Section: New Results

Locomotion analysis

Synchronous imitation of human motion by a humanoid robot

Participants : Mehdi Benallegue, Pierre-Brice Wieber.

Interactions between humans and robots require that each one is able to understand and interpret each other's actions. From the point of view of the robot, this means: (i) to move in a way that can be naturally interpreted by humans and (ii) to be able to understand the humans' actions. Studies in Neuroscience in the case of interactions between humans indicate that these two abilities might be tightly linked in the human's brain: we understand actions when we map the observed action onto our motor representation of the same action  [57] . In this work, we consider that the “motor representation” of a task is the control law, and “mapping an observed action” means finding the corresponding control parameter, in an observer-based approach.

Considering a correspondance between two different control laws can be seen a modeling error. This modeling error can be seen as an unknown arbitrary perturbation on the modeled system, or an unknown input sent to the observed system. We developed an observer that can cancel the effects of unknown inputs on the dynamics of discrete time linear systems with unknown inputs. To do so, the observer has to satisfy a delayed invertibility condition and use delayed outputs. In other words, the observer has to wait for several measurement after a given instant to collect enough data to reconstruct the state at that instant.

Hierarchic QP solver

Participant : Pierre-Brice Wieber.

We are working in collaboration with the LAAS-CNRS and the CEA-LIST on solving multi-objective Quadratic Programs with Lexicographic ordering: Hierarchic QPs [25] . The focus this year has been on the regularization of the problem when the Quadratic Program approaches singularities. There is indeed a problem of discontinuity of the solution when reaching such singularities, what's not a rare event in robotic applications. This discontinuity has been related to the fact that Lexicograpically ordered QPs correspond to the limit of weighed multi-objective QPs when weights go singular, and that the regularization is itself a weighting of objectives which goes to a limit when approaching singularities, and those two limit processes interfere. The solution we found so far is to cancel the first limiting process and move back from strict Hierarchic QPs to weighted QPs staying at a small distance from singularity [33] . But this solution is not really satisfying and we have to find a better one.

Numerical modeling of muscle contraction under FES

Participant : Pierre-Brice Wieber.

We have been working in collaboration with the EPI DEMAR in Montpellier on modeling muscle contraction under Functional Electric Stimulation (FES). With respect to the literature in the domain, our contributions are mostly linked to the model of the contractile element, through the introduction of the recruitment at the fibre scale, formalizing the link between FES parameters, recruitment and Calcium signal paths. The resulting controlled model is able to reproduce both short term (twitch) and long term (tetanus) responses. It also matches some of the main properties of the dynamic behaviour of muscles, such as the Hill force-velocity relationship or the instantaneous stiffness of the Mirsky-Parmley model. The specific contribution of the BIPOP team has been on the numerical implementation of the contraction model as a Linear Complementarity Problem (LCP) allowing fast and precise numerical simulations [22] .

Modeling of human balance in public transports

Participants : Pierre-Brice Wieber, Zohaib Aftab.

In our ongoing collaboration with the IFSTTAR (previously INRETS) on modeling human balance in public transports, we have aggregated biomechanical studies and numerical models proposed in robotics, and compared how they match or mismatch in situations of strong perturbations requiring a step to recover balance. We began developing a specific Model Predictive Control scheme for the prediction of recovery step locations with adaptive step timings, reproducing various balance recovery strategies as observed in humans. Initial results for stepping predictions have been validated against a balance recovery scenario found in the literature [45] .

Model Predictive Control for Biped Walking

Participants : Pierre-Brice Wieber, Andrei Herdt, Jory Lafaye, François Keith.

We improved our Linear MPC-based walking motion generator by incorporating explicitly the robot's kinematic constraints: polyhedral constraints on the position of the CoM ensure the kinematic feasibility of the generated walking motions for arbitrary vertical motions of the CoM. This more precise kinematic model within the LMPC allowed considering toe rotations in a safer way, considerably improving energy efficiency, naturalness of the motion, and maximal speed.

We proposed a formulation of dynamic constraints for 3D motion through simple bounds on the variables, leading to faster resolution of the corresponding optimization problem. This allowed generating three-dimensional walking on non-planar ground in real-time. Thanks to specifically enforcing leg compliance, this scheme managed additionally to reproduce the natural profiles of the CoM and the contact forces observed in human walking.

We finally refined our numerical scheme for solving Linear MPC problems in walking motion generation. We switched the underlying QP solver to enable reductions of the number of iterations through warmstart, non-empty initial active sets, and obtaining feasible solutions at each iterations, what considerably reduced the computation time, allowing 1 ms feedback loops [30] , [32] .