3 ms·

I don`t know much about physics engines, but I do know that there are much more sophisticated integrators than x = x + v*dt. For example schemes for gravity tha

I don`t know much about physics engines, but I do know that there are much more sophisticated integrators than x = x + v*dt. For example schemes for gravity that keeps energy invariant. Are any used in physics engines? I assume yes.

rhaps0dy 6y ago
The naive integrator you're describing, just taking the differential equations and substituting dt by a small, non-zero time step, is known as the Euler integrator [1]. As you say, this scheme does not keep energy invariant. In fact, energy may grow unbounded if you use it. The simplest integrator that "preserves" energy is the Symplectic Euler [2] integrator, which reads: v <- v + F/m * dt x <- x + v * dt The key is that you use the updated velocity to update the position in the same time step. In this scheme, energy isn't exactly preserved, it varies a little bit from one time step to the next. However, you can guarantee that the energy fluctuations will remain bounded, if your time step is below a (fairly large) critical value. Reading the code of this blog post, the first version of the Step (at the beginning of the blogpost) uses the Symplectic Euler scheme, and the second version (towards the end) uses the naive Euler scheme. Bullet, a popular physics engine, uses the Symplectic Euler method [3]. A popular, more sophisticated, scheme is the Runge-Kutta integrator, which comes with variants that preserve the energy and ones that don't. The MuJoCo simulator, used in robotics research, uses the Runge-Kutta method [4] [1] https://en.wikipedia.org/wiki/Euler_method https://en.wikipedia.org/wiki/Euler_method [2] https://en.wikipedia.org/wiki/Semi-implicit_Euler_method https://en.wikipedia.org/wiki/Semi-implicit_Euler_method [3] https://github.com/bulletphysics/bullet3/issues/2513 https://github.com/bulletphysics/bullet3/issues/2513 [4] http://www.mujoco.org/book/computation.html http://www.mujoco.org/book/computation.html
- IainWinter 6y ago
  Oh you're correct, I was trying to make the code more readable for the blog post and didn't even think of that! Thanks for the heads up I would of never caught that, and good info about different integrators, not going to pretend that I knew all of it. It makes sense though because the velocity used for the position would be a frame behind. I was looking into the Runge-Kutta method, but it looked too complex for what I was doing.
  - sgillen 6y ago
    Fixed step Rk4 is really not very complex once you understand numerical integration. Not that you need it, plus it will be about 4x slower.
  - rhaps0dy 6y ago
    You're very welcome! Excellent job with the video too :)
- sgillen 6y ago
  I think all of OpenAI’s gym environments use Mujoco with rk4, but the engine supports both rk4 and semi implicit euler, and the default is actually Euler. Seems like that is the default for almost all robot focused sims too, see some benchmarks here: https://leggedrobotics.github.io/SimBenchmark/ https://leggedrobotics.github.io/SimBenchmark/
rzzzt 6y ago
For a university assignment, we used the velocity Verlet method: https://en.wikipedia.org/wiki/Verlet_integration#Velocity_Verlet https://en.wikipedia.org/wiki/Verlet_integration#Velocity_Ve... Each displayed object was backed by a simplified "physics 3D model" (small number of points of mass, fully connected with edges), and after applying forces and moving points, we checked the length of each connecting link and made sure that they didn't stretch or compress too much. Rigs of Rods has a similar model: https://en.wikipedia.org/wiki/Rigs_of_Rods https://en.wikipedia.org/wiki/Rigs_of_Rods

rhaps0dy 6y ago

The naive integrator you're describing, just taking the differential equations and substituting dt by a small, non-zero time step, is known as the Euler integrator [1]. As you say, this scheme does not keep energy invariant. In fact, energy may grow unbounded if you use it. The simplest integrator that "preserves" energy is the Symplectic Euler [2] integrator, which reads: v <- v + F/m * dt x <- x + v * dt The key is that you use the updated velocity to update the position in the same time step. In this scheme, energy isn't exactly preserved, it varies a little bit from one time step to the next. However, you can guarantee that the energy fluctuations will remain bounded, if your time step is below a (fairly large) critical value. Reading the code of this blog post, the first version of the Step (at the beginning of the blogpost) uses the Symplectic Euler scheme, and the second version (towards the end) uses the naive Euler scheme. Bullet, a popular physics engine, uses the Symplectic Euler method [3]. A popular, more sophisticated, scheme is the Runge-Kutta integrator, which comes with variants that preserve the energy and ones that don't. The MuJoCo simulator, used in robotics research, uses the Runge-Kutta method [4] [1] https://en.wikipedia.org/wiki/Euler_method https://en.wikipedia.org/wiki/Euler_method [2] https://en.wikipedia.org/wiki/Semi-implicit_Euler_method https://en.wikipedia.org/wiki/Semi-implicit_Euler_method [3] https://github.com/bulletphysics/bullet3/issues/2513 https://github.com/bulletphysics/bullet3/issues/2513 [4] http://www.mujoco.org/book/computation.html http://www.mujoco.org/book/computation.html

IainWinter 6y ago
Oh you're correct, I was trying to make the code more readable for the blog post and didn't even think of that! Thanks for the heads up I would of never caught that, and good info about different integrators, not going to pretend that I knew all of it. It makes sense though because the velocity used for the position would be a frame behind. I was looking into the Runge-Kutta method, but it looked too complex for what I was doing.
- sgillen 6y ago
  Fixed step Rk4 is really not very complex once you understand numerical integration. Not that you need it, plus it will be about 4x slower.
- rhaps0dy 6y ago
  You're very welcome! Excellent job with the video too :)
sgillen 6y ago
I think all of OpenAI’s gym environments use Mujoco with rk4, but the engine supports both rk4 and semi implicit euler, and the default is actually Euler. Seems like that is the default for almost all robot focused sims too, see some benchmarks here: https://leggedrobotics.github.io/SimBenchmark/ https://leggedrobotics.github.io/SimBenchmark/

rzzzt 6y ago

For a university assignment, we used the velocity Verlet method: https://en.wikipedia.org/wiki/Verlet_integration#Velocity_Verlet https://en.wikipedia.org/wiki/Verlet_integration#Velocity_Ve... Each displayed object was backed by a simplified "physics 3D model" (small number of points of mass, fully connected with edges), and after applying forces and moving points, we checked the length of each connecting link and made sure that they didn't stretch or compress too much. Rigs of Rods has a similar model: https://en.wikipedia.org/wiki/Rigs_of_Rods https://en.wikipedia.org/wiki/Rigs_of_Rods