I'll try answering your two questions simply first. If these responses are too simple or miss the mark, let me know, and I'll edit the response.

1) What are the state propagation vector and State Transition Matrix (STM)?

The state propagation vector is simply the position & velocity at a given time.

The STM is a matrix that captures the sensitivity of the propagation to the initial state. So, it answers the question "If I change my starting x-coordinate by 5 meters, how much will my final position and velocity change?"

2) How can I use the STM to improve convergence on new Halo Orbits?

You can use the STM to achieve faster convergence on new Halo orbits by mapping the change you need at the Y-axis crossing back to the starting state. (E.g. if you arrive at the crossing with a +2 Z velocity, you can use the STM to compute a different initial state that will have a Z velocity reduced by about 2. (subject to linearization errors) Dr. Davis from CU Boulder (CCAR) provides the following handout in the Interplanetary Mission Design grad-course she teaches:

http://ccar.colorado.edu/imd/2015/documents/SingleShootingHandout.pdf

I'll try answering your two questions simply first. If these responses are too simple or miss the mark, let me know, and I'll edit the response.

1) What are the state propagation vector and State Transition Matrix (STM)?

The state propagation vector is simply the position & velocity at a given time.

The STM is a matrix that captures the sensitivity of the propagation to the initial state. So, it answers the question "If I change my starting x-coordinate by 5 meters, how much will my final position and velocity change?"

2) How can I use the STM to improve convergence on new Halo Orbits?

You can use the STM to achieve faster convergence on new Halo orbits by mapping the change you need at the Y-axis crossing back to the starting state. (E.g. if you arrive at the crossing with a +2 Z velocity, you can use the STM to compute a different initial state that will have a Z velocity reduced by about 2. (subject to linearization errors) Dr. Davis from CU Boulder (CCAR) provides the following handout in the Interplanetary Mission Design grad-course she teaches:

http://ccar.colorado.edu/imd/2015/documents/SingleShootingHandout.pdf

More over, here is the summary of a project on Halo orbits which includes a number of useful figures: http://ccar.colorado.edu/asen5050/projects/projects_2012/dowling/introduction.html

I'll try answering your two questions simply first. If these responses are too simple or miss the mark, let me know, and I'll edit the response.

1) What are the state propagation vector and State Transition Matrix (STM)?

The state propagation vector is simply the position & velocity at a given time.

The STM is a matrix that captures the sensitivity of the propagation to the initial state. So, it answers the question "If I change my starting x-coordinate by 5 meters, how much will my final position and velocity change?"

2) How can I use the STM to improve convergence on new Halo Orbits?

You can use the STM to achieve faster convergence on new Halo orbits by mapping the change you need at the Y-axis crossing back to the starting state. (E.g. if you arrive at the crossing with a +2 Z velocity, you can use the STM to compute a different initial state that will have a Z velocity reduced by about 2. (subject to linearization errors) A good breakdown of the technique is here:

http://ccar.colorado.edu/asen5050/projects/projects_2012/dowling/introduction.html

Just look for the section called Single-Shooting Differential Corrector.

I'll try answering your two questions simply first. If these responses are too simple or miss the mark, let me know, and I'll edit the response.

1) What are the state propagation vector and State Transition Matrix (STM)?

The state propagation vector is simply the position & velocity at a given time.

The STM is a matrix that captures the sensitivity of the propagation to the initial state. So, it answers the question "If I change my starting x-coordinate by 5 meters, how much will my final position and velocity change?"

2) How can I use the STM to improve convergence on new Halo Orbits?

You can use the STM to achieve faster convergence on new Halo orbits by mapping the change you need at the Y-axis crossing back to the starting state. (E.g. if you arrive at the crossing with a +2 Z velocity, you can use the STM to compute a different initial state that will have a Z velocity reduced by about 2. (subject to linearization errors) Dr. Davis from CU Boulder (CCAR) provides the following handout in the Interplanetary Mission Design grad-course she teaches:

http://ccar.colorado.edu/imd/2015/documents/SingleShootingHandout.pdf