try quatternions

It depends on how you want the rigid body to move. 

If you interpolate the yaw-pitch-roll angles, the interpolated orientation trajectory can be counter-intuitive.  The trajecotry also depends on the frame in which you describe the orientation of the rigid body. You can also express the end orientation with respect to the start orientation using yaw-pitch-roll angles and interpolate these. This is sometimes called three-axis interpolation. 

The orientation trajectory can also be interpolated along one rotation axis, with a constant rotational velocity.   This is sometimes a one-axis interpolation or a SLERP, this can be computed geometrically or using quaternions.  This motion is what most people find most intuitive. 

However, these motions only correspond to the natural motion of rigid bodies with specific inertia properties.  If you'll need a motion corresponding to a free floating rigid body, you'll need the Euler equations for the dynamics of rigid bodies.

Thank you for your responses Dr. Lapointe and Dr. Aertbelien.

The three-axis interpolation that you mention is what I was thinking of, but, as you say, it is counter-intuitive. I am not familiar with quaternions, so I'll have to do some work.

This is a kinematic analysis, so the dynamic equations of motion won't be considered.

Adwait, this is a spatial displacement. A 2 position spatial displacement as you have described, has an axis about which the rigid body could be displaced through pure rotation (pitch yaw roll for eg). This axis is called the screw axis and is the basis for screw theory. Quaternions are only a convenient notation that capture the orientation from R^3 to R^4 (if its spherical displacement). Dual quaternions are what you would technically need to capture the positions and orientations of rigid bodies from R^3 to R^8 in case of spatial displacements. Quaternions are highly advantageous as their algebra allows simple representations for displacement operations. However, you are not bound to use quaternions. You can simply express the screw axis in euclidean space and find the orientation of this rigid body at any point in this 2 position trajectory. I would suggest you turn your attention to screw theory. Hope this helps!!

Quaternions or screw theory are the names of the methods to interpolate rigid-body rotation in different branches of engineering. The underlying mathematics turns out to be the same.

Both approaches regard that the rigid-body motion can be decomposed in finite and distinct translations and rotations. This is true for most objects so long so they do not deform in motion. For example, outer-space satellites can be considered as rigid bodies and their motion can be decomposed. This simplifies the interpolation such that the next location of satellite can be predicted in proper orientation.

Try the method of successive screw displacements, for direct kinematics by using the given joint variables.