Do you want to know how robots work? Are you interested in robotics as a career? Are you willing to invest the effort to learn fundamental mathematical modeling techniques that are used in all subfields of robotics?
If so, then the “Modern Robotics: Mechanics, Planning, and Control” specialization may be for you. This specialization, consisting of six short courses, is serious preparation for serious students who hope to work in the field of robotics or to undertake advanced study. It is not a sampler.
In Course 2 of the specialization, Robot Kinematics, you will learn to solve the forward kinematics (calculating the configuration of the “hand” of the robot based on the joint values) using the product-of-exponentials formula. Your efforts in Course 1 pay off handsomely, as forward kinematics is a breeze with the tools you’ve learned.
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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 nn to theta_nθn, then move joint n-1n−1 to theta_{n-1}θn−1, etc., until you move joint 1 to theta_1θ1.
Q2. Consider the screw axis mathcal{S}_iSi used in the PoE formula. Which of the following is true?
Q3. When the robot is at an arbitrary configuration thetaθ, does the screw axis corresponding to motion along joint ii, represented in {s}, depend on theta_{i-1}θi−1?
Q1. When the robot is at an arbitrary configuration thetaθ, does the screw axis corresponding to motion along joint ii, represented in {b}, depend on theta_{i-1}θi−1?
Q2. When the robot arm is at its home (zero) configuration, the axis of joint 3, a revolute joint, passes through the point (3,0,0)(3,0,0) in the {b} frame. The axis of rotation is aligned with the hat{{rm z}}_{{textrm b}}z^b-axis of the {b} frame. What is the screw axis mathcal{B}_3B3?
Q1. In the image below, imagine a frame {c} on the axis of joint 2 and aligned with the {s} frame. What is the screw axis of joint 1 expressed in the frame {c}?
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
right]⎣⎢⎡1.114.447.772.225.558.883.336.669.99⎦⎥⎤.
Q2. Referring back to Question 1, determine the screw axes mathcal{S}_iSi 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], i.e., 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
right]⎣⎢⎡1.114.447.772.225.558.883.336.669.99⎦⎥⎤
Q3. Referring back to Question 1, determine the screw axes mathcal{B}_iBi in {b} when the robot is in its zero position. Again L = 1L=1. Give the axes as a matrix with the form left[mathcal{B}_1, mathcal{B}_2, dots, mathcal{B}_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
right]⎣⎢⎡1.114.447.772.225.558.883.336.669.99⎦⎥⎤.
Q4. Referring back to Question 1 and 2, given L = 1L=1 and joint variable values theta = (-pi/2, pi/2, pi/3, -pi/4, 1, pi/6)θ=(−π/2,π/2,π/3,−π/4,1,π/6), use the function {tt FKinSpace}FKinSpace in the given software to find the end-effector configuration T in SE(3)T∈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
right]⎣⎢⎡1.114.447.772.225.558.883.336.669.99⎦⎥⎤.
Q5. Referring back to Question 1 and 3, given L = 1L=1 and joint variable values theta = (-pi/2, pi/2, pi/3, -pi/4, 1, pi/6)θ=(−π/2,π/2,π/3,−π/4,1,π/6), use the function {tt FKinBody}FKinBody in the given software to find the end-effector configuration T in SE(3)T∈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
right]⎣⎢⎡1.114.447.772.225.558.883.336.669.99⎦⎥⎤.
Q1. True or false? The Jacobian matrix depends on the joint variables.
Q2. True or false? The Jacobian matrix depends on the joint velocities.
Q3. True or false? Row ii of the Jacobian corresponds to the end-effector velocity when joint ii moves at unit speed and all other joints are stationary.
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?
Q5. In general, a sphere (or hypersphere, meaning a sphere in more than 3 dimensions) of possible joint velocities maps through the Jacobian to
Q6. Assume a three-dimensional end-effector velocity. At a singularity, the volume of the ellipsoid of feasible end-effector velocities becomes
Q7. At a singularity,
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=JT(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?
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? Select all that apply.
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? Select all that apply.
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{V}_bVb, what is the dimension of the subspace of joint velocities (in the 8-dimensional joint velocity space) that create the desired twist?
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 nn joints, then the body Jacobian J_bJb is 6 times n6×n. We can break J_bJb into two sub-Jacobians, the angular and linear Jacobians:
J_b = left[
JbωJbv
right].Jb=[JbωJbv].
What is the dimension of J_{bv}J_{bv}^{rm T}JbvJbvT, which is used to generate the linear component of the manipulability ellipsoid?
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? Select all that apply.
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? Select all that apply.
Q1. A 3R planar open-chain robot is shown below.
Suppose the tip generates a wrench that can be expressed in the space frame {s} 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 {s} frame. 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.
Important: Remember that the wrench applied by the robot end-effector has zero moment in the {s} frame. No other frame is defined in the problem. In particular, 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
right]⎣⎢⎡1.112.223.33⎦⎥⎤.
Q2. The 4R planar open-chain robot below has an end-effector frame {b} at its tip.
Considering only the planar twist components (omega_{bz}, v_{bx}, v_{by})(ωbz,vbx,vby) of the body twist mathcal{V}_bVb, the body Jacobian is
Jb(θ)=⎡⎣1L3s4+L2s34+L1s234L4+L3c4+L2c34+L1c2341L3s4+L2s34L4+L3c4+L2c341L3s4L4+L3c410L4⎤⎦
where s23=sin(θ2+θ3), etc.
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=π/2,θ4=−π/2. The joints generate torques to create the wrench mathcal{F}_b = (0,0,10, 10,10,0)Fb=(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
right]⎣⎢⎢⎢⎡1.112.223.334.44⎦⎥⎥⎥⎤.
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_sJs 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
right]⎣⎢⎡1.114.447.772.225.558.883.336.669.99⎦⎥⎤.
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_bJb 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
right]⎣⎢⎡1.114.447.772.225.558.883.336.669.99⎦⎥⎤.
Q6. The kinematics of the 7R WAM robot are given in Section 4.1.3 in the textbook. The numerical body Jacobian J_bJb when all joint angles are pi/2π/2 is
J_b = left[
001−0.105−0.8890−10000.006−0.1050100.00600.889001−0.045−0.8440−10000.00600100.00600001000
right]Jb=⎣⎢⎢⎢⎢⎢⎢⎢⎡001−0.105−0.8890−10000.006−0.1050100.00600.889001−0.045−0.8440−10000.00600100.00600001000⎦⎥⎥⎥⎥⎥⎥⎥⎤
Extract the linear velocity portion J_vJv (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_vJv. 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
right]⎣⎢⎡1.112.223.33⎦⎥⎤.
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.
Q1. Consider the point (x,y) = (0,2)(x,y)=(0,2). What is {rm atan2}(y,x)atan2(y,x), measuring the angle from the xx-axis to the vector to the point (x,y)(x,y)?
Q2. What are advantages of numerical inverse kinematics over analytic inverse kinematics? Select all that apply.
Q1. Let f(theta)f(θ) be a nonlinear function of thetaθ mapping an nn-dimensional space (the dimension of thetaθ) to an mm-dimensional space (the dimension of ff). We want to find a theta_dθd, which may not be unique, that satisfies x_d = f(theta_d)xd=f(θd), i.e., x_d – f(theta_d) = 0xd−f(θd)=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)xd≈f(θ0)+J(θ0)(θd−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?
Q2. We want to solve the linear equation Ax = bAx=b where AA is a 3×2 matrix, xx is a 2-vector, and bb is a 3-vector. For a randomly chosen AA matrix and vector bb, how many solutions xx can we expect?
Q3. We want to solve the linear equation Ax = bAx=b, where
A = left[
142536
right]A=[142536]
and b = [7 ;;8]^{rm T}b=[78]T. Since xx is a 3-vector and bb is a 2-vector, we can expect a one-dimensional set of solutions in the 3-dimensional space of possible xx values. The following are all solutions of the linear equation. Which is the solution given by x = A^dagger bx=A†b? (You should be able to tell by inspection, without using software.)
Q4. If we would like to find an xx satisfying Ax = bAx=b, but AA is “tall” (meaning it has more rows than columns, i.e., the dimension of bb is larger than the dimension of xx), then in general we would see there is no exact solution. In this case, we might want to find the x^*x∗ that comes closest to satisfying the equation, in the sense that x^*x∗ minimizes|Ax^* – b|∥Ax∗−b∥ (the 2-norm, or the square root of the sum of the squares of the vector). This solution is given by x^* = A^dagger bx∗=A†b. Which of the two answers below satisfies this condition if
A = left[
12
right], ;; b = left[
34
right]?A=[12],b=[34]?
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 {b}, we write it as mathcal{V}_bVb. Which of the following is a correct expression?
Q1. Use Newton-Raphson iterative numerical root finding to perform two steps of finding the root of
f(x,y) = left[
x2−9y2−4
right]f(x,y)=[x2−9y2−4]
when your initial guess is (x^0,y^0) = (1,1)(x0,y0)=(1,1). Give the result after two iterations (x^2,y^2)(x2,y2) 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
right]⎣⎢⎡1.112.223.33⎦⎥⎤.
Q2.
Referring to the figure above, find the joint angles theta_d = (theta_1,theta_2,theta_3)θd=(θ1,θ2,θ3) that put the 3R robot’s end-effector frame {b} at
T(theta_d) = T_{sd} = left[
−0.5850.81100−0.811−0.5850000100.0762.60801
right]T(θd)=Tsd=⎣⎢⎢⎢⎡−0.5850.81100−0.811−0.5850000100.0762.60801⎦⎥⎥⎥⎤
relative to the {s} frame, where linear distances are in meters. (The {s} 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 meter. 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=(π/4,π/4,π/4)=(0.7854,0.7854,0.7854), and tolerances epsilon_omega = 0.001ϵω=0.001 (0.057 degrees) and epsilon_v = 0.0001ϵv=0.0001 (0.1 mm). Give theta_dθd 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}Tsd, but we are looking for the solution that is “close” to the initial guess theta^0 = (pi/4,pi/4,pi/4)θ0=(π/4,π/4,π/4), i.e., 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
right]⎣⎢⎡1.112.223.33⎦⎥⎤.
Q1. Which of the following statements is true about closed-chain and parallel robots? Select all that apply.
Q1. The inverse Jacobian J^{-1}J−1 for a parallel robot maps the end-effector twist mathcal{V}V to the actuated joint velocities dot{theta}θ˙, and therefore the inverse Jacobian has nn rows (if there are nn actuators) and 6 columns (since a twist is 6-dimensional).
If the twist mathcal{V}V consists of a 1 in the ii‘th element and zeros in all other elements, then what is the corresponding vector of actuated joint velocities dot{theta}θ˙?
Q2. For the 3xRRR planar parallel mechanism shown below, let phiϕ be the orientation of the end-effector frame and p in mathbb{R}^2p∈R2 be the vector p expressed in fixed frame coordinates. Let a_i in mathbb{R}^2ai∈R2 be the vector a_ii expresed in fixed frame coordinates and b_i in mathbb{R}^2bi∈R2 be the vector b_ii expressed in the moving body frame coordinates. Define vector text{d}_i = text{p} + Rtext{b}_{i} – text{a}_{i}di=p+Rbi−ai for i = 1, 2, 3i=1,2,3, where
R = left[begin{array}{cc}cosphi & -sinphi \sinphi & cosphi \end {array}right].R=[cosϕsinϕ−sinϕcosϕ].
Derive a set of independent equations relating (phi, p)(ϕ,p) and (theta_1, theta_2, theta_3)(θ1,θ2,θ3). Which of the following is correct?
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Where are the answers?, there are only questions.