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Orbital synchronization of homogeneous mechanical systems with one degree of underactuation

A methodology for asymptotic orbital synchronization of a set of homogeneous mechanical systems with one degree of underactuation is proposed. A reference model generating a reference orbit is first constructed under the virtual holonomic constraints approach. Then, the ℋ∞ synthesis is designed to, first, estimate the state vector of the homogeneous systems, provided that only position measurements are available, and second, to drive the state of the systems to that of the reference model. Due to the ℋ∞ synthesis properties, the homogeneous systems converge asymptotically to the reference orbit when there are no disturbances affecting the systems nor the measurements. When disturbances are present, robustness is guaranteed provided that the ℒ2 gain of the closed‐loop systems remains lower than an attenuation level γ . Numerical results for a set of underactuated cart‐pendulums corroborate the proposed methodology.