# Linear Guidance law into X-Y axis

Im currently writing a model for a guidance law named Maintaining Guidance law.

The law looks as following: $$a_{M} = [K_1,K_2]*[\dot{\lambda_T} ,\dot{\lambda_D}]'$$

Since im using a linear model I use the following calculations:

$$\ddot{y} = \ddot{y_T}-\ddot{y_M}=a_T-a_M$$ $$\dot{y} = \dot{y_T}-\dot{y_M}$$ $$y = y_T-y_M$$

Where: $$\dot{\lambda} = \frac{y+\dot{y}*t_{go}}{V_{c}*t_{go}^2}$$

And K1 and K2 are vectors calculated before.

The engagement looks as following:

I tried to transform it into X-Y axis however I got stuck in few things I couldnt figure out as:

1) In this model, $\dot{\lambda}$ depands of $t_{go}$ which becomes small close to the end and makes $\dot{\lambda}$ huge ---> $a_M$ is huge ---> $V_M$ is not constant as taken in linearized models.

2) The same problems occurs when I use $v_{c}$. In all the linear model $V_c=V_{M}+V_T$ where the velocities are constant. How to use it then in my model since its not constant/almost constant.

Is there any tips of how to transform it to X-Y graph but still keep it linear?

Thank you!