Example Trajectory
+A visualization of how the polynomial zonotope trajectory representation is used to construct the polynomial zonotope forward occupancy for a two link arm in 2D. + A desired trajectory for each joint is shown in black in the top two rows. + The same desired trajectory is depicted in each column. + The planning time horizon is represented as a finite set of polynomial zonotopes (Sec. VIII-A1). + In each column, a single time interval is highlighted in the top two rows and the corresponding time instance is highlighted in the visualization in the bottom row. + The top two rows of the first column depict a finite set of polynomial zonotopes generated by uniformly buffering a desired trajectory by the ultimate bound (Lem. 16). + The bottom row of the first column depicts the forward occupancy of the robot over this entire set (Lem. 18). + The second column depicts a family of desired trajectories that are overapproximated by a finite set of polynomial zonotopes and the corresponding forward occupancy of the robot over this entire set. + The third column depicts a family of desired trajectories buffered by the ultimate bound that are overapproximated by a finite set of polynomial zonotopes and the corresponding forward occupancy of the robot over this entire set. + The fourth column depicts a subset of the sets illustrated in the third column generated by slicing in a specific trajectory parameter (Sec. VIII-A4). + Note that the desired trajectory always remains within polynomial zonotopes.
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