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Multidimensional obstacle avoidance

https://math.stackexchange.com/questions/4146255/multidimensional-obstacle-avoidance-in-ode

For some time, I studied this question more closely and came to the conclusion that it is best to assign its own constraint for each selected coordinate. Here are the results.

I choose those system of ODE:

$F=\begin{cases} \dot{x}=g_x+\frac{d(U_{rep_1})}{dx} \\ \dot{g_x}=-g_x+\frac{df}{dx} \\ \dot{y}=g_y+\frac{d(U_{rep_2})}{dy} \\ \dot{g_y}=-g_y+\frac{df}{dy} \end{cases}$

where $x,y,g_x,g_y$ - variables;

$f=e^{-(x^2+y^2)}$

$U_{rep_1}=\begin{cases} \frac{1}{2}(\frac{1}{\rho_1}-\frac{1}{\rho_0})^2, & \mbox{if } \rho_1<=\rho_0\mbox{} \\ 0, & \mbox{if } \rho_1>\rho_0\mbox{} \end{cases}$

$U_{rep_2}=\begin{cases} \frac{1}{2}(\frac{1}{\rho_2}-\frac{1}{\rho_0})^2, & \mbox{if } \rho_2<=\rho_0\mbox{} \\ 0, & \mbox{if } \rho_2>\rho_0\mbox{} \end{cases}$

$\rho_1=\sqrt{(x-\psi_1)^2}-d_{obs}$

$\rho_2=\sqrt{(y-\psi_2)^2}-d_{obs}$

$\psi_1=\delta +2 e^{-t T}$

$\psi_2=\delta +\frac{1}{2.2}e^{-t T}$

$\rho_0 = 0.15, T = 0.5, \delta = 0.35,d_{obs}=0.1$

The figure below shows the results of numerical simulations from Mathematica. It can be seen that the selected variables cannot cross the selected barrier.

enter image description here

The following is interesting. When a variable touches the barrier, it kind of "sticks" to it, holds on for a while, and then "unsticks". At this time, $\frac{d(U_{rep_1})}{dx}$ and $\frac{d(U_{rep_2})}{dy}$ is not zero.

enter image description here

In general, I ask for help with this task - how to quickly "detach" the variable from the barrier? Are there any tricks? Maybe experts in differential equations and dynamical systems can help. I will be glad to help.

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