The Problem

A mobile robot $($equipped with wheels$)$ is pushing a shopping trolley in order to arrive to the house of its human the owner. The robot travels along in one dimension $x$ and pushes the trolley towards the same dimension.

The robot applies a force $F_R$ to the shopping trolley to make it to move ahead. It is assumed that due to the mechanics of the robot and the trolley, the connection of the hand of the robot and the trolley acts as a spring with stiffness $z.$ The mass of the robot is $m_R$ and the mass of the shopping trolley is $m_{tr}.$ The rolling friction coefficient is $.$

The application of the second law of Newton derives the following equations for the dynamic system, describing the motion of the robot pushing the shopping trolley:

$F_R-z(x_R-x_{tr})=m_R*x_R^{}+*m_R*g*x_R^{}$

$z*(x_R-x_{tr})=m_{tr}*x_{tr}^{}+*m_{tr}*g*x_{tr}^{}$

where $-$ denotes multiplication. $x_R$ is the displacement of the robot and $x_{tr}$ is the displacement of the shopping trolley. $g=9\mathrm{.}81\frac{m}{s^2}$ is the gravity. For the purposes of the simulation of the system here, use $F_R=2,=0\mathrm{.}05,z=1,m_R=2,m_{tr}=1.$

$3$

$1$ Simulink Model

Implement a Simulink block diagram model of the robot pushing the shopping trolley dynamic system described in equations $(1).$

$2$ Control of the Simulink Model

Extend the Simulink model you developed in Section $1$ with a PID controller, so that the speed of the robot follows the desired speed shown in Figure $1.$ The total simulation time is $1000$ secs and the robot starts with a speed of $0,$ increasing its speed to $1$ after $200$ secs and keeping this speed for $500$ secs before coming to a rest $($speed equals to $0)$ at time $t=700$ secs$.$