Issues in the real-time computation of optimal control

Under appropriate conditions, the dynamics of a control system governed by ordinary differential equations can be formulated several ways: differential inclusion, control parameterization, flatness parameterization, higher-order inclusions and so on. A plethora of techniques have been proposed for each of these formulations but they are typically not portable across equivalent mathematical formulations. Further complications arise as a result of configuration and control constraints such as those imposed by obstacle avoidance or control saturation. In this paper, we present a unified framework for handling the computation of optimal controls where the description of the governing equations or that of the path constraint is not a limitation. In fact, our method exploits the advantages offered by coordinate transformations and harnesses any inherent smoothness present in the optimal system trajectories. We demonstrate how our computational framework can easily and efficiently handle different cost formulations, control sets and path constraints. We illustrate our ideas by formulating a robotics problem eight different ways that includes a differentially flat formulation subject to control saturation. This example establishes the loss of convexity in the flat formulation as well as its ramifications on computation and optimality. In addition, a numerical comparison of our unified approach to a recent technique tailored for control-affine systems reveals that we get about 30% improvement in the performance index.