Kinematics in robotics

1. 06/12/2025
1
Day 03
Spatial Descriptions

2. 2
Points and Vectors
06/12/2025
 point : a location in space
 vector : magnitude (length) and direction between two points
p
q
v

3. 3
Coordinate Frames
06/12/2025
 choosing a frame (a point and two perpendicular vectors of
unit length) allows us to assign coordinates
0
p
0
q









1
2
0
0
0
q
p
v
0
x̂
0
ŷ
0
o
 
0







5
.
1
2
0
q







5
.
2
4
0
p

4. 4
Coordinate Frames
06/12/2025
 the coordinates change depending on the choice of frame
1
p
1
q









2
1
1
1
1
q
p
v
1
x̂
1
ŷ
1
o
 
1







2
5
.
0
1
q







4
5
.
0
1
p

5. 5
Dot Product
06/12/2025
 the dot product of two vectors













n
u
u
u
u

2
1













n
v
v
v
v

2
1
v
u
v
u
v
u
v
u
v
u T
n
n 




 
2
2
1
1
u
v


cos
u

cos
v
u
v
u 


6. 6
Translation
06/12/2025
 suppose we are given o1 expressed in {0}
1
x̂
1
ŷ
1
o
 
1
0
x̂
0
ŷ
0
o
 
0







0
3
0
1
o

7. 7
Translation 1
06/12/2025
 the location of {1} expressed in {0}
1
x̂
1
ŷ
1
o
 
1
0
x̂
0
ŷ
0
o
 
0























0
3
0
0
0
3
0
0
0
1
0
1 o
o
d

8. 8
Translation 1
06/12/2025
1. the translation vector can be interpreted as the location
of frame {j} expressed in frame {i}
i
j
d

9. 9
Translation 2
06/12/2025
 p1
expressed in {0}
1
x̂
1
ŷ
1
o
 
1
0
x̂
0
ŷ
0
o
 
0























1
4
1
1
0
3
1
0
1
0
p
d
p







1
1
1
p
a point expressed
in frame {1}

10. 10
Translation 2
06/12/2025
2. the translation vector can be interpreted as a coordinate
transformation of a point from frame {j} to frame {i}
i
j
d

11. 11
Translation 3
06/12/2025
 q0
expressed in {0}
0
x̂
0
ŷ
0
o
 
0























1
2
1
1
0
3
0
0
p
d
q







1
1
0
p 






1
2
0
q

12. 12
Translation 3
06/12/2025
3. the translation vector can be interpreted as an operator
that takes a point and moves it to a new point in the same
frame
d

13. 13
Rotation
06/12/2025
 suppose that frame {1} is rotated relative to frame {0}
1
x̂
1
ŷ
0
x̂
0
ŷ
1
0 o
o 


sin

cos

14. 14
Rotation 1
06/12/2025
 the orientation of frame {1} expressed in {0}
1
x̂
1
ŷ
0
x̂
0
ŷ
1
0 o
o 












0
1
0
1
0
1
0
1
0
1
y
y
y
x
x
y
x
x
R

15. 15
Rotation 1
06/12/2025
1. the rotation matrix can be interpreted as the orientation
of frame {j} expressed in frame {i}
i
j
R

16. 16
Rotation 2
06/12/2025
 p1
expressed in {0}
1
x̂
1
ŷ
0
x̂
0
ŷ
1
0 o
o 












 


1
1
cos
sin
sin
cos
1
0
1
0




p
R
p







1
1
1
p

17. 17
Rotation 2
06/12/2025
2. the rotation matrix can be interpreted as a coordinate
transformation of a point from frame {j} to frame {i}
i
j
R

18. 18
Rotation 3
06/12/2025
 q0
expressed in {0}
0
x̂
0
ŷ
0
o











 


1
1
cos
sin
sin
cos
0
0




p
R
q
0
q
0
p

19. 19
Rotation 3
06/12/2025
3. the rotation matrix can be interpreted as an operator
that takes a point and moves it to a new point in the same
frame
R

20. 20
Properties of Rotation Matrices
06/12/2025
 RT
= R-1
 the columns of R are mutually orthogonal
 each column of R is a unit vector
 det R = 1 (the determinant is equal to 1)

21. 21
Rotation and Translation
06/12/2025
1
x̂
1
ŷ
1
o
 
1
0
x̂
0
ŷ
0
o
 
0

22. 22
Rotations in 3D
06/12/2025




















0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
z
z
z
y
z
x
y
z
y
y
y
x
x
z
x
y
x
x
R