ex11

[alezana]

Draft of ex11 based on ex09.

For full testing you need to sync with the dg-commons spaceship model, you can start with some boilerplate, goal definition, evaluation metrics and so on. Please open a new branch ex11

Todos:

• [ ] code
• [ ] docs (ask for the dynamic model Yixiao)

[zipping-suger]

Dynamic model for Spaceship:

1. Position Dynamics: $\frac{dx}{dt} = v_x$ $\frac{dy}{dt} = v_y$

2. Orientation Dynamics: $\frac{d\psi}{dt} = \dot{\psi}$

3. Fuel Dynamics: $\frac{dm}{dt} = -CT \cdot F{\text{thrust}}$

4. Velocity Dynamics: $\frac{dvx}{dt} = \frac{1}{m} \cdot \cos(\delta + \psi) \cdot F{\text{thrust}}$ $\frac{dvy}{dt} = \frac{1}{m} \cdot \sin(\delta + \psi) \cdot F{\text{thrust}}$

5. Angular Velocity Dynamics: $\frac{d\psi}{dt} = -\frac{l_r}{Iz} \cdot \sin(\delta) \cdot F{\text{thrust}}$ $\frac{d\delta}{dt} = \dot{\delta}$

   Where:

   • $\delta$ is the angle of the thrusters with respect to the rocket.
   • $\psi$ is the angle of the rocket with respect to the x-axis.
   • $C_T$ is the thrust coefficient.
   • $l_r$ is the distance from the thruster to the center of mass.

[zipping-suger]

[alezana]

@zipping-suger see comment in the other repo, regarding the picture could you add the velocity vectors and underline that they are in body frame?

[zipping-suger]

Sorry the previous formulation is wrong. I forgot the Coriolis term. It should be:

Dynamics:

• $\frac{dx}{dt} = v_x \cdot \cos(\psi) -v_y \cdot \sin(\psi)$
• $\frac{dy}{dt} = v_x \cdot \sin(\psi) + v_y \cdot \cos(\psi)$
• $\frac{d\psi}{dt} = \dot{\psi}$
• $\frac{dvx}{dt} = \frac{1}{m} \cdot \cos(\delta) \cdot F{\text{thrust}} \textcolor{red}{+ \dot{\psi} \cdot v_y}$
• $\frac{dvy}{dt} = \frac{1}{m} \cdot \sin(\delta) \cdot F{\text{thrust}} \textcolor{red}{- \dot{\psi} \cdot v_x}$
• $\frac{d\dot{\psi}}{dt} = -\frac{1}{I} \cdot lr \cdot \sin(\delta) \cdot F{\text{thrust}}$
• $\frac{d\delta}{dt} = v_{\delta}$
• $\frac{dm}{dt} = -kl \cdot F{\text{thrust}}$

Where:

• $x$ and $y$ are the positions in the world frame.
• $v_x$ and $v_y$ are the velocities in the body frame of the spacecraft.
• $\delta$ is the angle of the thrusters with respect to the rocket.
• $\psi$ is the angle of the rocket with respect to the x-axis.
• $C_T$ is the thrust coefficient.
• $l_r$ is the distance from the thruster to the center of mass.
• $F_{\text{thrust}}$ is the force generated by the thruster.
• $I$ is the moment of inertia of the rocket.
• $k_l$ is the fuel consumption coefficient.