A robust adaptive velocity observer for mechanical systems transformed in cascade form

This note presents a continuous adaptive velocity observer for mechanical systems whose dynamic model motion does not allow for the cancellation of gyroscopic forces after coordinates transformation. Its design consists of three components: (i) for a class of systems, we propose a decomposition to render its model in cascade form; (ii) we give necessary and sufficient conditions to include unknown external disturbances and unknown friction parameters in the design, as parameter friction generates complications as it creates products of unknown terms, parameters and states; and (iii) we formulate an adaptive velocity observer based on the Immersion and Invariance (I&I) technique, which ensures velocity estimation while parameter estimators remain bounded. Finally, we analytically study several examples with more than two degrees of freedom and present simulations for a four-degree Scara manipulator.