Enh/liquid-motors-linear-algebra - #364

Great debugging work @MateusStano. I'll attempt to use your observations to fix the trajectory difference and get back to you ASAP.

I'll first review the motor mass PR and then get back to this.

Pending TO-DOs:

• Remove unnecessary implementations.
• Create a toggle to use older equations of motion.
• Merge beta/v1 to this branch.
• Review equations of motion based Stano's feedback.

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• I_11_dot (from uDotVariableMassNumpyFree) is different from TiDot (from standard uDot). From what I understand, these two should be the same, or am I wrong?

For a moment, I agreed with your hypothesis. But it is not true because I_11_dot and TiDot are relative to different origins. Since there is the m*d² term when translating inertia components, and both the mass and the distance are time varying functions, the relationship between I_11_dot and TiDot is non-trivial.

Is the multiplication and division of * self.rocket.motor.mass / total_mass correct here? It seems that this was done to simplify the expression of v_dot, but I am not sure if the other expressions that use r_CM_z were also changed.

Also r_CM_z is meant to be the CM position relative to CDM in the body frame. This multiplication and division makes it so r_CM_z represents something else, or am I wrong?

I have verified that the implementation is in fact correct. The logic is:

$r_{CM} \cdot m_{CM} = r_{CDM} \cdot m_{CDM} + r_{CPM} \cdot m_{CPM}$

where $CM$ is for Center of Mass, $CDM$ is for Center of Dry Mass and $CPM$ is for Center of Propellant Mass. Keep in mind that $m_{CDM}$ is the total dry mass (constant) and $m_{CPM}$ is the propellant mass (time-varying).

Since, for the equations of motion, the origin is fixed back to the $CDM$, then $r_{CDM} = 0$. This simplifies the equation to:

$r_{CM} = r_{CPM} \cdot m_{CPM} / m_{CM} $

I believe this settle the debate about the weird mass multiplication and division in the code.

With all EOMs doubts addressed, @MateusStano, feel free to review my last commits and merge.

Seems like I had an issue before with the beta v1 merge (user error). Just merged it again, and the liquid motor branch as well. Everything working perfectly now, with all tests passing. No code changes, in the scope of this PR, were needed.

@MateusStano, please skim through and merge when you feel ready. Let me know if you have any questions. For reviewing porpuses, I would recommend reviewing each commit individually.