Interactive Dynamics for Physics Simulations

This is a test made for the Constraint Based Simulator. It provides a constraint satisfaction physics simulator with automatic differentiation.

Screenshots

Usage

Setup

cd code
python3 -m venv venv
source venv/bin/activate
python3 -m pip install -r requirements.txt

Running

cd code
python3 main.py

Adding functionality

Generate a new Constraint subclass and use it in a case.

Design

Math for a single particle

This explanation is meant to complement the references, please read that first. This explains how to compute the derivates with respect to time and position.

$C_p(t)$ is a constraint function on a set of particles $p$ so that:

$$ \exists t / C_p(t) = 0 \land \exists \dot{C_p} $$

To obtain the constraint as a function of particles positions (instead of time) we use:

$$ \widetilde{C}_p(x(t)) = C_p(t) $$

We don't have the single particle $x(t)$ function analitically, but we can use an approximation (a Taylor approximation at $t = 0$):

$$ \widetilde{C}_p(\widetilde{x}(t)) \approx C_p(t) $$

$$ \widetilde{x}(t) = x + t * v + \frac{1}{2} * t^2 * a \approx x(t) $$

$$ \widetilde{x}(0) = x_t \land \dot{\widetilde{x}}(0) = v_t \land \ddot{\widetilde{x}}(0) = a_t $$

And that lets us compute the derivatives ($p_i \in p$ and $C^i$ is a constraint):

$$ C = \begin{bmatrix} C^0(0) & \cdots & C^m(0) \end{bmatrix}^T $$

$$ \dot{C} = \begin{bmatrix} \frac{\partial C^0} {\partial t}(0) & \cdots & \frac{\partial C^m} {\partial t}(0) \end{bmatrix}^T $$

$$ J = \begin{bmatrix} \frac{\partial C_ {p_0}^0}{\partial x_1}(0) & \frac{\partial C_ {p_0}^0}{\partial x_2}(0) & \cdots & \frac{\partial C_ {p_n}^0}{\partial x_1}(0) & \frac{\partial C_ {p_n}^0}{\partial x_2}(0) \\ \vdots & & \vdots & & \vdots \\ \frac{\partial C_ {p_0}^m}{\partial x_1}(0) & \frac{\partial C_ {p_0}^m}{\partial x_2}(0) & \cdots & \frac{\partial C_ {p_n}^m}{\partial x_1}(0) & \frac{\partial C_ {p_n}^m}{\partial x_2}(0) \\ \end{bmatrix} $$

$$ \dot{J} = \begin{bmatrix} \frac{\partial \dot{C}_ {p_0}^0}{\partial x_1}(0) & \frac{\partial \dot{C}_ {p_0}^0}{\partial x_2}(0) & \cdots & \frac{\partial \dot{C}_ {p_n}^0}{\partial x_1}(0) & \frac{\partial \dot{C}_ {p_n}^0}{\partial x_2}(0) \\ \vdots & & \vdots & & \vdots \\ \frac{\partial \dot{C}_ {p_0}^m}{\partial x_1}(0) & \frac{\partial \dot{C}_ {p_0}^m}{\partial x_2}(0) & \cdots & \frac{\partial \dot{C}_ {p_n}^m}{\partial x_1}(0) & \frac{\partial \dot{C}_ {p_n}^m}{\partial x_2}(0) \\ \end{bmatrix} $$

We can use the result:

$$ \ddot{C} = (J W J^T) {\lambda}^T + \dot{J} \dot{\widetilde{x}} + J W \ddot{\widetilde{x}} = - k_s C - k_d \dot{C} $$

And compute λ such that:

$$ (J W J^T) {\lambda}^T + \dot{J} \dot{\widetilde{x}} + J W \ddot{\widetilde{x}} + k_s C + k_d \dot{C} = 0 $$

We compute using an approximate least squares method.

Thanks

• Interactive Dynamics by Andrew Witkin, Michael Gleicher and William Welch
• An Introduction to Physically Based Modeling: Constrained Dynamics by Andrew Witkin
• Constrained dynamics by Karen Liu
• JAX by The JAX Authors
• pygame by the pygame community

License

MIT License

Copyright (c) 2023 EmmanuelMess

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