Car Acceleration Integrator Example

car_integration.py

python
import matplotlib.pyplot as plt
import numpy as np

def acceleration_curve(velocity):
    """
    Realistic car acceleration that decreases with velocity.
    Models: engine power limits, air resistance, gearing
    """
    max_accel = 10.0  # m/s^2 at low speed
    drag_coefficient = 0.05
    # Acceleration decreases as velocity increases
    return max_accel * np.exp(-drag_coefficient * velocity)

def simulate_car(dt, total_time):
    """Simulate car with given timestep"""
    time = 0
    velocity = 0
    position = 0
    
    times = [0]
    velocities = [0]
    positions = [0]
    
    while time < total_time:
        # Get current acceleration
        accel = acceleration_curve(velocity)
        
        # Euler integration
        velocity += accel * dt
        position += velocity * dt
        time += dt
        
        times.append(time)
        velocities.append(velocity)
        positions.append(position)
    
    return times, velocities, positions

# Simulate two cars
total_time = 10.0  # seconds

# Car A: updates every 1 second
times_A, vels_A, pos_A = simulate_car(dt=1.0, total_time=total_time)

# Car B: updates every 2 seconds
times_B, vels_B, pos_B = simulate_car(dt=2.0, total_time=total_time)

# Car C: "ground truth" with very small timestep
times_C, vels_C, pos_C = simulate_car(dt=0.01, total_time=total_time)

# Print results
print(f"After {total_time} seconds:")
print(f"Car A (dt=1.0s): Position = {pos_A[-1]:.2f}m, Velocity = {vels_A[-1]:.2f}m/s")
print(f"Car B (dt=2.0s): Position = {pos_B[-1]:.2f}m, Velocity = {vels_B[-1]:.2f}m/s")
print(f"Car C (dt=0.01s, 'truth'): Position = {pos_C[-1]:.2f}m, Velocity = {vels_C[-1]:.2f}m/s")
print(f"\nCar B is ahead of Car A by: {pos_B[-1] - pos_A[-1]:.2f}m")
print(f"Car B's error vs truth: {pos_B[-1] - pos_C[-1]:.2f}m")
print(f"Car A's error vs truth: {pos_A[-1] - pos_C[-1]:.2f}m")

# Plot results
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 8))

# Position over time
ax1.plot(times_C, pos_C, 'k-', linewidth=2, label='Ground Truth (dt=0.01s)', alpha=0.7)
ax1.plot(times_A, pos_A, 'bo-', label='Car A (dt=1.0s)', markersize=8)
ax1.plot(times_B, pos_B, 'rs-', label='Car B (dt=2.0s)', markersize=8)
ax1.set_xlabel('Time (s)')
ax1.set_ylabel('Position (m)')
ax1.set_title('Drag Race: Position vs Time')
ax1.legend()
ax1.grid(True, alpha=0.3)

# Velocity over time
ax2.plot(times_C, vels_C, 'k-', linewidth=2, label='Ground Truth (dt=0.01s)', alpha=0.7)
ax2.plot(times_A, vels_A, 'bo-', label='Car A (dt=1.0s)', markersize=8)
ax2.plot(times_B, vels_B, 'rs-', label='Car B (dt=2.0s)', markersize=8)
ax2.set_xlabel('Time (s)')
ax2.set_ylabel('Velocity (m/s)')
ax2.set_title('Velocity vs Time')
ax2.legend()
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('euler_drag_race.png', dpi=150, bbox_inches='tight')
print("\nPlot saved as 'euler_drag_race.png'")
plt.show()