Enhancing computational efficiency of iLQR and DDP via the parametric representation of control inputs

Efficiently solving nonlinear optimal control problems is crucial in trajectory planning and model predictive control. This can be achieved by utilizing differential dynamic programming (DDP) and iterative linear quadratic regulator (iLQR), which have recently gained attention. As these algorithms partition the problem into subproblems at each time step, they exhibit linear complexity of one iteration in the length of the prediction horizon. While these methodologies are computationally efficient, industrial applications demand further improvements in computational efficiency, primarily due to the limitations of embedded CPUs. The parametric representation of control inputs has been widely adopted to reduce the dimensionality of decision variables in optimal control problems. However, the subproblem partitioning inherent in DDP and iLQR presents challenges for directly incorporating this representation. In this study, we present a computationally efficient algorithm that integrates a parametric representation of control inputs into DDP- or iLQR-like algorithms. We exemplified a scenario in which parametric representation was introduced by considering interior-point DDP and iLQR, which could handle nonlinear inequality constraints. The effectiveness of this approach for practical applications was demonstrated through a series of numerical experiments. In particular, these numerical experiments mainly focused on key real-world problems, such as trajectory planning for forklifts and optimal excavation trajectory planning for an excavator. Regarding trajectory planning for forklifts and excavators, we achieved a maximum reduction of about 70% in the total computation time.