Need help with this please in system dynamics Consider the D

Consider the DC motor-driven wheeled mobile robot shown in figure, in which m 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 shows the simplified drive system, including the equivalent electrical circuit of the DC motor, the gears, and the driving wheel. The motor parameter values are: armature inductance L_a = 0.001 H, resistance R_a = 2.6 ohm, back emf constant K_b = 0.008 V-s/rad (remember, v_b = K_b omega), and torque constant K_m = 0.008 N-m/A (remember, T_m = K_m i_a). The mass moment of inertia of the motor can be negligible Also, neglect the viscous damping of the motor and the load (C = C_m = 0). The gear ratio N = theta_m/theta = T/T_m = 1.37. The wheel and the axle mechanism converts the rotational motion to translation, and the wheel radius is 0.00635 m. The mass of the cart is 0.455 kg. Derive the equations of motion of the system. Assuming zero initial conditions, find the transfer function X(s)/V_s(s). Create a Simulink model using the differential equation in pan (a) and plot the displacement output x(t) and robot velocity dot x(t) when the voltage applied to the DC motor is a pulse function, v_a(t) = 1 V for 1 lessthanorequalto t lessthanorequalto 2 s. Create a Simulink model using the transfer functions obtained in part (b) and plot the displacement output x(t) and robot velocity dot x(t) when the voltage applied to the DC motor is a pulse function, v_a(t) = 1 V for 1 lessthanorequalto t lessthanorequalto 2 s.

Solution

T - Tl = J dw/dt

V = E + ir

E = k*flux*speed

torque = k*flux*current

q/q1 = 2/2.5