Derive the equations of motion of the system - create a

Question

Consider the DC motor-driven wheeled mobile robot shown in figure, in which in is the mass of the wheeled mobile robot, r is the radius of the driving wheel, and T is the torque delivered to the wheeled mobile robot by the DC motor. For simplicity, the motion is restricted to one spatial dimension. The figure also shoves the simplified drive system, including the equivalent electrical circuit of the DC motor, the gears, and the driving wheel. The motor parameter values art. armature inductance La = 0.001 H, resistance Ra = 2.6 Ω, back emf constant K_b = 0.008 N-m/A (remember, t_m, Kna). and torque constant K_{t =} 0.008 N-m/A (remember, Tom, K14. The mass moment of inertia of the motor 5x10⁴ kg-nt2.111e gear ratio N (LAO la 1.5. The wheel and the axle mechanism converts the rotational motion to translation, and the wheel radius is 0.006 m. The mass of the cart is 0.5 kg. The mass moment of inertia of the wheel 1st 3 x10⁴ kg-m². Viscous damping of the load C = 10^-5 N m-s. Neglect the viscous damping of the motor (Cm = 0). Assume that all gear contact fortes are taken by shalt bearings.

a. Derive the equations of motion of the system.

b. Create a Simulink model using the differential equations found in part (a) and plot the displacement output x(t) and robot velocity x.(t) for 0 ≤ t ≤ 5s. The voltage applied to the DC motor is a pulse function, va(t) = 4V for 0 ≤ t ≤ 1s.