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https://github.com/emmanuelmess/interactive-dynamics-physics-simulations

This is a test made for the Constraint Based Simulator. It provides a constraint satisfaction physics simulator with automatic differentiation.
https://github.com/emmanuelmess/interactive-dynamics-physics-simulations

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

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README 

        
#  Interactive Dynamics for Physics Simulations



This is a test made for the [Constraint Based Simulator](https://github.com/EmmanuelMess/ConstraintBasedSimulator). It provides a constraint satisfaction physics simulator with automatic differentiation.



## Screenshots



 






## Usage



### Setup

```bash

cd code

python3 -m venv venv

source venv/bin/activate

python3 -m pip install -r requirements.txt

```



### Running

```bash

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](https://dl.acm.org/doi/pdf/10.1145/91394.91400) by Andrew Witkin, Michael Gleicher and William Welch

* [An Introduction to Physically Based Modeling: Constrained Dynamics](https://www.cs.cmu.edu/~baraff/pbm/constraints.pdf) by Andrew Witkin

* [Constrained dynamics](https://sites.cc.gatech.edu/classes/AY2017/cs7496_fall/slides/ConstrDyn.pdf) by Karen Liu

* [JAX](https://jax.readthedocs.io/en/latest/notebooks/quickstart.html) by The JAX Authors

* [pygame](https://www.pygame.org) by the pygame community



## License



```text

MIT License



Copyright (c) 2023 EmmanuelMess



Permission is hereby granted, free of charge, to any person obtaining a copy

of this software and associated documentation files (the "Software"), to deal

in the Software without restriction, including without limitation the rights

to use, copy, modify, merge, publish, distribute, sublicense, and/or sell

copies of the Software, and to permit persons to whom the Software is

furnished to do so, subject to the following conditions:



The above copyright notice and this permission notice shall be included in all

copies or substantial portions of the Software.



THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR

IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,

FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE

AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER

LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,

OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE

SOFTWARE.

```