End effector•End effector - the last coordinate system of

figure•Located in joint N.•But usually, we want to specify it in base

coordinates.

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End effectorA transformation from the link N to the

base :

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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 ?

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End effector•The origin of a coordinate frame

relative to some base coordinate frame is specified by the translation :

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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

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End effector•Orientation•The roll, pitch and yaw

transformation is then expressed :

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End effector•Finally, the transformation from a coordinate

frame to the base frame is expressed :

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End effector

We obtain directly the translation vector :

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End effector

We can obtain the yaw angle :Because :

arctan is π-periodic. Let’s use our function arctan2 to get the right angle.

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End effectorKnowing the yaw angle, we can obtain the pitch angle :

Because :

Again, let’s use our function arctan2 :

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End effector

We can obtain the roll angle :

Because :

Again, let’s use our function arctan2 :

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21

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sin cos sinarctan arctan arctancos cos cos

mm

2 21 11( , )arctan m m

End effectorLet’s define the state vector

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End effectorAs previously shown,

The state vector is composed of elements of this matrix. It’s also a function of joint parameters :

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