Abstract-We propose a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. Our main contribution is to introduce a state (position and velocity) observer that is invariant under any changes of the configuration coordinates. The observer dynamics equations, as the Euler-Lagrange equations, are intrinsic. The design method uses the Riemannian structure defined by the kinetic energy on the configuration manifold. The local convergence is proved by showing that the Jacobian of the observer dynamics is negative definite (contraction) for a particular metric defined on the state-space, a metric derived from the kinetic energy and the observer gains. From a practical point of view, such intrinsic observers can be approximated, when the estimated configuration is close to the true one, by an explicit set of differential equations involving the Riemannian curvature tensor. These equations can be automatically generated via symbolic differentiations of the metric and potential up to order two. Numerical simulations for the ball and beam system, an example where the scalar curvature is always negative, show the effectiveness of such approximation when the measured positions are noisy or include high frequency neglected dynamics.

For dynamics x = f(z) with output y = h ( z ) invariant with respect to a transformation group G,. we pefine invariant asymptotic observer of the form 2 = f(i, y) where y = h(s) is the measured output and i: an estimation of the unmeasured state x. Such a definition is motivated by a class of chemical reactors, treated in details, when the transformation group corresponds to unit changes and the output y to ratio of concentrations. We propose a constructive method that guaranties automatically the ohserver invariance j : = f(2, y): it is based on invariant vector fields and scalar functions, called invariant estimation errors, that can he computed via the Cartan moving frame method. The observer convergence remains, in the general case, an open problem. But for the class of chemical reactors considered here, the invariant observer convergence is proved by showing that, in a Killing metric associated to the action of G, the symmetric part of the Jacobian matrix af/aj: is definite negative (contraction).