aircraft pursuit dynamic system state update

Question

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 $\psi_e$.

Answer 1

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.

Answer 2

Following on from phoque, go back to you first four equations and rewrite them with each and every variable expressed one step forward in terms of the current state plus delta. You will need to do a few simple calcs to get some of the deltas. And you may want to think about how you want to update position. For instance do you want to use a delta based on the old velocity, the new one, or an average?

You can avoid averages by using two different time step phases. Calc. Velocities at 1.0, 2.0, 3.0 delta t's and positions at 0.5, 1.5, 2.5 delta t's etc.

Pro tip. The sketch that conveys the problem the clearest in terms of the variables is never the one that has the variables in the most convenient arrangement for computation. Get used to this. Draw two diagrams. The first should have all angles based on the common reference frame - Earth in this case, XYZ = ENU). All angles must have 1 and only 1 arrow. And they go from +X to +Y in the XY plane, +Y to +Z in the YZ plane, and +Z to +X in the ZX plane. This is for all Right Hand Rule coordinate systems. For instance, the angle labna evader should be from the +X and run ccw to the line of sight segment. If what you really want in the end is that minus phi, you can assign that a new name in the second sketch.

Do this and your autopilot won't unexpectedly turn the plane upside down when it crosses the equator.