We introduce Crocoddyl (Contact RObot COntrol by Differential DYnamic Library), an open-source framework tailored for efficient multi-contact optimal control. Crocoddyl efficiently computes the state trajectory and the control policy for a given predefined sequence of contacts. Its efficiency is due to the use of sparse analytical derivatives, exploitation of the problem structure, and data sharing. It employs differential geometry to properly describe the state of any geometrical system, e.g. floating-base systems. We have unified dynamics, costs, and constraints into a single concept-action-for greater efficiency and easy prototyping. Additionally, we propose a novel multipleshooting method called Feasibility-prone Differential Dynamic Programming (FDDP). Our novel method shows a greater globalization strategy compared to classical Differential Dynamic Programming (DDP) algorithms, and it has similar numerical behavior to state-of-the-art multiple-shooting methods. However, our method does not increase the computational complexity typically encountered by adding extra variables to describe the gaps in the dynamics. Concretely, we propose two modifications to the classical DDP algorithm. First, the backward pass accepts infeasible state-control trajectories. Second, the rollout keeps the gaps open during the early "exploratory" iterations (as expected in multiple-shooting methods). We showcase the performance of our framework using different tasks. With our method, we can compute highly-dynamic maneuvers for legged robots (e.g. jumping, front-flip) in the order of milliseconds.

This paper presents a generic and efficient approach to generate dynamically consistent motions for under-actuated systems like humanoid or quadruped robots. The main contribution is a walking pattern generator, able to compute a stable trajectory of the center of mass of the robot along with the angular momentum, for any given configuration of contacts (e.g. on uneven, sloppy or slippery terrain, or with closed-gripper). Unlike existing methods, our solver is fast enough to be applied as a model-predictive controller. We then integrate this pattern generator in a complete framework: an acyclic contact planner is first used to automatically compute the contact sequence from a 3D model of the environment and a desired final posture; a stable walking pattern is then computed by the proposed solver; a dynamically-stable whole-body trajectory is finally obtained using a second-order hierarchical inverse kinematics. The implementation of the whole pipeline is fast enough to plan a step while the previous one is executed. The interest of the method is demonstrated by real experiments on the HRP-2 robot, by performing long-step walking and climbing a staircase with handrail support.

We present a new open-source torque-controlled legged robot system, with a low cost and low complexity actuator module at its core. It consists of a low-weight high torque brushless DC motor and a low gear ratio transmission suitable for impedance and force control. We also present a novel foot contact sensor suitable for legged locomotion with hard impacts. A 2.2 kg quadruped robot with a large range of motion is assembled from 8 identical actuator modules and 4 lower legs with foot contact sensors. To the best of our knowledge, it is the most lightest force-controlled quadruped robot. We leverage standard plastic 3D printing and off-theshelf parts, resulting in light-weight and inexpensive robots, allowing for rapid distribution and duplication within the research community. In order to quantify the capabilities of our design, we systematically measure the achieved impedance at the foot in static and dynamic scenarios. We measured up to 10.8 dimensionless leg stiffness without active damping, which is comparable to the leg stiffness of a running human. Finally, in order to demonstrate the capabilities of our quadruped robot, we propose a novel controller which combines feedforward contact forces computed from a kino-dynamic optimizer with impedance control of the robot center of mass and base orientation. The controller is capable of regulating complex motions which are robust to environmental uncertainty. *

International audienceThe contribution of this work is to show that real-time nonlinear model predictive control (NMPC) can be implemented on position controlled humanoid robots. Following the idea of " walking without thinking " , we propose a walking pattern generator that takes into account simultaneously the position and orientation of the feet. A requirement for an application in real-world scenarios is the avoidance of obstacles. Therefore the paper shows an extension of the pattern generator that directly considers the avoidance of convex obstacles. The algorithm uses the whole-body dynamics to correct the center of mass trajectory of the underlying simplified model. The pattern generator runs in real-time on the embedded hardware of the humanoid robot HRP-2 and experiments demonstrate the increase in performance with the correction