Practical Parameterization of Rotations Using the Exponential Map
(1) This paper presents formulae for computing, differentiating, and 
integrating three DOF rotations with the exponential map.
(2) The exponential map maps a vector in R3 describing the axis and 
magnitude of a three DOF rotation to the corresponding rotation.
Several graphics researchers have applied it with limited success to 
interpolation of orientations, but it has been virtually ignored with 
respect to the other operations mentioned above.
(3)This paper shows that the formulation is numerically stable in the face 
of machine precision issues, and that for most applications all 
singularities in the map can be avoided through a simple technique of 
dynamic reparameterization. This paper also demonstrate how to use the 
exponential map to solve both the "freely rotating body" problem, and the 
important ball-and-socket joint required to accurately model shoulder and 
hip joints in articulated figures.
(4) Compared with Euler angles and quaternions, the exponential map has the 
following advantages: robustness, small state vectors, lack of explicit 
constraints, good modeling capabilities, simplicity of solving ODE's, and 
good interpolation behavior.
(5) This limitations of the exponential map are as follows:

1). cannot be used for spacetime optimizations of tumbling bodies.

2). no simple formula for combining rotations in R3 akin to quaternion 
multiplication in S3 or matrix multiplication
in SO(3).

3). impossible to simply interpolate between successive state snapshots 
output by a dynamics simulation or inverse kinematics engine.