Nonlinear model-based (NMB) control methods have been shown (both in theory and in practice) to provide the most advanced control performance for highly nonlinear hydraulic manipulators. In these methods, the inverse dynamics of a system are used to proactively generate the system actuation forces from the desired motion dynamics. To model the inverse dynamics in articulated systems, the Lagrange dynamics and the Newton-Euler dynamics are the most common methods.

In hydraulic cylinder actuated manipulators, a linear motion of the cylinder can be converted to a rotational joint motion between two links, creating closed-chain structures in the system. In Lagrange-dynamics-based control methods, the closed-chain structures are typically treated as an open-chain structure, which may raise the question of inaccurate system modeling. Contrary, the virtual decomposition control (VDC) approach is the first rigorous NMB control method to take full advantage of Newton-Euler dynamics, allowing to address the system nonlinear dynamics without imposing additional approximations.

In VDC, the actuated closed-chain structures can be virtually decomposed to open chain structures. To address the dynamics between the decomposed open chains, three specific terms (namely two load distribution factors and an internal force vector) need to be addressed. However, analytical solutions for these terms cannot be found in the literature. This paper provides the detailed solutions for these terms, which are further needed in a high-precision control of hydraulic robotic manipulators.

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