机器人学导论(克雷格)第二章作业答案
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2、1
solution:
According to the equation of pure transition transformation,the new point after transition is as follows:
2、3
solution:
According to the constraint equations:
Thus,the matrix should be like this:
2、4
Solution:
= 2、7 Solution:
According to the equation of pure rotation transformation , the new coordinates are as follows: 2、9 Solution:
Acording to the equations for the bined transformations ,the new coordinates are as follows:
010051
05110
0030010310(,90)(5,3,6)(,90)001060
1004900
0110
00111A B
P Rot z Trans Rot x P -⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥-⎢
⎥⎢⎥⎢⎥⎢⎥⎢⎥
=⨯⨯⨯==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥
⎢
⎥⎢⎥⎢⎥⎢⎥⎢⎥
⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦o o
A
B
Transformations relative to the reference frame
Transformations relative to the current frame
2、10
P=Trans(5,3,6)Rot(x,90)Rot(a,90) P
1 0 0 5 1 0 0 0 0 -1 0 0
2 = 0 1 0
3 0 0 -1 0 1 0 0 0 3
0 0 1 6 0 1 0 0 0 0 1 0 5 0 0 0 1 0 0 0 1 0 0 0 1 1 2 = -2 8 1 2、12
0、369 -0、766 -0、601
0、819 0 -2、947 0、439 0、643 -5、38 0 0 0 1 0 -0、39 -3、82
1 0 -6 0 0、9
2 -3、79 0 0 0 1 2、14
a) For spherical coordinates we have (for posihon )
1)r·cos γ·sin β = 3、1375
2)r·sin γ·sin β = 2、195
3)r·cos β = 3、214
I) Assuming sin β is posihve, from a and b →γ=35°
from b and c →β=50°
units
from c → r=5
II) If sin β were negative、Then
units
γ=35°β=50°r=5
Since orientation is not specified, no more information is available to check the results、
b) For case I, substifate corresponding values of sinβ, cosβ, sinγ, cosγ
and r in sperical coordinates to get:
0、5265 -0、5735 0、6275 3、1375
Tsph(r,β,γ)=Tsph(35,50,5)= 0、3687 0、819 0、439 2、195
-0、766 0 0、6428 3、
214
0 0 0 1 2、16
Solution:
According to the equations given in the text book, we can get the Euler angles as follows:
①
Which lead to :
②
③
2、18
Solution:
①Since the hand will be placed on the object, we can obtain this: Thus:
②No,it can’t、
If so,the element at the position of the third row and the second column
should be 0、However, it isn’t、
x=5,y=1,z=0
According to the equations of the euler angles:
2、21
(a)
(b)
# θ d a α0-1 0 0 1-2 0 180 2-H 0 0 (c)
= =
= =
(d)
2、22
(a)
(b)
(c)
= =
= = = (d)
2、23
(a)
(b)
# θ d a α
1 0 0 90
2 0 4 0 1 -90
4 0 90
5 0 0 -90
6 5 0 0
(c)