## Group 10 Project 1.2 ### Final presentation Attempt to go to Titan and back with the lowest cost of fuel and as fast as possible. _Fabio Barbero, Krzysztof Cybulski, Cavid Karca, Margarita Naryzhnyaya, Elliot Doe, Michael Balzer_ kutt.it/group10 --- ## Goal of the project - 1 ### Making a Solar System ---- ## Mathematical model ### Phase 1 _Gravitational force_ $$F = G \cdot\frac{m_1\cdot m_2}{d^2}$$ _Euler's method_ $$\vec v_{t+1} = \vec v_t + \vec a_{t}\cdot \Delta t$$ $$\vec x_{t+1} = \vec x_t + \vec v_{t}\cdot \Delta t + \frac{\vec a_{t} \cdot \Delta t}{2}$$ ---- ### Phase 2, 3: Leapfrog | Basic | Kick-drift-kick form | | -------- | -------- | |$$\vec v_{t+\frac{1}{2}}=\vec v_{t-\frac{1}{2}}+\vec a_t\Delta t$$ $$\vec x_{t+1}=\vec x_t+\vec v_{t+\frac{1}{2}}\Delta t$$|$$\vec v_{t+\frac{1}{2}} = \vec v_t + \vec a_t \frac{\Delta t}{2}$$ $$\vec x_{t+1} = \vec x_t + \vec v_{t + \frac{1}{2}}\Delta t$$ $$\vec v_{t+1} = \vec v_{t + \frac{1}{2}} + \vec a_{t+1} \frac{\Delta t}{2}$$| ---- ### Phase 2, 3: Verlet $$\vec x_{n+1}=2\cdot\vec x_{n-1}+\vec a \cdot \Delta t^2$$ ---- ### Energy drift ![](/uploads/upload_09bcb24c84149c8b2b9fffaccfd79f54.png) ---- ### Runtime of Leapfrog, Euler and Verlet methods ![](/uploads/upload_d5d46d67f067cdd4036060e0affd14cc.png) ---- ## Precomputing positions <!-- .slide: data-background="/uploads/sapproximator1.gif"--> ![](/uploads/sapproximator1.gif) ---- ## Visualisation 2D ![](/uploads/solarsystem1-1.gif) ---- ## Visualisation 3D ![](/uploads/horizon_3d.gif) --- ## Central body simulation - Rotation - Atmosphere - Air resistance ---- ## Rotation - Planets orbit the sun with a certain speed and on rotate around their axis - Useful to know the location where the landing should take place - Only taken into account when landing on celestial bodies ---- ## Rotation ![](/uploads/upload_ab6ce30f367187104cc1e6236810fac8.gif) ---- ## Atmosphere ### Wind on Titan - Wind load formula: $$A \cdot p \cdot v^2 \cdot 0.5 \cdot C$$ Where $A$ is the projected area $p$ is the pressure $v$ is the velocity of the wind $C$ is the drag coefficient ---- ## Wind function ![](/uploads/upload_e2bc78087d8276f73e95fb01890c5b4f.png) ---- ## Wind demo ![](/uploads/upload_6bad403385c807e4cf2a6fd0e224d9c9.gif) ---- ## Air resistance Air resistance: $$F = 0.5 \cdot p(a) \cdot C \cdot A \cdot v^2$$ $p(a)$ pressure in terms of altitude $C$ drag coefficient $A$ area spacecraft $v$ speed of spacecraft ---- ### Pressure function ![](/uploads/upload_e55f385ceaa713e5a5e3a0345c54f9cc.png) --- ## Space voyage - consists of different spacecraft phases - combined they represent a space mission ---- ## Spacecraft ![](/uploads/upload_bf0188ad9ee3e4c300c1bf55907aa80b.png) ---- ## Spacecraft ![](/uploads/RocketBreakOf.gif) --- ## Switching from 2D to 3D --- ## Launching from Earth - using a multistage Rocket to enter stable orbit - simple feedback controller - adjusting angle depending on altitude ---- ## Hohmann transfer $v_{p} = \sqrt{GM\left(\frac{2}{r} - \frac{1}{a} \right)}$ ![](/uploads/hohmann_departure.gif) ---- ## Interplanetary travel ![](/uploads/earth_to_titan.gif) --- ## Landing on Titan ---- ### Open-loop Controller A controller that only gets time as the input. - This is doable with constant gravity. Realistic gravity makes precomputation significantly more complex. - Simple implementation with constant gravity. ---- ### Open-Loop controller pre-computation ![](/uploads/upload_5aee55a4b614660b4ce0fdc33accf684.png) --- ### Closed-loop Controller - AKA feedback controller - Used to determine the thrust based on the current position and velocity. - PID-Controller ---- ### PID-Controller - Proportional - Integral - Derivative Combined, they form the PID-controller. ---- ### Proportional control $$P_{out}=K_p\cdot e(t)$$ $P_{out}$ = Proportional correction $K_p$ = Adjustable constant for proportional controller $e$ = error $t$ = time ---- ### Integral control $$I_{out} = K_i\int _{0}^{t} e(\tau)\, d\tau$$ $I_{out}$ = Integral correction $K_i$ = Adjustable constant for integral controller $\tau$ = Variable of the integral ---- ### Derivative control $$D_{out} = K_d\cdot \frac{de(t)}{dt}$$ $D_{out}$ = Derivative correction $K_d$ = Adjustable constant for derivative controller $\tau$ = Variable of the integral --- ## Back to Earth - Similar procedure, but reversed --- ## Fuel - Kerosene (or RP-1, rocket propellant 1) is used as fuel for the spaceship. - Kerosene is a bipropellant. - Liquid Oxygen (LOX) used as oxidizer for the fuel. ---- ### Mass flow rate Mdot is the mass flow rate of the spacecraft. $$\dot{m} = r\cdot V\cdot A$$ $r$ = Density of the fuel $V$ = Velocity of the fuel $A$ = Area of the nozzle ---- ### Mass flow rate $$\boldsymbol {F = \dot{m}\cdot V_e + (P_e - P_0)\cdot A_e}$$ $F$ = Thrust in N $\dot{m}$ = Mass-flow rate in kg/s $V_e$ = Exit velocity $P_e$ = Exit pressure $p_0$ = free stream pressure $A_e$ = Exit area --- ## Fuel use | Real Ariane 5 | Modified | | -------- | -------- | |![](/uploads/upload_825ea05e6a254e30e47d8a8433fc3258.png)|![](/uploads/upload_8e88dcfb660753f8563b99f201b897a6.png)| --- ### Landing results ![](/uploads/upload_69175aca04b187909e5cd14c3b43cd38.png) ---- ### Landing results ![](/uploads/upload_a8a74efceb6da5b828e7182651bef73d.png) --- # Conclusion ---- # Questions?
{"slideOptions":{"theme":"solarized"}}