The dynamics of the system were simulated in MATLAB using kinematic relationships, Lagrangian Dynamics, and Euler's method for numerical integration. Results can be seen below:

Slowed-down videos of the forward and backward directions are shown below:

Masses of important components (left gear, right gear, arm, lid, coin) were measured using a scale. The spring constant was calculated by measuring the force of the spring at a known displacement using the scale Each gear's inertia were modeled using the thin disc formula. The arm and lid were treated as slender rods, and the coin as a point mass. Damping components for the gears and the lid were arbitrarily assigned. After some research, the suction cup was modeled simply as a time-delay, as the force imparted by it would not impact the dynamics of the system.

The Potential and Kinetic Energy of the system are shown by the following equations:

\begin{align}
&PE = m_{arm} g l_{arm} sin(180-\theta_{arm}) + m_{lid} g l_{lid} sin(180-\theta_{lid}) + \frac{1}{2} k \Delta_{\theta_{arm}}
\\
&KE = \frac{1}{2}I_{arm}\omega_{arm}^2 + \frac{1}{2}m_{coin}v_{coin}^2 + \frac{1}{2}I_{gear_1}\omega_{gear_1}^2 + \frac{1}{2}I_{gear_2}\omega_{gear_2}^2 + \frac{1}{2}*I_{lid}\omega_{arm}^2;
\end{align}

After solving the Euler Lagrange equation and substituting the appropriate kinematic relationships between accelerations of the components, the acceleration of the arm is found to be:

\begin{align}
&\alpha_{arm} = -(\Delta_{\theta_{arm}}k - \Tau_{external} + b_{gears}\omega_{arm} + b_{lid}\omega_{lid} + \frac{1}{2}g l_{arm} m_{arm}cos(180 - \theta_{arm}) - \frac{1}{2} g l_{lid} m_{lid} cos(180 - \theta_{lid}/(ILid*Warm2Lid^2 + IGear2*Warm2Gear2^2 + mCoin*rToCoin^2 + IArm + IGear1);
\end{align}