(1) In equation (11) what is the problem of using minimum

norm acceleration in order to minimize the torque? And why (in term of the relation between the

torque and the acceleration?

(2) How could ρ (ρ is the radius of curvature of homogeneous solution space curve) effect in torque

and acceleration near internal singular configurations as shown below in S1, S2 and S3? What is the

value of ρ near internal singular configurations? Write the equations that show this effect?

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This question uses Paper 9: “Calculation of Repeatable Control Strategies for Kinematically Redundant Manipulators”

A) Clearly explain why the augmenting row �⃑�𝑣 should be the result of a gradient.

B) Given that �⃑�𝑣 is the result of a gradient, is any �⃑�𝑣 valid? Why or why not? Are there multiple choices for �⃑�𝑣 that yield the same result, valid and/or not valid?

2

3

1) The generalized inverse, 𝐺𝐺, satisfies �̇�𝜃 = 𝐺𝐺�̇�𝑥 and 𝐽𝐽𝐺𝐺 = 𝐼𝐼, where 𝐽𝐽 is the Jacobian and 𝐼𝐼 is the identity matrix, for non-singular matrices. The generalized inverse has the form 𝐺𝐺 = 𝐽𝐽+ + 𝑛𝑛�𝑗𝑗𝑤𝑤𝑇𝑇. Describe how choosing the correct 𝑤𝑤 will reduce 𝐺𝐺 to the pseudo inverse solution, 𝐽𝐽+.

2) Given 𝑤𝑤 = −(𝐽𝐽+)𝑇𝑇𝒗𝒗𝑛𝑛�𝐽𝐽∙𝒗𝒗

, describe what the 𝑣𝑣 vector should be in order for 𝑤𝑤 to be equal to 0. How

should you choose the 𝑣𝑣 vector to ensure that 𝑤𝑤 is equal to 0? Explain your method of choosing 𝑣𝑣 and how it is related to 𝐽𝐽+𝑇𝑇. (Hint: Draw the pictures of �̇�𝜃 space and �̇�𝑥 space and how the various 𝐽𝐽 matrices we’ve learned about transform vectors between the two spaces.)

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1. In paper 9, 𝐺𝐺 = 𝐽𝐽+ + 𝑛𝑛�𝐽𝐽𝑤𝑤𝑇𝑇 ○1 E

A but �̇�𝜃 = 𝐽𝐽+�̇�𝑥 + (𝐼𝐼 − 𝐽𝐽+𝐽𝐽)𝓏𝓏A○2 E

A what is the difference between equation A○1 E

A and A○2 E

A? (consider what the equation A○2 E

A could do but equationA○1 E

A can’t do).

2. To make the repeatability control strategy, we need to build an augmented matrix

𝐽𝐽𝑣𝑣 = �𝐽𝐽…𝑉𝑉𝑇𝑇� , which need to combine an 𝑉𝑉𝑇𝑇 vector to the Jacobian. The V vector could be

represented as 𝑉𝑉 = ∑ 𝑎𝑎𝑖𝑖𝑣𝑣𝑖𝑖𝑁𝑁𝑖𝑖=1 , where the 𝑣𝑣𝑖𝑖 is the base vector of a basis. To avoid the

algorithmic singularities, What is the main two constraints of choosing the V vector?

3. What is the shortcoming of the nearest optimal repeatable control strategy(NORCS)?

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Paper 9 (1) For a PPR manipulator, if we want to keep the joints in the middle of their joint

limits

1 1 π 4⎡⎣

⎤⎦

T

, we can optimize the cost function

g(θ ) = (d1 −1)2 + (d2 −1)2 + (θ3 −π 4)2

Calculate the gradient ∇g

(2) The null vector of a PPR manipulator is nJ =

1

2sinθ3 −cosθ3 1⎡

⎣⎤⎦

T

.

Calculate the augmenting vector v = ∇(∇g ⋅nJ ) .

(3) If d1=0.5, d2=0.5, θ3 = π 3 , calculate Jv−1 and G using the above the augmenting

vector v .

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Homework problem 6 (1) Explain the physical meaning of the equation below :

𝑣 = ∇(∇𝑔 ∙ 𝑛)) (2) What is the shortcoming when applying augmenting technique? Brief explain this

shortcoming.

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In order to choose an optimal repeatable control strategy, one can easily invert an

augmenting vector into Jacobian as mentioned in article 9:

𝐽𝑣 = [𝐽

𝑣𝑇]

where v is a null vector of 𝐺𝑇. And G which is the generalized inversed Jacobian is

given by

𝐺 = 𝐽+ + �̂�𝐽𝑤𝑇

where �̂�𝐽 is a unit length null vector of J and where w uniquely determines G.

1. What is the constraint between augmenting vector v and J? How does one calculate

this augmenting vector?

2. What is the disadvantage of using this technique and the condition of causing it?

3. According to the experiment data of performing NUSAM and NORCS in the paper,

which algorithm needs more computations in terms of higher-dimensional subspaces?

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Consider the 2 cases of planar robot below. Assume that the end-effectors have reached the desired position. And there is a moving object with the velocity shown in the figure below. Is it necessary for the end-effectors to move to avoid the object for the two cases? If not, is there velocity or acceleration we need to cope with to realize the obstacle avoidance with no end effector motion? Derive the equation representing velocity or acceleration and explain how to cope with it.

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Homework Question

a) Give the equations for joint angle and end effector accelerations and torque due to minimum norm acceleration and how are they changed due to kinematic redundancy(end effector is stationary)

b) How does the acceleration along the homogenous solution affect the torque requirements.

c) Explain about this .

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Question:

According paper 9

1) G = ?

2) Consider the planar manipulator which consists of two orthogonal prismatic joints

and a third joint.

a) J = ?

b) n̂J = ?

c) w = ?

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1. Explain the positive feedback scenario encountered in local torque minimization of

redundant manipulators. Give an example.

2. State two conditions that must be satisfied by basis used in NUSAN technique.

3. For a given gramian matrix M, find the optimal coefficients.

M = [

4 2 2.4 2.62 1 1.2 1.32.42.6

1.21.3

1.441.56

1.561.69

]

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1) Consider a planar robot with three revolute joints. The curves below represent the set ofhomogeneous solutions (or null curves) for one such robot. These curves represent the possible re-configuration for each end-effector position or configuration of the robot.

a) Draw out the any configuration that corresponds to the origin of the figure shownbelow.

b) Each curve in this figure represents the null curves (homogeneous solutions) that corresponds tothe configuration of the robot. In the figure shown below, there will be one homogeneous solutioncurve (or one null curve) which will be a straight line along the θ1 axis. What configuration doesthis straight line along the θ1 axis represent ? Draw one such configuration. Explain why it is astraight line rather than ellipses or ellipsoids like the ones in the figure below.

c) These curves that are shown below, break apart sometimes to form two sets of curves. What doesthis mean (or) under what circumstances does these curves break apart.

2) Given a choice, which optimization is better ? a) Local or Global ? b) Also, under what circumstances can global optimization be used ?

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Homework #6 Problem Problem (Please recall the paper 7):

(a) Please show how you obtain the equation below according to the paper?

�̇�𝜃 = 𝐽𝐽+�̇�𝑥 + [ 𝐽𝐽2(𝐼𝐼 − 𝐽𝐽+𝐽𝐽)]+(�̇�𝑥2 − 𝐽𝐽2𝐽𝐽+�̇�𝑥)

(b) Consider the techniques in (a), what are the defects if the homogeneous joint velocity or homogeneous solution is used for the above equation?

(c) Please show how you obtain the equation below which gives the resulting joint angle

acceleration required to maintain the desired configuration of the end effector?

�̈�𝜃 = 𝐽𝐽+��̇�𝑥 − 𝐽𝐽�̇̇�𝜃� + (𝐼𝐼 − 𝐽𝐽+𝐽𝐽)�̈�𝜑

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i) What is the motivation for the Paper 7 “Kinetic Limitations on the Use of Redundancy in Robotic Manipulators”?

ii) What is Coriolis and Centrifugal effect? iii) What is the problem with the solution that locally minimizes norm of the torque in the

below equation?

�̈� = 𝐽+ (�̈� − 𝐽�̇�) − [𝐻(𝐼 − 𝐽+𝐽)]+�̃�

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Given J = [1 0 −

1

2

0 1√3

2

] and v = [0.25

1.1401−0.1124

]. Find Jv-1 from the inverse of Jv and

from 𝐽𝑣−1 = [G ⋮

�̂�𝐽

�̂�𝐽∗𝑣]. Show they produce the same matrix.

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1. Give the dynamics equation of a manipulator and explain the H, c, and g terms in the

equation.

2. Why is the inertia matrix strictly positive definite?

3. Why is there infinite accelerations near singular configurations?

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