In this repo, I am simulating a mass-spring system to be later used for modeling robotic legs using SLIP model.

Here is a simple demo of the function:

The function simulates a mass attached to a mass-less spring. There are two phases:

• Swing phase: In this phase, the mass is just free falling. The equations governing this phase are

$$ \ddot{y} = -g ,\ \ddot{x} = 0 $$

The spring has its nominal length $l_0$.

• Stance phase: In this phase, the spring is touching the ground. When the spring is compressed, it exerts a force on the mass which will eventaully cuase a lift-off.
  During the stance phase, the dynamics of the system are calculated using the Lagrangian method. The kinetic energy of the system is

$$ K = \frac{1}{2} m (\ddot{x}^2 + \ddot{y}^2) $$

and the potential energy of the system, which consists of gravitational potential energy and spring potential energy is

$$ P = mgy + \frac{1}{2} k (l - l_0)^2 $$

Enforcing the mass to be above the ground at all times, we get the following Lagrangian

$$ \mathcal{L} = K - P + \lambda y $$

This leads to the unconstrained stance phase dynamics

$$ \begin{align} \ddot{x} = \frac{k}{m} (l - l_0) \frac{x - x_c}{l} \ddot{y} = \frac{k}{m} (l - l_0) \frac{y}{l} \end{align} $$

meaning this is the dynamics during the stance phase if the mass itself isn't touching the ground. If the mass is touching the ground, an upward force of $mg$ is exerted on it and $y = \dot{y} = \ddot{y} = 0$.

Switching between stance and swing phase happens at touchdown and at lift-off. Touchdown happens when

$$ y < l_0 \cos(\theta) \quad \text{and} \quad \dot{y} < 0 $$

where $x_c$ is the contact poin and $\theta$ is the foot placement angle. Lift-off happens when

$$ l = \sqrt{(x - x_c)^2 + y^2} = l_0 \quad \text{and} \quad \dot{y} > 0 $$

This is the enitre logic implemented in the code.

TO BE FURTHER COMPLETED.