الروبوتات الحديثة, دورة 2: Robot Kinematics Quizzes & الإجابات - كورسيرا

مرحبا بك في Robot Kinematics في Modern Robotics Course 2, where precision meets innovation in الروبوتات. Discover our engaging الإختبارات وخبير الإجابات that shed light on the principles that govern robot motion and positioning. These quizzes serve as a gateway to understanding the complex mechanics of robot kinematics, from forward and reverse kinematics to motion path design.

Whether you are a الروبوتات enthusiast who wants to deepen your knowledge or a student who wants to understand the complexity of إنسان آلي اقتراح, this collection provides valuable information on fundamental aspects of robot kinematics. Join us on a journey of discovery as we explore the dynamics of إنسان آلي motion and unlock the potential for accurate and efficient إنسان آلي عمليات. Let’s embark on this enlightening journey together as we explore robot kinematics and its role in shaping the future of الروبوتات and automation.

لغز 01: Lecture Comprehension, Product of Exponentials Formula in the Space Frame (الفصل 4 عبر 4.1.2)

Q1. True or false? The PoE formula in the space frame only correctly calculates the end-effector configuration if you first put the robot at its zero configuration, then move joint nن to \theta_nθn​, then move joint n-1ن−1 to \theta_{n-1}θn−1​, إلخ, until you move joint 1 to \theta_1θ1​.

• صحيح.
• خاطئة.

Q2. Consider the screw axis \mathcal{S}_iSأنا​ used in the PoE formula. Which of the following is true?

• \mathcal{S}_iSأنا​ represents the screw axis of joint iأنا, expressed in the end-effector frame {ب}, when the robot is at its zero configuration.
• \mathcal{S}_iSأنا​ represents the screw axis of joint iأنا, expressed in the end-effector frame {ب}, when the robot is at an arbitrary configuration \thetaθ.
• \mathcal{S}_iSأنا​ represents the screw axis of joint iأنا, expressed in the space frame {الصورة}, when the robot is at its zero configuration.
• \mathcal{S}_iSأنا​ represents the screw axis of joint iأنا, expressed in the space frame {الصورة}, when the robot is at an arbitrary configuration \thetaθ.

Q3. When the robot is at an arbitrary configuration \thetaθ, does the screw axis corresponding to motion along joint iأنا, represented in {الصورة}, depend on \theta_{i-1}θi−1​?

• لا.
• نعم فعلا.

لغز 02: Lecture Comprehension, Product of Exponentials Formula in the End-Effector Frame (الفصل 4.1.3)

Q1. When the robot is at an arbitrary configuration \thetaθ, does the screw axis corresponding to motion along joint iأنا, represented in {ب}, depend on \theta_{i-1}θi−1​?

• لا.
• نعم فعلا.

Q2. When the robot arm is at its home (zero) يجب أن يكون مهندسو الحلول قادرين على تسهيل قرارات التصميم عبر التطوير, the axis of joint 3, a revolute joint, passes through the point (3,0,0)(3,0,0) في ال {ب} الإطار. The axis of rotation is aligned with the \hat{{\rm z}}_{{\textrm b}}z^b​-axis of the {ب} الإطار. What is the screw axis \mathcal{ب}_3B3​?

• (0, 0, 1, -3, 0, 0)(0,0,1,−3,0,0)
• (0, 0, 1, 0, -3, 0)(0,0,1,0,−3,0)
• (0, 0, 1, 0, 0, -3)(0,0,1,0,0,−3)

لغز 03: Lecture Comprehension, Forward Kinematics Example

Q1. في الصورة أدناه, imagine a frame {ج} on the axis of joint 2 and aligned with the {الصورة} الإطار. What is the screw axis of joint 1 expressed in the frame {ج}?

• (0, 0, 1, 0, 10, 0)(0,0,1,0,10,0)
• (0, 0, 1, 0, 0, 10)(0,0,1,0,0,10)

لغز 04: الفصل 4, Forward Kinematics

Q1. The URRPR spatial open chain robot is shown below in its zero position.

For L = 1L=1, determine the end-effector zero configuration MM. The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.

Write the matrix in the answer box and click “Run”:

[[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​⎦⎥⎤​.

• 1
• [[0,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,1]]

Q2. Referring back to Question 1, determine the screw axes \mathcal{S}_iSأنا​ in {0} when the robot is in its zero position. Again L = 1L=1. Give the axes as a 6×6 matrix with the form \left[\mathcal{S}_1, \mathcal{S}_2, \dots, \mathcal{S}_6 \right][S1​,S2​,...,S6​], أي, each column is a screw axis. The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.

Write the matrix in the answer box and click “Run”:

[[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​⎦⎥⎤​

• 1
• [[0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0]]

Q3. Referring back to Question 1, determine the screw axes \mathcal{ب}_iBأنا​ in {ب} when the robot is in its zero position. Again L = 1L=1. Give the axes as a matrix with the form \left[\mathcal{ب}_1, \mathcal{ب}_2, \dots, \mathcal{ب}_6 \right][B1​,B2​,...,B6​]. The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.

Write the matrix in the answer box and click “Run”:

[[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​⎦⎥⎤​.

• 1
• [[0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0],[0,0,0,0,0,0]]

Q4. Referring back to Question 1 و 2, given L = 1L=1 and joint variable values \theta = (-\pi/2, \pi/2, \pi/3, -\pi/4, 1, \pi/6)θ=(-الأب/2,الأب/2,الأب/3,-الأب/4,1,الأب/6), use the function {\tt FKinSpace}FKinSpace in the given software to find the end-effector configuration T \in SE(3)تي∈SE(3). The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.

Write the matrix in the answer box and click “Run”:

[[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​⎦⎥⎤​.

• 1
• [[0,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,1]]

Q5. Referring back to Question 1 و 3, given L = 1L=1 and joint variable values \theta = (-\pi/2, \pi/2, \pi/3, -\pi/4, 1, \pi/6)θ=(-الأب/2,الأب/2,الأب/3,-الأب/4,1,الأب/6), use the function {\tt FKinBody}FKinBody in the given software to find the end-effector configuration T \in SE(3)تي∈SE(3). The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.

Write the matrix in the answer box and click “Run”:

[[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​⎦⎥⎤​.

• 1
• [[0,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,1]]

أسبوع 02: الروبوتات الحديثة, دورة 2: Robot Kinematics Coursera Quiz Answers

لغز 02: Lecture Comprehension, Velocity Kinematics and Statics (الفصل 5 مقدمة)

Q1. True or false? The Jacobian matrix depends on the joint variables.

• صحيح.
• خاطئة.

Q2. True or false? The Jacobian matrix depends on the joint velocities.

• صحيح.
• خاطئة.

Q3. True or false? Row iأنا of the Jacobian corresponds to the end-effector velocity when joint iأنا moves at unit speed and all other joints are stationary.

• صحيح.
• خاطئة.

Q4. Consider a square Jacobian matrix that is usually full rank. At a configuration where one row of the Jacobian becomes a scalar multiple of another row, is the robot at a singularity?

• نعم فعلا.
• لا.

Q5. بشكل عام, a sphere (or hypersphere, meaning a sphere in more than 3 dimensions) of possible joint velocities maps through the Jacobian to

• a sphere (or hypersphere).
• a polyehdron.
• an ellipsoid (or hyperellipsoid).

Q6. Assume a three-dimensional end-effector velocity. At a singularity, the volume of the ellipsoid of feasible end-effector velocities becomes

• zero.
• infinite.

Q7. At a singularity,

• some end-effector forces become impossible to resist by the joint forces and torques.
• some end-effector forces can be resisted even with zero joint forces or torqu

لغز 02: Lecture Comprehension, Statics of Open Chains (الفصل 5.2)

Q1. If the wrench -\mathcal{F}−F is applied to the end-effector, to stay at equilibrium the robot must apply the joint forces and torques \tau = J^{\rm T}(\theta) \mathcal{F}τ=Jتي(θ)F to resist it. If the robot has 4 one-dof joints, what is the dimension of the subspace of 6-dimensional end-effector wrenches that can be resisted by \tau = 0τ=0?

• 2-الأبعاد.
• At least 2-dimensional.
• 4-الأبعاد.
• At least 4-dimensional.

لغز 03: Lecture Comprehension, Singularities (الفصل 5.3)

Q1. Consider a robot with 7 joints and a space Jacobian with a maximum rank of 6 over all configurations of the robot. At the current configuration, the rank of the space Jacobian is 5. Which of the following statements is true? اختر كل ما ينطبق.

• The robot is redundant with respect to the task of generating arbitrary end-effector twists.
• The robot is kinematically deficient with respect to the task of generating arbitrary end-effector twists.
• The robot is at a singularity.

Q2. Consider a robot with 7 joints and a space Jacobian with a maximum rank of 3 over all configurations of the robot. At the current configuration, the rank of the space Jacobian is 3. Which of the following statements is true? اختر كل ما ينطبق.

• The robot is redundant with respect to the task of generating arbitrary end-effector twists.
• The robot is at a singularity.
• The space Jacobian is “fat.”

Q3. Consider a robot with 8 joints and a body Jacobian with rank 6 at a given configuration. For a given desired end-effector twist \mathcal{خامسا}_bVب​, what is the dimension of the subspace of joint velocities (in the 8-dimensional joint velocity space) that create the desired twist?

• 2
• 0
• The desired twist cannot be generated.

لغز 04: Lecture Comprehension, Manipulability (الفصل 5.4)

Q1. It’s more useful to visualize the manipulability ellipsoid using the body Jacobian than the space Jacobian, since the body Jacobian measures linear velocities at the origin of the end-effector frame, which has a more intuitive meaning than the linear velocity at the origin of the space frame. If the robot has nن joints, then the body Jacobian J_bJب​ is 6 \times n6×ن. We can break J_bJب​ into two sub-Jacobians, the angular and linear Jacobians:

J_b = \left[

JبωJبالخامس

\حق].Jب​=[Jبω​Jبالخامس​​].

What is the dimension of J_{bv}J_{bv}^{\rm T}Jbv​JbvT​, which is used to generate the linear component of the manipulability ellipsoid?

• 3 \times 33×3
• 6 \times 66×6
• n \times nن×ن

Q2. Consider a robot with a full rank Jacobian as it approaches a singular configuration. As it approaches a singular configuration, what happens to the manipulability ellipsoid? اختر كل ما ينطبق.

• The length of one principal axis approaches zero.
• The length of one principal axis approaches infinity.
• The interior “volume” of the ellipsoid approaches zero.
• The interior “volume” of the ellipsoid approaches infinity.

Q3. Consider a robot with a full rank Jacobian as it approaches a singular configuration. As it approaches the singular configuration, what happens to the force ellipsoid? اختر كل ما ينطبق.

• The length of one principal axis approaches zero.
• The length of one principal axis approaches infinity.
• The interior “volume” of the ellipsoid approaches zero.
• The interior “volume” of the ellipsoid approaches infinity.

لغز 05: الفصل 5, Velocity Kinematics and Statics

Q1. A 3R planar open-chain robot is shown below.

Suppose the tip generates a wrench that can be expressed in the space frame {الصورة} as a force of 2 N in the \hat{{\rm x}}_{{\rm s}}x^s​ direction, with no component in the \hat{{\rm y}}_{{\rm s}}y^​s​ direction and zero moment in the {الصورة} الإطار. What torques must be applied at each of the joints? Positive torque is counterclockwise (the joint axes are out of the screen, so positive rotation about the joints is counterclockwise). Give the torque values in the form (\tau_1, \tau_2, \tau_3)(τ1​,τ2​,τ3​). The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.

مهم: Remember that the wrench applied by the robot end-effector has zero moment in the {الصورة} الإطار. No other frame is defined in the problem. خاصه, no frame is defined at the tip of the robot.

Write the vector in the answer box and click “Run”:

[1.11,2.22,3.33] for \left[

1.112.223.33

\حق]⎣⎢⎡​1.112.223.33​⎦⎥⎤​.

• 1
• 2
• 3
• 4
• [0,0,0]
• # Edit the answer above this line! Do not edit below this line!
• print ‘Your answer has been recorded as’, Your_Answer()

Q2. The 4R planar open-chain robot below has an end-effector frame {ب} at its tip.

Considering only the planar twist components (\omega_{bz}, v_{bx}, v_{بواسطة})(ωبض​,الخامسبس​,الخامسبو​) of the body twist \mathcal{خامسا}_bVب​, the body Jacobian is

Jب(θ)=⎡⎣1L3s4+L2s34+L1s234L4+L3c4+L2c34+L1c2341L3s4+L2s34L4+L3c4+L2c341L3s4L4+L3c410L4⎤⎦

where s23=sin(θ2+θ3), إلخ.

Suppose L_1 = L_2 = L_3 = L_4 = 1L1​=L2​=L3​=L4​=1 and the chain is at the configuration \theta_1=\theta_2=0, \theta_3=\pi/2, \theta_4=-\pi/2θ1​=θ2​=0,θ3​=الأب/2,θ4​=−الأب/2. The joints generate torques to create the wrench \mathcal{F}_b = (0,0,10, 10,10,0)Fب​=(0,0,10,10,10,0) at the last link. What are the torques at each of the joints? Give the torque values in the form (\tau_1, \tau_2, \tau_3, \tau_4)(τ1​,τ2​,τ3​,τ4​). The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.

Write the vector in the answer box and click “Run”:

[1.11,2.22,3.33,4.44] for \left[

1.112.223.334.44

\حق]⎣⎢⎢⎢⎡​1.112.223.334.44​⎦⎥⎥⎥⎤​.

• 1
• [0,0,0,0]

Q3. The RRP robot is shown below in its zero position.

Its screw axes in the space frame are

S1=⎡⎣⎢⎢⎢⎢⎢⎢⎢001000⎤⎦⎥⎥⎥⎥⎥⎥⎥, S2=⎡⎣⎢⎢⎢⎢⎢⎢⎢100020⎤⎦⎥⎥⎥⎥⎥⎥⎥, S3=⎡⎣⎢⎢⎢⎢⎢⎢⎢000010⎤⎦⎥⎥⎥⎥⎥⎥⎥.

Use the function {\tt JacobianSpace}JacobianSpace in the given software to calculate the 6×3 space Jacobian J_sJالصورة​ when \theta =(90^\circ, 90^\circ, 1)θ=(90∘,90∘,1). The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.

Write the matrix in the answer box and click “Run”:

[[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​⎦⎥⎤​.

• 1
• [[0,0,0],[0,0,0],[0,0,0],[0,0,0],[0,0,0],[0,0,0]]

Q5. Referring back to Question 3, the screw axes in the body frame are

B1=⎡⎣⎢⎢⎢⎢⎢⎢⎢010300⎤⎦⎥⎥⎥⎥⎥⎥⎥, B2=⎡⎣⎢⎢⎢⎢⎢⎢⎢−100030⎤⎦⎥⎥⎥⎥⎥⎥⎥, B3=⎡⎣⎢⎢⎢⎢⎢⎢⎢000001⎤⎦⎥⎥⎥⎥⎥⎥⎥.

Use the function {\tt JacobianBody}JacobianBody in the given software to calculate the 6×3 body Jacobian J_bJب​ when \theta =(90^\circ, 90^\circ, 1)θ=(90∘,90∘,1). The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.

Write the matrix in the answer box and click “Run”:

[[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​⎦⎥⎤​.

• 1
• [[0,0,0],[0,0,0],[0,0,0],[0,0,0],[0,0,0],[0,0,0]]

Q6. The kinematics of the 7R WAM robot are given in Section 4.1.3 in the textbook. The numerical body Jacobian J_bJب​ when all joint angles are \pi/2الأب/2 هو

J_b = \left[

001−0.105−0.8890−10000.006−0.1050100.00600.889001−0.045−0.8440−10000.00600100.00600001000

\حق]Jب​=⎣⎢⎢⎢⎢⎢⎢⎢⎡​001−0.105−0.8890​−10000.006−0.105​0100.00600.889​001−0.045−0.8440​−10000.0060​0100.00600​001000​⎦⎥⎥⎥⎥⎥⎥⎥⎤​

Extract the linear velocity portion J_vJالخامس​ (joint rates act on linear velocity). Calculate the directions and lengths of the principal semi-axes of the three-dimensional linear velocity manipulability ellipsoid based on J_vJالخامس​. Give a unit vector, with at least 2 decimal places for each element in this vector, to represent the direction of the longest principal semi-axis.

Write the vector in the answer box and click “Run”:

[1.11,2.22,3.33] for \left[

1.112.223.33

\حق]⎣⎢⎡​1.112.223.33​⎦⎥⎤​.

• 1
• [0,0,0]

Q7. Referring back to Question 5 and its result, give the length, with at least 2 decimal places, of the longest principal semi-axis of that three-dimensional linear velocity manipulability ellipsoid.

أسبوع 03: الروبوتات الحديثة, دورة 2: Robot Kinematics Coursera Quiz Answers

لغز 01: Lecture Comprehension, Inverse Kinematics of Open Chains (الفصل 6 مقدمة)

Q1. Consider the point (س,و) = (0,2)(س,و)=(0,2). ما هو {\rm atan2}(و,س)atan2(و,س), measuring the angle from the xس-axis to the vector to the point (س,و)(س,و)?

• 0
• \pi/2الأب/2
• -\pi/2−الأب/2

Q2. What are advantages of numerical inverse kinematics over analytic inverse kinematics? اختر كل ما ينطبق.

• It can be applied to open-chain robots with arbitrary kinematics.
• It requires an initial guess at the solution.
• It returns all possible inverse kinematics solutions.

لغز 02: Lecture Comprehension, Numerical Inverse Kinematics (الفصل 6.2, جزء 1 من 2)

Q1. Let f(\theta)F(θ) be a nonlinear function of \thetaθ mapping an nن-dimensional space (the dimension of \thetaθ) to an mم-dimensional space (the dimension of fF). We want to find a \theta_dθد​, which may not be unique, that satisfies x_d = f(\theta_d)سد​=F(θد​), أي, x_d – f(\theta_d) = 0سد​−F(θد​)=0. If our initial guess at a solution is \theta^0θ0, then a first-order Taylor expansion approximation of f(\theta)F(θ) at \theta^0θ0 tells us

x_d \approx f(\theta^0) + J(\theta^0)(\theta_d – \theta^0)سد​≈F(θ0)+J(θ0)(θد​−θ0)

where J(\theta^0)J(θ0) is the matrix of partial derivatives \partial f/\partial \theta∂F/∂θ evaluated at \theta^0θ0. Which of the following is a good next guess \theta^1θ1?

• \theta^1 = \theta^0 + J^\dagger(\theta^0) (x_d – f(\theta^0))θ1=θ0+J†(θ0)(xd​−F(θ0))
• \theta^1 = \theta^0 – J^\dagger(\theta^0) (x_d – f(\theta^0))θ1=θ0-J†(θ0)(xd​−F(θ0))
• \theta^1 = J^{-1}(\theta^0) (x_d – f(\theta^0))θ1=J−1(θ0)(xd​−F(θ0))

Q2. We want to solve the linear equation Ax = bAx=ب where Aا is a 3×2 matrix, سس is a 2-vector, and bب is a 3-vector. For a randomly chosen Aا matrix and vector bب, how many solutions xس can we expect?

• لا شيء.
• واحد.
• More than one.

Q3. We want to solve the linear equation Ax = bاس=ب, أين

A = \left[

142536

\حق]ا=[14​25​36​]

and b = [7 \;\;8]^{\rm T}ب=[78]تي. Since xس is a 3-vector and bب is a 2-vector, we can expect a one-dimensional set of solutions in the 3-dimensional space of possible xس القيم. The following are all solutions of the linear equation. Which is the solution given by x = A^\dagger bس=ا†ب? (You should be able to tell by inspection, without using software.)

• (-1.06, -3.89, 5.28)(−1.06,−3.89,5.28)
• (-3.06, 0.11, 3.28)(−3.06,0.11,3.28)
• (-5.06, 4.11, 1.28)(−5.06,4.11,1.28)

Q4. If we would like to find an xس satisfying Ax = bاس=ب, but Aا is “tall” (meaning it has more rows than columns, أي, the dimension of bب is larger than the dimension of xس), then in general we would see there is no exact solution. في هذه الحالة, we might want to find the x^*س∗ that comes closest to satisfying the equation, in the sense that x^*س∗ minimizes\|Ax^* – b\|∥اس∗−ب∥ (the 2-norm, or the square root of the sum of the squares of the vector). This solution is given by x^* = A^\dagger bس∗=ا†ب. Which of the two answers below satisfies this condition if

A = \left[

12

\حق], \;\; b = \left[

34

\حق]?ا=[12​],ب=[34​]?

• x^* = 2.2س∗=2.2
• x^* = 1س∗=1

لغز 03: Lecture Comprehension, Numerical Inverse Kinematics (الفصل 6.2, جزء 2 من 2)

Q1. To adapt the Newton-Raphson root-finding method to inverse kinematics when the desired end-effector configuration is represented as a transformation matrix X_d \in SE(3)Xd​∈SE(3), we need to express the error between T_{sb}(\theta^i)Tsb​(θi) (the forward kinematics, where \theta^iθi is our current guess at a joint solution) and X_dXd​. One expression of this error is the twist that takes the the robot from T_{sb}(\theta^i)Tsb​(θi) to X_dXd​ in unit time. When this twist is expressed in the end-effector frame {ب}, we write it as \mathcal{خامسا}_bVب​. Which of the following is a correct expression?

• \mathcal{خامسا}_b = {\rm log} (T_{sb}^{-1}(\theta^i) X_d)خامساب​=log(Tsb−1​(θi)Xd​)
• [\mathcal{خامسا}_b] = {\rm log} (T_{sb}^{-1}(\theta^i) X_d)[خامساب​]=log(Tsb−1​(θi)Xd​)
• \mathcal{خامسا}_b = {\rm exp} (T_{sb}^{-1}(\theta^i) X_d)خامساب​=exp(Tsb−1​(θi)X

لغز 04: الفصل 6, Inverse Kinematics

Q1. Use Newton-Raphson iterative numerical root finding to perform two steps of finding the root of

F(س,و) = \left[

س2−9و2−4

\حق]F(س,و)=[س2−9و2−4​]

when your initial guess is (x^0,y^0) = (1,1)(س0,و0)=(1,1). Give the result after two iterations (x^2,y^2)(س2,و2) with at least 2 decimal places for each element in the vector. You can do this by hand or write a program.

Write the vector in the answer box and click “Run”:

[1.11,2.22,3.33] for \left[

1.112.223.33

\حق]⎣⎢⎡​1.112.223.33​⎦⎥⎤​.

• 1
• [0,0]

Q2.

Referring to the figure above, find the joint angles \theta_d = (\theta_1,\theta_2,\theta_3)θد​=(θ1​,θ2​,θ3​) that put the 3R robot’s end-effector frame {ب} في

تي(\theta_d) = T_{SD} = \left[

−0.5850.81100−0.811−0.5850000100.0762.60801

\حق]تي(θد​)=تيالصورةد​=⎣⎢⎢⎢⎡​−0.5850.81100​−0.811−0.58500​0010​0.0762.60801​⎦⎥⎥⎥⎤​

relative to the {الصورة} الإطار, where linear distances are in meters. (ال {الصورة} frame is located at joint 1, but it is drawn at a different location for clarity.) The robot is shown at its home configuration, and the screw axis for each joint points toward you (out of the screen). The length of each link is 1 متر. Your solution should use either {\tt IKinBody}IKinBody or {\tt IKinSpace}IKinSpace, the initial guess \theta^0 = (\pi/4,\pi/4,\pi/4) = (0.7854, 0.7854, 0.7854)θ0=(الأب/4,الأب/4,الأب/4)=(0.7854,0.7854,0.7854), and tolerances \epsilon_\omega = 0.001ϵω​=0.001 (0.057 درجات) and \epsilon_v = 0.0001ϵالخامس​=0.0001 (0.1 مم). Give \theta_dθد​ as a vector with at least 2 decimal places for each element in the vector. (Note that there is more than one solution to the inverse kinematics for T_{SD}تيالصورةد​, but we are looking for the solution that is “close” to the initial guess \theta^0 = (\pi/4,\pi/4,\pi/4)θ0=(الأب/4,الأب/4,الأب/4), أي, the solution that will be returned by {\tt IKinBody}IKinBody or {\tt IKinSpace}IKinSpace.)

Write the vector in the answer box and click “Run”:

[1.11,2.22,3.33] for \left[

1.112.223.33

\حق]⎣⎢⎡​1.112.223.33​⎦⎥⎤​.

• 1
• [0,0,0]

أسبوع 04: الروبوتات الحديثة, دورة 2: Robot Kinematics Coursera Quiz Answers

لغز 01: Lecture Comprehension, Kinematics of Closed Chains (الفصل 7)

Q1. Which of the following statements is true about closed-chain and parallel robots? اختر كل ما ينطبق.

• For a given set of positions of the actuated joints, there may be more than one configuration of the end-effector.
• Closed-chain robots are a subclass of parallel robots.
• Some joints may be unactuated.
• The inverse kinematics for a parallel robot are generally easier to compute than its forward kinematics.
• Parallel robots are sometimes chosen instead of open-chain robots for their larger workspace.

لغز 02: الفصل 7, Kinematics of Closed Chains

Q1. The inverse Jacobian J^{-1}J−1 for a parallel robot maps the end-effector twist \mathcal{خامسا}V to the actuated joint velocities \dot{\theta}θ˙, and therefore the inverse Jacobian has nن صفوف (if there are nن actuators) و 6 الأعمدة (since a twist is 6-dimensional).

If the twist \mathcal{خامسا}V consists of a 1 in the iأنا‘th element and zeros in all other elements, then what is the corresponding vector of actuated joint velocities \dot{\theta}θ˙?

• The iأنا‘th row of J^{-1}J−1.
• The iأنا‘th column of J^{-1}J−1.

Q2. For the 3xRRR planar parallel mechanism shown below, let \phiϕ be the orientation of the end-effector frame and p \in \mathbb{R}^2ص∈R2 be the vector p expressed in fixed frame coordinates. Let a_i \in \mathbb{R}^2اأنا​∈R2 be the vector a_iأنا​ expresed in fixed frame coordinates and b_i \in \mathbb{R}^2بأنا​∈R2 be the vector b_iأنا​ expressed in the moving body frame coordinates. Define vector \text{د}_i = \text{ص} + R\text{ب}_{أنا} – \text{ا}_{أنا}دأنا​=p+Rبأنا​−aأنا​ for i = 1, 2, 3أنا=1,2,3, أين

R = \left[\begin{مجموعة مصفوفة}{نسخة}\cos\phi & -\sin\phi \\\sin\phi & \cos\phi \\\end {مجموعة مصفوفة}\حق].R=[cosϕsinϕ​−sinϕcosϕ​].

Derive a set of independent equations relating (\phi, ص)(ϕ,ص) و (\theta_1, \theta_2, \theta_3)(θ1​,θ2​,θ3​). Which of the following is correct?

• ({ص} + R{ب}_{أنا} - {ا}_{أنا})^2 = 2L^2(1 + \cos\theta_{أنا}), i = 1, 2, 3.(ص+Rbi​−إلى​)2=2L2(1+cosθi​),أنا=1,2,3.
• ({ص} + R{ب}_{أنا} - {ا}_{أنا})^\intercal({ص} + R{ب}_{أنا} - {ا}_{أنا}) = 2L^2(1 – \sin\theta_{أنا}), i = 1, 2, 3.(ص+Rbi​−إلى​)⊺(ص+Rbi​−إلى​)=2L2(1−sinθi​),أنا=1,2,3.
• ({ص} + R{ب}_{أنا} - {ا}_{أنا})^\intercal({ص} + R{ب}_{أنا} - {ا}_{أنا}) = 2L^2(1 – \cos\theta_{أنا}), i = 1, 2, 3.(ص+Rbi​−إلى​)⊺(ص+Rbi​−إلى​)=2L2(1−cosθi​),أنا=1,2,3.
• ({ص} + R{ب}_{أنا} - {ا}_{أنا})^\intercal({ص} + R{ب}_{أنا} - {ا}_{أنا}) = 2L^2(1 + \cos\theta_{أنا}), i = 1, 2, 3.(ص+Rbi​−إلى​)⊺(ص+Rbi​−إلى​)=2L2(1+cosθi​),أنا=1,2,3.