On the Quasi-Linearization of the Equations of Motion of Simple Mechanical Systems

A simple mechanical system is said be quasi-linearizable if there is a linear transformation of velocity that eliminates all terms quadratic in the velocity from the equations of motion. It is well-known that controller/observer synthesis becomes tractable when the dynamics of a mechanical system are in quasi-linearized form. The quasi-linearization property is equivalent to the property that the Lie algebra of Killing vector fields is pointwise equal to the tangent space to the configuration manifold with the metric induced by the mass tensor of the mechanical system. We show conditions for full quasi-linearization and partial quasi-linearization, the latter of which is for systems that are not quasi-linearizable. A sufficient condition for full quasi-linearizability is that the Riemannian manifold be locally symmetric. On two dimensional manifolds, the constant scalar curvature condition is necessary and sufficient for full quasi-linearizability. The two conditions extend the zero Riemannian curvature condition by Bedrossian and Spong.