F23_ROB1_HW3_Solution
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Rensselaer Polytechnic Institute *
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Course
4480
Subject
Mechanical Engineering
Date
Dec 6, 2023
Type
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6
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Homework 3
For each problem show all steps and explain your reasoning. Inclusion of necessary steps is required to
receive full credit.
For this homework, the Dofbot will be the reference arm. This is the robot that will be used for the project.
A sketch of the arm with rotation axes marked is given in figure
1
.
h hs hs aYo ny JL Note h S paralel to z h3 hy are parallel to h h3 hs iS Parallel to X h5 Figure 1.
Dofbot Sketch
Table 1.
Lengths shown in Dofbot sketch. Note: Values are in
millimeters
ℓ
0
61
ℓ
1
43.5
ℓ
2
82.85
ℓ
3
82.85
ℓ
4
73.85
ℓ
5
54.57
The values provided were measured from the .step file (provided on Piazza).
Lengths
ℓ
1
, ℓ
2
, ℓ
3
, ℓ
4
are
measured between servo centers.
ℓ
0
is from the top side of the base plate to the center of servo 1.
ℓ
5
is
from the center of servo 5 to the inner edge of the gripper.
The center of the servo is defined as shown in figure
2
.
Figure 2.
Rough servo sketch with ”center” indicated by a red star.
1
2
For each problem where you must define subproblems and associated variables, use the
following rules (where SP is used as shorthand for subproblem):
•
SP1:
R
(
k, q
)
P
1
=
P
2
– Variables to provide:
k, P
1
, P
2
•
SP2:
R
(
k
1
, q
1
)
P
1
=
R
(
k
2
, q
2
)
P
2
– Variables to provide:
k
1
, k
2
, P
1
, P
2
•
SP3:
d
=
||
P
2
−
R
(
k, q
)
P
1
||
– Variables to provide:
d, k, P
1
, P
2
•
SP4:
h
T
R
(
k, q
)
P
=
d
– Variables to provide:
h, d, k, P
Question 1.
Given
R
=
R
x
(
α
)
R
y
(
β
)
R
z
(
γ
), provide a method to find
α, β, γ
using subproblems. For each
case, state the angle(s) to be found, the subproblem used, and the variables that would be supplied to the
subproblem function. [20pts]
Solution:
There are two possible solutions methods:
METHOD 1
First get
β
e
T
x
Re
z
=
e
T
x
R
x
(
α
)
R
y
(
β
)
R
z
(
γ
)
e
z
=
e
T
x
R
y
(
β
)
e
z
Solve for
β
using subproblem 4. The variables to provide are:
h
=
e
x
k
=
e
y
P
=
e
z
d
=
e
T
x
Re
z
Then find
α
Re
z
=
R
x
(
α
)
R
y
(
β
)
R
z
(
γ
)
e
z
=
R
x
(
α
)
R
y
(
β
)
e
z
Solve for
α
using subproblem 1. The variables to provide are:
k
=
e
x
P
1
=
R
y
(
β
)
e
z
P
2
=
Re
z
Lastly, find
γ
e
T
x
R
=
e
T
x
R
x
(
α
)
R
y
(
β
)
R
z
(
γ
) =
e
T
x
R
y
(
β
)
R
z
(
γ
)
R
T
e
x
=
R
z
(
−
γ
)
R
y
(
−
β
)
e
x
= rot(
−
e
z
, γ
)
R
y
(
−
β
)
e
x
Solve for
γ
using subproblem 1. The variables to provide are:
k
=
−
e
z
P
1
=
R
y
(
−
β
)
e
x
P
2
=
R
T
e
x
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
METHOD 2
3
Find
α
and
β
simultaneously
Re
z
=
R
x
(
α
)
R
y
(
β
)
R
z
(
γ
)
e
z
=
R
x
(
α
)
R
y
(
β
)
e
z
R
x
(
−
α
)
Re
z
=
R
y
(
β
)
e
z
rot(
−
e
x
, α
)
Re
z
= rot(
e
y
, β
)
e
z
Solve for
α
and
β
using subproblem 2. The variables to provide are:
k
1
=
−
e
x
k
2
=
e
y
P
1
=
Re
z
P
2
=
e
z
Then find
γ
R
=
R
x
(
α
)
R
y
(
β
)
R
z
(
γ
)
R
y
(
−
β
)
R
x
(
−
α
)
R
=
R
z
(
γ
)
R
y
(
−
β
)
R
x
(
−
α
)
Re
x
=
R
z
(
γ
)
e
x
(Could also use
e
y
. Cannot use
e
z
.)
R
y
(
−
β
)
R
x
(
−
α
)
Re
x
= rot(
e
z
, γ
)
e
x
Solve for
γ
using subproblem 1. The variables to provide are:
k
=
e
z
P
1
=
e
x
P
2
=
R
y
(
−
β
)
R
x
(
−
α
)
Re
x
Question 2.
Given a desired
P
0
T
and
R
0
T
, we can solve for
q
i
(where
i
= 1
,
2
,
3
,
4
,
5) for the Dofbot using
subproblems. Let
L
1
=
ℓ
0
+
ℓ
1
and
L
4
=
ℓ
4
+
ℓ
5
.
From our forward kinematics we have:
R
0
T
=
R
z
(
q
1
)
R
y
(
−
q
2
)
R
y
(
−
q
3
)
R
y
(
−
q
4
)
R
x
(
−
q
5
) :=
R
z
(
q
1
)
R
y
(
−
θ
)
R
x
(
−
q
5
)
P
0
T
=
L
1
e
z
+
R
z
(
q
1
)
R
y
(
−
q
2
) (
ℓ
2
e
x
+
R
y
(
−
q
3
) (
−
ℓ
3
e
z
−
R
y
(
−
q
4
)
L
4
e
x
))
=
L
1
e
z
+
R
z
(
q
1
)
R
y
(
−
q
2
)
ℓ
2
e
x
−
R
z
(
q
1
)
R
y
(
−
q
2
−
q
3
)
ℓ
3
e
z
−
R
z
(
q
1
)
R
y
(
−
θ
)
L
4
e
x
Note: Each of the following parts (except f) asks three questions: Fill in the placeholders,
specify which subproblem to use, and define the variable values that would be provided to
the subproblem function. For example, if the subproblem is SP0, state
k
= [
·
]
,
P
1
= [
·
]
,
P
2
= [
·
]
.
[30pts]
2.a
) First we will get
θ
from
R
0
T
. What vectors (
e
x
, e
y
, e
z
, e
T
x
, e
T
y
, e
T
z
) belong in slots
A
and
B
?
A
R
0
T
B
=
A
rot(
−
e
y
, θ
)
B
Which subproblem would be used to find
θ
?
What would be provided to the subproblem function?
Solution:
Substitute
e
T
z
for A,
e
x
for B.
Solve using subproblem 4.
Values needed by the function:
h
=
e
z
k
=
−
e
y
P
=
e
x
d
=
e
T
z
R
0
T
e
x
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4
2.b
) Next we will get
q
1
from
R
0
T
. What vector (
e
x
, e
y
, e
z
, e
T
x
, e
T
y
, e
T
z
) belongs in slot
A
?
R
0
T
A
= rot(
e
z
, q
1
)rot(
−
e
y
, θ
)
A
Which subproblem would be used to find
q
1
?
What would be provided to the subproblem function?
Solution:
Substitute
e
x
for A.
Solve using subproblem 1.
Values needed by the function:
k
=
e
z
P
1
=
R
y
(
−
θ
)
e
x
P
2
=
R
0
T
e
x
2.c
) Now we will get
q
5
from
R
0
T
. What vector (
e
x
, e
y
, e
z
, e
T
x
, e
T
y
, e
T
z
) belongs in slot
A
?
A
T
R
0
T
=
A
T
rot(
−
e
y
, θ
)rot(
−
e
x
, q
5
)
R
T
0
T
A
= rot(
e
x
, q
5
)rot(
e
y
, θ
)
A
Which subproblem would be used to find
q
5
?
What would be provided to the subproblem function?
Solution:
Substitute
e
z
for A.
Solve using subproblem 1.
Values needed by the function:
k
=
e
x
P
1
=
R
y
(
θ
)
e
z
P
2
=
R
T
0
T
e
z
2.d
) We can get
q
3
from
P
0
T
after some manipulation. What operator belongs in slot
O
?
P
0
T
−
L
1
e
z
=
R
z
(
q
1
)
R
y
(
−
q
2
)
ℓ
2
e
x
−
R
z
(
q
1
)
R
y
(
−
q
2
)
R
y
(
−
q
3
)
ℓ
3
e
z
−
R
z
(
q
1
)
R
y
(
−
θ
)
L
4
e
x
Group known terms on the left-hand side
R
z
(
−
q
1
) (
P
0
T
−
L
1
e
z
) +
R
y
(
−
θ
)
L
4
e
x
=
R
y
(
−
q
2
) (
ℓ
2
e
x
−
R
y
(
−
q
3
)
ℓ
3
e
z
)
Let
P
′
=
R
z
(
−
q
1
) (
P
0
T
−
L
1
e
z
) +
R
y
(
−
θ
)
L
4
e
x
P
′
=
R
y
(
−
q
2
) (
ℓ
2
e
x
−
R
y
(
−
q
3
)
ℓ
3
e
z
)
O
P
′
O
=
O
ℓ
2
e
x
−
rot(
−
e
y
, q
3
)
ℓ
3
e
z
O
Which subproblem would be used to find
q
3
?
What would be provided to the subproblem function (write your answer in terms of
P
′
where
applicable)?
Solution:
Substitute
||
(the norm operator) for O.
Solve using subproblem 3.
Values needed by the function:
d
=
||
P
′
||
P
2
=
ℓ
2
e
x
k
=
−
e
y
P
1
=
ℓ
3
e
z
5
2.e
) Now we can get
q
2
from
P
0
T
. What vector (
e
x
, e
y
, e
z
, e
T
x
, e
T
y
, e
T
z
) belongs in slot
A
?
P
′
= rot(
−
A
, q
2
) (
ℓ
2
e
x
−
R
y
(
−
q
3
)
ℓ
3
e
z
)
Which subproblem would be used to find
q
2
?
What would be provided to the subproblem function (write your answer in terms of
P
′
where
applicable)?
Solution:
Substitute
e
y
for A.
Solve using subproblem 1.
Values needed by the function:
k
=
−
e
y
P
1
=
ℓ
2
e
x
−
R
y
(
−
q
3
)
ℓ
3
e
z
P
2
=
P
′
2.f
) Finally, we can get
q
4
from our definition of
θ
. What belongs in slot A?
q
4
=
A
Solution:
Substitute
θ
−
q
2
−
q
3
for A.
Question
3.
Complete
”Dofbot
IK
-
Subproblems”
on
MATLAB
Grader
in
Homework
3
based
on
answers
to
question
2.
[20pts]
Solution:
See
uploaded
file
invkin_subproblems_Dofbot.m
Question
4.
Define
t he
a rm
J acobian
f or
t he
D ofbot.
Write
your
a nswer
i n
t erms
o f
h
i
,
R
0
i
,
a nd
P
iT
.
[10pts]
Solution:
Use
our
definition
from
class:
{
J
T
}
0
=
h
1
R
01
h
2
R
02
h
3
R
03
h
4
R
04
h
5
h
×
1
R
01
P
1
T
(
R
01
h
2
)
×
R
02
P
2
T
(
R
02
h
3
)
×
R
03
P
3
T
(
R
03
h
4
)
×
R
04
P
4
T
(
R
04
h
5
)
×
R
05
P
5
T
Question
5.
Complete
”Dofbot
Jacobian”
on
MATLAB
Grader
in
Homework
3
based
on
answers
to
question
4.
[20pts]
Solution:
See
uploaded
file
Jacobian_Dofbot.m Question
Bonus
for
4000-level,
Required
for
6000-level.
Given
the
Jacobian
for
q
where
q
i
=
90
o
is
equal
to:
{
J
(
q
)
}
0
=
3
777
777
777
777
777
777
777
777
55
0
1
1
1
0
0
0
0
0
0
1
0
0
0
1
0
0
0
0
0
0
−
0
.
2941
−
0
.
2113
−
0
.
1284
0
0
0
0
0
0
44
888
888
888
888
888
888
888
888
66
What can you say about rotation around and translation along each axis (feasible/infeasible, what joints
to use for instantaneous motion)? [10pts]
Solution:
•
Rotation about
⃗x
0
: Feasible using joints 2, 3, 4
•
Rotation about
⃗
y
0
: Infeasible
•
Rotation about
⃗
z
0
: Feasible using joints 1, 5
6
•
Translation along
⃗x
0
: Infeasible
•
Translation along
⃗
y
0
: Feasible using joints 2, 3, 4
•
Translation along
⃗
z
0
: Infeasible
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Find M, the magnitude of the moment, and 0, the angle between r and the force F.
Express your answers in pound-feet and degrees to three significant figures separated by a comma.
ΑΣΦ
?
vec
M,0 =
lb · ft,degrees
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Can someone please help me to solve the following question following all instructions and drawing needed table and diagrams and showing all work neatly. Please and Thank you!!
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Help with this would be great, thanks!
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help here would be great, thanks!
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The first image is a practice problem I am working out. The second image is showing how I've worked it out so far. But when I try to do the math to find what T=, I can't get it right on my calculator.
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Help!!! Answer all parts correctly!! Please
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