1. End effector End effector - the last coordinate system of figure Located in joint N. But usually, we want to specify it in base coordinates. 1

2. End effector A transformation from the link N to the base : 2

3. End effector We can also express it as three rotations (around each of the coordinate axes) followed by a translation How can we establish a relation with the other expression ? 3

4. End effector The origin of a coordinate frame relative to some base coordinate frame is specified by the translation : 4

5. End effector Any 3D orientation relative to some base coordinate frame can be specified by : three rotations, one around each of the coordinate axes. We do them in this order : around x, y, z. 5

6. End effector 6

7. Orientation The roll, pitch and yaw transformation is then expressed : 7

8. End effector Finally, the transformation from a coordinate frame to the base frame is expressed : 8

9. End effector We obtain directly the translation vector : 9

10. End effector We can obtain the yaw angle : Because : arctan is π-periodic. Let’s use our function arctan 2 to get the right angle. 10

11. End effector Knowing the yaw angle, we can obtain the pitch angle : Because : Again, let’s use our function arctan 2 : 11

12. End effector We can obtain the roll angle : Because : Again, let’s use our function arctan 2 : 12

13. End effector Let’s define the state vector 13

14. End effector As previously shown, The state vector is composed of elements of this matrix. It’s also a function of joint parameters : 14