High-order inverse dynamics is important for the trajectory tracking control of the elastic-joint manipulators. Usually it is undesirable to establish third-order or fourth-order inverse dynamics through the symbolic Lagrangian method, because of its complicated customization and higher computational complexity, which is not suitable for real-time control for multi-joint manipulators. Based on the Subsystem of Newton-Euler formulation and the recently proposed elastic joint Newton-Euler algorithm (EJNEA), a novel recursive algorithm (Sub-EJNEA) is proposed in this paper. The Sub-EJNEA obtains a linear complexity , and has a more compact form and clear physical meaning. Based on the Sub-EJNEA, a recursive control method is proposed, which can be seen as an enhancement of the feedback linearization (FL-E) control method. When facing with friction and external disturbances, a combined controller FL-E-RISE is proposed. Finally, to verify the correctness of the recursive algorithm and the effectiveness of the proposed control schemes, simulation results with the Baxter manipulator are presented.
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