I have been doing a lot of reading up on kinematic calibration and here is what I found:
A kinematic model should meet three basic requirements for
1) Completeness: A complete model must have enough parameters to describe any possible deviation of the actual kinematic parameters from the nominal values.
2) Continuity: Small changes in the geometric structure of the robot must
correspond to small changes in the kinematic parameters. In
mathematics, the model is a continuous function of the kinematic
3) Minimality: The kinematic model must include only a
minimal number of parameters. The error model for the kinematic
calibration should not have redundant parameters.
While DH parameters are complete and minimal, they are not continuous. In addition, there is a singularity when two consecutive joints have parallel axes. From :
Our assumption is that small variations in the position and orientation of two consecutive links can be modeled by small variations of the link parameters. This assumption is violated if we use the Denavit and Hartenberg link geometry characterization when the two consecutive joints have parallel or near parallel axes.
This has led a number of researchers to propose alternative models. Namely the Hayati model , Veitschegger and Wu’s model , Stone and
Sanderson’s S-model , and the "Complete and Parametrically Continuous" (CPC) model .
These models typically involve adding parameters. Which creates redundancy that has to be dealt with. Or they are specifically tailored to the geometry of their robot. Which eliminates generality.
One alternative is the Product of Exponentials formulation . The kinematic parameters of in POE model vary smoothly with changes in joint axes and can handle kinematic singularities naturally. However, due to the use of joint twists, this method is not minimal. This led Yang et al.  to propose a POE formulation with only 4 parameters per joint which is minimal, continuous, complete, and general. They do this by choosing joint frames very specifically. (Which actually vaguely resemble D-H frames).
: Ruibo He; Yingjun Zhao; Shunian Yang; Shuzi Yang, "Kinematic-Parameter Identification for Serial-Robot Calibration Based on POE Formula," in Robotics, IEEE Transactions on , vol.26, no.3, pp.411-423, June 2010
: Hayati, S.A., "Robot arm geometric link parameter estimation," in Decision and Control, 1983. The 22nd IEEE Conference on , vol., no., pp.1477-1483, - Dec. 1983
: W. Veitschegger and C. Wu, “Robot accuracy analysis based on kinematics,” IEEE Trans. Robot. Autom., vol. RA-2, no. 3, pp. 171–179, Sep.
: H. Stone and A. Sanderson, “A prototype arm signature identification
system,” in Proc. IEEE Conf. Robot. Autom., Apr. 1987, pp. 175–182.
: H. Zhuang, Z. S. Roth, and F. Hamano, “A complete and parametrically
continuous kinematic model for robot manipulators,” IEEE Trans. Robot.
Autom., vol. 8, no. 4, pp. 451–463, Aug. 1992.
: I. Chen, G. Yang, C. Tan, and S. Yeo, “Local POE model for robot kinematic calibration,” Mech. Mach. Theory, vol. 36, no. 11/12, pp. 1215–
: Xiangdong Yang, Liao Wu, Jinquan Li, and Ken Chen. 2014. A minimal kinematic model for serial robot calibration using POE formula. Robot. Comput.-Integr. Manuf. 30, 3 (June 2014), 326-334.