This study presents a recursive algorithm for solving the regularised least squares problem for online identification of rigid body dynamic model parameters with emphasis on the physical consistency of estimated inertial parameters. One of the geometric approaches is to use a regulariser that represents how close the pseudo-inertia matrix is to a given reference on the feasible manifold in the regression problem. The proposed extension enables memory-efficient online learning in addition to the benefits of geometry-aware convex regularisation using the log-determinant divergence of the pseudo-inertia matrix. Also, the recursive version endows the estimator with the capability to deal with time-variation of parameters by introducing an optional forgetting mechanism. The characteristics of the recursive regularised least squares algorithm is demonstrated using the MIT Cheetah 3 leg swinging experiment dataset and compared to the existing batch optimisation method.