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.

*= x_dot**

**v***+ y_dot**

**i**

**j****= x_ddot***

*a**+ yddot**

**i**

**j**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.