Robot
Kinematics and
Position Analysis
Fo r w ar d K i n em atics Eq u ati on s
fo r Po s i tion & O r i en tation
Translation
Revision
𝟏 𝟎 𝟎 𝒅𝒙
𝟎 𝟏 𝟎 𝒅𝒚
𝑻=
𝟎 𝟎 𝟏 𝒅𝒛
𝟎 𝟎 𝟎 𝟏
Rotation about x-axis
1 0
0
𝑅𝑜𝑡 𝑥 , 𝜃 = 0 𝐶𝜃 −𝑆𝜃
0 𝑆𝜃
𝐶𝜃
Rotation about y-axis
𝐶𝜃 0 𝑆𝜃
𝑅𝑜𝑡 𝑦 , 𝜃 = 0
1 0
−𝑆𝜃 0 𝐶𝜃
Rotation about z-axis
𝐶𝜃 −𝑆𝜃 0
𝑅𝑜𝑡 𝑧 , 𝜃 = 𝑆𝜃
𝐶𝜃 0
0
0
1
W.r.t fixed reference
frame -> PRECombined
Transformation
MULTIPLY
W.r.t. moving frame ->
POST-MULTIPLY
Exercise 1
A p o i n t P i n s p a c e i s d e f i n e d a s BP = [ 2 , 3 , 5 ] T r e l a t i v e t o f r a m e B , w h i c h i s a t t a c h e d t o
the origin of the reference frame A and is parallel to it. Apply the following
t r a n s f o r m a t i o n s t o f r a m e B a n d f i n e AP .
1.
Rotate 90o about x-axis
2.
Rotate 90o about the local a-axis
3.
Translate 3-units about the y-axis, 6-units about z-axis and 5-units about the xaxis
2
−2
Answer:
8
1
Exercise 2
A frame U B was moved along its own n -axis a distance of 5 units and then rotated about its o -axis an angle of 60 o ,
followed by rotation of 60 o about z-axis. It was then
translated about it’s a -axis by 3-units and finally rotated
about x-axis by 45 o .
Calculate the total transformation performed.
0.25
0.918
Answer:
−0.306
0
−0.866
0.354
0.354
0
0.433
0.177
0.884
0
3.8
3.59
5.71
1
Inverse
Transformation
Matrix
𝑇 −1
𝑛𝑥
𝑜
= 𝑥
𝑎𝑥
0
𝑛𝑦
𝑜𝑦
𝑎𝑦
0
𝑛𝑧
𝑜𝑧
𝑎𝑧
0
ഥ 𝑛
− 𝑃.
ഥ
ഥ 𝑜ത
− 𝑃.
ഥ 𝑎
− 𝑃.
ഥ
1
Calculate the inverse of the
following transformation matrix
Exercise 3
0.527
0.369
𝑇=
−0.766
0
−0.574
0.819
0
0
0.628
0.439
0.643
0
2
5
3
1