Modern Robotics, पाठ्यक्रम 1: Foundations of Robot Motion Quizzes & उत्तर- कौरसेरा

की दुनिया में यात्रा करें रोबोटिक with interesting quizzes and expert answers on Foundations of Robot गति में Modern रोबोटिक, पाठ्यक्रम 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 रोबोटिक enthusiast looking to deepen your knowledge or a student looking to explore the exciting field of रोबोटिक, 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.

प्रश्न पूछना 01: Lecture Comprehension, Degrees of Freedom of a Rigid Body (अध्याय 2 के माध्यम से 2.1)

Q1. Which of the following are possible elements of robots in this specialization? Select all that apply.

• Rigid bodies.
• कोमल, flexible bodies.
• Joints.

Q2. The number of degrees of freedom of a robot is (select all that apply):

• 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 nएन-dimensional space has mएम total degrees of freedom. How many of these mएम degrees of freedom are angular (not linear)? Select all that apply. (This is consistently one of the most incorrectly answered questions in this course, so think about it carefully!)

• m-nएममें प्राप्त संख्या जोड़ेंएन
• एन(n-1)/2एन(एन-1)/2
• Neither of the above.

प्रश्न पूछना 02: Lecture Comprehension, Degrees of Freedom of a Robot (अध्याय 2.2)

Q1. Consider a joint between two rigid bodies. Each rigid body has mएम degrees of freedom (m=3एम=3 for a planar rigid body and m=6एम=6 for a spatial rigid body) in the absence of any constraints. The joint has fएफ degrees of freedom (जैसे, f=1एफ=1 for a revolute joint or f=3एफ=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 mएम and fएफ

Q2. Consider a mechanism consisting of three spatial rigid bodies (including ground, N=4घर्षण से कितना काम होता है=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.

प्रश्न पूछना 03: अध्याय 2 के माध्यम से 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, है 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 nएन such arms grasping the object. What is the number of degrees of freedom of this system? Your answer should be a mathematical expression including nएन. Examples of mathematical expressions including nएन are 4*n-74∗एन−7 or n/3एन/3

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

क्यू 7. 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. भी, 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.)

सप्ताह 02: Modern Robotics, पाठ्यक्रम 1: Foundations of Robot Motion Quiz Answers

प्रश्न पूछना 01: Lecture Comprehension, Configuration Space Topology (अध्याय 2.3.1)

Q1. To deform one nएन-dimensional space into another topologically equivalent space, which operations are you allowed to use? Select all that apply.

• Stretching
• Cutting.
• Gluing.

Q2. True or false? An nएन-dimensional space can be topologically equivalent to an mएम-dimensional space, where m \neq nएम​=एन.

• सत्य.
• असत्य.

प्रश्न पूछना 02: Lecture Comprehension, Configuration Space Representation (अध्याय 2.3.2)

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

सत्य.

असत्य

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

प्रश्न पूछना 02: Lecture Comprehension, Configuration and Velocity Constraints (अध्याय 2.4)

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

• सत्य.
• असत्य.

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

• सत्य.
• असत्य.

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?)

प्रश्न पूछना 03: Lecture Comprehension, Task Space and Workspace (अध्याय 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.

• सत्य.
• असत्य.

प्रश्न पूछना 04: अध्याय 2.3 के माध्यम से 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) एक्स=cosमैं1​+2cos(मैं1​+मैं2​)

y = \sin \theta_1 + 2 \sin (\theta_1 + \theta_2)लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है=sinमैं1​+2sin(मैं1​+मैं2​)

(दूसरे शब्दों में, संपर्क 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.)

• [ए,बी] \बार [ए,बी] \times S^1[ए,बी]×[ए,बी]×एस1
• [ए,बी] \times \mathbb{आर}^1 \times S^1[ए,बी]×R1×एस1
• [ए,बी] \बार [ए,बी] \times \mathbb{आर}^1[ए,बी]×[ए,बी]×R1
• \mathbb{आर}^2 \times S^1R2×एस1

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

• \mathbb{आर}^3 \times S^3R3×एस3
• \mathbb{आर}^3 \times T^3R3×टी3
• \mathbb{आर}^3 \times T^2 \times S^1R3×टी2×एस1
• \mathbb{आर}^3 \times S^2 \times S^1R3×एस2×एस1

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{आर}^3 \times T^{10}R3×टी10
• \mathbb{आर}^3 \times S^2 \times T^8R3×एस2×टी8
• \mathbb{आर}^3 \times S^3 \times T^7R3×एस3×टी7
• \mathbb{आर}^4 \times S^2 \times T^7R4×एस2×टी7

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{आर}^2 \times S^2 \times S^1 \times [ए,बी]R2×एस2×एस1×[ए,बी]
• \mathbb{आर}^2 \times S^3 \times [ए,बी]R2×एस3×[ए,बी]
• \mathbb{आर}^2 \times T^3 \times [ए,बी]R2×टी3×[ए,बी]
• \mathbb{आर}^3 \times T^3R3×टी3

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

(1+ \cos q_1) \दूरसंचार विभाग{क्यू}_1 + (2+ \sin q_2) \दूरसंचार विभाग{क्यू}_2 + (\cos q_1+ \sin q_2 + 3) \दूरसंचार विभाग{क्यू}_3 = 0.(1+cosक्यू1​)क्यू˙​1​+(2+sinक्यू2​)क्यू˙​2​+(cosक्यू1​+sinक्यू2​+3)क्यू˙​3​=0.

• Holonomic
• Nonholonomic

क्यू 7. 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) पास होना? (Enter an integer number.)

सप्ताह 03: Modern Robotics, पाठ्यक्रम 1: Foundations of Robot Motion Quiz Answers

प्रश्न पूछना 01: Lecture Comprehension, Introduction to Rigid-Body Motions (अध्याय 3 के माध्यम से 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, लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है, and z axes of a coordinate frame, क्रमश?

• Thumb, अनुक्रमणिका, middle
• Middle, अनुक्रमणिका, thumb
• अनुक्रमणिका, 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 {बी}, इसलिये

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

प्रश्न पूछना 02: Lecture Comprehension, Rotation Matrices (अध्याय 3.2.1, भाग 1 का 2)

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

• the rows are the x, लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है, z axes of {ए} written in {बी} coordinates.
• the columns are the x, लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है, z axes of {ए} written in {बी} coordinates.
• the rows are the x, लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है, z axes of {बी} written in {ए} coordinates.
• the columns are the x, लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है, z axes of {बी} written in {ए} 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​, अर्थात।, R_{ab}^{-1}Rab−1​, है (select all that apply):

• -R_{ab}में प्राप्त संख्या जोड़ेंRab​
• R_{ab}^{\rm T}RabT​
• R-Iआरमें प्राप्त संख्या जोड़ेंमैं
• R_{ba}Rba​

Q4. Multiplication of SO(3)इसलिए(3) rotation matrices is (select all that apply):

• associative.
• commutative.

प्रश्न पूछना 03: Lecture Comprehension, Rotation Matrices (अध्याय 3.2.1, भाग 2 का 2)

Q1. Which of the following is equivalent to R_{ac}Rac​, the representation of the orientation of the {कीबोर्ड संशोधक चाबियों का एक समूह है जिसका उपयोग अन्य कुंजियों के व्यवहार को बदलने के लिए किया जा सकता है} frame relative to the {ए} frame? Select all that apply

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

Q2. The matrix

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

1000010−10

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

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

• R_{sb} आरRsb​आर
• R R_{sb}RRsb​

Q3. The matrix

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

1000010−10

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

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

• R_{sb}आरRsb​आर
• R R_{sb}RRsb​

प्रश्न पूछना 04: Lecture Comprehension, Angular Velocities (अध्याय 3.2.2)

Q1. Our representation of the three-dimensional orientation uses an implicit representation (a 3×3 SO(3) matrix with 9 नंबर), but our usual representation of the angular velocity uses only three numbers, अर्थात।, 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, वहीं दूसरी ओर, 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{आर}^3R3
• इसलिए(3)इसलिए(3)
• एक मुक्त-शरीर आरेख की आवश्यकता है(3)एक मुक्त-शरीर आरेख की आवश्यकता है(3)

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

• \mathbb{आर}^3R3
• इसलिए(3)इसलिए(3)
• एक मुक्त-शरीर आरेख की आवश्यकता है(3)एक मुक्त-शरीर आरेख की आवश्यकता है(3)

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

• \mathbb{आर}^3R3
• इसलिए(3)इसलिए(3)
• एक मुक्त-शरीर आरेख की आवश्यकता है(3)एक मुक्त-शरीर आरेख की आवश्यकता है(3)

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

• 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 pमैं×पी can be written [\ओमेगा] पी[मैं]पी, कहां [\ओमेगा][मैं] है

• the SO(3)इसलिए(3) representation of \omegaमैं.
• the skew-symmetric so(3)एक मुक्त-शरीर आरेख की आवश्यकता है(3) representation of \omegaमैं.

प्रश्न पूछना 05: Lecture Comprehension, Exponential Coordinates of Rotation (अध्याय 3.2.3, भाग 1 का 2)

Q1. The orientation of a frame {डी} relative to a frame {कीबोर्ड संशोधक चाबियों का एक समूह है जिसका उपयोग अन्य कुंजियों के व्यवहार को बदलने के लिए किया जा सकता है} can be represented by a unit rotation axis \hat{\ओमेगा}मैं^ and the distance \thetaमैं rotated about the axis. If we rotate the frame {कीबोर्ड संशोधक चाबियों का एक समूह है जिसका उपयोग अन्य कुंजियों के व्यवहार को बदलने के लिए किया जा सकता है} by \thetaमैं about the axis \hat{\ओमेगा}मैं^ expressed in the {कीबोर्ड संशोधक चाबियों का एक समूह है जिसका उपयोग अन्य कुंजियों के व्यवहार को बदलने के लिए किया जा सकता है} frame, we end up at {डी}. The vector \hat{\ओमेगा}मैं^ has 3 numbers and \thetaमैं है 1 संख्या, but we only need 3 नंबर, the exponential coordinates \hat{\ओमेगा} \thetaमैं^मैं, to represent {डी} relative to {कीबोर्ड संशोधक चाबियों का एक समूह है जिसका उपयोग अन्य कुंजियों के व्यवहार को बदलने के लिए किया जा सकता है}, इसलिये

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

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? Select all that apply.

• 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{एक्स}(टी) = Bx(टी)एक्स˙(टी)लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता हैBx(टी), where xएक्स is a vector and Bबी is a constant square matrix, is solved as x(टी) = e^{Bt} एक्स(0)एक्स(टी)लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता हैeBtx(0), where the matrix exponential e^{Bt}eBt की तरह परिभाषित किया गया है

• 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+….

प्रश्न पूछना 06: Lecture Comprehension, Exponential Coordinates of Rotation (अध्याय 3.2.3, भाग 2 का 2)

Q1. The solution to the differential equation \dot{पी}(टी) = \hat{\ओमेगा} \times p(टी) लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है [\hat{\ओमेगा}] पी(टी)पी˙​(टी)लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता हैमैं^×पी(टी)लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है[मैं^]पी(टी) is p(टी) = e^{[\hat{\ओमेगा}\theta]}पी(0)पी(टी)लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता हैइ[मैं^मैं]पी(0), where p(0)पी(0) is the initial vector and p(टी)पी(टी) is the vector after it has been rotated at the angular velocity \hat{\ओमेगा}मैं^ for time t=\thetaटीलाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता हैमैं (where \hat{\ओमेगा}\thetaमैं^मैं are the exponential coordinates). You can think of R = e^{[\hat{\ओमेगा}\theta]}आरलाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता हैइ[मैं^मैं] as the rotation operation that moves p(0)पी(0) to p(टी) = p(\theta)पी(टी)लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता हैपी(मैं).

Which of the following statements is correct? Select all that apply.

• R_{sb’} = R_{sb} e^{[\hat{\ओमेगा}\theta]}Rsb′​=Rsb​इ[मैं^मैं] represents the orientation of a new frame {b’} relative to {एस} after the frame {बी} has been rotated by \thetaमैं about an axis w represented in the {बी} frame as \hat{\ओमेगा}मैं^.
• R_{sb’} = R_{sb} e^{[\hat{\ओमेगा}\theta]}Rsb′​=Rsb​इ[मैं^मैं] represents the orientation of a new frame {b’} relative to {एस} after the frame {बी} has been rotated by \thetaमैं about an axis w represented in the {एस} frame as \hat{\ओमेगा}मैं^.
• R_{sb’} = e^{[\hat{\ओमेगा}\theta]} R_{sb} Rsb′​=इ[मैं^मैं]Rsb​ represents the orientation of a new frame {b’} relative to {एस} after the frame {बी} has been rotated by \thetaमैं about an axis w represented in the {बी} frame as \hat{\ओमेगा}मैं^.
• R_{sb’} = e^{[\hat{\ओमेगा}\theta]} R_{sb} Rsb′​=इ[मैं^मैं]Rsb​ represents the orientation of a new frame {b’} relative to {एस} after the frame {बी} has been rotated by \thetaमैं about an axis w represented in the {एस} frame as \hat{\ओमेगा}मैं^.

Q2. The simple closed-form solution to the infinite series for the matrix exponential when the matrix is an element of so(3)एक मुक्त-शरीर आरेख की आवश्यकता है(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)इसलिए(3)) and the skew-symmetric representation of the exponential coordinates (elements of so(3)एक मुक्त-शरीर आरेख की आवश्यकता है(3)), which can also be thought of as the so(3)एक मुक्त-शरीर आरेख की आवश्यकता है(3) representation of the angular velocity followed for unit time. Which of the following statements is correct? Select all that apply.

• exp: एक मुक्त-शरीर आरेख की आवश्यकता है(3) \rightarrow SO(3)एक मुक्त-शरीर आरेख की आवश्यकता है(3)→इसलिए(3)
• exp: इसलिए(3) \rightarrow so(3)इसलिए(3)→एक मुक्त-शरीर आरेख की आवश्यकता है(3)
• लकड़ी का लट्ठा: एक मुक्त-शरीर आरेख की आवश्यकता है(3) \rightarrow SO(3)एक मुक्त-शरीर आरेख की आवश्यकता है(3)→इसलिए(3)
• लकड़ी का लट्ठा: इसलिए(3) \rightarrow so(3)इसलिए(3)→एक मुक्त-शरीर आरेख की आवश्यकता है(3)

प्रश्न पूछना 07: अध्याय 3 के माध्यम से 3.2, Rigid-Body Motions

Q1. In terms of the \hat{एक्स}_{\textrm{एस}}एक्स^s​, \hat{लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है}_{\textrm{एस}}लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है^​s​, \hat{जेड}_{\textrm{एस}}जेड^s​ coordinates of a fixed space frame {एस}, the frame {ए} has its \hat{एक्स}_{\textrm{ए}}एक्स^a​-axis pointing in the direction (0,0,1)(0,0,1) and its \hat{लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है}_{\textrm{ए}}लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है^​a​-axis pointing in the direction (1,0,0)(1,0,0), and the frame {बी} has its \hat{एक्स}_{\textrm{बी}}एक्स^b​-axis pointing in the direction (1,0,0)(1,0,0) and its \hat{लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है}_{\textrm{बी}}लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है^​b​-axis pointing in the direction (0,0,-1)(0,0,-1). Draw the {एस}, {ए}, तथा {बी} फ्रेम्स, similar to examples in the book and videos (जैसे, आकृति 3.7 in the book), for easy reference in this question and later questions.

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

यदि आपका उत्तर है

\बाएं[

147258369

\अधिकार]⎣⎢⎡​147​258​369​⎦⎥⎤​

उदाहरण के लिए, 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}आरएसबी−1​. All elements of this matrix should be integers.

यदि आपका उत्तर है

\बाएं[

147258369

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

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

Q4. Referring back to Question 1, let R = R_{sb}आरलाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता हैRsb​ be considered as a transformation operator consisting of a rotation about \hat{एक्स}एक्स^ by -90^\circ−90∘. Calculate R_1 = R_{sa} आरआर1​=Rsa​आर, and think of R_{sa}Rsa​ as the representation of the initial orientation of {ए} relative to {एस}, आरआर as a rotation operation, and R_1आर1​ as the new orientation of {ए} after performing the rotation. The new orientation R_1आर1​ corresponds to the orientation of the new {ए} frame relative to {एस} after rotating the original {ए} frame by -90^\circ−90∘ about which axis?

• The \hat{एक्स}_{\textrm{ए}}एक्स^a​-axis of the {ए} frame.
• The \hat{एक्स}_{\textrm{एस}}एक्स^s​-axis of the {एस} frame.

Q5. Referring back to Question 1, use R_{sb}आरएसबी​ to change the representation of the point p_b = (1,2,3)^\intercalपीबी​=(1,2,3)⊺ (में {बी} coordinates) प्रति {एस} coordinates. All elements of this vector should be integers.

यदि आपका उत्तर है

\बाएं[

123

\अधिकार]⎣⎢⎡​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 {एस} coordinates. Calculate q = R^\intercal_{sb} p_sक्यूलाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता हैRsb⊺​ps​. Is qक्यू a representation of p in {बी} coordinates?

• हां.
• नहीं.

क्यू 7. Referring back to Question 1, an angular velocity wवू is represented in {एस} as \omega_s = (3,2,1)^\intercalमैंएस​=(3,2,1)⊺. What is its representation \omega_aमैंए​? All elements of this vector should be integers.

यदि आपका उत्तर है

\बाएं[

123

\अधिकार]⎣⎢⎡​123​⎦⎥⎤​

you should enter

[1,2,3]

in the text box below and click “Run.”

क्यू 8. Referring back to Question 1, calculate the matrix logarithm [\hat{\ओमेगा}]\theta[मैं^]मैं of R_{sa}Rsa​ by hand. (You may verify your answer with software.) Extract and enter the rotation amount \thetaमैं in radians with at least two decimal places.

• 1
• 0

प्रश्न 9. Calculate the matrix exponential corresponding to the exponential coordinates of rotation \hat{\ओमेगा}\theta = (1,2,0)^\intercalमैं^मैंलाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है(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 तथा 2, and click “Run.”

प्र10. Write the 3 \times 33×3 skew-symmetric matrix corresponding to \omega = (1,2,0.5)^\intercalमैंलाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है(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 तथा 2, and click “Run.”

प्रश्न 11. Use the function {\tt MatrixExp3}MatrixExp3 in the given software to calculate the rotation matrix R \in SO(3)आर∈एसO(3) corresponding to the matrix exponential of

[\hat{\ओमेगा}] \theta = \left[

0−0.510.50−2−120

\अधिकार].[मैं^]मैं=⎣⎢⎡​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 तथा 2, and click “Run.”

प्र12. Use the function {\tt MatrixLog3}MatrixLog3 in the given software to calculate the matrix logarithm [\hat{\ओमेगा}] \theta \in so(3)[मैं^]मैं∈एसहे(3) of rotation matrix

R = \left[

0−1000−1100

\अधिकार].आर=⎣⎢⎡​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 तथा 2, and click “Run.”

सप्ताह 04: Modern Robotics, पाठ्यक्रम 1: Foundations of Robot Motion Quiz Answers

प्रश्न पूछना 01 : Lecture Comprehension, Homogeneous Transformation Matrices (अध्याय 3 के माध्यम से 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 स्थिरांक?

• 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? Select all that apply.

• 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 {बी} frame is p_b \in \mathbb{आर}^3pb​∈R3. To find the representation of this point in the {ए} frame, we could write T_{ab} p_bTab​pb​, but there is a dimension mismatch; p_bpb​ has only 3 अवयव, 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 {बी} relative to {ए}? Select all that apply.

• 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

प्रश्न पूछना 02 : Lecture Comprehension, Twists (अध्याय 3.3.2, भाग 1 का 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, ए screw axis \mathcal{एस} लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है (\mathcal{एस}_\omega, \mathcal{एस}_v) \in \mathbb{आर}^6S=(एसमैं​,एसवी​)∈R6. The screw axis is a normalized representation of the direction of motion, and \dot{\theta}मैं˙ represents how fast the body moves in that direction of motion, so that the twist is given by \mathcal{वी} = \mathcal{एस}\दूरसंचार विभाग{\theta} \in \mathbb{आर}^6V=Sमैं˙∈R6. The normalized screw axis for full spatial motions is analogous to the normalized (unit) angular velocity axis for pure rotations.

The pitch hएच of the screw axis is defined as the ratio of the linear speed over the angular speed. Which of the following is true? Select all that apply.

• If the pitch hएच is infinite, then \mathcal{एस}_\omega = 0Sमैं​=0 and \|\mathcal{एस}_v\| = 1∥Sवी​∥=1.
• If the pitch hएच is infinite, फिर \|\mathcal{एस}_\omega\| = 1∥Sमैं​∥=1 and \mathcal{एस}_vSवी​ is arbitrary.
• If the pitch hएच is finite, then \mathcal{एस}_\omega = 0Sमैं​=0 and \|\mathcal{एस}_v\| = 1∥Sवी​∥=1.
• If the pitch hएच is finite, फिर \|\mathcal{एस}_\omega\| = 1∥Sमैं​∥=1 and \mathcal{एस}_vSवी​ 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 {बी} has an \hat{{\rm x}}_bx^बी​-axis pointing outward (away from the center of the turntable), a \hat{{\rm y}}_by^​बी​-axis pointing in the direction the turntable is moving at your location (the direction your eyes are looking), and a \hat{{\rm z}}_bz^बी​-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{एस} लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है (\mathcal{एस}_\omega, \mathcal{एस}_v)S=(एसमैं​,एसवी​) and the twist \mathcal{वी} लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है (\ओमेगा,वी)वी=(मैं,वी) expressed in your body frame {बी}? All angular velocities are in radians/second and all linear velocities are in meters/second.

• \mathcal{एस} लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है (0, 0, 0.1, 0, 0.3, 0), \;\; \mathcal{वी} लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है (0, 0, 0.01, 0, 0.03, 0)S=(0,0,0.1,0,0.3,0),वी=(0,0,0.01,0,0.03,0)
• \mathcal{एस} लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है (0, 0, 1, 0, 3, 0), \;\; \mathcal{वी} लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है (0, 0, 0.1, 0, 0.3, 0)S=(0,0,1,0,3,0),वी=(0,0,0.1,0,0.3,0)
• \mathcal{एस} लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है (1, 0, 0, 0, 3, 0), \;\; \mathcal{वी} लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है (0.1, 0, 0, 0, 0.3, 0)S=(1,0,0,0,3,0),वी=(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? Select all that apply.

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

प्रश्न पूछना 03 : Lecture Comprehension, Twists (अध्याय 3.3.2, भाग 2 का 2)

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

• 3×3
• 4×4
• 6×6

Q2. A 3-vector angular velocity \omegaमैं can be represented in matrix form as [\ओमेगा][मैं], an element of so(3)एक मुक्त-शरीर आरेख की आवश्यकता है(3), the set of 3×3 skew-symmetric matrices. Analogously, a 6-vector twist \mathcal{वी} लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है (\ओमेगा,वी)वी=(मैं,वी) can be represented in matrix form as [\mathcal{वी}][वी], an element of se(3)se(3). What is the dimension of [\mathcal{वी}][वी]?

• 3×3
• 4×4
• 6×6

प्रश्न पूछना 04 : Lecture Comprehension, Exponential Coordinates of Rigid-Body Motion (अध्याय 3.3.3)

Q1. Although we use six numbers to represent a screw \mathcal{एस} लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है (\mathcal{एस}_\omega,\mathcal{एस}_v)S=(एसमैं​,एसवी​), the space of all screws is only 5-dimensional. आप अभी भी आधे घर में नहीं हैं?

• \mathcal{एस}_\omegaSमैं​ must be unit length.
• \mathcal{एस}_vSवी​ must be unit length.
• Either \mathcal{एस}_\omegaSमैं​ or \mathcal{एस}_vSवी​ must be unit length.

Q2. A transformation matrix T_{ab}Tab​, representing {बी} relative to {ए}, can be represented using the 6-vector exponential coordinates \mathcal{एस}\thetaSमैं, where \mathcal{एस}S is a screw axis (represented in {ए} coordinates) and \thetaमैं is the distance followed along the screw axis that displaces {ए} प्रति {बी}. Which of the following is correct? Select all that apply.

• T_{ab} = e^{\mathcal{एस}\theta}Tab​=इएसमैं
• T_{ab} = e^{[\mathcal{एस}]\theta}Tab​=इ[एस]मैं
• T_{ab} = e^{[\mathcal{एस}\theta]}Tab​=इ[एसमैं]
• T_{ab} = e^{\mathcal{एस}[\theta]}Tab​=इएस[मैं]

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

• इसलिए(3)
• एक मुक्त-शरीर आरेख की आवश्यकता है(3)
• SE(3)
• se(3)

Q4. T_{ab’} = T_{ab} e^{[\mathcal{एस}\theta]}Tab′​=Tab​इ[एसमैं] is a representation of the new frame {b’} (relative to {ए}) achieved after {बी} has followed

• the screw axis \mathcal{एस}एस, expressed in {बी} coordinates, a distance \thetaमैं.
• the screw axis \mathcal{एस}एस, expressed in {ए} coordinates, a distance \thetaमैं.

Q5. T_{ab’} = e^{[\mathcal{एस}\theta]} T_{ab}Tab′​=इ[एसमैं]Tab​ is a representation of the new frame {b’} (relative to {ए}) achieved after {बी} has followed

• the screw axis \mathcal{एस}एस, expressed in {बी} coordinates, a distance \thetaमैं.
• the screw axis \mathcal{एस}एस, expressed in {ए} coordinates, a distance \thetaमैं.

Q6. Which of the following statements is true? Select all that apply.

• The matrix exponential maps [\mathcal{एस}\theta] \in se(3)[एसमैं]∈se(3) to a transformation matrix T \in SE(3)टी∈SE(3), where Tटी is the representation of the frame (relative to {एस}) that is achieved by following the screw \mathcal{एस}एस (expressed in {एस}) a distance \thetaमैं from the identity configuration (अर्थात।, a frame initially coincident with {एस}).
• The matrix exponential maps [\mathcal{वी}] \in se(3)[वी]∈se(3) to a transformation matrix T \in SE(3)टी∈SE(3), where Tटी is the representation of the frame (relative to {एस}) that is achieved by following the twist \mathcal{वी}वी (expressed in {एस}) for unit time from the identity configuration (अर्थात।, a frame initially coincident with {एस}).
• 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).

प्रश्न पूछना 05 : Lecture Comprehension, Wrenches (अध्याय 3.4)

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

• \mathcal{एफ}_a = (m_a,f_a)एफए​=(ma​,fa​)
• \mathcal{एफ}_a = (f_a,m_a)एफए​=(fa​,ma​)

Q2. We know that the power associated with a wrench and twist pair (\mathcal{एफ},\mathcal{वी})(एफ,वी) does not depend on whether they are represented in the frame {ए} जैसा (\mathcal{एफ}_a,\mathcal{वी}_a)(एफए​,वीए​) or the frame {बी} जैसा (\mathcal{एफ}_b,\mathcal{वी}_b)(एफबी​,वीबी​). इसलिए, we can write \mathcal{एफ}_a^{\rm T} \mathcal{वी}_a = \mathcal{एफ}_b^{\rm T} \mathcal{वी}_bFएT​Vए​=FबीT​Vबी​ and then use which identity to derive the equation \mathcal{एफ}_a = [{\rm Ad}_{T_{ba}}]^{\rm T} \mathcal{एफ}_bFए​=[AdTba​​]TFबी​ relating the representations \mathcal{एफ}_aFए​ and \mathcal{एफ}_bFबी​? (भी, remember the matrix identity (अब)^{\rm T} = B^{\rm T} A^{\rm T}(अब)T=बीटीएT.)

• \mathcal{वी}_a = T_{ab} \mathcal{वी}_bVए​=Tab​Vबी​
• \mathcal{वी}_a = T_{ba} \mathcal{वी}_bVए​=Tba​Vबी​
• \mathcal{वी}_a = [{\rm Ad}_{T_{ba}}] \mathcal{वी}_bVए​=[AdTba​​]वीबी​
• \mathcal{वी}_a = [{\rm Ad}_{T_{ab}}] \mathcal{वी}_bVए​=[AdTab​​]वीबी​

प्रश्न पूछना 06 : Chapters 3.3 तथा 3.4, Rigid-Body Motions

Q1. In terms of the \hat{एक्स}_{\textrm{एस}}एक्स^s​, \hat{लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है}_{\textrm{एस}}लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है^​s​, \hat{जेड}_{\textrm{एस}}जेड^s​ coordinates of a fixed space frame {एस}, the frame {ए} has its \hat{एक्स}_{\textrm{ए}}एक्स^a​-axis pointing in the direction (0,0,1)(0,0,1) and its \hat{लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है}_{\textrm{ए}}लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है^​a​-axis pointing in the direction (-1,0,0)(−1,0,0), and frame {बी} has its \hat{एक्स}_{\textrm{बी}}एक्स^b​-axis pointing in the direction (1,0,0)(1,0,0) and its \hat{लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है}_{\textrm{बी}}लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है^​b​-axis pointing in the direction (0,0,-1)(0,0,-1). The origin of {ए} is at (0,0,1)(0,0,1) में {एस} and the origin of {बी} is at (0,2,0)(0,2,0). Draw the {एस}, {ए}, तथा {बी} फ्रेम्स, similar to examples in the book and videos, for easy reference in this question and later questions.

Write the transformation matrix T_{sa}टीएसए​. 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

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

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

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

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

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

Q4. Referring back to Question 1, let T = T_{sb}टीलाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता हैटीएसबी​ be considered as a transformation operator consisting of a rotation about \hat{एक्स}एक्स^ by -90^\circ−90∘ and a translation along \hat{लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है}लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है^​ by 2 इकाइयों. Calculate T_1 = T T_{sa}टी1​=टीटीएसए​, and think of T_{sa}टीएसए​ as the representation of the initial configuration of {ए} relative to {एस}, टीटी as a transformation operation, and T_1टी1​ as the new configuration of {ए} after performing the transformation. Are the rotation axis \hat{एक्स}एक्स^ and translation axis \hat{लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है}लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है^​ of the transformation Tटी properly considered to be expressed in the frame {एस} or the frame {ए}?

1 बिंदु

• The frame {एस}.
• The frame {ए}.

Q5. Referring back to Question 1, use T_{sb}टीएसबी​ to change the representation of the point p_b = (1,2,3)^\intercalपीबी​=(1,2,3)⊺ (में {बी} coordinates) प्रति {एस} 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

\अधिकार]⎣⎢⎡​123​⎦⎥⎤​.

Q6. Referring back to Question 1, choose a point p represented by p_s = (1,2,3)^\intercalपीएस​=(1,2,3)⊺ in {एस} coordinates. Calculate q = T_{sb} p_sक्यूलाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता हैटीएसबी​पीएस​. Is qक्यू a representation of p in {बी} coordinates?

1 बिंदु

• हां
• नहीं

क्यू 7. Referring back to Question 1, a twist \mathcal{वी}V is represented in {एस} जैसा {\mathcal V}_s = (3,2,1,-1,-2,-3)^\intercalVएस​=(3,2,1,−1,−2,−3)⊺. What is its representation {\mathcal V}_aVए​? 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

\अधिकार]⎣⎢⎢⎢⎢⎢⎢⎢⎡​123456​⎦⎥⎥⎥⎥⎥⎥⎥⎤​.

क्यू 8. Referring back to Question 1, calculate the matrix logarithm [{\mathcal S}]\theta[एस]मैं of T_{sa}Tsa​. Write the rotation amount \thetaमैं in radians with at least 2 decimal places.

प्रश्न 9. Calculate the matrix exponential corresponding to the exponential coordinates of rigid-body motion {\mathcal S}\theta = (0,1,2,3,0,0)^\intercalSमैंलाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है(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

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

प्र10. Referring back to Question 1, use T_{sb}टीएसबी​ to change the representation of the wrench Fबीलाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है(1,0,0,2,1,0)⊺ (में {बी} coordinates) प्रति {एस} 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

\अधिकार]⎣⎢⎡​123​⎦⎥⎤​.

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

T = \left[

0100−100000103011

\अधिकार].टी=⎣⎢⎢⎢⎡​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

\अधिकार]⎣⎢⎡​147​258​369​⎦⎥⎤​.

प्र12. Write the se(3)एसइ(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

\अधिकार]⎣⎢⎡​147​258​369​⎦⎥

प्रश्न 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{एस} लाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है (1,0,0)एस^=(1,0,0) in the direction of the screw axis, located at the point p = (0,0,2)पीलाभ कमाने के लिए मुझे इस वस्तु का कितना हिस्सा बेचने की आवश्यकता है(0,0,2), with pitch h = 1एच=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

\अधिकार]⎣⎢⎡​123​⎦⎥⎤​.

प्र14. Use the function {\tt MatrixExp6}MatrixExp6 in the given software to calculate the homogeneous transformation matrix T \in SE(3)टी∈एसइ(3) corresponding to the matrix exponential of

[एस]मैं=⎡⎣⎢⎢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

\अधिकार]⎣⎢⎡​147​258​369​⎦⎥⎤​.

प्रश्न 15. Use the function {\tt MatrixLog6}MatrixLog6 in the given software to calculate the matrix logarithm [एस]मैं∈एसइ(3) of the homogeneous transformation matrix

T = \left[

0100−100000103011

\अधिकार].टी=⎣⎢⎢⎢⎡​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

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