Created a 3D physical constraint-based rope simulation for my team's game, Roped Together. The mathematical methods used were ispired by Sebastian Lague's use of Verlet Integration in his rope simulation.
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The system works by having a series of points constrained to be a fixed distance apart. Gravity is then applied, then the points fall, but since they can only be a certain distance apart, they behave like a rope.
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I added additional functionalities to the simulation to turn it into a game mechanic. All the points in the rope are in a list, and can be pinned to any other Unity Engine game object. From there, I added functions to wind up the rope at the waist of the character by pinning every few points to objects attached them. While climbing, the character lets out portions of the rope and pins points to clip game objects as the character climbs, and when the character falls, they are pinned to a point on the rope, causing it to catch them.