In a special machine tool, the carriage is equipped with a reversible motor, as shown in the figure. Thus, the

carriage can be moved between the two limit positions A and $B,$ which are detected by the two proximity

sensors PS$1$ and PS$2,$ correspondingly. The carriage lies initially at the left position $($A$)$ and we want it to be

moved according to the following specifications:

a$.$ By pressing instantly button b$1$ the carriage moves right to position $B.$

b$.$ When the carriage reaches point $B($with a signal from PS$2),$ it stops moving.

c$.$ With the carriage at position $B,$ the instant pressing of button b$1$ causes it to move left. When the

carriage reaches position A $($with a signal from PS$1),$ it stops. The steps $($a$)$ and $($c$)$ can be repeated

as many times as we want.

d$.$ If the button b $2$ is pressed during the carriage's movement in any direction, then the carriage stops

at its current location. By pressing button b$1$ again, the carriage continues moving in the same

direction before it had been stopped.

Design an automation circuit to satisfy the described specifications. $($Hint: Use the Classical Control

Method$).(13$Marks$)$