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 $\psi_e$.
$\ddot{\lambda}$
and you get $\ddot{\lambda}$. I partially did it for you, the rest is your task. $\endgroup$