1. Direct Kinematics

2. Link Description

3. 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

4. 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.

5. 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

6. Kinematics Parameters of a link
Link length
Link twist

7. What are the kinematics parameters of this link?
 = 450

8. Kinematics Parameters of a link
Link offset d
Joint angle  

9. 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.

10. 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).

13. 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)

14. DH table: i ai i di i 1 90 1 2 d2 3 L2 3