examples include combustion and rusting.3. What are the three main types of chemical bonds, and givean example of each.答案：The three main types of chemical bonds are ionic bonds, covalent bonds, and metallic bonds. An ionic bond is formed when electrons are transferred from one atom to another, resulting in the formation of oppositely charged ions. An example is the bond between sodium (Na) and chloride (Cl) in table salt (NaCl). A covalent bond is formed when two atoms share electrons, as seen in water (H2O) where hydrogen atoms share electrons with oxygen. Metallic bonds occur in metals, where a "sea" of delocalized electrons is shared among positively charged metal ions, as in sodium metal。

1、Which of the following is a type of white blood cell primarily involved in the immune response against viral infections?A. ErythrocytesB. PlateletsC. LymphocytesD. Neutrophils(答案：C)2、The process by which a drug is absorbed, distributed, metabolized, and excreted within the body is known as:A. PharmacodynamicsB. PharmacokineticsC. BioavailabilityD. Drug tolerance(答案：B)3、Which organ is responsible for filtering blood and removing waste products to form urine?A. LiverB. KidneyC. LungD. Heart(答案：B)4、The study of the structure and function of cells is called:A. HistologyB. CytologyC. AnatomyD. Physiology(答案：B)5、Which of the following hormones is released by the pancreas and plays a crucial role in regulating blood sugar levels?A. InsulinB. ThyroxineC. AdrenalineD. Glucagon (Note: correct term is Glucagon-like peptide-1, but for simplicity, 'Glucagon' is used here)(答案：A)6、The medical term used to describe the inflammation of a joint is:A. ArthritisB. CarditisC. NephritisD. Dermatitis(答案：A)7、Which of the following imaging techniques uses high-frequency sound waves to produce images of internal body structures?A. X-rayB. CT scanC. UltrasoundD. MRI(答案：C)8、The process of converting food into energy that the body can use is known as:A. DigestionB. MetabolismC. AbsorptionD. Assimilation(答案：B)9、Which vitamin is essential for maintaining healthy bones and teeth, and is primarily obtained from sunlight exposure?A. Vitamin AB. Vitamin CC. Vitamin DD. Vitamin K(答案：C)10、The branch of medicine that deals with the prevention, diagnosis, and treatment of diseases of the heart and blood vessels is called:A. NeurologyB. CardiologyC. DermatologyD. Gastroenterology(答案：B)。

高中英语世界著名科学家单选题50题1. Albert Einstein was born in ____.A. the United StatesB. GermanyC. FranceD. England答案：B。

解析：Albert Einstein（阿尔伯特·爱因斯坦）出生于德国。

本题主要考查对著名科学家爱因斯坦国籍相关的词汇知识。

在这几个选项中，the United States是美国，France是法国，England是英国，而爱因斯坦出生于德国，所以选B。

2. Isaac Newton is famous for his discovery of ____.A. electricityB. gravityC. radioactivityD. relativity答案：B。

解析：Isaac Newton 艾萨克·牛顿）以发现万有引力gravity）而闻名。

electricity是电，radioactivity是放射性，relativity 是相对论，这些都不是牛顿的主要发现，所以根据对牛顿主要成就的了解，选择B。

3. Marie Curie was the first woman to win ____ Nobel Prizes.A. oneB. twoC. threeD. four答案：B。

解析：Marie Curie 居里夫人）是第一位获得两项诺贝尔奖的女性。

这题主要考查数字相关的词汇以及对居里夫人成就的了解，她在放射性研究等方面的贡献使她两次获得诺贝尔奖，所以选B。

4. Thomas Edison is well - known for his invention of ____.A. the telephoneB. the light bulbC. the steam engineD. the computer答案：B。

解析：Thomas Edison（ 托马斯·爱迪生）以发明电灯（the light bulb）而闻名。

小学上册英语第四单元寒假试卷(有答案)英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.My dad is very ________.2. A __________ (长期计划) can support sustainable gardening.3.My mom is a ________.4.The dog wagged its ______ (尾巴) when it saw me. It was very ______ (兴奋).5.I can create a _________ (玩具动物) out of clay.6.中国的________ (history) 充满了起伏与变迁。

7.She has a _____ (cat/dog) at home.8.The cake is ___ (frosted).9.The atmosphere is essential for ______ life on Earth.10.The ancient Chinese invented _____ paper.11.My sister sings in the ________ (合唱团).12. A _____ is a group of stars that is visible in the night sky.13.I like to ______ pictures. (draw)14. A tiny ___ (小龙) appears in stories.15.What do we call the primary color that mixes with blue to create violet?A. RedB. YellowC. GreenD. Orange答案:A16. A molecule that can donate protons is called an ______.17.The process of ______ involves the gradual breakdown of rocks.18.The ______ (水分) in soil is crucial for growth.19.My cousin is a great __________ (志愿者).20.I love reading mystery books. My favorite author is __________.21.The chemical symbol for chromium is __________.22.The flowers are _____ and colorful. (bright)23. A __________ is a mixture that can be separated by evaporation.24.The Sun is a medium-sized star in the ______ galaxy.25.The process of making glass involves heating sand to a high _____.26.The pufferfish can _________ itself. (膨胀)27.The __________ (历史的冲击) prompts reflection.28. A ____ is a friendly creature that enjoys company.29.What is the capital city of the Philippines?A. ManilaB. CebuC. DavaoD. Quezon City答案: A30.An alligator lives in _________ (沼泽地).31. A __________ (分子间力) influences the physical properties of a substance.32. A chemical reaction requires a change in ______.33.We can _____ (graft) plants to improve growth.34.The ice cream is _____ melting. (slowly)35.My dad tells _______ jokes.36.The __________ (地震) shook the buildings.37.The gecko can stick to _______ (墙壁).38.We need to ________ the house.39. A ________ (树枝) can break if it is too heavy.40.Certain plants can ______ (生存) in low-nutrient soils.41.I can ________ (lead) a team effectively.42.She is _______ (reading) a novel.43.The __________ (历史的跨文化交流) fosters understanding.44.What do we call a story that explains the origins of something?A. MythB. FableC. LegendD. Fairy tale答案: A45.The Age of Exploration began in the _______ century.46.The chemical process that occurs in our bodies to release energy is called ______.47.I see a _____ (bird/fish) in the tree.48.He is a mechanic, ______ (他是一名机械师), fixing cars.49.The _____ (枝条) can be pruned for better growth.50.The ostrich is the world's largest ______ (鸟).51.Burning wood produces __________.52.She is studying to be a ________.53.The playground is full of ______.54.My cousin is passionate about __________ (艺术).55.I see a _____ (caterpillar) on the leaf.56.I saw a _______ (小松鼠) eating an acorn.57.My __________ (玩具名) always makes me laugh when I __________ (动词).58. A ________ (植物采集) can be educational.59._______ are important for the environment.60.The chemical formula for ammonium sulfate is __________.61. Carta was signed in __________ (1215), limiting the power of the king. The Magn62.The goldfish swims gracefully in the _________ (水).63.The _____ (植物生物) is varied and complex.64.My cousin always visits us during ____.65.He has a pet ___ . (fish)66.My brother has a remote-controlled _____ (飞机).67.What is the main language spoken in the USA?A. SpanishB. EnglishC. FrenchD. Chinese答案: B68.The weather is _____ today. (nice)69.I like ________ (吃) ice cream.70.We have a ________ (meeting) after school.71.What do you call a group of wolves?A. PackB. HerdC. FlockD. Swarm答案: A72.I love to eat ___. (cake)73.I love learning new languages because it helps me understand different _______ (文化).74.Light from ancient stars provides clues about the early _______.75.My dad loves to ________ (烹饪).76.The _____ (fish/bird) is swimming.77.The ice cream truck is ______ (coming) down the street.78.Pandas mainly eat _________ (竹子).79.The hamster runs on its _______ every night.80.What is the name of the famous clock tower in London?A. Big BenB. Eiffel TowerC. Leaning Tower of PisaD. Statue of Liberty答案:A81.My uncle is a __________ (建筑师).82.I always _______ (帮助) my parents.83.My friend is a ______. He wants to be an astronaut.84.His favorite animal is a ________.85.The _____ (星星) twinkle at night.86.The teacher, ______ (老师), inspires us to dream big.87.In art class, we made ________ (手工艺品) using old toys. It was ________ (有趣的) to create something new!88.The rabbit has powerful _______ (后腿) for jumping.89.The police officer, ______ (警察), keeps the community safe.90. A lever can increase the ______ applied to an object.91.The _____ (ocean) is blue.92.The nurse provides care and support in _____ (医院).93.My mom enjoys doing ____ (puzzles).94.I enjoy _______ (看书) at the library.95.The __________ is cool and refreshing during the summer. (海洋)96.What is the name of the place where we go to watch movies?A. TheaterB. MuseumC. Concert HallD. Library答案: A97.What do we call the process of making food from sunlight, carbon dioxide, and water?A. PhotosynthesisB. RespirationC. FermentationD. Digestion答案: A. Photosynthesis98.Chemical reactions can be classified as ______ or physical changes.99. A mixture that is not uniform throughout is called a ______ mixture.100.My uncle is a skilled ____ (sculptor).。

【引用】代数几何学习经验ZZ 和大家分享这篇日志，我的看法是：原文地址：代数几何学习经验ZZ原文作者：Kaka Abel代数几何学习经验(ZZ)古典代数几何起源于19世纪末，20世纪初得到充分的发展。

这篇帖子没有借助任何参考书目，仅仅是我头脑中的记忆堆积出来的，因此，如果有不同理解，或者我讲错了，请见谅。

因为我忘了很多了。

古典代数几何的发展主要是仿射簇和投射簇的研究，以及后来渐渐发展的代数簇。

直到现在，代数簇理论仍然是非常有用的方法，所以喜欢代数几何的不要盲目的崇尚现代代数几何理论，因为概型的直观性要大大少于代数簇。

最先引起我们注意的是仿射簇(affine variety)，用几何的语言叙述，那是affine space An里面由一些代数方程的公共零点集(zero locus set)。

因此我们考虑An上的代数方程构成的多项式环k[x1,.xn]及其理想，容易定义V：{I/I为理想}-An为公共零点集，I：An-{I/I为理想}为生成理想。

(k为代数闭域！)我们得到的第一个重要理论是nullstellensatz定理(零点定理)：I(V(I))=rad(I)(即I取radical)这个使得我们将理想和代数集一一对应。

另一个较弱的形式是说对于任何极大理想m，k[V]/m总是k的代数扩域，由于我们已经假设k是代数闭的，因此k[V]/m同构于k，所以任何仿射簇V，k[V]总是k和m的直和(作为k模)，这是我们研究局部性质的基础。

我们不能总是将V作为嵌入在放射空间的子集来看待，我们需要更本质更内蕴的方法。

(我认为这是很重要的数学思想，寻找内蕴的性质)现在大部分参考书采用的方法是给与一个structure sheaf来定义。

于是，我们说一个affine variety，总是指一个ringed space(具有层结构的拓扑空间)。

通过一系列形式推导(具体看任何一本参考书)，我们得到了一个很漂亮的最基本的定理：affine variety范畴反变(contravariant)等价于affine k-algebra范畴。

Survey of Theory and Steering Laws of Single-GimbalControl Moment GyrosHaruhisa KurokawaNational Institute of Advanced Industrial Science and Technology,Tsukuba,Ibaraki 305-8564JapanDOI:10.2514/1.27316A geometric theory regarding singularity problems of single gimbal control moment gyrosystems is outlined.Most control moment gyrosystems have impassable and inescapable singular states that obstruct continuous changes of the total angular momentum vector.The remaining problem of the theory regarding degenerate singularity is described with an unproved conjecture.Based on the theory,various steering laws such as gradient methods and singularity robust methods for the pyramid-type control moment gyrosystem,and those for variable speed control moment gyros are surveyed and analyzed.Problems of the attitude control by these steering laws are examined by geometrical analyses without numerical simulation.NomenclatureC =Jacobian matrixc i =torque direction vector of the i th CMG unit ^c =unit eigenvector in H spaceg i =gimbal direction vector of the i th CMG unit H =total angular momentum vector of the system h i =normalized angular momentum vector of the i th CMG unitn =number of CMG units in the system p i =1=q iQ =quadratic form at a singular state q i =u h iT =output torque vector of the system t =timeu =singular vector,unit normal to all c i at a singular state"i =sign at a singular state=Gaussian curvature of a singular surface j =singular value of the matrix C =( 1;...; n ),a CMG statei =gimbal angle of the i th CMG uniti =orthonormal basis in tangent space of space !=gimbal rate vector obtained by a steering lawSuperscriptsN =null vectorS =vector at a singular state T =torque producing vectort =transpose of vector and matrixI.IntroductionASINGLE-GIMBAL control moment gyro (CMG)system is regarded as an effective torque generator for attitude control both for large space stations and small agile satellites because of its large torque ampli fication capability.Three or more single-gimbal CMG units are necessary for three-axis attitude control.A steering law gives each unit ’s motion,usually each unit ’s gimbal rate,to generate a required torque,which is provided by an attitude ing an appropriate steering law,the CMG system is expected to make a rapid change of its angular momentum vector to reach its maximum for maneuvering and to make a precise control of torque for pointing and tracking.Singularity is the most serious problem for these tasks.The possible output torque does not cover the 3-D space at a singular state.Points on the border of the range of the angular momentum vector are trivially singular.Other than those,any CMG system has singular states,which form surfaces inside the border,and such singular states must be avoided.Various singularity avoidance methods have been studied.The pseudoinverse solution has been tried as an exact method for steering law calculation [1];additional null solutions were used for singularity avoidance using a gradient method [2–4].Even with various objective functions as singularity measures,such gradient methods were insuf ficiently successful,especially for the pyramid-type (four-skew)system.Then,varieties of singularity robust (SR)inverse methods were proposed,which attempted singularity avoidance by allowing torque error [5–9].Recently,variable-speed CMGs (VSCMG),which were once studied as integrated power and attitude control devices [10],have been revived for their singularity avoidance capability [11–15].Mathematical studies of CMG singularity are essential for design and evaluation of a steering law,as well as for selection and evaluation of system con figurations.The work by Margulies and Aubrun [16],which has offered a general theory,has remained theReceived 15August 2006;accepted for publication 6March 2007.Copyright ©2007by the American Institute of Aeronautics and Astronautics,Inc.All rights reserved.Copies of this paper may be made for personal or internal use,on condition that the copier pay the $10.00per-copy fee to the Copyright Clearance Center,Inc.,222Rosewood Drive,Danvers,MA 01923;include the code 0731-5090/07$10.00in correspondence with the CCC.Haruhisa Kurokawa received B.E.and M.E.degrees in precision machinery engineering,and a Dr.Eng.degree in aeronautics and astronautics from the University of Tokyo,respectively,in 1978,1981,and 1997.He currently heads the Distributed System Design Research Group,Intelligent Systems Research Institute,National Institute of Advanced Industrial Science and Technology (AIST),Japan.His main research subjects are kinematics of mechanisms,distributed autonomous systems,and nonlinear control.His current research theme is a modular robotic system.J OURNAL OF G UIDANCE ,C ONTROL ,AND D YNAMICS Vol.30,No.5,September –October 20071331D o w n l o a d e d b y H U A Z H O N G U N I VE R S I T Y OF S C I E N C E o n A p r i l 1, 2015 | h t t p ://a r c .a i a a .o r g | D O I : 10.2514/1.27316most important and most referred paper.In contrast,works by Tokar published originally in Russian have been virtually ignored,even though they were published during the same year as [16]and presented various theoretical ideas and calculation results [17–20].About ten years were spent to catch up to the results presented by Tokar through studies following the paper by Margulies and Aubrun [21–27].Some confusion still remains in relation to the theory regarding terminology and interpretations;for this reason,theoretical studies have not been used effectively for analyses and evaluation of recent steering law studies.This paper is intended as a survey and analysis of steering laws based on the theory of CMG singularities.After a summary of basic terms and equations in Sec.II,the various research streams of theory will be brie fly surveyed.Because theory itself is a basis of steering law analyses,a general and uni fied theory will be outlined after this survey.A remaining problem regarding a degenerate state will be formulated.Section IV presents a survey of steering laws.Because all steering laws have been analyzed and evaluated using simulations and experiments in their original papers,geometric analyses and qualitative evaluations of those steering laws ’motions are provided in Sec.V.Section VI gives a guide for future studies.Mathematical details and remaining problems are brie fly explained in the Appendix.II.Basic MathematicsVariables and equations,most of which are based on [16],are de fined in this section.The same notation as those in [16]are used to the greatest extent possible,although most expressions are produced using vector algebra.Satellite dynamics and the effects of gimbal acceleration are not considered unless stated otherwise.A.General DescriptionA general system,especially a 3-D redundant system,is con-sidered consisting of n (>3)single-gimbal CMG units.The angular momentum of all CMG units is set to unity.The system state is de fined by the set of all gimbal angles,each of which is denoted by i for i 1;...;n .No limits are assigned to the angles.Three mutually orthogonal unit vectors for each CMG unit are de fined as a gimbal vector g i ,an angular momentum vector h i ,and a torque vector c i ,wherec i @h i =@ i g i h i(1)The gimbal vectors are fixed and the others are dependent upon thegimbal angles.In this paper,the same symbol is used to represent a vector in the Euclidean space as well as its representation in a column vector based on the CMG system ’s coordinate basis.The system con figuration is de fined by the arrangement of the g i .Many con figurations have been investigated,most of which are categorized as follows:1)Independent-type con figuration:all g i are different and no three of them are coplanar.2)Coplanar con figuration:all g i are on the same plane.3)Multiparallel type con figuration:g i are grouped sharing the same directions.As examples,a pyramid (four-skew)con figuration is of the first category and a roof-type (two-speed)con figuration belongs to the latter two.The total angular momentum H and the output torque T are given byH HX n i 1h i(2)TXic id i =dt C d =dt(3)where is the state variable, 1;...; n ,and d =dt d 1=dt;...;d n =dt t is an n -dimensional column vector.The 3 n matrix C is the Jacobian of Eq.(2).Hereafter, and H are,respectively,called a state and a point.B.Steering LawA steering law obtains the gimbal rates,! !1;...;!n t ,with which the resulting torque in Eq.(3)is equal to the torque command T com .For n >3,the general solution is given as! C t CC t 1T com !N(4)The first term in the right is the Moor –Penrose pseudoinverse solution,which gives the minimal norm solution.The second term,!N ,is a solution of the homogeneous equation as C !N 0;0;0 t ;it is called a null vector .(The term null motion is used as a finite change of the state,keeping H the same.)It is noteworthy that two kinds of torque are not shown in Eqs.(3)and (4),which must be included in the precise equation of the attitude control problem.One is the reaction torque by acceleration of gimbal rotation.It is usually ignored for steering laws in practice,as described in Sec.V.A.The other is the gyro-effect torque caused by the vehicle ’s rotation.This torque is usually included in the dynamics equation of the attitude control system;consequently,it is not ignored but included in T com in Eq.(4).C.Singular State and SurfaceWhen the system is singular,that is,det CC t 0,all torque vectors c i become coplanar and the possible outputs T of Eq.(3)do not span 3-D space.The rank of C can become one if all g i are on the same plane,as is true of a coplanar system.Hereinafter,an independent-type con figuration is assumed.For that reason,the rank does not reduce to one.All singular H as points in 3-D space form a continuous and mostly smooth surface called a singular surface .It includes the angular momentum envelope (the envelope for short)and an internal singular surface.The singular surface and its corresponding are calculable using the methods described in [16,26].Let all variables at a singular state be superscripted by S .At a singular state S ,a unit vector u normal to all c S i along with related variables are de fined asu c S i 0(5)q i u h S i(6)p i 1=q i (7)"i 8<: ;q i >00;q i 0 ;q i <0(8)These equations do not uniquely de fine the variables,because there are two choices of u .Though unique de finition is not necessary for geometric analysis,a condition is assumed hereafter for simpler explanation in the next section so that the number of positive "i is not less than that of negative ones.Also see the footnote in Sec III.C.The Gaussian curvature of the singular surface is described as1= 1=2X iXjp i p j c S i c S ju 2(9)where a b c is a box product of three vectors such thata b c a b c [16,28].It is noteworthy that the range of H is assumed to be simply connected for any con figuration without a hole or a tunnel.This is not trivial and has not been proved yet [16](see Appendix).The1332KUROKAWAD o w n l o a d e d b y H U A Z H O N G U N I VE R S I T Y OF S C I E N C E o n A p r i l 1, 2015 | h t t p ://a r c .a i a a .o r g | D O I : 10.2514/1.27316Gaussian curvature and the principal curvatures have an important meaning,as described later,but other properties of the surface described by differential geometric tools have not been related clearly to singularity analysis and steering laws.For such mathematics,[24,26]provide good explanations.D.Singular Value DecompositionThe matrix C can be represented using its singular values i and unitary transformations U and V ,as follows:C U 100000 200...000 3002435V t(10)Throughout this paper,the following is assumed:1 2 3 0(11)At a nonsingular state, 3≠0and the unitary matrices U and V canbe constructed byU ^c 1^c 2^c 3 ;V ’T 1’T 2’T 3’N 1’N 2...’N n 3(12)where three ^ci are the eigenvectors corresponding to the singular values i and which make up an orthonormal basis in the Euclidean space,and all ’i make up a corresponding orthonormal basis of the tangent space of .The three ’T i (i 1,2,3)are Moor –Penrosepseudoinverse solutions,and ’Ni (i 1;...;n 3)are null vectors in Eq.(4).At a singular state,the minimal singular value 3becomes 0and the two matrices in Eq.(12)become U ^c T 1^c T 2u ;V ’T 1’T 2’N1’N 2...’N n 2 (13)III.Theory of Singular SurfaceThe intended goal of the theory was first to obtain the size andshape of the workspace of a CMG system for selection of a con figuration.Margulies and Aubrun established a general method described in Sec.II.C,as well as a general formulation of a steering law and null vectors [16].They introduced a quadratic form,which this author followed to formulate a classi fication of a singular state according to whether it is escapable by a null motion [21].Bedrossian et al.[24]and Wie [26]pursued a more detailed investigation of the differential geometric properties of a singular surface related to escapability.During the same year as the work by Margulies and Aubrun,Tokar presented an impassable singularity theory based on the same quadratic form and evaluated various con figurations [17–20].Because the original paper of those were in Russian and their English translations used different terminology,such as a “gyrostabilizer ”or a “gyrodyne ”for a CMG,those works did not become well-known in the West.Nearly ten years were spent by the aforementioned researchers to match Tokar ’s results.Two terms,impassability and escapability,were used inde-pendently for similar properties of a singular state.In the following,their de finitions and the theory of CMG singularity will be summarized.Remaining problems regarding a degenerate state,which [24,26]emphasized as important for escapability analysis,will be formulated.A.Quadratic Form,Passability,and EscapabilityIn [16],a quadratic form Q d at a singular state is de fined as a second-order in finitesimal change of H along the u direction by an in finitesimal displacement d (see Appendix):Q d 2 u H Xq i d i 2(14)An in finitesimal displacement d is an element of the n -dimensional tangent space of the space.Three special subspacesexist:the null space,its complementary space,and the singularly constrained space.(An expression of bases of the three spaces is shown in [23].)The null space f d N g is n 2-dimensional andspanned by f ’N i g in Eq.(13);its complementary space f d Tg is 2-D and spanned by f ’T i g .The singularly constrained space is 2-D and itselement,d S,keeps the system singular.Two spaces of d N and d T are suf ficient to cover the whole d space,but two spaces of d N and d S are not always so.For the explanations in this section,we presume the following property at a singular state:Surface Regularity :The singular surface is regular,that is,smooth,at H S ,which implies that d H d S span the surface,and that d S and d N are independent,and that they all span n -dimensional space.This condition is assumed in [16]without any discussion.Actually,this holds at most singular states for an independent-type con figuration (see Sec.III.D).By this condition,any d and the corresponding Q d are decomposable as [25,28]d d S d N(15)Q d Q d S Q d N(16)∗The first term of the right-hand side in Eq.(16),Q d S ,is the second fundamental form of the singular surface (see Appendix).See [16]for another expression.The last term of Eq.(16),Q d N ,gives passability of the singular surface as follows:Because the first term on the right is attributable to the curved singular surface,the last term represents second-order in finitesimal displacement of H to u direction from the singular surface.If this quadratic form is de finite,that is,if it is the same sign for any d N ,any motion is restricted to the same side of the surface in H space.Because the quadratic form and its derivatives are continuous with respect to S ,H S of de finite form make up a certain area of the surface,through which it is not possible to pass from one side to the other if is in the (in finitesimal)neighborhood of corresponding singular states.In this sense,such an area of the surface with a de finite form is termed impassable [20].Similarly,an area of the surface with an inde finite form is passable .Another aspect of this form category is escapability [21,24,26].There exist directions in the d N space (null space),along which the quadratic form is zero if a singular state is passable (i.e.,hyperbolic)having an inde finite Q d N .As the motion by d N keeps H on the singular surface and d N is independent from d S under surface regularity condition,escape from a passable singular state on a regular surface is always possible.In contrast,no escape is possible at an elliptic (i.e.,impassable)singular state.Therefore,passability and escapability are compatible where the surface is regular.Although other terms such as “elliptic/hyperbolic ”[16,21,24,26]and “de finite/inde finite ”[22,23]have also been used;the terms “impassable/passable ”are used in this paper because they were de fined earliest.B.Passability and Surface CurvatureA relation is apparent between passability and the local shape of asingular surface.The signature of Q d is the sum of the signatures of the two quadratic forms Q d S and Q d N by Sylvester ’s law of inertia.Because the signature of Q d is f "i g and that of Q d S includes signs of two principal curvatures,there can be only three impassable surface types [23].Type 1:If all "i are positive,Q d N is always positive de finite.The H of this type is on the envelope and the Gaussian curvature is positive;consequently,the surface is convex to u .∗It is helpful to represent d S and d N ,respectively,as column vectors d S 2R 2and d N 2R n 2based on the orthonormal basis of the two subspaces.The two quadratic forms are expressed as d St Q S d S and d Nt Q N d N ,where Q S and Q N are 2 2and n 2 n 2 matrices.KUROKAWA 1333D o w n l o a d e d b y H U A Z H O N G U N I VE R S I T Y OF S C I E N C E o n A p r i l 1, 2015 | h t t p ://a r c .a i a a .o r g | D O I : 10.2514/1.27316Type 2:If all "i but one are positive and the Gaussian curvature is negative,Q d N is positive de finite.Part of this type surface is on the envelope,but the remaining part is internal.The surface is curved as a hyperbolic saddle surface.Type 3:If all "i but two are positive and the signature of the first quadratic form is ; ,Q d N is positive de finite.The surface of this type is fully internal and is concave to u ,and its Gaussian curvature is positive.Note that a positive does not always imply impassability because the signature of Q d S can be ; with positive .†The preceding explanation ignores the case in which u is parallel to g i ,that is,"i q i 0.Equations (15)and (16)hold even in this case,and one signature of Q d S is 0,hence 0.Therefore,the signature of Q d N does not contain 0and a similar discussion is possible.C.Impassable Surface ExampleBased on this classi fication method and by various methods in [16],impassable surface regions are calculable.Any independent-type con figuration has an internal impassable surface that is distinct from the envelope because a type 2surface extends smoothly inside from the envelope.The symmetric six-unit system has only type 2internal surfaces very near its envelope [20].However,systems with n 4or 5have both type 2and type 3surfaces extending far inside.For the symmetric pyramid con figuration in Fig.1,impassable surface patches are obtained as in Fig.2.(A similar figure is shown in [20].)Impassable surfaces appear similar to connected strips forming a parallelepiped.A continuous curved line with analytical expressions is obtained to represent this framework of surface strips [25].Edges of the strip are shaped as folds;they are borders of impassable and passable regions (see a cross section in Fig.3a).On the folding border,1= in Eq.(9)is 0,and surface regularity does not hold.Figure 2a is a part of the internal impassable surfaces.All are obtained by successive 1=4rotations about the Z axis based on the system ’s symmetry (Fig.2b).As for multiparallel type con figurations,some differences exist in analysis,as described next.However,impassable surfaces of such systems are also de fined by Q d N and can be calculated.Impassable surfaces of the roof-type system are the point at the origin and two unit circles [22].Unlike independent-type con figurations,there are systems with no internal impassable surface with n 6.‡D.Regularity Condition and Degenerate StateFor the examinations in Secs.III.A and III.B,an independent type of con figuration and surface regularity are assumed.Although passability and escapability are de finable for any con figuration,we require additional analysis for exceptional cases.Moreover,if the regularity condition does not hold,that is,one d S direction becomes included in the null space,and if a finite null motion along this d S maintains singularity,escape by this null motion is not possible.Bedrossian et al.[24]discussed this problem and termed this state a degenerate state.A degenerate state is a special case in which surface regularity breaks.Examples of degenerate states that have been identi fied are all 2-D systems and multiparallel-type con figurations [24].Singularity and surface geometry of 2-D systems and multiparallel-type con figura-tions differ from those of independent-type con figurations.For example,a singular surface corresponding to u parallel to g i is a unit circle for an independent-type con figuration,but it is a circular plate for a multiparallel-type con figuration.Moreover,2-D systems withn 3and most multiparallel type systems with n 6have no internal impassable state.For an independent-type system,surface regularity breaks on the borderline of an impassable surface and a passable surface.Through calculations,no degenerate state has been found for various independent-type con figurations.This engenders the conjecture 1in the Appendix that no degenerate state exists for an independent-type con figuration.Because most multiparallel type systems have no problem of impassability,it is expected that degenerate states cause a problem only for the roof-type system.To construct a comprehensive theory,these must be proved in a future study (see Appendix).E.Inverse Manifold and Topological AnalysisPassability is de fined using in finitesimal analysis;it remains unclear how far the effect of an impassable surface extends.Inverse manifolds and their change clearly show the meaning of impassability and expected H deviation to avoid singularity [25,27].Through consideration of the topological connection of manifolds,a global problem of the pyramid-type system was found.1.Inverse ManifoldA steering law is a method to obtain gimbal rates that correspond to a given torque command:the change of H .A system ’s behaviorcanFig.1Symmetric pyramid-type system.The pyramid is half of a regular hexahedron for cos 2 1= 3.g 1g 4g 3g 2XY ZF QHGPW LV A a) Calculation resultsb) Total impassable surface structureFig.2Impassable surface of the symmetric pyramid-type system.a)Impassable singular points are calculated with " ; ; ; and ; ; ; for various u scattered on the Gaussian sphere.There is an analytical expression along line ABCDE.The surface patches around AB and inside BCDEB are of type 2,and that inside CFDC is of type 3.The cross section near *is shown in Fig.3a.b)All lines symmetrically transformed from the curved line ABCDE are shown with a half-cut envelope.†If n 4and the total signature is ; ; ; ,the surface with signature ( )is also impassable and positive is the suf ficient condition for type 3.In this case,it is better for consistent de finition to rede fine u by u and "by "so that u represents the direction in which H motion is impossible.‡The proofs of passability of multiparallel type con figurations in [23,28]were based on surface curvature,irrespective of surface regularity;hence they are not correct.An identical conclusion,however,is obtained by direct evaluation of Q d N .For example,a six-unit system as a pair of three-unit planar systems has no internal impassable surface because each planar subsystem has no internal impassable state as a 2-D system.1334KUROKAWAD o w n l o a d e d b y H U A Z H O N G U N I VE R S I T Y OF S C I E N C E o n A p r i l 1, 2015 | h t t p ://a r c .a i a a .o r g | D O I : 10.2514/1.27316be viewed by a path of that corresponds to a path of H if time is ignored.All possible paths of are visible by inverse images of the mapping of Eq.(2).The inverse image is a set of subspaces that are mutually disjoint.Each subspace is either an n 3-dimensional manifold or a similar complex,which is hereafter termed a null-motion manifold or a manifold for short.The shapes of manifolds in the neighborhood of a singular state are characterized by setting the quadratic form Q d N constant.It resembles either a superellipsoid or a superhyperbolic surface.The number of disjoint manifolds is held constant when H moves without encountering a singular surface.Figure 3a shows part of a cross section of the singular surface of the symmetric pyramid-type system in Fig.1.A manifold is topo-logically a loop in the 4-D torus space.Along with the H motion described by ABCDE,the manifold bifurcates at B,and one of them terminates at D (Fig.3b).Once the system state is on such a terminating manifold in the triangular region,there is no way to reach the other manifold by any null motion other than driving H out of the region through a passable surface.This example illustrates the effect of an impassable surface covering a region surrounded by singular surfaces.The shape and the size of an impassable surface are important for evaluation of possible error in the H path.2.Global ProblemTopological theory does not allow one-to-one continuous mappings from H to over the range of H [29].This fact is not always a problem for a steering law.If an H workspace without an impassable surface is considered,a gradient method works effec-tively.In this case,a discontinuous and one-to-multi mapping gives local maxima of the objective function for a gradient method.The question is the size of such workspace.A workspace without an impassable surface can be almost equal to the envelope in the case of the symmetric six unit system,but it must be reduced greatly in the case of the pyramid-type system.Apparently,the problem of impassability is not suf ficiently serious to exclude all impassable surfaces from the workspace when viewed with the aforementioned manifolds.Local avoidance of the impassable singularity shown in Fig.3b seems possible if an appropriate manifold is selected before entering the triangular region [27].However,if a reduced H workspace shown in Fig.4is considered,the aforementioned possibility was eliminated first by simulation results [30],and subsequently by geometrical analysis [25].This problem,hereafter called a global problem,holds that no steering law for the symmetric pyramid-type system can continue to exactly generate arbitrary torque inside the workspace shown in Fig.4.IV.Survey of Steering LawsVarious steering laws have been proposed in the literature.Although most steering laws were formulated for a general system,the symmetric pyramid con figuration is mainly assumed in the following unless stated otherwise.Steering laws are grouped as exact methods,of fline planning,an SR inverse family,and methods for variable-speed CMGs.A.Steering Law by Exact Solution1.Moor –Penrose Pseudoinverse Steering LawThe pseudoinverse solution,which is the first term on the right in Eq.(4),is orthogonal to the tangent space of the null-motion manifold.It has often been claimed that this method tends to drive the system to approach a singular state [3].There is actually no such tendency to approach or to escape from a singular state from a continuous control perspective because the motion of by the pseudoinverse solution remains on a curve that is normal to the manifolds.Discrete time control,however,induces the system to approach a passable state as follows:The shapes of manifolds near a passable singular state are hyperbolic;consequently,their normal curves are also hyperbolic and convex to the singular state.Stepwise change of i moves the system state away from the normal curve and nearer to the singular state.2.Gradient MethodVarious proposed steering laws with singularity avoidance are categorized as gradient methods [2–4].They use the solution of Eq.(4);a null motion is determined to increase (or decrease)an objective function,which represents distance (or similarity)to singular states such as det CC t ,the minimum eigenvalue of C ,or the condition number of C .Any passable singular state is avoidable and escapable by any such method,and no impassable one is escapable.Therefore,this method is effective for systems such as three double-gimbal CMGs or multiparallel type systems that have no internal impassable singularity.On the other hand,because of the global problem,no exact type steering law including a gradient method is effective for systems with a few CMG units,such as the pyramid-type system and the roof-type system.3.Steering Laws for the Roof-Type SystemStudies of the roof-type CMG system were made more than 15years ago [31,32].Those steering laws were based on decomposition of H into planar subsystems.Because singularity problems were better clari fied for the roof-type system,those steering laws were designed to avoid most of the internal singular states through the use of exact methods.The remaining singular states were not escapable [31],or were avoided using a discontinuous motion [32].4.Preferred Angle Setting and Constrained Steering LawAfter simulation results by Bauer [30]indicating the global problem,Vadali et al.proposed preferred gimbal angles to assure singularity avoidance for command torque in a speci fied direction [33].Such a method can be effective either for maneuvering or for pointing under the condition that the disturbance torque is roughly estimated along a certain direction.Even for maneuvering,its problem is that it is not always able to reach the preferred angles by null motion.For pointing,even when disturbance torque is in a certain direction,the CMG system controlled by a usual steeringlaw,AB Impassable surfaceb) Change of manifoldFig.3Path of H and change of inverse manifold.Manifolds are obtained by computer calculations and illustrated in 2-D space.XZYWorkspace candidateFig.4Spherical workspace candidate.The sphere entirely surrounds a hexahedral frame structure made of impassable surface strips.KUROKAWA 1335D o w n l o a d e d b y H U A Z H O N G U N I VE R S I T Y OF S C I E N C E o n A p r i l 1, 2015 | h t t p ://a r c .a i a a .o r g | D O I : 10.2514/1.27316。