Computing the local orbital frame vector directions in a geocentric reference frame In the context of the study of a cubesat (little satellite), I am asked to define the local orbital frame and compute the local orbital frame vector directions as a function of time :

Here the context :



Anyone could give me some tracks or links to find the answer of part 2.1 Local orbital frame ? this would be kind.

Regards
I would interpret this as the Local Vertical, Local Horizontal (LVLH) frame (see, e.g., Vallado and McClain, Fundamentals of Astrodynamics and Applications, 4th ed., p. 156). This is also known as the RIC frame (radial, in-track, cross-track) or the RTN frame (radial, transverse, normal).


The radial or local vertical vector $R$ is just the position vector from the center of the Earth to the spacecraft.
The in-track, transverse, or local horizontal vector $I$ is the vector in the orbital plane that is both prograde and perinduclar to the radial vector. It is not, in general, the velocity vector. (The in-track and velocity vectors are coincident if and only if the flight path angle is zero, i.e., you have one of a) a circular orbit, b) the spacecraft is at apoapsis, or c) the spacecraft is at periapsis). You can construct it by applying the Graham-Schmidt Process to the position and velocity vectors; $I = v - \frac{R\cdot v}{R \cdot R}R$. 
The cross-track or normal vector $C = R \times I$ completes the right-handed set.