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Piktul: Resolved Rate Position Movement
This module explores trajectory generation using the Jacobian. The objective here is to create a trajectory for the arm to follow, at only the position level. As the piktul manipulator space has equal dimension to the $R^3$ output position space, tracking a Euclidean position is feasible. It is possible to specify the 3D position as a function of time and get it to follow the trajectory, presuming that the trajectory is within the reachable workspace (for position).
Create a resolved rate trajectory generation member function that will do so for the piktul. Call the function genPositionTrajectory where it is assumed that some kind of position function of time and linear velocity function of time is given in a structure built with traj.position, traj.velocity, and traj.tspan fields. The output of the function should be a set of joint angles versus time that will move the arm to follow the specified trajectories. It can be
in the output format of ode45 and should be in a structure with the fields as per: jtraj.alpha and jtraj.time.
Furthermore, write a function called followJointTrajectory that takes in a structure whose fields are alpha and time much like the output of genPositionTrajectory. The function
will then command the arm to follow the specified trajectory. Doing this should be like a combination of the arm display as a movie code plus the joint angle command code for the piktul from earlier homeworks. You should also write a piktul member function called posJacobian that computes the linear velocity of the end effector (in world coordinates,
not body coordinates) as a function of the \alpha coordinates. For completeness, you should include all $\alpha$ coordinates in it (the last two will just have zero columns since they don't contribute to linear movement).