วิทยาการหุ่นยนต์สมัยใหม่, NFT ที่สมบูรณ์ 1: พื้นฐานของแบบทดสอบการเคลื่อนไหวของหุ่นยนต์ & คำตอบ – Coursera

Take a journey into the world of วิทยาการหุ่นยนต์ with interesting quizzes and expert answers on Foundations of Robot Motion ใน Modern คอมพิวเตอร์วิทัศน์ด้วย OpenCV, NFT ที่สมบูรณ์ 1.Discover the fundamental principles that govern robot motion, from kinematics to control algorithms that will shape the future of automation and technology. These quizzes are designed to provide a solid foundation in the fundamental concepts of robot motion, and offer insights into the mechanics and mathematics underlying robot motion.

Whether you are a วิทยาการหุ่นยนต์ enthusiast looking to deepen your knowledge or a student looking to explore the exciting field of วิทยาการหุ่นยนต์, this collection provides valuable information on the fundamentals of robot motion. Join us as we navigate the landscape of modern robotics, unravel the complexities of robot motion, and lay the groundwork for advanced robotic applications. Let’s embark together on this enlightening journey as we delve into the fascinating world of robotic motion and its implications for technology and innovation.

แบบทดสอบ 01: Lecture Comprehension, Degrees of Freedom of a Rigid Body (Chapter 2 ผ่าน 2.1)

Q1. Which of the following are possible elements of robots in this specialization? เลือกทั้งหมดที่ใช้.

• Rigid bodies.
• อ่อน, flexible bodies.
• Joints.

Q2. The number of degrees of freedom of a robot is (เลือกทั้งหมดที่ใช้):

• the dimension of its configuration space.
• the number of real numbers needed to specify its configuration.
• the number of points on the robot.
• the number of joints of the robot.
• the number of bodies comprising the robot.
• the number of freedoms of the bodies minus the number of independent constraints between the bodies

Q3. The number of degrees of freedom of a planar rigid body i

Q4. The number of degrees of freedom of a spatial rigid body is

Q5. A rigid body in nทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก-dimensional space has mม total degrees of freedom. How many of these mม degrees of freedom are angular (not linear)? เลือกทั้งหมดที่ใช้. (This is consistently one of the most incorrectly answered questions in this course, so think about it carefully!)

• m-nม−ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก
• ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก(n-1)/2ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก−1)/2
• Neither of the above.

แบบทดสอบ 02: Lecture Comprehension, Degrees of Freedom of a Robot (Chapter 2.2)

Q1. Consider a joint between two rigid bodies. Each rigid body has mม degrees of freedom (m=3ม=3 for a planar rigid body and m=6ม=6 for a spatial rigid body) in the absence of any constraints. The joint has fฉ degrees of freedom (เช่น., f=1ฉ=1 for a revolute joint or f=3ฉ=3 for a spherical joint). How many constraints does the joint place on the motion of one rigid body relative to the other? Write your answer as a mathematical expression in terms of mม and fฉ

Q2. Consider a mechanism consisting of three spatial rigid bodies (including ground, N=4นู๋=4) and four joints: one revolute, one prismatic, one universal, and one spherical. According to Grubler’s formula, how many degrees of freedom does the mechanism have?

Q3. A mechanism that is incapable of motion has zero degrees of freedom. In some circumstances, Grubler’s formula indicates that the number of degrees of freedom of a mechanism is negative. How should that result be interpreted?

• The constraints implied by the joints must not be independent.
• The number of joints, the degrees of freedom of those joints, or the number of rigid bodies must have been counted incorrectly.

แบบทดสอบ 03: Chapter 2 ผ่าน 2.2, Configuration Space

Q1. Using the methods for determining the number of degrees of freedom of a rigid body in 3-dimensional space from the book and the video, find the number of degrees of freedom of a rigid body in a conceptual 4-dimensional space. Your answer should be an integer

Q2. Referring back to Question 1, indicate how many of the total degrees of freedom are angular (rotational). Your answer should be an integer

Q3. Assume your arm, from your shoulder to your palm, มี 7 degrees of freedom. You are carrying a tray like a waiter, and you must keep the tray horizontal to avoid spilling drinks on the tray. How many degrees of freedom does your arm have while satisfying the constraint that the tray stays horizontal? Your answer should be an integer

Q4. Four identical SRS arms are grasping a common object as shown below.

Find the number of degrees of freedom of this system while the grippers hold the object rigidly (no relative motion between the object and the last links of the SRS arms). Your answer should be an integer

Q5. Referring back to Question 4, suppose there are now a total of nทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก such arms grasping the object. What is the number of degrees of freedom of this system? Your answer should be a mathematical expression including nทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก. Examples of mathematical expressions including nทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก are 4*n-74∗ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก−7 or n/3ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก/3

Q6. Referring back to Question 4 และ 5, suppose the revolute joint in each of the nทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก arms is now replaced by a universal joint. What is the number of degrees of freedom of the overall system? Your answer should be a mathematical expression including nทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก. Examples of mathematical expressions including nทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก are 4*n-74∗ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก−7 or n/3ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก/3

Q7. Use the planar version of Grubler’s formula to determine the number of degrees of freedom of the mechanism shown below. Your answer should be an integer. (Remember that a single joint can only connect two rigid bodies, so if you see more than two connecting at a single point, there must be more than one joint there. อีกด้วย, the two blocks in the channels are only allowed to move prismatically in those channels, and one of the joints is labeled “P” for prismatic. You will need to identify all the other joints, and links.)

สัปดาห์ 02: วิทยาการหุ่นยนต์สมัยใหม่, NFT ที่สมบูรณ์ 1: Foundations of Robot Motion Quiz Answers

แบบทดสอบ 01: Lecture Comprehension, Configuration Space Topology (Chapter 2.3.1)

Q1. To deform one nทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก-dimensional space into another topologically equivalent space, which operations are you allowed to use? เลือกทั้งหมดที่ใช้.

• Stretching
• Cutting.
• Gluing.

Q2. True or false? An nทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก-dimensional space can be topologically equivalent to an mม-dimensional space, where m \neq nม​=ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก.

• จริง.
• เท็จ.

แบบทดสอบ 02: Lecture Comprehension, Configuration Space Representation (Chapter 2.3.2)

Q1.True or false? An explicit parametrization uses fewer numbers to represent a configuration than an implicit representation.

จริง.

เท็จ

Q2. A kk-dimensional space is represented by 7 coordinates subject to 3 independent constraints. What is kk?

แบบทดสอบ 02: Lecture Comprehension, Configuration and Velocity Constraints (Chapter 2.4)

Q1. True or false? A nonholonomic constraint implies a configuration constraint.

• จริง.
• เท็จ.

Q2. True or false? A Pfaffian velocity constraint is necessarily nonholonomic.

• จริง.
• เท็จ.

Q3. A wheel moving in free space has the six degrees of freedom of a rigid body. If we constrain it to be upright on a plane (no “leaning”) and to roll without slipping, how many holonomic and nonholonomic constraints is the wheel subject to?

• Two holonomic constraints and two nonholonomic constraints.
• Three holonomic constraints and zero nonholonomic constraints.
• Zero holonomic constraints and three nonholonomic constraints.
• One holonomic constraint and two nonholonomic constraints.

Q4. How many degrees of freedom does the upright wheel on the plane have? (What is the minimum number of coordinates needed to describe its configuration?)

แบบทดสอบ 03: Lecture Comprehension, Task Space and Workspace (Chapter 2.5)

Q1. If the task is to control the orientation of a spaceship simulator, but not its position, how many degrees of freedom does the task space have?

Q2. True or false? The workspace depends on the robot’s joint limits but the task space does not.

• จริง.
• เท็จ.

แบบทดสอบ 04: Chapter 2.3 ผ่าน 2.5, Configuration Space

Q1. The tip coordinates for the two-link planar 2R robot of figure below are given by

x = \cos \theta_1 + 2 \cos (\theta_1 + \theta_2) NS=cosฉัน1​+2cos(ฉัน1​+ฉัน2​)

y = \sin \theta_1 + 2 \sin (\theta_1 + \theta_2)Y=sinฉัน1​+2sin(ฉัน1​+ฉัน2​)

(กล่าวอีกนัยหนึ่ง, ลิงค์ 1 has length 1 and link 2 has length 2.) The joint angles have no limits.

Which of the following best describes the shape of the robot’s workspace (the set of locations the endpoint can reach)?

• A circle and its interior.
• A circle only (not including the interior).
• Annulus or ring (the area between two concentric bounding circles).

Q2. The chassis of a mobile robot moving on a flat surface can be considered as a planar rigid body. Assume that the chassis is circular, and the mobile robot moves in a square room. Which of the following could be a mathematical description of the C-space of the chassis while it is confined to the room? (See Chapter 2.3.1 for related discussion.)

• [เอ,ข] \ครั้ง [เอ,ข] \times S^1[เอ,ข]×[เอ,ข]×NS1
• [เอ,ข] \times \mathbb{Active Directory}^1 \times S^1[เอ,ข]×R1×NS1
• [เอ,ข] \ครั้ง [เอ,ข] \times \mathbb{Active Directory}^1[เอ,ข]×[เอ,ข]×R1
• \mathbb{Active Directory}^2 \times S^1R2×NS1

Q3. Which of the following is a possible mathematical description of the C-space of a rigid body in 3-dimensional space?

• \mathbb{Active Directory}^3 \times S^3R3×NS3
• \mathbb{Active Directory}^3 \times T^3R3×ตู่3
• \mathbb{Active Directory}^3 \times T^2 \times S^1R3×ตู่2×NS1
• \mathbb{Active Directory}^3 \times S^2 \times S^1R3×NS2×NS1

Q4. A spacecraft is a free-flying rigid body with a 7R arm mounted on it. The joints have no joint limits. Give a mathematical description of the C-space of this system. (See Chapter 2.3.1 for related discussion.)

• \mathbb{Active Directory}^3 \times T^{10}R3×ตู่10
• \mathbb{Active Directory}^3 \times S^2 \times T^8R3×NS2×ตู่8
• \mathbb{Active Directory}^3 \times S^3 \times T^7R3×NS3×ตู่7
• \mathbb{Active Directory}^4 \times S^2 \times T^7R4×NS2×ตู่7

Q5. A mobile robot is moving on an infinite plane with an RPR robot arm mounted on it. The prismatic joint has joint limits, but the revolute joints do not. Give a mathematical description of the C-space of the chassis (which can rotate and translate in the plane) plus the robot arm. (See Chapter 2.3.1 for related discussion.)

• \mathbb{Active Directory}^2 \times S^2 \times S^1 \times [เอ,ข]R2×NS2×NS1×[เอ,ข]
• \mathbb{Active Directory}^2 \times S^3 \times [เอ,ข]R2×NS3×[เอ,ข]
• \mathbb{Active Directory}^2 \times T^3 \times [เอ,ข]R2×ตู่3×[เอ,ข]
• \mathbb{Active Directory}^3 \times T^3R3×ตู่3

Q6. Determine whether the following differential constraint is holonomic or not (nonholonomic). See the example in Chapter 2.4.

(1+ \cos q_1) \dot{ถาม}_1 + (2+ \sin q_2) \dot{ถาม}_2 + (\cos q_1+ \sin q_2 + 3) \dot{ถาม}_3 = 0.(1+cosถาม1​)ถาม˙​1​+(2+sinถาม2​)ถาม˙​2​+(cosถาม1​+sinถาม2​+3)ถาม˙​3​=0.

• Holonomic
• Nonholonomic

Q7. The task is to carry a waiter’s tray so that it is always horizontal (orthogonal to the gravity vector), but otherwise free to move in any other direction. How many degrees of freedom does the task space (the C-space of a horizontal tray) มี? (Enter an integer number.)

สัปดาห์ 03: วิทยาการหุ่นยนต์สมัยใหม่, NFT ที่สมบูรณ์ 1: Foundations of Robot Motion Quiz Answers

แบบทดสอบ 01: Lecture Comprehension, Introduction to Rigid-Body Motions (Chapter 3 ผ่าน 3.1)

Q1. Which do we typically use to represent the C-space of a rigid body?

• Explicit parametrization (minimum number of coordinates).
• Implicit representation.

Q2. By the right-hand rule, which fingers of your right hand correspond to the x, Y, and z axes of a coordinate frame, ตามลำดับ?

• Thumb, อธิบายประเภทของเวิร์กโหลดข้อมูลหลัก · อธิบายข้อมูลแบทช์ · อธิบายข้อมูลการสตรีม · อธิบายความแตกต่างระหว่างแบตช์และข้อมูลการสตรีม · อธิบายลักษณะของข้อมูลเชิงสัมพันธ์ อธิบายแนวคิดหลักของการวิเคราะห์ข้อมูล · อธิบายการสร้างภาพข้อมูล, middle
• Middle, อธิบายประเภทของเวิร์กโหลดข้อมูลหลัก · อธิบายข้อมูลแบทช์ · อธิบายข้อมูลการสตรีม · อธิบายความแตกต่างระหว่างแบตช์และข้อมูลการสตรีม · อธิบายลักษณะของข้อมูลเชิงสัมพันธ์ อธิบายแนวคิดหลักของการวิเคราะห์ข้อมูล · อธิบายการสร้างภาพข้อมูล, thumb
• Index, middle, thumb

Q3. When your thumb points along an axis of rotation, positive rotation about the axis is defined by the direction your fingers curl if you use which thumb?

• Right thumb
• Left thumb

Q4. When we refer to a frame attached to a moving body, we always consider a stationary frame {ข}, เพราะ

• the motion of all other frames is expressed relative to {ข}.
• {ข} is the stationary frame that is coincident (at a particular instant) with the frame attached to the moving body.

แบบทดสอบ 02: Lecture Comprehension, Rotation Matrices (Chapter 3.2.1, ส่วนหนึ่ง 1 ของ 2)

Q1. For the rotation matrix R_{ba}Rba​ representing the frame {เอ} relative to {ข},

• the rows are the x, Y, z axes of {เอ} written in {ข} coordinates.
• the columns are the x, Y, z axes of {เอ} written in {ข} coordinates.
• the rows are the x, Y, z axes of {ข} written in {เอ} coordinates.
• the columns are the x, Y, z axes of {ข} written in {เอ} coordinates.

Q2. NS 3 \times 33×3 rotation matrix is an implicit representation of spatial orientations consisting of 9 numbers subject to how many independent constraints

Q3. The inverse of a rotation matrix R_{ab}Rab​, เช่น., R_{ab}^{-1}Rab−1​, เป็น (เลือกทั้งหมดที่ใช้):

• -R_{ab}−Rab​
• R_{ab}^{\rm T}RabT​
• R-IActive Directory−ผม
• R_{ba}Rba​

Q4. Multiplication of SO(3)ดังนั้น(3) rotation matrices is (เลือกทั้งหมดที่ใช้):

• associative.
• commutative.

แบบทดสอบ 03: Lecture Comprehension, Rotation Matrices (Chapter 3.2.1, ส่วนหนึ่ง 2 ของ 2)

Q1. Which of the following is equivalent to R_{ac}Rac​, the representation of the orientation of the {สนับสนุนเหตุการณ์การเล่าเรื่องที่อธิบายไว้ในพระคัมภีร์ไบเบิลพระธรรม} frame relative to the {เอ} กรอบ? เลือกทั้งหมดที่ใช้

• R_{ab}R_{bc}Rab​Rbc​
• R_{ab}R_{cb}^{\rm T}Rab​RcbT​
• (R_{bc}^{\rm T} R_{ab}^{\rm T})^{\rm T}(RbcT​RabT​)ตู่
• R_{ad} R_{db} R_{bc}Rad​Rdb​Rbc​

Q2. The matrix

ร = {\rm Rot}(\hat{{\rm x}},90^\circ) = \left[

1000010−10

\ขวา]Active Directory=Rot(x^,90∘)=⎣⎢⎡​100​001​0−10​⎦⎥⎤​

represents the orientation R_{sa}Rsa​ of a frame {เอ} that has been achieved by rotating the {NS} frame by 90 degrees about its \hat{{\rm x}}x^-axis. ตอนนี้, given a matrix R_{sb}Rsb​ representing the orientation of {ข} relative to {NS}, which of the following represents the orientation of a frame (relative to {NS}) that was initially aligned with {ข}, but then rotated about the {ข}-frame’s \hat{{\rm x}}x^-axis by 90 องศา?

• R_{sb} Active DirectoryRsb​Active Directory
• R R_{sb}RRsb​

Q3. The matrix

ร = {\rm Rot}(\hat{{\rm x}},90^\circ) = \left[

1000010−10

\ขวา]Active Directory=Rot(x^,90∘)=⎣⎢⎡​100​001​0−10​⎦⎥⎤​

represents the orientation R_{sa}Rsa​ of a frame {เอ} that has been achieved by rotating the {NS} frame by 90 degrees about its \hat{{\rm x}}x^-axis. ตอนนี้, given a matrix R_{sb}Rsb​ representing the orientation of {ข} relative to {NS}, which of the following represents the orientation of a frame (relative to {NS}) that was initially aligned with {ข}, but then rotated about the {NS}-frame’s \hat{{\rm x}}x^-axis by 90 องศา

• R_{sb}Active DirectoryRsb​Active Directory
• R R_{sb}RRsb​

แบบทดสอบ 04: Lecture Comprehension, Angular Velocities (Chapter 3.2.2)

Q1. Our representation of the three-dimensional orientation uses an implicit representation (a 3×3 SO(3) matrix with 9 ตัวเลข), but our usual representation of the angular velocity uses only three numbers, เช่น., an explicit parametrization of the three-dimensional velocity space. Why do we use an implicit representation of the orientation but an explicit parametrization of the angular velocity?

• There is no natural implicit representation of an angular velocity.
• The space of angular velocities can be equated to a “flat” 3d space (a linear vector space) tangent to the curved 3d surface of orientations at any given time, so it can be globally represented by 3 numbers without singularities. The space of orientations, ในทางกลับกัน, is not flat, and cannot be globally represented by 3 numbers without a singularity.

Q2. A rotation matrix is an element of which space?

• \mathbb{Active Directory}^3R3
• ดังนั้น(3)ดังนั้น(3)
• ดังนั้น(3)ดังนั้น(3)

Q3. An angular velocity is an element of which space?

• \mathbb{Active Directory}^3R3
• ดังนั้น(3)ดังนั้น(3)
• ดังนั้น(3)ดังนั้น(3)

Q4. The 3×3 skew-symmetric matrix representation of an angular velocity is an element of which space

• \mathbb{Active Directory}^3R3
• ดังนั้น(3)ดังนั้น(3)
• ดังนั้น(3)ดังนั้น(3)

Q5. If an angular velocity is represented as \omega_bωb​ in the body frame {ข}, what is the representation of the same angular velocity in the space frame {NS}?

• R_{sb} \omega_bRsb​ωb​
• R_{bs} \omega_bRbs​ωb​
• \omega_b R_{sb}ωb​Rsb​
• \omega_b R_{bs}ωb​Rbs​

Q6. The cross-product \omega \times pโอ้×พี can be written [\โอเมก้า] พี[โอ้]พี, ที่ไหน [\โอเมก้า][โอ้] เป็น

• the SO(3)ดังนั้น(3) representation of \omegaโอ้.
• the skew-symmetric so(3)ดังนั้น(3) representation of \omegaโอ้.

แบบทดสอบ 05: Lecture Comprehension, Exponential Coordinates of Rotation (Chapter 3.2.3, ส่วนหนึ่ง 1 ของ 2)

Q1. The orientation of a frame {ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก} relative to a frame {สนับสนุนเหตุการณ์การเล่าเรื่องที่อธิบายไว้ในพระคัมภีร์ไบเบิลพระธรรม} can be represented by a unit rotation axis \hat{\โอเมก้า}โอ้^ and the distance \thetaฉัน rotated about the axis. If we rotate the frame {สนับสนุนเหตุการณ์การเล่าเรื่องที่อธิบายไว้ในพระคัมภีร์ไบเบิลพระธรรม} by \thetaฉัน about the axis \hat{\โอเมก้า}โอ้^ expressed in the {สนับสนุนเหตุการณ์การเล่าเรื่องที่อธิบายไว้ในพระคัมภีร์ไบเบิลพระธรรม} กรอบ, we end up at {ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก}. The vector \hat{\โอเมก้า}โอ้^ has 3 numbers and \thetaฉัน เป็น 1 ตัวเลข, but we only need 3 ตัวเลข, the exponential coordinates \hat{\โอเมก้า} \thetaโอ้^ฉัน, to represent {ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก} relative to {สนับสนุนเหตุการณ์การเล่าเรื่องที่อธิบายไว้ในพระคัมภีร์ไบเบิลพระธรรม}, เพราะ

• though we use 3 numbers to represent \hat{\โอเมก้า}โอ้^, \hat{\โอเมก้า}โอ้^ actually only represents a point in a 2-dimensional space, the 2-dimensional sphere of unit 3-vectors.
• the choice of \thetaฉัน is not independent of \hat{\โอเมก้า}โอ้^.

Q2. One reason we use 3×3 rotation matrices (an implicit representation) to represent orientation is because it is a good global representation: there is a unique orientation for each rotation matrix, and vice-versa, and there are no singularities in the representation. In what way does the 3-vector of exponential coordinates fail these conditions? เลือกทั้งหมดที่ใช้.

• There could be more than one set of exponential coordinates representing the same orientation.
• Some orientations cannot be represented by exponential coordinates.

Q3. The vector linear differential equation \dot{NS}(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก) = Bx(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก)NS˙(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก)=Bx(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก), where xNS is a vector and BNS is a constant square matrix, is solved as x(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก) = e^{Bt} NS(0)NS(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก)=eBtx(0), where the matrix exponential e^{Bt}eBt ถูกกำหนดเป็น

• the sum of an infinite series of matrices of the form (Bt)^0 + Bt + (Bt)^2/2! + (Bt)^3/3!\ldots(Bt)0+Bt+(Bt)2/2!+(Bt)3/3!….
• the sum of an infinite series of matrices of the form Bt + Bt/2 + Bt/3 + \ldotsBt+Bt/2+Bt/3+….

แบบทดสอบ 06: Lecture Comprehension, Exponential Coordinates of Rotation (Chapter 3.2.3, ส่วนหนึ่ง 2 ของ 2)

Q1. The solution to the differential equation \dot{พี}(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก) = \hat{\โอเมก้า} \times p(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก) = [\hat{\โอเมก้า}] พี(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก)พี˙​(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก)=โอ้^×พี(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก)=[โอ้^]พี(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก) is p(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก) = e^{[\hat{\โอเมก้า}\theta]}พี(0)พี(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก)=อี[โอ้^ฉัน]พี(0), where p(0)พี(0) is the initial vector and p(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก)พี(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก) is the vector after it has been rotated at the angular velocity \hat{\โอเมก้า}โอ้^ for time t=\thetaทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก=ฉัน (where \hat{\โอเมก้า}\thetaโอ้^ฉัน are the exponential coordinates). You can think of R = e^{[\hat{\โอเมก้า}\theta]}Active Directory=อี[โอ้^ฉัน] as the rotation operation that moves p(0)พี(0) to p(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก) = p(\theta)พี(ทำไม Tundra Biome จึงเป็น Biome ที่หนาวที่สุดในโลก)=พี(ฉัน).

Which of the following statements is correct? เลือกทั้งหมดที่ใช้.

• R_{sb’} = R_{sb} e^{[\hat{\โอเมก้า}\theta]}Rsb′​=Rsb​อี[โอ้^ฉัน] represents the orientation of a new frame {b’} relative to {NS} after the frame {ข} has been rotated by \thetaฉัน about an axis w represented in the {ข} frame as \hat{\โอเมก้า}โอ้^.
• R_{sb’} = R_{sb} e^{[\hat{\โอเมก้า}\theta]}Rsb′​=Rsb​อี[โอ้^ฉัน] represents the orientation of a new frame {b’} relative to {NS} after the frame {ข} has been rotated by \thetaฉัน about an axis w represented in the {NS} frame as \hat{\โอเมก้า}โอ้^.
• R_{sb’} = e^{[\hat{\โอเมก้า}\theta]} R_{sb} Rsb′​=อี[โอ้^ฉัน]Rsb​ represents the orientation of a new frame {b’} relative to {NS} after the frame {ข} has been rotated by \thetaฉัน about an axis w represented in the {ข} frame as \hat{\โอเมก้า}โอ้^.
• R_{sb’} = e^{[\hat{\โอเมก้า}\theta]} R_{sb} Rsb′​=อี[โอ้^ฉัน]Rsb​ represents the orientation of a new frame {b’} relative to {NS} after the frame {ข} has been rotated by \thetaฉัน about an axis w represented in the {NS} frame as \hat{\โอเมก้า}โอ้^.

Q2. The simple closed-form solution to the infinite series for the matrix exponential when the matrix is an element of so(3)ดังนั้น(3) (a skew-symmetric 3×3 matrix) is called what?

• Ramirez’s formula.
• Rodrigues’ formula.
• Robertson’s formula.

Q3. The matrix exponential and the matrix log relate a rotation matrix (an element of SO(3)ดังนั้น(3)) and the skew-symmetric representation of the exponential coordinates (elements of so(3)ดังนั้น(3)), which can also be thought of as the so(3)ดังนั้น(3) representation of the angular velocity followed for unit time. Which of the following statements is correct? เลือกทั้งหมดที่ใช้.

• exp: ดังนั้น(3) \rightarrow SO(3)ดังนั้น(3)→ดังนั้น(3)
• exp: ดังนั้น(3) \rightarrow so(3)ดังนั้น(3)→ดังนั้น(3)
• log: ดังนั้น(3) \rightarrow SO(3)ดังนั้น(3)→ดังนั้น(3)
• log: ดังนั้น(3) \rightarrow so(3)ดังนั้น(3)→ดังนั้น(3)

แบบทดสอบ 07: Chapter 3 ผ่าน 3.2, Rigid-Body Motions

Q1. In terms of the \hat{NS}_{\textrm{NS}}NS^s​, \hat{Y}_{\textrm{NS}}Y^​s​, \hat{z}_{\textrm{NS}}z^s​ coordinates of a fixed space frame {NS}, the frame {เอ} has its \hat{NS}_{\textrm{เอ}}NS^a​-axis pointing in the direction (0,0,1)(0,0,1) and its \hat{Y}_{\textrm{เอ}}Y^​a​-axis pointing in the direction (1,0,0)(1,0,0), and the frame {ข} has its \hat{NS}_{\textrm{ข}}NS^b​-axis pointing in the direction (1,0,0)(1,0,0) and its \hat{Y}_{\textrm{ข}}Y^​b​-axis pointing in the direction (0,0,-1)(0,0,−1). Draw the {NS}, {เอ}, และ {ข} เฟรม, similar to examples in the book and videos (เช่น., รูป 3.7 in the book), for easy reference in this question and later questions.

Write the rotation matrix R_{sa}Active DirectoryNSเอ​. All elements of this matrix should be integers.

ถ้าคำตอบของคุณคือ

\ซ้าย[

147258369

\ขวา]⎣⎢⎡​147​258​369​⎦⎥⎤​

ตัวอย่างเช่น, you should just type

[[1,2,3],[4,5,6],[7,8,9]]

in the answer box below. (You can just modify the matrix that is currently written there.) Then click “Run.” You will not get any immediate feedback; the grade will be given when you submit the whole quiz.

Q2 .Referring to your drawing from Question 1, write R_{sb}^{-1}Active DirectoryNSข−1​. All elements of this matrix should be integers.

ถ้าคำตอบของคุณคือ

\ซ้าย[

147258369

\ขวา]⎣⎢⎡​147​258​369​⎦⎥⎤​

you should just type

[[1,2,3],[4,5,6],[7,8,9]]

in the answer box below. (You can just modify the matrix that is currently written there.) Then click “Run.” You will not get any immediate feedback; the grade will be given when you submit the whole quiz.

Q3 .Referring to your drawing from Question 1, write R_{ab}Active Directoryเอข​. All elements of this matrix should be integers.

Write your matrix in the answer box below, using the format mentioned in questions 1 และ 2, and click “Run.”

Q4. Referring back to Question 1, let R = R_{sb}Active Directory=Rsb​ be considered as a transformation operator consisting of a rotation about \hat{NS}NS^ by -90^\circ−90∘. Calculate R_1 = R_{sa} Active DirectoryActive Directory1​=Rsa​Active Directory, and think of R_{sa}Rsa​ as the representation of the initial orientation of {เอ} relative to {NS}, Active DirectoryActive Directory as a rotation operation, and R_1Active Directory1​ as the new orientation of {เอ} after performing the rotation. The new orientation R_1Active Directory1​ corresponds to the orientation of the new {เอ} frame relative to {NS} after rotating the original {เอ} frame by -90^\circ−90∘ about which axis?

• The \hat{NS}_{\textrm{เอ}}NS^a​-axis of the {เอ} กรอบ.
• The \hat{NS}_{\textrm{NS}}NS^s​-axis of the {NS} กรอบ.

Q5. Referring back to Question 1, use R_{sb}Active DirectoryNSข​ to change the representation of the point p_b = (1,2,3)^\intercalพีข​=(1,2,3)⊺ (ใน {ข} coordinates) ถึง {NS} coordinates. All elements of this vector should be integers.

ถ้าคำตอบของคุณคือ

\ซ้าย[

123

\ขวา]⎣⎢⎡​123​⎦⎥⎤​

you should enter

[1,2,3]

in the text box below and click “Run.”

Q6. Referring back to Question 1, choose a point p represented by p_s = (1,2,3)^\intercalps​=(1,2,3)⊺ in {NS} coordinates. Calculate q = R^\intercal_{sb} p_sถาม=Rsb⊺​ps​. Is qถาม a representation of p in {ข} coordinates?

• ใช่.
• เลขที่.

Q7. Referring back to Question 1, an angular velocity wว is represented in {NS} as \omega_s = (3,2,1)^\intercalโอ้NS​=(3,2,1)⊺. What is its representation \omega_aโอ้เอ​? All elements of this vector should be integers.

ถ้าคำตอบของคุณคือ

\ซ้าย[

123

\ขวา]⎣⎢⎡​123​⎦⎥⎤​

you should enter

[1,2,3]

in the text box below and click “Run.”

Q8. Referring back to Question 1, calculate the matrix logarithm [\hat{\โอเมก้า}]\theta[โอ้^]ฉัน of R_{sa}Rsa​ by hand. (You may verify your answer with software.) Extract and enter the rotation amount \thetaฉัน in radians with at least two decimal places.

• 1
• 0

คำถามที่ 9. Calculate the matrix exponential corresponding to the exponential coordinates of rotation \hat{\โอเมก้า}\theta = (1,2,0)^\intercalโอ้^ฉัน=(1,2,0)⊺. The maximum allowable error for any matrix element is 0.01, so give enough decimal places where necessary.

Write your matrix in the answer box below, using the format mentioned in questions 1 และ 2, and click “Run.”

Q10. Write the 3 \times 33×3 skew-symmetric matrix corresponding to \omega = (1,2,0.5)^\intercalโอ้=(1,2,0.5)⊺. Confirm your answer using the function {\tt VecToso3}VecToso3 in the given software.

Write your matrix in the answer box below, using the format mentioned in questions 1 และ 2, and click “Run.”

คำถามที่ 11. Use the function {\tt MatrixExp3}MatrixExp3 in the given software to calculate the rotation matrix R \in SO(3)Active Directory∈NSเธ(3) corresponding to the matrix exponential of

[\hat{\โอเมก้า}] \theta = \left[

0−0.510.50−2−120

\ขวา].[โอ้^]ฉัน=⎣⎢⎡​0−0.51​0.50−2​−120​⎦⎥⎤​.

The maximum allowable error for any matrix element is 0.01, so give enough decimal places where necessary.

Write your matrix in the answer box below, using the format mentioned in questions 1 และ 2, and click “Run.”

Q12. Use the function {\tt MatrixLog3}MatrixLog3 in the given software to calculate the matrix logarithm [\hat{\โอเมก้า}] \theta \in so(3)[โอ้^]ฉัน∈NSโอ(3) of rotation matrix

R = \left[

0−1000−1100

\ขวา].Active Directory=⎣⎢⎡​0−10​00−1​100​⎦⎥⎤​.

The maximum allowable error for any matrix element is 0.01, so give enough decimal places where necessary.

Write your matrix in the answer box below, using the format mentioned in questions 1 และ 2, and click “Run.”

สัปดาห์ 04: วิทยาการหุ่นยนต์สมัยใหม่, NFT ที่สมบูรณ์ 1: Foundations of Robot Motion Quiz Answers

แบบทดสอบ 01 : Lecture Comprehension, Homogeneous Transformation Matrices (Chapter 3 ผ่าน 3.3.1)

Q1. A 4×4 transformation matrix (element of SE(3)SE(3)) consists of a rotation matrix, a 3-vector, and a row consisting of three zeros and a one. What is the purpose of the row of 4 ค่าคงที่?

• This row is a historical artifact.
• This row allows simple matrix operations for useful calculations.

Q2. Which of the following are possible uses of a transformation matrix? เลือกทั้งหมดที่ใช้.

• Displace (rotate and translate) a frame.
• Displace a vector.
• Change the frame of reference of a vector.
• Represent the position and orientation of one frame relative to another.

Q3. The representation of a point p in the {ข} frame is p_b \in \mathbb{Active Directory}^3pb​∈R3. To find the representation of this point in the {เอ} กรอบ, we could write T_{ab} p_bTab​pb​, but there is a dimension mismatch; p_bpb​ has only 3 ส่วนประกอบ, but T_{ab}Tab​ is 4×4. How do we alter p_bpb​ to allow this matrix operation?

• Put a 1 in the last row of p_bpb​, making it a 4-element column vector, and otherwise ignore the last row in your interpretation of the 4-vector.
• Put a 0 in the last row of p_bpb​, making it a 4-element column vector, and otherwise ignore the last row in your interpretation of the 4-vector.

Q4. Which of these is a valid calculation of T_{ab}Tab​, the configuration of the frame {ข} relative to {เอ}? เลือกทั้งหมดที่ใช้.

• T_{ac} T_{cb}Tac​Tcb​
• T_{cb} T_{ac}Tcb​Tac​
• T_{ac} T^{-1}_{dc} T_{db}Tac​Tdc−1​Tdb​
• (T_{bc} T_{ca})^{-1}(Tbc​Tca​)−1

แบบทดสอบ 02 : Lecture Comprehension, Twists (Chapter 3.3.2, ส่วนหนึ่ง 1 ของ 2)

Q1. Any instantaneous spatial velocity of a rigid body is equivalent to the motion of the body if it were simultaneously translating along, and rotating about, เอ screw axis \mathcal{NS} = (\mathcal{NS}_\omega, \mathcal{NS}_v) \in \mathbb{Active Directory}^6S=(NSโอ้​,NSโวลต์​)∈R6. The screw axis is a normalized representation of the direction of motion, and \dot{\theta}ฉัน˙ represents how fast the body moves in that direction of motion, so that the twist is given by \mathcal{วี} = \mathcal{NS}\dot{\theta} \in \mathbb{Active Directory}^6V=Sฉัน˙∈R6. The normalized screw axis for full spatial motions is analogous to the normalized (unit) angular velocity axis for pure rotations.

The pitch hชม. of the screw axis is defined as the ratio of the linear speed over the angular speed. Which of the following is true? เลือกทั้งหมดที่ใช้.

• If the pitch hชม. is infinite, then \mathcal{NS}_\omega = 0Sโอ้​=0 and \|\mathcal{NS}_v\| = 1∥Sโวลต์​∥=1.
• If the pitch hชม. is infinite, แล้ว \|\mathcal{NS}_\omega\| = 1∥Sโอ้​∥=1 and \mathcal{NS}_vSโวลต์​ is arbitrary.
• If the pitch hชม. is finite, then \mathcal{NS}_\omega = 0Sโอ้​=0 and \|\mathcal{NS}_v\| = 1∥Sโวลต์​∥=1.
• If the pitch hชม. is finite, แล้ว \|\mathcal{NS}_\omega\| = 1∥Sโอ้​∥=1 and \mathcal{NS}_vSโวลต์​ is arbitrary.

Q2. You are sitting on a horizontal rotating turntable, like a merry-go-round at an amusement park. It rotates counterclockwise when viewed from above. Your body frame {ข} has an \hat{{\rm x}}_bx^ข​-axis pointing outward (away from the center of the turntable), a \hat{{\rm y}}_by^​ข​-axis pointing in the direction the turntable is moving at your location (the direction your eyes are looking), and a \hat{{\rm z}}_bz^ข​-axis pointing upward. The turntable is rotating at 0.1 radians per second, and you are sitting 3 meters from the center of the turntable. What is the screw axis \mathcal{NS} = (\mathcal{NS}_\omega, \mathcal{NS}_v)S=(NSโอ้​,NSโวลต์​) and the twist \mathcal{วี} = (\โอเมก้า,โวลต์)วี=(โอ้,โวลต์) expressed in your body frame {ข}? All angular velocities are in radians/second and all linear velocities are in meters/second.

• \mathcal{NS} = (0, 0, 0.1, 0, 0.3, 0), \;\; \mathcal{วี} = (0, 0, 0.01, 0, 0.03, 0)S=(0,0,0.1,0,0.3,0),วี=(0,0,0.01,0,0.03,0)
• \mathcal{NS} = (0, 0, 1, 0, 3, 0), \;\; \mathcal{วี} = (0, 0, 0.1, 0, 0.3, 0)S=(0,0,1,0,3,0),วี=(0,0,0.1,0,0.3,0)
• \mathcal{NS} = (1, 0, 0, 0, 3, 0), \;\; \mathcal{วี} = (0.1, 0, 0, 0, 0.3, 0)S=(1,0,0,0,3,0),วี=(0.1,0,0,0,0.3,0)

Q3. A twist or a screw axis can be represented in any frame. Which of the following statements are true? เลือกทั้งหมดที่ใช้.

• A spatial twist is a representation of the twist in the space frame {NS}, and it does not depend on a body frame {ข}.
• A body twist is a representation of the twist in the body frame {ข}, and it does not depend on a space frame {NS}.

แบบทดสอบ 03 : Lecture Comprehension, Twists (Chapter 3.3.2, ส่วนหนึ่ง 2 ของ 2)

Q1. What is the dimension of the matrix adjoint representation [{\rm Ad}_T][Adตู่​] of a transformation matrix Tตู่ (an element of SE(3)SE(3))?

• 3×3
• 4×4
• 6×6

Q2. A 3-vector angular velocity \omegaโอ้ can be represented in matrix form as [\โอเมก้า][โอ้], an element of so(3)ดังนั้น(3), the set of 3×3 skew-symmetric matrices. Analogously, a 6-vector twist \mathcal{วี} = (\โอเมก้า,โวลต์)วี=(โอ้,โวลต์) can be represented in matrix form as [\mathcal{วี}][วี], an element of se(3)se(3). What is the dimension of [\mathcal{วี}][วี]?

• 3×3
• 4×4
• 6×6

แบบทดสอบ 04 : Lecture Comprehension, Exponential Coordinates of Rigid-Body Motion (Chapter 3.3.3)

Q1. Although we use six numbers to represent a screw \mathcal{NS} = (\mathcal{NS}_\omega,\mathcal{NS}_v)S=(NSโอ้​,NSโวลต์​), the space of all screws is only 5-dimensional. ทำไม?

• \mathcal{NS}_\omegaSโอ้​ must be unit length.
• \mathcal{NS}_vSโวลต์​ must be unit length.
• Either \mathcal{NS}_\omegaSโอ้​ or \mathcal{NS}_vSโวลต์​ must be unit length.

Q2. A transformation matrix T_{ab}Tab​, เป็นตัวแทน {ข} relative to {เอ}, can be represented using the 6-vector exponential coordinates \mathcal{NS}\thetaSฉัน, where \mathcal{NS}S is a screw axis (represented in {เอ} coordinates) and \thetaฉัน is the distance followed along the screw axis that displaces {เอ} ถึง {ข}. Which of the following is correct? เลือกทั้งหมดที่ใช้.

• T_{ab} = e^{\mathcal{NS}\theta}Tab​=อีNSฉัน
• T_{ab} = e^{[\mathcal{NS}]\theta}Tab​=อี[NS]ฉัน
• T_{ab} = e^{[\mathcal{NS}\theta]}Tab​=อี[NSฉัน]
• T_{ab} = e^{\mathcal{NS}[\theta]}Tab​=อีNS[ฉัน]

Q3. The matrix representation of the exponential coordinates \mathcal{NS}\theta \in \mathbb{Active Directory}^6Sฉัน∈R6 is [\mathcal{NS}\theta][NSฉัน]. What space does [\mathcal{NS}\theta][NSฉัน] belong to?

• ดังนั้น(3)
• ดังนั้น(3)
• SE(3)
• se(3)

Q4. T_{ab’} = T_{ab} e^{[\mathcal{NS}\theta]}Tab′​=Tab​อี[NSฉัน] is a representation of the new frame {b’} (relative to {เอ}) achieved after {ข} has followed

• the screw axis \mathcal{NS}NS, expressed in {ข} coordinates, a distance \thetaฉัน.
• the screw axis \mathcal{NS}NS, expressed in {เอ} coordinates, a distance \thetaฉัน.

Q5. T_{ab’} = e^{[\mathcal{NS}\theta]} T_{ab}Tab′​=อี[NSฉัน]Tab​ is a representation of the new frame {b’} (relative to {เอ}) achieved after {ข} has followed

• the screw axis \mathcal{NS}NS, expressed in {ข} coordinates, a distance \thetaฉัน.
• the screw axis \mathcal{NS}NS, expressed in {เอ} coordinates, a distance \thetaฉัน.

Q6. ข้อความใดต่อไปนี้เป็นจริง? เลือกทั้งหมดที่ใช้.

• The matrix exponential maps [\mathcal{NS}\theta] \in se(3)[NSฉัน]∈se(3) to a transformation matrix T \in SE(3)ตู่∈SE(3), where Tตู่ is the representation of the frame (relative to {NS}) that is achieved by following the screw \mathcal{NS}NS (expressed in {NS}) a distance \thetaฉัน from the identity configuration (เช่น., a frame initially coincident with {NS}).
• The matrix exponential maps [\mathcal{วี}] \in se(3)[วี]∈se(3) to a transformation matrix T \in SE(3)ตู่∈SE(3), where Tตู่ is the representation of the frame (relative to {NS}) that is achieved by following the twist \mathcal{วี}วี (expressed in {NS}) for unit time from the identity configuration (เช่น., a frame initially coincident with {NS}).
• The matrix log maps an element of se(3)se(3) to an element of SE(3)SE(3).
• The matrix log maps an element of SE(3)SE(3) to an element of se(3)se(3).
• There is a one-to-one mapping between twists and elements of se(3)se(3).

แบบทดสอบ 05 : Lecture Comprehension, Wrenches (Chapter 3.4)

Q1. A wrench \mathcal{F}_aFเอ​ consists of a linear force f_a \in \mathbb{Active Directory}^3fa​∈R3 and a moment m_a \in \mathbb{Active Directory}^3ma​∈R3, both expressed in the frame {เอ}. How do we usually write the wrench?

• \mathcal{F}_a = (m_a,f_a)Fเอ​=(ma​,fa​)
• \mathcal{F}_a = (f_a,m_a)Fเอ​=(fa​,ma​)

Q2. We know that the power associated with a wrench and twist pair (\mathcal{F},\mathcal{วี})(F,วี) does not depend on whether they are represented in the frame {เอ} เช่น (\mathcal{F}_a,\mathcal{วี}_a)(Fเอ​,วีเอ​) or the frame {ข} เช่น (\mathcal{F}_b,\mathcal{วี}_b)(Fข​,วีข​). ดังนั้น, we can write \mathcal{F}_a^{\rm T} \mathcal{วี}_a = \mathcal{F}_b^{\rm T} \mathcal{วี}_bFเอT​Vเอ​=FขT​Vข​ and then use which identity to derive the equation \mathcal{F}_a = [{\rm Ad}_{T_{ba}}]^{\rm T} \mathcal{F}_bFเอ​=[AdTba​​]TFข​ relating the representations \mathcal{F}_aFเอ​ and \mathcal{F}_bFข​? (อีกด้วย, remember the matrix identity (AB)^{\rm T} = B^{\rm T} A^{\rm T}(AB)T=NSตู่NST.)

• \mathcal{วี}_a = T_{ab} \mathcal{วี}_bVเอ​=Tab​Vข​
• \mathcal{วี}_a = T_{ba} \mathcal{วี}_bVเอ​=Tba​Vข​
• \mathcal{วี}_a = [{\rm Ad}_{T_{ba}}] \mathcal{วี}_bVเอ​=[AdTba​​]วีข​
• \mathcal{วี}_a = [{\rm Ad}_{T_{ab}}] \mathcal{วี}_bVเอ​=[AdTab​​]วีข​

แบบทดสอบ 06 : Chapters 3.3 และ 3.4, Rigid-Body Motions

Q1. In terms of the \hat{NS}_{\textrm{NS}}NS^s​, \hat{Y}_{\textrm{NS}}Y^​s​, \hat{z}_{\textrm{NS}}z^s​ coordinates of a fixed space frame {NS}, the frame {เอ} has its \hat{NS}_{\textrm{เอ}}NS^a​-axis pointing in the direction (0,0,1)(0,0,1) and its \hat{Y}_{\textrm{เอ}}Y^​a​-axis pointing in the direction (-1,0,0)(−1,0,0), and frame {ข} has its \hat{NS}_{\textrm{ข}}NS^b​-axis pointing in the direction (1,0,0)(1,0,0) and its \hat{Y}_{\textrm{ข}}Y^​b​-axis pointing in the direction (0,0,-1)(0,0,−1). The origin of {เอ} is at (0,0,1)(0,0,1) ใน {NS} and the origin of {ข} is at (0,2,0)(0,2,0). Draw the {NS}, {เอ}, และ {ข} เฟรม, similar to examples in the book and videos, for easy reference in this question and later questions.

Write the transformation matrix T_{sa}ตู่NSเอ​. All elements of this matrix should be integers.

Enter your matrix in the answer box (just modify the matrix already shown there) and click “Run.” Your answer will not be evaluated until you submit the quiz.

[[1,2,3,4],[5,6,7,8],[9,10,11,12],[0,0,0,1]] for \left[

1590261003711048121

\ขวา]⎣⎢⎢⎢⎡​1590​26100​37110​48121​⎦⎥⎥⎥⎤​.

Q2. Referring back to Question 1, write T_{sb}^{-1}ตู่NSข−1​. All elements of this matrix should be integers.

Enter your matrix in the answer box (just modify the matrix already shown there) and click “Run.” Your answer will not be evaluated until you submit the quiz.

[[1,2,3,4],[5,6,7,8],[9,10,11,12],[0,0,0,1]] for \left[

1590261003711048121

\ขวา]⎣⎢⎢⎢⎡​1590​26100​37110​48121​⎦⎥⎥⎥⎤​

Q3. Referring back to Question 1, write T_{ab}ตู่เอข​. All elements of this matrix should be integers.

Enter your matrix in the answer box (just modify the matrix already shown there) and click “Run.” Your answer will not be evaluated until you submit the quiz.

[[1,2,3,4],[5,6,7,8],[9,10,11,12],[0,0,0,1]] for \left[

1590261003711048121

\ขวา]⎣⎢⎢⎢⎡​1590​26100​37110​48121​⎦⎥⎥⎥⎤​.

Q4. Referring back to Question 1, let T = T_{sb}ตู่=ตู่NSข​ be considered as a transformation operator consisting of a rotation about \hat{NS}NS^ by -90^\circ−90∘ and a translation along \hat{Y}Y^​ by 2 หน่วย. Calculate T_1 = T T_{sa}ตู่1​=ตู่ตู่NSเอ​, and think of T_{sa}ตู่NSเอ​ as the representation of the initial configuration of {เอ} relative to {NS}, ตู่ตู่ as a transformation operation, and T_1ตู่1​ as the new configuration of {เอ} after performing the transformation. Are the rotation axis \hat{NS}NS^ and translation axis \hat{Y}Y^​ of the transformation Tตู่ properly considered to be expressed in the frame {NS} or the frame {เอ}?

1 จุด

• The frame {NS}.
• The frame {เอ}.

Q5. Referring back to Question 1, use T_{sb}ตู่NSข​ to change the representation of the point p_b = (1,2,3)^\intercalพีข​=(1,2,3)⊺ (ใน {ข} coordinates) ถึง {NS} coordinates. All elements of this vector should be integers.

Enter your vector in the answer box (just modify the vector already shown there) and click “Run.” Your answer will not be evaluated until you submit the quiz.

[1,2,3] for \left[

123

\ขวา]⎣⎢⎡​123​⎦⎥⎤​.

Q6. Referring back to Question 1, choose a point p represented by p_s = (1,2,3)^\intercalพีNS​=(1,2,3)⊺ in {NS} coordinates. Calculate q = T_{sb} p_sถาม=ตู่NSข​พีNS​. Is qถาม a representation of p in {ข} coordinates?

1 จุด

• ใช่
• เลขที่

Q7. Referring back to Question 1, a twist \mathcal{วี}V is represented in {NS} เช่น {\mathcal V}_s = (3,2,1,-1,-2,-3)^\intercalVNS​=(3,2,1,−1,−2,−3)⊺. What is its representation {\mathcal V}_aVเอ​? All elements of this vector should be integers.

Enter your vector in the answer box (just modify the vector already shown there) and click “Run.” Your answer will not be evaluated until you submit the quiz.

[1,2,3,4,5,6] for \left[

123456

\ขวา]⎣⎢⎢⎢⎢⎢⎢⎢⎡​123456​⎦⎥⎥⎥⎥⎥⎥⎥⎤​.

Q8. Referring back to Question 1, calculate the matrix logarithm [{\mathcal S}]\theta[NS]ฉัน of T_{sa}Tsa​. Write the rotation amount \thetaฉัน in radians with at least 2 decimal places.

คำถามที่ 9. Calculate the matrix exponential corresponding to the exponential coordinates of rigid-body motion {\mathcal S}\theta = (0,1,2,3,0,0)^\intercalSฉัน=(0,1,2,3,0,0)⊺. The maximum allowable error for any matrix element is 0.01, so give enough decimal places where necessary.

Enter your matrix in the answer box (just modify the matrix already shown there) and click “Run.” Your answer will not be evaluated until you submit the quiz.

[[1.11,2.22,3.33],[4.44,5.55,6.66],[7.77,8.88,9.99]] for \left[

1.114.447.772.225.558.883.336.669.99

\ขวา]⎣⎢⎡​1.114.447.77​2.225.558.88​3.336.669.99​⎦⎥⎤​.

Q10. Referring back to Question 1, use T_{sb}ตู่NSข​ to change the representation of the wrench Fข=(1,0,0,2,1,0)⊺ (ใน {ข} coordinates) ถึง {NS} coordinates. All elements of this vector should be integers.

Enter your vector in the answer box (just modify the vector already shown there) and click “Run.” Your answer will not be evaluated until you submit the quiz.

[1,2,3] for \left[

123

\ขวา]⎣⎢⎡​123​⎦⎥⎤​.

คำถามที่ 11. Use the function {\tt TransInv}TransInv in the given software to calculate the inverse of the homogeneous transformation matrix

T = \left[

0100−100000103011

\ขวา].ตู่=⎣⎢⎢⎢⎡​0100​−1000​0010​3011​⎦⎥⎥⎥⎤​.

All elements of this matrix should be integers.

Enter your matrix in the answer box (just modify the matrix already shown there) and click “Run.” Your answer will not be evaluated until you submit the quiz.

[[1,2,3],[4,5,6],[7,8,9]] for \left[

147258369

\ขวา]⎣⎢⎡​147​258​369​⎦⎥⎤​.

Q12. Write the se(3)NSอี(3) matrix corresponding to the twist V=(1,0,0,0,2,3)⊺. All elements of this matrix should be integers. Confirm your answer using the function {\tt VecTose3}VecTose3 in the given software.

Enter your matrix in the answer box (just modify the matrix already shown there) and click “Run.” Your answer will not be evaluated until you submit the quiz.

[[1,2,3],[4,5,6],[7,8,9]] for \left[

147258369

\ขวา]⎣⎢⎡​147​258​369​⎦⎥

ไตรมาสที่ 13. Use the function {\tt ScrewToAxis}ScrewToAxis in the given software to calculate the normalized screw axis representation S of the screw described by a unit vector \hat{NS} = (1,0,0)NS^=(1,0,0) in the direction of the screw axis, located at the point p = (0,0,2)พี=(0,0,2), with pitch h = 1ชม.=1. All elements of this vector should be integers.

Enter your vector in the answer box (just modify the vector already shown there) and click “Run.” Your answer will not be evaluated until you submit the quiz.

[1,2,3] for \left[

123

\ขวา]⎣⎢⎡​123​⎦⎥⎤​.

คำถามที่ 14. Use the function {\tt MatrixExp6}MatrixExp6 in the given software to calculate the homogeneous transformation matrix T \in SE(3)ตู่∈NSอี(3) corresponding to the matrix exponential of

[NS]ฉัน=⎡⎣⎢⎢01.570800−1.570800000002.3562−2.356210⎤⎦⎥⎥.

All elements of this matrix should be integers.

Enter your matrix in the answer box (just modify the matrix already shown there) and click “Run.” Your answer will not be evaluated until you submit the quiz.

[[1,2,3],[4,5,6],[7,8,9]] for \left[

147258369

\ขวา]⎣⎢⎡​147​258​369​⎦⎥⎤​.

Q15. Use the function {\tt MatrixLog6}MatrixLog6 in the given software to calculate the matrix logarithm [NS]ฉัน∈NSอี(3) of the homogeneous transformation matrix

T = \left[

0100−100000103011

\ขวา].ตู่=⎣⎢⎢⎢⎡​0100​−1000​0010​3011​⎦⎥⎥⎥⎤​.

The maximum allowable error for any matrix element is 0.01, so give enough decimal places where necessary.

Enter your matrix in the answer box (just modify the matrix already shown there) and click “Run.” Your answer will not be evaluated until you submit the quiz.

[[1.11,2.22,3.33],[4.44,5.55,6.66],[7.77,8.88,9.99]] for \left[

1.114.447.772.225.558.883.336.669.99

\ขวา]⎣⎢⎡​1.114.447.77​2.225.558.88​3.336.669.99​⎦⎥⎤​.