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This study examines the mechanical systems with an arbitrary number of passive (non-actuated) degrees of freedom and proposes an analytical method for computing coefficients of a linear controlled system, solutions of which approximate dynamics transverse to a feasible motion. This constructive procedure is based on a particular choice of coordinates and allows explicit introduction of a moving Poincaré section associated with a nontrivial finite-time or periodic motion. In these coordinates, transverse dynamics admits analytical linearization before any control design. If the forced motion of an underactuated mechanical system is periodic, then this linearization is an indispensable and constructive tool for stabilizing the cycle and for analyzing its orbital (in)stability. The technique is illustrated with two challenging examples. The first one is stabilization of a circular motions of a spherical pendulum on a puck around its upright equilibrium. The other one is creating stable synchronous oscillations of an arbitrary number of planar pendula on carts around their unstable equilibria.Index Terms-Moving Poincaré section, orbital stability, synchronisation of mechanical system, transverse linearization, underactuated mechanical systems, virtual holonomic constraints.

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A lack of sufficient actuation power as well as the presence of passive degrees of freedom are often serious constraints for feasible motions of a robot. Installing passive elastic mechanisms in parallel with the original actuators is one of a few alternatives that allows for large modifications of the range of external forces or torques that can be applied to the mechanical system. If some motions are planned that require a nominal control input above the actuator limitations, then we can search for auxiliary spring-like mechanisms complementing the control scheme in order to overcome the constraints. The intuitive idea of parallel elastic actuation is that spring-like elements generate most of the nominal torque required along a desired trajectory, so the control efforts of the original actuators can be mainly spent in stabilizing the motion. Such attractive arguments are, however, challenging for robots with non-feedback linearizable non-minimum phase dynamics that have one or several passive degrees of freedom. We suggest an approach to resolve the apparent difficulties and illustrate the method with an example of an underactuated planar double pendulum. The results are tested both in simulations and through experimental studies.