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## End effector

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**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**End effector**A transformation from the link N to the base : 2**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**End effector**• The originof a coordinate frame relative to some base coordinate frame is specified by the translation : 4**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**End effector**• Orientation • The roll, pitch and yaw transformation is then expressed : 7**End effector**• Finally, thetransformation from a coordinate frame to the base frame is expressed : 8**End effector**We obtain directly the translation vector : 9**End effector**We can obtain the yaw angle : Because : arctan is π-periodic. Let’s use our function arctan2 to get the right angle. 10**End effector**Knowing the yaw angle, we can obtain the pitch angle : Because : Again, let’s use our function arctan2 : 11**End effector**We can obtain the roll angle : Because : Again, let’s use our function arctan2 : 12**End effector**Let’s define the state vector 13**End effector**As previously shown, The state vector is composed of elements of this matrix. It’s also a function of joint parameters : 14