Gear predictor / corrector

[graffy76]

Hello,

I am working on a personal project that models the movement of pedestrians in large crowds. I am using something called the "social force model" and came across a master's thesis which describes an improved implementation. The model constructs a very large force equation to model the way people try to avoid each other / stationary objects as they navigate busy areas. In this case, a very large force equation to describe the acceleration of each pedestrian is developed (F=ma, assuming unit mass).

As I read the thesis, the authors described that they implemented the model using Gear's predictor/corrector to do the integration. The problem is that the Gear method requires first and second derivatives of the force equation (dV/dt), yet the equation itself is not explicitly described in terms of time.

I'm rusty on my phyics / diff eq's, so I suspect this is simpler than it appears to me. The thesis can be found at:

http://public.rz.fh-wolfenbuettel.de/~apel/files/thesis.pdf"

The model is described on pages 21-26 and the Gear implementation is on 28 and 29.

Again, this can't be as hard as it appears to me. Any suggestions?

 

[jvicens]

Hello,

Thank you for sharing your project and the master's thesis. I am familiar with the social force model and have also come across the Gear method in my own research. I have taken a look at the thesis and can offer some suggestions for implementing the model using the Gear method.

Firstly, the Gear method is a numerical integration technique that requires the first and second derivatives of the force equation. In this case, the force equation is given by F=ma, which can be written as a second-order differential equation in terms of acceleration (a=dV/dt). To use the Gear method, you will need to rewrite the force equation in terms of acceleration and then take the first and second derivatives of this equation.

Secondly, the force equation in the social force model is not explicitly described in terms of time. However, you can use the position and velocity of each pedestrian at different time steps to calculate the acceleration. This can be done by using the equations given in the thesis on pages 21-26, which describe the forces acting on each pedestrian.

Lastly, the Gear method requires initial values for the position, velocity, and acceleration of each pedestrian. These values can be obtained from the initial conditions of your simulation or from experimental data.

I hope this helps in implementing the social force model using the Gear method. If you have any further questions, please feel free to reach out to me. Best of luck with your project!