I am reading the following paper about the attitude control of aircraft:
http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1271671
The quaternion represents the relative orientation of two coordinate systems and can be represented as:
where $\|\hat{e}\|$ is the normalized rotational axis (not norm), and $\theta$ is the transformation angle.
Now my question is :
If I say the spacecraft is required to change its orientation (attitude) with $q(t_0)=q_0$ and $q(t_f)=q_f$. What does this mean?
I can understand if I say the spacecraft is require to have its position with $x(t_0)=x_0$; however, how to say attitude with $q(t_0)=q_0$? Does it mean we see the aircraft as a coordinate system and it is doing transformation?
When you are talking about a spacecraft's attitude, you are always talking about the rotational orientaion with respect to a known frame. Typically this frame is some Earth frame like J2000, but nothing stops the attitude from being described wrt the frame of another vehicle. Be careful to note we are talking about the rotational orientation only! Thus the aircraft can be moving translationally, but so long as the rotational difference is not affected, we would say the attitude is not affected.
Just as we may describe the attitude wrt to J2000 using a direction cosine matrix (e.g. an element of $SO_3$), we can describe the attitude wrt the aircraft by a similar matrix which is in general time varying. The unit quaternions come into play because they are a double cover of $SO_3$, the Lie group of all rotation (aka attitude) matrices; In particular for each $A\in SO_3$ there is a unit quaternion $q$ such that $\pm q\mapsto A$ so we can talk about attitude in terms of the unit quaternions exactly the same way we can using DCMs.
The version of the unit quaternions you wrote,
$$q = \begin{bmatrix} \cos\frac{\theta}{2}\\ \mathbf{e}\sin\frac{\theta}{2}\ \end{bmatrix},$$
are called the Euler Symmetric Parameters and result from parameterizing the unit quaternions with yet another representation of the attitude, which is the axis-angle representation. Here $\mathbf{e}$ is the eigenvector of $A$ whose eigenvalue is 1 (Euler's theorem implies this always exists for $A\in SO_3$) and $\theta$ is the angle about $\mathbf{e}$ which produces an equivalent rotation to $A$.
Now for your main question:
It means that at time $t_0$ the s/c has a certain attitude, say $A_0$, equivalently $q_0$, or equivalently $\mathbf{e}_0,\theta_0$ wrt to the aircraft, but since the attitude state (like the translational state) is generally time varying at time $t_f$ you will have a different attitude $A_f$, equivalently $q_f$, or equivalently $\mathbf{e}_f,\theta_f$ wrt to the aircraft. To make the attitude at time $t_f$ what you want it to be (e.g. the right DCM, quaternion, axis and angle) you need to perform a rotational maneuver which will be actuated by control torques from reaction wheels, small rockets, magnetic torque rods, etc. depending on the spacecraft.
Some words on that paper:
If you are reading about attitude control for fun, everything's fine, but you should know this paper represents an element in the set of rather bizarre IEEE papers trying to reinvent the wheel on attitude control. Attitude control using standard techniques, or even fancy nonlinear techniques, has been around for years and always relies on simple, closed-loop methods. Attitude is estimated from vector measurements (usually positions of stars, the moon, sun, earth, etc.) using a standard attitude determination algorithm (TRIAD, QUEST, ESOQ, FOAM) which is combined with feedback from gyros or other instruments to estimate the current rotation rate. This is fed into a well-tuned PID controller which computes required torques to actuate using reaction wheels or small rockets and the process is repeated in a closed-loop until the desired attitude state is accomplished. You can read more about spacecraft attitude determination and control (ADCS or sometimes ACS) systems in books like Sidi's Spacecraft Dynamics and Control, Wertz's Spacecraft Attitude Determination and Control, or Markley and Crassidis' Fundamentals of Spacecraft Attitude Determination and Control. For more general references on spacecraft dynamics and attitude representations see Hughes' Spacecraft attitude dynamics or (if you dare) Kane's Spacecraft dynamics.