机器人学导论(克雷格)第二章作业答案

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2.1

solution:

Accordingtotheequationofpuretransitiontransformation,thenewpointaftertransitionisasfollows:

2.3

solution:

Accordingtotheconstraintequations:

Thus,thematrixshouldbelikethis:

2.4

Solution:

=

2.7

Solution:

Accordingtotheequationofpurerotationtransformation,thenewcoordinatesareasfollows:

2.9

Solution:

Acordingtotheequationsforthecombinedtransformations,thenewcoordinatesareasfollows:

Transformationsrelativetothereferenceframe

Transformationsrelativetothecurrentframe

B

2.10

A

P=Trans<5,3,6>Rot<x,90>Rot<a,90>P

100510000-1002

=010300-1010003

0016010000105

0001000100011

2

=-2

8

1

2.12

-1

0.5270.369-0.766-0.601

T1=-0.5740.8190-2.947

0.6280.4390.643-5.38

0001

-1

0.920-0.39-3.82

T2=010-6

0.3900.92-3.79

0001

2.14

Forsphericalcoordinateswehave〔forposihon>

r·cosγ·sinβ=3.1375

r·sinγ·sinβ=2.195

r·cosβ=3.214

I>Assumingsinβisposihve,fromaandb→γ=35°

units

frombandc→β=50°

fromc→r=5

units

II>Ifsinβwerenegative.Then

γ=35°β=50°r=5

Sinceorientationisnotspecified,nomoreinformationisavailabletochecktheresults.

b>ForcaseI,substifatecorrespondingvaluesofsinβ,cosβ,sinγ,cosγ andrinspericalcoordinatestoget:

0.5265-0.57350.62753.1375

Tsph<r,β,γ>=Tsph<35,50,5>=0.36870.8190.4392.195

-0.76600.64283.214

0001

2.16

Solution:

Accordingtotheequationsgiveninthetextbook,wecangettheEuleranglesasfollows:

Whichleadto:

2.18

Solution:

Sincethehandwillbeplacedontheobject,wecanobtainthis:

Thus:

No,itcan’t.

Ifso,theelementatthepositionofthethirdrowandthesecondcolumnshouldbe0.However,itisn’t.

x=5,y=1,z=0

Accordingtotheequationsoftheeulerangles:

2.21

<a>

<b>

#

θ

d

a

α

0-1

0

0

1-2

0

180

2-H

0

0

<c>

==

==

<d>

2.22

<a>

<b>

#

θ

d

a

α

0-1

+90

0

0

90

1-2

0

0

-90

=

3-H

0

0

0