International audienceHierarchical least-square optimization is often used in robotics to inverse a direct function when multiple incompatible objectives are involved. Typical examples are inverse kinematics or dynamics. The objectives can be given as equalities to be satisfied (e.g. point-to-point task) or as areas of satisfaction (e.g. the joint range). This two-part paper proposes a complete solution to resolve multiple least-square quadratic problems of both equality and inequality constraints ordered into a strict hierarchy. Our method is able to solve a hierarchy of only equalities ten time faster than the classical method and can consider inequalities at any level while running at the typical control frequency on whole-body size problems. This generic solver is used to resolve the redundancy of humanoid robots while generating complex movements in constrained environment. In the first part, we establish the mathematical bases underlying the hierarchical problem and propose a dedicated solver. When only equalities are involved, the solver amounts to the classical solution used to handle redundancy in inverse kinematics in a far more efficient way. It is able to handle inequalities at any priority levels into a single resolution scheme, which avoids the high number of iterations encountered with cascades of solvers. A simple example is given to illustrate the interest of our approach. In the second part, we detail the implementation of the solver and its application to inverse kinematics. In particular, we explicit the solver complexity and prove the continuity and the stability of the resulting control laws. Finally, we experimentally demonstrate its efficiency in the context of generating robot motions

The goal of this paper is to demonstrate the capacity of Model Predictive Control to generate stable walking motions without the use of predefined foot steps. Building up on well-known Model Predictive Control schemes for walking motion generation, we show that a minimal modification of these schemes allows designing an online walking motion generator which can track a given reference speed of the robot and decide automatically the foot step placement. Simulation results are proposed on the HRP-2 humanoid robot, showing a significant improvement over previous approaches.

Abstract-A humanoid walking robot is a highly nonlinear dynamical system that relies strongly on contact forces between its feet and the ground in order to realize stable motions, but these contact forces are unfortunately severely limited. Model Predictive Control, also known as Receding Horizon Control, is a general control scheme specifically designed to deal with such contrained dynamical systems, with the potential ability to react efficiently to a wide range of situations. Apart from the question of computation time which needs to be taken care of carefully (these schemes can be highly computation intensive), the initial question of which optimal control problems should be considered to be solved online in order to lead to the desired walking movements is still unanswered. A key idea for answering to this problem can be found in the ZMP Preview Control scheme. After presenting here this scheme with a point of view slightly different from the original one, we focus on the problem of compensating strong perturbations of the dynamics of the robot and propose a new Linear Model Predictive Control scheme which is an improvement of the original ZMP Preview Control scheme.

Summary. In this overview paper, we first survey numerical approaches to solve nonlinear optimal control problems, and second, we present our most recent algorithmic developments for real-time optimization in nonlinear model predictive control.In the survey part, we discuss three direct optimal control approaches in detail: (i) single shooting, (ii) collocation, and (iii) multiple shooting, and we specify why we believe the direct multiple shooting method to be the method of choice for nonlinear optimal control problems in robotics. We couple it with an efficient robot model generator and show the performance of the algorithm at the example of a five link robot arm. In the real-time optimization part, we outline the idea of nonlinear model predictive control and the real-time challenge it poses to numerical optimization. As one solution approach, we discuss the real-time iteration scheme.

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