# Coefficients for linear tangent steering law

Given certain simplifying assumptions, a "linear tangent" steering law is optimal for orbital insertion of a launch vehicle:

$$\tan \psi = a\cdot t + b$$

i.e. the tangent of the flight path angle changes linearly from the point at which insertion guidance starts until circular orbit is achieved. My question is then how to compute the correct values for $a$ and $b$.

Assuming that an open-loop pitch guidance program is used for the initial ascent, linear-tangent guidance will take over from a given velocity vector and position, and our goal is to reach a given altitude with the correct horizontal velocity for circular orbit:

Is there a more direct way to determine the $a$ and $b$ coefficients than by a trial-and-error search algorithm? My application is for a generalized launch-to-LEO simulation tool.

• Hey Russel, did you ever figure out how to calculate the coefficients A and B for given target altitude and velocity? I'm struggling to find an answer myself. – user36480 Jan 7 at 6:06

The whole mathematical apparatus describing the rocket flight — gravity, rocket equation — it's a set of functions that take some parameters allowing you to model the process. If we set the thrust vector constant, we end with a complete equation of motion where we just plug parameters (Earth standard gravity, vehicle mass, $I_{SP}$ etc.) and we can numerically integrate it into the future. But if we want thrust to change direction, we must describe how it changes: parametrize it. We must have a function that models this change using some constants, so that we can integrate the trajectory using given constants OR retrieve them given the a trajectory. LTG (linear tangent steering law) is such a function — it only allows you to model control over the vehicle.