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The Newton's laws hold in an inertial frame. However the equations of motion involve the moment of inertia tensor which in turn depends on the orientation of the body and varies with time. This make it solution to the motion of a rigid body problem impossible. This difficulty is absent in the body fixed axes, the moment of inertia tensor depends only on the the geometry of the problem. So whether we use space axes ,or the body axes, depends on the problem to be solved, we use axes which makes the solution of the problem simpler.
Let us assume that an observer is observing the motion of a body from an inertial frame $k_i$.The Newton's Laws apply in the inertial frame and the EOM for most general motion take the form
\[\frac{d}{dt}\vec{L}=\vec{T}\]
where $\vec T$ is the torque, and $\vec{L}={\mathbf I}\cdot\vec{\Omega}$ is the angular momentum. ${\mathbf I}$ denotes the moment of inertia tensor of the body whose components are to computed w.r.t.the inertial frame of the observer. As the body is moving, the components of the moment of inertia tensor will change with time; they depend not only on the geometry and mass distributor of the body but also on the location and orientation of the body relative the coordinate frame of the observer. Thus dependence on $\vec{\Omega}$ is contained in ${\mathbf I}$ in an implicit fashion. This makes solution of EOM in the observer frame impossible. It therefore turns out to be useful to introduce another frame $K_b$ of reference which is fixed in the body and has origin at the same point as the inertial frame $K_i$.
As computed in the body frame $K_b$, the moment of inertia tensor components depend only on the geometry and mass distribution of the body and are time independent. However care must be exercised in writing EOM in $K_b$ as it is a non inertial frame and Newton's Laws do not apply. In fact as seen from the body frame, the body remains stationary and hence $\vec{\Omega_b}=0$. In order to take the best advantage of both the frames the following procedure is adopted.The EOM are written in the inertial frame $K_i$ but the components of moment of inertia tensor $I_i$ and angular velocity $\vec{\Omega_i}$ are evaluated by transforming to the body frame.
Thus we need to deal with two sets of coordinate frames $K_i$ and $K_b$ and need to transform quantities from one to the other. The orientation of rigid body is specified by the body fixed axes.
The six generalised coordinates, used frequently, are the position coordinates of a selected point in the body. This point is also taken as the common origin of the two sets of coordinate axes $K_i$ and $K_b$. The remaining three generalised coordinates will be a set of convenient parameters which describe orientation of the axes $K_b$ w.r.t the axes $K_i$.
