Using the link lengths for the forward kinematics, work out the inverse kinematics for the piktul end-effector. That is, given $(x_e^*, y_e^*, z_e^*, \theta_e^*)$, figure out a joint configuration $\vec \alpha$ that places the manipulator end-effector there..

The height part is easy, since that is determined by $\alpha_1$. The planar position is determined by $\alpha_2$ and $\alpha_3$, which is what will be used for that part. The last missing component is the orientation of the gripper. But we know that the end-effector orientation, in vector form, is: \begin{equation} \theta_e = \alpha_2 + \alpha_3 + \alpha_4 \end{equation} so once $\alpha_2$ and $\alpha_3$ have been determined, we can solve for $\alpha_4$ given the desired orientation $\theta_e^*$, \begin{equation} \alpha_4 = \theta_e^* - \alpha_2 + \alpha_3. \end{equation} The real tricky part of this all, is to determine whether the actual configuration is achievable or not, and to output as much when that is the case. Remember what the actuator limits are for the joint states.

Validity of the results can be done by applying the forward kinematics routine to the found joint angles. It should return the configuration requested. A more fun option would be to use the piktul display code to see if it really works out. To visualize the target end-effector configuration, plot it using the SE(3) class with the proper up-conversion of the desired configurations, or the SE(2) class but the height won’t be proper.

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