3DOF Rocket Dynamics for an LQR controller

Question

I'm trying to define a simplified model of a rocket to implement an LQ regulator. What I need is the translational dynamics on the earth's inertial frame and the rotational dynamics in the body frame. After obtaining the non linear model, I'll need to linearized it to get the state-space equations:

$$ \begin{cases} \dot{x}=Ax+Bu \\ y = Cx+Du \end{cases} $$

The simplified hypothesis are:

• the mass of the rocket is constant
• the rotation of the rocket $\theta\in [-5 \quad 5]deg$
• the flight is subsonic
• the centre of pressure $CP$ is lower than the centre of mass $CA$
• the gravity is constant
• the rocket is not flexible

Here is a schematic rappresentation of the system

Where:

• $x_b$ and $z_b$ are the body frame axis
• $x$ and $z$ are the earth frame axis
• $\theta$ is the rotation relative to the earth frame
• $\alpha$ is the angle of attack due to the relative speed body-wind
• $\gamma$ is the thruster gimbal angle
• $F_g$ is the gravity force [$N$]
• $F_A$ is the resulting vector of the aerodynamic forces (drag and lift) [$N$]
• $F_p$ is the thruster throttle [$N$]
• $r_{CP}$ distance of the centre of pressure from the centre of mass
• $r_T$ distance of the thruster form the centre of mass

The assuming that the wind angle is $\phi_w$ and the motion direction of the rocket is $\phi_m$ we can say that $\alpha=\phi_w + \phi_m$. $F_A = L-D$ with $L$ the lift force and $D$ the drag force. So i came up with is this:

$$ \begin{cases} m\ddot{x_b}=-D\cos{\alpha}+L\sin{\alpha}-F_g\cos{\theta}+F_p\cos{\gamma}\\ m\ddot{z_b}=-D\sin{\alpha}+L\cos{\alpha}-F_g\sin{\theta}+F_p\sin{\gamma}\\ I\ddot{\theta}=(L\cos{\alpha}-D\sin{\alpha})r_{CP}-F_p\sin{\gamma}r_T \end{cases} $$

So now I'm stuck, I need the model in the earth coordinate system so $(x,z)$, I tried the rotation matrix with no success.

Can anyone give me some hint, help, or a solution?

Answer 1

you can do that easily by:

1-modeling the equations in simulink environments (using the basic blocs) 2-Then, write a Matlab function which takes the xb,zb, theta double dot, etc.. as input and then write the transformation matrix without the second raw.

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