Homework answers / question archive / HW Assignment 6 (Total: 100 points + 20 extra credits) Z0 X0 X3 Z3 X3 Z3 Due: 9 AM, Mon

HW Assignment 6 (Total: 100 points + 20 extra credits) Z0 X0 X3 Z3 X3 Z3 Due: 9 AM, Mon

Mechanical Engineering

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HW Assignment 6 (Total: 100 points + 20 extra credits)

Z0

X0

X3

Z3

X3

Z3

Due: 9 AM, Mon. 12/6/2021

X0

Z0

X3

Z3

L

1. Find the Jacobian matrix of each robotic arm with three degrees of freedom shown below using the DH parameters and the given ?30, the transformation matrix. L and H are assumed as known parameters. (40 pts)

H

X2

Z2

X1

Z1

X2

Z2

X1, Z2

Z1

Z1

X1

X0

Z0

Z4

Z2

X2

Z3

X3,X4

Z5

X5

Z1

X1, Z2

X0

Z3

X3

Z4

X4

L

L

d2var

X2

θ2var

θ1var

θ1var

d3var

d2var

d3var

d3var

d1var

Z0, Z1

X0, X1

1. Cartesian Robot (10 points)

Cartesian Robot
Joint # (i)	ai	αi	di	θi
1 ( {0} -> {1} )	0	π/2	d1var	0
2 ( {1} -> {2} )	0	π/2	d2var	π/2
3 ( {2} -> {3} )	0	0	d3var+ L	0

?30=[ 0010−10100?3???+?−?2????1??? 0 0 0 1]

2. Cylindric Robot (15 points)

?30=[ ??1???0??1?????1???0−??1???010??1???(?3???+?)−??1???(?3???+?)?2???0 0 0 1]

Cylindric Robot
Joint # (i)	ai	αi	di	θi
1 ( {0} -> {1} )	0	0	0	θ1var
2 ( {1} -> {2} )	0	π/2	d2var	0
3 ( {2} -> {3} )	0	0	d3var + L	0

3. Polar Robot (15 points)

?30=[ −??1?????2?????1?????1?????2???−??1?????2???−??1?????1?????2?????2???0??2?????1?????2???(?3???+?)??1?????2???(?3???+?)?+??2???(?3???+?) 0 0 01]

Polar Robot
#	ai	αi	di	θi
1	0	π/2	H	θ1var
2	0	π/2	0	θ2var + π/2
3	0	0	d3var + L	0

2. Find the Jacobian matrix of the 4DOF manipulator on a fire truck below in a symbolic form using the given mfile (Assignment6_Fall2021.m file). D, H, L, and W are assumed as known parameters. (30 pts)

D-H Parameters
Joint # (i)	ai	αi	di	θi
1 ( {0} -> {1} )	-D	π/2	H	θ1var
2 ( {1} -> {2} )	0	π/2	0	θ2var + π/2
3 ( {2} -> {3} )	0	-π/2	d3var + L	0
4 ( {3} -> {4} )	W	0	0	θ4var - π/2

3. Find the Jacobian matrix of the LabVolt robot arm below in a symbolic form using the given mfile (Assignment6_Fall2021.m file). a2, a3, d1, and d5 are assumed as known parameters. (30 pts)

(Z1, Z2, and Z3 are pointing into the page)

 

a3

a2

θ1var

 

 

LabVolt 5150
Joint # (i)	ai [mm]	αi [° or radian]	di [mm]	θi [° or radian]
1 ( {0} -> {1} )	0	90° or π/2	d1	θ1var
2 ( {1} -> {2} )	a2	0	0	θ2var
3 ( {2} -> {3} )	a3	0	0	θ3var
4 ( {3} -> {4} )	0	90° or π/2	0	θ4var
5 ( {4} -> {5} )	0	0	d5	θ5var

4. (Optional for Extra Credit) Describe physical meanings of the 5-th column of manipulator Jacobian from Prob. 3) as detail as possible. (20 pts)