Section: New Results

Robotics

We are working in collaboration with the LAAS-CNRS and the CEA-LIST on solving multi-objective Quadratic Programs with Lexicographic ordering: Hierarchic QPs  [47] . The focus this year has been on enabling fast computations in the case of time-varying Hierarchic QPs through warm-starting the active set method. This has been possible by developing an active set method for lexicographic multi-objective ordering  [44] , [45] . The main difference with respect to classical active set methods is in the use of a “lexicographic” (sometimes called “multi-dimensional”) Lagrange multiplier.

Zohaib Aftab finished his PhD thesis in collaboration with IFSTTAR (previously INRETS) on modeling human balance in public transports. A Model Predictive Control scheme has been developed for the prediction of recovery motions, including ankle and hip strategies as well as stepping with adaptive step locations and timings  [37] . This MPC scheme has been validated against a balance recovery scenario found in the biomechanics literature  [38] .

In collaboration with the DLR in Munich, we designed an MPC scheme for biped walking based on the “Capture Point”. This is just a simple change of variable $\xi =x+\frac{1}{\omega }\dot{x}$ that transforms the second-order dynamics of the Center of Mass $x$ of the robot into a cascade of two first-order dynamics, one stable and one unstable. This MPC scheme has been evaluated successfully on the DLR biped robot  [49] .

Since fast computations are always a key objective for feedback controllers, we designed a change of variable in the underlying QP in order to expose the specific structure between time-varying and time-invariant parts of the Hessian matrix and compute its Cholesky decomposition in an efficient way by pre-computing the decomposition of the time-invariant part.