On Global, Closed-Form Solutions to Parametric Optimization Problems for Robots With Energy Regeneration

Parametric optimization problems are considered for serial robots with regenerative drive mechanisms. A subset of the robot joints are conventional, in the sense that external power is used for actuation. Other joints are energetically self-contained passive systems that use (ultra)capacitors for energy storage. Two different electrical interconnections are considered for the regenerative drives, a distributed and a star configuration. The latter allows for direct electric energy redistribution among joints, a novel idea shown in this paper to enable higher energy utilization efficiencies. Closed-form expressions are found for the optimal manipulator parameters (link masses, link lengths, etc.) and drive mechanism parameters (gear ratios, etc.) that maximize regenerative energy storage between any two times, given motion trajectories. A semi-active virtual control strategy previously proposed is used to achieve asymptotic tracking of trajectories. Optimal solutions are shown to be global and unique. In addition, closed-form expressions are provided for the maximum attainable energy. This theoretical maximum places limits on the amount of energy that can be recovered. The results also shed light on the comparative advantages of the star and distributed configurations. A numerical example with a double inverted pendulum and cart system is provided to demonstrate the results.