kinematics of machunary

1. An Introduction to
Robot Kinematics
Renata Melamud

2. Other basic joints
Spherical Joint
3 DOF ( Variables - 1, 2, 3)
Revolute Joint
1 DOF ( Variable - )
Prismatic Joint
1 DOF (linear) (Variables - d)

3. We are interested in two kinematics topics
Forward Kinematics (angles to position)
What you are given: The length of each link
The angle of each joint
What you can find: The position of any point
(i.e. it’s (x, y, z) coordinates
Inverse Kinematics (position to angles)
What you are given: The length of each link
The position of some point on the robot
What you can find: The angles of each joint needed to obtain
that position

4. Quick Math Review
Dot Product:
Geometric Representation:
A
B
θ
cosθ
B
A
B
A 

Unit Vector
Vector in the direction of a chosen vector but whose magnitude is 1.
B
B
uB 






y
x
a
a






y
x
b
b
Matrix Representation:
y
y
x
x
y
x
y
x
b
a
b
a
b
b
a
a
B
A 
















B
B
u

5. Quick Matrix Review
Matrix Multiplication:
An (m x n) matrix A and an (n x p) matrix B, can be multiplied since
the number of columns of A is equal to the number of rows of B.
Non-Commutative Multiplication
AB is NOT equal to BA
   
   























dh
cf
dg
ce
bh
af
bg
ae
h
g
f
e
d
c
b
a
Matrix Addition:
   
   























h
d
g
c
f
b
e
a
h
g
f
e
d
c
b
a

6. Basic Transformations
Moving Between Coordinate Frames
Translation Along the X-Axis
N
O
X
Y
Px
VN
VO
Px = distance between the XY and NO coordinate planes






 Y
X
XY
V
V
V 





 O
N
NO
V
V
V 






0
P
P x
P
(VN,VO)
Notation:

7. N
X
P
VN
VO
Y O

NO
O
N
X
XY
V
P
V
V
P
V 






 

Writing in terms of
XY
V NO
V

8. X
N
VN
VO
O
Y
Translation along the X-Axis and Y-Axis










 O
Y
N
X
NO
XY
V
P
V
P
V
P
V







Y
x
XY
P
P
P

9. 


































o
V
n
V
θ)
cos(90
V
cosθ
V
sinθ
V
cosθ
V
V
V
V NO
NO
NO
NO
NO
NO
O
N
NO
NO
V
o
n Unit vector along the N-Axis
Unit vector along the N-Axis
Magnitude of the VNO vector
Using Basis Vectors
Basis vectors are unit vectors that point along a coordinate axis
N
VN
VO
O
n
o

10. Rotation (around the Z-Axis)
X
Y
Z
X
Y

V
VX
VY






 Y
X
XY
V
V
V 





 O
N
NO
V
V
V
 = Angle of rotation between the XY and NO coordinate axis

11. X
Y

V
VX
VY

Unit vector along X-Axis
x
x
V
cosα
V
cosα
V
V NO
NO
XY
X




NO
XY
V
V 
Can be considered with respect to
the XY coordinates or NO coordinates
V
x
)
o
V
n
(V
V O
N
X





(Substituting for VNO using the N and O
components of the vector)
)
o
x
V
n
x
V
V O
N
X



 (
)
(
)
)
)
(sinθ
V
(cosθ
V
90))
(cos(θ
V
(cosθ
V
O
N
O
N






12. Similarly….
y
V
α)
cos(90
V
sinα
V
V NO
NO
NO
Y





y
)
o
V
n
(V
V O
N
Y





)
o
y
(
V
)
n
y
(
V
V O
N
Y




)
)
)
(cosθ
V
(sinθ
V
(cosθ
V
θ))
(cos(90
V
O
N
O
N





So….
)
) (cosθ
V
(sinθ
V
V O
N
Y


)
) (sinθ
V
(cosθ
V
V O
N
X

 





 Y
X
XY
V
V
V
Written in Matrix Form











 







 O
N
Y
X
XY
V
V
cosθ
sinθ
sinθ
cosθ
V
V
V
Rotation Matrix about the z-axis

13. X1
Y1

VXY
X0
Y0
VNO
P











 














 O
N
y
x
Y
X
XY
V
V
cosθ
sinθ
sinθ
cosθ
P
P
V
V
V
(VN,VO)
In other words, knowing the coordinates of a point (VN,VO) in some coordinate
frame (NO) you can find the position of that point relative to your original
coordinate frame (X0Y0).
(Note : Px, Py are relative to the original coordinate frame. Translation followed by
rotation is different than rotation followed by translation.)
Translation along P followed by rotation by 

14. 










 














 O
N
y
x
Y
X
XY
V
V
cosθ
sinθ
sinθ
cosθ
P
P
V
V
V
HOMOGENEOUS REPRESENTATION
Putting it all into a Matrix



















 























1
V
V
1
0
0
0
cosθ
sinθ
0
sinθ
cosθ
1
P
P
1
V
V
O
N
y
x
Y
X



















 












1
V
V
1
0
0
P
cosθ
sinθ
P
sinθ
cosθ
1
V
V
O
N
y
x
Y
X
What we found by doing a
translation and a rotation
Padding with 0’s and 1’s
Simplifying into a matrix form









 

1
0
0
P
cosθ
sinθ
P
sinθ
cosθ
H y
x
Homogenous Matrix for a Translation in
XY plane, followed by a Rotation around
the z-axis