Modern Robotics, 课程 1: Foundations of Robot Motion Quizzes & 答案 – Coursera

Take a journey into the world of 机器人 with interesting quizzes and expert answers on Foundations of Robot 运动 在 Modern 机器人, 课程 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 机器人 enthusiast looking to deepen your knowledge or a student looking to explore the exciting field of 机器人, 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.

测验 01: Lecture Comprehension, Degrees of Freedom of a Rigid Body (章节 2 通过 2.1)

第一季度. Which of the following are possible elements of robots in this specialization? 选择所有符合条件的.

• Rigid bodies.
• 柔软的, flexible bodies.
• Joints.

Q2. The number of degrees of freedom of a robot is (选择所有符合条件的):

• 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

第四季度. The number of degrees of freedom of a spatial rigid body is

Q5. A rigid body in nn-dimensional space has m米 total degrees of freedom. How many of these m米 degrees of freedom are angular (not linear)? 选择所有符合条件的. (This is consistently one of the most incorrectly answered questions in this course, so think about it carefully!)

• m-n米 - n
• n(n-1)/2n(n−1)/2
• Neither of the above.

测验 02: Lecture Comprehension, Degrees of Freedom of a Robot (章节 2.2)

第一季度. Consider a joint between two rigid bodies. Each rigid body has m米 degrees of freedom (m=3米=3 for a planar rigid body and m=6米=6 for a spatial rigid body) in the absence of any constraints. The joint has fF degrees of freedom (例如, f=1F=1 for a revolute joint or f=3F=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 m米 and fF

Q2. Consider a mechanism consisting of three spatial rigid bodies (including ground, N=4ñ=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.

测验 03: 章节 2 通过 2.2, Configuration Space

第一季度. 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, 具有 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

第四季度. 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 nn such arms grasping the object. What is the number of degrees of freedom of this system? Your answer should be a mathematical expression including nn. Examples of mathematical expressions including nn are 4*n-74∗n−7 or n/3n/3

Q6. Referring back to Question 4 和 5, suppose the revolute joint in each of the nn 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 nn. Examples of mathematical expressions including nn are 4*n-74∗n−7 or n/3n/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. 也, 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.)

周 02: Modern Robotics, 课程 1: Foundations of Robot Motion Quiz Answers

测验 01: Lecture Comprehension, Configuration Space Topology (章节 2.3.1)

第一季度. To deform one nn-dimensional space into another topologically equivalent space, which operations are you allowed to use? 选择所有符合条件的.

• Stretching
• Cutting.
• Gluing.

Q2. 对或错? An nn-dimensional space can be topologically equivalent to an m米-dimensional space, where m \neq n米​=n.

• 真正.
• 假.

测验 02: Lecture Comprehension, Configuration Space Representation (章节 2.3.2)

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

真正.

假

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

测验 02: Lecture Comprehension, Configuration and Velocity Constraints (章节 2.4)

第一季度. 对或错? A nonholonomic constraint implies a configuration constraint.

• 真正.
• 假.

Q2. 对或错? A Pfaffian velocity constraint is necessarily nonholonomic.

• 真正.
• 假.

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.

第四季度. 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?)

测验 03: Lecture Comprehension, Task Space and Workspace (章节 2.5)

第一季度. 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. 对或错? The workspace depends on the robot’s joint limits but the task space does not.

• 真正.
• 假.

测验 04: 章节 2.3 通过 2.5, Configuration Space

第一季度. 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) X=cosθ1​+2cos(θ1​+θ2​)

y = \sin \theta_1 + 2 \sin (\theta_1 + \theta_2)和=sinθ1​+2sin(θ1​+θ2​)

(换一种说法, 链接 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.)

• [一个,b] \时 [一个,b] \times S^1[一个,b]×[一个,b]×小号1
• [一个,b] \times \mathbb{[R}^1 \times S^1[一个,b]×R1×小号1
• [一个,b] \时 [一个,b] \times \mathbb{[R}^1[一个,b]×[一个,b]×R1
• \mathbb{[R}^2 \times S^1R2×小号1

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

• \mathbb{[R}^3 \times S^3R3×小号3
• \mathbb{[R}^3 \times T^3R3׍3
• \mathbb{[R}^3 \times T^2 \times S^1R3׍2×小号1
• \mathbb{[R}^3 \times S^2 \times S^1R3×小号2×小号1

第四季度. 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{[R}^3 \times T^{10}R3׍10
• \mathbb{[R}^3 \times S^2 \times T^8R3×小号2׍8
• \mathbb{[R}^3 \times S^3 \times T^7R3×小号3׍7
• \mathbb{[R}^4 \times S^2 \times T^7R4×小号2׍7

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{[R}^2 \times S^2 \times S^1 \times [一个,b]R2×小号2×小号1×[一个,b]
• \mathbb{[R}^2 \times S^3 \times [一个,b]R2×小号3×[一个,b]
• \mathbb{[R}^2 \times T^3 \times [一个,b]R2׍3×[一个,b]
• \mathbb{[R}^3 \times T^3R3׍3

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

(1+ \cos q_1) \点{q}_1 + (2+ \sin q_2) \点{q}_2 + (\cos q_1+ \sin q_2 + 3) \点{q}_3 = 0.(1+cosq1​)q˙​1​+(2+sinq2​)q˙​2​+(cosq1​+sinq2​+3)q˙​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) 有? (Enter an integer number.)

周 03: Modern Robotics, 课程 1: Foundations of Robot Motion Quiz Answers

测验 01: Lecture Comprehension, Introduction to Rigid-Body Motions (章节 3 通过 3.1)

第一季度. 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, 和, and z axes of a coordinate frame, 分别?

• Thumb, 指数, 中间
• Middle, 指数, thumb
• 指数, 中间, 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

第四季度. When we refer to a frame attached to a moving body, we always consider a stationary frame {b}, 因为

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

测验 02: Lecture Comprehension, Rotation Matrices (章节 3.2.1, 部分 1 的 2)

第一季度. For the rotation matrix R_{ba}Rba​ representing the frame {一个} relative to {b},

• the rows are the x, 和, z axes of {一个} written in {b} coordinates.
• the columns are the x, 和, z axes of {一个} written in {b} coordinates.
• the rows are the x, 和, z axes of {b} written in {一个} coordinates.
• the columns are the x, 和, z axes of {b} written in {一个} coordinates.

Q2. 该 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​, 即, R_{ab}^{-1}Rab−1​, 是 (选择所有符合条件的):

• -R_{ab} - Rab​
• R_{ab}^{\rm T}RabT​
• R-I[R - 一世
• R_{ba}Rba​

第四季度. Multiplication of SO(3)所以(3) rotation matrices is (选择所有符合条件的):

• associative.
• commutative.

测验 03: Lecture Comprehension, Rotation Matrices (章节 3.2.1, 部分 2 的 2)

第一季度. Which of the following is equivalent to R_{ac}Rac​, the representation of the orientation of the {C} frame relative to the {一个} 框架? 选择所有符合条件的

• 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​)Ť
• R_{广告} R_{db} R_{bc}Rad​Rdb​Rbc​

Q2. The matrix

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

1000010−10

\对][R=Rot(x^,90∘)=⎣⎢⎡​100​001​0−10​⎦⎥⎤​

represents the orientation R_{sa}Rsa​ of a frame {一个} that has been achieved by rotating the {小号} frame by 90 degrees about its \hat{{\rm x}}x^-axis. 现在, given a matrix R_{sb}Rsb​ representing the orientation of {b} relative to {小号}, which of the following represents the orientation of a frame (relative to {小号}) that was initially aligned with {b}, but then rotated about the {b}-frame’s \hat{{\rm x}}x^-axis by 90 度?

• R_{sb} [RRsb​[R
• R R_{sb}RRsb​

Q3. The matrix

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

1000010−10

\对][R=Rot(x^,90∘)=⎣⎢⎡​100​001​0−10​⎦⎥⎤​

represents the orientation R_{sa}Rsa​ of a frame {一个} that has been achieved by rotating the {小号} frame by 90 degrees about its \hat{{\rm x}}x^-axis. 现在, given a matrix R_{sb}Rsb​ representing the orientation of {b} relative to {小号}, which of the following represents the orientation of a frame (relative to {小号}) that was initially aligned with {b}, but then rotated about the {小号}-frame’s \hat{{\rm x}}x^-axis by 90 度

• R_{sb}[RRsb​[R
• R R_{sb}RRsb​

测验 04: Lecture Comprehension, Angular Velocities (章节 3.2.2)

第一季度. Our representation of the three-dimensional orientation uses an implicit representation (a 3×3 SO(3) matrix with 9 数字), but our usual representation of the angular velocity uses only three numbers, 即, 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, 另一方面, 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{[R}^3R3
• 所以(3)所以(3)
• 所以(3)所以(3)

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

• \mathbb{[R}^3R3
• 所以(3)所以(3)
• 所以(3)所以(3)

第四季度. The 3×3 skew-symmetric matrix representation of an angular velocity is an element of which space

• \mathbb{[R}^3R3
• 所以(3)所以(3)
• 所以(3)所以(3)

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

• 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 pω×p can be written [\它们还被证明可以预防抑郁症并提高老年人的认知功能] p[ω]p, 哪里 [\它们还被证明可以预防抑郁症并提高老年人的认知功能][ω] 是

• the SO(3)所以(3) representation of \omegaω.
• the skew-symmetric so(3)所以(3) representation of \omegaω.

测验 05: Lecture Comprehension, Exponential Coordinates of Rotation (章节 3.2.3, 部分 1 的 2)

第一季度. The orientation of a frame {d} relative to a frame {C} can be represented by a unit rotation axis \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}ω^ and the distance \thetaθ rotated about the axis. If we rotate the frame {C} by \thetaθ about the axis \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}ω^ expressed in the {C} 框架, we end up at {d}. The vector \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}ω^ has 3 numbers and \thetaθ 是 1 数, but we only need 3 数字, the exponential coordinates \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能} \thetaω^θ, to represent {d} relative to {C}, 因为

• though we use 3 numbers to represent \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}ω^, \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}ω^ actually only represents a point in a 2-dimensional space, the 2-dimensional sphere of unit 3-vectors.
• the choice of \thetaθ is not independent of \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}ω^.

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? 选择所有符合条件的.

• 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{X}(吨) = Bx(吨)X˙(吨)=Bx(吨), where xX is a vector and B乙 is a constant square matrix, is solved as x(吨) = e^{Bt} X(0)X(吨)=eBtx(0), where the matrix exponential e^{Bt}eBt 被定义为

• 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+....

测验 06: Lecture Comprehension, Exponential Coordinates of Rotation (章节 3.2.3, 部分 2 的 2)

第一季度. The solution to the differential equation \dot{p}(吨) = \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能} \times p(吨) = [\hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}] p(吨)p˙​(吨)=ω^×p(吨)=[ω^]p(吨) is p(吨) = e^{[\hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}\theta]}p(0)p(吨)=Ë[ω^θ]p(0), where p(0)p(0) is the initial vector and p(吨)p(吨) is the vector after it has been rotated at the angular velocity \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}ω^ for time t=\theta吨=θ (where \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}\thetaω^θ are the exponential coordinates). You can think of R = e^{[\hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}\theta]}[R=Ë[ω^θ] as the rotation operation that moves p(0)p(0) to p(吨) = p(\theta)p(吨)=p(θ).

Which of the following statements is correct? 选择所有符合条件的.

• R_{sb’} = R_{sb} e^{[\hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}\theta]}Rsb′​=Rsb​Ë[ω^θ] represents the orientation of a new frame {b’} relative to {小号} after the frame {b} has been rotated by \thetaθ about an axis w represented in the {b} frame as \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}ω^.
• R_{sb’} = R_{sb} e^{[\hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}\theta]}Rsb′​=Rsb​Ë[ω^θ] represents the orientation of a new frame {b’} relative to {小号} after the frame {b} has been rotated by \thetaθ about an axis w represented in the {小号} frame as \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}ω^.
• R_{sb’} = e^{[\hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}\theta]} R_{sb} Rsb′​=Ë[ω^θ]Rsb​ represents the orientation of a new frame {b’} relative to {小号} after the frame {b} has been rotated by \thetaθ about an axis w represented in the {b} frame as \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}ω^.
• R_{sb’} = e^{[\hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}\theta]} R_{sb} Rsb′​=Ë[ω^θ]Rsb​ represents the orientation of a new frame {b’} relative to {小号} after the frame {b} has been rotated by \thetaθ about an axis w represented in the {小号} frame as \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}ω^.

Q2. The simple closed-form solution to the infinite series for the matrix exponential when the matrix is an element of so(3)所以(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)所以(3)) and the skew-symmetric representation of the exponential coordinates (elements of so(3)所以(3)), which can also be thought of as the so(3)所以(3) representation of the angular velocity followed for unit time. Which of the following statements is correct? 选择所有符合条件的.

• exp: 所以(3) \rightarrow SO(3)所以(3)→所以(3)
• exp: 所以(3) \rightarrow so(3)所以(3)→所以(3)
• 日志: 所以(3) \rightarrow SO(3)所以(3)→所以(3)
• 日志: 所以(3) \rightarrow so(3)所以(3)→所以(3)

测验 07: 章节 3 通过 3.2, Rigid-Body Motions

第一季度. In terms of the \hat{X}_{\textrm{小号}}X^s​, \hat{和}_{\textrm{小号}}和^​s​, \hat{和}_{\textrm{小号}}和^s​ coordinates of a fixed space frame {小号}, the frame {一个} has its \hat{X}_{\textrm{一个}}X^a​-axis pointing in the direction (0,0,1)(0,0,1) and its \hat{和}_{\textrm{一个}}和^​a​-axis pointing in the direction (1,0,0)(1,0,0), and the frame {b} has its \hat{X}_{\textrm{b}}X^b​-axis pointing in the direction (1,0,0)(1,0,0) and its \hat{和}_{\textrm{b}}和^​b​-axis pointing in the direction (0,0,-1)(0,0,−1). Draw the {小号}, {一个}, 和 {b} 框架, similar to examples in the book and videos (例如, 数字 3.7 in the book), for easy reference in this question and later questions.

Write the rotation matrix R_{sa}[R小号一个​. All elements of this matrix should be integers.

如果你的答案是

\剩下[

147258369

\对]⎣⎢⎡​147​258​369​⎦⎥⎤​

例如, 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}[R小号b−1​. All elements of this matrix should be integers.

如果你的答案是

\剩下[

147258369

\对]⎣⎢⎡​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}[R一个b​. All elements of this matrix should be integers.

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

第四季度. Referring back to Question 1, let R = R_{sb}[R=Rsb​ be considered as a transformation operator consisting of a rotation about \hat{X}X^ by -90^\circ−90∘. Calculate R_1 = R_{sa} [R[R1​=Rsa​[R, and think of R_{sa}Rsa​ as the representation of the initial orientation of {一个} relative to {小号}, [R[R as a rotation operation, and R_1[R1​ as the new orientation of {一个} after performing the rotation. The new orientation R_1[R1​ corresponds to the orientation of the new {一个} frame relative to {小号} after rotating the original {一个} frame by -90^\circ−90∘ about which axis?

• The \hat{X}_{\textrm{一个}}X^a​-axis of the {一个} 框架.
• The \hat{X}_{\textrm{小号}}X^s​-axis of the {小号} 框架.

Q5. Referring back to Question 1, use R_{sb}[R小号b​ to change the representation of the point p_b = (1,2,3)^\intercalpb​=(1,2,3)⊺ (在 {b} coordinates) 至 {小号} coordinates. All elements of this vector should be integers.

如果你的答案是

\剩下[

123

\对]⎣⎢⎡​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 {小号} coordinates. Calculate q = R^\intercal_{sb} p_sq=Rsb⊺​ps​. Is qq a representation of p in {b} coordinates?

• 是.
• 没有.

Q7. Referring back to Question 1, an angular velocity w瓦 is represented in {小号} as \omega_s = (3,2,1)^\intercalω小号​=(3,2,1)⊺. What is its representation \omega_aω一个​? All elements of this vector should be integers.

如果你的答案是

\剩下[

123

\对]⎣⎢⎡​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{\它们还被证明可以预防抑郁症并提高老年人的认知功能}]\theta[ω^]θ of R_{sa}Rsa​ by hand. (You may verify your answer with software.) Extract and enter the rotation amount \thetaθ in radians with at least two decimal places.

• 1
• 0

Q9. Calculate the matrix exponential corresponding to the exponential coordinates of rotation \hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}\theta = (1,2,0)^\intercalω^θ=(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 和 2, and click “Run.”

辅酶Q10. Write the 3 \times 33×3 skew-symmetric matrix corresponding to \omega = (1,2,0.5)^\intercalω=(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 和 2, and click “Run.”

Q11. Use the function {\tt MatrixExp3}MatrixExp3 in the given software to calculate the rotation matrix R \in SO(3)[R∈小号该(3) corresponding to the matrix exponential of

[\hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}] \theta = \left[

0−0.510.50−2−120

\对].[ω^]θ=⎣⎢⎡​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 和 2, and click “Run.”

Q12. Use the function {\tt MatrixLog3}MatrixLog3 in the given software to calculate the matrix logarithm [\hat{\它们还被证明可以预防抑郁症并提高老年人的认知功能}] \theta \in so(3)[ω^]θ∈小号哦(3) of rotation matrix

R = \left[

0−1000−1100

\对].[R=⎣⎢⎡​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 和 2, and click “Run.”

周 04: Modern Robotics, 课程 1: Foundations of Robot Motion Quiz Answers

测验 01 : Lecture Comprehension, Homogeneous Transformation Matrices (章节 3 通过 3.3.1)

第一季度. 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 常数?

• 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? 选择所有符合条件的.

• 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 {b} frame is p_b \in \mathbb{[R}^3pb​∈R3. To find the representation of this point in the {一个} 框架, we could write T_{ab} p_bTab​pb​, but there is a dimension mismatch; p_bpb​ has only 3 组件, 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.

第四季度. Which of these is a valid calculation of T_{ab}Tab​, the configuration of the frame {b} relative to {一个}? 选择所有符合条件的.

• 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

测验 02 : Lecture Comprehension, Twists (章节 3.3.2, 部分 1 的 2)

第一季度. 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, 一个 screw axis \mathcal{小号} = (\mathcal{小号}_\omega, \mathcal{小号}_v) \in \mathbb{[R}^6S=(小号ω​,小号v​)∈R6. The screw axis is a normalized representation of the direction of motion, and \dot{\theta}θ˙ represents how fast the body moves in that direction of motion, so that the twist is given by \mathcal{V} = \mathcal{小号}\点{\theta} \in \mathbb{[R}^6V=Sθ˙∈R6. The normalized screw axis for full spatial motions is analogous to the normalized (单元) angular velocity axis for pure rotations.

The pitch hH of the screw axis is defined as the ratio of the linear speed over the angular speed. Which of the following is true? 选择所有符合条件的.

• If the pitch hH is infinite, then \mathcal{小号}_\omega = 0Sω​=0 and \|\mathcal{小号}_v\| = 1∥Sv​∥=1.
• If the pitch hH is infinite, 然后 \|\mathcal{小号}_\omega\| = 1∥Sω​∥=1 and \mathcal{小号}_vSv​ is arbitrary.
• If the pitch hH is finite, then \mathcal{小号}_\omega = 0Sω​=0 and \|\mathcal{小号}_v\| = 1∥Sv​∥=1.
• If the pitch hH is finite, 然后 \|\mathcal{小号}_\omega\| = 1∥Sω​∥=1 and \mathcal{小号}_vSv​ 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 {b} has an \hat{{\rm x}}_bx^b​-axis pointing outward (away from the center of the turntable), a \hat{{\rm y}}_by^​b​-axis pointing in the direction the turntable is moving at your location (the direction your eyes are looking), and a \hat{{\rm z}}_bz^b​-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{小号} = (\mathcal{小号}_\omega, \mathcal{小号}_v)S=(小号ω​,小号v​) and the twist \mathcal{V} = (\它们还被证明可以预防抑郁症并提高老年人的认知功能,v)V =(ω,v) expressed in your body frame {b}? All angular velocities are in radians/second and all linear velocities are in meters/second.

• \mathcal{小号} = (0, 0, 0.1, 0, 0.3, 0), \;\; \mathcal{V} = (0, 0, 0.01, 0, 0.03, 0)S=(0,0,0.1,0,0.3,0),V =(0,0,0.01,0,0.03,0)
• \mathcal{小号} = (0, 0, 1, 0, 3, 0), \;\; \mathcal{V} = (0, 0, 0.1, 0, 0.3, 0)S=(0,0,1,0,3,0),V =(0,0,0.1,0,0.3,0)
• \mathcal{小号} = (1, 0, 0, 0, 3, 0), \;\; \mathcal{V} = (0.1, 0, 0, 0, 0.3, 0)S=(1,0,0,0,3,0),V =(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? 选择所有符合条件的.

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

测验 03 : Lecture Comprehension, Twists (章节 3.3.2, 部分 2 的 2)

第一季度. What is the dimension of the matrix adjoint representation [{\rm Ad}_T][AdŤ​] of a transformation matrix TŤ (an element of SE(3)SE(3))?

• 3×3
• 4×4
• 6×6

Q2. A 3-vector angular velocity \omegaω can be represented in matrix form as [\它们还被证明可以预防抑郁症并提高老年人的认知功能][ω], an element of so(3)所以(3), the set of 3×3 skew-symmetric matrices. Analogously, a 6-vector twist \mathcal{V} = (\它们还被证明可以预防抑郁症并提高老年人的认知功能,v)V =(ω,v) can be represented in matrix form as [\mathcal{V}][V], an element of se(3)se(3). What is the dimension of [\mathcal{V}][V]?

• 3×3
• 4×4
• 6×6

测验 04 : Lecture Comprehension, Exponential Coordinates of Rigid-Body Motion (章节 3.3.3)

第一季度. Although we use six numbers to represent a screw \mathcal{小号} = (\mathcal{小号}_\omega,\mathcal{小号}_v)S=(小号ω​,小号v​), the space of all screws is only 5-dimensional. 为什么?

• \mathcal{小号}_\omegaSω​ must be unit length.
• \mathcal{小号}_vSv​ must be unit length.
• Either \mathcal{小号}_\omegaSω​ or \mathcal{小号}_vSv​ must be unit length.

Q2. A transformation matrix T_{ab}Tab​, 代表 {b} relative to {一个}, can be represented using the 6-vector exponential coordinates \mathcal{小号}\thetaSθ, where \mathcal{小号}S is a screw axis (represented in {一个} coordinates) and \thetaθ is the distance followed along the screw axis that displaces {一个} 至 {b}. Which of the following is correct? 选择所有符合条件的.

• T_{ab} = e^{\mathcal{小号}\theta}Tab​=Ë小号θ
• T_{ab} = e^{[\mathcal{小号}]\theta}Tab​=Ë[小号]θ
• T_{ab} = e^{[\mathcal{小号}\theta]}Tab​=Ë[小号θ]
• T_{ab} = e^{\mathcal{小号}[\theta]}Tab​=Ë小号[θ]

Q3. The matrix representation of the exponential coordinates \mathcal{小号}\theta \in \mathbb{[R}^6Sθ∈R6 is [\mathcal{小号}\theta][小号θ]. What space does [\mathcal{小号}\theta][小号θ] belong to?

• 所以(3)
• 所以(3)
• SE(3)
• se(3)

第四季度. T_{ab’} = T_{ab} e^{[\mathcal{小号}\theta]}Tab′​=Tab​Ë[小号θ] is a representation of the new frame {b’} (relative to {一个}) achieved after {b} has followed

• the screw axis \mathcal{小号}小号, expressed in {b} coordinates, a distance \thetaθ.
• the screw axis \mathcal{小号}小号, expressed in {一个} coordinates, a distance \thetaθ.

Q5. T_{ab’} = e^{[\mathcal{小号}\theta]} T_{ab}Tab′​=Ë[小号θ]Tab​ is a representation of the new frame {b’} (relative to {一个}) achieved after {b} has followed

• the screw axis \mathcal{小号}小号, expressed in {b} coordinates, a distance \thetaθ.
• the screw axis \mathcal{小号}小号, expressed in {一个} coordinates, a distance \thetaθ.

Q6. Which of the following statements is true? 选择所有符合条件的.

• The matrix exponential maps [\mathcal{小号}\theta] \in se(3)[小号θ]∈se(3) to a transformation matrix T \in SE(3)Ť∈SE(3), where TŤ is the representation of the frame (relative to {小号}) that is achieved by following the screw \mathcal{小号}小号 (expressed in {小号}) a distance \thetaθ from the identity configuration (即, a frame initially coincident with {小号}).
• The matrix exponential maps [\mathcal{V}] \in se(3)[V]∈se(3) to a transformation matrix T \in SE(3)Ť∈SE(3), where TŤ is the representation of the frame (relative to {小号}) that is achieved by following the twist \mathcal{V}V (expressed in {小号}) for unit time from the identity configuration (即, a frame initially coincident with {小号}).
• 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).

测验 05 : Lecture Comprehension, Wrenches (章节 3.4)

第一季度. A wrench \mathcal{F}_aF一个​ consists of a linear force f_a \in \mathbb{[R}^3fa​∈R3 and a moment m_a \in \mathbb{[R}^3ma​∈R3, both expressed in the frame {一个}. How do we usually write the wrench?

• \mathcal{F}_a = (m_a,f_a)F一个​=(ma​,fa​)
• \mathcal{F}_a = (f_a,m_a)F一个​=(fa​,ma​)

Q2. We know that the power associated with a wrench and twist pair (\mathcal{F},\mathcal{V})(F,V) does not depend on whether they are represented in the frame {一个} 如 (\mathcal{F}_a,\mathcal{V}_a)(F一个​,V一个​) or the frame {b} 如 (\mathcal{F}_b,\mathcal{V}_b)(Fb​,Vb​). 因此, we can write \mathcal{F}_a^{\rm T} \mathcal{V}_a = \mathcal{F}_b^{\rm T} \mathcal{V}_bF一个T​V一个​=FbT​Vb​ and then use which identity to derive the equation \mathcal{F}_a = [{\rm Ad}_{T_{ba}}]^{\rm T} \mathcal{F}_bF一个​=[AdTba​​]TFb​ relating the representations \mathcal{F}_aF一个​ and \mathcal{F}_bFb​? (也, remember the matrix identity (从)^{\rm T} = B^{\rm T} A^{\rm T}(从)T=乙Ť一个T.)

• \mathcal{V}_a = T_{ab} \mathcal{V}_bV一个​=Tab​Vb​
• \mathcal{V}_a = T_{ba} \mathcal{V}_bV一个​=Tba​Vb​
• \mathcal{V}_a = [{\rm Ad}_{T_{ba}}] \mathcal{V}_bV一个​=[AdTba​​]Vb​
• \mathcal{V}_a = [{\rm Ad}_{T_{ab}}] \mathcal{V}_bV一个​=[AdTab​​]Vb​

测验 06 : 章节 3.3 和 3.4, Rigid-Body Motions

第一季度. In terms of the \hat{X}_{\textrm{小号}}X^s​, \hat{和}_{\textrm{小号}}和^​s​, \hat{和}_{\textrm{小号}}和^s​ coordinates of a fixed space frame {小号}, the frame {一个} has its \hat{X}_{\textrm{一个}}X^a​-axis pointing in the direction (0,0,1)(0,0,1) and its \hat{和}_{\textrm{一个}}和^​a​-axis pointing in the direction (-1,0,0)(−1,0,0), and frame {b} has its \hat{X}_{\textrm{b}}X^b​-axis pointing in the direction (1,0,0)(1,0,0) and its \hat{和}_{\textrm{b}}和^​b​-axis pointing in the direction (0,0,-1)(0,0,−1). The origin of {一个} is at (0,0,1)(0,0,1) 在 {小号} and the origin of {b} is at (0,2,0)(0,2,0). Draw the {小号}, {一个}, 和 {b} 框架, similar to examples in the book and videos, for easy reference in this question and later questions.

Write the transformation matrix T_{sa}Ť小号一个​. 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

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

Q2. Referring back to Question 1, write T_{sb}^{-1}Ť小号b−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

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

Q3. Referring back to Question 1, write T_{ab}Ť一个b​. 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

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

第四季度. Referring back to Question 1, let T = T_{sb}Ť=Ť小号b​ be considered as a transformation operator consisting of a rotation about \hat{X}X^ by -90^\circ−90∘ and a translation along \hat{和}和^​ by 2 单位. Calculate T_1 = T T_{sa}Ť1​=ŤŤ小号一个​, and think of T_{sa}Ť小号一个​ as the representation of the initial configuration of {一个} relative to {小号}, ŤŤ as a transformation operation, and T_1Ť1​ as the new configuration of {一个} after performing the transformation. Are the rotation axis \hat{X}X^ and translation axis \hat{和}和^​ of the transformation TŤ properly considered to be expressed in the frame {小号} or the frame {一个}?

1 观点

• The frame {小号}.
• The frame {一个}.

Q5. Referring back to Question 1, use T_{sb}Ť小号b​ to change the representation of the point p_b = (1,2,3)^\intercalpb​=(1,2,3)⊺ (在 {b} coordinates) 至 {小号} 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

\对]⎣⎢⎡​123​⎦⎥⎤​.

Q6. Referring back to Question 1, choose a point p represented by p_s = (1,2,3)^\intercalp小号​=(1,2,3)⊺ in {小号} coordinates. Calculate q = T_{sb} p_sq=Ť小号b​p小号​. Is qq a representation of p in {b} coordinates?

1 观点

• 是
• 没有

Q7. Referring back to Question 1, a twist \mathcal{V}V is represented in {小号} 如 {\mathcal V}_s = (3,2,1,-1,-2,-3)^\intercalV小号​=(3,2,1,−1,−2,−3)⊺. What is its representation {\mathcal V}_aV一个​? 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

\对]⎣⎢⎢⎢⎢⎢⎢⎢⎡​123456​⎦⎥⎥⎥⎥⎥⎥⎥⎤​.

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

Q9. Calculate the matrix exponential corresponding to the exponential coordinates of rigid-body motion {\mathcal S}\theta = (0,1,2,3,0,0)^\intercalSθ=(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

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

辅酶Q10. Referring back to Question 1, use T_{sb}Ť小号b​ to change the representation of the wrench Fb=(1,0,0,2,1,0)⊺ (在 {b} coordinates) 至 {小号} 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

\对]⎣⎢⎡​123​⎦⎥⎤​.

Q11. Use the function {\tt TransInv}TransInv in the given software to calculate the inverse of the homogeneous transformation matrix

T = \left[

0100−100000103011

\对].Ť=⎣⎢⎢⎢⎡​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

\对]⎣⎢⎡​147​258​369​⎦⎥⎤​.

Q12. Write the se(3)小号Ë(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

\对]⎣⎢⎡​147​258​369​⎦⎥

Q13. 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{小号} = (1,0,0)小号^=(1,0,0) in the direction of the screw axis, located at the point p = (0,0,2)p=(0,0,2), with pitch h = 1H=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

\对]⎣⎢⎡​123​⎦⎥⎤​.

Q14. Use the function {\tt MatrixExp6}MatrixExp6 in the given software to calculate the homogeneous transformation matrix T \in SE(3)Ť∈小号Ë(3) corresponding to the matrix exponential of

[小号]θ=⎡⎣⎢⎢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

\对]⎣⎢⎡​147​258​369​⎦⎥⎤​.

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

T = \left[

0100−100000103011

\对].Ť=⎣⎢⎢⎢⎡​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

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