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 contact planner for complex legged locomotion tasks: standing up, climbing stairs using a handrail, crossing rubble and getting out of a car. The need for such a planner was shown at the Darpa Robotics Challenge, where such behaviors could not be demonstrated (except for egress). Current planners suffer from their prohibitive algorithmic complexity, because they deploy a tree of robot configurations projected in contact with the environment. We tackle this issue by introducing a reduction property: the reachability condition. This condition defines a geometric approximation of the contact manifold, which is of low dimension, presents a Cartesian topology, and can be efficiently sampled and explored. The hard contact planning problem can then be decomposed into two sub-problems: first, we plan a path for the root without considering the whole-body configuration, using a sampling-based algorithm; then, we generate a discrete sequence of whole-body configurations in static equilibrium along this path, using a deterministic contact-selection algorithm. The reduction breaks the algorithm complexity encountered in previous works, resulting in the first interactive implementation of a contact planner (open source). While no contact planner has yet been proposed with theoretical completeness, we empirically show the interest of our framework: in a few seconds, with high success rates, we generate complex contact plans for various scenarios and two robots, HRP-2 and HyQ. These plans are validated either in dynamic simulations, or on the real HRP-2 robot.

Synthesizing legged locomotion requires planning one or several steps ahead (literally): when and where, and with which effector should the next contact(s) be created between the robot and the environment? Validating a contact candidate implies a minima the resolution of a slow, non-linear optimization problem, to demonstrate that a Center Of Mass (CoM) trajectory, compatible with the contact transition constraints, exists.We propose a conservative reformulation of this trajectory generation problem as a convex 3D linear program, CROC (Convex Resolution Of Centroidal dynamic trajectories). It results from the observation that if the CoM trajectory is a polynomial with only one free variable coefficient, the non-linearity of the problem disappears. This has two consequences. On the positive side, in terms of computation times CROC outperforms the state of the art by at least one order of magnitude, and allows to consider interactive applications (with a planning time roughly equal to the motion time). On the negative side, in our experiments our approach finds a majority of the feasible trajectories found by a non-linear solver, but not all of them. Still, we demonstrate that the solution space covered by CROC is large enough to achieve the automated planning of a large variety of locomotion tasks for different robots, demonstrated in simulation and on the real HRP-2 robot, several of which were rarely seen before.Another significant contribution is the introduction of a Bezier curve representation of the problem, which guarantees that the constraints of the CoM trajectory are verified continuously, and not only at discrete points as traditionally done. This formulation is lossless, and results in more robust trajectories. It is not restricted to CROC, but could rather be integrated with any method from the state of the art.