This paper presents a new geometric and recursive algorithm for analytically computing the forward dynamics of heavy-duty parallel-serial mechanisms. Our solution relies on expressing the dynamics of a class of linearly-actuated parallel mechanism to a lower dimensional dual Lie algebra to find an analytical solution for the inverse dynamics problem. Thus, by applying the articulated-body inertias method, we successfully provide analytic expressions for the total wrench in the linear-actuator reference frame, the linear acceleration of the actuator, and the total wrench exerted in the base reference frame of the closed loop. This new formulation allows to backwardly project and assemble inertia matrices and wrench bias of multiple closed-loops mechanisms. The final algorithm holds an O(n) algorithmic complexity, where $n$ is the number of degrees of freedom (DoF). We provide accuracy results to demonstrate its efficiency with 1-DoF closed-loop mechanism and 4-DoF manipulator composed by serial and parallel mechanisms. Additionally, we release a URDF multi-DoF code for this recursive algorithm.

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