To implement the `Visualization` class in full detail for the double pendulum simulation, we can use the `matplotlib` library to animate the pendulum's motion. The `ControlSystem` class will be a bit more abstract, as control systems can vary widely; however, I will provide a simple proportional controller as an example.

First, let's implement the `Visualization` class:

```python
import matplotlib.pyplot as plt
import matplotlib.animation as animation

class Visualization:
def __init__(self, lengths):
self.lengths = lengths
self.fig, self.ax = plt.subplots()
self.line, = self.ax.plot([], [], 'o-', lw=2)

def _init_animation(self):
"""Initialize the background of the animation."""
self.ax.set_xlim(-sum(self.lengths), sum(self.lengths))
self.ax.set_ylim(-sum(self.lengths), sum(self.lengths))
self.line.set_data([], [])
return self.line,

def _update_animation(self, frame):
"""Update the animation for each frame."""
x1, y1 = self.lengths[0] * np.sin(frame[0]), -self.lengths[0] * np.cos(frame[0])
x2, y2 = x1 + self.lengths[1] * np.sin(frame[1]), y1 - self.lengths[1] * np.cos(frame[1])
self.line.set_data([0, x1, x2], [0, y1, y2])
return self.line,

def display_pendulum_motion(self, data):
"""Animate the double pendulum motion."""
ani = animation.FuncAnimation(self.fig, self._update_animation, frames=data, init_func=self._init_animation,
blit=True, interval=25)
plt.show()

# Example usage:
if __name__ == "__main__":
lengths = [1, 1] # Lengths of the pendulum arms
# Simulated data for demonstration purposes (list of [angle1, angle2] pairs)
data = [(np.sin(i / 10.0), np.sin(i / 10.0 + np.pi / 4)) for i in range(100)]
visualization = Visualization(lengths)
visualization.display_pendulum_motion(data)
```

In the `Visualization` class, we have a method `display_pendulum_motion` which takes the simulation data (a list of `[angle1, angle2]` pairs) and animates the pendulum using `matplotlib.animation`. The `_init_animation` and `_update_animation` methods are used to set up and update the animation for each frame, respectively.

Now, let's implement the `ControlSystem` class with a simple proportional controller:

```python
class ControlSystem:
def __init__(self, kp):
self.kp = kp # Proportional gain

def apply_control(self, current_angle, target_angle):
"""
Apply proportional control to the pendulum system.

:param current_angle: The current angle of the pendulum
:param target_angle: The desired target angle for the pendulum
:return: Control force to be applied
"""
error = target_angle - current_angle
control_force = self.kp * error
return control_force

# Example usage:
if __name__ == "__main__":
kp = 2.0 # Proportional gain
target_angle = 0.0 # Target angle for the pendulum
current_angle = 0.1 # Current angle of the pendulum

control_system = ControlSystem(kp)
control_force = control_system.apply_control(current_angle, target_angle)
print(f"Control Force: {control_force}")
```

The `ControlSystem` class has a method `apply_control` which takes the current angle and target angle as inputs and calculates the control force based on the proportional gain `kp`. This is a simple example; in practice, control systems may include more complex algorithms such as PID controllers or state-space controllers.

**Required Packages:**
- `matplotlib` for the `Visualization` module. Install it using `pip` if not already installed:
```bash
pip install matplotlib
```
- `numpy` for mathematical computations. It is assumed to be already installed from the previous code sections.

Please note that the provided visualization code is for a simple 2D animation. More sophisticated 3D visualization or real-time rendering might require different libraries like `PyOpenGL` or `VTK`.

To run the double pendulum simulation using the classes we've implemented, we need to tie everything together in a `main` method. This method will:

1. Collect user input for initial conditions and parameters.
2. Configure the simulation.
3. Run the physics engine to calculate the motion.
4. Store the data at each time step.
5. Handle any potential errors.
6. Visualize the motion.
7. Perform analysis on the simulation data.

Here's how you can implement the `main` method:

```python
def main():
try:
# Initialize the input interface and collect user input
input_interface = InputInterface()
input_interface.collectUserInput()

# Extract the parameters from the input interface
initial_angles = input_interface.initialAngles
lengths = input_interface.lengths
masses = input_interface.masses
g = input_interface.gravitationalAcceleration

# Convert initial angles from degrees to radians for computation
initial_conditions = [np.radians(angle) for angle in initial_angles] + [0, 0] # [theta1, theta2, omega1, omega2]

# Initialize the physics engine with the parameters
physics_engine = PhysicsEngine(lengths, masses, g)

# Set the simulation parameters
time_step = 0.01 # Time step for the simulation
total_time = 10.0 # Total simulation time

# Run the physics engine to calculate the motion
motion = physics_engine.calculate_motion(initial_conditions, (0, total_time), time_step)

# Initialize data storage
data_storage = DataStorageManagement()

# Store the data at each time step
for i, state in enumerate(motion):
data_storage.storeData(i * time_step, state)

# Initialize the visualization
visualization = Visualization(lengths)

# Extract only the angle data for visualization
angle_data = motion[:, :2] # Assuming motion is a NumPy array with columns [theta1, omega1, theta2, omega2]

# Visualize the motion
visualization.display_pendulum_motion(angle_data)

# Initialize the analysis tools
analysis_tools = AnalysisTools(lengths, masses, g)

# Perform energy analysis on the simulation data
energies = [analysis_tools.calculate_total_energy(state) for state in motion]

# Log the analysis results
error_handling = ErrorHandlingLogging()
error_handling.log_event("Simulation completed successfully.")
error_handling.log_event(f"Total Energy: {energies}")

except Exception as e:
error_handling = ErrorHandlingLogging()
error_handling.handle_error(e)

if __name__ == "__main__":
main()
```

Please note that the `motion` variable is assumed to be a NumPy array with each row corresponding to the state of the system at a given time step. The state consists of the angles and angular velocities `[theta1, omega1, theta2, omega2]`. The `display_pendulum_motion` method in the `Visualization` class is designed to take an array of angles for the two pendulum arms.

In the example above, I've also included the error handling mechanism. If any exception occurs during the simulation, it will be caught, and an error message will be logged.

Before running the `main` method, make sure all the classes have been properly defined in your script, and you have the required packages installed:

```bash
pip install numpy matplotlib
```

Running the `main` method should now execute the entire simulation, from collecting user input to visualizing the motion and logging the results.