[ ]

cosd (θ) −sind(θ) 0

1. Rz (θ )= sin ⁡d (θ) cosd ⁡(θ) 0
0 0 1

[ ]
0.866 −0.5 0
Rz (30) = 0.5 0.866 0
0 0 1

AT =[ Rz(30),[10;5;0];0 0 0 1]

[ ]
0..866 −0.5 0 10
T 0.5 0.866 0 5
A = 0 0 1 0
(by using matlab)
0 0 0 1

[]
3
B
P =7
0

[ ]
9.098
A T B
P = A * P = 12.562
0

[ ]
cosd ( θ ) −sind ( θ ) 0
2. θ ¿
Rz( = sin d (θ ) cosd ( θ ) 0
0 0 1

[ ]
0.866 −0.5 0
Rz(30)= 0.5 0.866 0
0 0 1

[]
0
A
P =2
1
0

[ ]
1
A A
P =Rz(30 ¿∗¿ P = 1.732
2 1
0

[ ]
0.866 −0.5 0 10
0.5 0.866 0 5
3. T=
0 0 1 0
0 0 0 1
 []
3
A
P1 = 7
0

[ ]
9.098
A A
P2 =T* P1 = 12.562
0

[ ]
0.866 −0.5 0 4
0.5 0.866 0 3
4. T BA=
00 0 1 0
0 0 0 1
B A
T A=inv(T B )

[ ]
0.866 0.5 0 −4.964
−0.5 0.866 0 −4.964
=
0 0 1 0
0 0 0 1

5. P=3i+5j=2k

[]
3
P= 5
2

(a) with a scale factor of 2

[]
6
P= 10
4

(b) if it were to describe a direction as a unit vector

λ=√ 32 +52 +22=6.16

[ ]
0.487
1
Punit = *P = 0.811
λ
0.324

[ ]
0.371
0.557
6. Q unit =
qz
0

The unit vector’s length must be 1.

λ=√ 0.3712+ 0.5572 +q 2z =1 ,q z =0.743

 [ ]
0.371
Qunit = 0.557
0.743
0

[ ]
1.855
2.785
P=5*Qunit =
3.715
0

[ ]
? 0 ? 5
0.707 ? ? 3
7. F=
? ? 0 2
0 0 0 1

[ ]
nx ox ax px
n oy ay py
F= y
nz o z a z pz
0 0 0 1

n x ox +n y o y +n z o z=0

n x (0)+ n y (0.707)+n z o z=0

n x ax +n y a y + nz a z =0

n x ax +a y (0.707)+ n z (0)=0

a x o x + a y o y +a z o z =0

a x (0)+a y o y +(0)o z=0

2 2 2
n x + n y +n z =1
2 2 2
n x + 0.707 +n z=1
2 2 2
o x +o y + o z=1
2 2 2
o y + 0 +o z =1

a 2x +a2y + a2z =1
2 2 2
a x +a y +0 =1

Solving these six equations with matlab,

 n x =a x =± 0.707 , n z=o y =0 , o z =1 ,a y =−0.707

[ ] [ ]
0.707 0 0.707 5 −0.707 0 −0.707 5
0.707 0 −0.707 3 0.707 0 −0.707 3
F1= and F1=
0 1 0 2 0 1 0 2
0 0 0 1 0 0 0 1

[ ]
? 0 ? 3
0.5 ? ? 9
8. F=
0 ? 0 7
0 0 0 1

[ ]
nx ox ax px
n oy ay py
F= y
nz o z a z pz
0 0 0 1
2 2 2
n x + n y +n z =1
2
n x + 0.25=1 , n x =0.866

n x ox +n y o y +n z o z=0

0.866(0)+0.5*o y +(0)*o z = 0, o y =0

o 2x +o 2y + o2z=1 , o z =1

n x ax +n y a y + nz a z =0

0.866 a x +0.5 a y +0∗a z =0

a x∗0+ a y ∗0+a z∗1=0 ,a z = 0

a 2x +a2y + a2z =1

a x =0.5 , a y =−0.866
 [ ]
0.866 0 0.5 3
0.5 0 −0.866 9
F=
0 1 0 7
0 0 0 1

[ ]
0.527 0−0.574 0.628 5
0.369 0.819 0.439 3
9. F=
−0.766 0 0.643 8
0 0 0 1

[ ][ ]
1 0 0 d x nx ox ax px
0 1 0 d y ny oy ay py
F new =¿Trans(d x , d y , d z )* F old = *
0 0 1 d z nz o z a z pz
0 0 0 1 0 0 0 1

[ ]
nx ox ax p x +d x
n oy ay p y+ d x
= y
nz o z a z p z +d z
0 0 0 1

[ ][ ]
1 0 0 0 0.527 0−0.574 0.628 5
F new =¿Trans(0,10,5)* F old =
0 1 0 10 0.369 0.819 0.439 3
*
0 0 1 5 −0.766 0 0.643 8
0 0 0 1 0 0 0 1

[ ]
0.527 0−0.574 0.628 5
0.369 0.819 0.439 13
=
−0.766 0 0.643 18
0 0 0 1

[][ ][ ][ ][ ][ ]
px 1 0 0 2 1 0 0 2 2
10. p y = 0 cosd (θ) −sind(θ) * 3 = 0 cosd (90) −sind (90) * 3 = −4
pz 0 sind (θ) cosd (θ) 4 0 sind (90) cosd (90) 4 3

[]
7
11. P0= 3
1

P=Trans(4,-3,7)*Rot(y,90)*Rot(z,90)*P0
 [ ][ ][ ][ ][ ]
1 0 0 4 0 0 1 0 0 −1 0 0 7 5
0 1 0 −3 0 1 0 0 1 0 0 0 3 4
P= * * * =
0 0 1 7 −1 0 0 0 0 0 1 0 1 10
0 0 0 1 0 0 0 1 0 0 0 1 1 1

[]
7
12. P0= 3
1

P= Rot(y,90)*Trans(4,-3,7)* Rot(z,90)*P0

[ ][ ][ ][ ][ ]
0 0 1 0 1 0 0 4 0 −1 0 0 7 8
0 1 0 0 0 1 0 −3 1 0 0 0 3 4
P= ∗ * * =
−1 0 0 0 0 0 1 7 0 0 1 0 1 −1
0 0 0 1 0 0 0 1 0 0 0 1 1 1

[]
7
13. P0= 3
1

P= Rot(a,90)*Trans(4,-3,7)* Rot(o,90)*P0

[ ][ ][ ][ ][ ]
0 −1 0 0 1 0 0 4 0 0 1 0 7 0
1 0 0 0 0 1 0 −3 0 0 0 0 3 5
P= ∗ * * =
0 0 1 0 0 0 1 7 −1 0 1 0 1 0
0 0 0 1 0 0 0 1 0 0 0 1 1 1
U
14. (a)T B =¿ Rot(z,90)* Rot(x,90)*Trans(0,0,3)* Trans(0,5,0)

[]
1
(b) P0= 5
4
U U B
P =T B ∗P

[ ][ ][ ][ ][ ][ ]
0 −1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 7
1 0 0 0 0 0 −1 0 0 1 0 0 0 1 0 5 5 1
P =0 0 1 0∗0 1 0 0*0 0
U
* * =
1 3 0 0 1 0 4 10
0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1

15. T= Trans(4,0,0)* Rot(y,90)* Rot(o,30)* )* Trans(5,0,0)

[ ][ ][ ][ ]
1 0 0 4 0 0 1 0 0.866 0 0.5 0 1 0 0 5
0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0
T= ∗ ∗ ∗
0 0 1 0 −1 0 1 0 −0.5 0 0.866 0 0 0 1 0
0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
 [ ]
−0.5 0 0.866 1.5
0 1 0 0
T=
−0.866 0 −0.5 −4.33
0 0 0 1

[ ]
1 0 0 0
0 0.766 −0.643 0
16. Rot(x,40)=
0 0.643 0.766 0
0 0 0 1

[ ]
1 0 0 0
−1
Rot ( x , 40) =inv ( Rot ( x , 40 ) )= 0 0.766 0.643 0
0 −0.643 0.766 0
0 0 0 1

[ ]
0.5 0 0.866 3
0.866 0 −0.5 2
17. T=
0 1 0 5
0 0 0 1

[ ]
0.5 0.866 0 −3.23
−1 0 0 1 −5
T =inv(T)= 0.866 −0.5 0 −1.598
0 0 0 1

[ ]
0 0 −1 3
T 5
= 0 −1 0 0
18. cam
−1 0 0 5
0 0 0 1

[ ]
0 −1 0 0
5 1 0 0 0
T H=
0 0 1 4
0 0 0 1

[ ]
0 0 1 2
1 0 0 2
T cam
obj =
0 1 0 4
0 0 0 1
 [ ]
1 0 0 0
0 1 0 0
T HE =
0 0 1 3
0 0 0 1

T R5 ∗T 5H∗T HE ∗T Eobj =T 5R∗T 5cam∗T cam

obj

−1 −1

T Eobj =T HE ∗T 5H ∗T 5cam∗T cam

obj

[ ]
1 0 0 0
0 1 0 0
=inv ( T E ) =
−1
H H
TE
0 0 1 −3
0 0 0 1

[ ]
0 1 0 0
T 5H =inv ( T 5H )= −1 0 0 0
−1

0 0 1 −4
0 0 0 1

[ ]
−1 0 0 −2
T Eobj = 0 1 0 1
0 0 −1 −4
0 0 0 1

[ ]
1 0 0 3
R 0 1 0 4
19. T P=
0 0 1 7
0 0 0 1

T RP =T cyl (r , α , l)=Trans ( 0,0 , l )∗Rot ( z , α )∗Trans ( 4,0,0 )

[ ][ ][ ]
1 0 0 0 cos ( α ) −sin ( α ) 0 0 1 0 0 r
0 1 0 0 sin ( α ) cos ( α ) 0 0∗0 1 0 0
T RP = ∗
0 0 1 l 0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1 0 0 0 1

[ ]
cos ⁡(α ) −sin ⁡(α ) 0 r cos ⁡(α )
sin ⁡( α ) cos ⁡( α ) 0 r sin ⁡( α )
=
0 0 1 l
0 0 0 1
 [ ][ ]
cos ( α ) −sin ( α ) 0 rcos ( α ) cos ⁡( −α ) −sin ⁡(−α ) 0 0
R sin ( α ) cos ( α ) 0 rsin ( α ) sin ⁡(−α ) cos ⁡(−α ) 0 0
T P∗Rot ( a ,−α )= ∗
0 0 1 l 0 0 1 0
0 0 0 1 0 0 0 1

[ ]
1 0 0 r cos ⁡(α )
0 1 0 r sin ⁡( α )
=
0 0 1 l
0 0 0 1

[ ][ ]
3 r cos ⁡( α )
20. T cyl = 4 = r sin ⁡(α ) (On cylindrical robot)
7 l

r cos ( α )=3 , r sin ( α )=4 ,l=7

4
Tan(α ¿= and α =53.1
3

r=5

[ ][ ]
−2.394 r cos ⁡( α )
21. T cyl = 6.578 = r sin ⁡( α )
9 l

r cos ( α )=−2.394 , r sin ( α )=6.578 , l=9

6.578
Tan(α ¿= and α =180−70=110
−2.394

r=7

[ ][ ]
cos ( α ) −sin ( α ) 0 rcos ( α ) −0.342 −0.9397 0 −2.394
R sin ( α ) cos ( α ) 0 rsin ( α ) 0.9397 −0.342 0 6.578
T =
P =
0 0 1 l 0 0 1 9
0 0 0 1 0 0 0 1

22. rSβCγ=3, rSβSγ=4, rCβ=7(On spherical robot)

4
Tanγ= , γ=53.1 or 233.1
3

Sγ=0.8 or -0.8
Cγ=0.6 or -0.6
 3
rSβ= =5∨−5
0.6

rCβ=7 and β=35.5 or -35.5

r = 8.6

[ ][ ]
nx o x a x px 0.354 −0.674 0.649 4.33
R n o y ay py 0.505 0.722 0.475 2.5
23. T P= y ∗
n z o z az p z −0.788 0.16 0.595 8
0 0 0 1 0 0 0 1

Φ A= ATAN 2 ( n y , n x )= ATAN 2 ( 0.505,0 .354 ) =55∨235

Φ A= ATAN 2 ¿

Φ A= ATAN 2 ¿

¿ 15∨195
p x =4.33 , p y =2.5 , pz =8

[ ]
0.354 −0.674 0.649 4.33
R 0.505 0.722 0.475 2.5
24. T P= =T cyl ( r , α , l )∗RPY (Φa , Φ o , Φ n)
−0.788 0.16 0.595 8
0 0 0 1

r cos ( α )=4.33 ,r sin ( α )=2.5 , l=8 , α =30

α +Φa =55 ,Φ a=25

sin ( α )=0.5 , r=5

p z=8

[ ][ ]
nx ox ax p x 0.579 −0.548 −0.604 5
R n oy ay p y 0.540 0.813 −0.22 7
25. T H= y ∗
nz o z a z p z 0.611 −0.199 0.766 3
0 0 0 1 0 0 0 1

Φ= ATAN 2 ( a y , a x )= ATAN 2 (−0.22 ,−0.604 )=20∨200

Ψ =ATAN 2¿

¿ 20∨200
θ=ATAN 2¿
 [ ] [ ]
C1 −S 1 0 a1 C 1 C2 −S 2 0 a2 C 2
S C1 0 a1 S 1 S2 C 2 0 a2 S 2
26. A1= 1 and A2=
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
0
T H = A 1∗A2=¿ ¿

C 1 C 2−S1 S 2=C 12 =C(θ 1+θ 2) and S1 C2 + C1 S 2=S12=S(θ 1+θ 2)

[ ]
C12 −S12 0 a 2 C 12+a 1 C 1
0 S C 12 0 a2 S 2 +a1 S 1
T H = 12
0 0 1 0
0 0 0 1

[ ] [ ]
C1 −S 1 0 a1 C 1 −S2 0 C2 0
S C1 0 a1 S 1 C 0 S2 0
27. A1= 1 ∧ A 2= 2 ∧¿
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1

[ ]
C3 −S 3 0 0
S C3 0 0
A3 = 3
0 0 1 d3
0 0 0 1

T 0H =¿

C 1 C 2−S1 S 2=C 12∧S 1 C 2 +C 1 S2=S 12

[ ]
−S 12 C 3 S12 S 3 C 12 C12 d3 + a1 C 1
C C C 12 S 3 S12 S12 d3 + a1 S1
T 0H = 12 3
S3 C3 0 0
0 0 0 1

[ ] [ ] [ ]
C1 0 S1 0 C2 −S 2 0 a2 C2 C3 −S 3 0 a3 C3
28. A 1 = S1 0 −C 1 0 ,A =
2
S2 C 2 0 a 2 S2 , A =
3
S3 C3 0 a3 S3
0 1 0 0 0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1 0 0 0 1
 [ ] [ ] [ ]
C 4 0 −S 4 a 4 C4 C 5 −S5 0 a5C5 C 6 −S 6 0 a6 C6
S 0 C 4 a4 S 4 S5 C5 0 a5 S 5 S6 C 6 0 a6 S6
A 4= 4 , A 5= , A 6=
0 −1 1 0 0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1 0 0 0 1

C 1 C 2−S1 S 2=C 12 =C(θ 1+θ 2) and S1 C2 + C1 S 2=S12=S(θ 1+θ 2)

T H = A1 A 2 A 3 A 4 A 5 A 6
R

R
T H=

[
C1 ( C234 C5 C6 −S 234 S6 )−S1 S 5 C 6 C 1 ( −C234 C5 C6 −S 234 S6 ) + S 1 S5 C6 C 1 ( C 234 S5 + S1 C5 ) C 1 ( C 234 a 4 +a3 C 23+ a2 C
C1 ( C 234 C 5 C 6−S234 S 6 ) +S 1 S 5 C 6 C 1 (−C 234 C 5 C6 −S 234 S6 ) −S 1 S5 C 6 S1 ( C234 S 5−C 1 C 5 ) S1 ( C 234 a 4 +a3 C23 +a 2 C
S 234 C 5 C 6 +C234 S 6 −S 234 C5 C 6+C 234 S 6 S234 S 5 0
0 0 0 1

[ ]
1 0 0 l1
0 1 0 l2 A A A
30. T UH = T U0 *T 0H = * 1 2 3
0 0 1 0
0 0 0 1