Squash-Box Feasibility Driven Differential Dynamic Programming
J. Marti-Saumell, J. SolĂ , C. Mastalli and Angel Santamaria-Navarro
IEEE/RSJ International Conference on Intelligent Robots and Systems, Las Vegas (online), USA, 2020.

Recently, Differential Dynamic Programming (DDP) and other similar algorithms have become the solvers of choice when performing non-linear Model Predictive Control (nMPC) with modern robotic devices. The reason is that they have a lower computational cost per iteration when compared with off-the-shelf Non-Linear Programming (NLP) solvers, which enables its online operation. However, they cannot handle constraints, and are known to have poor convergence capabilities. In this paper, we propose a method to solve the optimal control problem with control bounds through a squashing function (i.e., a sigmoid, which is bounded by construction). It has been shown that a naive use of squashing functions damage the convergence rate. To tackle this, we first propose toadd a quadratic barrier that avoids the difficulty of the plateau produced by the sigmoid. Second, we add an outer loop that adapts both the sigmoid and the barrier; it makes the optimal control problem with the squashing function converge to theoriginal control-bounded problem. To validate our method,we present simulation results for different types of platforms including a multi-rotor, a biped, a quadruped and a humanoid robot.