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:
1002350
10358(,,)00147110
00111trans
x y z old P Trans d d d P ⎡⎤⎡⎤⎡⎤
⎢⎥⎢⎥⎢⎥⎢
⎥⎢⎥⎢⎥=⨯==⎢⎥⎢⎥⎢⎥
⎢⎥⎢⎥⎢⎥
⎣⎦⎣⎦⎣⎦
2.3 solution:
According to the constraint equations:
0;0;01
n a n o a o n •=•=•==r r r r r r r Thus,the matrix should be like this:
0015001
51003100301020102000
1000
1or --⎡⎤⎡⎤⎢⎥⎢⎥-⎢
⎥⎢
⎥⎢⎥⎢⎥--⎢
⎥⎢
⎥⎣⎦⎣⎦
2.4 Solution:
X Y Z
P P P ⎛⎫ ⎪ ⎪ ⎪⎝⎭=cos 0sin 010sin 0cos θθθθ⎛⎫ ⎪ ⎪ ⎪-⎝
⎭0n a P P P ⎛⎫
⎪
⎪ ⎪⎝⎭
2.7 Solution:
According to the equation of pure rotation transformation , the
new coordinates are as follows:
10022(,45)0
3402
22new
P rot x P ⎡⎤⎡⎤
⎢⎥
⎢⎥⎡⎤⎢
⎢⎢⎥⎢⎢=⨯==⎢⎥⎢⎢⎢⎥⎢⎥⎢⎥⎣⎦⎢⎥⎢⎥
⎢⎥⎢⎥⎣
⎦⎣⎦
o 2.9 Solution:
Acording to the equations for the combined transformations ,the
new coordinates are as follows:
B
010051
00
05110
0030010310(,90)(5,3,6)(,90)001060
1004900011000111A B
P Rot z Trans Rot x P -⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥-⎢
⎥⎢⎥⎢⎥⎢⎥⎢⎥
=⨯⨯⨯==⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥
⎢
⎥⎢⎥⎢⎥⎢⎥⎢⎥
⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦
o o
Transformations relative to the reference frame
Transformations relative to the current frame
2.10
A
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 0 0 1
T2 = 0 1 0 -6
0.39 0 0.92 -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,
sin γ, cos γ 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:
arctan 2(,)arctan 2(,)y x y x a a or a a Φ=--
Which lead to :
21535or Φ=o o
arctan 2(,)0180x y x y n S n C o S o C or ψ=-Φ+Φ-Φ+Φ=o o
arctan 2(,)5050x y z a C a S a or θ=Φ+Φ=-o o