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# Rotation angle of the second degree of freedom

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The serial manipulator shown in figure 1 (in the attachment) is set in a configuration such that the pose of the effector with respect to a world reference frame, Rworld, is defined by the composed homogeneous transformation matrix, Qgripper/world, that follows (defined with respect to evolving reference frames):

{Please refer to the attachment for the matrix.}

On the other hand, the pose of the robot base with respect to the world reference frame, Rworld, is defined by the composed homogeneous transformation matrix, Qbase/world, that follows (defined with respect to evolving reference frames):

{Please refer to the attachment for the matrix.}

Moreover, the Denavit-Hartenberg parameters of the manipulator in this configuration are defined as follows:

{Please refer to the attachment for the table.}

Determine the rotation angle of the second degree of freedom, theta2, that must be used to bring the effector to the specified configuration.

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#### Solution Preview

The conversion matrix T(i-1,i) is given as:

t11 = cos(Qi); t12=-cos(ai).sin(Qi); t13 =sin(ai).sin(Qi); t14=l.cos(Qi)
Qi== read as theta(i) ai== read as alpha(i)
t21 = sin(Qi); ...

#### Solution Summary

Solution compares given Qbase/world, and T(world,4) matrices to get the required result.

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