Leveraging Control Inputs to Enforce Constraints in Differential Dynamic Programming for Nonlinear Optimization*

Differential Dynamic Programming (DDP) has become a popular strategy for optimizing nonlinear dynamic systems due to its algorithmic efficiency and precision in complex control tasks. The goal to integrate inequality constraints into DDP has sparked considerable interest, aiming to extend its utility to more demanding situations with strict operational constraints. This study provides an extension to the conventional control-limited DDP framework, introducing a methodology that leverages control inputs during the backward pass to incorporate inequality constraints. Our methodology enhances the efficiency of DDP and expedites its convergence in a variety of scenarios. We present our method in two variants: the first handles inequality constraints that are functions of both state and control variables, and the second leverages the concept of relative degree from nonlinear control theory to handle inequality constraints that are solely dependent on state variables. Through simulations on an inverted pendulum and a nonholonomic 2D car, we benchmark our approach against established methods such as Constrained DDP (CDDP) and primal-dual interior-point DDP (IPDDP). The results showcase our method’s superior convergence rate and trajectory efficiency, particularly highlighting the efficacy of employing control inputs for constraint enforcement.