As collective motion plays a crucial role in modern day robotics and engineering, it seems appealing to seek inspiration from nature, which abounds with examples of collective motion (starling flocks, fish schools etc.). However, the individual-level mechanisms that give rise to group level collective behavior are not yet well-understood. Therefore, it seems to be a relevant effort to explore the underlying strategies and control laws governing collective motion, because it is not only beneficial from the perspective of engineering adaptation and exploitation, but it also furthers our basic scientific understanding. This approach towards understanding and reverse-engineering a particular aspect of the nature involves exploration of the underlying strategies and steering control laws governing collective motion, and this quest cannot be pursued without analyzing appropriate parameters of motion (namely curvature, speed, lateral acceleration etc.). This talk will introduce novel methods for “parameter extraction” from “sampled observations of a trajectory”. Drawing influence from optimal control theory, we tackle this problem of trajectory reconstruction by the method of regularization. Using generative models with inputs, states and outputs, and suitable penalty functionals of input (to impose smoothness), we formulate this problem as one of finding optimal inputs. For the choice of a triple integrator model, with jerk as input, and a penalty functional depending quadratically on jerk, we propose an analytical solution for the trajectory reconstruction problem. The resulting algorithm is fast, with complexity of the order of sample size. Alternatively, we can describe the trajectory using a nonlinear generative model evolving on a Lie group, and by appropriate choice of control inputs this problem can be reformulated as a data smoothing problem in a sub-Riemannian setting. By noting left invariance of the underlying dynamics, we develop the initial framework (a version of the maximum principle) towards obtaining a semi-analytical solution.