Solving Sub-Problem for OpenAI Lunar Lander v2: Seeking Advice

[genxium]

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

• 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

and the "to find variables (in red)" seem non-trivial to solve for.

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

 

[Filip Larsen]

At a glance it appears your equations can be simplified a bit by expressing the force and torque equations using body coordinates in which input acceleration and gravity are rotated by $\theta$.