About linearisation of equations of motion rigid body

Question

I am modeling a system of solid bodies. Consider $\theta \approx 0$ and $\chi \approx 0$. At a certain moment I get the following formula for the angular velocity: $$ \omega = \begin{bmatrix} 0 &-\sin(\theta) &\cos(\chi) \cos(\theta)\\ 0 &\cos(\theta) &\cos(\chi) \sin(\theta)\\ 1 &0 &\sin(\chi)\end{bmatrix} \cdot \begin{bmatrix} \dot{\theta}\\ \dot{\chi} \\ \dot{\phi} \end{bmatrix}$$

and I am interested in a linearisation of the equations. Then I write $$ \omega \approx \begin{bmatrix} 0 &-\theta &1 \\ 0 &1 & \theta\\ 1 &0 & \chi\end{bmatrix} \cdot \begin{bmatrix} \dot{\theta}\\ \dot{\chi} \\ \dot{\phi} \end{bmatrix} $$ but the equation is still nonlinear. Also consider the inertia matrix in body coordinate $I_B$ and in world coordinate $I_C = R^T I_B R$. The equations of motion are non-linear especially when writing the angular momentum $L = I_C \cdot \omega$ and it's derivative $\dot{L} = \omega \times (I_C \omega) + I_C\ddot{\omega}$. How can I obtain the linerized equations of motion? Is it ok if I proceed as follows: $$ \omega \approx \begin{bmatrix} \dot{\theta}\\ \dot{\chi} \\ \dot{\phi} \end{bmatrix} \hspace{1cm} \dot{L} \approx I_B\cdot \begin{bmatrix} \ddot{\theta}\\ \ddot{\chi} \\ \ddot{\phi} \end{bmatrix}$$ That is pretty much ignoring everything ... and saying that $\omega$ and $\dot{L}$ have those coordinates in world frame?

Answer 1

Linearization of rigid body motion happens in velocity space (like you show) where a small angle approximation is taken. The problem is that even for small angles, the solution is highly non-linear.

Take a look at the work at NASA which provide some insight into an analytical solution for symmetric bodies with one dominant motion

• Solution of Euler's Equations of Motion and Eulerian Angles for near symmetric rigid bodies subject to constant moments pdf link.

Analytic expressions are found for Euler's Equations of Motion and for the Eulerian Angles for both symmetric and near symmetric rigid bodies under the influence of arbitrary constant body-fixed torques. These solutions provide the body-fixed angular velocities and the attitude of the body, respectively, as functions of time. They are of special interest in applications to spinning spacecraft (such as the Galileo Spacecraft to be launched in 1984) because they include the effect of time-varying spin rate. Thus they can be applied to spin-up and spin-down maneuvers as well as to error analysis for thruster misalignments. The solutions are given for arbitrary initial conditions in terms of Fresnel, Sine and Cosine Integrals. Numerical integration of the governing differential equations has verified that the approximate analytic solutions are very accurate in many physical situations of interest.

• Analytical Solutions for a Spinning Rigid Body Subject to Time Varying Body-Fixed Torques pdf link

Analytic solutions are derived for the general attitude motion of a near-symmetric rigid body subject to time-varying torques in terms of certain integrals. A methodology is presented for evaluating these integrals in closed form. We consider the case of constant torque about the spin axis and of transverse torques expressed in terms of polynomial functions of time. For an axisymmetric body with constant axial torque, the resulting solutions of Euler's equations of motion are exact. The analytic solutions for the Eulerian angles are approximate owing to a small angle assumption, but these apply to a wide variety of practical problem