**Original manuscript:** 2002/09/09

A simple mechanical system is a triple (*Q*,*g*,*V*) where
*Q* is a configuration space, *g* is a Riemannian metric on
*Q*, and *V* is the potential energy. The Lagrangian associated
with a simple mechanical system is defined by the kinetic energy minus the
potential energy. The equations of motion given by the Euler-Lagrange
equations for a simple mechanical system without potential energy can be
formulated as an affine connection control system. If these systems are
underactuated then they do not provide a controllable linearization about
their equilibrium points. Without a controllable linearization it is not
entirely clear how one should deriving a set of controls for such systems.

There are recent results that define the notion of kinematic controllability and its required set of conditions for underactuated systems. If the underactuated system in question satisfies these conditions, then a set of open-loop controls can be obtained for specific trajectories. These open-loop controls are susceptible to unmodeled environmental and dynamic effects. Without a controllable linearization a feedback control is not readily available to compensate for these effects.

This report considers linearizing affine connection control systems with zero potential energy along a reference trajectory. This linearization yields a linear second-order differential equation from the properties of its integral curves. The solution of this differential equation measures the variations of the system from the desired reference trajectory. This second-order differential equation is then written as a control system. If it is controllable then it provides a method for adding a feedback law. An example is provided where a feedback control is implemented.

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Last Updated: Fri Oct 27 08:42:33 2017