## Controlling the position of a motorized cart x(t) with a remote control slider position p(t):

### engineering

##### Description

Controlling the position of a motorized cart x(t) with a remote control slider position p(t):

As described in past notes, the velocity v(t) of a motorized cart (m/sec) can be modeled as

where m = 100 kg is the mass, b = 20 N sec/m is the coefficient of a friction or drag force that opposes the motion (proportional to the velocity), and f(t) is the externally applied force to the cart (in N). The position x(t) of the cart is related to the velocity by

The force produced by an electric motor and a transmission system is related to the position of a mechanical slider p(t) (cm) for remote control by

where a = 1 sec-1 is the “lag coefficient” and g = 10 N / (cm-s) is a gain that describes the conversion from slider position to force applied to the cart (through the transmission system that converts the motor torque to translational force).

The slider position can be positive or negative, resulting in force applied in the positive or negative direction. This is like a remote-controlled car, where negative force will slow down the car if it is moving in the positive direction.