Geometric Criteria for the Quasi-Linearization of the Equations of Motion of Mechanical Systems

A linear transformation of velocity for a mechanical system is said to quasi-linearize the equations of motion of the system if it eliminates all terms quadratic in the velocity. It is well-known that controller/observer synthesis becomes tractable when the dynamics of a mechanical system are in quasi-linearized form. In this technical note, we show that 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 Riemannian metric induced by the mass tensor of the mechanical system. A sufficient condition for this property is that the Riemannian manifold be locally symmetric. We further show that a necessary and sufficient condition for quasi-linearizability on 2-D Riemannian manifolds is that the scalar curvature is constant. The above results extend the zero Riemannian curvature condition that has been extensively applied since its introduction in 1992. Moreover, the local symmetricity condition and the constant scalar curvature condition can be easily verified using differentiation.