# aircraft pursuit dynamic system state update

Planar Evasive Aircrafts Maneuvers Using Reinforcement Learning

$$d=\sqrt{(x_m-x_e)^2+(y_m-y_e)^2}$$

$$\dot{d} = V_e \cos (\lambda-\Psi_e) - V_m \cos(\lambda-\Psi_m)$$

$$\lambda = \arctan\frac{x_e - x_m}{y_e - y_m}$$

$$\dot{\lambda} = \frac{V_m\sin(\lambda - \Psi_m) + V_e\sin(\lambda-\Psi_e)}{d}$$

In the state equations, $d$ is the distance between the objects $m$ and $e$. $\lambda$ is the line of sight angle between $m$ and $e$. $V_e$ and $V_m$ are constant scalars. $\Psi$ is the heading angle of an object. The picture of the geometry can be seen below.

I am interested in the discrete time state update $x(k+1) = x(k) + \delta t \times x(k)$. I have solved $d$ dotdot and $\lambda$ dotdot above. I can just plug in $\dot{psi}$ = acceleration / velocity and other variables except for $\psi_m = arctan(\dot{y_m}/\dot{x_m})$ and $\psi_e = arctan(\dot{y_e}/\dot{x_e})$, which I cannot solve using the four state variables. Please advise me how to solve for heading angles $\psi_m$ and $\psi_e$.

• Also please see how to use mathjax to format math on se.
– user10585
Jun 4, 2018 at 10:59
• Also is $x(t)$ the position or position and orientation of which object? To start with?
– user10585
Jun 4, 2018 at 11:04
• Also what is $V_m$?
– user10585
Jun 4, 2018 at 11:10
• This site supports Latex, just type in $\ddot{\lambda}$ and you get $\ddot{\lambda}$. I partially did it for you, the rest is your task. Jun 5, 2018 at 22:07

So far there is no solution for updating the states directly. I suspect the authors of the paper update the states $x_m, y_m, \dot{x}_m, \dot{y}_m, \psi_m$ at each step using the acceleration input and then recompute the states $d, \lambda$, etc every time.