الروبوتات الحديثة, دورة 3: مسابقات ديناميكيات الروبوت & الإجابات - كورسيرا
Join us for a fascinating exploration of robot dynamics في دورة 3, where the complex interaction of forces and motion shapes إنسان آلي behaviour. Immerse yourself in our engaging quizzes and expert answers that shed light on the principles governing the dynamic behaviour of robotic أنظمة. These quizzes serve as a gateway to understanding the complex ديناميات that govern إنسان آلي motion from acceleration to torque and beyond.
Whether you are a الروبوتات enthusiast who wants to deepen your understanding or a student who wants to understand the complexity of robot dynamics, this collection provides valuable insight into fundamental aspects of إنسان آلي اقتراح. Join us on a journey of discovery as we unravel the ديناميات من إنسان آلي behaviour and unlock the potential for precise and efficient إنسان آلي عمليات.
لغز 01: Lecture Comprehension, Lagrangian Formulation of Dynamics (الفصل 8 عبر 8.1.2, جزء 1 من 2)
Q1. The Lagrangian for a mechanical system is
- the kinetic energy plus the potential energy.
- the kinetic energy minus the potential energy.
Q2. To evaluate the Lagrangian equations of motion,
\tau_i = \frac{د}{dt} \فارك{\partial \mathcal{L}}{\partial \dot{\theta}_i} – \frac{\partial \mathcal{L}}{\partial \theta_i}τأنا=درد∂θ˙أنا∂L−∂θأنا∂L,
you must be able to take derivatives, such as the partial derivative with respect to a joint variable or velocity or a total derivative with respect to time. وبالتالي, the product rule and chain rules for derivatives will be useful. (If you have forgotten them, you can refresh your memory with any standard reference, including Wikipedia.) Which of the answers below represents the time derivative of 2 \theta_1 \cos(4 \theta_2)2θ1cos(4θ2), where \theta_1θ1 and \theta_2θ2 are functions of time?
- -8 \نقطة{\theta}_1 \sin(4 \theta_2)−8θ˙1sin(4θ2)
- 2 \نقطة{\theta}_1 \cos (4 \theta_2) - 8 \theta_1 \sin(4 \theta_2) \نقطة{\theta}_22θ˙1cos(4θ2)−8θ1sin(4θ2)θ˙2
- 2 \نقطة{\theta}_1 \cos (4 \theta_2) - 2 \theta_1 \sin(4 \theta_2) \نقطة{\theta}_22θ˙1cos(4θ2)−2θ1sin(4θ2)θ˙2
- 2 \نقطة{\theta}_1 \cos (4 \theta_2) + 2 \theta_1 \cos(4 \theta_2) \نقطة{\theta}_22θ˙1cos(4θ2)+2θ1cos(4θ2)θ˙2
Q3. The equations of motion for a robot can be summarized as
\tau = M(\theta) \ddot{\theta} + ج(\theta,\نقطة{\theta}) + ز(\theta)τ=M(θ)θ¨+ج(θ,θ˙)+ز(θ).
If the equation for the first joint is
\tau_1 = term1 + term2 + term3 + ...τ1=رالبريدصم1+رالبريدصم2+رالبريدصم3+...
Q4. which of the following terms, written in terms of (\theta,\نقطة{\theta},\ddot{\theta})(θ,θ˙,θ¨), يستطع ليس be one of the terms on the right-hand side of the equation? (The value kك is not a function of (\theta,\نقطة{\theta},\ddot{\theta})(θ,θ˙,θ¨) and could represent constants like link lengths, masses, or inertias, as needed to get correct units.) اختر كل ما ينطبق.
- k\ddot{\theta}_2 \cos(\theta_1)kθ¨2cos(θ1)
- k\ddot{\theta}_1 \dot{\theta}_1kθ¨1θ˙1
- k\sin \theta_3كsinθ3
- k\dot{\theta}_1 \dot{\theta}_2 \sin \theta_2kθ˙1θ˙2sinθ2
- k \dot{\theta}_1 \sin \theta_2kθ˙1sinθ2
لغز 02: Lecture Comprehension, Lagrangian Formulation of Dynamics (الفصل 8 عبر 8.1.2, جزء 2 من 2)
Q1. Which of the following could be a centripetal term in the dynamics?
- k \dot{\theta}_1^2kθ˙12
- k \dot{\theta}_1 \dot{\theta}_2kθ˙1θ˙2
Q2. Which of the following could be a Coriolis term in the dynamics?
- k \dot{\theta}_1^2kθ˙12
- k \dot{\theta}_1 \dot{\theta}_2kθ˙1θ˙2
Q3. One form of the equations of motion is
\tau = M(\theta)\ddot{\theta} + \نقطة{\theta}^{\rm T} \Gamma(\theta) \نقطة{\theta} + ز(\theta).τ=M(θ)θ¨+θ˙TΓ(θ)θ˙+ز(θ).
Which of the following is true about \Gamma(\theta)Γ(θ)? اختر كل ما ينطبق.
- \Gamma(\theta)Γ(θ) is zero if the mass matrix MM has no dependence on \thetaθ.
- \Gamma(\theta)Γ(θ) is an n \times nن×ن مصفوفة, where nن is the number of joints.
- \Gamma(\theta)Γ(θ) depends on M(\theta)M(θ) and \dot{\theta}θ˙.
Lecture Comprehension, Understanding the Mass Matrix (الفصل 8.1.3)لغز 03:
Q1. Which of these is a possible mass matrix M(\theta)M(θ) for a two-joint robot at a particular configuration \thetaθ? اختر كل ما ينطبق.
- \اليسار[
- 200−1
- \حق][200−1]
- \اليسار[
- 4123
- \حق][4123]
- \اليسار[
- 3112
- \حق][3112]
- \اليسار[
- 2221
- \حق][2221]
Q2. True or false? If you grab the end-effector of a robot and try to move it around by hand, the apparent mass (لك) depends on the configuration of the robot.
- صحيح.
- خاطئة.
Q3. True or false? إذا قمت بتطبيق (by hand) a linear force to the end-effector of a robot, the end-effector will accelerate in the same direction as the applied force.
- Always true.
- Always false.
- Sometimes true, sometimes false.
لغز 04: Lecture Comprehension, Dynamics of a Single Rigid Body (الفصل 8.2, جزء 1 من 2)
Q1. How is the center of mass of a rigid body defined?
- The point at the geometric centroid of the body.
- The point at the mass-weighted (or density-weighted) centroid of the body.
Q2. If the body consists of a set of rigidly connected point masses, with a frame {ب} at the center of mass, what is the wrench \mathcal{F}_bFب needed to generate the acceleration \dot{{\mathcal V}}_bV˙ب when the body’s current twist is \mathcal{خامسا}_bVب?
- The acceleration \dot{{\mathcal V}}_bV˙ب defines the linear acceleration of each point mass in the inertial frame {ب}. The linear component f_bfb of \mathcal{F}_bFب is the sum of the individual vector forces needed to cause those point-mass accelerations (using f=maF=ma), and the moment m_bmb is the sum of the moments the individual linear forces create in {ب}.
- سويا, \mathcal{خامسا}_bVب and \dot{{\mathcal V}}_bV˙ب define the linear acceleration of each point mass in the inertial frame {ب}. The linear component f_bfb of \mathcal{F}_bFب is the sum of the individual vector forces needed to cause those point-mass accelerations (using f=maF=ma), and the moment m_bmb is the sum of the moments the individual linear forces create in {ب}.
Q3. What is the kinetic energy of a rotating rigid body?
- \فارك{1}{2} \omega_b^{\rm T} \mathcal{أنا}_b \omega_b21ωbTIبωb
- \فارك{1}{2} \mathcal{أنا}_b \omega_b^221Iبωb2
Q4. True or false? For a given body, there is exactly one orientation of a frame at the center of mass that yields a diagonal rotational inertia matrix.
- صحيح.
- خاطئة.
لغز 05: Lecture Comprehension, Dynamics of a Single Rigid Body (الفصل 8.2, جزء 2 من 2)
Q1. The spatial inertia matrix is the 6×6 matrix
\mathcal{G}_B = \left[
أناب00مأنا
\حق]Gب=[أناب00mأنا].
What is the maximum number nن من فريدة من نوعها nonzero values the spatial inertia could have? بعبارات أخرى, even though the 6×6 matrix has 36 entries, we only need to store nن numbers to represent the spatial inertia matrix.
- 6
- 7
- 10
- 12
Q2. التعبير [\omega_1][\omega_2]-[\omega_2][\omega_1[ω1][ω2]-[ω2][ω1 is the so(3)وبالتالي(3) 3×3 skew-symmetric matrix representation of the cross product of two angular velocities, \omega_1 \times \omega_2 = [\omega_1]\omega_2 \in \mathbb{R}^3ω1×ω2=[ω1]ω2∈R3. An analogous expression for twists is [\mathcal{خامسا}_1][\mathcal{خامسا}_2]-[\mathcal{خامسا}_2][\mathcal{خامسا}_1][V1][V2]-[V2][V1], the 4×4 se(3)se(3) representation of the Lie bracket of \mathcal{خامسا}_1V1 and \mathcal{خامسا}_2V2, sometimes written [{\rm ad}_{\mathcal{خامسا}_1}] \mathcal{خامسا}_2 \in \mathbb{R}^6[adV1]V2∈R6. Which of the following statements is true? اختر كل ما ينطبق.
- The matrix [{\rm ad}_{\mathcal{خامسا}_1}][adV1] is an element of se(3)se(3).
- [{\rm ad}_{\mathcal{خامسا}_1}] \mathcal{خامسا}_2 = -[{\rm ad}_{\mathcal{خامسا}_2}] \mathcal{خامسا}_1[adV1]V2=−[adV2]V1
Q3. The dynamics of a rigid body, in a frame at the center of mass {ب}, can be written
\mathcal{F}_b = \mathcal{G}_b \dot{\mathcal{خامسا}}_b – [{\rm ad}_{\mathcal{خامسا}_b}]^{\rm T} \mathcal{G}_b \mathcal{خامسا}_bFب=GبV˙ب−[adVب]TGبVب.
If \mathcal{خامسا}_b = (\omega_b,v_b) = (0,v_b)خامساب=(ωb,vb)=(0,vb) and \dot{\mathcal{خامسا}}_b = (\نقطة{\أوميغا}_b, \نقطة{الخامس}_b) = (0,0)V˙ب=(ω˙ب,الخامس˙ب)=(0,0), which of the following is true?
- \mathcal{F}_bFب is zero.
- \mathcal{F}_bFب is nonzero.
- Either of the above could be true.
Q4. The dynamics of a rigid body, in a frame at the center of mass {ب}, can be written
\mathcal{F}_b = \mathcal{G}_b \dot{\mathcal{خامسا}}_b – [{\rm ad}_{\mathcal{خامسا}_b}]^{\rm T} \mathcal{G}_b \mathcal{خامسا}_bFب=GبV˙ب−[adVب]TGبVب.
If \mathcal{خامسا}_b = (\omega_b,v_b) = (\omega_b,0)خامساب=(ωb,vb)=(ωb,0) and \dot{\mathcal{خامسا}}_b = (\نقطة{\أوميغا}_b, \نقطة{الخامس}_b) = (0,0)V˙ب=(ω˙ب,الخامس˙ب)=(0,0), which of the following is true?
- \mathcal{F}_bFب is zero.
- \mathcal{F}_bFب is nonzero.
- Either of the above could be true.
لغز 06: الفصل 8 عبر 8.3, Dynamics of Open Chains
Q1. Consider an iron dumbbell consisting of a cylinder connecting two solid spheres at either end of the cylinder. The density of the dumbbell is 5600 kg/m^33. The cylinder has a diameter of 4 cm and a length of 20 سم. Each sphere has a diameter of 20 سم. Find the approximate rotational inertia matrix \mathcal{أنا}_bIب in a frame {ب} at the center of mass with z-axis aligned with the length of the dumbbell. Your entries should be written in units of kg-m^2, and the maximum allowable error for any matrix entry 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]⎣⎢⎡1.114.447.772.225.558.883.336.669.99⎦⎥⎤.
- 1
- [[0,0,0],[0,0,0],[0,0,0]]
Q2. The equations of motion for a particular 2R robot arm can be written M(\theta)\ddot{\theta} + ج(\theta,\نقطة{\theta}) + ز(\theta) = \tauM(θ)θ¨+ج(θ,θ˙)+ز(θ)=τ. The Lagrangian \mathcal{L}(\theta,\نقطة{\theta})L(θ,θ˙) for the robot can be written in components as
\mathcal{L}(\theta,\نقطة{\theta}) = \mathcal{L}^1(\theta,\نقطة{\theta}) + \mathcal{L}^2 (\theta,\نقطة{\theta}) + \mathcal{L}^3 (\theta,\نقطة{\theta}) +\ldotsL(θ,θ˙)=L1(θ,θ˙)+L2(θ,θ˙)+L3(θ,θ˙)+...
One of these components is \mathcal{L}^1 = \mathfrak{م} \نقطة{\theta}_1 \dot{\theta}_2 \sin\theta_2 L1=mθ˙1θ˙2sinθ2.
Find the right expression for the component of the joint torque \tau^1_1τ11 at joint 1 corresponding to the component \mathcal{L}^1L1.
- \tau^1_1 = \mathfrak{م} \ddot{\theta_2} \sin \theta_2 – \mathfrak{م} \dot \theta_2^2 \cos \theta_2τ11=mθ2¨sinθ2−mθ˙22cosθ2
- \tau^1_1 = \mathfrak{م} \ddot{\theta_2} \sin \theta_2 + \mathfrak{م} \dot \theta_2^2 \cos \theta_2τ11=mθ2¨sinθ2+mθ˙22cosθ2
- \tau^1_1 = \mathfrak{م} \ddot{\theta_2} \cos \theta_2 + \mathfrak{م} \dot \theta_2^2 \sin \theta_2τ11=mθ2¨cosθ2+mθ˙22sinθ2
Q3. Referring back to Question 2, find the right expression for the component of joint torque \tau^1_2τ21 at joint 2 corresponding to the component \mathcal{L}^1L1.
- \tau^1_2 = \mathfrak{م} \ddot{\theta_2} \sin \theta_2 + \mathfrak{م} \dot \theta_2^2 \cos \theta_2τ21=mθ2¨sinθ2+mθ˙22cosθ2
- \tau^1_2 = \mathfrak{م} \ddot{\theta_1} \sin \theta_2τ21=mθ1¨sinθ2
- \tau^1_2 = \mathfrak{م} \ddot{\theta_1} \sin \theta_2 + \mathfrak{م} \dot \theta_1 \dot \theta_2 \cos \theta_2τ21=mθ1¨sinθ2+mθ˙1θ˙2cosθ2
Q4. For a given configuration \thetaθ of a two-joint robot, the mass matrix is
M(\theta) = \left[
3با12
\حق],M(θ)=[3با12],
which has a determinant of 36-ab36−ab and eigenvalues \frac{1}{2} (15 \pm \sqrt{81 + 4 a b})21(15±81+4ab). What constraints must aا and bب satisfy for this to be a valid mass matrix? اختر كل ما ينطبق.
- ا < 6ا<6
- ب > 6ب>6
- ا > با>ب
- a = bا=ب
- ا<با<ب
- ا < \sqrt 6ا<6
Q5. An inexact model of the UR5 mass and kinematic properties is given below:
M_{01} = \left[
100001000010000.0891591
\حق], M_{12} = \left[
00−10010010000.280.1358501
\حق], M_{23} = \left[
1000010000100−0.11970.3951
\حق], M01=⎣⎢⎢⎢⎡100001000010000.0891591⎦⎥⎥⎥⎤,M12=⎣⎢⎢⎢⎡00−10010010000.280.1358501⎦⎥⎥⎥⎤,M23=⎣⎢⎢⎢⎡1000010000100−0.11970.3951⎦⎥⎥⎥⎤,
M_{34} = \left[
00−1001001000000.142251
\حق], M_{45} = \left[
10000100001000.09301
\حق], M_{56} = \left[
100001000010000.094651
\حق], M34=⎣⎢⎢⎢⎡00−1001001000000.142251⎦⎥⎥⎥⎤,M45=⎣⎢⎢⎢⎡10000100001000.09301⎦⎥⎥⎥⎤,M56=⎣⎢⎢⎢⎡100001000010000.094651⎦⎥⎥⎥⎤,
M_{67} = \left[
100000−10010000.082301
\حق],M67=⎣⎢⎢⎢⎡100000−10010000.082301⎦⎥⎥⎥⎤,
G_1 = {\tt diag}([0.010267495893,0.010267495893, 0.00666,3.7,3.7,3.7]),G1=diag([0.010267495893,0.010267495893,0.00666,3.7,3.7,3.7]),
G_2 = {\tt diag}([0.22689067591,0.22689067591,0.0151074,8.393,8.393,8.393]),G2=diag([0.22689067591,0.22689067591,0.0151074,8.393,8.393,8.393]),
G_3 = {\tt diag}([0.049443313556,0.049443313556,0.004095,2.275,2.275,2.275]),G3=diag([0.049443313556,0.049443313556,0.004095,2.275,2.275,2.275]),
G_4 = {\tt diag} ([0.111172755531 ,0.111172755531 ,0.21942, 1.219, 1.219 ,1.219]),G4=diag([0.111172755531,0.111172755531,0.21942,1.219,1.219,1.219]),
G_5 = {\tt diag} ([0.111172755531 ,0.111172755531, 0.21942 ,1.219 ,1.219 ,1.219]),G5=diag([0.111172755531,0.111172755531,0.21942,1.219,1.219,1.219]),
G_6 = {\tt diag} ([0.0171364731454 ,0.0171364731454, 0.033822 ,0.1879 ,0.1879, 0.1879]),G6=diag([0.0171364731454,0.0171364731454,0.033822,0.1879,0.1879,0.1879]),
{\tt Slist} = \left[
001000010−0.08915900010−0.08915900.425010−0.08915900.8172500−1−0.109150.8172500100.00549100.81725
\حق].Slist=⎣⎢⎢⎢⎢⎢⎢⎢⎡001000010−0.08915900010−0.08915900.425010−0.08915900.8172500−1−0.109150.8172500100.00549100.81725⎦⎥⎥⎥⎥⎥⎥⎥⎤.
Here are three versions for these UR5 parameters:
Given
θ=⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢0الأب/6الأب/4الأب/3الأب/22الأب/3⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥,θ˙=⎡⎣⎢⎢⎢⎢⎢⎢⎢0.20.20.20.20.20.2⎤⎦⎥⎥⎥⎥⎥⎥⎥,θ¨=⎡⎣⎢⎢⎢⎢⎢⎢⎢0.10.10.10.10.10.1⎤⎦⎥⎥⎥⎥⎥⎥⎥,g=⎡⎣00−9.81⎤⎦,Ftip=⎡⎣⎢⎢⎢⎢⎢⎢⎢0.10.10.10.10.10.1⎤⎦⎥⎥⎥⎥⎥⎥⎥,
use the function {\tt InverseDynamics}InverseDynamics in the given software to calculate the required joint forces/torques of the robot. 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] for \left[
1.112.223.33
\حق]⎣⎢⎡1.112.223.33⎦⎥⎤.
1
[0,0,0,0,0,0]
RunReset
أسبوع 02: الروبوتات الحديثة, دورة 3: Robot Dynamics Coursera Quiz Answers
لغز 01: Lecture Comprehension, Forward Dynamics of Open Chains (الفصل 8.5)
Q1. To derive the mass matrix M(\theta)M(θ) of an nن-joint open-chain robot, how many times would we need to invoke the recursive Newton-Euler inverse dynamics algorithm?
- 1 زمن.
- نن مرات.
- It is not possible to derive the mass matrix from the Newton-Euler inverse dynamics.
Q2. When calculating the mass matrix M(\theta)M(θ) using the Newton-Euler inverse dynamics, which of these quantities must be set to zero? اختر كل ما ينطبق.
- \thetaθ
- \نقطة{\theta}θ˙
- \ddot{\theta}θ¨
- The gravitational constant.
- The end-effector wrench \mathcal{F}_{{\rm tip}}Ftip.
لغز 02: Lecture Comprehension, Dynamics in the Task Space (الفصل 8.6)
Q1. Converting the joint-space dynamics to the task-space dynamics requires an invertible Jacobian, as well as the relationships \mathcal{خامسا} = J(\theta)\نقطة{\theta}الخامس =J(θ)θ˙ and \dot{\mathcal{خامسا}} = J\ddot{\theta} + \نقطة{J}\نقطة{\theta}V˙=Jθ¨+J˙θ˙, to find \Lambda(\theta)Λ(θ) and \eta(\theta,\mathcal{خامسا})η(θ,خامسا) in \mathcal{F} = \Lambda(\theta) \نقطة{\mathcal{خامسا}} + \eta(\theta,\mathcal{خامسا})F=Λ(θ)V˙+η(θ,خامسا).
Why do you suppose we left the dependence on \thetaθ, instead of writing it as a dependence on the end-effector configuration T \in SE(3)تي∈SE(3), which would seem to be more aligned with our task-space view?
- Either Tتي or \thetaθ could be used; there is no reason to prefer one to the other.
- The inverse kinematics of an open-chain robot does not necessarily have a unique solution, so we may not know the robot’s full configuration, and therefore the mass properties, given just Tتي.
لغز 03: Lecture Comprehension, Constrained Dynamics (الفصل 8.7)
Q1. A serial-chain robot has nن links and actuated joints, but it is subject to kك independent Pfaffian velocity constraints of the form A(\theta)\نقطة{\theta}=0ا(θ)θ˙=0. These constraints partition the nن-dimensional \tauτ space into orthogonal subspaces: a space of forces CC that do not create any forces against the constraints, and a space of forces Bب that do not cause any motion of the robot. What is the dimension of each of these spaces?
- CC هو (n-k)(ن-ك)-dimensional and Bب is kك-الأبعاد.
- CC is nن-dimensional and Bب is kك-الأبعاد.
- CC is kك-dimensional and Bب هو (n-k)(ن-ك)-الأبعاد.
- CC is kك-dimensional and Bب is nن-الأبعاد.
Q2. Let the constrained dynamics of a robot be \tau = M(\theta)\ddot{\theta} + ح(\theta,\نقطة{\theta}) + A^{\rm T}(\theta)\lambdaτ=M(θ)θ¨+ح(θ,θ˙)+اتي(θ)λ, where \lambda \in \mathbb{R}^kλ∈Rك. Let P(\theta)P(θ) be the matrix, as discussed in the video, that projects an arbitrary \tau \in \mathbb{R}^nτ∈Rن to P(\theta)\tau \in CP(θ)τ∈C, where the space CC is the same CC from the previous question. Then what is P(\theta) \tauP(θ)τ? اختر كل ما ينطبق.
- M(\theta)\ddot{\theta} + ح(\theta,\نقطة{\theta}) + A^{\rm T}(\theta)\lambdaM(θ)θ¨+ح(θ,θ˙)+اتي(θ)λ
- P(\theta)(M(\theta)\ddot{\theta} + ح(\theta,\نقطة{\theta}))P(θ)(M(θ)θ¨+ح(θ,θ˙))
- P(\theta)(M(\theta)\ddot{\theta} + ح(\theta,\نقطة{\theta})) +A^{\rm T}(\theta)\lambda)P(θ)(M(θ)θ¨+ح(θ,θ˙))+اتي(θ)λ)
لغز 04: Lecture Comprehension, Actuation, Gearing, and Friction (الفصل 8.9)
Q1. What is the typical reason for putting a gearhead on a motor for use in a robot?
- To increase torque (simultaneously reducing the maximum speed).
- To increase speed (simultaneously reducing the maximum torque).
Q2. Compared to a “direct drive” robot that is driven by motors without gearheads (G=1G=1), increasing the gear ratios has what effect on the robot’s dynamics? اختر كل ما ينطبق.
- The mass matrix M(\theta)M(θ) is increasingly dominated by the apparent inertias of the motors.
- The mass matrix M(\theta)M(θ) is increasingly dominated by off-diagonal terms.
- The mass matrix M(\theta)M(θ) is increasingly dominated by constant terms that do not depend on the configuration \thetaθ.
- The robot is capable of higher speeds but lower accelerations.
- The significance of velocity-product (Coriolis and centripetal) terms diminishes.
لغز 05: الفصل 8.5-8.7 و 8.9, Dynamics of Open Chains
Q1. A robot system (UR5) يعرف ب
M_{01} = \left[
100001000010000.0891591
\حق], M_{12} = \left[
00−10010010000.280.1358501
\حق], M_{23} = \left[
1000010000100−0.11970.3951
\حق], M01=⎣⎢⎢⎢⎡100001000010000.0891591⎦⎥⎥⎥⎤,M12=⎣⎢⎢⎢⎡00−10010010000.280.1358501⎦⎥⎥⎥⎤,M23=⎣⎢⎢⎢⎡1000010000100−0.11970.3951⎦⎥⎥⎥⎤,
M_{34} = \left[
00−1001001000000.142251
\حق], M_{45} = \left[
10000100001000.09301
\حق], M_{56} = \left[
100001000010000.094651
\حق], M34=⎣⎢⎢⎢⎡00−1001001000000.142251⎦⎥⎥⎥⎤,M45=⎣⎢⎢⎢⎡10000100001000.09301⎦⎥⎥⎥⎤,M56=⎣⎢⎢⎢⎡100001000010000.094651⎦⎥⎥⎥⎤,
M_{67} = \left[
100000−10010000.082301
\حق],M67=⎣⎢⎢⎢⎡100000−10010000.082301⎦⎥⎥⎥⎤,
G_1 = {\tt diag}([0.010267495893,0.010267495893, 0.00666,3.7,3.7,3.7]),G1=diag([0.010267495893,0.010267495893,0.00666,3.7,3.7,3.7]),
G_2 = {\tt diag}([0.22689067591,0.22689067591,0.0151074,8.393,8.393,8.393]),G2=diag([0.22689067591,0.22689067591,0.0151074,8.393,8.393,8.393]),
G_3 = {\tt diag}([0.049443313556,0.049443313556,0.004095,2.275,2.275,2.275]),G3=diag([0.049443313556,0.049443313556,0.004095,2.275,2.275,2.275]),
G_4 = {\tt diag} ([0.111172755531 ,0.111172755531 ,0.21942, 1.219, 1.219 ,1.219]),G4=diag([0.111172755531,0.111172755531,0.21942,1.219,1.219,1.219]),
G_5 = {\tt diag} ([0.111172755531 ,0.111172755531, 0.21942 ,1.219 ,1.219 ,1.219]),G5=diag([0.111172755531,0.111172755531,0.21942,1.219,1.219,1.219]),
G_6 = {\tt diag} ([0.0171364731454 ,0.0171364731454, 0.033822 ,0.1879 ,0.1879, 0.1879]),G6=diag([0.0171364731454,0.0171364731454,0.033822,0.1879,0.1879,0.1879]),
{\tt Slist} = \left[
001000010−0.08915900010−0.08915900.425010−0.08915900.8172500−1−0.109150.8172500100.00549100.81725
\حق].Slist=⎣⎢⎢⎢⎢⎢⎢⎢⎡001000010−0.08915900010−0.08915900.425010−0.08915900.8172500−1−0.109150.8172500100.00549100.81725⎦⎥⎥⎥⎥⎥⎥⎥⎤.
Here are three versions for these UR5 parameters above:
Given
\theta = \left[
0الأب/6الأب/4الأب/3الأب/22الأب/3
\حق]θ=⎣⎢⎢⎢⎢⎢⎢⎢⎡0الأب/6الأب/4الأب/3الأب/22الأب/3⎦⎥⎥⎥⎥⎥⎥⎥⎤,
\dot \theta = \left[
0.20.20.20.20.20.2
\حق]θ˙=⎣⎢⎢⎢⎢⎢⎢⎢⎡0.20.20.20.20.20.2⎦⎥⎥⎥⎥⎥⎥⎥⎤,
\ddot{\theta} = \left[
0.10.10.10.10.10.1
\حق]θ¨=⎣⎢⎢⎢⎢⎢⎢⎢⎡0.10.10.10.10.10.1⎦⎥⎥⎥⎥⎥⎥⎥⎤,
\mathfrak{ز} = \left[
00−9.81
\حق]g=⎣⎢⎡00−9.81⎦⎥⎤,
\mathcal{F}_{\نص{tip}} = \left[
0.10.10.10.10.10.1
\حق]Ftip=⎣⎢⎢⎢⎢⎢⎢⎢⎡0.10.10.10.10.10.1⎦⎥⎥⎥⎥⎥⎥⎥⎤
use the function {\tt MassMatrix}MassMatrix in the given software to calculate the numerical inertia matrix of the robot. The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.
Use Python syntax to express a matrix in the answer box:
[[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]⎣⎢⎡1.114.447.772.225.558.883.336.669.99⎦⎥⎤.
- 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]]
Q2. Referring back to Question 1, for the same robot system and condition, use the function {\tt VelQuadraticForces}VelQuadraticForces in the given software to calculate the Coriolis and centripetal terms in the robot’s dynamics. The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.
Use Python syntax to express a vector in the answer box:
[1.11,2.22,3.33] for \left[
1.112.223.33
\حق]⎣⎢⎡1.112.223.33⎦⎥⎤.
- 1
- [0,0,0,0,0,0]
Q3. Referring back to Question 1, for the same robot system and condition, use the function {\tt GravityForces}GravityForces in the given software to calculate the joint forces/torques required to overcome gravity. The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.
Use Python syntax to express a vector in the answer box:
[1.11,2.22,3.33] for \left[
1.112.223.33
\حق]⎣⎢⎡1.112.223.33⎦⎥⎤.
- 1
- [0,0,0,0,0,0]
Q4., Referring back to Question 1, for the same robot system and condition, use the function {\tt EndEffectorForces}EndEffectorForces in the given software to calculate the joint forces/torques required to generate the wrench \mathcal{F}_{{\rm tip}}Ftip. The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.
Use Python syntax to express a vector in the answer box:
[1.11,2.22,3.33] for \left[
1.112.223.33
\حق]⎣⎢⎡1.112.223.33⎦⎥⎤.
- 1
- [0,0,0,0,0,0]
Q5. Referring back to Question 1, for the same robot system and condition plus the known joint forces/torques
\tau = \left[
0.0128−41.1477−3.78090.03230.03700.1034
\حق] τ=⎣⎢⎢⎢⎢⎢⎢⎢⎡0.0128−41.1477−3.78090.03230.03700.1034⎦⎥⎥⎥⎥⎥⎥⎥⎤,
use the function {\tt ForwardDynamics}ForwardDynamics in the given software to find the joint acceleration. The maximum allowable error for any number is 0.01, so give enough decimal places where necessary.
Use Python syntax to express a vector in the answer box:
[1.11,2.22,3.33] for \left[
1.112.223.33
\حق]⎣⎢⎡1.112.223.33⎦⎥⎤.
- 1
- [0,0,0,0,0,0]
Q6. Assume that the inertia of a revolute motor’s rotor about its central axis is 0.005 kg m^22. The motor is attached to a zero-inertia 200:1 gearhead. If you grab the gearhead output and spin it by hand, what is the inertia you feel?
- 200 kg m^22
- 1 kg m^22
- 0.005 kg m^22
أسبوع 03: الروبوتات الحديثة, دورة 3: Robot Dynamics Coursera Quiz Answers
لغز 03: Lecture Comprehension, Point-to-Point Trajectories (الفصل 9 عبر 9.2, جزء 1 من 2)
Q1. A point robot moving in a plane has a configuration represented by (س,و)(س,و). The path of the robot in the plane is (1+ 2\cos (\pi s), 2\sin(\pi s)), \; s \in [0,1](1+2cos(πs),2sin(πs)),الصورة∈[0,1]. What does the path look like?
- An ellipse.
- A sine wave.
- A semi-circle.
- A circle.
Q2. Referring back to Question 1, assume the time-scaling of the motion along the path is s = 2t, \; t \in [0, 1/2]الصورة=2ر,ر∈[0,1/2]. At time tر, أين 0 \leq t \leq 0.50≤ر≤0.5, what is the velocity of the robot (\نقطة{س},\نقطة{و})(س˙,و˙)?
- (-4 \pi \sin(2 \pi t), 4 \pi \cos(2 \pi t))(−4الأبsin(2πt),4الأبcos(2πt))
- (-2\sin(2 \pi t), 2\cos(2 \pi t))(−2sin(2πt),2cos(2πt))
Q3. True or false? For a trajectory \theta(الصورة(ر))θ(الصورة(ر)), the acceleration \ddot{\theta}θ¨ is \frac{d\theta}{ds}\ddot{الصورة}dsdθالصورة¨.
- صحيح
- خاطئة
Q4. Let \mathcal{خامسا}_sVالصورة be the spatial twist that takes X_{الصورة,{\rm start}}Xs,start to X_{الصورة,{\rm end}}Xs,end in unit time. Which is an expression for the constant screw path that takes X_{الصورة,{\rm start}}Xs,start (at s=0الصورة=0) to X_{الصورة,{\rm end}}Xs,end (at s=1الصورة=1)?
- \exp([\mathcal{خامسا}_s s]) X_{الصورة,{\rm start}}, \; s \in [0,1]exp([خامساالصورةالصورة])Xs,start,الصورة∈[0,1]
- X_{الصورة,{\rm start}}\exp([\mathcal{خامسا}_s s]), \; s \in [0,1]Xs,startexp([خامساالصورةالصورة]),الصورة∈[0,1]
لغز 02: Lecture Comprehension, Point-to-Point Trajectories (الفصل 9 عبر 9.2, جزء 2 من 2)
Q1. For a fifth-order polynomial time scaling s(ر)الصورة(ر), t \in [0,تي]ر∈[0,تي], what is the form of \ddot{الصورة}(ر)الصورة¨(ر)?
- Third-order polynomial
- Fourth-order polynomial
- Fifth-order polynomial
لغز 03: Lecture Comprehension, Polynomial Via Point Trajectories (الفصل 9.3)
Q1. True or false? Third-order polynomial interpolation between via points ensures that the path remains inside the convex hull of the via points.
- صحيح
- خاطئة
Q2. A robot has 3 joints and it follows a motion interpolating 6 نقاط: a start point, an end point, و 4 other via points. The interpolation is by cubic polynomials. How many total coefficients are there to describe the motion of the 3-DOF robot over the motion consisting of 5 شرائح?
- 60
- 30
Q3. Referring again to Question 2, imagine we constrain the position and velocity of each DOF at the beginning and end of the trajectory, and at each of the 4 intermediate via points, we constrain the position (so the robot passes through the via points) but only constrain the velocity and acceleration to be continuous at each via point. Then how many total constraints are there on the coefficients describing the joint motions for all motion segments?
- 60
- 30
لغز 04: الفصل 9 عبر 9.3, Trajectory Generation
Q1. Consider the elliptical path in the (س,و)(س,و)-plane shown below. The path starts at (0,0)(0,0) and proceeds clockwise to (1.5,1)(1.5,1), (3,0)(3,0), (1.5,-1)(1.5,−1), and back to (0,0)(0,0). Choose the appropriate function of s \in [0,1]الصورة∈[0,1] to represent the path.
- س = 3 (1 – \cos 2 \pi s)س=3(1−cos2πs)
- y = \sin 2 \pi sو=sin2πs
- س = 1.5 (1 – \cos 2 \pi s)س=1.5(1−cos2πs)
- y = \sin 2 \pi sو=sin2πs
- س = 1.5 (1 – \cos s)س=1.5(1−cosالصورة)
- y = \sin sو=sinالصورة
- x = \cos 2 \pi sس=cos2πs
ص = 1.5 (1 – \sin 2 \pi s)و=1.5(1−sin2πs
Q2. Find the fifth-order polynomial time scaling that satisfies s(تي) = 1الصورة(تي)=1 and s(0) = \dot{الصورة}(0) = \ddot{الصورة}(0) = \dot{الصورة}(تي) = \ddot{الصورة}(تي) = 0الصورة(0)=الصورة˙(0)=الصورة¨(0)=الصورة˙(تي)=الصورة¨(تي)=0.
Your answer should be only a mathematical expression, a polynomial in tر, with coefficients involving Tتي. (Don’t bother to write “s(ر) = الصورة(ر)=”, just give the right-hand side.
Q3. If you want to use a polynomial time scaling for point-to-point motion with zero initial and final velocity, التسريع, and jerk, what would be the minimum order of the polynomial
Q4. Choose the correct acceleration profile \ddot{الصورة}(ر)الصورة¨(ر) for an S-curve time scaling.
- ا
- ب
- C
- د
Q5. Given a total travel time T = 5تي=5 and the current time t = 3ر=3, use the function {\tt QuinticTimeScaling}QuinticTimeScaling in the given software to calculate the current path parameter sالصورة, with at least 2 decimal places, corresponding to a motion that begins and ends at zero velocity and acceleration
Q6. Use the function {\tt ScrewTrajectory}ScrewTrajectory in the given software to calculate a trajectory as a list of N=10N=10 SE(3)SE(3) matrices, where each matrix represents the configuration of the end-effector at an instant in time. The first matrix is
X_{{\rm start}} = \left[
1000010000100001
\حق]Xstart=⎣⎢⎢⎢⎡1000010000100001⎦⎥⎥⎥⎤
and the 10th matrix is
X_{{\rm end}} = \left[
0100001010001231
\حق].Xend=⎣⎢⎢⎢⎡0100001010001231⎦⎥⎥⎥⎤.
The motion is along a constant screw axis and the duration is T_f = 10تيF=10. The parameter {\tt method}method equals 3 for a cubic time scaling. Give the 9th matrix (one before X_{{\rm end}}Xend) in the returned trajectory. The maximum allowable error for any matrix entry is 0.01, so give enough decimal places where necessary.
Use Python syntax to express a matrix in the answer box:
[[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]⎣⎢⎡1.114.447.772.225.558.883.336.669.99⎦⎥⎤.
- 1
- [[0,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,1]]
Q7. Referring back to Question 6, use the function {\tt CartesianTrajectory}CartesianTrajectory in the MR library to calculate another trajectory as a list of N=10N=10 SE(3)SE(3) matrices. Besides the same X_{{\rm start}}Xstart, X_{{\rm end}}Xend, T_fتيF and N = 10N=10, we now set {\tt method}method to 5 for a quintic time scaling. Give the 9th matrix (one before X_{{\rm end}}Xend) in the returned trajectory. The maximum allowable error for any matrix entry is 0.01, so give enough decimal places where necessary.
Use Python syntax to express a matrix in the answer box:
[[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]⎣⎢⎡1.114.447.772.225.558.883.336.669.99⎦⎥⎤.
- 1
- [[0,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,1]]
RunReset
أسبوع 04: الروبوتات الحديثة, دورة 3: Robot Dynamics Coursera Quiz Answers
لغز 01: Lecture Comprehension, Time-Optimal Time Scaling (الفصل 9.4, جزء 1 من 3)
Q1. When a robot travels along a specified path \theta(الصورة)θ(الصورة), torque or force limits at each actuator place bounds on the path acceleration \ddot{الصورة}الصورة¨. The constraints due to actuator iأنا can be written
\tau_i^{{\rm min}}(الصورة,\نقطة{الصورة}) \leq m_i(الصورة) \ddot{الصورة} + c_i(الصورة)\نقطة{الصورة}^2 + g_i(الصورة) \leq \tau_i^{{\rm max}}(الصورة,\نقطة{الصورة})τأناmin(الصورة,الصورة˙)≤مأنا(الصورة)الصورة¨+جأنا(الصورة)الصورة˙2+زأنا(الصورة)≤τأناmax(الصورة,الصورة˙).
What is one reason \tau_i^{{\rm min}}τimin and \tau_i^{{\rm max}}τimax might depend on \dot{الصورة}الصورة˙?
- The positive torque available from an electric motor typically increases as its positive velocity increases.
- The positive torque available from an electric motor typically decreases as its positive velocity increases.
Q2. At a particular state along the path, (الصورة,\نقطة{الصورة})(الصورة,الصورة˙), the constraints on \ddot{الصورة}الصورة¨ due to the actuators at the three joints of a robot are: L_1 = -10, U_1 = 10L1=−10,ال1=10; L_2 = 3, U_2 = 12L2=3,ال2=12; and L_3 = -2, U_3 = 5L3=−2,ال3=5. في (الصورة,\نقطة{الصورة})(الصورة,الصورة˙), what is the range of feasible accelerations \ddot{الصورة}الصورة¨?
- 3 \leq \ddot{الصورة} \leq 53≤الصورة¨≤5
- -10 \leq \ddot{الصورة} \leq 12−10≤الصورة¨≤12
Q3. If the robot is at a state (الصورة,\نقطة{الصورة})(الصورة,الصورة˙) where no feasible acceleration \ddot{الصورة}الصورة¨ exists that satisfies the actuator force and torque bounds, ماذا حدث
- One or more of the actuators is damaged.
- The robot leaves the path.
- The robot must begin to decelerate, \ddot{الصورة}<0الصورة¨<0.
لغز 02: Lecture Comprehension, Time-Optimal Time Scaling (الفصل 9.4, جزء 2 من 3)
Q1. Consider the figure below, showing 4 motion cones at different states in the (الصورة,\نقطة{الصورة})(الصورة,الصورة˙) الفراغ.
Which cone corresponds to U(الصورة,\نقطة{الصورة})=4, L(الصورة,\نقطة{الصورة})=-3ال(الصورة,الصورة˙)=4,L(الصورة,الصورة˙)=−3?
- ا
- ب
- C
- د
Q2. Considering the figure in Question 1, which cone corresponds to U(الصورة,\نقطة{الصورة})=4, L(الصورة,\نقطة{الصورة})=5ال(الصورة,الصورة˙)=4,L(الصورة,الصورة˙)=5?
- ا
- ب
- C
- د
Q3. Which cone corresponds to U(الصورة,\نقطة{الصورة})=5, L(الصورة,\نقطة{الصورة})=2ال(الصورة,الصورة˙)=5,L(الصورة,الصورة˙)=2?
- ا
- ب
- C
- د
Q4. Which cone corresponds to U(الصورة,\نقطة{الصورة})=-2, L(الصورة,\نقطة{الصورة})=-6ال(الصورة,الصورة˙)=−2,L(الصورة,الصورة˙)=−6?
- ا
- ب
- C
- د
Q5. Assume a time scaling s(ر) = \frac{1}{2}t^2الصورة(ر)=21ر2. How is this time scaling written as \dot{الصورة}(الصورة)الصورة˙(الصورة)? (Note that this particular time scaling does not satisfy \dot{الصورة}(1) = 0الصورة˙(1)=0.)
- \نقطة{الصورة} = \sqrt{2الصورة}الصورة˙=2الصورة.
- \نقطة{الصورة} = \frac{1}{2}s^2الصورة˙=21الصورة2.
لغز 03: الفصل 9.4, Trajectory Generation
Q1. Four candidate trajectories (ا, ب, C, and D) are shown below in the (الصورة,\نقطة{الصورة})(الصورة,الصورة˙) plane. Select all of the trajectories that cannot be correct, regardless of the robot’s dynamics. ملحوظة: It is OK for the trajectory to begin and end with nonzero velocity. (This is consistently one of the most incorrectly answered questions in this course, so think about it carefully!)
- ا
- ب
- C
- د
Q2. Four candidate motion cones at \dot{الصورة} = 0الصورة˙=0 (ا, ب, ج, and d) في ال (الصورة,\نقطة{الصورة})(الصورة,الصورة˙) plane are shown below. Which of these motion cones cannot be correct for any robot dynamics? (Do not assume that the robot can hold itself statically at the configuration.)
- ا
- ب
- ج
- د
3.
سؤال 3
We have been assuming forward motion on a path, \dot s > 0الصورة˙>0. What if we allowed backward motion on a path, \dot s < 0الصورة˙<0? This question involves motion cones in the (الصورة, \dot s)(الصورة,الصورة˙)-plane when both positive and negative values of \dot sالصورة˙ are available. Assume that the maximum acceleration is U(الصورة, \dot s) = 1ال(الصورة,الصورة˙)=1 (constant over the (الصورة, \dot s)(الصورة,الصورة˙)-plane) and the maximum deceleration is L(الصورة, \dot s) = -1L(الصورة,الصورة˙)=−1. For any constant sالصورة, which of the following are the correct motion cones at the five points where \dot sالصورة˙ takes the values \{-2, -1, 0, 1, 2\}{−2,−1,0,1,2}?
1 نقطة
ا
ب
C
د
4.
سؤال 4
Referring back to Question 3, assume the motion starts at (الصورة, \dot s) = (0, 0)(الصورة,الصورة˙)=(0,0) and follows the maximum acceleration Uال for time tر. Then it follows the maximum deceleration LL for time 2t2ر. Then it follows Uال for time tر. Which of the following best represents the integral curve?
1 نقطة
ا
ب
C
د
5.
سؤال 5
Below is a time-optimal time scaling \dot{الصورة}(الصورة)الصورة˙(الصورة) with three switches between the maximum and minimum acceleration allowed by the actuators. Also shown are example motion cones, which may or may not be correct.
Without any more information about the dynamics, which motion cones must be incorrect (أي, the motion cone is inconsistent with the optimal time scaling)? Select all that are incorrect (there may be more than one).
1 نقطة
- ا
- ب
- C
- د
- E
- F
- G
- H
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