Impact Dynamics 2D Dice Simulation

Overview

A 2D physics simulation modeling impact dynamics between two planar rigid bodies: a diamond-shaped box that can translate and rotate, and a cross-shaped “jack” that bounces inside under gravity with perfectly elastic collisions.

System Description

The system uses six generalized coordinates: $q = [x_1, y_1, \theta_1, x_2, y_2, \theta_2]^T$ where subscript 1 is the jack and subscript 2 is the box.

Equations of Motion

The Lagrangian $L = T - V$ where:

\[T = \frac{1}{2}M_1(\dot{x}_1^2 + \dot{y}_1^2) + \frac{1}{2}I_1\dot{\theta}_1^2 + \frac{1}{2}M_2(\dot{x}_2^2 + \dot{y}_2^2) + \frac{1}{2}I_2\dot{\theta}_2^2\] \[V = M_1 g y_1 + M_2 g y_2\]

Collision Detection

Each jack tip is expressed in the box body frame using $g_{B,tip} = g_{WB}^{-1} \cdot g_{W,tip}$. A collision occurs when $\phi \leq 0$ and $\dot{\phi} < 0$.

Results

• Jack falls under gravity and bounces off walls
• Impacts transfer momentum to the box, causing movement and rotation
• With $e = 1$, energy is conserved indefinitely
• Off-center impacts produce realistic rotational responses