# 3DOF Rocket Dynamics for an LQR controller

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?

• Cmon, just simple rocket science here ;-) – John Alexiou Feb 14 at 8:16
• So can you link me to a book where I can study this simple rocket science? – Fellarrusto Feb 20 at 21:44
• I was being "funny". No rocket science isn't easy. Anyway, you need to show what you tried and why it did not work out. In theory, if you work out the equations of motion in one coordinate system, you can rotate to another one. But if this is a 3D problem, then care must be taken to transform the mass moment of inertia matrix to the new coordinates. When to comes to integrating the equations, you need to be on an inertial reference frame. – John Alexiou Feb 20 at 23:04
• Actually, I've been able to transform in the earth reference frame, it was the easiest task xD. Now I have to tune all the constants and functions. I'll share the results. – Fellarrusto Feb 21 at 16:55