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I'm currently working on a pet project which is similar to the OpenAI Lunar Lander v2 (hence the problem is in a 2D context), and seeking help for a sub-problem that's been blocking me for a while.

At any instant of time, I'm to find

, given

To my understanding, the dynamics are (don't know whether I can use TeX here, tried wrapping with $$ and ## but preview didn't work)

Is there any advice for solving them at least numerically? An analytical solution will certainly be much appreciated but not a pursuit here.

At any instant of time, I'm to find

- F
_{e}: magnitude of main engine thrust, must be >0 - F
_{s}: magnitude of side engine thrust (>0 to point to the "right w.r.t. lander body", and < 0 to point to opposite direction), - φ: tilted angle of main engine nozzle w.r.t. lander body, should be within [-π/2, +π/2]

, given

- a
_{x}: the expected x-acceleration of COG (center of gravity, same below) - a
_{y}: the expected y-acceleration of COG - β
_{C}: the expected angular acceleration w.r.t axis through COG and perpendicular to the plane (this screen) - m: current mass of the whole lander
- I
_{C}: current moment of inertia of the whole lander w.r.t. same axis of β_{C} - θ: current tilted angle of the lander body w.r.t. the
**fixed**gravity direction - h: current distance of COG and the nozzle hinge as shown in the figure below
- g:
**fixed**gravity acceleration - H: total height of the lander body

To my understanding, the dynamics are (don't know whether I can use TeX here, tried wrapping with $$ and ## but preview didn't work)

- -F
_{e}*sin(θ+φ) + F_{s}*cosθ = a_{x}*m - F
_{e}*cos(θ+φ) + F_{s}*sinθ - g*m = a_{y}*m - F
_{e}*sinφ*h + F_{s}*(H-h) = β_{C}*I_{C}

Is there any advice for solving them at least numerically? An analytical solution will certainly be much appreciated but not a pursuit here.

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