الروبوتات الحديثة, دورة 1: Foundations of Robot Motion Quizzes & الإجابات - كورسيرا

Take a journey into the world of الروبوتات with interesting quizzes and expert answers on Foundations of Robot Motion في Modern علم الروبوتات, دورة 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 الروبوتات enthusiast looking to deepen your knowledge or a student looking to explore the exciting field of الروبوتات, 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.

لغز 01: Lecture Comprehension, Degrees of Freedom of a Rigid Body (الفصل 2 عبر 2.1)

Q1. Which of the following are possible elements of robots in this specialization? اختر كل ما ينطبق.

• Rigid bodies.
• ناعم, flexible bodies.
• Joints.

Q2. The number of degrees of freedom of a robot is (اختر كل ما ينطبق):

• 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 nن-dimensional space has mم total degrees of freedom. How many of these mم degrees of freedom are angular (not linear)? اختر كل ما ينطبق. (This is consistently one of the most incorrectly answered questions in this course, so think about it carefully!)

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

لغز 02: Lecture Comprehension, Degrees of Freedom of a Robot (الفصل 2.2)

Q1. Consider a joint between two rigid bodies. Each rigid body has mم degrees of freedom (m=3م=3 for a planar rigid body and m=6م=6 for a spatial rigid body) in the absence of any constraints. The joint has fF degrees of freedom (مثلا, 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 mم and fF

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

لغز 03: الفصل 2 عبر 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, لديها 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 nن such arms grasping the object. What is the number of degrees of freedom of this system? Your answer should be a mathematical expression including nن. Examples of mathematical expressions including nن are 4*n-74∗ن−7 or n/3ن/3

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

أسبوع 02: الروبوتات الحديثة, دورة 1: Foundations of Robot Motion Quiz Answers

لغز 01: Lecture Comprehension, Configuration Space Topology (الفصل 2.3.1)

Q1. To deform one nن-dimensional space into another topologically equivalent space, which operations are you allowed to use? اختر كل ما ينطبق.

• Stretching
• Cutting.
• Gluing.

Q2. True or false? An nن-dimensional space can be topologically equivalent to an mم-dimensional space, where m \neq nم​=ن.

• صحيح.
• خاطئة.

لغز 02: Lecture Comprehension, Configuration Space Representation (الفصل 2.3.2)

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

صحيح.

خاطئة

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

لغز 02: Lecture Comprehension, Configuration and Velocity Constraints (الفصل 2.4)

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

• صحيح.
• خاطئة.

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

• صحيح.
• خاطئة.

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?)

لغز 03: Lecture Comprehension, Task Space and Workspace (الفصل 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.

• صحيح.
• خاطئة.

لغز 04: الفصل 2.3 عبر 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) س=cosθ1​+2cos(θ1​+θ2​)

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

(بعبارات أخرى, حلقة الوصل 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.)

• [ا,ب] \مرات [ا,ب] \times S^1[ا,ب]×[ا,ب]×S1
• [ا,ب] \times \mathbb{R}^1 \times S^1[ا,ب]×R1×S1
• [ا,ب] \مرات [ا,ب] \times \mathbb{R}^1[ا,ب]×[ا,ب]×R1
• \mathbb{R}^2 \times S^1R2×S1

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×S3
• \mathbb{R}^3 \times T^3R3×تي3
• \mathbb{R}^3 \times T^2 \times S^1R3×تي2×S1
• \mathbb{R}^3 \times S^2 \times S^1R3×S2×S1

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{R}^3 \times T^{10}R3×تي10
• \mathbb{R}^3 \times S^2 \times T^8R3×S2×تي8
• \mathbb{R}^3 \times S^3 \times T^7R3×S3×تي7
• \mathbb{R}^4 \times S^2 \times T^7R4×S2×تي7

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 [ا,ب]R2×S2×S1×[ا,ب]
• \mathbb{R}^2 \times S^3 \times [ا,ب]R2×S3×[ا,ب]
• \mathbb{R}^2 \times T^3 \times [ا,ب]R2×تي3×[ا,ب]
• \mathbb{R}^3 \times T^3R3×تي3

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

(1+ \cos q_1) \نقطة{ف}_1 + (2+ \sin q_2) \نقطة{ف}_2 + (\cos q_1+ \sin q_2 + 3) \نقطة{ف}_3 = 0.(1+cosف1​)ف˙​1​+(2+sinف2​)ف˙​2​+(cosف1​+sinف2​+3)ف˙​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) يملك? (Enter an integer number.)

أسبوع 03: الروبوتات الحديثة, دورة 1: Foundations of Robot Motion Quiz Answers

لغز 01: Lecture Comprehension, Introduction to Rigid-Body Motions (الفصل 3 عبر 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, و, and z axes of a coordinate frame, على التوالي?

• Thumb, فهرس, وسط
• Middle, فهرس, thumb
• فِهرِس, وسط, 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 {ب}, لان

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

لغز 02: Lecture Comprehension, Rotation Matrices (الفصل 3.2.1, جزء 1 من 2)

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

• the rows are the x, و, z axes of {ا} written in {ب} coordinates.
• the columns are the x, و, z axes of {ا} written in {ب} coordinates.
• the rows are the x, و, z axes of {ب} written in {ا} coordinates.
• the columns are the x, و, z axes of {ب} written in {ا} coordinates.

Q2. ال 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​, أي, R_{ab}^{-1}Rab−1​, هو (اختر كل ما ينطبق):

• -R_{ab}-Rab​
• R_{ab}^{\rm T}RabT​
• R-IR-أنا
• R_{ba}Rba​

Q4. Multiplication of SO(3)وبالتالي(3) rotation matrices is (اختر كل ما ينطبق):

• associative.
• commutative.

لغز 03: Lecture Comprehension, Rotation Matrices (الفصل 3.2.1, جزء 2 من 2)

Q1. Which of the following is equivalent to R_{ac}Rac​, the representation of the orientation of the {ج} frame relative to the {ا} الإطار? اختر كل ما ينطبق

• 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​)تي
• R_{ad} R_{db} R_{bc}Rad​Rdb​Rbc​

Q2. The matrix

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

1000010−10

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

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

• R_{sb} RRsb​R
• R R_{sb}RRsb​

Q3. The matrix

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

1000010−10

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

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

• R_{sb}RRsb​R
• R R_{sb}RRsb​

لغز 04: Lecture Comprehension, Angular Velocities (الفصل 3.2.2)

Q1. Our representation of the three-dimensional orientation uses an implicit representation (a 3×3 SO(3) matrix with 9 أعداد), but our usual representation of the angular velocity uses only three numbers, أي, 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, من ناحية أخرى, 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
• وبالتالي(3)وبالتالي(3)
• وبالتالي(3)وبالتالي(3)

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

• \mathbb{R}^3R3
• وبالتالي(3)وبالتالي(3)
• وبالتالي(3)وبالتالي(3)

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

• \mathbb{R}^3R3
• وبالتالي(3)وبالتالي(3)
• وبالتالي(3)وبالتالي(3)

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

• 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×ص can be written [\أوميغا] ص[ω]ص, أين [\أوميغا][ω] هو

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

لغز 05: Lecture Comprehension, Exponential Coordinates of Rotation (الفصل 3.2.3, جزء 1 من 2)

Q1. The orientation of a frame {د} relative to a frame {ج} can be represented by a unit rotation axis \hat{\أوميغا}ω^ and the distance \thetaθ rotated about the axis. If we rotate the frame {ج} by \thetaθ about the axis \hat{\أوميغا}ω^ expressed in the {ج} الإطار, we end up at {د}. The vector \hat{\أوميغا}ω^ has 3 numbers and \thetaθ هو 1 رقم, but we only need 3 أعداد, the exponential coordinates \hat{\أوميغا} \thetaω^θ, to represent {د} relative to {ج}, لان

• though we use 3 numbers to represent \hat{\أوميغا}ω^, \hat{\أوميغا}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{\أوميغا}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? اختر كل ما ينطبق.

• 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{س}(ر) = Bx(ر)س˙(ر)=Bx(ر), where xس is a vector and Bب is a constant square matrix, is solved as x(ر) = e^{Bt} س(0)س(ر)=eBtx(0), where the matrix exponential e^{Bt}eBt يعرف ب

• 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+....

لغز 06: Lecture Comprehension, Exponential Coordinates of Rotation (الفصل 3.2.3, جزء 2 من 2)

Q1. The solution to the differential equation \dot{ص}(ر) = \hat{\أوميغا} \times p(ر) = [\hat{\أوميغا}] ص(ر)ص˙​(ر)=ω^×ص(ر)=[ω^]ص(ر) is p(ر) = e^{[\hat{\أوميغا}\theta]}ص(0)ص(ر)=البريد[ω^θ]ص(0), where p(0)ص(0) is the initial vector and p(ر)ص(ر) is the vector after it has been rotated at the angular velocity \hat{\أوميغا}ω^ for time t=\thetaر=θ (where \hat{\أوميغا}\thetaω^θ are the exponential coordinates). You can think of R = e^{[\hat{\أوميغا}\theta]}R=البريد[ω^θ] as the rotation operation that moves p(0)ص(0) to p(ر) = p(\theta)ص(ر)=ص(θ).

Which of the following statements is correct? اختر كل ما ينطبق.

• R_{sb’} = R_{sb} e^{[\hat{\أوميغا}\theta]}Rsb′​=Rsb​البريد[ω^θ] represents the orientation of a new frame {b’} relative to {الصورة} after the frame {ب} has been rotated by \thetaθ about an axis w represented in the {ب} frame as \hat{\أوميغا}ω^.
• R_{sb’} = R_{sb} e^{[\hat{\أوميغا}\theta]}Rsb′​=Rsb​البريد[ω^θ] represents the orientation of a new frame {b’} relative to {الصورة} after the frame {ب} has been rotated by \thetaθ about an axis w represented in the {الصورة} frame as \hat{\أوميغا}ω^.
• R_{sb’} = e^{[\hat{\أوميغا}\theta]} R_{sb} Rsb′​=البريد[ω^θ]Rsb​ represents the orientation of a new frame {b’} relative to {الصورة} after the frame {ب} has been rotated by \thetaθ about an axis w represented in the {ب} frame as \hat{\أوميغا}ω^.
• R_{sb’} = e^{[\hat{\أوميغا}\theta]} R_{sb} Rsb′​=البريد[ω^θ]Rsb​ represents the orientation of a new frame {b’} relative to {الصورة} after the frame {ب} has been rotated by \thetaθ about an axis w represented in the {الصورة} frame as \hat{\أوميغا}ω^.

Q2. The simple closed-form solution to the infinite series for the matrix exponential when the matrix is an element of so(3)وبالتالي(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)وبالتالي(3)) and the skew-symmetric representation of the exponential coordinates (elements of so(3)وبالتالي(3)), which can also be thought of as the so(3)وبالتالي(3) representation of the angular velocity followed for unit time. Which of the following statements is correct? اختر كل ما ينطبق.

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

لغز 07: الفصل 3 عبر 3.2, Rigid-Body Motions

Q1. In terms of the \hat{س}_{\textrm{الصورة}}س^s​, \hat{و}_{\textrm{الصورة}}و^​s​, \hat{ض}_{\textrm{الصورة}}ض^s​ coordinates of a fixed space frame {الصورة}, the frame {ا} has its \hat{س}_{\textrm{ا}}س^a​-axis pointing in the direction (0,0,1)(0,0,1) and its \hat{و}_{\textrm{ا}}و^​a​-axis pointing in the direction (1,0,0)(1,0,0), and the frame {ب} has its \hat{س}_{\textrm{ب}}س^b​-axis pointing in the direction (1,0,0)(1,0,0) and its \hat{و}_{\textrm{ب}}و^​b​-axis pointing in the direction (0,0,-1)(0,0,−1). Draw the {الصورة}, {ا}, و {ب} إطارات, similar to examples in the book and videos (مثلا, الشكل 3.7 in the book), for easy reference in this question and later questions.

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

If your answer is

\اليسار[

147258369

\حق]⎣⎢⎡​147​258​369​⎦⎥⎤​

فمثلا, 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}Rالصورةب−1​. All elements of this matrix should be integers.

If your answer is

\اليسار[

147258369

\حق]⎣⎢⎡​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}Rاب​. All elements of this matrix should be integers.

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

Q4. Referring back to Question 1, let R = R_{sb}R=Rsb​ be considered as a transformation operator consisting of a rotation about \hat{س}س^ by -90^\circ−90∘. Calculate R_1 = R_{sa} RR1​=Rsa​R, and think of R_{sa}Rsa​ as the representation of the initial orientation of {ا} relative to {الصورة}, RR as a rotation operation, and R_1R1​ as the new orientation of {ا} after performing the rotation. The new orientation R_1R1​ corresponds to the orientation of the new {ا} frame relative to {الصورة} after rotating the original {ا} frame by -90^\circ−90∘ about which axis?

• The \hat{س}_{\textrm{ا}}س^a​-axis of the {ا} الإطار.
• The \hat{س}_{\textrm{الصورة}}س^s​-axis of the {الصورة} الإطار.

Q5. Referring back to Question 1, use R_{sb}Rالصورةب​ to change the representation of the point p_b = (1,2,3)^\intercalصب​=(1,2,3)⊺ (في {ب} coordinates) إلى {الصورة} coordinates. All elements of this vector should be integers.

If your answer is

\اليسار[

123

\حق]⎣⎢⎡​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)^\intercalملاحظة​=(1,2,3)⊺ in {الصورة} coordinates. Calculate q = R^\intercal_{sb} p_sف=Rsb⊺​ملاحظة​. Is qف a representation of p in {ب} coordinates?

• نعم فعلا.
• لا.

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

If your answer is

\اليسار[

123

\حق]⎣⎢⎡​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{\أوميغا}]\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{\أوميغا}\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 و 2, and click “Run.”

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 و 2, and click “Run.”

سوف تحتاج إلى تحقيق ما لا يقل عن. Use the function {\tt MatrixExp3}MatrixExp3 in the given software to calculate the rotation matrix R \in SO(3)R∈Sال(3) corresponding to the matrix exponential of

[\hat{\أوميغا}] \theta = \left[

0−0.510.50−2−120

\حق].[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 و 2, and click “Run.”

س 12. Use the function {\tt MatrixLog3}MatrixLog3 in the given software to calculate the matrix logarithm [\hat{\أوميغا}] \theta \in so(3)[ω^]θ∈الصورةا(3) of rotation matrix

R = \left[

0−1000−1100

\حق].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 و 2, and click “Run.”

أسبوع 04: الروبوتات الحديثة, دورة 1: Foundations of Robot Motion Quiz Answers

لغز 01 : Lecture Comprehension, Homogeneous Transformation Matrices (الفصل 3 عبر 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 نظام الري الذكي?

• 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? اختر كل ما ينطبق.

• 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 {ب} frame is p_b \in \mathbb{R}^3pb​∈R3. To find the representation of this point in the {ا} الإطار, we could write T_{ab} p_bTab​pb​, but there is a dimension mismatch; p_bpb​ has only 3 المكونات, 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 {ب} relative to {ا}? اختر كل ما ينطبق.

• 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

لغز 02 : Lecture Comprehension, Twists (الفصل 3.3.2, جزء 1 من 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, ا screw axis \mathcal{S} = (\mathcal{S}_\omega, \mathcal{S}_v) \in \mathbb{R}^6S=(Sω​,Sالخامس​)∈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{خامسا} = \mathcal{S}\نقطة{\theta} \in \mathbb{R}^6V=Sθ˙∈R6. The normalized screw axis for full spatial motions is analogous to the normalized (وحدة) angular velocity axis for pure rotations.

The pitch hح of the screw axis is defined as the ratio of the linear speed over the angular speed. Which of the following is true? اختر كل ما ينطبق.

• If the pitch hح is infinite, then \mathcal{S}_\omega = 0Sω​=0 and \|\mathcal{S}_v\| = 1∥Sالخامس​∥=1.
• If the pitch hح is infinite, ثم \|\mathcal{S}_\omega\| = 1∥Sω​∥=1 and \mathcal{S}_vSالخامس​ is arbitrary.
• If the pitch hح is finite, then \mathcal{S}_\omega = 0Sω​=0 and \|\mathcal{S}_v\| = 1∥Sالخامس​∥=1.
• If the pitch hح is finite, ثم \|\mathcal{S}_\omega\| = 1∥Sω​∥=1 and \mathcal{S}_vSالخامس​ 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 {ب} has an \hat{{\rm x}}_bx^ب​-axis pointing outward (away from the center of the turntable), a \hat{{\rm y}}_by^​ب​-axis pointing in the direction the turntable is moving at your location (the direction your eyes are looking), and a \hat{{\rm z}}_bz^ب​-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{S} = (\mathcal{S}_\omega, \mathcal{S}_v)S=(Sω​,Sالخامس​) and the twist \mathcal{خامسا} = (\أوميغا,الخامس)الخامس =(ω,الخامس) expressed in your body frame {ب}? All angular velocities are in radians/second and all linear velocities are in meters/second.

• \mathcal{S} = (0, 0, 0.1, 0, 0.3, 0), \;\; \mathcal{خامسا} = (0, 0, 0.01, 0, 0.03, 0)S=(0,0,0.1,0,0.3,0),الخامس =(0,0,0.01,0,0.03,0)
• \mathcal{S} = (0, 0, 1, 0, 3, 0), \;\; \mathcal{خامسا} = (0, 0, 0.1, 0, 0.3, 0)S=(0,0,1,0,3,0),الخامس =(0,0,0.1,0,0.3,0)
• \mathcal{S} = (1, 0, 0, 0, 3, 0), \;\; \mathcal{خامسا} = (0.1, 0, 0, 0, 0.3, 0)S=(1,0,0,0,3,0),الخامس =(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? اختر كل ما ينطبق.

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

لغز 03 : Lecture Comprehension, Twists (الفصل 3.3.2, جزء 2 من 2)

Q1. What is the dimension of the matrix adjoint representation [{\rm Ad}_T][Adتي​] of a transformation matrix Tتي (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 [\أوميغا][ω], an element of so(3)وبالتالي(3), the set of 3×3 skew-symmetric matrices. Analogously, a 6-vector twist \mathcal{خامسا} = (\أوميغا,الخامس)الخامس =(ω,الخامس) can be represented in matrix form as [\mathcal{خامسا}][خامسا], an element of se(3)se(3). What is the dimension of [\mathcal{خامسا}][خامسا]?

• 3×3
• 4×4
• 6×6

لغز 04 : Lecture Comprehension, Exponential Coordinates of Rigid-Body Motion (الفصل 3.3.3)

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

• \mathcal{S}_\omegaSω​ must be unit length.
• \mathcal{S}_vSالخامس​ must be unit length.
• Either \mathcal{S}_\omegaSω​ or \mathcal{S}_vSالخامس​ must be unit length.

Q2. A transformation matrix T_{ab}Tab​, تمثل {ب} relative to {ا}, can be represented using the 6-vector exponential coordinates \mathcal{S}\thetaSθ, where \mathcal{S}S is a screw axis (represented in {ا} coordinates) and \thetaθ is the distance followed along the screw axis that displaces {ا} إلى {ب}. Which of the following is correct? اختر كل ما ينطبق.

• T_{ab} = e^{\mathcal{S}\theta}Tab​=البريدSθ
• T_{ab} = e^{[\mathcal{S}]\theta}Tab​=البريد[S]θ
• T_{ab} = e^{[\mathcal{S}\theta]}Tab​=البريد[Sθ]
• T_{ab} = e^{\mathcal{S}[\theta]}Tab​=البريدS[θ]

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

• وبالتالي(3)
• وبالتالي(3)
• SE(3)
• se(3)

Q4. T_{ab’} = T_{ab} e^{[\mathcal{S}\theta]}Tab′​=Tab​البريد[Sθ] is a representation of the new frame {b’} (relative to {ا}) achieved after {ب} has followed

• the screw axis \mathcal{S}S, expressed in {ب} coordinates, a distance \thetaθ.
• the screw axis \mathcal{S}S, expressed in {ا} coordinates, a distance \thetaθ.

Q5. T_{ab’} = e^{[\mathcal{S}\theta]} T_{ab}Tab′​=البريد[Sθ]Tab​ is a representation of the new frame {b’} (relative to {ا}) achieved after {ب} has followed

• the screw axis \mathcal{S}S, expressed in {ب} coordinates, a distance \thetaθ.
• the screw axis \mathcal{S}S, expressed in {ا} coordinates, a distance \thetaθ.

Q6. Which of the following statements is true? اختر كل ما ينطبق.

• The matrix exponential maps [\mathcal{S}\theta] \in se(3)[Sθ]∈se(3) to a transformation matrix T \in SE(3)تي∈SE(3), where Tتي is the representation of the frame (relative to {الصورة}) that is achieved by following the screw \mathcal{S}S (expressed in {الصورة}) a distance \thetaθ from the identity configuration (أي, a frame initially coincident with {الصورة}).
• The matrix exponential maps [\mathcal{خامسا}] \in se(3)[خامسا]∈se(3) to a transformation matrix T \in SE(3)تي∈SE(3), where Tتي is the representation of the frame (relative to {الصورة}) that is achieved by following the twist \mathcal{خامسا}خامسا (expressed in {الصورة}) for unit time from the identity configuration (أي, a frame initially coincident with {الصورة}).
• 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).

لغز 05 : Lecture Comprehension, Wrenches (الفصل 3.4)

Q1. A wrench \mathcal{F}_aFا​ 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 {ا}. How do we usually write the wrench?

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

Q2. We know that the power associated with a wrench and twist pair (\mathcal{F},\mathcal{خامسا})(F,خامسا) does not depend on whether they are represented in the frame {ا} مثل (\mathcal{F}_a,\mathcal{خامسا}_a)(Fا​,خامساا​) or the frame {ب} مثل (\mathcal{F}_b,\mathcal{خامسا}_b)(Fب​,خامساب​). وبالتالي, we can write \mathcal{F}_a^{\rm T} \mathcal{خامسا}_a = \mathcal{F}_b^{\rm T} \mathcal{خامسا}_bFاT​Vا​=FبT​Vب​ and then use which identity to derive the equation \mathcal{F}_a = [{\rm Ad}_{T_{ba}}]^{\rm T} \mathcal{F}_bFا​=[AdTba​​]TFب​ relating the representations \mathcal{F}_aFا​ and \mathcal{F}_bFب​? (أيضا, remember the matrix identity (من عند)^{\rm T} = B^{\rm T} A^{\rm T}(من عند)T=بتياT.)

• \mathcal{خامسا}_a = T_{ab} \mathcal{خامسا}_bVا​=Tab​Vب​
• \mathcal{خامسا}_a = T_{ba} \mathcal{خامسا}_bVا​=Tba​Vب​
• \mathcal{خامسا}_a = [{\rm Ad}_{T_{ba}}] \mathcal{خامسا}_bVا​=[AdTba​​]خامساب​
• \mathcal{خامسا}_a = [{\rm Ad}_{T_{ab}}] \mathcal{خامسا}_bVا​=[AdTab​​]خامساب​

لغز 06 : Chapters 3.3 و 3.4, Rigid-Body Motions

Q1. In terms of the \hat{س}_{\textrm{الصورة}}س^s​, \hat{و}_{\textrm{الصورة}}و^​s​, \hat{ض}_{\textrm{الصورة}}ض^s​ coordinates of a fixed space frame {الصورة}, the frame {ا} has its \hat{س}_{\textrm{ا}}س^a​-axis pointing in the direction (0,0,1)(0,0,1) and its \hat{و}_{\textrm{ا}}و^​a​-axis pointing in the direction (-1,0,0)(−1,0,0), and frame {ب} has its \hat{س}_{\textrm{ب}}س^b​-axis pointing in the direction (1,0,0)(1,0,0) and its \hat{و}_{\textrm{ب}}و^​b​-axis pointing in the direction (0,0,-1)(0,0,−1). The origin of {ا} is at (0,0,1)(0,0,1) في {الصورة} and the origin of {ب} is at (0,2,0)(0,2,0). Draw the {الصورة}, {ا}, و {ب} إطارات, similar to examples in the book and videos, for easy reference in this question and later questions.

Write the transformation matrix T_{sa}تيالصورةا​. 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

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

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

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

Q3. Referring back to Question 1, write T_{ab}تياب​. 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

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

Q4. Referring back to Question 1, let T = T_{sb}تي=تيالصورةب​ be considered as a transformation operator consisting of a rotation about \hat{س}س^ by -90^\circ−90∘ and a translation along \hat{و}و^​ by 2 وحدات. Calculate T_1 = T T_{sa}تي1​=تيتيالصورةا​, and think of T_{sa}تيالصورةا​ as the representation of the initial configuration of {ا} relative to {الصورة}, تيتي as a transformation operation, and T_1تي1​ as the new configuration of {ا} after performing the transformation. Are the rotation axis \hat{س}س^ and translation axis \hat{و}و^​ of the transformation Tتي properly considered to be expressed in the frame {الصورة} or the frame {ا}?

1 نقطة

• The frame {الصورة}.
• The frame {ا}.

Q5. Referring back to Question 1, use T_{sb}تيالصورةب​ to change the representation of the point p_b = (1,2,3)^\intercalصب​=(1,2,3)⊺ (في {ب} coordinates) إلى {الصورة} 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

\حق]⎣⎢⎡​123​⎦⎥⎤​.

Q6. Referring back to Question 1, choose a point p represented by p_s = (1,2,3)^\intercalصالصورة​=(1,2,3)⊺ in {الصورة} coordinates. Calculate q = T_{sb} p_sف=تيالصورةب​صالصورة​. Is qف a representation of p in {ب} coordinates?

1 نقطة

• نعم فعلا
• لا

Q7. Referring back to Question 1, a twist \mathcal{خامسا}V is represented in {الصورة} مثل {\mathcal V}_s = (3,2,1,-1,-2,-3)^\intercalVالصورة​=(3,2,1,−1,−2,−3)⊺. What is its representation {\mathcal V}_aVا​? 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

\حق]⎣⎢⎢⎢⎢⎢⎢⎢⎡​123456​⎦⎥⎥⎥⎥⎥⎥⎥⎤​.

Q8. Referring back to Question 1, calculate the matrix logarithm [{\mathcal S}]\theta[S]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

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

Q10. Referring back to Question 1, use T_{sb}تيالصورةب​ to change the representation of the wrench Fب=(1,0,0,2,1,0)⊺ (في {ب} coordinates) إلى {الصورة} 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

\حق]⎣⎢⎡​123​⎦⎥⎤​.

سوف تحتاج إلى تحقيق ما لا يقل عن. Use the function {\tt TransInv}TransInv in the given software to calculate the inverse of the homogeneous transformation matrix

T = \left[

0100−100000103011

\حق].تي=⎣⎢⎢⎢⎡​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

\حق]⎣⎢⎡​147​258​369​⎦⎥⎤​.

س 12. Write the se(3)الصورةالبريد(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

\حق]⎣⎢⎡​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{الصورة} = (1,0,0)الصورة^=(1,0,0) in the direction of the screw axis, located at the point p = (0,0,2)ص=(0,0,2), with pitch h = 1ح=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

\حق]⎣⎢⎡​123​⎦⎥⎤​.

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

[S]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

\حق]⎣⎢⎡​147​258​369​⎦⎥⎤​.

س 15. Use the function {\tt MatrixLog6}MatrixLog6 in the given software to calculate the matrix logarithm [S]θ∈الصورةالبريد(3) of the homogeneous transformation matrix

T = \left[

0100−100000103011

\حق].تي=⎣⎢⎢⎢⎡​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

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