Set-point control based on an observer and globally contraction mapping

A set-point controller was developed for a parallel manipulator with unknown parameters when only the position signals are available. A linear observer was used to reconstruct the velocity signals. This pseudo velocity was the used in a set-point controller using PD (proportion pseudo derivative) plus estimated nominal gravity compensation which guarantees that the closed-loop system is globally stable. The algorithm modifies the observer parameter using contraction mapping to compensate for steady-state positioning errors due to estimates of the gravitational term. Every time the system reaches steady state, the observer parameter is iteratively adjusted to provide globally asymptotic stability. Simulations with a 6 degree of freedom parallel manipulator demonstrate that various parameter values will lead to different convergence velocities and output vibrations, as the controller ensures that the system outputs converge into arbitrarily small neighborhoods of the desired equilibrium point.