Progressive Learning: Part II — Stable Trajectory Synthesis Optimization for Non-Collocated Systems

The progressive learning theory described in the companion paper, Part I, is applied to trajectory control of non-collocated systems, and implemented on a high speed direct-drive robot. The key issue in progressive learning is how to design a series of reference inputs that gradually excite the system so that the stability conditions are met throughout the learning process. For trajectory tracking, the system is required to visit a set of specified points in space whole the frequency contents of the trajectory must satisfy the stability conditions. Namely, learning must begin with low excitation frequencies, and then the frequency band must be expanded as leaning proceeds. To generate such a series of trajectories, we first represent a given geometric path by interpolating specified points using a Fourier series and them assign a tracking speed. Starting with a slow tracking speed and gradually increasing the speed, the while frequency spectrum extends towards the higher frequency range, while the system tracks the same path in space. First the trajectory synthesis problem is formally formulated. Conditions for a trajectory to go through a set of specified points in space as well as to guarantee stable learning are provided. Moreover, the convergence speed of the learning process is accommodated by selecting appropriate excitation frequencies and magnitudes. The proposed method is then applied to a high-speed direct-drive robot with non-collocated sensors and actuators. A tunable reduced-order observer is obtained in order to incorporate directly measurable state variables into the full state feedback. Simulations using the parameter values of the actual robot are conducted, and effectiveness and feasibility of the proposed method are demonstrated through simulations.