python - Car Following Model with Optimal Velocity function - overtake occurence -

i creating project on car simulations , i've run problem. when run code, cars overtake 1 another.

i have spent few days trying figure out why happening , still have no idea. optimal velocity function should set (de)acceleration, overtake not happen, reason, still allows cars overatake each other , not decelerate fast enough.

could me, or push me in direction should looking ?

here optimal velocity function:

def optimal_velocity_function(dx, d_safe, v_max): vx_opt = v_max * (np.tanh(dx - d_safe) + np.tanh(d_safe)) return vx_opt     

now using within euler's method solve ode:

def euler_method(x, v, n_cars, h, tau, d_safe, v_max): # euler method used solve ode # returns new position of car , new velocity dv = np.zeros(n_cars)  j in range(n_cars - 1):         dv[j] = tau ** (-1) * (optimal_velocity_function(x[j+1] - x[j],   d_safe, v_max) - v[j])  dv[n_cars - 1] = tau ** (-1) * (v_max - v[n_cars - 1])  # speed of first car  v_new = v + h * dv x_new = x + h * v_new  return [x_new, v_new]     

and here rest of model, generating starting values , iterating using functions above.

def optimal_velocity_model(n, n_cars, d_0, v_0, h, tau, d_safe, v_max): global x_limit, canvas, xx, vv  car_positions = np.linspace(0, n_cars, n_cars) x = np.array(sorted(np.random.random(n_cars) + car_positions))  # generation of cars minimal distance x = x * d_0 v = np.random.random(n_cars) + v_0  # generating initial speeds around v_0 xx = np.zeros([n_cars, n])  # matrix of locations vv = np.zeros([n_cars, n])  # matrix of velocities  in range(n):     xx[:, i] = x     vv[:, i] = v     [x, v] = euler_method(x, v, n_cars, h, tau, d_safe, v_max)  x_limit = xx.max()  # interval in cars return     

thanks lot. j.

i think getting bit lost here maths , not focusing on physics of problem. using tanhh on delta x safe distance generating soft speed transition might not guarantee overtaking inhibited.

my train of thought like:

• firstly assess relative speed of 2 vehicles approaching safety distance

• define minimum deceleration given vehicle properties can stop within distance

• use tanhh or other function model transition between aggressive smooth braking or viceversa.

• update speed integrating acceleration.

to calculate deceleration can think of simple approach keep use of tanh :

deceleration=max_dec(relative_speed)*[1+tanh(d_safe-dx)] 

hope can guide further!