Direct Kinematics
Link
Description
The concept of Direct Kinematics
• Choosing wisely the coordinate systems on
the links
• If the wise choice was made, each link can
be represented with 4 parameters
• When the parameters are found, the
transformation matrices between the links
can be found from a closed formula
DK Algorithm
• 1) Draw sketch
• 2) Identify and number robot links. Base = 0, Last = n
• 3) Draw axis Zi for joint i. For rotating joint, Zi is the
rotation axis. For prismatic (translating) joint, Zi can
merge with the DOF axis or be perpendicular to it.
• 5) Determine joint length ai-1 between Zi-1 and Zi
• 6) Draw axis Xi-1 along the shortest distance between
Zi-1 and Zi. If the distance is 0, choose the direction of
Xi-1 to be a normal to the plane that they create.
DK Algorithm (2)
• 7) Determine joint twist i-1 measured around
Xi-1 (between Zi-1 and Zi)
• 8) Determine the joint offset di
• 9) Determine joint angle i around Zi
• 10) Write DH table
• 11+12) Write link transformations and calculate
the common transformation
Kinematics Parameters of a link
Link length
ai 1
Link twist
 i 1
What are the kinematics parameters of this link?
• a=7
•  = 450
Kinematics Parameters of a link
• Link offset d
• Joint angle 
Summary of the link parameters in
terms of link frames
•
•
•
•
ai = the distance from Zi to Zi+1 measured along Xi
i = the angle between Zi and Zi+1 measured about Xi
di = the distance from Xi-1 to Xi measured along Zi
i = the angle between Xi-1 and Xi measured about Zi
• We usually choose ai > 0 since it corresponds to a
distance;
• However, i , di , i are signed quantities.
There is no unique attachment of frames to links:
• 1. When we align Zi axis with joint axis i, two
choices of the Zi direction.
• 2. When we have intersecting joint axes (ai=0),
two choices of the Xi direction, corresponding to
choice of signs for the normal to the plane
containing Zi and Zi+1.
• 3. When axes i and i+1 are parallel, the choice of
origin location for {i} is arbitrary (generally
chosen in order to cause di to be zero).
i 1
i 1
i
i 1
R
P T P T T T T P
i
R Q P i
Q P i
i 1
i
i 1
R
T T T T T
R
Q
P R
R i
T  RX  i 1 DX ai 1 RZ i DZ di 
i 1
i
T  Screw X  i 1 , ai 1 Screw Z di ,i 
i 1
i
cos( i )

sin(  ) cos( )
i
i 1
i 1

T

i
 sin(  i ) sin(  i 1 )

0

 sin(  i )
0
cos( i ) cos( i 1 )
 sin(  i 1 )
cos( i ) sin(  i 1 )
cos( i 1 )
0
0

 sin(  i 1 )d i 
cos( i 1 )d i 

1

ai 1
Three link Arm : RPR mechanism
• “Cylindrical” robot – 2 joints analogous to polar
coordinates when viewed from above.
• Schematic: point – axes intersection; prismatic joint
at minimal extension
• Find coordinate systems and a, , d,  (i=3)
DH table:
i
ai
i
di
i
0
1
0
0
0
90
0
1
2
3
0
0
d2
L2
0
3
cos(1 )  sin( 1 )
 sin(  ) cos( )
1
1
0

1T 
 0
0

0
 0
0 0
0 0

1 0

0 1
1
0
1

T

2
0

0
0 

0  1  d2

1 0
0 

0 0
1 
0
0
cos( 3 )  sin(  3 )
 sin(  ) cos( )
3
3
2

T

3
 0
0

0
 0
0
0 0

1 L2 

0 1
0
T  T T T ...
0
N
0
1
1 2
2 3
N 1
N
T