Cartesian vending machine

A common type of machine for the vending of bottled drinks is the Cartesian-drive machine shown below. Here, a carriage for the drink bottles moves from a starting point on the lower left side of the machine to pick up the drink bottle, with the carriage then delivering the bottle to a pick-up location for the bottle and a subsequent return of the carriage to its original starting location, readying for the next sale. The position of the carriage is set through linear actuation along two mutually perpendicular xy-axes, as shown below right.

This mechanism prescribes the x(t) and y(t) components of position for the carriage. The time derivatives of these two components of position produce the Cartesian components of velocity and acceleration of the bottle carriage:
        v = x_dot*i + y_dot*j
        a = x_ddot*i + yddot*j
Shown below is a video recording of the motion of the vending machine carriage as it moves through the different phases of its motion. Tracking software was used to capture the xy-components of position for the carriage. From this data,  x(t) vs. t, y(t) vs. t and y(t) vs. x(t) plots are constructed and shown to the right of this video. The y(t) vs. x(t) plot shows the path taken by the carriage during its motion.

Through the joint kinematics descriptions relating the Cartesian and path components of kinematics we would be able to quantify the rate of change of speed of the carriage and the radius of curvature of the path taken by the carriage.