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 to add 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 the original 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.

Optimal control is a widely used tool for synthesizing motions and controls for user-defined tasks under physical constraints. A common approach is to formulate it using direct multiple-shooting and then to use off-the-shelf nonlinear programming solvers that can easily handle arbitrary constraints on the controls and states. However, these methods are not fast enough for many robotics applications such as realtime humanoid motor control. Exploiting the sparse structure of optimal control problem, such as in Differential Dynamic Programming (DDP), has proven to significantly boost the computational efficiency, and recent works have been focused on handling arbitrary constraints. Despite that, DDP has been associated with poor numerical convergence, particularly when considering long time horizons. One of the main reasons is due to system instabilities and poor warm-starting (only controls). This paper presents control-limited Feasibility-driven DDP (Box-FDDP), a solver that incorporates a direct-indirect hybridization of the control-limited DDP algorithm. Concretely, the forward and backward passes handle feasibility and control limits. We showcase the impact and importance of our method on a set of challenging optimal control problems against the Box-DDP and squashing-function approach.

Differential dynamic programming (DDP) is a direct single shooting method for trajectory optimization. Its efficiency derives from the exploitation of temporal structure (inherent to optimal control problems) and explicit roll-out/integration of the system dynamics. However, it suffers from numerical instability and, when compared to direct multiple shooting methods, it has limited initialization options (allows initialization of controls, but not of states) and lacks proper handling of control constraints. In this work, we tackle these issues with a feasibility-driven approach that regulates the dynamic feasibility during the numerical optimization and ensures control limits. Our feasibility search emulates the numerical resolution of a direct multiple shooting problem with only dynamics constraints. We show that our approach (named Box-FDDP) has better numerical convergence than Box-DDP$$^+$$ + (a single shooting method), and that its convergence rate and runtime performance are competitive with state-of-the-art direct transcription formulations solved using the interior point and active set algorithms available in Knitro. We further show that Box-FDDP decreases the dynamic feasibility error monotonically—as in state-of-the-art nonlinear programming algorithms. We demonstrate the benefits of our approach by generating complex and athletic motions for quadruped and humanoid robots. Finally, we highlight that Box-FDDP is suitable for model predictive control in legged robots.

Non-linear model predictive control (nMPC) is a powerful approach to control complex robots (such as humanoids, quadrupeds or unmanned aerial manipulators (UAMs)) as it brings important advantages over other existing techniques. The full-body dynamics, along with the prediction capability of the optimal control problem (OCP) solved at the core of the controller, allows to actuate the robot in line with its dynamics. This fact enhances the robot capabilities and allows, e.g., to perform intricate maneuvers at high dynamics while optimizing the amount of energy used. Despite the many similarities between humanoids or quadrupeds and UAMs, full-body torque-level nMPC has rarely been applied to UAMs.This paper provides a thorough description of how to use such techniques in the field of aerial manipulation. We give a detailed explanation of the different parts involved in the OCP, from the UAM dynamical model to the residuals in the cost function. We develop and compare three different nMPC controllers: Weighted MPC, Rail MPC and Carrot MPC, which differ on the structure of their OCPs and on how these are updated at every time step. To validate the proposed framework, we present a wide variety of simulated case studies. First, we evaluate the trajectory generation problem, i.e., optimal control problems solved offline, involving different kinds of motions (e.g., aggressive maneuvers or contact locomotion) for different types UAMs. Then, we assess the performance of the three nMPC controllers, i.e., closed-loop controllers solved online, through a variety of realistic simulations. For the benefit of the community, we have made available the source code related to this work.