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In the last two videos we learned how to define straight-line paths and then time scale them to get trajectories. If we want more flexibility to design the shape of the path, as well as the speed with which it is executed, we could specify a set of configurations through which we would like the robot to transit. These configurations are called via points. We also specify the times at which the robot should achieve each of these via points. We then solve for a smooth trajectory that passes through the via points at the specified times. The choice of the via points and times allows us to shape the path and trajectory. In this case, we solve directly for a trajectory; we do not first find a path and then time scale it. Letamp;#39;s consider motion in an n-dimensional joint space. For joint i, moving between via points j and j-plus-one, we could define the motion as a third-order polynomial of time. We then apply four terminal constraints, the initial and final position and the initial an
